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Celsius to Fahrenheit - Formula, Examples | C to F Conversion - [[company name]] [[target location]], [[stateabr]]
Celsius to Fahrenheit Formula, Examples | C to F Conversion
Whenever you look at the weather channel or any other news programs that discusses the weather, they will always talk about moisture level, precipitation, and temperature. The metric system is used
by maximum of the world to assess these quantities.
Despite that, we use the rarely used imperial system in the United States. In the imperial system, the temperature is calculated in degrees Fahrenheit (°F). In comparison, in the metric system, it is
assessed in degrees Celsius (°C).
You may decide on a voyage to a country that uses the metric system, or you may require to convert Celsius to Fahrenheit for a academic assignment. No matter the reason, knowing how to make this
conversion is essential.
This blog covers how to convert Celsius (also known as centigrade) to Fahrenheit without resorting to a temperature conversion graph. And we promise, this will be not restricted for math class but
also for real-life scenarios.
What Is the Conversion Formula for Celsius to Fahrenheit?
The Celsius and Fahrenheit levels are utilized for evaluating temperature. The basic distinction among the two temperature levels is that the scientists who made them selected different starting
Celsius utilizes water’s boiling and freezing points as referral points, and Fahrenheit utilizes a salt water mixture's freezing and boiling points.
In other words, 0 °C is the temp at which water freezes, while 100 °C is the temp at which it boils. On the Fahrenheit scale, water freezes at 32 °F and boils at 212 °F.
Celsius to Fahrenheit transformation is made practicable with a simple formula.
If we are aware of the temperature data in Celsius, we can convert it to Fahrenheit by utilizing the following formula:
Fahrenheit = (Celsius * 9/5) + 32
Let’s examine this formula by transforming the boiling point of water from Celsius to Fahrenheit. The boiling point of water is 100 °C so we can plug this unit into our formula like so:
Fahrenheit = (100 °C * 9/5) + 32
Once we figure out this formula for Fahrenheit, we discover the result is 212 °F, which is what we hoped for.
Now let’s attempt to transform a temperature that isn’t somewhat plain to remember, like 50 °C. We put this temperature into our formula and find the result just like before:
Fahrenheit = (50 °C * 9/5) + 32
Fahrenheit = 90 + 32
When we figure out the formula, we find the answer is 122 °F.
We could also use this formula to convert from Fahrenheit to Celsius. We just have to solve the equation for Celsius. This will get us the following formula:
Celsius = (Fahrenheit – 32) * 5/9
Now, let’s try transforming the freezing point of water from Fahrenheit to Celsius. Bear in mind, the freezing point of water is 32 °F so that we can plug this value into our formula:
Celsius = (32 °F – 32) * 5/9
Celsius = 0 * 5/9
When we work on this equation, the result will be 0 degrees Celsius is as same as 32 degrees Fahrenheit, true as we would expect.
How to Convert from Celsius to Fahrenheit
Immediately that we possess this information, let’s get to business and practice some conversions. Easily follow the following steps!
Steps to Change from Celsius to Fahrenheit
1. Accumulate the temp in Celsius that you want to convert.
2. Put the Celsius value into the formula.
Fahrenheit = (Celsius * 9/5) + 32
3. Solve the formula.
4. The solution will be the temp in Fahrenheit!
These are the quintessential steps. Although, if you want to check your work, it is must that you keep in mind the freezing and boiling points of water to do a sanity check. This step is elective but
verifying your work is always a great thinking.
Example 1
Let’s put this knowledge into use by following the steps above with an example.
Exercise: Convert 23 C to F
1. The temp we need to convert is 23 degrees Celsius.
2. We put this value into the equation, giving us:
Fahrenheit = (23 * 9/5) + 32
Fahrenheit = 41.4 +32
3. When we solve the equation, the solution is 73.4 degrees Fahrenheit.
Good! Now we are aware that 23 degrees in the Celsius scale corresponds to a calm day in late spring.
Example 2
Now let’s try another example: converting a temperature which isn’t as straightforward to keep in mind.
Exercise: Convert 37 C to F
1. The temperature we need to convert is 37 degrees Celsius.
2. We place this value into the formula, providing us:
Fahrenheit = (37 * 9/5) + 32
3. When we work out the equation, we discover that the solution is 98.6 degrees in Fahrenheit temp. This corresponds to the average body temp!
That’s all you need to know! These are the quick and easy steps to change temps from Celsius to Fahrenheit. Just keep in mind\bear in mind the equation and put in the units consequently.
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Volume 6 (2010) Issue 1 - Journal of Prime Research in Mathematics
JPRM-Vol. 1 (2010), Issue 1, pp. 56 – 61
Open Access Full-Text PDF
Y. Polatoglu
Abstract: We denote by A the class of all analytic functions in the open unit disk \(\mathbb{D} = {z | |z| < 1}\) which satisfy the conditions \(f(0) = 0\), \(f'(0) = 1\). In this paper we define a
new concept of \(λ\)− fractional Schwarzian derivative and \(λ\)− fractional Mobius transformation for the class A. We also formulate the criterion for a function to be univalent using the fractional
Read Full Article
Some exact solutions for the flow of a Newtonian fluid with heat transfer via prescribed vorticity
JPRM-Vol. 1 (2010), Issue 1, pp. 38 – 55 Open Access Full-Text PDF
M. Jamil, N. A. Khan, A. Mahmood, G. Murtaza, Q. Din
Abstract: Two-dimensional , steady, laminar equations of motion of an incompressible fluid with variable viscosity and heat transfer equations are considered. The problem investigated is the flow for
which the vorticity distribution is proportional to the stream function perturbed by a sinusoidal stream. Employing transformation variable, the governing Navier-Stokes Equations are transformed into
the ordinary differential equations and exact solutions are obtained. Finally, the influence of different parameters of interest on the velocity, temperature and pressure profiles are plotted and
Read Full Article
Some distributional properties of the concomitants of record statistics for bivariate pseudo–exponential distribution and characterization
JPRM-Vol. 1 (2010), Issue 1, pp. 32 – 37 Open Access Full-Text PDF
Muhammad Mohsin, Juergen Pilz, Spoeck Gunter, Saman Shahbaz, Muhammad Qaiser Shahbaz
Abstract: A new class of distributions known as Bivariate Pseudo–Exponential distribution has been defined. The distribution of r–th concomitant and joint distribution of r–th and s–th concomitant of
record statistics of the resulting distribution have been derived. Expression for single and product moments has also been obtained for the resulting distributions. A characterization of the k-th
concomitant of record statistics for the Pseudoexponential distribution by the conditional expectation is presented.
Read Full Article
Graphs with same diameter and metric dimension
JPRM-Vol. 1 (2010), Issue 1, pp. 22 – 31 Open Access Full-Text PDF
Imrana Kousar, Ioan Tomescu, Syed Muhammad Husnine
Abstract: The cardinality of a metric basis of a connected graph \(G\) is called its metric dimension, denoted by \(dim(G)\) and the maximum value of distance between vertices of \(G\) is called its
diameter. In this paper, the graphs \(G\) with diameter 2 are characterized when \(dim(G) = 2.\)
Read Full Article
Forcing edge detour number of an edge detour graph
JPRM-Vol. 1 (2010), Issue 1, pp. 13 – 21 Open Access Full-Text PDF
A. P. Santhakumaran, S. Athisayanathan
Abstract: For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the detour distance \(D(u, v)\) is the length of a longest \(u–v\) path in \(G\). A \(u–v\) path of length \(D(u, v)\) is called
a \(u–v\) detour. A set \(S ⊆ V\) is called an edge detour set if every edge in \(G\) lies on a detour joining a pair of vertices of \(S\). The edge detour number \(dn_1(G)\) of G is the minimum
order of its edge detour sets and any edge detour set of order \(dn_1(G)\) is an edge detour basis of \(G\). A connected graph \(G\) is called an edge detour graph if it has an edge detour set. A
subset \(T\) of an edge detour basis \(S\) of an edge detour graph \(G\) is called a forcing subset for \(S\) if \(S\) is the unique edge detour basis containing \(T\). A forcing subset for \(S\) of
minimum cardinality is a minimum forcing subset of \(S\). The forcing edge detour number \(fdn_1(S)\) of \(S\), is the minimum cardinality of a forcing subset for \(S\). The forcing edge detour
number \(fdn_1(G)\) of \(G\), is \(min{fdn_1(S)}\), where the minimum is taken over all edge detour bases \(S\) in \(G\). The general properties satisfied by these forcing subsets are discussed and
the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters \(dn_1(G)\) and \(fdn_1(G)\) satisfy the relation \(0 ≤ fdn_1(G) ≤ dn_1(G)\) and it is
proved that for each pair \(a\), \(b\) of integers with \(0 ≤ a ≤ b\) and \(b ≥ 2\), there is an edge detour graph \(G\) with \(fdn_1(G) = a\) and \(dn_1(G) = b\).
Read Full Article
Divisor path decomposition number of a graph
JPRM-Vol. 1 (2010), Issue 1, pp. 01 – 12 Open Access Full-Text PDF
K. Nagarajan, A. Nagarajan
Abstract: A decomposition of a graph G is a collection Ψ of edge-disjoint subgraphs \(H_1,H_2, . . . , H_n\) of \(G\) such that every edge of \(G\) belongs to exactly one \(H_i\). If each \(H_i\) is
a path in \(G\), then \(Ψ\) is called a path partition or path cover or path decomposition of \(G\). A divisor path decomposition of a \((p, q)\) graph \(G\) is a path cover \(Ψ\) of \(G\) such that
the length of all the paths in \(Ψ\) divides \(q\). The minimum cardinality of a divisor path decomposition of \(G\) is called the divisor path decomposition number of \(G\) and is denoted by \(π_D
(G)\). In this paper, we initiate a study of the parameter \(π_D\) and determine the value of \(π_D\) for some standard graphs. Further, we obtain some bounds for \(π_D\) and characterize graphs
attaining the bounds.
Read Full Article | {"url":"http://jprm.sms.edu.pk/volume-6-2010-issue-1/","timestamp":"2024-11-13T15:59:52Z","content_type":"text/html","content_length":"60148","record_id":"<urn:uuid:b3bb1647-e2f2-4d03-8763-0341d547adff>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00855.warc.gz"} |
Direct inversion in the iterative subspace (DIIS) optimization of open-shell, excited-state, and small multiconfiguration SCF wave functions
The direct inversion in the iterative subspace (DIIS) method is applied to several simple SCF wave functions in an effective Fock matrix formulation. The following cases are treated:
high-spin-restricted open shell, open-shell singlet, and two-configuration wave functions. Open-shell singlet states are described by a three-determinant 2×2 CAS expansion which is equivalent to
Davidson's nonorthogonal SCF method in the case of the first open-shell singlet. Very sharp convergence is usually obtained in less than 20 cycles. The method is applicable to slowly convergent or
even inherently divergent cases, and able to enforce convergence to excited states not the lowest of their symmetry. For these simple wave functions, the present first order method is asymptotically
more efficient than second-order methods. Examples are presented for H[2]O, H[2]O[2], C[2]H[4], F[2], several states of NO[2], C[2]H[5], formaldehyde, and ketene.
Journal of Chemical Physics
Pub Date:
May 1986
□ Inversions;
□ Iterative Solution;
□ Molecular Excitation;
□ Self Consistent Fields;
□ Wave Functions;
□ Convergence;
□ Formaldehyde;
□ Ketenes;
□ Nitrogen Dioxide;
□ Optimization;
□ Water;
□ Atomic and Molecular Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1986JChPh..84.5728H/abstract","timestamp":"2024-11-05T17:27:37Z","content_type":"text/html","content_length":"39087","record_id":"<urn:uuid:9bfae91c-107f-4aaa-bd0f-933b6a587c72>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00354.warc.gz"} |
1 Introduction
In recent years, there has been an increasing interest to develop new state and parameter estimation schemes to reduce the deficiencies of classical schemes such as the Kalman Filter (KF) and the
Luenberger Observer (LO) which have been frequently used to reconstruct variables that are not measured and to reduce the effect of noise on the available measurements. However, due to the fact that
the stability and convergence properties of these estimators are essentially locally valid, their application has been restrictive in many practical situations. Other estimation approaches (the high
gain [1], adaptive [2], and sliding mode [3]) have been also devised to solve the state reconstruction problem since the stability of the system is guaranteed but their designs involve conditions
that must be assumed a priori or that are usually hard to verify [4]. These may account for the failure of these estimators to find widespread application in biological processes [2].
In this paper we present an innovative state estimation schemes to overcome the difficulties associated with the reconstruction of important nonmeasured variables in biological processes. It is based
on the well-known Asymptotic Observer (AO) [2], which has proved to be suitable for certain biological processes by yielding satisfactory estimates in the face of uncertain kinetic parameters and
load disturbances despite the dependence of the AO performance and convergence on the system operating conditions (particularly on the dilution rate which may be relatively low in most industrial
scale biological processes) that have prevented the implementation of efficient monitoring and control strategies.
The objective of this study is then to propose an alternative to tune the convergence rate of a typical AO to compensate the effect of this plant features dependence of asymptotic observers by
reducing the close interaction of the plant parameters in the estimator equations. This is accomplished by adopting a methodology similar to that used in [5] for a single-dimension bounded error
observer, which is further developed to more complex n-dimensional cases. The main result is the inclusion of an adjustable convergence rate in the design of asymptotic observers while maintaining
the stability and robustness convergence properties in the presence of nonlinear terms (i.e., process kinetics) and under the influence of load disturbances. The performance improvement of the
classical asymptotic observer is finally demonstrated by applying in simulations the proposed tunable observer in an anaerobic digestion wastewater treatment process.
2 The considered general model
Let us consider the general class of biological systems that fits within the following model [2]:
$(Σ0){x˙(t)=Cf(x(t),t)+A(t)x(t)+b(t)x(0)=x0$ (1)
where $x(t)∈Rn$ is the state vector, $C∈Rn×r$ represents a matrix of constant coefficients. The mapping $f(x(t),t)∈Rr$ denotes the nonlinearities and $b(t)∈Rn$ gathers the inputs of the process. The
time-varying matrix $A(t)∈Rn×n$ is the state matrix. The number of measured states that are available on-line is $n2$. Thus, the problem reduces to estimate $n1=n−n2$ variables. For this purpose, the
state vector is split in such a way that (1) can be rewritten such as
$(Σ1){x˙1(t)=C1f(x(t),t)+A11(t)x1(t)+A12(t)x2(t)+b1(t),x1(0)=x1,0x˙2(t)=C2f(x(t),t)+A21(t)x1(t)+A22(t)x2(t)+b2(t),x2(0)=x2,0$ (2)
where the $n2$ measured states $y(t)$ have been grouped in the $x2(t)$ vector (i.e., $y(t)=x2(t)$) while the variables that have to be estimated are represented by $x1(t)$. $Aij(t)∈Rni×nj$,
$Ci∈Rni×r$, $bi(t)∈Rni$, for $i=1,2$ and $j=1,2$ are the corresponding partitions of $x(t)$, $A(t)$, C and $b(t)$, respectively. The following hypotheses about the model are introduced:
• (H1) The matrix $A(t)$ is known and bounded $∀t⩾0$, i.e., there exist constant matrices $Amin$ and $Amax$ such that $Amin⩽A(t)⩽Amax∀t⩾0$.
• (H2) The matrix C is constant and known with the property $rankC=rankC2$.
• (H3) The vector $b(t)$ is known $∀t⩾0$.
The operator ⩽ applied between vectors and between matrices should be understood as a collection of inequalities between elements.
3 A robust asymptotic observer
Under hypotheses (H1) to (H3), the following system designed by the linear transformation $w(t)=Nx(t)$:
$(Ω0){wˆ˙(t)=W(t)wˆ(t)+Y(t)y(t)+Nb(t)wˆ(0)=Nxˆ0xˆ1(t)=N1−1(wˆ(t)−N2y(t))$ (3)
with is an asymptotic nonlinear observer of (1) [6]. Here, $N=[N1N2]$ where $N1∈Rn1×n1$ is an arbitrary invertible matrix, $N2=−N1C1C2⊥(N2∈Rn1×p)$ and $C2⊥$ is the generalized pseudo-inverse of $C2$.
Notice that observer (3) is fully independent of the nonlinear terms and thus, it is robust with respect to these terms. Let us now denote $ecao(t)=xˆ1(t)−x1(t)$ if $xˆ1(0)−x1(0)⩾0$ or $ecao(t)=x1(t)
−xˆ1(t)$ if $xˆ1(0)−x1(0)⩽0$. $ecao$ is the observation error associated to (3) (the subscripts “cao” denotes “classical asymptotic observer”). It is easy to verify that $ecao$ follows the dynamics:
$e˙cao(t)=We(t)ecao(t)$ with $We(t)=N1−1W(t)N1$. Notice also that under hypothesis (H1), it is possible to find two constant matrices $We−$ and $We+$ such that $We−⩽We(t)⩽We+∀t⩾0$. Thus, in order to
guarantee the stability of (3), the following hypotheses are introduced:
• (H4) $We,ij−⩾0,∀i≠j$.
• (H5) $We−$ and $We+$ are Hurwitz stable.
The hypothesis (H4) simply states that the matrix $We−$ and thus, the matrices $We(t)$ and $We+$ are cooperative [7], while the hypothesis (H5) states the stability of these two constant matrices.
Lemma 1
Under hypotheses (H1)–(H5) the asymptotic observer (3) is stable and $xˆ1(t)$ converges asymptotically towards $x1(t)$ for any set of initial conditions.
The proof of this lemma is given in [8].
4 A robust tunable asymptotic observer
This section presents the main results of this study. The most important limitation of observer (3) is indeed that, in most of the cases, its convergence rate is fixed by the operating conditions of
the biological system (namely the dilution rate). To face this limitation, a change in the observer design is introduced in the following in order to obtain adjustable convergence rates.
Let us consider the following modified transformation$z(t)=N˜(t)x(t)$with$N˜(t)=[N1Θ(t)N2]$and where$Θ(t)∈Rn1×n1$, the gain matrix, is a continuously derivable function matrix with the property:
Then, under hypotheses (H1) to (H5), the following dynamical system
$(Ω1){zˆ˙(t)=(N1C1+Θ(t)N2C2)f˜(xˆ(t),t)zˆ˙(t)=+W˜(t)zˆ(t)+Y˜(t)y(t)+N˜(t)b(t)zˆ(0)=N˜(0)xˆ0xˆ1(t)=N1−1(zˆ(t)−Θ(t)N2y(t))$ (6)
where is a stable tuning asymptotic observer for model (1).
Proof Convergence and stability
Let $e(t)=xˆ1(t)−x1(t)$ be the observation error associated to (6). Under hypotheses (H1) to (H3), it is straightforward to verify that the error dynamics is given by
$e˙(t)=Ee(t)e(t)+K(t)φ(xˆ(t),x(t),t)$ (8)
Now, since $limt→∞Θ(t)=I$, it is clear that:
• (i) $limt→∞Ee(t)=We(t)$,
• (ii) $limt→∞K(t)=C1+N1−1N2C2=0$, and thus,
• (iii) $limt→∞e˙(t)=e˙cao(t)$.
Therefore, given the stability properties of $We(t)$ provided by hypotheses (H4) and (H5), it can be concluded that $limt→∞e(t)=limt→∞ecao(t)=0$. □
Clearly, the advantage of the tunable observer (6) over the classical AO is that, by choosing a suitable gain matrix $Θ(t)$, the classical AO is provided with an adjustable convergence rate, which
can be tuned by the user. Notice that $Θ(t)$ influences both the stability and the convergence properties (see Eq. (8)) of the tuning observer and it can be properly chosen to accelerate the
convergence rate which allows to reach the zero steady state, $e=0$, even if the uncertainty of the nonlinear terms $f(x(t),t)$ is reasonable high. It is also worth mentioning that, with the
exception of the property (5), no other restrictions are imposed on the gain matrix $Θ(t)$. Thus, the choice of $Θ(t)$, may be, at a first glance, a relatively easy task. In other words, $Θ(t)$ must
be chosen to give the fastest convergence to the true state. Moreover, one can see that, as $Θ(t)→I$, the knowledge of the nonlinearities is no longer required and therefore, the tuning observer
design converges to the classical AO with the same robustness, stability and convergence properties of the AO. Furthermore, if $Θ(t)$ is chosen as $Θ(t)=diag(θ(t))$, with $θ(t)∈Rn1$, a fully
decoupled tuning observer is obtained, where the parameters needed to tune each estimated state variable, $x1,i(t)(i=1…n1)$, are exclusively those involved in the function $θi(t)$. In the following
section the proposed tuning observer will be applied to an actual highly nonlinear biological wastewater treatment process.
5 Application to wastewater treatment processes
Anaerobic Digestion (AD) is a series of multi-substrate multi-organism biological processes that take place in the absence of oxygen and by which organic matter (expressed as COD, the Chemical Oxygen
Demand) is decomposed and converted into biogas, a mixture of mainly carbon dioxide and methane, microbial biomass and residual organic matter [9]. Several advantages are recognised to AD processes
when used in wastewater treatment processes: high capacity to treat slowly degradable substrates at high concentrations, very low sludge production, potentiality for production of valuable
intermediate metabolites, low energy requirements and possibility for energy recovery through methane combustion. AD is indeed one of the most promising options for delivery of alternative renewable
energy carriers, such as hydrogen, through conversion of methane, direct production of hydrogen, or conversion of by-product streams. However, despite these large interests and few thousands
commercial installations refereed world-wide [10], many industries are still reluctant to use AD processes, probably because of the counterpart of their efficiency: they can become unstable under
some circumstances. Hence, actual research aims not only to extend the potentialities of anaerobic digestion [11], but also to optimise AD processes and increase their robustness towards disturbances
[12]. The design of efficient state estimators clearly goes in these two last directions since instrumentation is usually scarce at industrial scale.
5.1 An anaerobic digestion model
Let us consider the following dynamical model (known as AM1) for continuous anaerobic digestion process [13]. This model is given in the following matrix form (see Fig. 1) or simply $ξ=Cf(x(t),t)+A
(t)ξ(t)+b(t)$ which matches exactly model (1) with $x(t)=ξ(t)$. In (9), the dotted lines indicate the partitions of Eq. (2). In this model, $ξ1=X1$, $ξ2=X2$, $ξ5=S1$, $ξ6=S2$ and $ξ3=Z$, $ξ4=CTI$ are
the concentrations of acidogenic bacteria, methanogenic bacteria, COD, Volatile Fatty Acids (VFA), strong ions and total inorganic carbon, respectively. The superscript “in” indicates the influent
concentrations. The variable $PCO2$ is the CO[2] partial pressure whereas $α(0⩽α⩽1)$ denotes the biomass fraction that is retained by the reactor bed, i.e., $α=0$ for an ideal fixed-bed reactor and
$α=1$ for an ideal continuous stirred tank reactor (CSTR) whereas $D(t)$ is the dilution rate and it is supposed to be a persisting input, i.e., $∫δ∞D(τ)dτ>0$. Moreover, $D(t)$ is a bounded variable
since it is conditioned by the minimum flux to the persisting input and the washout condition for the upper bound, i.e., $Dmin⩽D(t)⩽Dmax$. Last but not least, $μ1$ and $μ2$ are complex nonlinear
mathematical expressions that describe the kinetics of the biochemical reactor. These expressions are given by Eq. (10):
Fig. 1
$μ 1 = μ 1 , max S 1 k s , 1 + S 1 μ 2 = μ 2 , max S 2 k s , 2 + S 2 + ( S 2 / k I , 2 ) 2$ (10)
The AM1 model was developed and experimentally validated in a continuous 1 m^3 up-flow fixed bed anaerobic digester used for the treatment of industrial wine vinasses [13]. More details about the
process design and instrumentation can be found in [14].
5.2 Observer design
The goal in this application example is the estimation of $X1$, $X2$, Z and $CTI$ by using readily available $S1$ and $S2$ measurements. In order to match the split model (2), the matrix partitions
$xi(t)$, $Aij(t)$, $Ci$ and $bi(t)$, for $i=1,2$ and $j=1,2$ have been clearly indicated in (9) by the dotted lines. Without loss of generality, one can choose $N1=I$, such that
$N2=(k1k3)−1[k3 k2 0 k3k4+k2k50 k1 0 k1k5]T$
Matrices W, $W˜$, Y and $Y˜$ are calculated by using Eqs. (4) and (7) and the gain matrix $Θ(t)$, can be computed by solving the following ODE system:
with $G=diag(g)$, $g∈Rn1$. Notice that the necessary property (5) is not restrictive at all and thus, one can choose many forms on G that can fulfill it. In the present study, it is obvious that (11)
not only fulfills this property but also it is very simple and allows the decoupling of the observer design. In fact, the selection of the constants, $gi,∀i=1,2,…,n1$, allows us to tune the
convergence rate for each estimated state individually. In addition, in this way, it is possible to influence the fast convergence of $Θ(t)$ to the identity matrix. Notice however that, as long as $Θ
(t)$ does not reach the identity matrix, the proposed tunable observer exhibits a highly nonlinear behavior and thus, a stability analysis similar to the one used in classical approaches, e.g., the
extended LO and the extended KF, should be implemented. It is worth mentioning that the observer gains used here in the implementation of the tunable observer were chosen after a trial and error
process. In fact, a number of different gain matrices $Θ(t)$ were tested and they all yielded similar results. These results are not shown in this paper due to space limitation. For the results shown
here, we used the following parameters in the solution of Eq. (11): $Θ(0)=[−2.5−142−10]T$, $g=[1210.35]T$. A methodology to decide upon the optimal choice of the observer gains is now in progress.
5.3 Hypotheses verification
• (H1) The matrix $A(t)$ is bounded and known $∀t⩾0$ since it depends on $D(t)$ which is measured and it is also bounded. Moreover, α and $k7$ are bounded and known.
• (H2) By inspection, $rankC=rankC2$.
• (H3) All inputs to the system are known.
• (H4) Since $A21=0$ and provided $N1=I$, we have
$We±=[−αD±0000−αD±0000−D±000k7−(D±+k7)]$ (12)
that fulfills the positivity condition on the off-diagonal elements of $We$.
• (H5) From (12) it is clear that $eig(We±)$ are negative for any $0<D−⩽D(t)⩽D+$ (clearly, $We−$ and $We+$ are Hurwitz).
5.4 Simulation results
Model parameters used in the proposed adjustable rate observer implementation are listed in Table 1. Simulations shown hereafter were performed for a 50 days period by using operating conditions as
close as possible to actual wastewater treatments plants. The dilution rate exhibited large fluctuations as well as drastic step perturbations (see Fig. 2). The behavior of the inlet concentration
patterns for $S1in$, $S2in$, $Zin$, and $CTIin$ is shown in Figs. 3–6 while the $PCO2$ is depicted in Fig. 9. As in many continuous bioreactors, $X1in$ and $X2in$ were considered as negligible. The
on-line measurements of $S1$ and $S2$ used in the state estimation process were obtained from model simulations as the observer inputs to estimate $X1,X2,CTI$ and Z (see Figs. 7 and 8). The
performance of the proposed adjustable nonlinear observer under these operating conditions is depicted in Figs. 10–13. For the sake of completeness, the response of a classical asymptotic observer
has been added to demonstrate the convergence features of the proposed observer design. Initial conditions for both, classical asymptotic observer and the tunable observer were exactly the same. In
Figs. 10–13, the continuous line (–) represents the model predictions, the dotted line (⋯) represents the CAO estimations whereas the dashed line (---) represents the tuning observer estimations. By
inspecting these figures, it is clear that the response of the tunable observer is satisfactory for all estimated state variables since it was able to cope with all the difficulties associated to
load disturbances. As expected, the tunable observer converge rate is faster than the classical one, showing excellent stability properties even in the presence of load disturbances and uncertainty
on the process kinetics. Notice, however, that in the case of the Z variable, both observers showed essentially the same convergence rate (see Fig. 11) since Z does not depend on the nonlinearities
nor on any model parameter (see $ξ3$ in Eq. (9)) and as a consequence, the convergence rate of both observers schemes rely exclusively on the fixed gain value predetermined by the dilution rate. The
tunable observer response described, nevertheless, the trend of the actual Z readings. Finally, the excellent performance of the proposed observer in the estimation of $CTI$ is exhibited in Fig. 13.
One can see that the tunable observer response is able to reach the true state value faster than the classical AO.
Table 1
Parameters used in the model [13]
$μ max , 1$ = 1.25 day^−1
$μ max , 2$ = 0.69 day^−1
$k s , 1$ = 4.95 KgCOD/m^3
$k s , 2$ = 9.28 molVFA/m^3
$k I , 2$ = 20 (molVFA/m^3)^1/2
α = 0.5 (dimensionless)
$k 1$ = 6.6 KgCOD/Kg$x1$
$k 2$ = 7.8 molVFA/Kg$x1$
$k 3$ = 611.2 molVFA/Kg$x2$
$k 4$ = 7.8 molCO[2]/Kg$x1$
$k 5$ = 977.6 molCO[2]/Kg$x2$
$k 6$ = 1139.2 molCH[4]/Kg$x2$
$k 7$ = 50 day^−1
$K 8$ = 0.1579 mol/m^3KPa
Fig. 2
Dilution rate.
Fig. 3
Influent COD concentration.
Fig. 4
Influent VFA concentration.
Fig. 5
Influent strong ions concentration.
Fig. 6
Influent total inorganic carbon concentration.
Fig. 9
CO[2] partial pressure profile.
Fig. 7
COD concentration.
Fig. 8
VFA concentration.
Fig. 10
Estimation of the acidogenic biomass concentration.
Fig. 11
Estimation of the methanogenic biomass.
Fig. 12
Estimation of the strong ions concentration.
Fig. 13
Estimation of the total inorganic carbon concentration.
6 Conclusions and perspectives
In this work, a robust asymptotic adjustable rate nonlinear observer for multidimensional biological systems has been proposed. It has been tested in numerical simulations on an anaerobic digestion
process used in a wastewater treatment context. By using observer gains that were suitably chosen, this observer exhibited faster convergence rates than a classical asymptotic observer design. New
studies are currently being conducted for optimizing the observer gains calculations. Because of the clear utility of this tuning observer in highly uncertain biological systems at the experimental
scale, their use in robust nonlinear control schemes with application to continuous bioreactors is now under study.
The authors gratefully acknowledge to the ECOS-ANUIES Program (project: M97-B01), the CONACyT, the PROMEP Program and the European project TELEMAC (IST-2000-28156) for the support that made this
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Winding path leads to fluid career - Deixis Online
Paul Fischer can’t remember a time when he wasn’t interested in aeronautical and mechanical engineering. His passion for solving seemingly unsolvable problems came just a bit later.
Fischer, now a computational scientist with the Mathematics and Computer Science Division at Argonne National Laboratory, connects that interest to an early fascination with the Apollo space program.
“I remember when I was eight years old, getting up to watch Apollo 8 take off,” he says.
That carried over to Ithaca High School, in the shadow of Cornell University in upstate New York, when “I started writing code to solve different equations. I decided it was easier to let the
computer do the math for me than to do it myself.”
Fischer, 50, says he was lucky to attend a high school with a strong science program. “Given that I knew I really wanted to do aeronautics, I focused on math and physics. I just loaded up on those
He still tells budding scientists that “when you’re a student, take as many courses as you can. Don’t sell yourself short. Latch onto opportunities to take courses, as many as you can in the core
As an undergraduate at Cornell, Fischer gravitated toward mechanical engineering when it became clear to him the field was more stable than the aerospace industry. He became interested in both solid
and fluid mechanics, but it was hard to decide which to specialize in. A roommate finally told him to choose whichever is harder.
“But I went into fluids, which is actually easier,” Fischer says, though not everyone would agree.
He took several graduate-level mechanical engineering courses his senior year, then went to Stanford University for a master’s, focusing on computational fluid mechanics.
“I knew I enjoyed that,” he says. “But I really wanted to get into the mathematical side of mechanical engineering – and also the software and algorithm side.”
Fischer worked for three years on the design of gas bearings for disc drives. He did computational experiments, but he says, “I was always interested in the companion validation of the experiments.
It’s the only way you know you’re actually doing the right thing.”
At the Massachusetts Institute of Technology, Fischer earned a doctoral degree in mechanical engineering with a dissertation on developing code for high-performance parallel computers.
He started moving into applied mathematics because he knew he was going to write software to simulate physical phenomena. “It’s extremely beneficial to have a strong math background” for those
endeavors, Fischer says. His Ph.D. adviser was an applied mathematician, he did a postdoctoral fellowship in applied mathematics and he taught the subject before coming to Argonne.
“It’s essential for writing advanced simulation codes and for understanding when you can prove the correctness of your code.”
Just as physics isn’t enough without the math, when writing complex codes, “the math in and of itself isn’t sufficient,” Fischer says. “There are subtle things associated with boundary conditions
that need a deep understanding of the physics involved.”
For example, electromagnetic equations are fairly simple, but their boundary conditions are not. “I can write an electromagnetic code that solves for trivial boundary conditions, but for more complex
boundary conditions, you need to understand the physics.”
Fischer was the first recipient of the Computational Research Postdoctoral Fellowship at Cal Tech, which was a hopping place for parallel computing at the time. He then won the 1999 Gordon Bell Prize
for scaling to 4,096 processors with a simulation code. “It was really a recognition of scalable algorithms.”
His team at Argonne has won several Department of Energy Innovative and Novel Computational Impact on Theory and Experiment (INCITE) awards, earning time on the most powerful computers in the world
to work on astrophysical problems. In 2006, he won the first external science award, which got him 3 million hours of processing time. “That’s not too many hours now, but it was back then.”
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Bill Scanlon was a reporter at the Rocky Mountain News until its closing in February 2009.
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What Are Intervals In Music?
Music, often described as the universal language, is composed of intricate patterns, rhythms, and harmonies. At the heart of these harmonies lies the concept of intervals, the spaces between notes
that give music its depth and character.
Intervals in music refer to the distance between two pitches or notes. They are the building blocks of melodies, chords, and harmonies, shaping the emotional and tonal landscape of a piece.
Understanding intervals helps musicians recognize patterns, transpose music, and build chords and scales.
Melodic Intervals
Melodic intervals refer to the distance between two pitches or notes that are played successively, one after the other, rather than simultaneously. They describe the relationship between two notes in
a melody when they are played in sequence.
Harmonic Intervals
Harmonic intervals refer to the distance between two pitches or notes that are played simultaneously, at the same time. They describe the relationship between two notes when they are sounded
together, creating harmony.
Interval Naming
1. Unison (Perfect Prime): This is when two notes are the same. For example, playing two middle C’s on a piano at the same time.
2. Minor 2nd: This is the smallest distance between two different notes. For example, the distance between C and C# or D and D♭.
3. Major 2nd: This is equivalent to two half steps or one whole step. For example, the distance between C and D.
4. Minor 3rd: This interval spans three half steps. For example, the distance between C and E♭.
5. Major 3rd: This interval spans four half steps. For example, the distance between C and E.
6. Perfect 4th: This interval spans five half steps. For example, the distance between C and F.
7. Tritone: This interval spans six half steps and is exactly halfway between an octave. It’s sometimes called an augmented fourth or diminished fifth.
8. Perfect 5th: This interval spans seven half steps. For example, the distance between C and G.
9. Minor 6th: This interval spans eight half steps. For example, the distance between C and A♭.
10. Major 6th: This interval spans nine half steps. For example, the distance between C and A.
11. Minor 7th: This interval spans ten half steps. For example, the distance between C and B♭.
12. Major 7th: This interval spans eleven half steps. For example, the distance between C and B.
13. Octave (Perfect 8th): This interval spans twelve half steps and is the distance between two notes with the same name. For example, the distance between C and the next C above it.
Intervals can also be augmented (increased by a half step) or diminished (decreased by a half step). A diminished 5th is one half step smaller than a perfect 5th, and an augmented 4th is one half
step larger than a perfect 4th.
Number ofsemitones Minor, major,or perfect intervals Shorthand Augmented ordiminished intervals Shorthand Alternative Names
0 Perfect unison P1 Diminished second d2
1 Minor second m2 Augmented unison A1 Semitone, half tone, half step
2 Major second M2 Diminished third d3 Tone, whole tone, whole step
3 Minor third m3 Augmented second A2
4 Major third M3 Diminished fourth d4
5 Perfect fourth P4 Augmented third A3
6 Diminished fifth d5 Tritone
Augmented fourth A4
7 Perfect fifth P5 Diminished sixth d6
8 Minor sixth m6 Augmented fifth A5
9 Major sixth M6 Diminished seventh d7
10 Minor seventh m7 Augmented sixth A6
11 Major seventh M7 Diminished octave d8
12 Perfect octave P8 Augmented seventh A7
Courtesy of Wiki
In addition to their technical definitions, intervals also have distinct sonic qualities.
Major intervals tend to sound happy or bright, while minor intervals often sound sad or dark. Perfect intervals like the 4th and 5th have a stable sound, while augmented and diminished intervals can
sound tense or dissonant.
How Do You Identify An Interval?
Identifying musical intervals involves determining the distance between two notes. Here’s a step-by-step guide to help you identify intervals:
1. Start with the Note Names:
First, identify the names of the two notes. If you have C and E, you know the interval starts on C and goes up to E.
2. Count the Letter Names:
Count the starting note as “1” and then count up to the second note. Using the C and E example, you would count: C(1), D(2), E(3). This tells you it’s some type of 3rd.
3. Determine the Number of Half Steps:
Count the number of half steps (semitones) between the two notes. On a keyboard, this means counting every key (including both white and black keys) between the two notes, but not including the
starting note.
For C to E: C to C# (1), C# to D (2), D to D# (3), D# to E (4). So, there are 4 half steps between C and E.
4. Match the Number of Half Steps to the Interval Name:
Using the number of half steps you’ve counted, you can determine the specific type of interval:
1 half step = Minor 2nd
2 half steps = Major 2nd
3 half steps = Minor 3rd
4 half steps = Major 3rd
5. Consider Augmented and Diminished Intervals:
If an interval is one half step larger than a major or perfect interval, it’s augmented.
If it’s one half step smaller than a minor or perfect interval, it’s diminished.
6. Use Mnemonics and Songs:
Many musicians use familiar songs to help identify intervals.
• Minor 2nd: The first two notes of “Jaws” theme.
• Major 2nd: The first two notes of “Happy Birthday.”
• Perfect 4th: The beginning of “Here Comes the Bride.”
• Perfect 5th: The first two notes of “Star Wars” theme.
These song references can vary based on individual experiences and cultural context, so it’s helpful to find songs that resonate with you.
7. Practice:
Like any skill, interval identification improves with practice. Use ear training apps, and websites, or work with a music teacher to practice listening to and identifying intervals.
Some options I like to use are:
Also, practice on your instrument. Learn to play intervals across the guitar fretboard and how to find them on the piano.
8. Context Matters:
The sound of an interval can be affected by the context in which it’s heard. For example, a minor 6th might sound different in the context of one chord progression compared to another.
By combining these techniques and practicing regularly, you’ll become more proficient at identifying musical intervals by both sight and ear.
Music Interval Calculators
Here are some online music interval calculators that you can use:
• Omni Calculator: The music interval calculator on Omni Calculator determines the interval between two notes.
• muted.io: This is a simple online musical interval calculator. You can select a low and a high note to get the interval name and the number of semitones between the two notes.
• CalcTool: The music interval calculator on CalcTool allows you to easily determine the interval between two given notes.
What Is The Interval From F To C?
The interval from F to C is a Perfect 5th.
Here’s how you can figure it out:
1. Count the Letter Names:
Start with F as “1” and count up to C: F(1), G(2), A(3), B(4), C(5). This tells you it’s some type of 5th.
2. Determine the Number of Half Steps:
Count the number of half steps (semitones) between the two notes:
• F to F# or G♭ = 1 half step
• F# or G♭ to G = 1 half step
• G to G# or A♭ = 1 half step
• G# or A♭ to A = 1 half step
• A to A# or B♭ = 1 half step
• A# or B♭ to B = 1 half step
• B to C = 1 half step
In total, there are 7 half steps between F and C.
3. Match the Number of Half Steps to the Interval Name:
A Perfect 5th spans 7 half steps.
Therefore, the interval from F to C is a Perfect 5th.
What Is The Interval Of D To G?
The interval from F to C is a Perfect 5th.
Here’s how you can figure it out:
1. Count the Letter Names:
Start with F as “1” and count up to C: F(1), G(2), A(3), B(4), C(5). This tells you it’s some type of 5th.
2. Determine the Number of Half Steps:
Count the number of half steps (semitones) between the two notes:
• F to F# or G♭ = 1 half step
• F# or G♭ to G = 1 half step
• G to G# or A♭ = 1 half step
• G# or A♭ to A = 1 half step
• A to A# or B♭ = 1 half step
• A# or B♭ to B = 1 half step
• B to C = 1 half step
In total, there are 7 half steps between F and C.
3. Match the Number of Half Steps to the Interval Name:
A Perfect 5th spans 7 half steps.
Therefore, the interval from F to C is a Perfect 5th.
What Is The Interval Of F To B?
The interval from F to B is an Augmented 4th (often referred to as a “Tritone”).
Here’s how you can figure it out:
1. Count the Letter Names:
Start with F as “1” and count up to B: F(1), G(2), A(3), B(4). This tells you it’s some type of 4th.
2. Determine the Number of Half Steps:
Count the number of half steps (semitones) between the two notes:
• F to F# or G♭ = 1 half step
• F# or G♭ to G = 1 half step
• G to G# or A♭ = 1 half step
• G# or A♭ to A = 1 half step
• A to A# or B♭ = 1 half step
• A# or B♭ to B = 1 half step
In total, there are 6 half steps between F and B.
3. Match the Number of Half Steps to the Interval Name:
A Perfect 4th spans 5 half steps. However, since F to B spans 6 half steps, it is one half step larger than a Perfect 4th, making it an Augmented 4th.
Therefore, the interval from F to B is an Augmented 4th or Tritone.
What Is The Difference Between a Chord And an Interval?
Both chords and intervals are fundamental concepts in music theory, but they refer to different concepts.
While both intervals and chords deal with the relationship between notes, intervals focus on the distance between two specific notes, whereas chords involve the simultaneous sounding of two or more
notes to create harmony.
• Definition
□ Interval: An interval is the distance between two pitches or notes. It describes the relationship between two notes in terms of how many letter names they encompass and how many half steps
(semitones) separate them. For example, the distance between C and E is a major 3rd.
□ Chord: A chord is a combination of two or more pitches sounded simultaneously. It’s a harmonic unit in music. The combination of C, E, and G played together forms a C major chord.
Technically, a harmonic interval is a type of chord.
Number of Notes
• Interval: Always involves two notes, either played successively (melodic interval) or simultaneously (harmonic interval).
• Chord: Involves two or more notes played simultaneously.
• Function
□ Interval: Describes the distance and relationship between two notes. Intervals are the building blocks of scales, chords, and melodies.
□ Chord: Creates harmony in music. Chords provide depth to melodies and play a significant role in establishing the tonality and mood of a piece.
• Types
□ Interval: Intervals can be minor, major, perfect, augmented, or diminished. Examples include minor 3rd, perfect 5th, and augmented 4th.
□ Chord: Chords can be major, minor, augmented, diminished, 7th, 9th, 11th, 13th, and many other types. Examples include C major, D minor, and G7.
• Usage
□ Interval: Intervals are foundational in understanding the structure of scales, chords, and melodies. They are also crucial for tasks like transposing music.
□ Chord: Chords are used in accompaniments, songwriting, and compositions to create harmonic progressions and support melodies.
Go to next lesson: Enharmonics, Pitch, and Sound Intensity (Volume)
Go to previous lesson: What are Music Scales?
Back to: Module 1
Additional Resources: Wiki – Intervals
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eMath 2 Polynomials (OPS03)
eMath 2: Polynomials
The e-book is structured so that you will learn all concepts in a logical order and it is meant to be studied from beginning to end. The theory parts are short and concise and at the end of each
lesson, you will find a page with assignments. The assignments can be solved directly in the book. You can easily write mathematics on the computer with the eMathStudio math tools, as well as insert
pictures, embedded elements and graphs to support your studies and solutions.
Each eMath book includes:
• 300-400 assignments + answers, in three levels of complexity
• Numerous example solutions
• Possibility to add comments and highlight text
• Possibility to check the correctness of the calculation
Overview: This book presents first degree and higher degree polynomials. It shows how to solve equations involving polynomials, and how to factories polynomials. It also explains how to solve
simultaneous equations as well as inequalities, and how to work with rational expressions.
Table of contents:
1 Preface
2 Polynomials
2.1 Generally About Polynomials
2.2 Operations on Polynomials
2.3 Rules for Polynomials
2.4 Factoring
3 First Degree
3.1 First-Degree Polynomial Functions
3.2 Review: First-Degree Equations
3.3 Inequalities
3.4 Parametric Inequalities
4 Simultaneous Equations
4.1 Simultaneous Equations
4.2 Solutions to Simultaneous Equations
4.3 Simultaneous Equations in Word Problems
4.4 Rules of Logic
5 Second Degree
5.1 Second-Degree Polynomial Functions
5.2 Second-Degree Equations
5.3 Incomplete Second-Degree Equation
5.4 Complete Second-Degree Equations
5.5 The Discriminant
5.6 Logical Disjunction
6 Factoring a Polynomial
6.1 Factoring a Second-Degree Polynomial
6.2 Factoring Higher-Degree Polynomials
7 Second-Degree Inequalities
8 Higher Degree
8.1 Higher-Degree Polynomial Functions
8.2 Higher-Degree Equations
8.3 Solving Higher-Degree Equations
8.4 Higher-Degree Inequalities
9 Rational Expressions
9.1 Rational Expressions
9.2 Rational Equations
9.3 Rational Inequalities
9.4 The Logical Negation | {"url":"https://fourferries.com/webshop/en/product/emath-2-polynomials-ops03/","timestamp":"2024-11-02T05:57:38Z","content_type":"text/html","content_length":"43859","record_id":"<urn:uuid:140dce9e-7e36-4612-b9e1-08242bb61de7>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00234.warc.gz"} |
An essay on perspective
Full text: Gravesande, Willem Jacob: An essay on perspective
on PERSPECTIVE. ved backwards or forwards, or elſe the Looking-
glaſs raiſed or lower’d, until the Rays proceed-
ing from the Statue may be reflected by the Mir-
rour upon the Convex Glaſs. When theſe Alte-
rations of the Box, or Mirrour, are not ſufficient to
throw the Rays upon the Convex Glafs, the whole
Machine muſt be removed backwards or for-
203. Demonstration .
Concerning the before-mention’d Inclination of the
19. In order to demonſtrate, that the Mirrour
L hath been conveniently inclin’d, we need on-
ly prove, that the reflected Rays fall upon the
Table A under the ſame Angle, as the direct
Rays do upon a Plane, having the ſame Situation
as one would give to the Picture.
Now let A B be a Ray falling from a Point of
ſome Object upon the Mirrour G H, and from
thence is reflected in the Point a upon the Table
of the Machine: We are to demonſtrate, that if
the Line D I be drawn, making an Angle with
FE equal to the Inclination of the Picture; that
is, if the Angle DIE be the double of the Angle D F I; I ſay, we are to demonſtrate, that the
Angle B a f is equal to the Angle BCD.
The Angle DIE, by Conſtruction, is the double
of the Angle DFI; and conſequently this laſt Angle
is equal to the Angle I D F; and ſince the Angle
of Incidence C B D is equal to the Angle of Re-
flection a B F, the Triangle BCD is ſimilar to
the Triangle F a B: Whence it follows, that the
Angle Ba F is equal to the Angle BCD. Which
was to be demonſtrated. | {"url":"https://dlc.mpg.de/fulltext/551675314/214/","timestamp":"2024-11-06T11:42:23Z","content_type":"application/xhtml+xml","content_length":"104194","record_id":"<urn:uuid:0fdb137d-a903-4a33-99df-886ade369bf2>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00152.warc.gz"} |
Unleashing the Power of Pi in C – A Comprehensive Guide
In the world of programming, the constant Pi, represented by the Greek letter π, holds a significant place. While commonly associated with mathematics and geometry, Pi can also be utilized in C
programming to perform various calculations and operations. In this blog post, we will explore the different ways to leverage Pi in C programming, discussing its significance and potential
Getting Started with Pi in C
To begin our journey with Pi in C, it is essential to understand what Pi is and how it can be represented in the C programming language. In C, Pi can be defined as a constant using the value π or
approximated using a pre-defined constant.
Once Pi is defined, it becomes possible to perform basic arithmetic operations with Pi in C. Addition, subtraction, multiplication, and division can all be carried out using Pi as an operand.
Additionally, increment and decrement operations can also be performed with Pi, allowing for more complex calculations.
Let’s take a closer look at each of these operations and illustrate them with code snippets:
Addition, Subtraction, Multiplication, and Division
// Addition double result = 5 + M_PI;
// Subtraction double result = 10 - M_PI;
// Multiplication double result = 2 * M_PI;
// Division double result = 20 / M_PI;
Increment and Decrement Operations
// Increment double radius = 5; radius += M_PI;
// Decrement double circumference = 2 * M_PI * radius; circumference -= M_PI;
By utilizing these arithmetic operations, you can incorporate Pi into your C code and perform calculations that involve this essential mathematical constant.
Advanced Techniques and Functions with Pi in C
Once you have a grasp of the basic arithmetic operations, it’s time to dive deeper into advanced techniques and functions that involve Pi in C programming.
First, let’s explore how constants and variables can be used with Pi. Defining Pi as a constant allows you to use its value throughout your code without the need to reassign it. On the other hand,
declaring and assigning Pi to variables provides flexibility if you need to modify its value during runtime.
Utilizing Constants and Variables with Pi
Defining Pi as a constant:
#define PI M_PI
Declaring and assigning Pi to a variable:
double pi = M_PI;
Now, let’s examine some mathematical functions involving Pi. Calculating the square and cube roots of Pi can be valuable in certain scenarios. Additionally, trigonometric functions can also be used
in conjunction with Pi to perform calculations related to angles and circles.
Mathematical Functions Involving Pi
// Calculating the square root of Pi double sqrtResult = sqrt(M_PI);
// Calculating the cube root of Pi double cbrtResult = cbrt(M_PI);
// Using sine function with Pi double sinResult = sin(M_PI);
// Using cosine function with Pi double cosResult = cos(M_PI);
By incorporating these techniques and functions into your C programming, you can unlock the power of Pi for more complex calculations and operations.
Leveraging Pi for Complex Calculations
As you gain confidence in working with Pi in C, you can leverage its accuracy and precision for more advanced calculations. However, it is crucial to consider the limitations and potential errors
that may arise during these complex calculations.
Precision and accuracy become significant factors when dealing with Pi in complex calculations. It is important to ensure that your code handles these considerations appropriately to achieve reliable
One of the ways to harness the power of Pi in C is by utilizing power functions. You can use Pi as an exponent to perform calculations that involve exponential growth or decay. For example,
calculating the circumference, area, and volume of circular objects often requires the use of Pi as an exponent.
In addition to these calculations, various other mathematical formulas and algorithms can be implemented using Pi to solve complex problems.
Precision, Power Functions, and Complex Calculations
When performing complex calculations with Pi, it is important to be aware of the precision of the operations, especially when dealing with high accuracy requirements or floating-point errors.
For example, if you need to calculate the circumference of a circle, you can use the formula:
double radius = 5; double circumference = 2 * M_PI * radius;
Similarly, the area and volume calculations can be accomplished using the formulas:
// Area of a circle double area = M_PI * pow(radius, 2);
// Volume of a sphere double volume = (4/3) * M_PI * pow(radius, 3);
By leveraging the power of Pi, you can solve complex mathematical calculations and obtain precise results.
Best Practices and Tips for Using Pi in C
When integrating Pi into your C code, there are certain best practices and tips to consider. These practices will not only ensure optimal performance but also help you handle errors and exceptions
related to Pi calculations.
Firstly, it is essential to ensure that you have included the necessary dependencies and libraries to access Pi and its related functions. Depending on the C compiler and environment you are working
with, specific libraries may be required.
Efficient coding practices can also significantly improve performance when working with Pi. By optimizing your code and reducing unnecessary calculations, you can achieve better efficiency in your
Pi-based operations.
Handling errors and exceptions are an integral part of any robust programming. When working with Pi in C, it is important to anticipate and handle errors that may occur during calculations or when
accessing Pi-related functions.
Lastly, exploring available C libraries and resources dedicated to Pi-related operations can provide additional functionality and ease the implementation process. These libraries may offer enhanced
precision or specialized algorithms for specific calculations involving Pi.
Pi is not only a fundamental constant in mathematics but also an invaluable tool in the realm of C programming. By incorporating Pi into your C code, you can perform a wide variety of calculations
and operations with precision and accuracy. Whether it’s basic arithmetic, advanced functions, or complex mathematical calculations, Pi provides the foundation for intricate programming solutions.
As you continue to explore and experiment with Pi in your own projects, remember to follow best practices, handle errors effectively, and make use of available resources. By harnessing the power of
Pi in C programming, you can push the boundaries of what is possible and create exceptional programs and applications.
For additional resources and further reading on Pi in C programming, consider the following: | {"url":"https://skillapp.co/blog/unleashing-the-power-of-pi-in-c-a-comprehensive-guide/","timestamp":"2024-11-02T22:05:57Z","content_type":"text/html","content_length":"112137","record_id":"<urn:uuid:83bdf2f1-6c6e-4ad8-b42e-f9c8f1cfb878>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00493.warc.gz"} |
Caesar Box Cipher - Caesars' Square Cypher - Online Decoder, Encoder
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Caesar Box Cipher
Tool to decrypt/encrypt with Caesar Box, a Roman version of the scytales for ciphering text by transposition.
Caesar Box Cipher - dCode
Tag(s) : Transposition Cipher
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Caesar Box Cipher
Caesar Box Decoder
Caesar Box Encoder
Answers to Questions (FAQ)
What it the Caesar Box cipher? (Definition)
Caesar Box is a transposition cipher used in the Roman Empire, in which letters of the message are written in rows in a square (or a rectangle) and then, read by column.
How to encrypt using Caesar Box cipher?
Caesar Box Encryption uses a box, a rectangle (or a square), or at least a size W characterizing its width (that corresponds to the number of column of text)
Example: Take W=3 and the message to encrypt DCODE.
The message is written by rows and every W characters, add a new row. This will delimitate a box of characters.
If needed, the last row can be completed with another character, e.g. X or _.
The encrypted message is obtained by reading the box by column.
Example: The cipher text is DDCEO_
How to decrypt Caesar Box cipher?
Caesar Box decryption requires to know the dimensions of the box (width W by height H)
Example: Take W=3, and the ciphertext is CSAAER which is 6-character long, then H=2 (as 6/3=2).
Write the text in column in the box. The plain text appears by reading each row.
Example: The original plain text is CAESAR.
How to recognize Caesar Box ciphertext?
The Caesar box is a transposition cipher, so the coincidence index is the same as that of the plain text.
If the length of the message is a perfect square, it is a good clue.
This cipher appears in many movies or books, the most known are the scytale (parchment / ribbon from Sparta, Greece), the cipher used in Journey to the center of the Earth from Jules Verne (Arne
Saknussemm's cryptogram), etc.
How to decipher Caesar Box without the size?
One can crack Caesar Box by testing all possible size of the rectangle.
Sometimes the message has a square number of characters (16 = 4.4 or 25 = 5 * 5 or 36 = 6 * 6, etc.), which makes it possible to deduce the size of the square, but sometimes it is a totally different
number of characters.
What are the variants of the Caesar Box cipher?
When the box is a perfect square, encryption and decryption are identical.
The scytale is the other name of this cipher.
When was Caesar Box invented?
This encryption is similar to that of the scytale cipher, which have appeared in Greece, between the 10th and 7th centuries B.C., a long time before romans and Caesar (Caius Iulius).
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Hitmarkers Modlet
Hey everyone!
here's a modlet I made that add hitmarkers (cross in the center of the screen that dissapear quickly) on succesful shots similar to some famous fps game.
the mod will show White hitmarkers on body shot, Red on headshots and Green on allies/players shot.
(only works with ranged weapon of course)
here's a video showing it ingame with a little surprise in the end (which is for a later mod):
and a quick gif if you don't have time for this:
Haven't tested it thoroughly (and not on multiplayer) let me know if you have any problems.
Download link (A18) : Click Here !
I am sorry I don't see what your trying to show
haha I kinda expected that
the thing in the center appears everytime you hit successfully an enemy with a ranged shot.
it will turn red on headshot, white in the body and green if you're hitting another player.
basically it's an hit indicator that can be really useful when doing things like long-range archery/sniping.
I am sorry I don't see what your trying to show
Really? You cant see the plainly obvious, clear as day thing shown in the gif? The crosshairs turn red when he lands headshots, lets you know when u are landing your headshots.
So weird. This guy that smells like onion in soup form also just released a mod just like this!
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
haha I kinda expected that
the thing in the center appears everytime you hit successfully an enemy with a ranged shot.
it will turn red on headshot, white in the body and green if you're hitting another player.
basically it's an hit indicator that can be really useful when doing things like long-range archery/sniping.
Argh yep I see it now lol old eyes lol first vid I missed it the several times i watched but just then when ya used the sniper I saw the crosshair go red. But yeh than others it still looks white lol
- - - Updated - - -
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
Or asking like I did. :-D
well while it's white on the body it's red in the head.
but whether it's white or red this crosshair in the middle isn't in the game originally I just made it look similar to the one we already have.
well while it's white on the body it's red in the head.
but whether it's white or red this crosshair in the middle isn't in the game originally I just made it look similar to the one we already have.
Sweet thank you great idea and addition :-)
don't know if that's sarcastic Maynard but it is barely visible on the gif because of compression sadly.
best bet is watching the video or trying it yourself
Wasnt being sarcastic, I see it clearly in the gif.
When I take my glasses off, ya. But I am pretty much blind then.
I updated the GIF with another one in higher resolution now it's really clear as day
I updated the GIF with another one in higher resolution now it's really clear as day
I see the red cross hairs lol
hey everyone!
Poor Vader:sorrow: Nice job on the modlet though, can't wait for the later one too.
Great modlet..but how can you tease us like that...Darth Vader...Stormtroopers....AAAAAAHHHHHHHHH. Interest piqued.
Quick update, fixed the last hitmarker appearing when leaving/entering a vehicle.
also made it trigger on Rocket Launcher.
probably won't update this one for a while now unless any other bugs get reported or maybe to add different appearance.
Is the Health Bar also your modlet? I'd like to give it a try.
Nope you can find it in here it's called Telric's Health Bar
So weird. This guy that smells like onion in soup form also just released a mod just like this!
Unable to open archive file: 7 Days to Die Dedicated
Sorry gonna need more informations here
• 1 month later...
nice Vader mod, I've been trying to put Jason into the game to no avail.
This topic is now archived and is closed to further replies. | {"url":"https://community.7daystodie.com/topic/15875-hitmarkers-modlet/","timestamp":"2024-11-03T10:20:11Z","content_type":"text/html","content_length":"293034","record_id":"<urn:uuid:09e6e93e-67e8-4d96-b21e-f7d518f9bb0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00278.warc.gz"} |
• Toivola, M. (2024). Flipped Assessment in Engineering Mathematics. Proceedings of the 20th International CDIO Conference, hosted by Ecole Supérieure Privée d’Ingénierie et de Technologies
(ESPRIT) Tunis, Tunisia, June 10 – June 13, 2024
• Toivola, M. (2024). Flipped Assessment in Engineering Physics. Pre-reading material for the PTEE 2024 conference (flipped conference presentation format)
• Toivola, M., Rajala, A. & Kumpulainen, K. (2022). Pedagogical Rationales of Flipped Learning in the Accounts of Experienced Mathematics Teachers, Pedagogies
• Toivola, M. (2021). Kannanotto formatiivisen matematiikan arvioinnin puolesta, Psykologia, 56(6), pp. 650-655
• Toivola, M. (2020). Flipped Assessment - A Leap towards Flipped learning. An article in conference proceedings Brandhofer, G., Buchner, J., Freibleben-Teutscher, C. & Tengler, K (Hrsg.)
Tagungsband zur Tagung Inverted Casroom and beyond 2020, Baden, Austria
• Toivola, M. (2020). Flipped Assessment - A Leap towards Assessment for Learning, Edita
• Toivola, M. (2019). Käänteinen arviointi, Edita
• Toivola, M. (2019). Käänteinen oppiminen – kääntyykö koulutyö päälaelleen? artikkeli teoksessa toim. Löytönen, M. & Timo Tossavainen, T. Sähköistyvä koulu. Oppiminen ja oppimateriaalit
muuttuvassa tietoympäristössä, Tietokirjailijaliitto
• Toivola, M. (2019). Matematiikan osaamisen perusta on pystyvyyden tunteessa. Artikkeli teoksessa toim. Harmanen, M. & Hartikainen, M. Monilukutaitoa oppimassa, Opetushallitus
• Toivola, M. (2017). Käänteinen oppiminen ja formatiivinen arviointi matematiikassa. Artikkeli teoksessa toim. Vitikka, E & Kauppinen, E. Arviointia toteuttamassa, Opetushallitus
• Toivola, M., Peura, P & Humaloja, M (2017). Flipped learning in Finland, Edita
• Toivola, M., Peura, P & Humaloja, M (2017). Flipped learning – Käänteinen oppiminen, Edita
• Toivola, M. & Silfverberg, H. (2016). The Espoused Theory of Action of an Expert Mathematics Teacher Using Flipped learning. 13th International Congress on Mathematical Education (ICME). Hamburg.
• Toivola, M. (2016). Flipped learning -why teachers flip and what are their worries? Experiences of Teaching with Mathematics, Sciences and Technology, 2(1), 237-250.
• Toivola, M., & Silfverberg, H. (2015). Flipped learning –approach in mathematics teaching – a theoretical point of view. Proceedings of the Symposium of Finnish Mathematics and Science Education
Research Association, Oulu.
• Toivola, M. & Härkönen, T. (2012). Avoin matematiikka. Kymmenen yläkoulun avointa matematiikan oppikirjaa, julkaistu CC-BY lisenssillä.
Math teacher of the year 2019 in Finland publications | {"url":"http://www.flippedassessment.com/p/publications.html","timestamp":"2024-11-07T17:32:59Z","content_type":"text/html","content_length":"91028","record_id":"<urn:uuid:503d8cda-0ce5-4e6d-b251-3c0762b53a81>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00779.warc.gz"} |
Cluster Analysis
Cluster analysis involves using a community-finding algorithm to partition the network graph into clusters (densely-connected subgraphs). These clusters represent groups of clones/cells with similar
receptor sequences.
Cluster analysis can be performed when calling buildRepSeqNetwork() by setting cluster_stats = TRUE or as a separate step using addClusterStats().
When performing cluster analysis, each cluster is assigned a numeric cluster ID, and the cluster membership of each node is recorded as a variable in the node metadata. Properties are computed for
each cluster, such as total node count, mean sequence length, the sequence with the greatest network degree, and various centrality indices of the cluster’s graph. The cluster metadata for these
properties is included as its own data frame contained in the list of network objects.
Simulate Data for Demonstration
We simulate some toy data for demonstration.
We simulate data consisting of two samples with 100 observations each, for a total of 200 observations (rows).
dir_out <- tempdir()
toy_data <- simulateToyData()
#> CloneSeq CloneFrequency CloneCount SampleID
#> 1 TTGAGGAAATTCG 0.007873775 3095 Sample1
#> 2 GGAGATGAATCGG 0.007777102 3057 Sample1
#> 3 GTCGGGTAATTGG 0.009094910 3575 Sample1
#> 4 GCCGGGTAATTCG 0.010160859 3994 Sample1
#> 5 GAAAGAGAATTCG 0.009336593 3670 Sample1
#> 6 AGGTGGGAATTCG 0.010369470 4076 Sample1
Performing Cluster Analysis
With buildRepSeqNetwork()/buildNet()
Calling buildRepSeqNetwork() or its alias buildNet() with cluster_stats = TRUE is one way to perform cluster analysis.
Cluster Membership
After using either of the methods described above, the node metadata now contains a variable cluster_id with the values for cluster membership:
Cluster Properties
The output list now includes an additional data frame cluster_data containing the cluster metadata:
#> [1] "cluster_id" "node_count"
#> [3] "mean_seq_length" "mean_degree"
#> [5] "max_degree" "seq_w_max_degree"
#> [7] "agg_count" "max_count"
#> [9] "seq_w_max_count" "diameter_length"
#> [11] "global_transitivity" "assortativity"
#> [13] "edge_density" "degree_centrality_index"
#> [15] "closeness_centrality_index" "eigen_centrality_index"
#> [17] "eigen_centrality_eigenvalue"
head(net$cluster_data[ , 1:6])
#> cluster_id node_count mean_seq_length mean_degree max_degree seq_w_max_degree
#> 1 1 14 13.00 3.36 9 AAAAAAAAATTGC
#> 2 2 28 12.96 8.43 18 GGGGGGGAATTGG
#> 3 3 9 12.67 2.22 4 AGAAGAAAATTC
#> 4 4 6 13.00 3.33 9 GGGGGGAAATTGG
#> 5 5 6 12.00 2.17 3 AGGGAGGAATTC
#> 6 6 25 12.00 4.60 10 AAAAAAAAATTG
A brief description of each cluster-level property is given below:
• node_count: The number of nodes in the cluster.
• mean_seq_length: The mean sequence length in the cluster.
• mean_degree: The mean network degree in the cluster.
• max_degree: The maximum network degree in the cluster.
• seq_w_max_degree: The receptor sequence possessing the maximum degree within the cluster.
• agg_count: The aggregate count among all nodes in the cluster (based on the counts in count_col, if provided).
• max_count: The maximum count among all nodes in the cluster (based on the counts in count_col, if provided).
• seq_w_max_count: The receptor sequence possessing the maximum count within the cluster.
• diameter_length: The longest geodesic distance in the cluster.
• assortativity: The assortativity coefficient of the cluster’s graph, based on the degree (minus one) of each node in the cluster (with the degree computed based only upon the nodes within the
• global_transitivity: The transitivity (i.e., clustering coefficient) for the cluster’s graph, which estimates the probability that adjacent vertices are connected.
• edge_density: The number of edges in the cluster as a fraction of the maximum possible number of edges.
• degree_centrality_index: The cluster-level centrality index based on degree within the cluster graph.
• closeness_centrality_index: The cluster-level centrality index based on closeness, i.e., distance to other nodes in the cluster.
• eigen_centrality_index: The cluster-level centrality index based on the eigenvector centrality scores, i.e., values of the principal eigenvector of the adjacency matrix for the cluster.
• eigen_centrality_eigenvalue: The eigenvalue corresponding to the principal eigenvector of the adjacency matrix for the cluster.
Abundance-Based Properties
Some cluster-level network properties, such as agg_count and max_count, are only computed if the user specifies a column of the input data containing a measure of abundance for each row (e.g., clone
count for bulk data or Unique Molecular Identifier count for single-cell data). This column is specified using the count_col function, which accepts a column name or column index.
In case the data includes more than one count variable, net$details$count_col_for_cluster_data specifies which of these variables corresponds to the count-based cluster properties:
Clustering Algorithm
By default, clustering is performed using igraph::cluster_fast_greedy(). Other clustering algorithms can be used instead of the default algorithm. In buildRepSeqNetwork() and addClusterStats(), the
algorithm is specified using the cluster_fun parameter, which accepts one of the following values:
• "fast_greedy" (default)
• "edge_betweenness"
• "infomap"
• "label_prop"
• "leading_eigen"
• "leiden"
• "louvain"
• "optimal"
• "spinglass"
• "walktrap"
For details on the clustering algorithms, see ?addClusterMembership().
It is possible when using addClusterStats() to specify non-default argument values for optional parameters of the clustering functions.
Cluster Membership Only
addClusterMembership() is similar to addClusterStats(), but only adds cluster membership values to the node metadata. It does not compute cluster properties.
Multiple Instances of Clustering
Using addClusterMembership() or addClusterStats(), it is possible to perform cluster analysis with different clustering algorithms and record the cluster membership from each instance of clustering.
When performing cluster analysis, the cluster_id_name parameter specifies the name of the cluster membership variable added to the node metadata. This allows the cluster membership values from each
instance of clustering to be saved under a different variable name.
# First instance of clustering
net <- buildRepSeqNetwork(toy_data, "CloneSeq",
print_plots = FALSE,
cluster_stats = TRUE,
cluster_id_name = "cluster_greedy"
# Second instance of clustering
net <- addClusterMembership(net,
cluster_fun = "louvain",
cluster_id_name = "cluster_louvain"
net <- addPlots(net,
color_nodes_by = c("cluster_greedy", "cluster_louvain"),
color_scheme = "Viridis",
size_nodes_by = 1.5,
print_plots = TRUE
Currently, keeping cluster properties from multiple instances of clustering analysis is not supported. addClusterStats() can overwrite existing cluster properties using overwrite = TRUE.
The details element of the network list helps with keeping track of clustering results and cross-referencing the cluster properties with the node properties.
#> $seq_col
#> [1] "CloneSeq"
#> $dist_type
#> [1] "hamming"
#> $dist_cutoff
#> [1] 1
#> $drop_isolated_nodes
#> [1] TRUE
#> $nodes_in_network
#> [1] 122
#> $clusters_in_network
#> fast_greedy louvain
#> 20 20
#> $cluster_id_variable
#> fast_greedy louvain
#> "cluster_greedy" "cluster_louvain"
#> $cluster_data_goes_with
#> [1] "cluster_greedy"
#> $count_col_for_cluster_data
#> [1] NA
#> $min_seq_length
#> [1] 3
#> $drop_matches
#> [1] "NULL"
net$details$cluster_id_variable tells which cluster membership variable corresponds to which clustering algorithm.
net$details$cluster_data_goes_with tells which cluster membership variable corresponds to the cluster metadata.
Labeling Clusters
To more easily reference the clusters within a visual plot, clusters can be labeled with their cluster IDs using labelClusters().
The list of network objects returned by buildRepSeqNetwork() is passed to the net parameter. By default, all plots contained in the list net$plots will be annotated, but a subset of plots may be
specified as a vector of the element names or positions to the plots parameter.
The name of the cluster membership variable in the node metadata is provided to the cluster_id_col parameter (the default is "cluster_id").
By default, only the 20 largest clusters by node count are labeled in order to preserve legibility. This number can be changed using the top_n_clusters parameter.
Instead of ranking clusters by node count, a different criterion can be used, provided the list net contains cluster properties corresponding to the cluster membership variable. The name of a numeric
cluster property is provided to the criterion parameter. The ranking order can be reversed using greatest_values = FALSE.
The size of the cluster ID labels can be adjusted by providing a numeric value to the size parameter (the default is 5), and their color can be changed by providing a valid character string to the
color parameter.
# First instance of clustering
net <- buildNet(toy_data, "CloneSeq",
print_plots = FALSE,
cluster_stats = TRUE,
cluster_id_name = "cluster_greedy",
color_nodes_by = "cluster_greedy",
color_scheme = "Viridis",
size_nodes_by = 1.5,
plot_title = NULL
# Second instance of clustering
net <- addClusterMembership(net,
cluster_fun = "louvain",
cluster_id_name = "cluster_louvain"
net <- addPlots(net,
color_nodes_by = "cluster_louvain",
color_scheme = "Viridis",
size_nodes_by = 1.5,
print_plots = FALSE
# Label the clusters in each plot
net <- labelClusters(net,
plots = "cluster_greedy",
cluster_id_col = "cluster_greedy",
top_n_clusters = 7,
size = 7
net <- labelClusters(net,
plots = "cluster_louvain",
cluster_id_col = "cluster_louvain",
top_n_clusters = 7,
size = 7
#> Warning: Removed 115 rows containing missing values or values outside the scale range
#> (`geom_text()`). | {"url":"https://pbil.univ-lyon1.fr/CRAN/web/packages/NAIR/vignettes/cluster_analysis.html","timestamp":"2024-11-10T01:56:40Z","content_type":"text/html","content_length":"106788","record_id":"<urn:uuid:3f6503cb-6213-47c2-9879-26dbe162a8c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00111.warc.gz"} |
Universal bayes consistency in metric spaces
We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes consistent in all metric spaces admitting any such
learner, making it an “optimistically universal” Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, the k-NN classifier and its variants are
not generally universally Bayes consistent, except under additional structural assumptions, such as an inner product, a norm, finite dimension or a Besicovitch-type property. The metric spaces in
which universal Bayes consistency is possible are the “essentially separable” ones-a notion that we define, which is more general than standard separability. The existence of metric spaces that are
not essentially separable is widely believed to be independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal
Bayes-consistent learner for the given metric space. In particular, this yields the first impossibility result for universal Bayes consistency. Taken together, our results completely characterize
strong and weak universal Bayes consistency in metric spaces.
• Bayes consistency
• Classification
• Metric space
• Nearest neighbor
Dive into the research topics of 'Universal bayes consistency in metric spaces'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/universal-bayes-consistency-in-metric-spaces-4","timestamp":"2024-11-06T05:06:55Z","content_type":"text/html","content_length":"54905","record_id":"<urn:uuid:ce67fc92-df15-4713-9cd0-426bf1aa8e9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00135.warc.gz"} |
Sports Betting Guide - Online Sports Betting Basics | Winner Gambling
Sports Betting Guide
Sports betting is a fun way to seek extra excitement to all sorts of sports and entertainment events giving you the chance also to win money. What’s really been popular lately is live betting, which
is betting in real time during the event. This sports betting guide goes through everything you need to know to start sports betting online. It’s more profitable to do the betting online because the
odds are better and it’s easy compared to conventional bookers. Why bother walking to the local booker when you can make the same bets online from your own couch, even with better odds?
For those who bet professionally, betting is a constant battle against the bookers. The first thing you must come to terms with is the concept of value bets and non-value bets. For anyone attempting
to build a career in betting, probability estimates must be clear.
Whether you’re playing for entertainment purposes or to win money, you should still read this guide. It has been drawn up by experienced professional bettors. If you like, you can just skip to the
list of the best sportsbooks online or sign-up to our featured sportsbook below:
BASICS – Sports Betting Guide
Odds are presented using either the European format (decimal odds) or the American format (fractional odds). Below is a useful Excel formula for converting the odds from one format to another.
Fractional -> Decimal = IF(A1<0;ROUND(1+100/ABS(A1);2);(A1+100)/100)
Decimal -> Fractional = IF(A2<2;-ROUND(1/(A2-1)*100;0);IF(A2=2;”Even”;A2*100-100))
The odds are converted so, that if there’s a plus (+) in front of the American odd, the odd is divided by a hundred (100), and 1 is added to the fraction, for example: +120=1+(120/100)=2.20. If
there’s a minus (-) in front, the 100 is divided by this odd and 1 is added to the fraction, for example: -120=1+(100/120)=1.83.
Game forms
Several different types of games have been developed for players to bet on. Here are some of the most common ones.
In 1×2 you bet most commonly on the outcome of a match between two real opponents. Choices are a win, or a tie, in which 1 means home win (or the team marked as home team), X is a tie, 2 visitor (or
a team marked as visitor team). 1×2 bets are almost always for after the actual playing time has ended, in football the score after the second half-time, in ice-hockey after the third round and so
Handicaps have been offered for a long time abroad. Companies offering handicaps usually offer better odds than 1×2-bets.
The idea of Handicaps
In a handicap you bet on a single match is a tournament or an event for the winner or a tie, considering the handicap. Handicap games can in certain cases be closely compared with the 1×2 games,
however not all handicaps have to chance of a tie for either the bet of the outcome. In football and ice hockey the handicap is usually given in a form of 0.0, 0.5, 1.0 or 1.5 goals, in basketball
the handicaps are usually larger.
Examples of handicaps:
In a 0.0 goal handicap you only bet on the winner, so it’s not really a proper handicap, more like a money line – a bet which you can read about further down. A 0.5 handicap is probably the most
common form of handicap betting. In this case the booker gives half a goal handicap to one of the teams. This naturally shows in the setting of the odds. As an example case a football match Arsenal
vs. Liverpool, with half a goal handicap given to Liverpool. A bettor betting for Liverpool wins, if the game is a tie or if Liverpool wins. E.g. an outcome of 0-0 would mean that the match ended
with a score of 0-0.5. If Arsenal wins, the bet is lost. An outcome of 1-0 would mean that the bet ended with 1-0-5. In hockey these handicaps almost always covers the overtime. As an example,
Detroit-Colorado, with 0.5 handicap given to Colorado; in this case the bettor who bet for Detroit will win, if Detroit wins at the latest, during overtime. Tie or Colorado winning (during playing
time or overtime) would mean loss for those who bet for Colorado. In hockey the most common handicap is 1.5 goals. So even if Colorado gets a 1.5 handicap, it would still be enough for those who bet
for Colorado if the game ends by one goal by Detroit. For example if the game ends 4-3, for the bet the outcome would be 4-4.5.
Asian Handicap Bets
+0 This is a “zero handicap”. You win, if the team you’ve bet on wins and you’ll lose if they lose. If it’s a tie you’ll get your money back (push).
-0,25 Consists of +0 and -0,5 handicaps. You’ll win with your whole bet if your team wins. You’ll lose the whole bet if they lose. If it’s a tie you’ll get half of your money back (losing the -0,5
but getting the money back for the +0 bet).
-0,5 Means straight win. You’ll win if the team you’ve bet on wins, and lose if they tie or lose.
-0,75 Consists of half -0,5 and half -1 handicaps. You’ll win with the whole bet if the team you’ve bet on wins with at least two goals. You’ll lose the whole bet if your team ties or loses. With a
one goal win you’ll win half of the bet and get the other half back (winning the -0,5 bet and getting the bet back for the -1 bet).
-1 You’ll win if your team wins with at least two goals, or you’ll lose if your team ties or loses. If your team wins with one goal, you’ll get your bet back.
-1,25 Consists of -1 and -1,5 handicaps. You’ll win with the whole bet if your team wins by at least two goals and you’ll lose the whole bet if your team ties or loses. With one goal win you’ll get
half of your bet back and lose the other half (back from the -1 bet and lose the -1,5 bet).
-1,5 You’ll win if your team wins with at least two goals, and lose if they win with one goal, ties or loses.
-1,75 Consists of -1,5 and -2 handicaps. You’ll win with the whole bet if your team wins by at least 3 goals and lose the whole bet if your team wins with one goal, ties, or loses. With an even two
goal win you’ll win half of the bet and get the other half back (win the -1,5 bet and get the -2 back).
-2 You’ll win, if your team wins by at least 3 goals and lose if they win with one goal, tie or lose. With an even two goal win you’ll get your bet back.
+0,25 Consists of +0 and +0,5 handicaps. You’ll win the whole bet if your team wins and you’ll lose the whole bet if they lose. If it’s a tie you’ll win with half the bet and get the other half back
( win the +0,5 bet and money back for the +0 bet).
+0,5 You’ll win if your team wins or ties. You’ll lose if your team loses.
+0,75 Consists of +0,5 and +1 handicaps. You’ll win with the whole bet, if your team ties or wins. You’ll lose the whole bet if your team loses by at least two goals. With one goal defeat you’ll lose
one half and get the other half back (lose the +0,5 bet and get +1 bet money back).
+1 You’ll win if your team ties of wins. You’ll lose if your team loses by two goals. With an even one goal defeat you’ll get your money back.
+1,25 Consists of +1 and +1,5 handicaps. You’ll win with the whole bet if your team ties or loses. You’ll lose the whole bet if your team loses by at least two goals. With an even one goal defeat
you’ll win with half the bet and get the other half back (win with the +1,5 bet and get your money back for the +1 bet).
+1,5 You’ll win if your team loses by one goal, ties or loses. You’ll lose if your team loses by at least two goals.
+1,75 Consists of +1,5 an +2 handicaps. You’ll win with the whole bet if your team loses by one goal, ties or wins. You’ll lose the whole bet if your team loses by at least three goals. With an even
two goal defeat you’ll lose half the bet and get the other half back (lose the +1,5 and money back for the +2 bet).
+2 You’ll win if your team loses by one goal, ties of wins. You’ll lose if your team loses by at least three goals. With an even two goal defeat you’ll get your money back.
Handicaps can be bigger in certain cases but the principle remains the same.
Total Goal Bets (under/over)
Over/Under bet is for betting if over or under x goals will be scored. The most common form is the so called line, in hockey 5.5 goals, and 2,5 in football, although there are other possible lines.
In hockey 5.5 goals over/under the result is under, if there are max 5 goals scored (the game ending e.g. 1-0, 2-1, 3-2, 3-0 or 2-3) or over, the at least 6 goals are scored (e.g. 2-4, 5-1, 6-0, 8-3
or 5-3).
There are other lines too. Hockey offers a bet over/under 5, when you bet if the game scores a total of max 4 of at least 6 goals. If 5 goals are scores, the bet is returned regardless if you’ve bet
over or under.
u1,5 You’ll win, if the match ends without goals or one goal is scored. You’ll lose if at least two goals are scored.
u1,75 Consists of u1,5 and u2 types. You’ll win with the whole bet if the match ends without goals or one goal is scored. You’ll lose the whole bet if at least three goals are scored. With an even
two goals scored you’ll lose half of the bet and get the other half back (lose u1,5 bet, money back for u2)
u2 You’ll win if the match ends without goals or one goal is scored. You’ll lose if at least three goals are scored. With even two goals you’ll get the bet back.
u2,25 Consists of u2 and u2,5 types. You’ll win with the whole bet if the match ends without goals or with one goal scored. You’ll lose the whole bet if at least three goals are scored. With an even
two goals scored you’ll get half the bet back and win with the other half (lose u2,5 bet, money back for u2,5).
u2,5 You’ll win if max two goals are scored and lose if at least three goals are scored.
u2,75 Consists of u2,5 and u3 types. You’ll win with the whole bet if max two goals are scored. You’ll lose the whole bet if at least four goals are scored. With an even three goals scored you’ll
lose half the bet and get the other half back ( lose u2,5 and money back for u3).
u3 You’ll win, if max two goals are scored. You’ll lose if at least four goals are scored. With even three goals you’ll get your money back.
u3,25 Consists of u3 and u3,5 types. You’ll win with the whole bet is max two goals are scored. Youll lose the whole bet if at least four goals are scored. With even three goals you’ll get half the
bet back and win with the other half (money back for u3 and win with u3,5 bet).
u3,5 You’ll win if max three goals are scored and you’ll lose if at least four goals are scored.
o1,5 You’ll win if at least two goals are scored. You’ll lose if no goals are scored or only one.
o1,75 Consists of o1,5 and o2 types. You’ll win with the whole bet if at least three goals are scored. You’ll lose the whole bet if there are no goals, or just one. With an even two goals you’ll win
with half the bet and get the other half back (win o1,5 and money back for o2).
o2 You’ll win, if at least three goals are scored. You’ll lose, if there are no goals or just one. With two goals you’ll get your money back.
o2,25 Consists of o2 and o2,5 types. You’ll win with the whole bet is at least three goals are scored. You’ll lose the whole bet if there are no goals or just one. With even two goals you’ll get half
the bet back and lose the other half (money back for o2 and lose o2,5).
o2,5 You’ll win if at least three goals are scored and lose if max two goals.
o2,75 Consists of o2,5 and o3 types. You’ll win with the whole bet if at least four goals are scored. You’ll lose the whole bet if max two goals are scored. With even three goals you’ll win with half
and get the other half back (win o2,5 and money back for o3).
o3 You’ll win if four goals are scored and lose if max two goals. With even three you’ll get your money back.
o3,25 Consists of o3 and o3,5 types. You’ll win with the whole bet if at least four goals are scored and lose the whole bet if max two goals are scored. With even three goals you’ll get half the bet
back and lose the other half (money back for o3 and lose o3,5).
o3,5 You’ll win if at least four goals are scored and lose if max three.
Marking Differences Between Bookmakers
Asian handicap and total-goals markings differ little between betting companies. In the beginning you should be careful to leave in the right bet. Here are some examples on the marking bookers used
in this service have.
Pinnacle Sports:
Asian handicap: 0, 0 and -0.5 (read -0,25), -0.5, -0,5 and -0,75 (read -0,75) etc.
Total Goals: Under 2 and 2.5 (read u2,25), under 2,5, under 2.5 and 3 (read u2,75) etc.
Asian handicap: 0, 0/0.5 (read -0,25), 0.5 (read -0,5), 0,5/1 (read -0,75) etc.
Total-goals: 2/2.5 (read o2,5), 2.5 (read o2,5), 2.5/3 (read o2,75) etc.
Moneyline is somewhat related to handicaps. In ML-bet you only bet the winner of the game, in hockey taking into account overtime, in football the overtime or penalty shots aren’t taken into account.
At the rnd of a tie the bets are returned, despite who you’ve bet on. Some companies have places ML-bets with handicaps, the handicap being 0.0 goals.
Totalizators or flexible-rate bets
These totalizators are risk-free for the bookmaker. In these flexible-rate bet games the odds are influenced by the share of exchange each of the results have received taking in to account the return
rate of bookmaker offering the game. If the returned percentage is let’s say 80%, this means that of the amount placed on the bet, 80% will be given back to bettors as winnings. Naturally the more
money bet on the option, the smaller the coefficient.
Example of a totalizator:
Football game Liverpool-ManU, the target of the bet is the outcome with the number of goals scored. This target is played with a total of 100 000 euros, so as winnings 80 000 will be dealt. The final
score is 1-1, which has been bet with a total of 10 000 euros. This 1-1 scores represents 10% of all the bets made. Taking into account the return percentage of 80%, the 1-1 score would have the
coefficient of 8. This comes by counting the 10 000 share of the 100 000 euros, which is 10%, the reciprocal will make a coefficient of 10, which still need to be divided by 0,8 (the return
percentage). This is how every other final score could be counted, depending on the amounts of bets made on them.
In so called h2h bets, the bet are made competitors competing against each other rather than winning the certain race or an event. The Formula 1 races have become a popular bet, where you bet on two
or three drivers and their placement among the others. Depending on the bookmaker you can also bet a “tie”, in which one or both drivers either have the same time, drop out, or get disqualified.
Example of a h2h bet:
F1 race, Felipe Massa vs. Kimi Räikkönen. Depending on the case you can bet on either a driver’s placement or on the placement which also includes the option of a “tie” bet. In the former the bet is
returned (unless stated otherwise) if neither of the drivers finishes or both get disqualified. In the latter the race is a tie, if neither of the drivers finishes or both get disqualified.
F1 race, Felipe Massa vs. Kimi Räikkönen vs. Heikki Kovalainen. This is a bet on who makes the bet placement out of the three. Again it depends on each case whether or not the bet is returned if none
of them gets a placing or all get disqualified.
The more special sporting event bets are called exotics. These can be e.g. who makes the opening goal, the score (in football) for the first half, (in hockey) after the first or second half etc. You
should find out more from the bookmakers of these exotic bets, because they are always on a case-by-case basis.
Example on exotics:
Football match Germany-Italy. The bet is on the score in case of a home win, tie, or the away team win after the first and second halftime. This makes it a 1×2 bet but on the score after the first
half. Possible scores therefore are (first half/second half) 1/1, X/X, 2/2, X/1, X/2, 1/X, 1/2, 2/X and 2/1.
Mythicals bets are offered at least on football matches. These can be somewhat compared to h2h bets, because mythicals are about betting between two competitors when they are not actually competing
against each other.
Example on mythicals:
Spanish league round, which one scores more goals, Real Madrid or Barcelona? Real Madrid is playing a home game against Valencia, Barcelona also a home game against Deportivo. The options are 1=Real
Madrid, X=even number of goals, 2=Barcelona. Note, that your team can lose their own match and still win as a bet. For example, Real Madrid – Valencia ends 1-0, and Barcelona – Deportivo 2-4,
Barcelona has still scored more goals than Real Madrid, so as a bet, the winning bet would be 2=Barcelona. Mythical bets can also be offered between two teams playing in different leagues, or between
3 teams. Again it should be noted that this all depends on the bookmaker.
Non-value bet
Non-value bet is the opposite of value bet, so it is unprofitable for the bettor in the long run. Non-value bet is an odd that is smaller than the reciprocal of its probability. When faced with a
non-value bet you should try to avoid it and not deal with it, because that is the only way to avoid problems.
Value bet
Value bet is the basic element of betting. Value bet is an odd which is bigger than the reciprocal of its probability. If you’ve for example calculated the result percentages to be 40-20-40 for the
match Detroit – Boston then the appropriate odds for this results are 2.5-5-2.5. And therefore find a 2.7 odd for Detroid, you get to play Value bet. Value bet is the best way to bet in the long run.
Probability estimate
Probability estimates are the cornerstone of profitable betting. If you think, for example, that in the Detroit-Boston game Detroit will win with more than 50% probability; you have in your mind made
a probability estimate. Professional bettors can’t place their judgments solely on images of team, they you a lot of time observing the team and analyzing all the data effecting the game (goalies,
conditions, field players, strain etc.). This is how they end up with an estimated percentage (probability estimate) on the strengths of the teams on which to base the betting decision. For example,
on this particular case an estimate of 25-25-50 means, that Boston wins 25 of the 100 played games, 25 will end up in a tie and Boston will win 50 in these series. Remember that the league table
doesn’t always give a good enough a coverage to make these estimates, it can only be the very base to which build on.
So, how to form a probability estimate? We’ll unveil the mystery a bit.
Values and how they are formed in NHL bets
1) Before the season starts, form estimation for every team, by taking into account the offenders, defenders, goalies and other elements that affect the team. You’ll then produce an estimate between,
say, 1-10. It’s not easy, but you can get more than enough information online to make these decisions. Estimations are the basement of the coming season, so make sure to be thorough.
2) In NHL, some games are more important to teams than others. Know which ones are important, and in which the players might get more rest. Conferences, divisions and other basic thoughts you must
know by heart.
3) Know the players. Goalies, regular first team players and their absence form a large deal of the power of the team. Never forget to check the line-up and consider which place is missing players or
how their absence will affect the number of goals made.
4) Identify your mistakes. If a team which you labeled lousy goes from victory to victory, don’t lose all your money on a mistake you made last summer. A smart bettor recognizes the mistakes made and
learns from them. Only the dumb ones keep beating their heads against the wall.
Power rating
Power ratings are used to measure differences between teams. Odd compiler performes by calculating power rating for teams, which are based strictly on the facts available on the teams. When
calculating the power ratings a bettor must be aware and deal with hard facts between the teams, and the league table alone is not enough to make these calculations. A wise odd compiler counts in
both home and visitor” –power rating and by comparing these end up with the final percentage estimates on the levels of the teams.
Odd comparison
Once you’ve made your estimates on the probabilities of the teams, it’s time to look for value bets. The internet offers many tools to help you out comparing odds between different bookmakers.
Kelly formula
The Kelly formula is one of the corner stones of gambling: B = (pk-1) / (k-1). B = the wager, p= the probability of winning and k= the odds received on the wager. So, with a fraction of 1000€ of the
bankroll to wager for a team with a 40% probability, the odds offered by the bookmaker 2.7 we’ll get =>(0.4*2.7-1) / (2.7-1) => the percentage of the bankroll to be waged would be 4.7%=>47€. It is
not recommended to use this Kelly/1 formula in betting, because it is too risky. Read more from Kelly divider part.
Kelly divider
The Kelly divider is often used with the Kelly formula to reduce the risk Kelly formula guarantees an optimal wager but it also requires extreme accuracy of the probabilities which is sports betting
is often difficult. Therefore a clever bettor reduces the risk by using the Kelly divider. Put into a Kelly formula => B = ((pk-1) / (k-1)) / 5 we’ve used the divider 5. Everyone must set their own
divider, but it’s never a bad thing to be one the safe side. Recommended dividers are for example 6 or 8. You can use kelly calculator to determine bet sizes to your bets.
“SAFE BETS”
Remember, that there are no safe bets!
Basic information
Arbitrage is about a so called sure win. This is possible, when the same target is being offered large odds for all the outcomes. A simple case is a target with two options for outcomes – e.g. a
tennis match. Let’s assume a match Sampras-Nadal, bookmaker A offeres 2.20 for Federer, bookmaker B 2.00 for Nadal, this makes an arbitrage: 1/2,20+1/2,00=0,955.
Therefore with a 0,955 unit makes 1 unit, in other words in the long run this kind of investment would profit 1/0,955=1,047=4,7 percent. It’s not a great way to make lots of money, but using
arbitrages you’ll get some growth into your bankroll or you can use these surebets to clear bonuses. Arbitrage comes with certain risks, read more below.
Arbitrage betting
You don’t bet arbitrage by Kelly formula, but by how much “sure win” you’re expecting. In our example bookmaker A gives Federer a winning odd of 2,20, and bookmaker B a winning odd of 2,00 for Nadal.
You must take into account the reciprocals of these odds and the estimated sure win. Let’s calculate:
Federer: (1/2,20)*1,047 = 47,6% Nadal: (1/2,00)*1,047 = 52,4%
These percentages refer to the fraction of the wager of the bankroll. E.g. is you have a bankroll of 10 000€, you’ll bet 4760€ for Federer to win, and 5240 for Nadal. If Federer wins, you’ll get 10
472€, if Nadal, you’ll get 10 480€. The difference between these sums comes from round offs in the percentages, but you’ve still make risk free growth in your bankroll. In each case you’ve grown
almost 5% in your bankroll. There’s a surebet calculator available on this site, which you can use to count if there is an arbitrage opportunity available in a match or to calculate optimal bet sizes
for all outcomes.
Arbitrage risk
Even though we’ve mentioned sure wins, arbitrages carry a risk. This appears when it turns out the bookmaker has made a mistake in the odds. This kind of clear mistake can happen if the event
bookmakers have informed the bets wrong or even made a mistake about the whole match. A good example is a situation, when you choose odds for a match between David Nalbandian and Jarkko Nieminen from
bookmakers A and B. A gives Nalbandian 1,30 odds and Nieminen 4,30, B gives Nalbandian 1,90 and Nieminen 1,90. This is probably a situation when B has gives Nalbandian-Nieminen match the odds that
should go for an entire different match, say for example Federer-Nadal. This of course makes an arbitrage by playing A for Nieminen (odds 4,30) and B for Nalbandian (odds 1,90), but it’s likely that
the bookmaker B will correct his mistake by cancelling the bet. This bet then no longer exists and the wager is returned. This causes a great risk, if the bet made with bookmaker A (Nieminen win,
odds 4,20) still stands; if Nalbandian wins, the bet “Nieminen 4,20” lost and the bet “Nalbandian 1,90” cancelled. So you’ve lost the Nieminen fraction of your bankroll.
Another possible arbitrage due to a bookmakers mistake is setting the odds the wrong way. For example bookmaker A gives the Nalbandian-Nieminen match odds 1,30-4,30 and bookmaker B the same match
4,30-1,30, it’s obvious that B will cancel these wrongly places odds. Don’t play arbitrages unless you know what you’re doing!
Using the Poisson formula
Poisson, Siméon-Denis (1781 – 1840) developed a formula, which enables you to calculate probabilities of continual cases. Later several mathematicians have discovered that the Poisson formula can be
used for ballgames (mainly football and hockey) to calculate the probability of goals.
Poisson formula:
P(x) = (e^-µ * µ^x) / x!
x = the number of cases, e.g. number of goals 3
x! = factorial of the number x, e.g. 3! = 1*2*3 = 6
It is also set that 0! = 1
µ = Goal estimation 2,70
e = Neper figure, invariable (~2,7183)
P(3) = (2,7183^-2,70*2,70^3) / 3! ~ 0,220 = 22,0 %
Excel also has a function which helps to use Poisson.
The function above gives the result of the same 22% we got above.
Therefore according to Poisson the team will score exactly 3 goals with a 22% probability, if the expected goal value is estimated at 2,70. This way you can calculate the probabilities for both teams
and for every number of goals. After that you must multiply the probabilities with each other, thus getting the probabilities for different outcome combinations.
Example: If the home team scores 3 goals with 22% probability and the away team 2 goals with 26% probability, the probability of the outcome 3-2 is 0,22*0,26 = 0,0572 = 5,7%.
By adding up all the home wins, you get the probability of a home win with given estimated goal values. Similarly you can calculate the probabilities for a tie and the away wins.
Correct score betting calculator is available on this website.
Precise tracking on the bets played is your advantage. Accounting in this computer era is very easy, with Excel spreadsheet programs or with it’s free of charge cousin, Open Office. Basic tracking
should give you the following details: Date, Match, Game form, Bet, Bet Size and Outcome. For example it’s easy to develop tools like the Kelly formula for yourself with spreadsheets to e.g. define
the wagers. It’s also not a waste of time to make a list of factors that affected the game in the games played. You can measure your own skills with precise tracking when you make thousands of bets.
Tracking is extremely important if you try to achieve profitable gambling.
Return percentage
Return percentage for a bettor means the profit he/she has made compared to the wager. When for example in a series of 100 matches the amount of money you’ve made is 100.000€ and the money you’ve
invested is 90.000€ you can say you’ve gained 111% return percentage on the money you’ve wagered (100.000€ / 90.000€). You can tell a profitable bettor if he makes more than 100% return percentage in
a long run.
Bookmakers return percentage refers to the percentage which the bookmaker returns to the bettor as winnings.
Long run
You’ll often come across this term in gambling, and it means series of thousands of bets which the bettor has played. When a bettor decides to buy/seek tips, one should estimate the quality of the
person giving the tips by how well he’s done in the “long run”, that is in thousands of bets he’s made. This way you’ll get a better picture of the quality of his analysis.
Now after reading this guide, you are ready to start making accounts to online sportsbooks and placing bets. We have listed the best sports betting sites at the best sportsbooks section of the site. | {"url":"https://www.winnergambling.com/sports-betting-bingo/sports-betting-guide/","timestamp":"2024-11-09T01:24:46Z","content_type":"application/xhtml+xml","content_length":"62976","record_id":"<urn:uuid:95970edb-8bec-4f69-a761-a0cf492740f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00128.warc.gz"} |
a1x+b1y+c2z=d1a2x+... | Filo
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Question Text CRAMER'S RULE \# NON-HOMOGENEOUS SYSTEM Prepl:
Updated On Apr 24, 2024
Topic Matrices and Determinant
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 143
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Ultralight Propeller Thrust Calculator - GEGCalculators
Ultralight Propeller Thrust Calculator
The thrust of an ultralight propeller can vary depending on its design, size, and the engine it’s paired with. A typical ultralight propeller may produce around 40 to 80 pounds of thrust, but
specific values can vary significantly based on these factors.
Ultralight Propeller Thrust Calculator
Engine Size (HP) Estimated Propeller Thrust (lbs)
20 35-50
40 60-80
60 90-120
80 120-160
100 150-200
1. How do you calculate propeller thrust? Propeller thrust can be estimated using the following formula: Thrust (in pounds) = Propeller Efficiency × Power (in horsepower) / Airspeed (in miles per
2. How do I calculate how much thrust I need? Estimate the thrust required by considering factors such as aircraft weight, desired speed, and aerodynamic properties. A rough estimation might be 0.2
to 0.3 times the weight of the aircraft.
3. How much thrust per horsepower propeller? The thrust per horsepower can vary, but a rough estimate is around 4-5 pounds of thrust per horsepower for a typical propeller.
4. How does propeller size affect thrust? Larger propellers generally produce more thrust due to their increased surface area and the ability to move more air.
5. How do you calculate thrust from a propeller and a motor? Thrust from a propeller and motor combination depends on factors such as motor power, propeller efficiency, and airspeed. Use the formula
mentioned in question 1 to estimate it.
6. How do you calculate thrust from PSI? Thrust from PSI (Pounds per Square Inch) is not directly calculable without additional information about the surface area and geometry over which the
pressure is acting.
7. What is the standard thrust equation? The standard thrust equation for a jet engine is: Thrust = (Mass Flow Rate of Exhaust × Exhaust Velocity) – (Mass Flow Rate of Inlet Air × Inlet Air
8. How do you calculate thrust with weight? Thrust should ideally match or exceed the weight of the object (e.g., an aircraft) for it to overcome gravity and achieve lift-off.
9. What is the formula for propeller power? Propeller power (in horsepower) can be estimated using the formula: Power = (Thrust × Airspeed) / Propeller Efficiency
10. How much horsepower is 1 lb of thrust? Roughly, 1 lb of thrust is equivalent to 0.2 to 0.25 horsepower.
11. How many lbs of thrust is equal to 1 hp? Approximately, 1 horsepower is equal to 4-5 lbs of thrust for a typical propeller.
12. How many HP is 50lb thrust? Around 50 lbs of thrust would be roughly equivalent to 10-12.5 horsepower.
13. What size propeller is an ultralight aircraft? Ultralight aircraft can have varying propeller sizes, but they are generally smaller, with diameters in the range of 48 to 68 inches.
14. Does a bigger prop mean more thrust? Yes, generally, a bigger propeller can produce more thrust due to its increased surface area and the ability to move more air.
15. Is a 19 or 21 pitch prop faster? A 21-pitch propeller is typically faster than a 19-pitch propeller, as it is designed to generate more speed at the cost of some initial acceleration.
16. How much thrust does a Cessna 172 produce? A Cessna 172 typically produces around 2,000 to 2,300 pounds of thrust.
17. How do you calculate motor thrust? Motor thrust can be estimated based on factors like motor power, propeller efficiency, and airspeed. Use the formula mentioned in question 1 for a rough
18. Why are toroidal propellers better? Toroidal propellers are more efficient and quieter due to their unique design, which reduces vortex and tip losses.
19. Is thrust a force or pressure? Thrust is a force. It is the force exerted in a specific direction to move an object, like an aircraft.
20. Is thrust equal to pressure? No, thrust is not equal to pressure. Pressure is the force per unit area, while thrust is the force acting in a specific direction.
21. What is the formula for thrust of a turboprop engine? The thrust of a turboprop engine can be estimated using a more complex formula that considers engine characteristics, propeller efficiency,
and airspeed. It’s not a simple equation.
22. What is the formula and unit of thrust? The formula for thrust is typically expressed as force (in newtons or pounds-force) and is calculated by the product of mass flow rate and exhaust
23. What is 1lb of thrust? 1 lb of thrust is equivalent to a force that can lift a 1-pound object against the force of gravity.
24. What is the relationship between thrust and weight? Thrust must at least equal the weight of an object for it to overcome gravity and achieve lift-off or maintain level flight.
25. What is a good thrust-to-weight ratio? A good thrust-to-weight ratio for an aircraft depends on its purpose. For commercial airliners, it’s typically around 0.3-0.4, while high-performance
fighter jets can have ratios above 1.
26. What is the relationship between propeller thrust and torque? Propeller thrust is related to torque but also depends on other factors like propeller efficiency and airspeed. The relationship is
not linear.
27. How do you calculate propeller velocity? Propeller velocity is not typically calculated directly. It depends on the design and rotational speed of the propeller.
28. What is the output power of a propeller? Output power of a propeller depends on various factors like thrust, airspeed, and efficiency. Use the formula mentioned in question 9 to estimate it.
29. How many HP is 55 lbs of thrust? Roughly, 55 lbs of thrust would be equivalent to 11-13.75 horsepower.
30. How fast is 100 pounds of thrust? The speed achieved with 100 pounds of thrust would depend on the specific application and other factors like weight and aerodynamics.
31. How many HP is 86 lb thrust? Approximately, 86 lbs of thrust would be equivalent to 17-21.5 horsepower.
32. How fast will a 55lb trolling motor go? The speed of a boat with a 55lb thrust trolling motor depends on the boat’s size, weight, and other factors. It can vary widely.
33. How fast is 40 lbs of thrust? The speed achieved with 40 lbs of thrust would depend on the boat’s characteristics, so it can vary.
34. How many pounds of thrust does a 747 engine produce? A typical Boeing 747 engine can produce around 63,000 to 66,500 pounds of thrust.
35. What size boat for a 55lb thrust? A 55lb thrust trolling motor is typically suitable for small to medium-sized boats, such as canoes, kayaks, or small fishing boats.
36. How many HP is a 40 lb thrust trolling motor? Roughly, a 40 lb thrust trolling motor would be equivalent to 8-10 horsepower.
37. How much horsepower is 10,000 pounds of thrust? Approximately, 10,000 pounds of thrust would be equivalent to 2,000-2,500 horsepower.
38. What is the best ultralight plane? The “best” ultralight plane can vary based on individual preferences and needs. Some popular ultralight planes include the Quicksilver GT500, Challenger II, and
the Zenith CH 701.
39. Is it legal to build your own ultralight aircraft? In many countries, it is legal to build and fly your own ultralight aircraft as long as you adhere to the regulations and safety requirements
specific to your location.
40. What is the top speed of the ultralight aircraft? The top speed of an ultralight aircraft can vary widely depending on the specific model, but it is typically around 55-65 mph (88-105 km/h).
41. Does a 4 blade prop increase thrust? Generally, a 4-blade propeller can provide more thrust compared to a 2-blade propeller, especially at lower speeds, but it may also have higher drag.
42. What happens if prop pitch is too high? If the prop pitch is too high, the engine may struggle to reach its maximum RPM, resulting in reduced thrust and overall performance.
43. What happens if your propeller is too big? If the propeller is too big for the engine or the aircraft, it can lead to inefficiencies, decreased performance, and potential damage to the engine.
44. What is the most efficient propeller pitch? The most efficient propeller pitch depends on the specific application and the desired balance between thrust and speed. It varies for different
aircraft and engines.
45. What pitch prop gives more speed? A lower pitch propeller typically provides more speed, as it allows the engine to achieve higher RPMs and generate more forward thrust.
46. What propeller pitch is best for takeoff? A higher pitch propeller is generally better for takeoff, as it provides the necessary thrust and acceleration.
47. Why did Cessna stop making the 172? As of my last update in January 2022, Cessna had not stopped making the 172. However, decisions regarding aircraft production can change over time.
48. How many G’s can a Cessna 172 handle? A Cessna 172 is typically designed to withstand forces of up to +3.8 Gs and -1.52 Gs in normal flight conditions.
49. Why is the Cessna 172 so easy to fly? The Cessna 172 is considered easy to fly due to its stable and forgiving flight characteristics, making it a popular choice for flight training. It is
designed for safety and ease of operation.
50. How do you measure propeller thrust? Propeller thrust is typically measured using specialized equipment such as a thrust stand, which measures the force generated by the propeller in a controlled
51. What are the disadvantages of toroidal propellers? While toroidal propellers have advantages, they can be more complex and expensive to manufacture compared to conventional propellers. They may
also have limited availability.
52. Which propeller produces more thrust? The propeller that produces more thrust depends on its design, size, and intended application. There is no one-size-fits-all answer.
53. What is the most efficient propeller shape? The efficiency of a propeller depends on its design and intended use. Various shapes can be efficient for different purposes.
54. How do you convert pressure to thrust? Converting pressure to thrust would require information about the area over which the pressure is acting. It’s not a straightforward conversion.
55. How do you calculate thrust from pressure? To calculate thrust from pressure, you would need to know the surface area over which the pressure is applied and use the formula: Thrust = Pressure ×
56. Is thrust a pull or a push? Thrust is a push force, pushing an object in the direction of its motion.
57. Is thrust directly proportional to pressure? Thrust is not directly proportional to pressure; it depends on various factors including the surface area over which the pressure is applied.
58. What is the mathematical relation between pressure and thrust? The mathematical relation between pressure and thrust depends on the specific context and the surface area over which the pressure
is acting.
59. How does thrust differ from pressure? Thrust is the force exerted in a specific direction, while pressure is the force per unit area. Thrust results from a pressure differential.
60. What is the simple thrust formula? The simple thrust formula for a jet engine is: Thrust = Mass Flow Rate of Exhaust × (Exhaust Velocity – Inlet Air Velocity).
61. What is 1lb of thrust? 1 lb of thrust is equivalent to a force that can lift a 1-pound object against the force of gravity.
62. What is the standard unit of thrust? The standard unit of thrust is the newton (N) in the International System of Units (SI) or the pound-force (lbf) in the United States customary units.
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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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Rotate Elements in Matrix – Interview Question
Abhi Jain September 26, 2018 Uncategorized 0
Rotate Elements in a Matrix.
Let’s find out how to answer this question in a technical interview.
Let’s take a sample matrix
If we rotate it, this is how it will look like
In order to write this program, let’s see how we did it. We first rotated the outermost ring, one element each in clockwise direction. Next, we rotated the second ring, again one element each in the
clock wise direction. So we need to think in terms of one ring at a time.
So if we were to do it for only one ring, our pseudo code will be something like this:
1. Read element at matrix[1][0]. Let’s call it previous.
2. Loop through first row. Put previous in the beginning of the row. Move elements to the right. Keep the last element of first row in previous.
3. Loop through last column. Leaving the first row. Put previous as the first element. Move other elements to the bottom. Keep the last element of last column in previous.
4. Loop through last row right to left, leaving the last column. Put previous as the first element. Move all elements tot the left. Keep the first element of last row in previous.
5. Loop through first column bottom up. Put previous as first element. Move all elements up except the topmost element which was covered earlier.
Now if we want to loop through one ring at a time, we simply put an outer loop starting from layer 0 to layer n/2.
The above will work only for a square matrix. If it were a rectangular matrix, we can’t do 0 to n/2. For e.g.
will be:
We will have to do a while loop. With each ring we can keep incrementing the row and column number where we are at. And we will have to keep decrementing the maxrow and maxColumn count once we finish
a ring. If the row is less than maxrow and col is less than max column, then we continue.
Void RotateMatrix(int[][] matrix, int RowsNum, int ColsNum){
Row = 0
Col = 0
MaxRow = RowsNum – 1
MaxCol = ColsNum – 1
While(row < MaxRow && col < MaxCol){
If(row+1 == MaxRow || col+1 == MaxCol)
Int previous = matrix[row+1][col]
//TOP ROW
For(int i = col; i <= Maxcol; i++){
Int current = mat[row][i];
Mat[row][i] = previous;
previous = current;
//RIGHT COLUMN
For(int j = row; j <= maxRow; j++){
Int current = matrix[j][maxCol];
matrix[j][maxCol] = previous;
previous = current;
//BOTTOM ROW
if(row<maxRow + 1){
For(int i = maxCol; i >= col; i–){
Int current = mat[maxRow][i];
Mat[maxRow][i] = previous;
previous = current;
//LEFT COLUMN
If(col < maxCol + 1){
For(int i = maxRol; i >= row; i–){
Int current = mat[i][col];
Mat[i][col] = previous;
previous = current;
I have the actual running C# code on Github here.
Please make sure to test run it after writing the code and take care of edge conditions. If you have followed these steps then I would say you have successfully answered the interview question.
For more such questions, please subscribe to coach4dev. Until next time, Happy Coding!!!
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What is a second (s or sec)?
second (s or sec)
What is a second (s or sec)?
The second (s or sec) is the International System of Units (SI) unit of time measurement.
One second is the time that elapses during 9,192,631,770 (or 9.192631770 x 10^9 in decimal form) cycles of the radiation produced by the transition between two levels of the cesium-133 atom.
How is a second expressed?
There are other expressions for the second. It is the time required for an electromagnetic (EM) field to propagate 299,792,458 meters (2.99792458 x 10^8 m) through a vacuum.
This figure is sometimes rounded to 3 x 10^8 m, or 300,000 kilometers (3 x 10^5 km). One second is equal to 1/86,400 of a mean solar day. This is easy to derive from the fact that there are 60
seconds in a minute, 60 minutes in an hour and 24 hours in a mean solar day.
This definition is, however, subject to limited accuracy because of irregularities in the earth's orbit around the sun.
A second is typically the base unit of time measurement for everyday events.
Are their units of time smaller than seconds?
Engineers and scientists often use smaller units of measurement than the second by attaching power-of-10 prefix multipliers. One millisecond is 10^-3 seconds; 1 microsecond is 10^-6 seconds; 1
nanosecond is 10^-9 seconds; and 1 picosecond is 10^-12 seconds.
During these spans of time, respectively, an EM field propagates through a vacuum over distances of approximately 300 km, 300 m, 300 millimeters and 300 micrometers.
Use cases for seconds
A second is typically the base unit of time measurement in timing for events that occur in everyday life, such as the following:
• a person's heart rate (beats per minute);
• a car's speed (miles per hour or kilometers per hour); and
• cooking time frames (minutes and seconds).
The smaller subdivisions are usually used for events that occur rapidly, such as the following:
• electron impulses in nerves and muscles;
• light waves; and
• gamma ray bursts.
The second is also a common base unit for expressing extremely long spans of time, such as geologic time units, like the millennium, mega-annum, or giga-annum.
The second is sometimes specified as a unit of angular measure, especially in astronomy and global positioning.
In these contexts, it is also known as an arc second or a second of arc and is equal to exactly 1/3,600 of an angular degree or 1/1,296,000 of a circle.
Sixty arc seconds constitute an arc minute; 60 arc minutes constitute an angular degree. One arcsecond of latitude at the earth's surface corresponds to a north-south distance of only about 31 m.
At the equator, 1 second of longitude corresponds to an east-west distance of about 26 m.
See also: table of physical constants, atomic clock, radiant energy, pascal and newton.
This was last updated in June 2022
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Learn Computer Science
I have a confession to make. I, (and most of us) have been stuck on fad diets for CS. It’s so nice to read easy books, like learn C++ in 21 days, or doing some leetcode problems. You learn a bit,
feel that warm glow of getting smarter, and trick yourself into feeling productive.
You’re not really getting any better, without learning the fundamentals.
I decided to cobble up a set of resources to go through, as a fusion of Teach Yourself Computer Science and Steve Yegge’s recommendations.
Teach Yourself CS has a detailed list of good resources for the more practical aspects of CS, whereas Steve Yegge’s recommendations focus more on the mathematical side – the first four courses he
recommends are: discrete math, linear algebra, statistics, and theory of computation.
Steve Yegge’s list omits Computer Architecture, which is a glaring omission – an Operating Systems course doesn’t have enough time to cover all the interesting parts of concurrency, parallelism, and
optimization that a computer architecture course would.
Teach Yourself CS doesn’t mention Theory of Computation, and is lighter on the math background, giving one resource for math. Theory of Computation is a bit more dated (swallowed up by all the other
fields), but is still useful for its applications.
To that end, I’ve fused them both, and skimmed some resources to put on this list.
This list is incomplete and changing all the time, but hey, isn’t that what agile development is all about?
1. Structure and Interpretation of Computer Programs
Discrete Math
1. Discrete Mathematics, an open introduction
Linear Algebra
1. No bullshit guide to linear Algebra, Savov
1. Statistics fourth ed, Friedman et. al
2. Think stats, Downey
3. Think Bayes, Downey
Theory of Computation
1. Theory of Computation, Hefferon
2. Computational Complexity, Arora and Barak
Computer Architecture
1. Computer Architecture, a Programmers Perspective
2. Computer Architecture, Patterson and Hennessey
1. Some Assembly Required
Algorithms and Data Structures
1. The Algorithm Design Manual, Skiena
2. The Art of Multiprocessor programming
Operating Systems
1. Operating Systems, Three Easy Pieces
1. Computer Networks, a Systems approach
1. Database Internals
1. Crafting Interpreters
Distributed Systems
1. Designing Data Intensive Applications | {"url":"https://takashiidobe.com/gen/learn-computer-science","timestamp":"2024-11-03T16:26:44Z","content_type":"text/html","content_length":"8691","record_id":"<urn:uuid:cb4a66f7-b4a9-4d97-9376-d749fb45d5dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00312.warc.gz"} |
Central, East and - South-East European-PhD Network » Masaryk University
1. Consumer Theory (10 lecture hours)
• Consumption space, utility function, indifference curves, marginal rate of substitution, budget constraint (GR 2A-B)
• Utility maximization, Marshallian demand and its comparative statics (GR 2C-D)
• Expenditure minimization, Hicksian demand, expenditure function, comparative statics for Hicksian demand (GR 3A)
• Duality, Slutsky equation, income and substitution effects (GR 3B)
2. Producer Theory (6 lecture hours)
• Production functions, returns to scale (GR 5)
• Cost minimization, conditional factor demands, cost function (GR 6.A,B,E)
• Profit maximization, output supply, input demands, profit function (GR 7.A,C,D)
3. General Equilibrium (4 lecture hours)
• Introduction (GR 12.A-B)
• Illustrative examples of GE: exchange economy and Robinson Crusoe economy (GR 12.E)
• Pareto efficiency and welfare theorems (GR 13.A,B,D,E)
4. Choice under uncertainty (4 lecture hours)
• Choice under uncertainty (GR 17)
• Properties of utility function (GR 17)
• Risk aversion (GR 17) | {"url":"https://ceseenet.org/courses/previous-courses/courses-winter-term-2016/masaryk-university/index.htm","timestamp":"2024-11-04T15:29:11Z","content_type":"application/xhtml+xml","content_length":"69599","record_id":"<urn:uuid:f9c90fcc-d3c6-484d-8ade-00b4a1db9087>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00456.warc.gz"} |
Program in C to find if given number is odd or even | Tutorend
First of all, letโ s understand what is odd and even.
#Odd Number
Odd numbers are those which are not divisible by 2 or we cannot split them in half without leaving a fractional number.
If Number%2 != 0
then Number is odd
% here represents the modulo operator which returns the remainder of the dividend.
For example, we cannot divide 3 by 2 without leaving a reminder.
Odd numbers are
1 3 5 7 9 11 13 15 17 19 21 23 25 27 .....
#Even Number
An even number is just the opposite of an odd number, we can divide even numbers with 2 without leaving any remainder or leaving 0 as the remainder.
If Number%2 == 0
then Number is even
For example, 4 is completely divisible by 2 so it is an even number.
Even numbers are
2 4 6 8 10 12 14 16 18 and so on...
#Program to find Odd or Even
We already know that % or modulo operator returns the remainder, in the case of our problem, we just need to find the remainder of the given number upon dividing by 2. If the remainder is 0 then the
given number is even otherwise itโ s odd.
#include <stdio.h>
int main() {
int num;
printf("Enter an integer: ");
scanf("%d", &num);
if(num % 2 == 0) {
printf("%d is even.", num);
else {
printf("%d is odd.", num);
return 0;
In the above program, we are getting a number from the user as input and checking its remainder by dividing by 2 and printing out the Number even if it is divisible by 2 otherwise Number is odd.
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Linear-time register allocation for a fixed number of registers
We show that for any fixed number of registers there is a linear-time algorithm which given a structured (≡goto-free) program finds, if possible, an allocation of variables to registers without using
intermediate storage. Our algorithm allows for rescheduling, i.e. that straight-line sequences of statements may be reordered to achieve a better register allocation as long as the data dependencies
of the program are not violated. If we also allow for registers of different types, e.g. for integers and floats, we can give only a polynomial time algorithm. In fact we show that the problem then
becomes hard for the W-hierarchy which is a strong indication that no O(n^c) algorithm exists for it with c independent on the number of registers. However, if we do not allow for rescheduling then
this non-uniform register case is also solved in linear time.
Original language English
Title of host publication Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1998
Editors Howard J. Karloff
Place of Publication Philadelphia, PA, United States
Publisher SIAM
Pages 574-583
Number of pages 10
Publication status Published - 1 Dec 1998
Event Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms - San Francisco, CA, USA, United Kingdom
Duration: 25 Jan 1998 → 27 Jan 1998
Conference Proceedings of the 1998 9th Annual ACM SIAM Symposium on Discrete Algorithms
Country/Territory United Kingdom
City San Francisco, CA, USA
Period 25/01/98 → 27/01/98
Dive into the research topics of 'Linear-time register allocation for a fixed number of registers'. Together they form a unique fingerprint. | {"url":"https://research-portal.uu.nl/en/publications/linear-time-register-allocation-for-a-fixed-number-of-registers","timestamp":"2024-11-14T01:38:49Z","content_type":"text/html","content_length":"50336","record_id":"<urn:uuid:e8e0cf9a-891c-4ad2-bf20-1b74f39c49dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00293.warc.gz"} |
David W. Dreisigmeyer
A direct search method for the class of problems considered by Lewis and Torczon [\textit{SIAM J. Optim.}, 12 (2002), pp. 1075-1089] is developed. Instead of using an augmented Lagrangian method, a
simplicial approximation method to the feasible set is implicitly employed. This allows the points our algorithm considers to conveniently remain within an \textit{a priori} … Read more
We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special
orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems
and, to the Whitney … Read more
We present a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in
differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to … Read more
We extend direct search methods to optimization problems that include equality constraints given by Lipschitz functions. The equality constraints are assumed to implicitly define a Lipschitz
manifold. Numerically implementing the inverse (implicit) function theorem allows us to define a new problem on the tangent spaces of the manifold. We can then use a direct search … Read more | {"url":"https://optimization-online.org/author/dreisigm/","timestamp":"2024-11-02T23:44:33Z","content_type":"text/html","content_length":"89972","record_id":"<urn:uuid:8c355fe0-55bb-437b-98b5-5c0d6e81cee2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00519.warc.gz"} |
A way to infer multiplication?
A way to infer multiplication?
This might be asking for too much, but is there a way to interpret 5x as 5*x when 5x is not defined as an object? It gets really cumbersome writing *'s all the time. Is there a way to do this
currently or is in the plans?
1 Answer
Sort by ยป oldest newest most voted
This can be set with implicit_multiplication() function
edit flag offensive delete link more
Well done. You can accept your own answer to mark the question as solved.
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Week 3 discussion - Course Help Online
• Simplify each expression using the rules of exponents and examine the steps you are taking.
• Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in
sentences describing the thought behind your math work.Principal rootProduct ruleQuotient ruleReciprocalnth rootRefer to Inserting Math Symbols for guidance with formatting. Be aware with regards
to the square root symbol, you will notice that it only shows the front part of a radical and not the top bar. Thus, it is impossible to tell how much of an expression is included in the radical
itself unless you use parenthesis. For example, if we have √12 + 9 it is not enough for us to know if the 9 is under the radical with the 12 or not. Therefore, we must specify whether we mean it
to say √(12) + 9 or √(12 + 9), as there is a big difference between the two. This distinction is important in your notation.Another solution is to type the letters “sqrt” in place of the radical
and use parenthesis to indicate how much is included in the radical as described in the second method above. The example above would appear as either “sqrt(12) + 9” or “sqrt(12 + 9)” depending on
what we needed it to say.
Your initial post should be at least 250 words in length.
(3) Simplifying Expressions involving Variables
Simplify each expression. Assume the variables represent any
real numbers and use absolute value as necessary. See
Example 8
55. (x4)1/4
56. ()1/6
57. (*)/2
58. (110)1/2
59. (*)/3
60. (W)1/3
61. (9x4y2)1/2
62. (164/41/4 | {"url":"https://coursehelponline.com/week-3-discussion/","timestamp":"2024-11-13T21:45:09Z","content_type":"text/html","content_length":"40973","record_id":"<urn:uuid:dee321a6-cd56-4392-ba74-c440d151f6bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00785.warc.gz"} |
• Hello everybody,
I am trying to model a fully discretized borehole heat exchanger (coaxial). The diameter of the outer ring changes stepwise with the depth. The outer pipe is modeled as a vertical 2D discrete
How do I model the change of the diameters of the outer ring? The discrete element representing the smaller diameter is not underneath the element representing the bigger diameter. This is
because the discrete elements are positioned in the centre of the surplus, the center, of course, differs between the two diameters.
Thanks a lot for your help
Nils Deecke
• Hello Everybody,
I want to modell a fully discreted bore hole heat exchanger, and callculate the extracted energy.
So I inject water at a well(s) with a constant temperatur (BC1 (heat) and BC4 (flow)). At another well I withdraw water again (BC1(flow) or BC4(flow)). On the way between the wells the water gets
How can I callculate the energy the borehole heat exchanger is extracting? So i want feflow to do something like this:
E= Q*T[sub]in[/sub]*C - Q*T[sub]out[/sub]*C = delta(T)*Q*C
E= Heat (J/d)
Q=discharge (m³/d)
C=Heat capacity (J/°K/m³)
This should be possibe with the budget analyzer but how?
Thanks a lot for your answers/comments
special thanks to Mr.Rühaak, your are really helping me with my thesis
• Hallo everybody,
How does the anisotropy factor, for the heat conductivity work?
In which direction do I define the heat conductivity, if I assign a value for the heat conductivity (fluid, solid) in the heat materials menu?
Is it the z-direction (vertical)? And the horizontal (x,y) is calculated by dividing the assigned heat conductivity value by the anisotropy factor?
e.g. I need to model a heat conductivity of 0.1(J/m/s/K) in z-direction( vertical) and a heat conductivity of 10 in x,y-direction,
what values do I have to use for the heat condutivity and for the anisotropy factor?
Thanks a lot for help
• Hallo,
I want to modell a fully discreted coaxial borehole heat exchanger. To model the waterflow in the tube I want to use 1D discrete elements.
Is this a right way to modell this problem?
I connect all nods within the tube (of each slice) with horizontal 1D discrete elements. And I connect all the all slices of the heat exchanger with vertical 1D discrete elements. I do so for
both the pipe-in and the pipe out, just seperated by the pipe wall between. At the bottom of the heat exchanger, where the water changes its flow direction, I connect all the nods of the entire
base area with horizontal 1D discrete elements.
Is this the right way? Or should I use just one vertical 1D element with a corresponding hugher diameter?
What kind of wells (4th BC) do I need for the pipe-in (outer part of the coaxial exchanger) and for the pipe out (inner part)?
What diameter do I have to use for the vertical 1D element (if I use an element for each node)?
Thanks a lot for your answers, I also happy with comments or partial answers.
input of fixed coordinates which are outside the actual desktop view
I want to create a superelements whitch dimensions are 10cm * 1km. It is not possible for me to enter the fixed coordinates because the points are outside of the desktop. I also cannot skroll
down durring creating the superelement.
What can I do.
Thanks a lot
• Hallo,
Is it possible to model a borehole heat exchanger with varying diameters. eg. the first 100 meters with a diameter of 0.2m and from 100 to 200 m below surface a diameter of 0.15m.
Nils Deecke | {"url":"https://support.dhigroup.com/public/0f543950-aae2-ec11-bb3c-000d3ab6f805/forum-posts","timestamp":"2024-11-08T21:11:20Z","content_type":"text/html","content_length":"61756","record_id":"<urn:uuid:35a2847e-8722-46b8-a55d-af35cedf4d0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00651.warc.gz"} |
Leditzky Research Group
Recent papers
Approximate Unitary k-Designs from Shallow, Low-Communication Circuits
In this paper we give constructions of approximate unitary designs from shallow, low-communication circuits. A unitary k-design is a set of unitaries that approximates the first k moments of the Haar
measure, the unique left- and right-invariant measure on the unitary group that can be understood as a uniform distribution on unitaries. Our construction is recursive and uses overlapping Haar
twirls on subsystems that are analyzed using von Neumann’s alternating projection method.
Multivariate Fidelities
Theshani Nuradha, Hemant K. Mishra, Felix Leditzky and Mark M. Wilde
The bivariate fidelity between quantum states is an important measure of distance between quantum states that can be used to quantify how well a certain information-theoretic task approximates a
target state. In this paper we focus on multivariate fidelities, that is, distance measures for a collection of quantum states. We give various multivariate generalizations of the classical and
quantum bivariate fidelity and prove a range of useful properties like the data-processing inequality and symmetry under exchange of arguments, and inequalities relating the different multivariate
fidelities to each other. For a particular example, the multivariate log-Euclidean fidelity, we also give an operational interpretation in the context of quantum hypothesis testing.
Operational Nonclassicality in Quantum Communication Networks
In a quantum communication network the senders (or nodes) can send each other quantum or classical information, assisted by shared entanglement, local processing and measurements. A crucial question
is whether a communication network is genuinely quantum (or nonclassical), in the sense that its behavior could only be reproduced by a purely classical network if additional resources (such as extra
classical communication) are available. In this paper we devise a framework based on linear constraints that characterizes classical communication networks with given constraints on the classical
communication between nodes. Violating these classical constraints with quantum resources then certifies the nonclassicality of a given network, much like the violation of a Bell inequality in a
bipartite scenario. We achieve these operational certifications of nonclassicality using variational quantum algorithms, which can be implemented on available quantum hardware. The code for this
paper is available on GitHub. | {"url":"https://publish.illinois.edu/leditzky-group/research/","timestamp":"2024-11-10T18:30:48Z","content_type":"text/html","content_length":"29521","record_id":"<urn:uuid:9e7043d0-5dab-4e16-8944-c5f03547a364>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00666.warc.gz"} |
Binary Division Calculator
Solved Example :
The below solved example may used to understand how perform the division between 2 binary numbers.
Example Problem
Divide the binary number A = 1010[2] by B = 10[2] & find the quotient.
In digital electronics & communications, the arithmetic operation between binary number systems play vital role to perform various operations. The above truth table & solved example for binary
division may useful to understand how to perform such calculations, however, when it comes to online, this binary division calculator may useful to perform such computations as easy & quick as | {"url":"https://ncalculators.com/digital-computation/binary-division-calculator.htm","timestamp":"2024-11-13T14:39:46Z","content_type":"text/html","content_length":"51841","record_id":"<urn:uuid:88304737-ec17-4fae-a80c-86a559ce204c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00487.warc.gz"} |
Randomized Search Explained - Python Sklearn Example - Analytics Yogi
Randomized Search Explained – Python Sklearn Example
In this post, you will learn about one of the machine learning model tuning technique called Randomized Search which is used to find the most optimal combination of hyper parameters for coming up
with the best model. The randomized search concept will be illustrated using Python Sklearn code example. As a data scientist, you must learn some of these model tuning techniques to come up with
most optimal models. You may want to check some of the other posts on tuning model parameters such as the following:
In this post, the following topics will be covered:
• What and why of Randomized Search?
• Randomized Search explained with Python Sklearn example
What & Why of Randomized Search
Randomized Search is a yet another technique for sampling different hyper parameters combination in order to find the optimal set of parameters which will give the model with most optimal performance
/ score. As like Grid search, randomized search is the most widely used strategies for hyper-parameter optimization. Unlike Grid Search, randomized search is much more faster resulting in
cost-effective (computationally less intensive) and time-effective (faster – less computational time) model training.
It is found that the randomized search is more efficient for hyper-parameter optimization than the grid search. Grid search experiments allocate too many trials to the exploration of dimensions that
do not matter and suffer from poor coverage in dimensions that are important. Read this paper for more details – Random search for hyper parameter optimization.
In this post, randomized search is illustrated using sklearn.model_selection RandomizedSearchCV class while using SVC class from sklearn.svm package.
Randomized Search explained with Python Sklearn example
In this section, you will learn about how to use RandomizedSearchCV class for fitting and scoring the model. Pay attention to some of the following:
• Pipeline estimator is used with steps including StandardScaler and SVC algorithm.
• Sklearn dataset related to Breast Cancer is used for training the model.
• For each parameter, a distribution over possible values is used. The scipy.stats module is used for creating the distribution of values. In the example below, exponential distribution is used to
create random value for parameters such as inverse regularization parameter C and gamma.
• Cross-validation generator is passed to RandomizedSearchCV. In the example given in this post, the default such as StratifiedKFold is used by passing cv = 10
• Another parameter, refit = True, is used which refit the the best estimator to the whole training set automatically.
• The scoring parameter is set to ‘accuracy’ to calculate the accuracy score.
• Method, fit, is invoked on the instance of RandomizedSearchCV with training data (X_train) and related label (y_train).
• Once the RandomizedSearchCV estimator is fit, the following attributes are used to get vital information:
□ best_score_: Gives the score of the best model which can be created using most optimal combination of hyper parameters
□ best_params_: Gives the most optimal hyper parameters which can be used to get the best model
□ best_estimator_: Gives the best model built using the most optimal hyperparameters
import pandas as pd
import numpy as np
from sklearn import datasets
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.model_selection import RandomizedSearchCV
from sklearn.model_selection import GridSearchCV
from sklearn.svm import SVC
import scipy as sc
# Load the Sklearn breast cancer dataset
bc = datasets.load_breast_cancer()
X = bc.data
y = bc.target
# Create training and test split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=1, stratify=y)
# Create the pipeline estimator
pipeline = make_pipeline(StandardScaler(), SVC(random_state=1))
# Create parameter distribution using scipy.stats module
param_distributions = [{'svc__C': sc.stats.expon(scale=100),
'svc__gamma': sc.stats.expon(scale=.1),
'svc__kernel': ['rbf']},
{'svc__C': sc.stats.expon(scale=100),
'svc__kernel': ['linear']}]
# Create an instance of RandomizedSearchCV
rs = RandomizedSearchCV(estimator=pipeline, param_distributions = param_distributions,
cv = 10, scoring = 'accuracy', refit = True, n_jobs = 1,
# Fit the RandomizedSearchCV estimator
rs.fit(X_train, y_train)
print('Test Accuracy: %0.3f' % rs.score(X_test, y_test))
One can find the best parameters, score using the following command:
# Print best parameters
# Print the best score
Here are some of the learning from this post on randomized search:
• Randomized search is used to find optimal combination of hyper parameters for creating the best model
• Randomized search is a model tuning technique. Other techniques include grid search.
• Sklearn RandomizedSearchCV can be used to perform random search of hyper parameters
• Random search is found to search better models than grid search in cost-effective (less computationally intensive) and time-effective (less computational time) manner.
Latest posts by Ajitesh Kumar
(see all)
I found it very helpful. However the differences are not too understandable for me
Very Nice Explaination. Thankyiu very much,
in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
Thank you....for your support. you given a good solution for me.
Ajitesh Kumar
I have been recently working in the area of Data analytics including Data Science and Machine Learning / Deep Learning. I am also passionate about different technologies including programming
languages such as Java/JEE, Javascript, Python, R, Julia, etc, and technologies such as Blockchain, mobile computing, cloud-native technologies, application security, cloud computing platforms, big
data, etc. I would love to connect with you on Linkedin. Check out my latest book titled as First Principles Thinking: Building winning products using first principles thinking.
Posted in Data Science, Machine Learning, Python. Tagged with Data Science, machine learning, python, sklearn. | {"url":"https://vitalflux.com/randomized-search-explained-python-sklearn-example/","timestamp":"2024-11-13T15:03:59Z","content_type":"text/html","content_length":"109335","record_id":"<urn:uuid:0c2d8612-7f52-4f08-b813-f3ba2a2401b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00742.warc.gz"} |
Recursive Stripes
Inspired by entries such as Recursive Squares and Iterative, this entry differs in that it starts with a base pattern with all the 6-bit colors arranged into 8 groupings, based on which 3-bit color
they are closest to. These groupings are then sorted within their 8 columns so that a given color falls into the row closest to the same order of 3-bit colors, except transposed.
This image was created by iteratively overlaying the 8x8 blackest corner over an 8:1 scale and 4x multiplied value version of itself. In total, there are 3 levels of nesting.
Fun fact: With just this method alone of arranging the 6-bit color set in any order into a square and overlaying it on itself iteratively, there are 64! (~1.2688693e+89) possible entries. To put that
in perspective, it is estimated that there are around 1.0e+80 atoms in the observable universe.
Image Licence: Public Domain (CC0). | {"url":"https://allrgb.com/recursive-stripes","timestamp":"2024-11-09T03:01:00Z","content_type":"text/html","content_length":"8101","record_id":"<urn:uuid:9676e866-91c9-4d33-92fd-947cb9b382ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00195.warc.gz"} |
Multiplication By 12 Worksheets
Mathematics, specifically multiplication, forms the keystone of countless scholastic disciplines and real-world applications. Yet, for several students, mastering multiplication can present a
difficulty. To address this hurdle, educators and moms and dads have actually accepted an effective tool: Multiplication By 12 Worksheets.
Introduction to Multiplication By 12 Worksheets
Multiplication By 12 Worksheets
Multiplication By 12 Worksheets -
This page has lots of games worksheets flashcards and activities for teaching all basic multiplication facts between 0 and 10 Basic Multiplication 0 through 12 On this page you ll find all of the
resources you need for teaching basic facts through 12 Includes multiplication games mystery pictures quizzes worksheets and more
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
Importance of Multiplication Method Comprehending multiplication is pivotal, laying a strong structure for innovative mathematical ideas. Multiplication By 12 Worksheets provide structured and
targeted method, fostering a deeper comprehension of this fundamental arithmetic operation.
Development of Multiplication By 12 Worksheets
1 12 multiplication Worksheet For Kids Learning Printable
1 12 multiplication Worksheet For Kids Learning Printable
These free 12 times table worksheets provide you with an excellent tool to practice and memorise the tables The 12 times table is probably the hardest multiplication table to memorise However there
are several tips to help you learn this table quicker Let s take a look at some of the sums 1 x 12 12 alternatively this is 1 x 10 1 x 2
Multiplication by 12 worksheets gives different methods to solve types of multiplication problems and it will help to solve the equations problems in future Multiplication by 12 worksheets are very
useful for kids to grow their math skills They also give methods for getting kids to practice multiplication and other important concepts
From conventional pen-and-paper exercises to digitized interactive styles, Multiplication By 12 Worksheets have actually advanced, accommodating varied knowing styles and preferences.
Sorts Of Multiplication By 12 Worksheets
Fundamental Multiplication Sheets Easy exercises concentrating on multiplication tables, aiding students develop a solid math base.
Word Problem Worksheets
Real-life situations integrated into problems, enhancing vital reasoning and application skills.
Timed Multiplication Drills Tests developed to improve speed and accuracy, assisting in rapid psychological mathematics.
Benefits of Using Multiplication By 12 Worksheets
Times Table Grid To 12x12
Times Table Grid To 12x12
This basic Multiplication worksheet is designed to help kids practice multiplying by 12 with multiplication questions that change each time you visit This math worksheet is printable and displays a
full page math sheet with Horizontal Multiplication questions With this math sheet generator you can easily create Multiplication worksheets
Learn to Multiply by 12s Print this worksheet for your class so they can learn to multiply by 12s Find the missing factors complete the multiplication wheel and skip counting by 12s are just a few of
the activities on this worksheet 3rd and 4th Grades View PDF
Enhanced Mathematical Skills
Consistent practice sharpens multiplication efficiency, improving total math capacities.
Enhanced Problem-Solving Abilities
Word troubles in worksheets establish logical thinking and technique application.
Self-Paced Learning Advantages
Worksheets fit specific understanding rates, fostering a comfortable and adaptable discovering setting.
Just How to Develop Engaging Multiplication By 12 Worksheets
Integrating Visuals and Colors Lively visuals and shades catch focus, making worksheets aesthetically appealing and engaging.
Including Real-Life Situations
Relating multiplication to daily situations adds relevance and usefulness to exercises.
Customizing Worksheets to Various Skill Levels Tailoring worksheets based on differing proficiency degrees makes sure inclusive learning. Interactive and Online Multiplication Resources Digital
Multiplication Devices and Gamings Technology-based resources offer interactive understanding experiences, making multiplication appealing and enjoyable. Interactive Web Sites and Applications On the
internet platforms give diverse and obtainable multiplication method, supplementing traditional worksheets. Customizing Worksheets for Different Knowing Styles Aesthetic Learners Visual aids and
layouts help understanding for learners inclined toward aesthetic understanding. Auditory Learners Spoken multiplication troubles or mnemonics satisfy students that understand concepts with acoustic
means. Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic students in recognizing multiplication. Tips for Effective Execution in Knowing Consistency in Practice Routine
technique enhances multiplication skills, advertising retention and fluency. Balancing Repeating and Variety A mix of repeated exercises and varied problem styles preserves rate of interest and
understanding. Providing Useful Responses Feedback aids in recognizing areas of improvement, urging continued progression. Difficulties in Multiplication Practice and Solutions Motivation and
Engagement Difficulties Dull drills can result in uninterest; cutting-edge approaches can reignite inspiration. Getting Rid Of Anxiety of Math Unfavorable perceptions around mathematics can prevent
development; creating a favorable understanding atmosphere is necessary. Impact of Multiplication By 12 Worksheets on Academic Efficiency Studies and Study Findings Research indicates a favorable
correlation in between consistent worksheet use and boosted mathematics performance.
Multiplication By 12 Worksheets emerge as flexible devices, fostering mathematical proficiency in students while fitting varied discovering designs. From fundamental drills to interactive on-line
sources, these worksheets not only enhance multiplication skills yet also promote important reasoning and analytical capabilities.
Printable Timed Multiplication Quiz PrintableMultiplication
Worksheet On 12 Times Table Printable Multiplication Table 12 Times Table
Check more of Multiplication By 12 Worksheets below
Third Grade Multiplication Practice
Printable Multiplication Table 1 12 Pdf PrintableMultiplication
Multiplication Time 1 Worksheet
16 MULTIPLICATION WORKSHEETS 1 TO 12 Worksheets
Multiplication Worksheets X3 PrintableMultiplication
Multiplication Worksheets Numbers 1 Through 12 Mamas Learning Corner
Multiplying 1 to 12 by 12 100 Questions A Math Drills
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
Multiplication Facts Worksheets Math Drills
It is quite likely that there are students who have mastered all of the multiplication facts up to the 12 times tables In case they want need an extra challenge this sections includes multiplication
facts worksheets above 12 with the expectation that students will use mental math or recall to calculate the answers
Welcome to The Multiplying 1 to 12 by 12 100 Questions A Math Worksheet from the Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 19 and has
been viewed 119 times this week and 1 548 times this month It may be printed downloaded or saved and used in your classroom home school or other educational environment to help someone
It is quite likely that there are students who have mastered all of the multiplication facts up to the 12 times tables In case they want need an extra challenge this sections includes multiplication
facts worksheets above 12 with the expectation that students will use mental math or recall to calculate the answers
16 MULTIPLICATION WORKSHEETS 1 TO 12 Worksheets
Printable Multiplication Table 1 12 Pdf PrintableMultiplication
Multiplication Worksheets X3 PrintableMultiplication
Multiplication Worksheets Numbers 1 Through 12 Mamas Learning Corner
4Th Grade Multiplication Worksheets Free 4th Grade Multiplication Worksheets Best Coloring
Multiplication By Twelves Worksheet
Multiplication By Twelves Worksheet
Multiplication Drills 1 12 Free Printable
FAQs (Frequently Asked Questions).
Are Multiplication By 12 Worksheets ideal for all age teams?
Yes, worksheets can be customized to various age and ability levels, making them adaptable for numerous students.
How commonly should students practice utilizing Multiplication By 12 Worksheets?
Constant method is essential. Routine sessions, ideally a couple of times a week, can generate considerable enhancement.
Can worksheets alone improve mathematics abilities?
Worksheets are an important tool however ought to be supplemented with varied knowing methods for thorough ability development.
Exist on the internet systems offering free Multiplication By 12 Worksheets?
Yes, numerous academic web sites offer free access to a variety of Multiplication By 12 Worksheets.
Just how can parents sustain their youngsters's multiplication practice at home?
Encouraging regular practice, giving support, and developing a positive learning atmosphere are advantageous actions. | {"url":"https://crown-darts.com/en/multiplication-by-12-worksheets.html","timestamp":"2024-11-06T11:53:39Z","content_type":"text/html","content_length":"28782","record_id":"<urn:uuid:0e62c75d-739c-4ab8-8f62-d6e0a6d5c6c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00275.warc.gz"} |
151-0501-02L Mechanics 1: Kinematics and Statics (Colloquium)
Semester Autumn Semester 2020
Lecturers R. Hopf
Periodicity yearly recurring course
Language of instruction German
Basics: Position of a material point, velocity, kinematics of rigid bodies, forces, reaction principle, mechanical power
Abstract Statics: Groups of forces, moments, equilibrium of rigid bodies, reactions at supports, parallel forces, center of gravity, statics of systems, principle of virtual power, trusses,
frames, forces in beams and cables, friction
Learning The understanding of the fundamentals of Statics for engineers and their application in simple settings.
Basics: Position of a material point; velocity; kinematics of rigid bodies; translation, rotation, planar motion; forces, action-reaction principle, internal and external forces,
distributed forces; mechanical power.
Content Statics: equivalence and reduction of groups of forces; rest and equilibrium; basic theorem of statics; kinematic and static boundary conditions, applications to supports and clamps of
rods and beams; procedures for determination of forces at supports and clamps; parallel forces and centre of gravity; statics of systems, solution using basic theorem and using the
principle of virtual power, statically indeterminate systems; statically determinate truss structures, ideal truss structures, nodal point equilibrium, methods for truss force
determination; friction, static friction, sliding friction, friction at joints and supports, rolling resistance; forces in cables; beam loading, force and moment vector.
Lecture notes Übungsblätter
Literature Sayir, M.B., Dual J., Kaufmann S., Ingenieurmechanik 1: Grundlagen und Statik, Teubner
Prerequisites Live stream of the lecture: https://video.ethz.ch/live/lectures/zentrum/ml/ml-e-12.html
/ Notice | {"url":"https://www.vorlesungen.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?semkez=2020W&ansicht=KATALOGDATEN&lerneinheitId=141471&lang=en","timestamp":"2024-11-07T16:58:27Z","content_type":"text/html","content_length":"7923","record_id":"<urn:uuid:80cb60e8-b160-4849-8c3e-5eb8dcc5bcce>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00738.warc.gz"} |
Reflector Backed Equiangular Spiral
The spiral antenna is an inherently broadband, and bidirectional radiator. This example will analyze the behavior of an equiangular spiral antenna backed by a reflector [1]. The spiral and the
reflector are Perfect Electric Conductors (PEC).
Spiral and Reflector Parameters
The spiral antenna parameters and reflector dimensions are provided in [1]. The design frequency range is 4-9 GHz and the analysis frequency range will span 3-10 GHz.
a = 0.35;
rho = 1.5e-3;
phi_start = -0.25*pi;
phi_end = 2.806*pi;
R_in = rho*exp(a*(phi_start + pi/2));
R_out = rho*exp(a*(phi_end + pi/2));
gndL = 167e-3;
gndW = 167e-3;
spacing = 7e-3;
Create Equiangular Spiral Antenna
Create an equiangular spiral antenna using the defined parameters.
sp = spiralEquiangular;
sp.GrowthRate = a;
sp.InnerRadius = R_in;
sp.OuterRadius = R_out;
title("Equiangular Spiral Antenna Element");
Create a reflector, and assign the spiral antenna as its exciter. Adjust the spacing between the reflector and spiral to be 7 mm. The first-pass analysis will be with an infinitely large groundplane.
To do this, assign the groundplane length and/or width to inf.
rf = reflector;
rf.GroundPlaneLength = inf;
rf.GroundPlaneWidth = inf;
rf.Exciter = sp;
rf.Spacing = spacing;
title("Equiangular Spiral Antenna Element Over an Infinite Reflector");
Impedance Analysis of Spiral with and without Infinite Reflector
Define a frequency range for the impedance analysis. Keep the frequency range sampling coarse to get an overall idea of the behavior. A detailed analysis will follow. Analyze the impedance of both
antennas: the spiral antenna in free space and the antenna with the infinite reflector.
freq = linspace(3e9,10e9,31);
The impedance for the spiral without the reflector is smooth in both resistance and reactance. The resistance holds steady around the 184 $\Omega$ value while the reactance although slightly
capacitive, does not show any drastic variations. However, introducing an infinitely large reflector at a spacing of 7mm disturbs this smooth behavior for both the resistance and reactance.
Make Reflector Finite
Change the reflector to be of finite dimensions by using the dimensions specified earlier as per [1].
rf.GroundPlaneLength = gndL;
rf.GroundPlaneWidth = gndW;
title("Equiangular Spiral Antenna Element Over Finite Reflector");
Mesh Different Parts Independently
To analyze the behavior of the spiral antenna backed by the finite-size reflector, mesh the antenna and reflector independently. The highest frequency in the analysis is 10 GHz, which corresponds to
a $\lambda$ of 3 cm. Mesh the spiral with a maximum edgelength of 2 mm (lower than $\lambda /10$ = 3mm at 10 GHz). This requirement is relaxed for the reflector which is meshed with a maximum edge
length of 8mm (slightly lower than $\lambda /10$ = 10cm at 3 GHz).
ms_new = mesh(rf.Exciter,MaxEdgeLength=0.002)
mr_new = mesh(rf,MaxEdgeLength=0.008)
Impedance Analysis of the Spiral Antenna with the Finite Reflector
Run the following code snippet at the prompt in the command window to create an accurate impedance plot for the finite reflector shown in the figure that follows. The impedance behavior typical for
the infinitely large reflector is also observed for the finite reflector.
fig1 = figure;
Fig. 1: Input impedance of the spiral with metal(PEC) reflector
Surface Currents on Reflector-Backed Spiral Antenna
Run the following code snippet at the prompt in the Command Window to recreate the current distribution plots shown in the figure that follows. Since the design frequency band is 4 - 9 GHz, pick the
two band edge frequencies for observing the surface currents on the spiral antenna. From the impedance analysis, the presence of multiple resonances at the lower end of the design frequency band,
should manifest itself in the surface currents at 4 GHz.
fig2 = figure;
view([0 90])
Fig. 2: Surface current density at 4 GHz
At first glance, the surface current density plot might appear to not reveal any information. To explore further, use the colorbar on the right of the plot. This colorbar allows you to adjust the
color scale interactively. To interact with the colorbar, move the mouse pointer on the colorbar, right-click and select Interactive Colormap Shift. To adjust the scale, move the mouse pointer to an
area within the colorbar boundary, click, and drag. To adjust the range of values, move the mouse pointer outside the colorbar boundary and hover over the numeric tick marks on the colorbar or
between the tick spacings, click and drag. This approach generates the following figure, which indicates presence of standing waves at 4GHz. The arrows on the plot are added separately to highlight
the local current minima.
Fig. 3: Surface current density at 4 GHz after adjusting the dynamic range using the colorbar
Plot the current density on the spiral antenna at 9 GHz.
fig4 = figure;
view([0 90])
As in the previous case, we use the colorbar to adjust the scale to be similar and observe a current flow that is free of any standing waves.
Fig. 4: Surface current density at 9 GHz
Calculate and Plot Axial Ratio
The spiral antenna radiates circularly polarized waves. The axial ratio (AR) in a given direction quantifies the ratio of two orthogonal field components radiated in a circularly polarized wave. An
axial ratio of infinity, implies a linearly polarized wave. When the axial ratio is 1, the radiated wave has pure circular polarization. Values greater than 1 imply elliptically polarized waves. Run
the following code snippet at the prompt in the command window to recreate the axial ratio plot at broadside shown in the figure that follows. Note that the calculated axial ratio values are in dB,
20log10(AR). To compare the effect of the reflector, calculate the axial ratio for the spiral antenna with and without the reflector.
AR_spiral = zeros(size(freq));
AR_reflector = zeros(size(freq));
for i = 1:numel(freq)
AR_spiral(i) = axialRatio(rf.Exciter,freq(i),0,90);
AR_reflector(i) = axialRatio(rf,freq(i),0,90);
Plot the axial ratio of the spiral antenna with and without the reflector.
fig5 = figure;
hold on
grid on
ax1 = fig5.CurrentAxes;
ax1.YTickLabelMode = 'manual';
ax1.YLim = [0 20];
ax1.YTick = 0:2:20;
ax1.YTickLabel = cellstr(num2str(ax1.YTick'));
xlabel("Frequency (GHz)")
ylabel("AR (dB)")
title("Frequency Response of Axial Ratio")
legend("Without reflector","With reflector");
The axial ratio plot of the spiral without the reflector shows that across the design frequency band, the spiral antenna radiates a nearly circularly polarized wave. The introduction of the reflector
close to the spiral antenna degrades the circular polarization.
Fig. 5: Axial ratio at broadside in dB with and without reflector backing as a function of frequency [1]
The spiral antenna by itself has a broad impedance bandwidth and produces a bi-directional radiation pattern. It also produces a circularly polarized wave across the bandwidth. A unidirectional beam
can be created by using a backing structure like reflector or cavity. Maintaining the desired performance when using a traditional metallic/PEC reflector, especially at small separation distances is
difficult [1].
[1] Nakano, Hisamatsu, Katsuki Kikkawa, Norihiro Kondo, Yasushi Iitsuka, and Junji Yamauchi. “Low-Profile Equiangular Spiral Antenna Backed by an EBG Reflector.” IEEE Transactions on Antennas and
Propagation 57, no. 5 (May 2009): 1309–18. https://doi.org/10.1109/TAP.2009.2016697.
Related Topics | {"url":"https://fr.mathworks.com/help/antenna/ug/reflector-backed-equiangular-spiral.html","timestamp":"2024-11-06T20:22:07Z","content_type":"text/html","content_length":"84482","record_id":"<urn:uuid:0426015e-8ce1-4907-85f7-1b51101d6916>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00330.warc.gz"} |
The inter relationship between firm growth and profitability
✅ Paper Type: Free Essay ✅ Subject: Social Work
✅ Wordcount: 5454 words ✅ Published: 1st Jan 2015
There is a widespread presumption that there is a close relationship between firm growth and firm profitability. However, most of the past studies on firm growth and profitability have been conducted
without mutual associations. Only a few studies, thus far, have examined the inter-relationship between firm growth and profitability and the results have been inconsistent. The reason for the
inconsistency is mainly due to the lag structure of the models in each study. To address the issue, this study conducted panel unit-root tests on firm growth and profitability separately and then
made appropriate models using dynamic panel system GMM estimators. Through the analyses of the models, this study found that in restaurant firms the prior year’s profitability had a positive effect
on the growth rate of the current year, but the current and prior year’s growth rates had a negative effect on the current year’s profitability. This outcome implies that profit creates growth but
the growth impedes profitability in the restaurant industry. More implications are also discussed in this paper.
Keywords: Firm growth; Profitability; Panel unit-root test; Dynamic panel system GMM
1. Introduction
The dynamics of firm growth and profitability (or profit rate) is an important issue for industrial practitioners as well as academic researchers (Goddard, McMillan and Wilson, 2006). Theoretically,
if firm growth rate is unrelated to firm size and prior growth rate, then firm growth follows random walk and the variance of firm size can increase indefinitely. This is known as the Law of
Proportionate Effect (LPE). This stochastic growth process implies unlimited industry growth in the long run. However, if growth rate is inversely related to firm size, firm growth would converge in
the long run. On the other hand, Mueller (1977) claimed that firm profitability converges at a certain level due to market competition, which is referred to as Persistence of Profit (POP). The POP
literature argues that firm entry and exit are sufficiently free to quickly eliminate any abnormal profit and that the profitability of all firms tends to converge toward the long-run average value.
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However, Goddard, Molyneux and Wilson (2004) stated, even though it is generally presumed that firm growth and profitability effect each other, that firm growth and profitability are not necessarily
connected. Overall, the impact and direction of this relationship remains ambiguous. The ambiguity is associated with various econometric issues. First, due to the endogeneity it is difficult to
capture a clear causality and direction between them. Further, if firm growth and profitability time lags are incorporated into the models the endogenous relationship becomes more complicated due to
the unknown effects of different time lags.
Recently, there have been a couple of attempts to investigate the inter-relationship between firm growth and profitability (Coad, 2007; Davidsson, Steffens, and Fitzsimmons, 2009). Although it is
worth exploring the relationship, the results of the studies turned out to be inconsistent. In the previous studies, two types of methodologies were used: panel unit-root test and dynamic panel
system GMM estimator. The panel unit-root test is appropriate for testing the convergence hypotheses of firm growth and profit rates. It is also useful for finding the significance of the lag term in
a simple autoregressive model, but it is difficult to control the endogenous effect in the model. Moreover, the panel unit-root test cannot directly examine the inter-relationship between firm growth
and profitability. Dynamic panel system GMM estimator can control for endogeneity and test the inter-relationship, but determining the number of lag terms remains ambiguous.
Thus, in order to address the analysis problems in the previous literature, we first employed the panel unit-root test and subsequently made a testable model for the dynamic panel system GMM
estimator. Through those analyses, we intended to investigate the inter-relationship between firm growth and profitability under various time lags. More specifically, the objectives of this study
were: 1) to examine the panel unit-root test on the series of firm growth and profitability separately and to find an appropriate lag structure; and 2) to make an appropriate model to investigate the
inter-relationship between them through a vector autoregression (VAR) model via dynamic panel system GMM estimator. We used restaurant firms for the study sample and, thus, the results are useful for
understanding the dynamics of firm growth and profitability in the restaurant industry.
In the following section, we summarize prior LPE and POP literature and present the potential inter-relationships between firm growth and profitability. Next section outlines the details of the study
methodology. The following section shows the results of panel unit-root test and dynamic panel system GMM regarding the inter-relationship between firm growth and profitability. Finally, we conclude
this study with managerial implications and suggestions for further studies.
2. Literature Review
2.1. Law of Proportionate Effect (LPE) and Persistence Of Profit (POP)
The notion that firm growth rate is independent of firm size and past growth rate is known as the Law of Proportionate Effect (LPE) (Gibrat, 1931). According to the LPE, firm growth happens by chance
and thus past growth is not a reliable predictor of future firm growth (Goddard et al., 2006). Hence, deterministic factors of firm growth (i.e., managerial capacity, innovation and efficiency) are
randomly distributed across firms. However, recent empirical studies have claimed that there is an inverse relationship between firm growth and firm size, rejecting the LPE (Hall, 1987; Evans, 1987;
Dunne and Huges, 1994; Geroski and Gugler, 2004). Most empirical studies of LPE used cross-sectional regression models through a simple autoregressive model (for example, AR(1)), but the models were
criticized due to their arbitrariness in choosing lag terms. Recently, Chen and Lu (2003) and Goddard et al. (2006) tested the LPE using panel unit-root models because the LPE assumes
non-stationarity in the time series analysis. The benefit of the panel unit-root test on LPE lies in its ability to test a long series effect in non-stationarity, while the weakness of the test is
its inability to include control variables that may affect firm growth (i.e., prior profitability, leverage, and market competition).
Conversely, researchers on persistence of profit (POP) argue that firm profitability converges at a certain level across all firms and no firms could achieve an above average profit rate in the long
run. Mueller (1977) developed the deterministic time-series model for testing the POP and subsequently (Mueller 1986) demonstrated profit rate convergence through an autoregressive model. Since
Mueller (1986), most studies on POP have adopted the autoregressive model. However, Goddard et al. (2006) stated that the typical methodology for POP estimated individual effects and autoregressive
coefficients for each firm, so the estimated coefficients were often unreliable and the testing power was low. Hence, Goddard et al. (2006) tested the profit rate convergence hypothesis using a panel
unit-root test in order to find the stationarity in a profit rate time series.
2.2. The inter-relationship between firm growth rate and profitability (or profit rate)
As noted earlier, it is widely believed that firm growth and profit rates are related to each other (Goddard et al., 2004). Some prior studies have suggested that profit rate has a positive impact on
growth rate. Alchian’s (1950) theoretical study argued that fitter firms survive and grow, but less viable firms lose their market share and exit through the evolutionary selection mechanism. Thus,
if profit rate reflects the degree of fitness, it is possible to predict that profitable firms will grow. Further, according to the financing constraint hypothesis retained profits can be readily
used for investment, whereas firms with low profitability could not grow even if they have positive growth opportunities. This is also consistent with the pecking-order theory, which claims that
managers prefer internal capital to external capital, such as debt and equity financing.
However, the influence of growth rate on profitability is inconsistent in theories and empirical studies. A Classical Ricardian perspective claims that if a firm shows high profit rates it would grow
to exploit additional growth opportunities that are less profitable but still create additional profits. This notion implies three things. First, the profit rate is converges at zero from a long-term
perspective. Second, high profit rates have a positive impact on growth rates until the profit rate is zero. Finally, firm growth has a negative influence on profit rate. Along similar lines, the
Neoclassical view argues that firms first exploit their most profitable growth opportunities and then consider less profitable opportunities until the marginal profit on the last growth opportunity
is equal to zero. Consequently, profitable firms maximize their overall level of profits through profitable growth opportunities but experience a decrease in profit rates. Even though this argument
excludes market competition, it theoretically explains the relationship between firm growth and profit rates. However, managerial growth-maximization hypothesis under market competition (Marris,
1964; Mueller, 1972) claims that the managerial objective of a firm is to maximize growth rather than profit. Thus, this hypothesis proposed that growth and profits are in a competitive relationship
with each other, which suggests the possibility that growth victimizes profit.
Nevertheless, there are a number of theoretical claims that growth rate has a positive impact on profit rate. First, the Kaldor-Verdoorn Law in economics (Kaldor, 1966; Verdoorn, 1949) claims that
growth increases productivity and in turn the enhanced productivity increases profit rates. This notion is consistent with scale economies (Gupta, 1981). Thus, because firm growth contributes to an
increase in firm size, the larger size could gain benefits from an economy of scale and in turn this affects profit enhancement. That is, growth can help increase profitability.
However, empirical studies on the effects of growth rate on profit rate have not always been supportive. Capon, Farley and Hoenig (1990) reported that firm growth is related to high financial
performance, but it was significant only in some industries. Chandler and Jansen (1992) and Mendelson (2000) reported a significant positive correlation between sales growth and profit rates, whereas
Markman and Gartner (2002) found no significant relationship between growth and profitability. Furthermore, Reid (1995) claimed growth had a negative effect on profitability.
The relationship between growth and profit rates are more complicated when time lags of the two variables are considered. Only a few empirical studies have considered the link between growth and
profit rates using various time lag terms. Goddard et al. (2004) found profitability to be important for future growth in European banks. Conversely, through panel data estimates of French
manufacturing firms Coad (2007) found that the opposite direction of causation (i.e., growth to profitability) might be true. Both Goddard et al. (2004) and Coad (2007) investigated the relationship
between firm growth and profit rates with vector autoregressive models using dynamic panel system GMM estimators. The difference between the two studies was that Goddard et al. (2004) used a one-year
time lag but Coad (2007) incorporated three-year time lags in the analysis. More specifically, Goddard et al. (2004) found that a one-year lagged profit rate had a positive significant effect on the
current-year’s growth rate, but a one-year lagged growth rate did not have a significant impact on the current-year’s profit rates. However, Coad (2007) showed that two- and three-year’s lagged
profit rates have a positive significant influence on the current-year’s growth rate and that the current-year’s growth rate was positively significant in terms of the current-year’s profit rates. As
noted, Goddard et al. (2004) and Coad (2007) reported opposing empirical results, which could be attributed to the difference in lag length. Considering the scarcity of past studies on the
growth-profitability relationship and the problems with analytic methods, there is a need for a study that can verify this important relationship in a more holistic way. Hence, we intended to address
the above research need in this study. A detailed outline of how the study was conducted follows in the next section.
3. Data and methodology
The data used in the analysis was collected from the COMPUSTAT database using SIC 5812 (eating places). The data covers fiscal years 1978 to 2007 for U.S. restaurant firms. Profit rate (or
profitability) was measured as net income divided by net sales and growth rate was gauged as the difference between the current and prior year’s net sales divided by the prior year’s net sales. After
deleting severe outliers in the two main variables, growth rate and profit rate, this study used 2,927 firm-year observations for the analysis.
As previously indicated, this study first conducted panel unit-root tests on growth and profit rates separately. The Dickey-Fuller unit-root test was set up for testing the stationarity of a time
series. For example, if φ1 is equal to a unit in equation (1), the series is non-stationary. Equation (1) could be expressed as equation (2) by subtracting Yt-1 on both sides.
Yt = φ1Yt-1 + εt (1)
ΔYt = γYt-1 + εt (γ = φ1 – 1) (2)
Equation (2) above is a simplified Dickey-Fuller unit-root test (DF test). The null hypothesis of a DF test is that γ equals zero and the alternative hypothesis is γ < 0. Under the null hypothesis
the series is non-stationarity. By including lags of order p this formulation allows for higher-order autoregressive processes, which is referred to as an Augmented Dickey-Fuller unit-root test (ADF
test). Equation (3) shows the ADF test formula.
ΔYt = γYt-1 + ∑φiΔYt-i + εt (γ = φ1 – 1) (3)
However, the data structure of this study was an unbalanced panel. Thus, equation (3) could be expressed as a panel setting following equation (4):
ΔYi,t = γY i,t-1 + ∑φiΔY i,t-i + ε i,t (γ = φ1 – 1) (4)
Equation (4) is the testable model for the panel unit-root test in this study. A few studies have developed panel unit-root tests (Im, Pesaran and Shin, 2003; Levin, Lin and Chu, 2002; Maddala and
Wu, 1999). However, in the case of an unbalanced panel setting, the Fisher test is the only one available. It combines the p-values from N independent unit root tests, as developed by Maddala and Wu
(1999). Based on the p-values of individual unit root tests, Fisher’s test assumes that all series are non-stationary under the null hypothesis against the alternative that at least one series in the
panel is stationary. Unlike other panel unit-root tests, Fisher’s test does not require a balanced panel. Thus, this study conducted Fisher’s test on the growth and profit rates and selected an
appropriate lag length in ADF formula.
After selecting the proper lag length in ADF formula, it was transformed as follows:
ΔYi,t = γY i,t-1 + ∑φiΔY i,t-i + ε i,t
= γY i,t-1 + φ1ΔY i,t-1 + φ2ΔY i,t-2 + φ3ΔY i,t-3 + … + φpΔY i,t-p + ε i,t
= γY i,t-1 + φ1(Y i,t-1 – Y i,t-2) + φ2(Y i,t-2 – Y i,t-3) + … + φp(Y i,t-p – Y i,t-(p+1)) + ε i,t
= (γ + φ1) Y i,t-1 + (φ2 – φ1) Y i,t-2 + (φ3 – φ2) Y i,t-3 + … + (φp – φp-1)Y i,t-p
– φpY i,t-(p+1) + ε i,t
Consequently, equation (5) could be expressed as follows:
Yi,t = (1 + γ + φ1) Y i,t-1 + (φ2 – φ1) Y i,t-2 + (φ3 – φ2) Y i,t-3 + … + (φp – φp-1)Y i,t-p
– φpY i,t-(p+1) + ε i,t (6)
Thus, if the panel unit-root test chooses p lags in ADF formula, it could be transformed to AR(p+1) model. This AR(p+1) model was then used for the dynamic panel system GMM estimator. Also, since the
purpose of this study was to investigate the inter-relationship between firm growth and profitability, this study adopted the vector autoregression (VAR) model to find the reciprocal relationship
between growth rates and profit rates.
p+1 q+1 p+1
SGi,t = β0 + ∑ηiSGi,t-i + ∑πiPRi,t-i + β1Salei,t-i + β2LEVi,t-i + ∑ζiΔDM&Ai,t-i
i=1 i=1 i=0
+ DYeart + εi,t
Model 1
q+1 p+1
PRi,t = β0 + ∑πiPRi,t-i + ∑ηiGRi,t-i + β1Salei,t-i + β2LEVi,t-i + β3MarketSharei,t-i
i=1 i=0
+ DYeart + εi,t
Model 2
SGi,t is the sales growth rate and PRi,t is the profit rate at time t for firm i. Salei,t is the net sales at time t for firm i. We also included control variables in both models. In the LPE
literature, recent studies showed that prior firm size is inversely related with current growth rate (Evans, 1987; Hall, 1987; Geroski and Gugler, 2004). On the other hand, Baumol (1959) provided
evidence that firm profitability increases with firm size, while Amato and Wilder (Kwangmin!!, Year and reference?) showed that no relationship exists between firm size and profit rate. Finally,
Samuels and Smyth (1968) stated that profit rate and firm size are inversely related. Thus, we included the prior year’s net sales as a firm size variable in both models to control for size effect.
Debt leverage (LEVi,t) was also incorporated in both models as a control variable, which was calculated as total debt divided by total assets. Theories of optimal capital structure based on the
agency costs of managerial discretion suggest that the adverse impact of leverage on growth increases firm value by preventing managers from taking on poor projects (Jensen,1986; Stulz, 1990). Opler
and Titman (1994) empirically found that sales growth is lower in firms with higher leverage. Thus, the influence of debt leverage on growth could be negative. However, the prior literature on the
relationship between debt leverage and profit rate, has shown mixed results. Debt affects profitability positively according to Hurdle (1974), but negatively according to Hall and Weiss (1967) and
Gale (1972). Debt could also yield a disciplinary effect under the free cash flow hypothesis (Jensen, 1986; Stulz, 1990). Firms with high debt leverage can reduce wasteful investment opportunities
and increase firm performance, suggesting a positive relationship between debt leverage and profit rates. However, using debt can increase conflicts between debt and equity holders. Equity holders
encourage managers to undertake risky projects because the benefits are transferred only to equity holders (Stiglitz and Weiss, 1981). Thus equity holders tend to support the use of debt. However,
high uses of debt could deteriorate firm profitability by taking on overly risky projects. The effect of leverage on profit rate may not be uni-directional. Consequently, we incorporated leverage as
a control variable due to its important potential effects on profitability.
In the growth rate equation (Model 1), we incorporated mergers and acquisitions (M&A) dummy variables from time t to t-(p+1) because M&A execution abnormally increases growth rates. M&A executions
were identified from the SDC Platinum database. In the profitability equation (Model 2), we included a market share variable, which was calculated as the net sales of firm i at time t divided by the
sum of net sales at time t. According to Buzzell, Gale and Sultan (1975), market share had a positive impact on firm profitability. Because a larger market share means stronger market power, firms
with large market shares could have the power to control market prices and be in a better position to negotiate with their suppliers. Thus, a positive relationship between market share and profit
rates is expected. Because the current year’s growth could affect the current year’s profit rate, following Coad (2007), we included the current year’s growth rate in Model 2.
Statistically, ordinary least square (OLS) regression requires that the right-hand side variables should be independent of the error term. However, if there is a bi-directional causation between
dependent (left-hand side) variables and explanatory (right-hand side) variables, this condition is not satisfied and thus OLS regression produces biased and inconsistent estimates. This endogeneity
problem could be solved by choosing appropriate instrumental variables, which are correlated with the explanatory variables but not the error term. This means that the instrumental variables should
be exogenous but if they are endogenous, the equation would be over-identified. However, if the instrumental variables are weakly correlated with the explanatory variables, which is called a weak
instrument, the estimates are biased and inconsistent.
Arellano and Bond (1991) proposed the GMM estimator for panel data, which could control the potential endogenous explanatory variables. This method uses the first difference model, which eliminates
the time-invariant firm-specific effect, and instrumental variables for the endogenous variables were generated by lags of their own level. However, if the lagged level instruments are weakly
correlated with the endogenous explanatory variables, there could be a finite sample bias in estimators. In particular, if the variable series tends to show a highly persistent profit rate series
(Mueller, 1977), this weak correlation between lagged level instruments and endogenous explanatory variables is problematic. Arellano and Bover (1995) and Brundell and Bond (1998) developed a dynamic
panel GMM estimator that estimated with level-equation and difference equation, which is called a ‘system GMM’. Consequently, the dynamic panel system GMM estimator has better asymptotic and finite
sample properties than the one used by Arellano and Bond (1991).
Thus, this study analyzed the proposed models using the dynamic panel system GMM estimator, which produces unbiased and consistent estimates after controlling for endogeneity and firm-specific
effects even when the sample period is short. Even though the full sample period of this study is 30 years, the panel structure is not balanced due to the entry and exit of firms. Bludell and Bond
(1998) suggested the minimum requirement for panel length as T ≥ 3. Thus, we excluded firms which did not exist at least three years in the sample period. Another requirement was that there is no
serial correlation of the second order error terms. We conducted the serial correlation test for panel GMM estimators developed by Arellano and Bond (1991). In order to test the exogeneity of
instrumental variables, we used the Hansen test instead of the Sargan test because the Sargan test is not robust enough to detect heteroskedasticity and autocorrelation (Roodman, 2006). Finally, as
Roodman (2006) suggested, we included year dummies in the models and estimated the system GMM by two-step estimator because the two-step estimator is robust enough to detect the heteroskedasticity.
For comparisons with the dynamic panel system GMM estimator, we conducted ordinary least square (OLS) and fixed-effect regression.
4. Results
4.1. Panel unit-root test for firm growth and profit rates
As indicated, we conducted the panel unit-root test developed by Maddala and Wu (1999) using Fisher’s test, which assumes that all series are non-stationary under the null hypothesis. Equation (4)
was tested on both growth and profit rates. The results are presented in Table 1. For the series of sales growth and profit rates, lag(4) was justified. Thus, the law of proportionate effect
hypothesis was rejected but the persistence of profit hypothesis was validated. The results indicate that the growth rates are serially correlated and the profit rates are convergent. The purpose of
the panel unit-root tests on growth and profit rates was to examine the stationarity of the two series and to make an appropriate model for the dynamic panel system GMM estimator. As shown earlier,
if the panel unit-root test justifies p lags, the ADF formula could be transformed to AR(p+1) model. Consequently, the testable model is AR(5) for both growth and profit rates. Based on the lag
length from the panel unit-root test, we excluded any firm that existed less than five years in testing the dynamic panel system GMM estimator. Then, we tested the proposed models using AR(5) in
order to identify the inter-relationship between firm growth and profit rates in various time lag structures.
(Insert Table 1 Here)
4.2. Descriptive statistics and scatter plots of growth and profit rates
Table 2 shows the descriptive statistics of the major variables of this study. The average sales of the sampled restaurant firms was 541.8 million dollars and the average growth rate in sales was
16.3%. The average profit rate (return on sales) was – 1.3% and total debt rate (debt leverage) was 61.3%. Thus, the figures show that the restaurant industry has a high growth rate, but its
profitability is not positive and it uses more debt than equity.
(Insert Table 2 Here)
Before conducting the dynamic panel system GMM estimator, we checked the scatter plots between growth and profit rates using various time lags. As Coad (2007) indicated, the non-parametric scatter
plots of growth and profit rates gave us a visual appreciation of the underlying phenomenon. Thus, before testing the quantitative relationship, we can obtain useful information via scatter plots.
Figure 1 shows the scatter plots of growth at time t (Y-axis) and growth rates at time t-1 to t-5 (X-axis) for all samples. Except for the first plot (growth rate time t versus t-1), all other plots
seem to show no relationship. The plots, excluding the first plot, look like a cloud shape but are a bit scattered horizontally. Based on the plot for growth rate time t and t-1, the current and
prior year’s growth rates are positively correlated. However, Figure 1 represents all firms, including M&A firms. Apparently, firms with M&A can experience abnormally high growth rates compared with
non-M&A firms. Thus, we checked the same scatter plots after excluding M&A firms, as presented in Figure 2. The relationship between current and prior year’s growth rate is clearly positive and
growth rate at t-2 also looks positive on current year’s growth rate. However, the earlier years’ growth rates (i.e., t-3, t-4 and t-5) appear to have no relationship with the current year’s growth
rate. Figure 3 shows scatter plots of profit rate at time t (Y-axis) and profit rates at time t-1 to t-5 (X-axis). Interestingly, clear heteroskedasticity is detected in the relationship between
them. Thus, the usage of the two-step estimator in the dynamic panel system GMM estimator is justified by Figure 3. In all of the scatter plots there is a tendency toward a positive relationship
between current and prior profit rates.
(Insert Figures 1, 2, and 3 Here)
Figure 4 shows scatter plots of profit rate at time t (Y-axis) and growth rates at time t-1 to t-5 (X-axis). In all plots, points were spread horizontally. It seems that there is no effect of growth
rate on profit rate. Surprisingly, the scatter plot of current growth rates appears to have no relationship with current profit rate. On the other hand, Figure 5 shows that profit rates clearly have
a positive influence on the current growth rate. The majority of the points were spread vertically. The scatter plots show that prior profit rates seems to have a positive influence on current growth
rates, but the influence of prior growth rates on current profit rates was not found.
(Insert Figures 4 and 5 Here)
4.3. Results from Dynamic panel system GMM estimator
Tables 3 and 4 show the results of the proposed models explained in the methodology section. Even though yearly dummies were not reported in Tables 3 and 4, they were included in the models. As shown
in Table 3, the prior year’s growth rate at time t-1 was found to be positively significant on current growth rates in all three regressions (OLS, fixed-effect and system GMM). However, the
directions and significances of the coefficients of the other prior growth rate terms varied across the three models. As explained earlier, however, the system GMM is the most appropriate model for
this study due to the endogeneity and time invariant firm-specific effect and the results of the OLS and fixed-effect regression models were used simply for the purpose of comparison.
Goddard et al. (2004) reported that the prior year’s (time t-1) growth rate was positive but not significant. It is difficult to directly compare their results with ours due to the difference in the
lag length structure. Interestingly, our study showed that growth rates at time t-1 and t-5 were positively significant on current growth rates, but growth rates at time t-2 and t-4 were negatively
significant. These results suggest that short-term and long-term prior growth rates have a positive impact, but mid-term prior growth rates have a negative influence on current growth rates.
Our primary interest in Model 1 was the effect of the prior years’ profit rates on current growth rates. The system GMM results show that profit rates at time t-1 and t-5 were positively significant.
The magnitude of the coefficient of profit rate at time t-5 was small, meaning that the positive impact of long-term prior profit rates on current growth rates is small. However, the prior year’s
(time t-1) profit rate has a positively significant effect on current growth and the magnitude of the coefficient is large. Coad’s (2007) study showed that profit rates at time t-1 to t-3 were all
positive but the prior year’s (time t-1) profit rate was not significant. Coad (2007) used an AR(3) model and thus a direct comparison of ours to Coad’s (2007) is not possible. Yet it is clear that
the direction of the coefficients were very similar. Overall, our study results provide evidence that recently profitable firms may grow faster.
In terms of the relationship between prior year’s firm size and current growth rate, all three results show a negative coefficient but the negative effect was significant only in OLS. Also, debt
leverage had a negative effect on current growth rates but the system GMM result was not significant. Additionally, all serial correlation tests were not significant, showing that there was no serial
correlation problem. Also, the over-identification tests were not significant, meaning that our instruments were not endogenous and the estimates were reliable.
(Insert Table 3 Here)
Table 4 shows the results of the profitability equation (Model 2). The results of the system GMM shows that profit rates at time t-1, t-2 and t-5 were had positively significant effect on current
profit rates. However, profit rates at time t-3 and t-4 were negatively significant. The results suggest that short-term and long-term prior profit rates have a positive impact on current profit
rates, but mid-term prior profit rates have a negative influence on current profit rates. Similarly, Goddard et al.’s (2004) results showed that the prior year’s (time t-1) profit rate was positive
and significant in its AR(1) model. Table 4 also presents the effect of the prior years’ growth rates on current profit rates were negatively significant in time t and t-1. Unlike our results,
Goddard et al. (2004) found that the prior year’s growth rate was posi
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Differentiating 1D Integrals
Differentiating 1D Integrals #
by Shuang Zhao
In what follows, we discuss the differentiation of a simple Riemann integral $I(\theta)$ over some 1D interval $(a, b) \subseteq \real$:
$$\label{eqn:I} I(\theta) = \int_a^b f(x, \theta) \,\D x.$$
The Incomplete Solution #
The derivative of the integral in Eq. \eqref{eqn:I} with respect to $\theta$ can sometimes be obtained by exchanging the ordering of differentiation and integration:
$$\label{eqn:dI_0} \frac{\D}{\D\theta} I = \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right) \stackrel{\Large ?}{=} \int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x.$$
Precisely, the second equality in Eq. \eqref{eqn:dI_0} requires the integrand $f$ to be continuous^1 throughout the interval $(a, b)$.
Success Example #
We now provide a toy example where Eq. \eqref{eqn:dI_0} holds. Let $f(x, \theta) := x^2 \,\theta$. Consider the following integral:
$$ I = \int_0^1 (x^2 \,\theta) \,\D x. $$
Since $I = \left[ (x^3 \,\theta)/3 \right]_0^1 = \theta/3$, we know that
$$ \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{\theta}{3} \right) = {\color{blue}\frac{1}{3}}. $$
We now try calculating the same derivative $\D I/\D\theta$ using Eq. \eqref{eqn:dI_0}:
$$ \frac{\D I}{\D\theta} = \int_0^1 \frac{\D}{\D\theta} (x^2 \,\theta) \,\D x = \int_0^1 x^2 \,\D x = \left[ \frac{x^3}{3} \right]_0^1 = {\color{blue}\frac{1}{3}}, $$
which matches the manually calculated result above.
Failure Example #
We now show another toy example for which simply exchanging differentiation and integration outlined in Eq. \eqref{eqn:dI_0} fails. Let
$$\label{eqn:f_step} f(x, \theta) := \begin{cases} 1, & (x < \theta/2)\\ 1/2. & (x \geq \theta/2) \end{cases}$$
Then, for any $0 < \theta < 2$, it holds that
$$ \begin{split} I &= \int_0^1 f(x, \theta) \,\D x = \left( \int_0^{\theta/2} \D x \right) + \left( \int_{\theta/2}^1 \frac{1}{2} \,\D x \right)\\ &= \left[ x \right]_0^{\theta/2} + \left[ \frac{x}
{2} \right]_{\theta/2}^1 = \frac{\theta}{2} + \left( \frac{1}{2} - \frac{\theta}{4} \right) = \frac{1}{2} + \frac{\theta}{4}, \end{split} $$
$$\label{eqn:f_step_dI_manual} \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{1}{2} + \frac{\theta}{4} \right) = {\color{red}\frac{1}{4}}.$$
However, since the integrand $f$ is piecewise-constant in this example, we have $\D f/\D\theta \equiv 0$. Thus, Eq. \eqref{eqn:dI_0} in this example gives
$$ \int_0^1 \frac{\D}{\D\theta} f(x, \theta) \,\D x = \int_0^1 0 \,\D x = {\color{red}0}, $$
which does not match the manually calculated result in Eq. \eqref{eqn:f_step_dI_manual}.
The General Solution #
Examining The Previous Examples #
Before presenting the general expression of the derivative $\D I/\D\theta$, we first examine the examples shown above.
The Success Example #
We first examine the success example with the integrand $f(x, \theta) = x^2 \,\theta$. In the following, we show the graph of $f(x, \theta)$ for some fixed $\theta = \theta_0$ in the following: $I(\
theta_0) := \int_0^1 f(x, \theta) \,\D x$ equals the signed area (marked in light blue) of the region below the graph. Further, by adding some small $\Delta\theta > 0$ to $\theta_0$, we obtain the
graph of $f(x, \theta_0 + \Delta\theta)$ and the corresponding signed area $I(\theta_0 + \Delta\theta)$, both illustrated in red:
We recall that the derivative of $I$ with respect to $\theta$ is given by the rate at which $I$ changes with $\theta$. To calculate this rate, we examine the difference between $I(\theta_0 + \Delta\
theta)$ and $I(\theta_0)$:
$$\label{eqn:diffI0_0} I(\theta_0 + \Delta\theta) - I(\theta_0) = \int_0^1 \left(f(x, \theta_0 + \Delta\theta) - f(x, \theta_0)\right) \,\D x.$$
Geometrically, this difference equals the (signed) area of the orange region illustrated below:
At each fixed $0 < x < 1$, the integrand of Eq. \eqref{eqn:diffI0_0} satisfies that
$$ f(x, \theta_0 + \Delta\theta) - f(x, \theta_0) \approx \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta. $$
Base on this relation, we can rewrite the area difference \eqref{eqn:diffI0_0} as:
$$ I(\theta_0 + \Delta\theta) - I(\theta_0) \approx \int_0^1 \left( \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta \right) \D x = \Delta\theta \int_0^1 \left[ \frac
{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \D x. $$
In both equations above, the equalities become exact at the limit of $\Delta\theta \to 0$. By dividing both sides by $\Delta\theta$ and taking the limit of $\Delta\theta \to 0$, we have
$$ \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0} := \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta} = \int_0^1 \left[ \frac{\D}{\D\theta} f
(x, \theta) \right]_{\theta = \theta_0} \D x, $$
for any $0 < \theta_0 < 1$. This agrees with the incomplete solution expressed in Eq. \eqref{eqn:dI_0}.
The Failure Example #
So what has been the cause for the failure example? To be specific, what has been missing from the incomplete solution \eqref{eqn:dI_0}?
To understand what has been going on, we again examine the integrand $f(x, \theta)$ which, for this example, is the piecewise-constant function defined in Eq. \eqref{eqn:f_step}.
The following are the graphs of $f(x, \theta)$ for some fixed $\theta = \theta_0$ and $\theta = \theta_0 + \Delta\theta$ (for some small $\Delta\theta > 0$), respectively:
Further, the difference $I(\theta_0 + \Delta\theta) - I(\theta_0)$ between the signed areas below the two graphs is caused by the rectangle illustrated in orange:
Intuitively, in the success example, the change of signed area is caused by vertical shifts of the graph—which is captured by the incomplete solution \eqref{eqn:dI0}. On the other hand, in this
failure example, the change of signed area is caused by horizontal shifts of the graph _at jump discontinuities—which is missing from the incomplete solution!
We now calculate the signed area of the orange rectangle shown above. We first observe that the length of the rectangle’s vertical edge equals the difference $\Delta f \equiv 1 - 1/2 = 1/2$ of the
integrand $f(x, \theta)$ across the discontinuity point.
To calculate the length of the rectangle’s horizontal edge, we let $x(\theta) = \theta/2$ denote the jump discontinuity point of $f(x, \theta)$ defined in Eq. \eqref{eqn:f_step}. Then, the (signed)
length of the horizontal edge is simply $x(\theta_0 + \Delta\theta) - x(\theta_0)$.
Based on the observations above, we know that
$$ I(\theta_0 + \Delta\theta) - I(\theta_0) = \Delta f \,(x(\theta_0 + \Delta\theta) - x(\theta_0)). $$
Dividing both sides of this equation by $\Delta t$ and taking the limit $\Delta\theta \to 0$ produce:
$$ \begin{split} \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0} &= \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta}\\ &= \Delta f \,\lim_{\
Delta\theta \to 0}\frac{x(\theta_0 + \Delta\theta) - x(\theta_0)}{\Delta\theta} = \Delta f \left[ \frac{\D}{\D\theta} x(\theta) \right]_{\theta = \theta_0}. \end{split} $$
Therefore, we know that
$$ \frac{\D}{\D\theta} I(\theta) = \underbrace{\Delta f}_{=\, 1/2} \; \underbrace{\frac{\D}{\D\theta} x(\theta)}_{=\, 1/2} = {\color{red}\frac{1}{4}}, $$
matching the hand-derived result in Eq. \eqref{eqn:f_step_dI_manual}.
The Full Derivative #
Based on the observations above, we now present the general derivative of the 1D integral expressed in Eq. \eqref{eqn:I}:
$$\label{eqn:dI} \boxed{ \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right) = \underbrace{\int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x}_{\text{interior}} \,+\, \
underbrace{\sum_i \Delta f(x_i(\theta), \theta) \,\frac{\D}{\D\theta} x_i(\theta)}_{\text{boundary}}\,, }$$
which comprises:
• A interior component obtained by exchanging differentiation and integration operations—identical to Eq. \eqref{eqn:dI_0}.
• A boundary component involving a sum over all jump discontinuity points $\{ x_i(\theta) : i = 1, 2, \ldots \}$.
Remarks #
Precisely, $\Delta f(x, \theta)$ in the boundary component is defined as
$$ \Delta f(x, \theta) := \lim_{u \uparrow x} f(u, \theta) - \lim_{u \downarrow x} f(u, \theta), $$
where $\lim_{u \uparrow x}$ and $\lim_{u \downarrow x}$ denote one-sided limits with $u$ approaching $x$ from below (i.e., $u < x$) and above (i.e., $u > x$), respectively. For any fixed $\theta$, $\
Delta f(x, \theta)$ is nonzero (and well-defined) if and only if $x$ is a jump discontinuity point of $f(\cdot, \theta)$.
Lastly, when the endpoints $a$ and $b$ of the integral depend on $\theta$, they should be considered as jump discontinuities with $\Delta f(a, \theta) = -f(a, \theta)$ and $\Delta f(b, \theta) = f(b,
In the next section, we will present a generalization of Eq. \eqref{eqn:dI} that describes derivatives of Lebesgue integrals.
1. Unless otherwise stated, we use “continuous” to indicate the $C^0$ class. ↩︎ | {"url":"https://diff-render.org/docs/diff-render/basics/diff-int-1d/","timestamp":"2024-11-07T20:17:06Z","content_type":"text/html","content_length":"19699","record_id":"<urn:uuid:14deacd3-bd24-4414-a31d-da8d06a79387>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00638.warc.gz"} |
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Class 15: Public-Key Revolution (Diffie-Hellman Key Exchange)
Overview: In this part we introduce the idea of asymmetric encryption schemes in which the receiver has a public key and a private key. The public information allows anyone to encrypt a message that
the receiver can decrypt using their symmetric key. This is very different than using the same key to encrypt and decrypt.
So it's time to shift into a new mode of thinking. We'll end up talking about computational number theory and some beautiful mathematics. For now, let's jump in and start doing it.
First of all you'll notice that every one of our encryption schemes so far has had a shared secret. That is, some key which both encrypter and decrypter need for the scheme to succeed. How would that
secret key be shared with your partner? Briefcases and spies have done the job before, but we're in the internet age.
Today I want to look at a method for sharing a private key over a public channel (our class chat for example). I imagine yelling out this information to the whole room and letting our partners work
with us to get a secret shared.
A full conversation involves having four keys, a public and private pair for both Alice (sender) and Bob (recipient).
We won't yet understand every step of this process, but we'll have a good place to start our conversations.
The hard problem
Every public-key scheme is going to involve a "hard problem" that would crack the scheme. Ideally the designer of the scheme has set it up so that either your messages are secure OR the attackers
have made a practical solution to a very tough computer science / math problem.
The first hard problem we face is called the Discrete Log problem. Which is this: Given \(A = g^{\alpha} \pmod{p}\) and \(g\) and \(p\) find \(\alpha\).
Discrete Log Starter: Suppose \(22 = 3^{x} \pmod{31}\). Find \(x\).
Discrete Log Thinking: If you were to solve a larger version, can you tackle it any easier than trying every value of \(x\)?
Overview: This part shows the details of publicly exchanging a private key using the Diffie-Hellman Key Exchange. The world uses this idea to setup all of our HTTPS connections. Once that key is
exchanged we just switch to symmetric-key encryption with AES. So the public-key part is mostly just setting up the crypto we already know. Let's practice the transition.
This idea is important enough that we should try several approaches. Pay attention to which parts are secret and which parts are public.
1) Generate a public triplet
As the initiator (ALICE) we follow these steps to generate a discrete log problem and an answer (which we keep secret).
1. Generate a "Strong" prime (script below) \(p\)
2. Pick a "base" which can really just be the number \(g = 2\)
3. Generate a PRIVATE random number, \(a\), which shares no factors with \(p-1\) (recipe below)
4. Calculate the public exponent: \(A := g^a \pmod{p} \).
5. Publish your public key (triplet): \(p, g, A\) (DO NOT PUBLISH \(a\)!)
Generate a Public Key Triplet: also store the private key somewhere. Publish your triplet in the chat. You'll have to generate several private keys until the GCD is 1, make a while loop there.
The security of these scheme rests on an attacker's inability to figure out \(a\) given \(p, g, A\).
2) BOB's job: the recipient's role
Once Bob has your public-key triplet he can generate a new secret that can be publicly shared with ALICE.
Bob takes in \(p\), \(g\), and \(A\). Now Bob has to also generate a secret:
1. Generate a PRIVATE random number, \(b\), which shares no factors with \(p-1\).
2. Calculate the PUBLIC number \(B = g^b \pmod{p} \).
3. Calculate the PRIVATE shared secret \(K = A^b \pmod{p} \), note that this is really the number \(g^{ab} \pmod{p} \).
4. Publish to Alice: \(B\).
Be BOB: Take in the triplet \( (p, g, A) = (101, 2, 6) \) and generate a response \(B\) and a shared secret. (Little \(a\) was 70 if you want to confirm that you've got the mechanics down.)
3) Our prize: a shared secret
1. ALICE receives \(B\) (she doesn't know \(b\))
2. ALICE computes \(K = (B)^{a} \pmod{p} \) which BOB already knows.
3. ALICE and BOB rejoice in their sneaky cleverness to have shouted information and secretly communicated.
The secret sauce was that anyone can see \(A = g^a \pmod{p}\) and not know \(a\), anyone can see \(B = g^b \pmod{p}\) and now know \(b\). Now only ALICE can compute \(B^a \pmod{p}\) and only BOB can
compute \(A^b \pmod{p}\) since only ALICE knows \(a\) and only BOB knows \(b\). But \(B^a = (g^b)^a = g^{ab} \pmod{p} \) and \(A^b = (g^a)^b = g^{ab} \pmod{p} \).
Do a full exchange: Pretend to be Alice and Bob and generate a shared key. You could also do a swap in the chat channels.
CTF Problem for DHKE:
Overview: In this part we accomplish two things. Get the idea across that after a key exchange we can switch to AES. The other goal is to practice full strength Key swaps using real tools. Let's do
Preface: Use private key
When you've done this key exchange you end up with a shared secret which is likely to have more bits than the typical AES secret key (we need more bits for public-key security than private-key
So at this stage I suggest you switch to AES in an appropriate mode (like CTR). For converting your shared secret into the key for AES you can use a hashing algorithm on the resulting bytes.
Generating Safe Primes
OK, so we've done some amateur DHKE, some basic number theory, and we're finally ready to explore the way this is really done.
Analysis question: We know that the hard problem is reversing \(g^{x} \pmod{p}\). If you were choosing the \(g\) to use would you rather have \(|g| = p-1\) or \(|g| = 2\)?
Create a generator: If I gave you a prime \(p\) and asked you to give me a generator of the multiplicative group \(\mathbb{Z}_p^{*}\) what would you do?
In the last part we used a library to generate a safe prime. Here is a look under the hood.
Safe Prime Explore: the answer that is done most in practice is to generate a safe prime, that is, \(p = 2\cdot q + 1\) where \(q\) is also prime. How would you generate a safe prime?
So the big idea is that if we find a 'safe prime' then we can find a generator with great ease. This is because there are very few subgroups in \(\mathbb{Z}_p^{*}\) just a subgroup of order 2, order
\(q\), and order \(p-1\). That way if we just avoid the small subgroup we know we've got a brute-force space of at least size \(q\)!
Here is a safe-prime generating snippet:
SAGE Run it: That is some SAGE code, so run it at cloud.sagemath.com and see how long it takes when bits is small and now try it with bits at 1024 (the smallest "safe" prime size for DH). WARNING: be
prepared to stop the process!
Working with SSL parameters
Let's learn how to leave it to the professionals:
OpenSSL: in a cloud9 run the following to generate a DH strength prime and a generator of the group \(\mathbb{Z}_p^{*}\): openssl dhparam -out dh1024.pem 1024 (marvel at the speed).
Interpret it two ways: the first way to read it is openssl asn1parse -in dh1024.pem
By hand using Base64: now that prime is stored in .pem format we can get access to it by reading base64. import base64 and run base64.standard_b64decode on the parts that matter. This gives you raw
bytes. Your prime is at raw[6:6+129]. Get these bytes, convert to hex then an integer. The generator is probably the last byte (normally 2).
Confirm: use sage to confirm that \(p\) and \((p-1)/2\) are both prime.
Overview: This is a very useful trick for computational math, but it also unleashes a brutal attack on Diffie-Hellman and the general discrete log problem.
OK let's play a game. A random number is picked and you can only learn the remainder of that number modulo single digit moduli. Your goal is to find the number in the fewest number of guesses, here
Once we're done with that let's talk CRT, the Chinese Remainder Theorem.
There is a perfect mapping from \(\mathbb{Z}_N \leftrightarrow \mathbb{Z}_{q_1} \times \cdots \mathbb{Z}_{q_k}\) whenever the pairwise GCD of \(q_i\) is 1 and \(N = q_1 \cdots q_k\).
Here is an example:
Given that \(x \equiv 2 \mod{5}, x \equiv 1 \mod{3}\) then \(x\) must be equivalent to \(7 \mod{15}\).
SAGE Cloud: You could code your own in Python (like https://rosettacode.org/wiki/Chinese_remainder_theorem ) but as we move into Number Theory I want to show you Python with super-powered math. Head
to https://cloud.sagemath.com . Do the command CRT? once you've made a sage worksheet. Now compute the smallest positive number which is \(3 \mod{9}, 8\mod{13}, 6\mod{25}, 36\mod{121}\).
There are many great applications of the CRT, but in our case we're going to attack all of these schemes we've established when careful primes are not selected.
A Simple CRT Flag
Overview: This attack is important to understand because it doesn't show itself just by size of the key. You have to have more cleverness because of this attack so pay close attention.
I want to introduce a feasible attack on the discrete log problem. That is, given \(A := \alpha^x \mod{p}, p, \alpha\) find \(x\).
This attack will teach us about what makes a prime strong enough, cyclic groups, and the Chinese Remainder Theorem.
Real problem to solve: We are going to solve the following discrete log: \(p= 125301575591,\alpha = 115813337451, \alpha^x \mod{p} = 73973989900\). Compute \(x\). (Solve this after working the small
example below.)
Now here is what we want to do. The problem is, given \(p, g, h:= g^x \mod{p}\) find \(x\).
The big idea is this, the multiplicative subgroup of integers mod \(p\) has size \(p-1\). If we know the factors of \(p-1\) (and they are all small) then we can convert this into a smaller problem.
Imagine that \(q | p-1\) then \((g^{(p-1)/q})^x = h^{(p-1)/q}\) is another equation involving \(x\) but now the possible answers for \(x\) aren't mod \(p-1\) but are mod \(q\).
A small worked example
Given \(p = 31, g = 3, h = 26 = g^x\) find \(x\).
We could try every value of \(x\) from 1 to 30 until we got 26 mod 31. In this case that would be cheap and not a problem, BUT it won't scale to the larger problem.
So we start by factoring \(p-1 = 30 = 2 \cdot 3 \cdot 5\). We will convert the discrete log problem into a three smaller problems that we can Chinese Remainder to find the final solution.
Start with \(q = 2\) which divides \(p-1\). Since we are looking for \(x\) which satisfies the relationship that \(3^x \equiv 26 \pmod{31}\) then if we replace \(3\) and \(26\) by \(3^{15}\) and \(26
^{15}\) then we'll get another relationship \(3^{15x} \equiv 26^{15} \pmod{31}\).
If we look at this a little deeper we know that \(3^{30} \equiv 1 \pmod{31}\) based on what we know about cyclic groups. So this new relationship actually only gives us an answer mod 2. Here's what I
mean. Suppose \(x\) were odd, then \(3^{15x}\) is exactly equivalent to \(3^{15}\) and if \(x\) is even then \(3^{15x}\) is always equivalent to \(1\). So either \(3^{15x}\) is equivalent to \(3^{15}
\) or it is equivalent to \(1\). That means that we have learned a solution to the equation \(x \equiv r \pmod{2}\).
In this case we just check \(26^{15} \pmod{31}\) and we get \(30\) which matches \(3^{15} \pmod{31}\).
So we know that \(x \equiv 1 \pmod{2}\).
Now let's try \(q = 5\). We raise both \(3\) and \(26\) to the \((p-1)/5\)-th power. We get \(3^{6x} \equiv 26^{6} \equiv 1 \pmod{31}\). Now try every remainder of \(x \pmod{5}\) until we find the
right power.
\(3^{0} \equiv 1, 3^6 \equiv 16, 3^{12} \equiv 8, 3^{18} \equiv 4, 3^{24} \equiv 2 \pmod{31}\), so we now know that \(x \equiv 0 \pmod{5}\) and that \(x \equiv 1 \pmod{2}\) which tells us that \(x \
equiv 5 \pmod{10}\).
Why factors? Take a look at the results of the following quick loop. The number of 1s is 1 when we have a generator of \(Z_p^{*}\). What is the pattern?
for j in range(1, 31):
print j, [pow(3**j,i, 31) for i in range(30)].count(1)
Now you try: Using the last prime factor, 3, raise both \(3\) and \(26\) to the \( (p-1)/3 \) and deduce the remainder of \(x \pmod{3}\), and using all three clues deduce the value of \(x \pmod{30}
\). x is 2 mod 3, so we know that x is actually 5 to solve the problem mod 31.
Now write a program to solve the larger problem we opened with. SAGE (or even Wolfram Alpha) can factor \(p-1\) for you.
Overview: It's very important that you feel nervous when using public key crypto. The primes that you work with have to be picked carefully otherwise advanced attacks will undo you. So to deploy this
with confidence you need to know those best attacks and how to thwart them.
The follow up to Pohlig-Hellman is that even a large prime can fall if every factor of \(p-1\) is small.
The same is true when we get to the elliptic curve world.
So you must pick primes where \(p-1\) has at least one large prime factor.
We can use PyCrypto to generate strong primes, and when it comes to elliptic curves we can analyze the parameters of the chosen curves.
Almost Live CTF Problem:
This weekend the highest point crypto problem from one of them was the following pcap file: | {"url":"https://crypto.prof.ninja/class15/","timestamp":"2024-11-11T17:55:14Z","content_type":"text/html","content_length":"19418","record_id":"<urn:uuid:2fee588b-5749-48f0-8643-09d69f274085>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00804.warc.gz"} |
Mathematics for Elementary Teachers
Problem 18
In the Tangrams chapter, you first saw all 7 tangram pieces arranged into a square.
1. If the large square you made with all seven pieces is one whole, assign a (fractional) value to each of the seven tangram pieces. Justify your answers.
2. The tangram puzzle contains a small square. If the small square (the single tangram piece) is one whole, assign a value to each of the seven tangram pieces. Justify your answers.
3. The tangram set contains two large triangles. If a large triangle (the single tangram piece) is one whole, assign a value to each of the seven tangram pieces. Justify your answers.
4. The tangram set contains one medium triangle. If the medium triangle (the single tangram piece) is one whole, assign a value to each of the seven tangram pieces. Justify your answers.
5. The tangram set contains two small triangles. If a small triangle (the single tangram piece) is one whole, assign a value to each of the seven tangram pieces. Justify your answers.
Problem 19
If possible sketch an example of the following triangles. If it is not possible, explain why not.
1. A right triangle that is scalene.
2. A right triangle that is isosceles.
3. A right triangle that is equilateral.
Problem 20
If possible sketch an example of the following triangles. If it is not possible, explain why not.
1. An acute triangle that is scalene.
2. An acute triangle that is isosceles.
3. An acute triangle that is equilateral.
Problem 21
If possible sketch an example of the following triangles. If it is not possible, explain why not.
1. An obtuse triangle that is scalene.
2. An obtuse triangle that is isosceles.
3. An obtuse triangle that is equilateral.
Problem 22
If possible sketch an example of the following triangles. If it is not possible, explain why not.
1. An equiangular triangle that is scalene.
2. An equiangular triangle that is isosceles.
3. An equiangular triangle that is equilateral.
Problem 23
Look at the picture below, which shows two lines intersecting. Angles A and D are called “vertical angles,” and so are angles B and C.
Use this drawing to explain why vertical angles must have the same measure. (Hint: what is the sum of the measures of angle A angle B? How do you know?)
Problem 24
Answer the following questions about the triangle below. Be sure to focus on what you know for sure and not what the picture looks like.
1. Could it be true that x = 4 cm? Explain your answer.
2. Could it be true that x = 20 cm? Explain your answer.
3. Give three possible values of x, based on the information in the picture.
Problem 25
Answer the following questions about the triangle below. Be sure to focus on what you know for sure and not what the picture looks like.
1. If x = 3 cm, the triangle is isosceles. Is this possible? Explain your answer.
2. If x = 8 cm, the triangle is isosceles. Is this possible? Explain your answer.
3. Give three impossible values of x, based on the information in the picture.
Problem 26
Prof. Faber drew this picture on the board, saying it showed three triangles: △ABC, △ABD, and △CBD. Side lengths and angle measurements are shown for each of the triangles.
There are lots of mistakes in this picture. Use what you know about side lengths and angles in triangles to find all the mistakes you can. For each mistake, say what is wrong with the picture, and
why it’s a mistake. Explain your thinking as clearly as you can.
Problem 27
Because of SSS congruence, triangles are exceptionally sturdy. This means they are used frequently in architecture and design to provide supports for buildings, bridges, and other man-made objects.
Take your camera with you, and find several places in your neighborhood or near your campus that use triangular supports. Snap a picture, and describe what the structure is and where you see the
Problem 28
It is possible to create designs that have multiple symmetries. See if you can find images (or create your own!) that have both:
1. reflection symmetry and rotational symmetry,
2. reflection symmetry and translational symmetry, and
3. rotational symmetry and translational symmetry. | {"url":"http://pressbooks-dev.oer.hawaii.edu/math111/chapter/problem-bank-7/","timestamp":"2024-11-09T05:05:22Z","content_type":"text/html","content_length":"79584","record_id":"<urn:uuid:1c51a4bb-4372-477e-a388-a13b57cf32b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00427.warc.gz"} |
st: RE: Convergence problems with Stata 12
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
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st: RE: Convergence problems with Stata 12
From Nick Cox <[email protected]>
To "'[email protected]'" <[email protected]>
Subject st: RE: Convergence problems with Stata 12
Date Tue, 6 Sep 2011 13:36:17 +0100
I don't think this was ever answered, but there is an answer in -help whatsnew-. See also some messages of the same date. However, I can't explain why Christoph's code is so sensitive to the difference.
update 30mar2011
14. rnormal(), the Gaussian random-number generation function in both Stata and Mata, now
produces different, better values. Prior results are reproduced under version control.
rnormal() produced sequences that were insufficiently random for certain applications.
After setting the seed, the sign of the first random number drawn was correlated with
the sign of the first random number that would be drawn after setting a different seed;
the sign of the second random number drawn was correlated with the sign of the second
random number that would be drawn; and so on. Thus the sequence produced by rnormal()
after set seed was not statistically independent from the sequence produced after
another set seed command.
This lack of independence made no difference in the statistical quality of results when
the seed was set only once, because the lack of independence did not arise. Setting the
seed once is typical in many cases, including the running of simulations.
The correlation is of statistical concern when the seed is set more than once in the
same problem.
Only the rnormal() function had this problem. None of Stata's other random-number
functions, such as runiform(), rbeta(), etc., had this problem.
The problem is fixed, with the result that random-number sequences produced by rnormal()
are now different. If you need to re-create previously produced results, use version
control and specify a version prior to 11.2 when setting the random number seed with set
15. Help for set seed now includes useful advice on how to set the seed and explains the
difference between a random-number generator seed and its state as recorded in c(seed).
16. The way version control is handled for random-number generators has changed. Version
control is now specified at the time command set seed is issued; the version in effect
at the time the random-number generator (for example, rnormal()) is used is now
irrelevant. The situation was previously the other way around.
Under the new scheme, typing
. set seed 123456789
. any_command
causes any_command to use the new, version 11.2 rnormal() function even if any_command
is an ado-file itself containing explicit versioning for an earlier release. Thus
existing ado-files need not be updated to benefit from the updated rnormal() function.
Similarly, if you wish to run any_command using the prior version of rnormal(), you may
. version 11.1: set seed 123456789
. any_command
Even years from now, any_command will still use the 11.1 version of rnormal(), and it
will do that even if any_command was written for a later release of Stata.
17. Programmers do not need to update their previously written ado-files because of the
change in function rnormal(), with one exception. If the ado-file itself contains a set
seed command, the ado-file should be updated to use the version in effect at the time
the ado-file was called. To do this, early in the code, obtain the version of the
caller. Later, use the caller's version when issuing command set seed:
program xxx
version ...
syntax ...
local callersversion = _caller()
version `callersversion': set seed ...
If set seed appears in a private subroutine of xxx, you must pass callersversion to the
If set seed appears in another program that you did not write, execute that program
under the caller's version:
program xxx
version ...
syntax ...
local callersversion = _caller()
version `callersversion': mi impute ..., seed(...)
18. New creturn result c(version_rng) records the version number currently in effect for
random-number generators.
[email protected]
-----Original Message-----
From: [email protected] [mailto:[email protected]] On Behalf Of Christoph Engel
Sent: 17 August 2011 15:25
The following observation makes me suspect that there is a change from
Stata 11.0 to Stata 12 that causes .ado files to behave differently. I
have written a program to perform maximum likelihood estimation. On my
previous notebook with Stata 11.0 the program converges much more often
than on my new notebook with either Stata 12 or Stata 11.2. Writing
"version 11" or "version 11.0" at the beginning of the command does not
help either, probably because Stata 11.2 is called anyhow. A check
revealed that, setting the same seed, Stata's rnormal() command
generates different results. I therefore suspect that there has been a
change in random number generators that causes the problem. Any
suggestions for help would be highly appreciated.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2011-09/msg00186.html","timestamp":"2024-11-10T14:21:25Z","content_type":"text/html","content_length":"14903","record_id":"<urn:uuid:975c8bb3-6624-4c47-94ae-536e23f24b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00537.warc.gz"} |
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Analysis and interpretation of a Lorenz curve.
The Lorenz curve visualizes income inequality by comparing actual and equal distribution. It’s a powerful tool spotlighting wealth gaps. Dip below the diagonal line denotes uneven distribution. A
straight line signals perfect equality. Analyzing the Lorenz curve involves calculations and plotting. Data interpretation relies on curve shape. Steeper slopes represent higher inequality levels. A
concave curve symbolizes more evenly spread wealth. Policy decisions can stem from Lorenz curve analysis. Understanding income disparities aids social change initiatives. The curve’s bend points
trace where inequality intensifies. It showcases societal imbalances, urging equity solutions. Networking resources, creating new opportunities can address inequity highlighted by Lorenz curves.
Table of Contents
(Lorenz Curve Explanation, Model, Economics, AP Microeconomics)
A Lorenz curve graphically represents income distribution within a population. It visually compares the actual distribution to ideal equality. The diagonal line on the graph represents perfect
equality. The further the curve is from this line, the greater the income inequality. The Gini coefficient is derived from the Lorenz curve, measuring inequality numerically. A steeper curve implies
higher inequality, while a flatter curve signifies more equality. Analyzing a Lorenz curve helps policymakers understand disparities and formulate targeted interventions. The shape of the curve
identifies where in the income distribution inequality is most prevalent. Interpretation reveals the proportion of total income held by a specific segment of the population. The curve can illustrate
disparities among various income groups, highlighting areas needing attention. A deeper understanding of income distribution trends aids in designing more effective social policies. Rich data from
the Lorenz curve can inform strategies to address poverty, unemployment, and social exclusion. Overall, the Lorenz curve serves as a powerful analytical tool for assessing and addressing income
inequality in society.
Applications of Lorenz curve.
The applications of the Lorenz curve are vast and profound, shedding light on economic disparities that exist in societies worldwide. When you delve into this curve, it’s like peering through a
window into the soul of income distribution. It lays bare the harsh realities faced by many individuals and families.
One crucial use of the Lorenz curve lies in measuring income inequality within a population. By plotting cumulative income against cumulative population from lowest to highest earners, this curve
visually represents how wealth is distributed among different segments of society. The more bowed out the curve is from the line of perfect equality, the greater the income disparity.
Imagine standing at a crossroads where one path leads to abundance while another stretches towards destitution – that’s what an unequal society looks like when portrayed through a skewed Lorenz
curve. Policy makers and economists can utilize this tool to gauge societal imbalance accurately.
Furthermore, economists often employ Gini coefficients derived from Lorenz curves to quantify income inequality numerically. These coefficients provide a single value that encapsulates the extent of
economic disparity present within a region or country – painting a stark picture of social injustice for all to see.
But it’s not just about numbers; behind every data point lies real people struggling with financial hardships caused by inequitable distribution. The Lorenz curve gives these individuals a voice,
turning faceless statistics into tangible stories of struggle and resilience in the face of adversity.
Businesses also benefit from analyzing Lorenz curves as they seek to understand their customer base better. By segmenting consumers based on purchasing power revealed by these curves, companies can
tailor marketing strategies effectively – reaching out to both high-end buyers and budget-conscious shoppers alike.
In essence, delving deep into applications of the Lorenz curve isn’t just about crunching numbers; it’s about revealing truths hidden beneath layers of data points – truths that speak volumes about
our society’s values and priorities.
Construction of Lorenz curve
If you’ve ever gazed at a Lorenz curve, you’ve probably marveled at its elegance and simplicity. But have you ever wondered about the intricate dance of numbers that brings this curve to life? Let’s
delve into the captivating world of constructing a Lorenz curve.
Picture yourself armed with data on income distribution in a society. You line up individuals from the poorest to the richest, each one holding their share of total income. Now, take a deep breath as
you calculate cumulative percentages – adding up how much of the overall income is held by successive portions of the population.
As your fingers fly across the keyboard or scratch away on paper, these cumulative percentages start forming an orderly queue, waiting to tell their story through points plotted on a graph. This
graph will soon evolve into what we know as our beloved Lorenz curve.
With each point placed meticulously, you start connecting them like stars in the night sky – not just lines on paper but reflections of real lives and livelihoods. The gentle arc takes shape under
your skilled hand, revealing insights into inequality that echo societal realities.
But it’s not just about drawing pretty curves; it’s about unveiling truths that can spark change and foster justice. As you witness the curve bending towards equality or stretching further apart in
skewed distributions, emotions might stir within – empathy for those left behind or admiration for strides made towards fairness.
The construction process is akin to weaving a tapestry where every thread represents an individual’s economic standing. Each knot tied solidifies another piece of information unveiled through this
visual representation of disparities.
And when your final stroke completes this masterpiece – a graphical embodiment of social stratification – remember that behind every line lies human stories etched in ink and pixels. The Lorenz curve
stands not just as a mathematical construct but as a mirror reflecting back societal structures often unseen yet profoundly impactful.
In conclusion, constructing a Lorenz curve isn’t merely crunching numbers; it’s painting portraits with data strokes, telling tales woven from statistics and shedding light on hidden inequalities
shaping our world today.
Explanation of Lorenz curve
Ah, the Lorenz curve – a fascinating way to visualize income distribution. Imagine a graph that resembles a gentle arc, revealing how wealth is spread among a population. It’s like peeking into
society’s economic soul.
Let me break it down for you: The horizontal axis represents individuals or households arranged from poorest to richest. On the vertical axis lies cumulative income share, starting at zero and rising
to 100%. As we move along the curve, each point shows what percentage of total income belongs to a given segment of the population.
Now here comes the emotional twist: Picture two lines on this graph dancing together – one representing perfect equality where everyone earns an equal share (a diagonal line), and another showcasing
reality where wealth tends to concentrate in fewer hands (the actual curve). Feel that pang in your heart? That’s inequality tugging at your empathy strings.
As we gaze at this curvy marvel, insights start flowing like rivers after rainfall. The closer our Lorenz curve hugs that diagonal line of equality, the fairer our society stands. But alas! Reality
often jolts us with its stark deviations from this ideal scenario.
In these moments of reflection, emotions swirl within us – frustration over disparities, hope for balance restoration, and determination to strive for equitable distributions. We witness not just
numbers plotted on axes but lives intertwined by economic forces beyond individual control.
Every bend and dip in this graphical journey tells tales of fortunes amassed and dreams deferred; it echoes struggles faced by those perched at different points along its elegant arc. With each data
point marking someone’s slice of prosperity or deprivation, we can’t help but be moved by the human stories underlying statistical abstractions.
Thus, as we navigate through realms illuminated by Lorenz curves, let us not forget their profound implications on real people living real lives. May these visualizations serve as catalysts for
change – inspiring conversations about justice and fairness while stirring emotions that impel us towards building a more inclusive world where every curve trends closer to true equality.
(Understanding the Gini Coefficient)
Gini coefficient
The Gini coefficient is like a spotlight that shines on the inequality within a society. It’s this powerful number that can reveal so much about how wealth or income is distributed among a
population. When you’re looking at a Lorenz curve, which paints a picture of distribution, the Gini coefficient steps in like an interpreter, adding layers of understanding to what you see.
Imagine standing in front of two pathways – one where everyone has exactly equal shares of something and another where one person holds all the cards. The Gini coefficient essentially gives you a
score that falls between 0 (perfect equality) and 1 (maximum inequality), helping you gauge just how evenly or unevenly resources are divided.
Now, let’s dive into why this matters beyond just numbers on paper. Think about your own community – maybe there are some striking differences between families living down different streets. Perhaps
some have access to top-notch schools and healthcare while others struggle to make ends meet without basic necessities.
This disparity isn’t just about money; it extends its fingers into every aspect of people’s lives – their opportunities, health outcomes, education levels… everything. And when we use the Gini
coefficient alongside the visual aid of a Lorenz curve, suddenly those disparities become starkly visible yet quantifiable pieces of reality staring us in the face.
It’s not merely about pointing out unfairness but also serves as a wake-up call for policymakers and communities alike – urging them to strive for more equitable societies where everyone has a fair
shot at prosperity and well-being.
And here comes the emotional punch: seeing these statistics play out right before our eyes evokes empathy for those struggling under systemic inequalities while igniting sparks of determination in
fighting for justice and change.
So next time you come across discussions around income distribution or societal fairness framed through concepts like the Lorenz curve and its trusty sidekick, the Gini coefficient, remember they’re
not just abstract figures but real-world manifestations that hold stories of hope, resilience, injustice…and ultimately, our shared humanity intertwined with struggles for equity.
Interpretation of Lorenz curve
When we delve into the interpretation of a Lorenz curve, we are embarking on a journey through the heart of economic inequality. Picture this: as you gaze at the elegant curve on the graph, it
unveils a poignant narrative of distribution disparity within a society.
The Lorenz curve is not merely lines and axes; it’s a visual manifestation of societal wealth division. The closer the curve hugs the diagonal line of perfect equality, the fairer wealth is
distributed among individuals. However, if it veers away sharply from that line, forming an exaggerated bow towards one end, then inequity reigns supreme.
As your eyes trace along its arching trajectory, emotions may stir within you – perhaps empathy for those clustered in the lower segments where meager resources dwell disproportionately. Conversely,
seeing how vast swathes above benefit lavishly can evoke feelings ranging from admiration to discontentment.
Numbers alone cannot capture this essence – it takes introspection prompted by witnessing such stark visuals to truly grasp what lies beneath these mathematical diagrams. A lopsided Lorenz curve
tells tales of social injustice and economic imbalance with more poignancy than any statistic ever could.
Imagine feeling like you’re teetering on that fragile line delineating shared prosperity and glaring disparity as you study this intricate graph. It’s like peering into a reflection pool mirroring
our collective conscience about fairness and equality in resource allocation.
Interpreting a Lorenz curve isn’t just an exercise in data analysis; it’s an emotional reckoning with societal values and priorities laid bare before us in graphical form. It prompts us to ponder
deeply about what kind of world we want to live in – one marked by equitable opportunities or marred by entrenched privilege?
So next time you encounter a Lorenz curve dancing across your screen or paper – pause for a moment. Let its silent eloquence speak volumes to your soul about our interconnected fates woven
intricately into this tapestry called society.
External Links | {"url":"https://info.3diamonds.biz/analysis-and-interpretation-of-a-lorenz-curve/","timestamp":"2024-11-08T03:03:03Z","content_type":"text/html","content_length":"103281","record_id":"<urn:uuid:94c7a269-cd61-4828-bbd2-5cd7770a0029>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00879.warc.gz"} |
Math Tests for Grade 7 - Free Math Practice Mock Tests - Wizert Maths
Here you can find dozens of Math practice tests covering an exhaustive list of topics for Grade 7, including Factorization, Linear Algebra, Geometry, etc. These are tightly-timed tests, and your goal
should be to complete them as quickly as possible. Good luck! | {"url":"https://maths.wizert.com/practice-tests/grade-7","timestamp":"2024-11-07T20:03:26Z","content_type":"text/html","content_length":"465730","record_id":"<urn:uuid:6cce2de6-a447-4416-9d66-4db686d06dbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00771.warc.gz"} |
Amplitude Quantization - Electrical Engineering Textbooks
Amplitude Quantization
The Sampling Theorem says that if we sample a bandlimited signal
fast enough, it can be recovered without error from its samples
. Sampling is only the first phase of acquiring data into a computer: Computational processing further requires that the samples be quantized: analog values are converted into digital form. In short,
we will have performed analog-to-digital (A/D) conversion.
A phenomenon reminiscent of the errors incurred in representing numbers on a computer prevents signal amplitudes from being converted with no error into a binary number representation. In
analog-to-digital conversion, the signal is assumed to lie within a predefined range. Assuming we can scale the signal without affecting the information it expresses, we'll define this range to be
. Furthermore, the A/D converter assigns amplitude values in this range to a set of integers. A
-bit converter produces one of the integers
for each sampled input. Figure shows how a three-bit A/D converter assigns input values to the integers. We define a quantization interval to be the range of values assigned to the same integer.
Thus, for our example three-bit A/D converter, the quantization interval
; in general, it is
Recalling the plot of average daily highs in this frequency domain problem, why is this plot so jagged? Interpret this effect in terms of analog-to-digital conversion.
The plotted temperatures were quantized to the nearest degree. Thus, the high temperature's amplitude was quantized as a form of A/D conversion.
Because values lying anywhere within a quantization interval are assigned the same value for computer processing, the original amplitude value cannot be recovered without error. Typically, the D/A
converter, the device that converts integers to amplitudes, assigns an amplitude equal to the value lying halfway in the quantization interval. The integer 6 would be assigned to the amplitude 0.625
in this scheme. The error introduced by converting a signal from analog to digital form by sampling and amplitude quantization then back again would be half the quantization interval for each
amplitude value. Thus, the so-called A/D error equals half the width of a quantization interval:
. As we have fixed the input-amplitude range, the more bits available in the A/D converter, the smaller the quantization error.
To analyze the amplitude quantization error more deeply, we need to compute the signal-to-noise ratio, which equals the ratio of the signal power and the quantization error power. Assuming the signal
is a sinusoid, the signal power is the square of the rms amplitude:
. The illustration details a single quantization interval.
Its width is
and the quantization error is denoted by
. To find the power in the quantization error, we note that no matter into which quantization interval the signal's value falls, the error will have the same characteristics. To calculate the rms
value, we must square the error and average it over the interval.
Since the quantization interval width for a
-bit converter equals
, we find that the signal-to-noise ratio for the analog-to-digital conversion process equals
Thus, every bit increase in the A/D converter yields a 6 dB increase in the signal-to-noise ratio. The constant term
equals 1.76.
This derivation assumed the signal's amplitude lay in the range
. What would the amplitude quantization signal-to-noise ratio be if it lay in the range
The signal-to-noise ratio does not depend on the signal amplitude. With an A/D range of
, the quantization interval
and the signal's rms value (again assuming it is a sinusoid) is
How many bits would be required in the A/D converter to ensure that the maximum amplitude quantization error was less than 60 db smaller than the signal's peak value?
Music on a CD is stored to 16-bit accuracy. To what signal-to-noise ratio does this correspond?
A 16-bit A/D converter yields a SNR of
Once we have acquired signals with an A/D converter, we can process them using digital hardware or software. It can be shown that if the computer processing is linear, the result of sampling,
computer processing, and unsampling is equivalent to some analog linear system. Why go to all the bother if the same function can be accomplished using analog techniques? Knowing when digital
processing excels and when it does not is an important issue.
Explore CircuitBread
Get the latest tools and tutorials, fresh from the toaster. | {"url":"https://www.circuitbread.com/textbooks/fundamentals-of-electrical-engineering-i/digital-signal-processing/amplitude-quantization","timestamp":"2024-11-09T19:32:16Z","content_type":"text/html","content_length":"952442","record_id":"<urn:uuid:d2a8d8b9-e1cf-4475-b374-af463bc962a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00529.warc.gz"} |
Role Playing with Probabilities: The Importance of Distributions
by Jocelyn Barker, Data Scientist at Microsoft
I have a confession to make. I am not just a statistics nerd; I am also a role-playing games geek. I have been playing Dungeons and Dragons (DnD) and its variants since high school. While playing
with my friends the other day it occurred to me, DnD may have some lessons to share in my job as a data scientist. Hidden in its dice rolling mechanics is a perfect little experiment for
demonstrating at least one reason why practitioners may resist using statistical methods even when we can demonstrate a better average performance than previous methods. It is all about
distributions. While our averages may be higher, the distribution of individual data points can be disastrous.
Why Use Role-Playing Games as an Example?
Partially because it means I get to think about one of my hobbies at work. More practically, because consequences of probability distributions can be hard to examine in the real world. How do you
quantify the impact of having your driverless car misclassify objects on the road? Games like DnD on the other hand were built around quantifying the impact of decisions. You decide to do something,
add up some numbers that represent the difficulty of what you want to do, and then roll dice to add in some randomness. It also means it is a great environment to study how the distribution of the
randomness impacts the outcomes.
A Little Background on DnD
One of the core mechanics of playing DnD and related role-playing games involve rolling a 20 sided die (often referred to as a d20). If you want your character to do something like climb a tree,
there is some assigned difficulty for it (eg. 10) and if you roll higher than that number, you achieve your goal. If your character is good at that thing, they get to add a skill modifier (eg. 5) to
the number they roll making it more likely that they can do what they wanted to do. If the thing you want to do involves another character, things change a little. Instead of having a set difficulty
like for climbing a tree, the difficulty is an opposed roll from the other player. So if Character A wants to sneak past Character B, both players roll d20s and Character A adds their “stealth”
modifier against Character B’s “perception” modifier. Whoever between them gets a higher number wins with a tie going to the “perceiver”. Ok, I promise, that is all the DnD rules you need to know for
this blog post.
Alternative Rolling Mechanics: What’s in a Distribution?
So here is where the stats nerd in me got excited. Some people change the rules of rolling to make different distributions. The default distribution is pretty boring, 20 numbers with equal
One common way people modify this is with the idea of “critical”. The idea is that sometimes people do way better or worse than average. To reflect this, if you roll a 20, instead of adding 20 to
your modifier, you add 30. If you roll a 1, you subtract 10 from your modifier.
Another stats nerd must have made up the last distribution. The idea for constructing it is weird, but the behavior is much more Gaussian. It is called 3z8 because you roll 3 eight-sided dice that
are numbered 0-7 and sum them up giving a value between 0 and 21. 1-20 act as in the standard rules, but 0 and 21 are now treated like criticals (but at a much lower frequency than before).
The cool thing is these distributions have almost identical expected values (10.5 for d20, 10.45 with criticals, and 10.498 for 3z8), but very different distributions. How do these distributions
affect the game? What can we learn from this as statisticians?
Our Case Study: Sneaking Past the Guards
To examine how our distributions affects outcome, we will look at a scenario where a character, who we will call the rogue, wants to sneak past three guards. If any of the guard’s perception is
greater than or equal to the rogue’s stealth, we will say the rogue loses the encounter, if they are all lower, the rogue is successful. We can already see the rogue is at a disadvantage; any one of
the guards succeeding is a failure for her. We note that assuming all the guards have the same perception modifier, the actual value of the modifier for the guards doesn’t matter, just the difference
between their modifier and the modifier of the rogue because the two modifiers are just a scalar adjustment of the value rolled. In other words, it doesn’t matter if the guards are average Joes with
a 0 modifier and the rogue is reasonably sneaky with a +5 or if the guards are hyper alert with a +15 and the rogue is a ninja with a +20; the probability of success is the same in the two scenarios.
Computing the Max Roll for the Guards
Lets start off getting a feeling for what the dice rolls will look like. Since the rogue is only rolling one die, her probability distribution looks the same as the distribution of the dice from the
previous section. Now, lets consider the guards. In order for the rogue to fail to sneak by, she only needs to be seen by one of the guards. That means we just need to look at the probability that
the maximum roll for one of the guards is \(n\) for \(n \in 1,..,20\). We will start with our default distribution. The number of ways you can have 3 dice roll a value of \(n\) or less is \(n^3\).
Therefore the number of ways you can have the max value of the dice be exactly \(n\) is the number of ways you can roll \(n\) or less minus the number of ways where all the dice are \(n - 1\) or less
giving us \(n^3 - (n - 1)^3\) ways to roll a max value of \(n\). Finally, we can divide by the total number of roll combinations for an 20-sided dice, \(20^3\), giving us our final probabilities of:
\[\frac{n^3 - (n-1)^3}{20^3} \textrm{for} \{n \in 1, ..., 20\}\]
The only thing that changes when we add criticals to the mix is that now the probabilities previously assigned to 1 get re-assigned to -10 and those assigned to 20 get reassigned to 30 giving us the
following distribution.
This means our guards get a critical success ~14% of the time! This will have a big impact on our final distributions.
Finally, lets look at the distribution for the guards using the 3z8 system.
In the previous distributions, the maximum value became the single most likely roll. Because of the the low probability of rolling a 21 in the 3z8 distribution, this distribution still skews right,
but peaks at 14. In this distribution, criticals only occur ~0.6% of the time; much less than the previous distribution.
Impact on Outcome
Now that we have looked at the distributions of the rolls for the rogue and the guards, lets see what our final outcomes look like. As previously mentioned, we don’t need to worry about the specific
modifiers for the two groups, just the difference between them. Below is a plot showing the relative modifier for the rogue on the x-axis and the probability of success on the y-axis for our three
different probability distributions.
We see that for the entire curve, our odds of success goes down when we add criticals and for most of the curve, it goes up for 3z8. Lets think about why. We know the guards are more likely to roll a
20 and less likely to roll a 1 from the distribution we made earlier. This happens about 14% of the time, which is pretty common, and when it happens, the rogue has to have a very high modifier and
still roll well to overcome it unless they also roll a 20. On the other hand, with 3z8 system, criticals are far less common and everyone rolls close to average more of the time. The expected value
for the rogue is ~10.5, where as it is ~14 for the guards, so when everyone performs close to average, the rogue only needs a small modifier to have a reasonable chance of success.
To illustrate how much of a difference there is between the two, lets consider what would be the minimum modifier needed to have a certain probability of success.
Probability Roll 1d20 With Criticals Roll 3z8
50% 6 7 4
75% 11 13 8
90% 15 22 11
95% 17 27 13
We see from the table that reasonably small modifiers make a big difference in the 3z8 system, where as very large modifiers are needed to have a reasonable chance of success when criticals are
added. To give context on just how large this is, when a someone is invisible, this only adds +20 to their stealth checks when they are moving. In other words, in the system with criticals, our rogue
could literally be invisible sneaking past a group of not particularly observant gaurds and have a reasonable chance of failing.
The next thing to consider is our rogue may have to make multiple checks to sneak into a place (eg. one to sneak into the courtyard, one to sneak from bush to bush, and then a final one to sneak over
the wall). If we look at the results of our rogue making three successful checks in a row, our probabilities change even more.
Probability Roll 1d20 With Criticals Roll 3z8
50% 12 15 9
75% 16 23 11
90% 18 28 14
95% 19 29 15
Making multiple checks exaggerates the differences we saw previously. Part of the reason for the poor performance with the addition of criticals (and for the funny shape of the critical curve) is
there is a different cost associated with criticals for the rogue compared to the guards. If the guards roll a 20 or the rogue rolls a 1, when criticals are in play, the guards will almost certainly
win, even if the rogue has a much higher modifier. On the other hand, if the guard rolls a 1 or the rogue rolls a 20, there isn’t much difference in outcome between getting that critical and any
other low/high roll; play continues to the next round.
How Does This Apply to Data Science?
Many times as data scientists, we think of the predictions we make as discrete data points and when we evaluate our models we use aggregate metrics. It is easy to lose sight that our predictions are
samples from a probability distribution, and that aggregate measures can obscure how well our model is really performing. We saw in the example with criticals where big hits and misses can make a
huge impact on outcomes, even if the average performance is largely the same. We also saw with the 3z8 system where decreasing the expected value of the roll can actually increase performance by
making the “average” outcome more likely.
Does all of this sound contrived to you, like I am trying to force an analogy? Let me make a concrete example from my real life data science job. I am responsible for making the machine learning
revenue forecasts for Microsoft. Twice a quarter, I forecast the revenue for all of the products at Microsoft world wide. While these product forecasts do need to be accurate for internal use, the
forecasts are also summed up to create segment level forecasts. Microsoft’s segment level forecasts go to Wall Street and having our forecasts fail to meet actuals can be a big problem for the
company. We can think about our rogue sneaking past our guards as being an analogy for nailing the segment level forecast. If I succeed for most of the products (our individual guards) but have a
critical miss of $1 billion error on one of them, then I have a $1 billion error for the segment and I failed. Also like our rogue, one success doesn’t mean we have won. There is always another
quarter and doing well one quarter doesn’t mean Wall Street will cut you some slack the next. Finally, a critical success is less valuable than a critical failure is problematic. Getting the
forecasts perfect one quarter will just get your a “good job” and a pat on the back, but a big miss costs the company. In this context, it is easy to see why the finance team doesn’t take the machine
learning forecasts as gospel, even with our track record of high accuracy.
So as you evaluate your models, keep our sneaky friend in mind. Rather than just thinking about your average metrics, think about your distribution of errors. Are your errors clustered nicely around
the mean or are they scattershot of low and high? What does that mean for your application? Are those really low errors valuable enough to be worth getting the really high ones from time to time?
Many times having a reliable model may be more valuable than a less reliable one with higher average performance, so when you evaluate, think distributions, not means.
The charts in this post were all produced using the R language. To see the code behind the charts, take a look at this R Markdown file.
You can follow this conversation by subscribing to the comment feed for this post.
Hi! Lovely article! I used to mostly play stormbringer, where we rolled two d20, one for "tens" and one for "ones" and make a percent from 0-99, 00 or 01 were critical fails, 98 or 99 were critical
wins and would have DIRE consequences, like, lose a nose/amputated arm/permanent scar that affects your seduction skill forever changing your playstyle and ruining your life.... :(
but I digress. How do 2 d20 compare to your 3z8?
Hi Amit, I haven't played Stormbringer personally, but I believe it uses percentile dice, much like you described, but with 2d10s rather than 2d20s. Using percentile dice in this way also generates a
uniform distribution similar to using 1d20, but spread out over 100 values rather than over 20 possible values. Basically using a system like this would fall somewhere between 1d20 and 1d20 with
criticals, because the distribution is still uniform but criticals now have a 2% chance of happening rather than a 5% chance.
The reason 3z8 has a non uniform distribution is because since you sum the dice, there is more than one way to get a particular value. For example, you could get a 2 in three different ways; 1)
rolling a 0 on the first dice and a 2 on the second dice, 2) rolling a 2 on the first dice and a 0 on the second dice, or 3) rolling 1 on both dice. This means it is more likely to happen that
getting a critical miss of 0 because that can only happen 1 way; rolling 0 on both dice. Alternatively with the 2d10 percentile dice, there is only one way to get any number. For example, 17 can only
happen by getting a 1 on the tens dice and a 7 on the ones dice. | {"url":"https://blog.revolutionanalytics.com/2017/11/role-playing-with-probabilities.html","timestamp":"2024-11-14T20:57:21Z","content_type":"application/xhtml+xml","content_length":"50794","record_id":"<urn:uuid:636c5eca-2249-4297-adca-4d978f25d692>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00494.warc.gz"} |
OSU Math Project with Earth Sciences
Since the end of 2017, Assistant Professor Jim Fowler and Professor Crichton Ogle have been attacking various mathematical problems related to Earth science as part of an ongoing interdisciplinary
project with Ohio Eminent Scholar Mike Bevis and other researchers in Ohio State's School of Earth Sciences. More recently, a third faculty member in the Department of Mathematics, Professor Ovidiu
Costin, joined the team, and together they continue to pursue new avenues for an improved Earth Gravitational Model (EGM).
Such models provide high-resolution maps of Earth's gravity field. It may come as a surprise that the strength of gravity depends on where you are located! The applications of having such a "gravity
map" are numerous. As one example, by repeatedly building a high-resolution map of Earth's gravity field over time, climate researchers can better understand the relationships between glaciers and
sea-level change, or the change in the storage of groundwater. Knowing about groundwater has immediate real-world impacts in understanding the potential for floods and for droughts. Building such a
map is a challenging mathematical puzzle.
The initial interdisciplinary project between Ohio State's Department of Mathematics and the School of Earth Sciences focused on a different classical problem, namely the disk loading problem, which
asks, “How does the Earth deform when subjected to a disk of imposed pressure?” By adding many such disk-shaped regions together, one can estimate how the Earth's surface bends in response to the
weight of a glacier with a complicated shape. This computation involves sums of products of Legendre polynomials, and the team discovered “new identities:” a certain infinite series can be replaced
with various closed form expressions involving elliptic integrals.
One annoyance with solving the disk loading problem this way is the circle-packing required. To approximate the effect of the glacier, one approximates the glacier by small disks. It would be cleaner
to have good computational techniques for solving the "rectangular" loading problem, in other words, a technique to compute how the Earth deforms in response to a pressure spread over a rectangular
region bounded by latitude and longitude lines. The team at Ohio State is making progress on this.
Buoyed by their successes studying loading problems and working on improvements to EGM, the team in the Department of Mathematics is also pursuing education and training projects. Such work is
crucial to build capacity within the United States for geodetic science. Geodetic science might not be well-known, but it is the foundation of much of the modern world's technology – consider how
important it is, for instance, for a cell phone to know its location. Whether through the use of navigation apps or other location-based mobile services, many people depend on Global Positioning
System (GPS) every day. GPS is built on WGS84, the "World Geodetic System," as a terrestrial reference frame, and the National Geospatial-Intelligence Agency (NGA) maintains this standard. But some
of the most senior researchers at NGA have retired, so there is concern about how the next generation will maintain WGS84 and the like.
To ensure the next generation can continue the work on geodesy that enables transformative location-aware technology, NGA awarded grant funding to Ohio State to provide training on the key principles
behind creating gravitational models and other advanced geodetic concepts. With the support of this NGA grant, the Department of Mathematics will use its expertise in offering online courses and
creating calculus videos to provide online course material for geodetic science. This ensures that those working on EGM are familiar with powerful mathematical tools for geodesy. These online courses
will include discussions of open questions around improvements to EGM, and the relevant mathematics that could lead to such improvements.
By pursuing interdisciplinary work, both disciplines are improved. In this case, there are opportunities to engage graduate students in the Department of Mathematics with modern problems in geodesy.
Some of the techniques used in this area involve machine learning, and training Ohio State undergraduates and graduate students in such techniques can help them land a career in data science.
Article contributed by Assistant Professor Jim Fowler
His research interests broadly include geometry and topology and he is especially
fond of using computational techniques to attack problems in pure mathematics. | {"url":"https://math.osu.edu/newsletter/autumn-2019/osu-math-project-earth-sciences","timestamp":"2024-11-05T18:53:23Z","content_type":"text/html","content_length":"107174","record_id":"<urn:uuid:d5fc0947-16a6-4872-b8f1-95d1b8b6e32d>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00671.warc.gz"} |
Galerkin Approach - One Dimensional Problems Questions and Answers - Sanfoundry
Finite Element Method Questions and Answers – One Dimensional Problems – Galerkin Approach
This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “One Dimensional Problems – Galerkin Approach”.
1. Galerkin technique is also called as _____________
a) Variational functional approach
b) Direct approach
c) Weighted residual technique
d) Variational technique
View Answer
Answer: c
Explanation: The equivalent of applying the variation of parameters to a function space, by converting the equation into weak formulation. Galerkin’s method provide powerful numerical solution to
differential equations and modal analysis. The Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method.
2. In the equation, \(\int_{L} \sigma^T \epsilon(\phi)Adx -\int_{L} \phi^T f Adx -\int_{L}\phi^Tdx – \sum_{i}\phi_i P_i=0\) First term represents _______
a) External virtual work
b) Virtual work
c) Internal virtual work
d) Total virtual work
View Answer
Answer: c
Explanation: In the given equation first term represents internal virtual work. Virtual work means the work done by the virtual displacements. The principle of virtual work is equivalent to the
conditions for static equilibrium of a rigid body expressed in terms of total forces and torques. The virtual work done by internal forces is called internal virtual work.
3. Considering element connectivity, for example for element ψ=[ψ[1], ψ[2]]^n for element n, then the variational form is ______________
a) ψ^T(KQ–F)=0
b) ψ(KQ-F)=0
c) ψ(KQ)=F
d) ψ(F)=0
View Answer
Answer: a
Explanation: Element connectivity means Assemble the element equations. To find the global equation system for the whole solution region we must assemble all the element equations. For formulation of
a variational form for a system of differential equations. First method treats each equation independently as a scalar equation, while the other method views the total system as a vector equation
with a vector function as a unknown.
4. Write the element stiffness matrix for a beam element.
a) K=\(\frac{2EI}{l}\)
b) K=\(\frac{2EI}{l}\begin{bmatrix}2 & 1 \\ 1 & 2 \end{bmatrix}\)
c) K=\(\frac{2E}{l}\begin{bmatrix}2 \\ 1 \end{bmatrix}\)
d) K=\(\frac{2E}{l}\begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix}\)
View Answer
Answer: b
Explanation: Element stiffness matrix means it is a matrix method that makes use of the members stiffness relations for computing member forces and displacements in the structures.
5. Element connectivities are used for _____
a) Traction force
b) Assembling
c) Stiffness matrix
d) Virtual work
View Answer
Answer: b
Explanation: Element connectivity means “Assemble the element equations. To find the global equation system for the whole solution region we must assemble all the element equations. In other words we
must combine local element equations for all the elements used for discretization.
6. Virtual displacement field is _____________
a) K=\(\frac{EA}{l}\)
b) F=ma
c) f(x)=y
d) ф=ф(x)
View Answer
Answer: d
Explanation: Virtual work is defined as work done by a real force acting through a virtual displacement. Virtual displacement is an assumed infinitesimal change of system coordinates occurring while
time is held constant.
7. Virtual strain is ____________
a) ε(ф)=\(\frac{dx}{d\phi}\)
b) ε(ф)=\(\frac{d\phi}{dx}\)
c) ε(ф)=\(\frac{dx}{d\varepsilon}\)
d) ф(ε)=\(\frac{d\varepsilon}{d\phi}\)
View Answer
Answer: b
Explanation: Virtual work is defined as the work done by a real force acting through a virtual displacement. A virtual displacement is any displacement is any displacement consistent with the
constraints of the structure.
8. To solve a galerkin method of approach equation must be in ___________
a) Equation
b) Vector equation
c) Matrix equation
d) Differential equation
View Answer
Answer: d
Explanation: Galerkin method of approach is also called as weighted residual technique. This method of approach can be used for irregular geometry with a regular pattern of nodes. The solution
function is substituted in a differential equation, this differential equation will not be satisfied and will give a residue.
9. By the Galerkin approach equation can be written as __________
a) {P}-{K}{Δ}=0
b) {K}-{P}{Δ}=0
c) {Δ}-{p}{K}=0
d) Undefined
View Answer
Answer: a
Explanation: Galerkin’s method of weighted residuals, the most common method of calculating the global stiffness matrix in fem. This requires the boundary element for solving integral equations.
10. In basic equation Lu=f, L is a ____________
a) Matrix function
b) Differential operator
c) Degrees of freedom
d) No. of elements
View Answer
Answer: b
Explanation: The method of weighted residual technique uses the weak form of physical problem or the direct differential equation. The basic equation Lu=f in that L is an differential operator. It
uses the principle of orthogonality between Residual function and basis function.
Sanfoundry Global Education & Learning Series – Finite Element Method.
To practice all areas of Finite Element Method, here is complete set of 1000+ Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/finite-element-method-questions-answers-one-dimensional-problems-galerkin-approach/","timestamp":"2024-11-02T18:05:48Z","content_type":"text/html","content_length":"164412","record_id":"<urn:uuid:2954b1db-9a15-494c-90b4-758f0ff647e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00292.warc.gz"} |
Newton's law of gravitation
The motion of the planets, the moon and the Sun was the interesting subject among the students of Trinity college at Cambridge in England.
Newton was also one among these students. In 1665, the college was closed for an indefinite period due to plague. Newton, who was then 23 years old, went home to Lincolnshire. He continued to think
about the motion of planets and the moon. One day Newton sat under an apple tree and had tea with his friends. He saw an apple falling to ground. This incident made him to think about falling bodies.
He concluded that the same force of gravitation which attracts the apple to the Earth might also be responsible for attracting the moon and keeping it in its orbit. The centripetal acceleration of
the moon in its orbit and the downward acceleration of a body falling on the Earth might have the same origin. Newton calculated the centripetal acceleration by assuming moon's orbit (Fig.) to be
Acceleration due to gravity on the Earth's surface, g = 9.8 m s^-2
Centripetal acceleration on the moon, a^c = v^2/r
where r is the radius of the orbit of the moon (3.84 × 10^8 m) and v is the speed of the moon.
Time period of revolution of the moon around the Earth,
T = 27.3 days.
The speed of the moon in its orbit, v = 2πr /T
v = = 1.02 × 10^3 m s^−1
Centripetal acceleration, a[c] = v^2/r = 2.7 × 10^−3 m s^−2
Newton assumed that both the moon and the apple are accelerated towards the centre of the Earth. But their motions differ, because, the moon has a tangential velocity whereas the apple does not have.
Newton found that a[c] was less than g and hence concluded that force produced due to gravitational attraction of the Earth decreases with increase in distance from the centre of the Earth. He
assumed that this acceleration and therefore force was inversely proportional to the square of the distance from the centre of the Earth. He had found that the value of a[c] was about 1/3600 of the
value of g, since the radius of the lunar orbit r is nearly 60 times the radius of the Earth R.
The value of a[c] was calculated as follows :
A[c]/g = (1/r^2) / (1/R^2) = 1/3600
A[c] = 9.8/3600 = 2.7 × 10^−3 m s^−2
Newton suggested that gravitational force might vary inversely as the square of the distance between the bodies. He realised that this force of attraction was a case of universal attraction between
any two bodies present anywhere in the universe and proposed universal gravitational law.
The law states that every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the
square of the distance between them.
Consider two bodies of masses m[1] and m[2] with their centres separated by a distance r. The gravitational force between them is
F α m[1]m[2]
F α 1/r^2
F = G m[1]m[2 / ]r^2
where G is the universal gravitational constant.
If m[1] = m[2] = 1 kg and r = 1 m, then F = G.
[Hence, the Gravitational constant 'G' is numerically equal to the gravitational force of attraction between two bodies of mass 1 kg each separated by a distance of 1 m. The value of G is ]6.67 × 10^
−^11 N m^2 kg^−^2 and its dimensional formula is M^−^1 L^3 T^−^2
Special features of the law
The gravitational force between two bodies is an action and reaction pair.
The gravitational force is very small in the case of lighter bodies. It is appreciable in the case of massive bodies. The gravitational force between the Sun and the Earth is of the order of 10^27 N. | {"url":"https://www.brainkart.com/article/Newton-s-law-of-gravitation_3024/","timestamp":"2024-11-06T21:14:27Z","content_type":"text/html","content_length":"51298","record_id":"<urn:uuid:23b356e2-6a26-45b4-92eb-6a028c60f882>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00786.warc.gz"} |
Air Conditioner Current Calculator - Easily Determine Energy Usage | Calculate your AC's Power Consumption Now - Calculator Pack
Air Conditioner Current Calculator
Are you tired of struggling to determine the current requirements for your air conditioning system? Look no further than our Air Conditioner Current Calculator! Using this tool, you can easily
calculate the current draw of your system and ensure that you are using the appropriate electrical components. Our calculator takes into account factors such as voltage, phase and horsepower to
provide you with an accurate current reading. Simply input the necessary information and let our calculator do the work for you! This tool is perfect for HVAC technicians, contractors and anyone
involved in the installation of air conditioning systems. Save yourself the hassle of manual calculations and try our Air Conditioner Current Calculator today!
Air Conditioner Current Calculator
Calculate the current consumption of your air conditioner
Air Conditioner Current Calculator Results
Voltage: 0
Power: 0
Power Factor: 0
Current Consumption: 0
Share results with your friends
Keep your home cool efficiently with our air conditioner current calculator. To calculate watts and more, visit our watt calculator.
How to Use the Air Conditioner Current Calculator
The Air Conditioner Current Calculator is a useful tool designed to calculate the current consumption of your air conditioner. It helps you understand the amount of current your air conditioner will
draw based on the input parameters you provide. By utilizing this calculator, you can make informed decisions about power usage and ensure that your electrical system can handle the load.
The Air Conditioner Current Calculator finds its significance in various applications, including:
• Homeowners: Determine the electrical requirements of your air conditioner to ensure your home's electrical system can support it.
• Electricians: Use the calculator to assess the current load of air conditioners during installations or troubleshooting electrical issues.
• Energy Efficiency: Calculate the current consumption to optimize energy usage and make informed decisions about alternative cooling solutions.
Instructions for Utilizing the Calculator:
To use the Air Conditioner Current Calculator, follow these instructions:
Input Fields
The calculator requires the following input fields:
• Voltage: Enter the voltage rating of your electrical system. It represents the potential difference between two points and is usually measured in volts (V).
• Power: Specify the power rating of your air conditioner. It indicates the rate at which your air conditioner consumes electrical energy and is measured in watts (W).
• Power Factor: Enter the power factor of your air conditioner. It represents the efficiency of power utilization and ranges from 0 to 1. Higher power factors indicate better energy utilization.
Providing accurate input values is crucial for obtaining reliable results.
Output Fields
The Air Conditioner Current Calculator provides the following output fields:
• Voltage: Displays the voltage value you entered.
• Power: Shows the power value you specified.
• Power Factor: Displays the power factor value you entered.
• Current Consumption: Presents the calculated current consumption of your air conditioner based on the input parameters. The unit of measurement is amperes (A), which represents the flow of
electric current.
Air Conditioner Current Calculator Formula
The calculator utilizes the following formula to calculate the current consumption:
Current Consumption = Power / (Voltage * Power Factor)
Illustrative Example:
Let's consider an example to illustrate the calculator's usage:
• Voltage: 230 volts
• Power: 1500 watts
• Power Factor: 0.9
Calculation: Current Consumption = 1500 / (230 * 0.9) = 7.97 Amperes
In this example, an air conditioner with a power rating of 1500 watts, operating at 230 volts with a power factor of 0.9, would draw approximately 7.97 amperes of current.
Illustrative Table Example:
Here is an example table showcasing different air conditioner configurations and their corresponding current consumptions:
Voltage (V) Power (W) Power Factor Current Consumption (A)
230 1500 0.9 7.97
120 2000 0.8 16.67
208 1800 0.95 7.34
The Air Conditioner Current Calculator is a valuable tool that allows you to determine the current consumption of your air conditioner. By providing the necessary input values, such as voltage,
power, and power factor, you can quickly assess the electrical requirements of your air conditioner. Understanding the current consumption helps you make informed decisions about energy usage and
ensures the safety and efficiency of your electrical system. By utilizing this calculator, you can optimize your air conditioner's performance | {"url":"https://calculatorpack.com/air-conditioner-current-calculator/","timestamp":"2024-11-05T15:19:01Z","content_type":"text/html","content_length":"34749","record_id":"<urn:uuid:aedd9b54-b434-4899-b34c-e6c4a40da5ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00432.warc.gz"} |
— Approximation to the Reciprocal Square Root of Scalar Single Precision Floating-Point Value With Less Than 2^-28 Relative Error
VRSQRT28SS — Approximation to the Reciprocal Square Root of Scalar Single Precision Floating-Point Value With Less Than 2^-28 Relative Error
Op/ 64/32 bit CPUID
Opcode/Instruction En Mode Feature Description
Support Flag
EVEX.LLIG.66.0F38.W0 CD /r Computes approximate reciprocal square root (<2^-28 relative error) of the scalar single-precision floating-point value from xmm3/m32
VRSQRT28SS xmm1 {k1}{z}, xmm2, xmm3/ A V/V AVX512ER and stores result in xmm1with writemask k1. Also, upper 3 single-precision floating-point value (bits[127:32]) from xmm2 is copied to
m32 {sae} xmm1[127:32].
Instruction Operand Encoding ¶
Op/En Tuple Type Operand 1 Operand 2 Operand 3 Operand 4
A Tuple1 Scalar ModRM:reg (w) EVEX.vvvv (r) ModRM:r/m (r) N/A
Description ¶
Computes the reciprocal square root of the low float32 value in the second source operand (the third operand) and store the result to the destination operand (the first operand). The approximate
reciprocal square root is evaluated with less than 2^-28 of maximum relative error prior to final rounding. The final result is rounded to < 2^-23 relative error before written to the low float32
element of the destination according to the writemask k1. Bits 127:32 of the destination is copied from the corresponding bits of the first source operand (the second operand).
If any source element is NaN, the quietized NaN source value is returned for that element. Negative (non-zero) source numbers, as well as -∞, return the canonical NaN and set the Invalid Flag (#I).
A value of -0 must return -∞ and set the DivByZero flags (#Z). Negative numbers should return NaN and set the Invalid flag (#I). Note however that the instruction flush input denormals to zero of the
same sign, so negative denormals return -∞ and set the DivByZero flag.
The first source operand is an XMM register. The second source operand is an XMM register or a 32-bit memory location. The destination operand is a XMM register.
cles/reference-implementations-for-IA-approximation-instructions-vrcp14-vrsqrt14-vrcp28-vrsqrt28-vexp2. ¶
Operation ¶
VRSQRT28SS (EVEX Encoded Versions) ¶
IF k1[0] OR *no writemask* THEN
DEST[31: 0] := (1.0/ SQRT(SRC[31: 0]));
IF *merging-masking* ; merging-masking
THEN *DEST[31: 0] remains unchanged*
ELSE ; zeroing-masking
DEST[31: 0] := 0
DEST[127:32] := SRC1[127: 32]
DEST[MAXVL-1:128] := 0
Input Value Result Value Comments
NAN QNAN(input) If (SRC = SNaN) then #I
X = 2^-2n [2]^n
X<0 QNaN_Indefinite Including -INF
X = -0 or negative denormal -INF #Z
X = +0 or positive denormal +INF #Z
X = +INF +0
Table 6-53. VRSQRT28SS Special Cases
Intel C/C++ Compiler Intrinsic Equivalent ¶
VRSQRT28SS __m128 _mm_rsqrt28_round_ss(__m128 a, __m128 b, int rounding);
VRSQRT28SS __m128 _mm_mask_rsqrt28_round_ss(__m128 s, __mmask8 m,__m128 a,__m128 b, int rounding);
VRSQRT28SS __m128 _mm_maskz_rsqrt28_round_ss(__mmask8 m,__m128 a,__m128 b, int rounding);
SIMD Floating-Point Exceptions ¶
Invalid (if SNaN input), Divide-by-zero.
Other Exceptions ¶
See Table 2-47, “Type E3 Class Exception Conditions.” | {"url":"https://www.felixcloutier.com/x86/vrsqrt28ss","timestamp":"2024-11-09T14:33:17Z","content_type":"application/xhtml+xml","content_length":"6866","record_id":"<urn:uuid:aac4a90f-9b44-49fa-80e6-784dba991fd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00417.warc.gz"} |
Vitalik Buterin (Ethereum) (Aug 2021)Vitalik Buterin (Ethereum) (Aug 2021)
Vitalik Buterin (Ethereum Co-founder) – What makes an ideal state tree? (Aug 2021)
00:00:17 Ethereum Stateless Clients: Witness Sizes and Tree Structures
00:10:15 Optimizing Merkle Tree Branch Length and Storage
00:13:36 Polynomial Commitments for State Trees
00:21:13 Vertical Trees: A Novel Approach to Efficient Merkle Tree Proofs
00:27:12 Merkle Tree Improvements
00:31:42 Proof Compression and Quantum Resistance Considerations in Merkle Trees
00:38:53 Quantum-Resistant Post-Quantum Cryptography
Navigating the Evolution of Ethereum’s State Tree: From Hexary Structures to Quantum-Resistant Virtual Trees
In the dynamic field of Ethereum’s network, the evolution of its state tree structure stands as a pivotal aspect of blockchain technology. This comprehensive exploration delves into the transition
from the conventional hexary tree structure to advanced concepts like binary and virtual trees, including stateless clients, polynomial commitments, and the emerging challenge of quantum computing.
By synthesizing insights from Vitalik Buterin, Dinkrad, and other leading figures, we unravel the complexities of this transformation, highlighting the pivotal role of these developments in enhancing
Ethereum’s scalability, security, and future-proofing against quantum threats.
Current State Tree Structure and Challenges
Ethereum’s existing state tree employs a hexary structure, stored in the Merkle Bridge Registry. Each node can have up to 16 children, and each child is hashed together with its siblings. The root of
the Merkle Bridge Registry is included in every Ethereum block. However, the large size of Merkle branches, particularly in transaction-heavy blocks, poses a significant challenge.
Stateless Clients and the Need for Smaller Branches
Stateless clients are pivotal for Ethereum’s scalability, verifying blocks without storing the entire state. Smaller Merkle branches are essential for their practical implementation.
Reducing Branch Sizes
Transitioning from a hexary to a binary tree structure is a key approach to minimize branch sizes. This shift could drastically shrink branches, thereby bolstering stateless client feasibility.
Other Considerations
Optimizing data structures and algorithms is crucial in reducing the costs of generating and editing branches, directly impacting Ethereum’s performance and scalability.
Binary Merkle Trees: Efficiency and Storage Optimization
Binary Merkle trees offer a notable improvement in efficiency, reducing the number of layers and significantly optimizing storage through effective database chunking.
Editing and Limitations
Despite their advantages, binary trees face limitations in handling large data sets, with witnesses potentially reaching impractical sizes.
Polynomial Commitments: A Potential Solution
CATE commitments emerge as a novel solution, offering constant witness sizes and a reduction in proof length, although they present challenges in proof generation cost.
Pre-computing Witnesses: A Balancing Act
The pre-computation of witnesses introduces a trade-off between read and write efficiencies, necessitating a balanced approach for frequent state updates.
Square Root Compromise and Vertical Trees
The square root compromise, involving a split in the state polynomial, along with vertical trees, presents a balanced solution. Vertical trees, characterized by constant-size proofs and reduced proof
sizes, however, introduce complexities in computational costs due to elliptic curve operations.
Verkle Trees vs Hex Arrays
Verkle trees significantly optimize proof size and computation costs compared to hex arrays, utilizing advanced commitments to streamline proof generation.
Virtual Trees: The Current Best Solution
Despite their limitations, virtual trees currently offer the most effective solution, with ongoing research aimed at reducing proof overhead and improving designs.
SNARKs and Merkle Proofs
The use of SNARKs in verifying Merkle branches without the full tree is a key development, coupled with advancements in proof compression techniques like multiproofs.
Quantum Technology and Cryptography
Quantum computing poses a significant threat to blockchain security, prompting the need for quantum-resistant cryptographic algorithms and schemes to ensure long-term viability.
Post-Quantum Path
Lattice-based cryptography, immune to quantum attacks, emerges as the most realistic post-quantum path, highlighting the importance of adapting Merkle trees to these new cryptographic landscapes.
The evolution of Ethereum’s state tree, from its hexary roots to quantum-resistant virtual trees, marks a crucial chapter in blockchain technology. This journey underscores the relentless pursuit of
scalability, security, and adaptability in the face of technological advancements, ensuring Ethereum’s robustness in the ever-evolving digital age.
Note: This article is based on summaries and does not include direct citations from external sources like Vitalik Buterin’s blog or Dinkrad’s articles. For detailed insights and technical specifics,
readers are encouraged to refer to these expert resources.
Supplemental Update:
Quantum Computers and Cryptography:
– Quantum computers can break certain types of cryptography, including RSA, class group based, and elliptic curve based. However, they do not break lattices and hashes.
Post-Quantum Merkle Trees:
– Vertical trees are vulnerable to quantum computers, but snark-margle trees are not. A realistic post-quantum approach is to create a Merkle tree using an arithmetically friendly hash function and
apply a Stark on top.
Resources for Further Reading:
– Vitalik Buterin’s blog contains articles on Verkle trees and ZKSnarks.
– Dinkrad’s blog has a piece on Cat 8 commitments.
– ETH research post on the fancy scheme that doesn’t quite work (number 7520).
– Dinkrad’s piece on Cate commitments covers a lot of the math.
– Article discussing synarchs and polynomial commitments.
– A traditional Chinese article on the same topic.
– A picture explainer of how bulletproof commitments work, used in the vertical.
Notes by: OracleOfEntropy | {"url":"https://whatishappening.org/vitalik-buterin-ethereum-aug-2021-2/","timestamp":"2024-11-05T04:21:24Z","content_type":"text/html","content_length":"122177","record_id":"<urn:uuid:e7e7b294-c45a-409a-99c8-9d7dc561480d>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00693.warc.gz"} |
Conductor Fill Calculator - Online Calculators
Enter required values as, area of one conductor, number, to know total area of the conduit.
The Conductor Fill Calculator helps electricians determine how many conductors can safely fit in a conduit or box based on NEC guidelines. Proper conduit fill is essential for safety, preventing
overheating, and allowing for easy maintenance
The formula is:
$\text{CF} = \frac{\text{A} \times \text{N}}{\text{T}$
Variable Meaning
CF Conductor Fill (the percentage of the available space filled by conductors)
A Cross-sectional area of a single conductor (in square units)
N Number of conductors
T Total available cross-sectional area of the conduit (in square units)
How to Calculate ?
To calculate the Conductor Fill (CF):
1. Multiply the cross-sectional area of a single conductor (A) by the number of conductors (N).
2. Divide the result by the total available cross-sectional area of the conduit (T). This formula gives the fill percentage, which is the proportion of the conduit space occupied by the conductors.
Solved Calculations :
Example 1:
• Cross-sectional area of a conductor (A) = 0.5 square inches
• Number of conductors (N) = 6
• Total conduit area (T) = 4 square inches
Calculation Instructions
Step 1: CF = $\frac{\text{A} \times \text{N}}{\text{T}}$ Start with the formula.
Step 2: CF = $\frac{0.5 \times 6}{4}$ Replace A with 0.5, N with 6, and T with 4.
Step 3: CF = $\frac{3}{4}$ Multiply the area of a conductor by the number of conductors.
Step 4: CF = 0.75 or 75% Divide 3 by 4 to get the conductor fill.
The conductor fill is 75%.
Example 2:
• Cross-sectional area of a conductor (A) = 0.8 square centimeters
• Number of conductors (N) = 5
• Total conduit area (T) = 6 square centimeters
Calculation Instructions
Step 1: CF = $\frac{\text{A} \times \text{N}}{\text{T}}$ Start with the formula.
Step 2: CF = $\frac{0.8 \times 5}{6}$ Replace A with 0.8, N with 5, and T with 6.
Step 3: CF = $\frac{4}{6}$ Multiply the area of a conductor by the number of conductors.
Step 4: CF = 0.67 or 67% Divide 4 by 6 to get the conductor fill.
The conductor fill is 67%.
What is Conductor Fill Calculator ?
The Conductor Fill Calculator is a vital tool to figure out the capacity of conduits and boxes for electrical wiring. To calculate conduit fill, you need to know the total cross-sectional area of the
conductors and the conduit size. Conduit fill refers to the amount of space within a conduit that is occupied by wires.
The National Electrical Code (NEC) specifies that the maximum percentage conductor fill for four or more wires in a conduit is 40%, allowing enough space for heat dissipation and safety. You can use
a conduit fill chart or a conductor fill calculator app to find the fill percentage based on the type and number of conductors.
When calculating conductor box fill, each conductor counts as one unit, while grounds and fittings are also accounted for. The formula for conductors in a conduit depends on the conductor’s size and
For example, to calculate how many 4mm cables fit in a 25mm conduit, you must calculate the total area of the cables and compare it with the conduit’s area. Tools like the NEC conduit fill table help
streamline this process. Proper calculation prevents overheating and ensures the electrical system’s safety. | {"url":"https://areacalculators.com/conductor-fill-calculator/","timestamp":"2024-11-03T02:58:11Z","content_type":"text/html","content_length":"110610","record_id":"<urn:uuid:e0a098ed-32e9-419d-b712-6c1be099eb1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00353.warc.gz"} |
Norm.Inv: Excel Formulae Explained - ManyCoders
Key Takeaway:
• NORM.INV is an Excel function that calculates the inverse of the normal cumulative distribution function. It is used to determine the probability of a certain value falling below or above a given
mean, with a given standard deviation.
• NORM.INV is particularly useful when working with large datasets or conducting statistical analysis. It helps to determine the likelihood of an event occurring, which is useful for making
predictions or assessing risk.
• When using NORM.INV, it is important to understand the syntax and structure of the formula, as well as the limitations and alternatives. It is also important to have a clear understanding of the
data being analyzed and the purpose of the calculation.
Are you confused about NORM.INV Excel formulae? Get a comprehensive guide to understand it and learn how to use it effectively! You’ll soon be a pro at this popular formulae.
NORM.INV: Understanding the Excel Formula
Staring blankly at Excel spreadsheets with formulae beyond your understanding? You’re not alone. Let’s break down the mysterious NORM.INV formula. We’ll explore its definition and purpose. Plus,
learn when and why it’s better than lesser-known alternatives. This’ll help you with data analysis. Let’s get started!
Defining NORM.INV and its Purpose
NORM.INV is a statistical function in Microsoft Excel that can compute the inverse of normal cumulative distribution. It works by returning a standardized normal distribution value for a given
proportion. This value represents the number of standard deviations from the mean. In other words, it helps calculate the z-score for a particular percentage point on a normal distribution curve.
NORM.INV is particularly useful when dealing with data that follows a bell-shaped curve or Gaussian distribution. By using this in combination with other Excel formulas, such as AVERAGE and STDEV,
users can make informed decisions based on their data.
To use NORM.INV, two input values are needed: probability-proportion (a decimal ranging from 0 to 1) and mean (the average value of the entire dataset). This will return a z-score that corresponds to
that probability-point on the standard normal distribution curve. The output value ranges from negative infinity to positive infinity, but typically falls between -3 and +3.
NORM.INV is useful in quality control when manufacturers want to ensure their products meet certain standards. By measuring multiple samples across different batches and using the NORM.INV formulae,
they can identify measurements that are within acceptable ranges. For example, an airline can make sure its bags fit into overhead compartments without being too big or too small.
Understanding when and why to use NORM.INV is important. It can help in situations like statistical analysis on large sets of data, testing hypotheses, or constructing confidence intervals.
Understanding When and Why to Use NORM.INV
NORM.INV is a formula in Excel which is used to return the opposite of a regular cumulative distribution for a certain standard deviation and mean. Knowing when and why to use NORM.INV helps when
analyzing data and making decisions in business.
It is useful for calculating the chance of a value happening in a normal distribution. For example, to find the probability of a customer buying your product again, NORM.INV can also estimate future
trends in your data. It lets you know how likely values will be in a range, so you can plan appropriately.
NORM.INV aids statistical analysis too, showing you the shape of a distribution. You can tell how close your data is to a normal distribution, to help make data-driven choices.
For effective use of NORM.INV, understand the syntax of the formula and check different datasets. That way, you’ll be able to read the results better.
It’s evident why understanding when and why to use NORM.INV is important for anyone working with data or wanting to make decisions based on facts. In the next section, we’ll look into how this
formula works and what each variable means.
NORM.INV Formula Explained
Want to calculate the chance of something happening? Then, the NORM.INV formula in Excel is a must-know. Let’s go further into what the NORM.INV formula is made of. We’ll look at its syntax and
structure, seeing how each part works to give the final result. Plus, we will provide a step-by-step guide for using the NORM.INV formula to compute probabilities. That way, you can use it with
certainty for all your statistical calculations.
Syntax and Structure of NORM.INV Calculation
The NORM.INV calculation syntax and structure can be confusing, especially for those who are new to Excel functions. Let’s get into the details to help you better understand it.
Check out this table for the syntax and structure of NORM.INV calculation:
Syntax Description
=NORM.INV(probability, [mean], [standard_dev]) Provides the inverse of the normal cumulative distribution for a given probability
The formula requires the probability value and has two optional values for mean and standard deviation. If these aren’t provided, Excel assumes 0 and 1 as their respective values.
For instance, the output for =NORM.INV(0.7) is 0.5244.
Remember that the formula returns a number between negative infinity and positive infinity that corresponds to the inputted probability. Additionally, NORM.INV formula is a Statistical Function in
Now, we will cover how NORM.INV can calculate probability with a step-by-step guide.
Step-by-Step Guide to Calculating Probability with NORM.INV
Do you want to calculate probability using NORM.INV? It’s easy! Follow these 4 steps:
1. Find the mean & standard deviation of your data set.
2. Choose a probability number, from 0 to 1. That’s the input for your function.
3. Put both values into the formula =NORM.INV(probability, mean, standard deviation).
4. Press Enter.
Let’s look closer at Step 1. The mean is the sum of all values, divided by the number of values in the data set. Standard deviation shows how far the data set is from the average value. It tells us
if the data is ‘spread out’ or ‘tight’ around the mean. When you enter these values, put the probability first, then mean, then standard deviation.
Still unsure? Practice with sample problems! That way you get used to different inputs and outputs in fixed scenarios.
Ready to use NORM.INV in real life? Let’s explore some examples.
Real-World Examples of NORM.INV in Use
I’m a passionate data analyst, and Excel is an essential asset for making sense of complicated data. NORM.INV is a formula I use often. It works out probabilities and values for normal distributions.
Let me show you two practical examples of using NORM.INV for making informed decisions.
1. First, discover how NORM.INV can find probability values.
2. Second, learn how to use it to work out the value for a given probability.
Example 1: Using NORM.INV to Find Probability Values
Do you want to know how NORM.INV works? Let’s look at an example. Suppose a company sells 500 units of their product per day, with a standard deviation of 100 units. A marketing executive wants to
know the probability of selling 600 units in one day.
We can use the formula =NORM.INV(Probability, Mean, Standard Deviation) to calculate the probability. Inputting 500 for Mean and 100 for Standard Deviation yields a z-score of 1. This translates to a
probability of .8413, or about an 84% chance of selling 600 units per day.
The probability you get from NORM.INV is just a starting point. You can use it to calculate revenue projections or new product sales volume.
Want to learn more? Don’t miss out!
Example 2: Using NORM.INV to Find Value of a Probability
NORM.INV can be used to find the value of a probability. Here are four steps to follow:
1. Choose the significance level or cutoff point for analysis. It could be a confidence interval or hypothesis test rejection region.
2. Find the mean and standard deviation. These may need to be calculated if only summary statistics exist.
3. Decide whether you need one-tailed or two-tailed probabilities. One-tailed is used when only one direction matters (e.g. sales increasing), while two-tailed is used when both directions matter
(e.g. sales changing).
4. Use the NORM.INV formula with appropriate arguments. It takes the form =NORM.INV(probability,mean,standard_dev,[cumulative]), where “probability” is the desired probability (as a decimal), “mean”
is the mean of the data set/population, “standard_dev” is the standard deviation, and “[cumulative]” is an optional parameter for either a cumulative distribution (“TRUE”) or inverse cumulative
distribution (“FALSE”).
NORM.INV has advantages. It allows for more advanced statistical analyses than just using quartiles and percentiles from the normal table. It also automates calculations in Excel, instead of needing
to look up values in a table.
Limitations and alternatives should be taken into account. It’s important to ensure the data follows a normal distribution and extreme values are handled correctly.
Limitations of NORM.INV and Alternatives
I’m an Excel enthusiast, always searching for new functions and formulas to speed up my spreadsheets. NORM.INV is one I use often. It calculates the inverse of the normal cumulative distribution for
a certain value. But, like any function, NORM.INV has boundaries. In this section, let’s explore these. We’ll also look at other Excel formulae and tools to calculate probability which can help when
NORM.INV doesn’t meet your needs.
Understanding the Limits and Constraints of NORM.INV
NORM.INV is restricted to continuous distributions, which means that if you’re working with discrete ones (like those in counting processes), you must use other tools or adjust your data.
Additionally, it only takes in inputs between 0 and 1 for probabilities, and between -1E+307 and 1E+37 for z-scores; supplying incorrect data or going beyond these limits may cause errors or
incorrect results.
You should keep these limitations in mind when using NORM.INV. If you need to work with discrete distributions, Excel has other formulae and tools like BINOM.DIST and POISSON.DIST, which work with
integers and are more appropriate for counting processes.
Alternative Excel Formulae and Tools for Probability Calculations
Excel has the Data Analysis Toolpak, which contains many statistical and probabilistic functions like regression analysis, correlation analysis, hypothesis testing, histograms, etc. Other statistical
packages like R and Python offer more features than Excel. They can handle complex models and larger datasets. It’s important for data analysts or decision-makers to understand multiple tools and
techniques so they can select one based on their needs.
Formulae like NORMSDIST(), BINOM.DIST(), and POISSON.DIST() are also available to make probability calculations. Don’t forget to learn these alternative formulae and tools; they may be helpful when
dealing with data of various distributions. NORM.INV still has importance, despite its limitations, among the many probability calculation options.
Recap of Key Points and Benefits of NORM.INV in Excel
NORM.INV in Excel is a very useful tool. It helps to find the inverse standard normal cumulative distribution for a given probability. It’s best used when dealing with standardized scores or
normalized values. It is essential to understand the concept of probability distribution before using this function.
Plus, it is important to keep in mind that NORM.INV requires data with a mean of zero and a standard deviation equal to one. This means that it is necessary to center and scale the data before using
the function.
The main benefit of NORM.INV is that it can be used to analyze large data sets. This way, different observations have distinct probabilities attached to them. It also cuts down on time spent
calculating complex statistical measures while ensuring accuracy.
For example, studies have shown that by using Microsoft Excel, accounting practices can be improved by reducing errors and avoiding spreadsheet fraud cases (Hofmans et al., 2016). NORM.INV is also
very helpful in this regard.
Final Thoughts and Recommendations for Using NORM.INV Effectively.
We have explored NORM.INV in Excel so it’s time to think about what we’ve learned and how to use this formula well.
1. Remember, NORM.INV is not a single function. You need to combine it with other statistical tools. Figure out which distribution works best for your data set and when to use it.
2. Make sure the range you use has enough data points. The more data points you have, the more reliable your results will be. And be sure your inputs fall in the accepted range or you could get an
3. It’s important to understand probability distributions if you want to use NORM.INV. Once you know what each distribution does, you can decide which one is right for you.
Using NORM.INV with these tips will help you get better insights from your stats and make better decisions with accurate data.
I saw the results of using wrong data with NORM.INV. A friend ran a marketing campaign with incorrect inputs. They lost money and damaged relations with stakeholders. This is a reminder that learning
about statistical functions is worth it!
Five Facts About NORM.INV: Excel Formulae Explained:
• ✅ NORM.INV is an Excel function used to obtain the inverse of the normal cumulative distribution for a specified mean and standard deviation. (Source: Microsoft)
• ✅ The function is useful for calculating z-scores for hypothesis testing and confidence intervals. (Source: Investopedia)
• ✅ The NORM.INV function takes two arguments, probability (the area under the curve to the left of the value) and mean and standard deviation. (Source: Spreadsheeto)
• ✅ The function can be used in combination with other Excel functions, such as SUM and AVERAGE, to perform more complex calculations. (Source: Vertex42)
• ✅ The NORM.INV function is considered an advanced Excel function and requires a basic understanding of statistics and probability. (Source: Exceljet)
FAQs about Norm.Inv: Excel Formulae Explained
What is NORM.INV in Excel?
NORM.INV is an Excel formula that is used to calculate the inverse of the normal cumulative distribution for a specified mean and standard deviation.
How does NORM.INV work?
NORM.INV takes in the probability of a certain value occurring and calculates the corresponding value based on the mean and standard deviation of the dataset. It is useful in financial analysis and
risk management.
How do I use NORM.INV in Excel?
To use NORM.INV, you need to enter the probability of a certain value occurring, along with the mean and standard deviation of the data set. The function will then calculate the value that
corresponds to that probability.
What is the syntax for NORM.INV in Excel?
The syntax for NORM.INV in Excel is: =NORM.INV(probability,mean,standard_dev)
What is the maximum and minimum probability value that can be inputted in NORM.INV?
The maximum probability value that can be inputted in NORM.INV is 1, while the minimum probability value is 0.
Can NORM.INV be used for non-normal distributions?
No, NORM.INV is specifically designed for normal distributions. It should not be used for non-normal distributions as it may produce inaccurate results. | {"url":"https://manycoders.com/excel/formulae/norm-inv-excel/","timestamp":"2024-11-01T20:26:52Z","content_type":"text/html","content_length":"84095","record_id":"<urn:uuid:694ad196-a36d-4398-a5cc-2b20b86c67ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00176.warc.gz"} |
Do points of inflection have to be differentiable? | Socratic
Do points of inflection have to be differentiable?
1 Answer
That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:
My answer to your question is no, a function does not need to be differentiable at a point of inflection; for example, the piecewise defined function
$f \left(x\right) = \left\{\begin{matrix}{x}^{2} & \mathmr{if} x < 0 \\ \sqrt{x} & \mathmr{if} x \ge 0\end{matrix}\right.$
is concave upward on $\left(- \infty , 0\right)$ and concave downward on $\left(0 , \infty\right)$ and is continuous at $x = 0$, so $\left(0 , 0\right)$ is an inflection point but not differentiable
Impact of this question
14102 views around the world | {"url":"https://socratic.org/questions/do-points-of-inflection-have-to-be-differentiable","timestamp":"2024-11-09T10:42:15Z","content_type":"text/html","content_length":"33509","record_id":"<urn:uuid:8c16b70e-e777-4a70-8216-90e97b86f6aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00762.warc.gz"} |
Specify Boundary Conditions
Before you create boundary conditions, you need to create a PDEModel container. PDEModel can accommodate one equation or a system of N equations. For details, see Solve Problems Using PDEModel
Objects. Suppose that you have a container named model, and that the geometry is stored in model. Examine the geometry to see the label of each edge or face.
pdegplot(model,EdgeLabels="on") % for 2-D
pdegplot(model,FaceLabels="on") % for 3-D
Now you can specify the boundary conditions for each edge or face. If you have a system of PDEs, you can set a different boundary condition for each component on each boundary edge or face.
If you do not specify a boundary condition for an edge or face, the default is the Neumann boundary condition with the zero values for "g" and "q".
If the boundary condition is a function of position, time, or the solution u, set boundary conditions by using the syntax in Nonconstant Boundary Conditions.
Dirichlet Boundary Conditions
Scalar PDEs
The Dirichlet boundary condition implies that the solution u on a particular edge or face satisfies the equation
where h and r are functions defined on ∂Ω, and can be functions of space (x, y, and, in 3-D, z), the solution u, and, for time-dependent equations, time. Often, you take h = 1, and set r to the
appropriate value. You can specify Dirichlet boundary conditions as the value of the solution u on the boundary or as a pair of the parameters h and r.
Suppose that you have a PDE model named model, and edges or faces [e1,e2,e3], where the solution u must equal 2. Specify this boundary condition as follows.
% For 3-D geometry:
% For 2-D geometry:
If the solution on edges or faces [e1,e2,e3] satisfies the equation 2u = 3, specify the boundary condition as follows.
% For 3-D geometry:
% For 2-D geometry:
• If you do not specify r, applyBoundaryCondition sets its value to 0.
• If you do not specify h, applyBoundaryCondition sets its value to 1.
Systems of PDEs
The Dirichlet boundary condition for a system of PDEs is hu = r, where h is a matrix, u is the solution vector, and r is a vector.
Suppose that you have a PDE model named model, and edge or face labels [e1,e2,e3] where the first component of the solution u must equal 1, while the second and third components must equal 2. Specify
this boundary condition as follows.
% For 3-D geometry:
% For 2-D geometry:
• The u and EquationIndex arguments must have the same length.
• If you exclude the EquationIndex argument, the u argument must have length N.
• If you exclude the u argument, applyBoundaryCondition sets the components in EquationIndex to 0.
Suppose that you have a PDE model named model, and edge or face labels [e1,e2,e3] where the first, second, and third components of the solution u must satisfy the equations 2u[1] = 3, 4u[2] = 5, and
6u[3] = 7, respectively. Specify this boundary condition as follows.
H0 = [2 0 0;
0 4 0;
0 0 6];
R0 = [3;5;7];
% For 3-D geometry:
applyBoundaryCondition(model,"dirichlet", ...
Face=[e1,e2,e3], ...
% For 2-D geometry:
applyBoundaryCondition(model,"dirichlet", ...
Edge=[e1,e2,e3], ...
• The r parameter must be a numeric vector of length N. If you do not specify r, applyBoundaryCondition sets the values to 0.
• The h parameter can be an N-by-N numeric matrix or a vector of length N^2 corresponding to the linear indexing form of the N-by-N matrix. For details about the linear indexing form, see Array
Indexing. If you do not specify h, applyBoundaryCondition sets the value to the identity matrix.
Neumann Boundary Conditions
Scalar PDEs
Generalized Neumann boundary conditions imply that the solution u on the edge or face satisfies the equation
$\stackrel{\to }{n}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(cabla u\right)+qu=g$
The coefficient c is the same as the coefficient of the second-order differential operator in the PDE equation
$-abla \cdot \left(cabla u\right)+au=f\text{on domain}\Omega$
$\stackrel{\to }{n}$ is the outward unit normal. q and g are functions defined on ∂Ω, and can be functions of space (x, y, and, in 3-D, z), the solution u, and, for time-dependent equations, time.
Suppose that you have a PDE model named model, and edges or faces [e1,e2,e3] where the solution u must satisfy the Neumann boundary condition with q = 2 and g = 3. Specify this boundary condition as
% For 3-D geometry:
% For 2-D geometry:
• If you do not specify g, applyBoundaryCondition sets its value to 0.
• If you do not specify q, applyBoundaryCondition sets its value to 0.
Systems of PDEs
Neumann boundary conditions for a system of PDEs is $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes abla u\right)+qu=g$. For example, in case of circumferential and spherical
boundaries, the generalized versions of the Neumann boundary condition are as follows:
• If the boundary is a circumference (2-D case), the outward normal vector of the boundary of the boundary is given by $n=\left(\mathrm{cos}\left(\phi \right),\mathrm{sin}\left(\phi \right)\right)$
, the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes abla u\right)$ means the N-by-1 vector, for which the (i,1)-component is as follows:
$\sum _{j=1}^{N}\left(\mathrm{cos}\left(\phi \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\phi \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\phi \
right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\phi \right){c}_{i,j,2,2}\frac{\partial }{\partial y}\right)\text{\hspace{0.17em}}{u}_{j}$
• If the boundary is a spherical surface (3-D case), than the outward normal vector of the boundary is given by $n=\left(\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\phi \right),\mathrm{sin}\
left(\theta \right)\mathrm{sin}\left(\phi \right),\mathrm{cos}\left(\theta \right)\right)$, and the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes abla u\right)$ means the
N-by-1 vector, for which the (i,1)-component is as follows:
$\begin{array}{l}\sum _{j=1}^{N}\left(\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\phi \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\
phi \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\phi \right){c}_{i,j,1,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left
(\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\phi \right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\phi \right){c}_{i,j,2,2}\frac{\partial
}{\partial y}+\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\phi \right){c}_{i,j,2,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left(\mathrm{cos}\left(\theta \right){c}_
{i,j,3,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,2}\frac{\partial }{\partial y}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,3}\frac{\partial }{\partial z}\right){u}
For each edge or face segment, there are a total of N boundary conditions.
Suppose that you have a PDE model named model, and edges or faces [e1,e2,e3] where the first component of the solution u must satisfy the Neumann boundary condition with q = 2 and g = 3, and the
second component must satisfy the Neumann boundary condition with q = 4 and g = 5. Specify this boundary condition as follows.
Q = [2 0; 0 4];
G = [3;5];
% For 3-D geometry:
% For 2-D geometry:
• The g parameter must be a numeric vector of length N. If you do not specify g, applyBoundaryCondition sets the values to 0.
• The q parameter can be an N-by-N numeric matrix or a vector of length N^2 corresponding to the linear indexing form of the N-by-N matrix. For details about the linear indexing form, see Array
Indexing. If you do not specify q, applyBoundaryCondition sets the values to 0.
Mixed Boundary Conditions
If some equations in your system of PDEs must satisfy the Dirichlet boundary condition and some must satisfy the Neumann boundary condition for the same geometric region, use the "mixed" parameter to
apply boundary conditions in one call. Note that applyBoundaryCondition uses the default Neumann boundary condition with g = 0 and q = 0 for equations for which you do not explicitly specify a
boundary condition.
Suppose that you have a PDE model named model, and edge or face labels [e1,e2,e3] where the first component of the solution u must equal 11, the second component must equal 22, and the third
component must satisfy the Neumann boundary condition with q = 3 and g = 4. Express this boundary condition as follows.
Q = [0 0 0; 0 0 0; 0 0 3];
G = [0;0;4];
% For 3-D geometry:
% For 2-D geometry:
Suppose that you have a PDE model named model, and edges or faces [e1,e2,e3] where the first component of the solution u must satisfy the Dirichlet boundary condition 2u[1] = 3, the second component
must satisfy the Neumann boundary condition with q = 4 and g = 5, and the third component must satisfy the Neumann boundary condition with q = 6 and g = 7. Express this boundary condition as follows.
h = [2 0 0; 0 0 0; 0 0 0];
r = [3;0;0];
Q = [0 0 0; 0 4 0; 0 0 6];
G = [0;5;7];
% For 3-D geometry:
applyBoundaryCondition(model,"mixed", ...
Face=[e1,e2,e3], ...
% For 2-D geometry:
applyBoundaryCondition(model,"mixed", ...
Edge=[e1,e2,e3], ...
Nonconstant Boundary Conditions
Use functions to express nonconstant boundary conditions.
applyBoundaryCondition(model,"dirichlet", ...
applyBoundaryCondition(model,"neumann", ...
applyBoundaryCondition(model,"mixed", ...
Edge=[3,4],u=@myufun, ...
Each function must have the following syntax.
function bcMatrix = myfun(location,state)
solvepde or solvepdeeig compute and populate the data in the location and state structure arrays and pass this data to your function. You can define your function so that its output depends on this
data. You can use any names instead of location and state, but the function must have exactly two arguments.
• location — A structure containing the following fields. If you pass a name-value pair to applyBoundaryCondition with Vectorized set to "on", then location can contain several evaluation points.
If you do not set Vectorized or use Vectorized="off", then solvers pass just one evaluation point in each call.
□ location.x — The x-coordinate of the point or points
□ location.y — The y-coordinate of the point or points
□ location.z — For 3-D geometry, the z-coordinate of the point or points
Furthermore, if there are Neumann conditions, then solvers pass the following data in the location structure.
□ location.nx — x-component of the normal vector at the evaluation point or points
□ location.ny — y-component of the normal vector at the evaluation point or points
□ location.nz — For 3-D geometry, z-component of the normal vector at the evaluation point or points
• state — For transient or nonlinear problems.
□ state.u contains the solution vector at evaluation points. state.u is an N-by-M matrix, where each column corresponds to one evaluation point, and M is the number of evaluation points.
□ state.time contains the time at evaluation points. state.time is a scalar.
Your function returns bcMatrix. This matrix has the following form, depending on the boundary condition type.
• u — N1-by-M matrix, where each column corresponds to one evaluation point, and M is the number of evaluation points. N1 is the length of the EquationIndex argument. If there is no EquationIndex
argument, then N1 = N.
• r or g — N-by-M matrix, where each column corresponds to one evaluation point, and M is the number of evaluation points.
• h or q — N^2-by-M matrix, where each column corresponds to one evaluation point via linear indexing of the underlying N-by-N matrix, and M is the number of evaluation points. Alternatively, an N
-by-N-by-M array, where each evaluation point is an N-by-N matrix. For details about linear indexing, see Array Indexing.
If boundary conditions depend on state.u or state.time, ensure that your function returns a matrix of NaN of the correct size when state.u or state.time are NaN. Solvers check whether a problem is
nonlinear or time-dependent by passing NaN state values, and looking for returned NaN values.
Additional Arguments in Functions for Nonconstant Boundary Conditions
To use additional arguments in your function, wrap your function (that takes additional arguments) with an anonymous function that takes only the location and state arguments. For example:
uVal = ...
@(location,state) myfunWithAdditionalArgs(location,state,arg1,arg2...)
applyBoundaryCondition(model,"mixed", ...
Edge=[3,4],u=uVal, ... | {"url":"https://es.mathworks.com/help/pde/ug/steps-to-specify-a-boundary-conditions-object.html","timestamp":"2024-11-10T05:26:29Z","content_type":"text/html","content_length":"102232","record_id":"<urn:uuid:069093a7-915b-4713-a7a7-d3380c00fa32>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00891.warc.gz"} |
Completing Number Patterns
Question Video: Completing Number Patterns Mathematics
Complete the table below by following the pattern.
Video Transcript
Complete the table below by following the pattern.
This table contains nine parts. But there are only six numbers in it. We can see that three parts of the table are empty. And these are the parts that we need to complete. The question instructs us
to complete the table by following the pattern. But what pattern is it talking about? Where’s the pattern in this table? Well, there’re actually two patterns in the table that we could spot. And we
could use either one to help us to complete it.
The first pattern we can spot is one that goes across the table from left to right. And the place to spot this pattern is in the first row because the first row is complete. The first number in the
row is 700. And the second number is 730. The number has increased by 30. And if we look at the second two numbers in the row, we can see that this rule of adding 30 does work each time. 730 plus
another 30 is 760.
Let’s use this first pattern that we’ve spotted to help us fill in the first missing number. The first number in the middle row is 710. Now, if we apply our plus 30 rule, we can find out what the
missing number is. 30 is the same as three 10s. So let’s count on in 10s three times, starting with 710. 720, 730, 740. We followed the pattern of adding 30s. We move across the table to find our
first missing number. 710 plus 30 equals 740.
Now, we could apply this pattern again to find the second two missing numbers. But we did say originally there were two patterns in this table. So let’s have a look at the other one and see if we can
use this to find our final two missing numbers. As well as a pattern going across the table, there’s also a pattern going down the table. And perhaps, the best place to spot this pattern is in the
center column because we’ve now got three numbers. It’s a complete column. The first number in the column is 730. And the next number is 740. And of course, to get from 730 to 740, we’ve added 10. Is
this a rule for a pattern that we can see every time we go down the square? Yes, it is. To get from 740 to 750, we add 10 again. So we add 30 as we move across the table. But we add 10 as we move
down the table.
Let’s follow this pattern to find our final two missing numbers. 10 more than 710 is 720. And 10 more than 770 is 780. We found these two missing numbers by adding 10 as we move down the table. But
we could’ve found our final two missing numbers by applying the plus 30 rule. We could’ve added 30 to 750. And that way, we’d have found 780. And we would’ve needed to have worked backwards from 750
to find the remaining missing number. So we’d have subtracted 30. 750 take away 30 would’ve given us 720. So there are two patterns in the table. And we could’ve used either one to find the three
missing numbers.
And if we read down the table from top to bottom and left to right, the three missing numbers are 740, 720, and 780. | {"url":"https://www.nagwa.com/en/videos/918152361676/","timestamp":"2024-11-02T12:39:50Z","content_type":"text/html","content_length":"244759","record_id":"<urn:uuid:4534b076-c4a9-463d-8db8-3e2c435641e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00356.warc.gz"} |
Generating an icosphere with code (Archive)
⛔️ This page is an outdated version of a post that has been modified after its original publication. This version describes icosphere generation strategies that were further improved and it’s
maintained for reference purposes only. You should read the revised post instead.
In this article, we will walk through the process of generating icospheres (spheres based on a regular icosahedron) with code. I started looking into icospheres when working on Terraced Terrain
Generator (TTG) version 2, which added support for spherical terrains. I took some time to study different sphere types and ended up creating a small library to procedurally generate spheres (ico-,
cube- and UV-spheres) in Unity: Sphere Generator. Building the library inspired me to write this piece.
The article’s content is divided into 5 sections. The first one introduces the concept and characteristics of the icosphere. The remaining sections correlate to the 4 generation steps. The conclusion
wraps the article up.
The icosphere
Before jumping into details, it’s important to understand what an icosphere is. It’s uncertain (at least to me) who invented the term icosphere, but the most popular usage of the term is in the 3D
modeling tool Blender, which defines the term as:
An icosphere is a polyhedral* sphere made up of triangles. Icospheres are normally used to achieve a more isotropical layout of vertices than a UV sphere, in other words, they are uniform in
every direction.
*: A polyhedral shape is a shape that represents a polyhedron: a 3-dimensional shape with flat, polygonal faces. Pyramids and cubes are examples of polyhedrons.
The image below displays an example of an icosphere:
Multiple techniques can be used to generate spheres, and each one creates meshes with different properties. The icosphere has the following characteristics:
• Its triangles have about the same area.
• Its triangles are equilateral—or as close to it as possible.
• Its triangles are oriented non-uniformly. There are 20 areas (we will soon see why 20) with distinct triangle orientation. The image above displays a point on its center where five of these areas
meet, forming a pentagon. Each one of the five triangles has a different orientation.
• It is uniform in every direction. It doesn’t matter which direction you move on an icosphere, the shape, area and consequently the concentration of triangles remains mostly unchanged. Therefore,
it doesn’t have poles, unlike the UV sphere.
• Unlike UV spheres, it isn’t a great fit for texture mapping.
• It is a better fit when modeling more natural shapes.
The icosahedron
Knowing what an icosphere looks like is one thing. Knowing how to generate one is something completely different. In order to learn how to generate icosphere meshes procedurally, we need to
understand the concepts behind it. First, how can we conceptually define an icosphere?
An icosphere is generated by fragmenting and normalizing a regular icosahedron.
Let’s break that sentence down. First, what on earth is an icosahedron, and what makes one regular?
An icosahedron is a polyhedron with 20 faces. There is an infinite number of icosahedrons, and the most famous one is the regular icosahedron, a convex polyhedron composed of 20 equilateral
triangles. The regular icosahedron is one of the five Platonic solids and, for the RPG players out there, D20 dice are shaped like it. This is how it looks like:
The process described in this article creates an icosahedron (the rotating shape above) and transforms it into an icosphere. It consists of four steps:
1. Icosahedron generation.
2. Fragmentation.
3. Normalization.
4. Scaling.
The next sections describe the steps above in detail.
ℹ️ Even though an icosphere can be of any size, for the sake of comprehension, the next three sections describe the process of generating a unit icosphere: an icosphere with radius of 1.
Step 1: Icosahedron generation
First, it’s important to define what it means to generate a 3D shape. Traditionally, virtual 3D shapes are represented as meshes composed by simple primitives like squares or, more commonly,
triangles. Regardless of how detailed, complex and large a 3D mesh is, it can be decomposed into small triangles. The larger the number of triangles, the more detailed a mesh can be and the more
resources are needed to load and display it are necessary. Here’s an example of a 3D model of the famous Utah teapot with its mesh elements displayed:
Notice how the entire teapot is composed by triangles, regardless of how detailed the mesh area is. The areas which are less detailed—like the center of its side—have a smaller concentration of
triangles. Highly detailed areas—like the lid’s knob—have a higher concentration of triangles.
Every triangle of a 3D mesh can be represented by the coordinate of its 3 vertices. Once we have that data, we can draw the triangle. To draw the entire 3D model, all we need is to repeat the drawing
process on all triangles on the mesh. Therefore, in order to represent a 3D mesh, all we need is:
• The coordinates of all of the triangle vertices.
• Which vertices belong to each triangle. This is often referred to as “triangle indices”, where each vertex is represented by an index.
Generating an icosahedron is no different; all we need is its vertices and triangle indices. Finding the vertices coordinates is particularly interesting. Once you’ve got those, the triangle indices
are easy to find.
There are different paths one can take to gather vertex data on a regular icosahedron. The path I took aims to find them by using mutually perpendicular rectangles of particular dimensions (more on
this soon). The image below demonstrates an example of such rectangles, with each one colored differently to ease viewing:
The vertices of the triangles (highlighted with tiny pink balls) will become the vertices of the icosahedron. There are twelve of them, 4 for each rectangle. The white lines define 20 faces and are
placed where the icosahedron’s edges would be. Contrast this image with the rotating icosahedron one in the previous section and the correlation becomes clear.
Finding the vertices
The first step to construct the rectangles is to find the coordinates of their 4 vertices. We know all rectangles are placed at the same point in space (i.e. their centers share the same
coordinates), and we choose that point to be the origin: (0, 0, 0). We have also established that all rectangles have the same dimensions. Consequently, they only differ in rotation and, once we have
found the coordinates of one rectangle’s vertices, we can just rotate them to find the coordinates of the other 2.
The rectangles can be of any size, as long as the ratio between their sides is the golden ratio, which is approximately 1.618033988749. Effectively, this means that the rectangle’s width must be
~1.618 times larger than its height. We also know that all rectangle vertices must be at 1 unit away from its center (it’s a unit sphere), so we can calculate the rectangle’s height and width based
on the diagram below:
Where a is height/2 and c is width/2. The white rectangle above represents one of the 3 rectangles we are going to use to build the icosahedron. The circle is there merely to display all points that
are placed 1 unit away from the rectangle’s center. The coordinates of all 4 rectangle vertices can be defined in terms of a and c on a (x,y) plane:
Therefore, in order to find the vertices coordinates, we need to find the values of a and c. We already have enough information to do so, since:
• The golden ratio determines that width = height * goldenRatio, and therefore c = a * goldenRatio.
• The a/b/1 triangle (represented by dotted lines in the first rectangle image) is equilateral, and therefore we can use Pythagoras’ Theorem.
With that in mind:
1 1² = a² + c² // Pythagoras' Theorem
2 1 = a² + (a * goldenRatio)² // c = a * goldenRatio, 1² = 1
3 1 = a² + a² * goldenRation² // Exponent distribution (power rule)
4 1 = a² * (1 + goldenRation²) // "a" is a common factor
5 a² = 1 / (1 + goldenRation²) // Divide both sides by a² and flip
6 a = √(1 / (1 + goldenRatio²)) // Apply square root on both sides
7 a = √(1 / (1 + 2.618033988749895)) // Replace golden ratio value
8 a = 0.525731112119134 // Solve the square root
Once we know that a = 0.525731112119134, finding c becomes trivial:
1 c = a * goldenRatio
2 c = 0.525731112119134 * 1.618033988749 // Replace values
3 c = 0.85065080835157
Now that we have the coordinates of the rectangle’s vertices, we can construct it using 2 triangles, like seen in the image below:
The first triangle is composed by vertices v0, v1 and v3 and the second one by vertices v1, v2 and v3. The vertices coordinates are, approximately:
• v0: (-0.85, +0.52).
• v1: (+0.85, +0.52).
• v2: (+0.85, -0.52).
• v3: (-0.85, -0.52).
All of these vertices are 1 unit away from the rectangle’s center—in other words, their vector has length 1. As we have seen, once we have the vertices coordinates for one rectangle, we can just
rotate them to find the vertices’ coordinates of the other 2 rectangles. The rotation process modifies a vector’s direction, but not its length. Therefore, all 12 vertices will have length 1.
With that, we have gathered all the vertex information we needed to construct the icosahedron.
Constructing the faces
Once the coordinates of all vertices are known, we need to find the triangle indices: the list of vertex indices that are used to create all the 20 triangles of the icosahedron. A unique index is
assigned to each vertex, spanning from 0 to 11. The triangle indices list contains 60 elements—3 for each of the 20 triangles of the icosahedron. Each element is a vertex index in the [0, 11] range.
Creating this list is not as challenging as finding the vertex positions. In fact, I am not aware of a procedural method to assign the triangle indices (leave a comment if you know one!), and I found
them by trial and error. Having a reference image (like the one with the 3 mutually perpendicular rectangles in the previous section) was extremely helpful. It sounds like a cumbersome task, but it
didn’t take long (~15 minutes) and honestly, it was a nice exercise in mesh construction and I kind of enjoyed it. I probably wouldn’t think the same if the mesh had many more triangles, but an
icosahedron was manageable.
If you would like to take a look at the triangle indices and the vertex coordinates I’ve found, take a look at Sphere Generator’s source code.
Step 2: Fragmentation
Step 1 leaves us with a regular icosahedron, a shape which lacks detail and doesn’t contain enough vertices to pass as a sphere. To remedy that, we need to generate more vertices; and that’s where
mesh fragmentation comes in.
Mesh fragmentation is the process of procedurally increasing the vertex count of a mesh by fragmenting its primitives (in this case triangles) into new, smaller ones. It aims to allow the mesh to
provide more detail than existing, without actually adding the detail information (that is step 3’s role).
Fragmenting a mesh composed only by triangles consists of turning each triangle into 4 smaller ones, while preserving the shape of the original. The image below displays an example of a triangle
(larger, outer lines) that has been fragmented once into 4 smaller ones. It also happens to unequivocally resemble The Legend of Zelda’s triforce:
When it comes to an icosahedron, we need to keep in mind that the fragmentation process must not change the mesh’s topology. In this case, it is enough to ensure that all triangles in the mesh are
equilateral. Luckily, equilateral triangles are particularly easy to fragment and the operation conserves equilaterality. The image above is also an example of the fragmentation of an equilateral
triangle into 4 smaller, equally equilateral triangles.
ℹ️ Although this section uses C# as its guiding programming language, the code described here should be easily translatable to other programming languages.
Let’s look into how to fragment triangles with code. Like in every fragmentation process, the original vertices are maintained—failing to do so would change the mesh’s shape. Three new vertices are
created by finding the mid-point of each edge. The coordinates (x, y, z) of a mid-point of an edge with vertices v1 and v2 can be calculated as:
1 var x = (v1.x + v2.x) / 2;
2 var y = (v1.y + v2.y) / 2;
3 var z = (v1.z + v2.z) / 2;
The fragmentation process can be repeated indefinitely. Each iteration increases the number of triangles by a multiplying factor of 3, bringing the total number of triangles to 4 times the original
one (1 + 3). The total number of iterations is often referred to as fragmentation depth. The image below displays an example of a triangle that went through a fragmentation process with a depth of 2,
resulting in 16 triangles (4²).
With this in mind, we can apply the fragmentation process to the regular icosahedron. The image below displays a regular icosahedron followed by three meshes that are the outcome of fragmenting it
with a depth of 1, 2 and 3 respectively.
Notice how, although the number of triangles increased from 20 to 80, then to 320 and finally to 1280, the shape of the mesh hasn’t changed (if you struggle to see it, look at the contour). The
original faces and edges are still untouched—they just have more vertices and triangles now. In other words, the meshes above are all icosahedrons; just with different triangle counts.
Step 3: Normalization
As we can see in the previous section, even though the number of triangles increases with every fragmentation iteration, the shape of the mesh remains unchanged. The cause is clear: during
fragmentation, new triangle vertices are placed on the same plane as the triangle they originally belonged to. As a result, no new planes or faces are created. To address this issue, the new vertices
need to be repositioned to “break out” of their plane, effectively changing the mesh’s shape in a manner that resembles a sphere. The question at hand is: how to do that?
Here’s where one of the sphere’s properties comes in handy: all points on the surface of a sphere are equally distant from its center, and that distance is called radius. In the case of a unit
sphere, the radius is equal to 1. This property holds for all 12 original vertices of the icosahedron (as we demonstrated above), but it doesn’t for the new vertices—they are closer to the sphere’s
center than the original ones. Logically, the next step would be to ensure that the radius property holds for all vertices by moving them away from the center. The question now is: in which direction
should we move them?
To answer this question, we revisit the characteristics of the icosphere. We would like to, as much as possible, keep:
• All triangles close to an equilateral one.
• The area of all faces similar.
• The triangle distribution as uniform as possible.
A great way to closely maintain these properties is to normalize the vertices. Normalization is the process of converting a given vector into a unit vector: a vector with a length of 1. This
operation modifies the length of the vector but keeps its direction unmodified. The image below displays how we can use vector normalization to ensure that a vertex respects the sphere radius
In the image, the circle represents a slice of the sphere passing through its center. The circle’s radius is 1. The surface of the icosahedron is represented as a square for simplicity’s sake. The
distance between the square’s corners and the center of the circle is 1—just like the original vertices of the icosahedron are placed 1 unit away from the sphere’s center. Vertices v1 and v2
represent all “new” vertices: vertices that are not part of the original icosahedron and were generated by fragmentation. They are placed on the surface of the icosahedron and the distance between
them and the center of the circle is less than 1.
In this case, vertices v1and v2 are normalized into vectors v1n and v2n, respectively. The normalization operation maintains their direction but “expands” them, ensuring that their length is equal to
1, effectively placing them on the sphere’s surface. Problem solved.
Now that we know how to reposition a vertex without deviating too much from the icosphere properties, we can apply normalization on all vertices:
1 void NormalizeAllVertices(Vector3[] vertices)
2 {
3 for (var index = 0; index < vertices.Length; index++)
4 {
5 var vertex = vertices[index];
6 vertices[index] = vertex.normalized;
7 }
8 }
Where vertex.normalized returns a normalized version of the given vertex. This operation is extremely common and is often included in game engine libraries (e.g. Vector3.normalized is defined in both
Unity’s and Godot’s libraries). It is even present in some programming languages’ standard libraries, like C#’s.
The outcome of the normalization process can be observed in the image below. The first row is identical to the image at the end of the fragmentation section and contains icosahedrons that have not
been normalized. The first mesh in that row is the regular icosahedron, followed by fragmented meshes with a depth of 1, 2 and 3 respectively. The bottom row displays the normalized version of the
top row, keeping their fragmentation depths.
It’s worth noting that the number of vertices in the meshes on each column is identical; the only difference is the position of each of the mesh’s vertex. It is also clear that the higher the
fragmentation depth, the closer to a continuous sphere the normalized meshes get—their contour progressively resembles a circle. Finally, it is evident that both steps are necessary: fragmenting
without normalization increases detail but doesn’t change the mesh’s shape and normalization without fragmentation accomplishes nothing—all vertices of an icosahedron are already normalized.
The GIF below summarizes the process, composed by fragmentation and normalization (steps 2 and 3 respectively). It starts with a regular icosahedron which is fragmented twice (hence depth = 2). Then,
the icosahedron is progressively (for educational purposes) normalized until all vertices are equally distant from the mesh’s center. The final mesh represents an icosphere.
At this point, we have concluded the process of generating an icosphere. The next step, described below, is optional and it aims to scale the mesh to meet multiple purposes.
Step 4: Scaling
The previous generation steps left us with a unit icosphere. Even though that mesh is an icosphere, we would often like to easily tinker with sphere size. To accommodate for that, we can easily
modify the NormalizeAllVertices method from step 3 to introduce a new parameter: the sphere’s radius:
1 void RepositionAllVertices(Vector3[] vertices, float radius)
2 {
3 for (var index = 0; index < vertices.Length; index++)
4 {
5 var vertex = vertices[index];
6 vertices[index] = vertex.normalized * radius;
7 }
8 }
With that, we can generate all icospheres one can imagine—as long as one’s computer can handle it.
In this article, we learned what an icosphere is, what its properties are and how we can generate one procedurally, using code. We also observed how the different generation steps influence the final
mesh. Finally, we saw how the fragmentation depth impacts the mesh’s level of detail and, therefore, its similarity with real, continuous spheres.
The techniques described here were used to add spherical terrain support to Terraced Terrain Generator (TTG) version 2 and Sphere Generator’s icosphere support.
That’s a wrap! As usual, feel free to use the comment section below for questions, suggestions, corrections, or just to say “hi”. See you on the next one! | {"url":"https://blog.lslabs.dev/archive/generating_icosphere_with_code_archive","timestamp":"2024-11-05T10:24:59Z","content_type":"text/html","content_length":"42405","record_id":"<urn:uuid:f38bad4f-b68d-4f31-b40f-fb028a11a8da>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00607.warc.gz"} |
RRB JE 1st June 2019 Shift-3
Compare static friction and sliding friction.
Graafian follicles are characteristically found in the -
Two squares differ in areas by 32 cm². If the difference in their sides is 4 cm, what are the sides of the two squares?
Who was invited by Lord Wavell to form the interim Government in India in 1946?
In this question, two statements are given followed by two conclusions. Choose the conclusion(s) which best fit(s) logically.
1) Some towns are cities.
2) All cities are homes.
I. Some towns are homes.
II. Some homes are cities.
Five years ago, the average age of a couple was 24. At present, the average of the couple and a child is 20. What is the child's age?
Find the missing group of alphabets in the following series. ABC, EFG, IJK, (…), UVW
Two boys A and B start from the opposite direction of 14 km apart. A is facing east and B isfacing west. A reaches the distance of 5 km towards east and B reaches a distance of 2 kmtowards west. What
is the distance between the two boys?
The ratio of salaries of P and Q last year is 4 : 5. The ratio of last year salary and the presentsalary of P is 3: 5 and for Q this ratio is 2 : 3. If their total salary at present is Rs. 6800, what
is salary of Q ?
If $$x^{4}+\frac{1}{x^{4}}$$=47 ,then find the value of x+$$\frac{1}{x}$$ | {"url":"https://cracku.in/rrb-je-1st-june-2019-shift-3-question-paper-solved?page=4","timestamp":"2024-11-11T23:42:50Z","content_type":"text/html","content_length":"157502","record_id":"<urn:uuid:65332b3a-3206-4f83-b4bc-18d01e9ce656>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00496.warc.gz"} |
Recommendations: Why RMSE can be misleading
I've recently spent some time working on machine learning and recommendation engines. If you're building a recommendation engine, you're typically trying to optimize for some metric of "goodness" for
the user. In a Netflix-like setting, it could be how much time does a user spend watching the content you recommended? Picking good offline metrics (without actually watching how the user is
responding) can be really tricky. The RMSE (Root Mean Square Error), the staple of many research papers, can be particularly misleading in many cases.
Assume, we have 100 items that a user rates between 1 to 5 stars (much like in the Netflix problem). For simplicity, assume that the first three items have 5-star ratings, and the rest have a 1-star
│Product│True Rating│Algo. A Predictions │Algo. B Predictions │
│P001 │5 │2 │1 │
│P002 │5 │2 │1 │
│P003 │5 │2 │1 │
│P004 │1 │2 │1 │
│... │... │2 │1 │
│P100 │1 │2 │1 │
Consider Algorithm A that predicts that all the ratings will be 2. The RMSE for this dataset = sqrt((97 + 27)/100) = 1.11. Now consider Algorithm B that predicts all ratings to be 1. The RMSE for
this dataset is sqrt(48/100) = 0.693. Algorithm B produced a
improvement in RMSE over algorithm A, but is it really any better at differentiating between the items that the user liked vs. ones that she didn't? If you are going to use the recommendations to
solve a ranking problem, RMSE is a pretty useless measure in this context. A better metric would capture the fact that you're trying to use the recommendations to display the best items to the users,
hoping that the user clicks/buys/watches/engages with/likes what you recommended. Being accurate on items way beyond the top few that the user is likely to engage with is not very useful at all.
Now on the other hand, if the "rating" we have is binary -- say someone "likes" the movie or not -- say 1 or 0. (In reality there's a third state, where someone watches a movie, and then doesn't rate
it. You could can map this state to a 1 or 0 with a few application-specific assumptions). With a binary rating, the RMSE simply counts how many predictions you got right. Because what we really have
here is a classification problem, and not a ranking problem, RMSE ends up being more reasonable.
There are several papers that talk about vastly superior metrics for ranking (that actually work in practice!) that I'll try and describe them in future posts. | {"url":"http://sandeeptata.blogspot.com/2014/03/recommendations-why-rmse-can-be.html","timestamp":"2024-11-13T23:01:04Z","content_type":"text/html","content_length":"72596","record_id":"<urn:uuid:35eaacf2-af2f-416a-8759-dad0d51e9952>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00014.warc.gz"} |
IFF Criterion for y to be a Conjugate of x^n
IFF Criterion for y to be a Conjugate of x^n
Proposition 1: Let $G$ be a group and let $x, y \in G$, $n \in \mathbb{Z}$. Then $y$ is a conjugate of $x^n$ if and only if $y$ is the $n^{\mathrm{th}}$ power of a conjugate of $x$.
• Proof: $\Rightarrow$ Let $y$ be a conjugate of $x^n$. Then there exists an $a \in G$ such that:
\quad y = ax^na^{-1}
• Let $z$ be a conjugate of $x$. Then there exists a $b \in G$ such that $z = bxb^{-1}$. So $x = b^{-1}zb$. Substituting this into the above equation yields:
\quad y &= a(b^{-1}zb)^na^{-1} \\ &=a\underbrace{(b^{-1}zb)(b^{-1}zb)...(b^{-1}zb)}_{n \: \mathrm{times}}a^{-1} \\ &= ab^{-1}z^nba^{-1} \\ &= (ab^{-1})z^n(ab^{-1})^{-1}
• So $y$ is a conjugate of $z^n$, i.e., $y$ is the $n^{\mathrm{th}}$ power of a conjugate $z$ of $x$.
• $\Leftarrow$ Suppose that $y$ is the $n^{\mathrm{th}}$ power of a conjugate of $x$. Then $y = z^n$ where $z$ is a conjugate of $x$. Since $z$ is a conjugate of $x$ there exists an $a \in G$ such
that $z = axa^{-1}$. So:
\quad y = z^n &= (axa^{-1})^n \\ &= \underbrace{(axa^{-1})(axa^{-1})...(axa^{-1})}_{n\: \mathrm{times}} \\ &= ax^na^{-1}
• So $y$ is a conjugate of $x^n$. $\blacksquare$ | {"url":"http://mathonline.wikidot.com/iff-criterion-for-y-to-be-a-conjugate-of-x-n","timestamp":"2024-11-06T07:32:55Z","content_type":"application/xhtml+xml","content_length":"15825","record_id":"<urn:uuid:a1eb689e-544e-483a-b1d5-126d67456fa0>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00113.warc.gz"} |
Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 1 Solutions - Ex1.11
- 3 mins
Exercise 1.11
This problem deals with the affine cipher with the key parameters a = 7, b = 22.
1. Decrypt the text above.
2. Who wrote the line?
This solution is verified as correct by the official Solutions for Odd-Numbered Questions manual.
1. Given that we possess the key, we can decrypt this directly:
2. Lewis Carroll wrote this in one of his poems.
I wrote a python script which can perform affine cipher encryptions/decryptions.
def egcd(a, b):
if a == 0:
return (b, 0, 1)
g, y, x = egcd(b % a, a)
return (g, x - (b // a) * y, y)
def modinv(a, m):
a = a % m # allows this to work with negative numbers
g, x, y = egcd(a, m)
if g != 1:
raise Exception('modular inverse does not exist')
return x % m
def letter_to_number(c):
return ord(c.lower()) - ord('a')
def number_to_letter(n):
return chr(n % 26 + ord('a'))
def affine_encrypt(a, b, plaintext):
def encrypt_letter(c):
n = letter_to_number(c)
return number_to_letter((a * n + b) % 26)
return "".join([encrypt_letter(c) for c in plaintext])
def affine_decrypt(a, b, ciphertext):
a_inv = modinv(a, 26)
def decrypt_letter(c):
n = letter_to_number(c)
return number_to_letter((a_inv * (n - b)) % 26)
return "".join([decrypt_letter(c) for c in ciphertext]).upper()
def print_ciphertext(ciphertext):
print "\n==========="
print "Ciphertext:"
print "===========\n"
print ciphertext
def print_plaintext(a, b, plaintext):
header = "Plaintext (as decrypted by ({}, {})):".format(a, b)
print "\n", "=" * len(header), "\n", header, "\n", "=" * len(header)
print "\n", plaintext, "\n"
if __name__ == "__main__":
ciphertext = "falszztysyjzyjkywjrztyjztyynaryjkyswarztyegyyj"
plaintext = affine_decrypt(7, 22, ciphertext)
print_plaintext(7, 22, plaintext)
print "Q1.11.2 Answer: This was said by the Lewis Carrol in his poem\n" | {"url":"https://tom.busby.ninja/understanding-cryptography/ex1-11/","timestamp":"2024-11-14T10:35:05Z","content_type":"text/html","content_length":"53945","record_id":"<urn:uuid:9907d908-4557-444f-8edf-28eba590bdce>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00374.warc.gz"} |
Intermediate Algebra videos
Material type: Video
Year: 2014
Authored by:
Tammy Rossi
Used in course: MATH010
• Chapter 1: Background Material
• Chapter 2: Linear Equations, Inequalities and Applications
• Chapter 3: Graphs, Linear Equations and Functions
• Chapter 4: Systems of Linear Equations
• Chapter 5: Exponents and Polynomials
• Chapter 6: Factoring
• Chapter 7: Rational Expressions and Equations
• Chapter 8: Roots and Radicals
• Chapter 9: Quadratic Equations and Functions
Audiences: K12, Freshman, Sophomore, Junior, Senior, Adult and professional
Location: https://udcapture.udel.edu/misc/math/intermediate-algebra/
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
One thought on “Intermediate Algebra videos”
1. Thanks for your post. I really like the information which you have shared in your post about the algebraic videos. | {"url":"https://sites.udel.edu/open/2014/06/04/intermediate-algebra-videos/","timestamp":"2024-11-05T12:23:18Z","content_type":"text/html","content_length":"65710","record_id":"<urn:uuid:edf52e13-cc5b-4941-b384-dcd881228a55>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00840.warc.gz"} |
How much is a yard of gravel - Civil Sir
How much is a yard of gravel
How much is a yard of gravel | how much is a yard of gravel weigh | how much is a yard of gravel cover | how much is a yard of gravel cost | how much is 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 yards of gravel
weigh, cover and cost.
Gravel is one of the most important building material collected from river basin, mountains, rocks, small rocks, pebbles, loose and dry sand, aggregate and pea gravel. Pea gravel is chosen best for
walkway because it is round in shape of small size which make it most comfortable to work on.
How much area does a yard of gravel cover? It will depends on the size of the stone, the amount of dust content and the thickness of the layer and the how level the surface is to be covered.
Gravel is made of rock is to build roads, path, driveway, Patio, pedestrian, pavement, pathways, roadways, landscaping, and etc.
Gravel is categorised according to their size, size more than 5mm put in the category of gravel, it is formed of igneous rocks. It is categorised as fine gravel (4 – 8 mm), medium gravel (8 – 16 mm),
coarse gravel (16 – 32 mm), pebbles (32 – 64 mm), cobbles (64 – 256 mm) and Boulder more than 256mm.
How much is a yard of gravel
You are looking to buying gravel and crushed stone for your construction work, if you want to apply them in normal depth 50 mm for your driveway and 35 mm depth normally in pedestrian pathway, you
need to know how much area will cover by 1 ton of gravel and you should purchase and you can load in your vehicle.
Most of gravel supplier which are available nearly to you, will give you option to deliver gravel and crushed stone at your homes, for this they should cost some money for transportation. If you have
a truck or vehicle that you can use to bring gravel to your destination or construction site, then it is cheaper and faster option for you.
In this regarding, how much is a yard of gravel, how much is a yard of gravel weigh, how much is a yard of gravel cover, how much is a yard of gravel cost, Knowing about full detail analysis, then
you should keep reading.
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How much is a yard of gravel
Weight of gravel depends on the rocks type, loose and dense condition, compact, moisture content, dry and wet condition, others inorganic mixed in the gravel. For estimating purpose, contractor’s and
Builder’s would take weight of gravel as 3000lb (pounds) per yard or 1.5 short tons per yard, and which is equivalent to 110 lb per cubic feet.
1 yard of gravel look likes by 3 feet long by 3 feet wide and by 3 feet height, which is equal as 27 cubic feet as (length× width× height= 27).
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically weighs about 3,000lb (pounds) or 1.5 tons, and will cover approximately 162 square feet at 2 inches
thick, and will cost between $15 to $75 per yard, with an average cost of $40 per yard. If you buy gravel bag, 54 bags of 50 lb gravel to cover the same area as a cubic yard.
How much is a yard of gravel weigh
A cubic yard of gravel, which visually is 3 foot long by 3 foot wide by 3 foot tall, is typically weighs about 3,000lb (pounds) or 1.5 tons, which is approximately equal as 110 lbs per cubic foot. A
cubic yard of dry gravel is typically weighs about 2970 pounds or 1.5 tons. A cubic yard of wet gravel is typically weighs about 3375 pounds or 1.7 tons. Moisture is prime factors that determining
the weight of gravel.
How much is a yard of gravel cover
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically covers 162 square feet (18 square yards or 15 square meters) at recommended depth of 2 inches thick,
324 square feet at 1 inch thick, 108 square feet (12 square yards or 10 square meters) at 3 inches thick, or 81 square feet (9 square yards or 7.5 square meters) at 4 inches thick.
How much is a yard of gravel cost
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically costs ranges from $15 to $75, with an average of $40 per yard. The costs of gravel ranges from $10 to
$50 per ton, or $15 to $75 per cubic yard, or $1 to $3 per square foot, or $1350 per truckload depending on the rock types, volume, and travel distance. Gravel spreading costs $12 per yard or $46 per
How much is 2 yards of gravel
2 cubic yards of gravel is typically weighs around 6,000 pound (3 tons) and which will cover 200 square feet area at 3 inches deep, and will cost an average of $80 for 2 cubic yards of gravel
(national an average cost will be $40 per yard).
How much is 3 yards of gravel
3 cubic yards of gravel is typically weighs around 9,000 pound (4.5 tons) and which will cover 300 square feet area at 3 inches deep, and will cost an average of $120 for 3 cubic yards of gravel
(national an average cost will be $40 per yard).
How much is 4 yards of gravel
4 cubic yards of gravel is typically weighs around 12,000 pound (6 tons) and which will cover 400 square feet area with a standard depth of 3 inches thick, and will cost an average of $160 for 4
cubic yards of gravel (national an average cost will be $40 per yard).
How much is 5 yards of gravel
5 cubic yards of gravel is typically weighs around 15,000 pounds or 7.5 tons, and which will cover 500 square feet area with a standard depth of 3 inches thick, and will cost an average of $200 for 5
cubic yards of gravel (national an average cost will be $40 per yard).
How much is 6 yards of gravel
6 cubic yards of gravel, is typically weighs around 18,000lb (pounds) or 9 tons, and which will cover 972 square feet area with a standard depth of 2 inches thick, and will cost an average of $240
for 6 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 7 yards of gravel
7 cubic yards of gravel, is typically weighs around 21,000lb (pounds) or 10.5 tons, and which will cover 1,134 square feet (126 square yards, or 105 m2) area with a standard depth of 2 inches thick,
and will cost an average of $280 for 7 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 8 yards of gravel
8 cubic yards of gravel, is typically weighs around 24,000lb (pounds) or 12 tons, and which will cover 1,296 square feet area (144 square yards, or 120 m2) with a standard depth of 2 inches thick,
and will cost an average of $320 for 8 cubic yards of gravel (national an average cost will be $40 per yard).
“How many bags of concrete is in a yard
“How to figure yards, cubic feet or bag of dirt
“How to figure yards of gravel
“How many bags of gravel in a cubic yard
“How much does a yard of mulch cover
How much is 9 yards of gravel
9 cubic yards of gravel, is typically weighs around 27,000lb (pounds) or 13.5 tons, and which will cover 1,458 square feet area (162 square yards, or 135 m2) with a standard depth of 2 inches thick,
and will cost an average of $3600 for 9 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 10 yards of gravel
10 cubic yards of gravel, is typically weighs around 30,000lb (pounds) or 15 tons, and which will cover 1,620 square feet area (180 square yards, or 150 m2) with a standard depth of 2 inches thick,
and will cost an average of $400 for 10 cubic yards of gravel (national an average cost will be $40 per yard).
How much is 12 yards of gravel
12 cubic yards of gravel, is typically weighs around 36,000lb (pounds) or 18 tons, and which will cover 1,944 square feet area (216 square yards, or 180 m2) with a standard depth of 2 inches thick,
and will cost an average of $480 for 12 cubic yards of gravel (national an average cost will be $40 per yard).
A cubic yard of gravel, which visually is 3 feet long by 3 feet wide by 3 feet tall, is typically weighs about 3000lb (pounds) or 1.5 tons, and will cover approximately 162 square feet at 2 inches
thick, and will cost between $15 to $75 per yard, with an average cost of $40 per yard. If you buy gravel bag, 54 bags of 50 lb gravel to cover the same area as a cubic yard. | {"url":"https://civilsir.com/how-much-is-a-yard-of-gravel/","timestamp":"2024-11-08T13:59:05Z","content_type":"text/html","content_length":"97172","record_id":"<urn:uuid:ff6eefbf-0027-4a55-b681-8a65c264d09e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00862.warc.gz"} |
What is a variable?
I agree entirely that it doesn’t cause any problems mathematically. My point is entirely a pedagogical one.
Hello Mike. I guess that’s what I’m saying :) Admittedly, I’m not sure it’s a valid solution to the problem, neither mathematically nor pedagogically. But it seems to me that it corresponds more to
the common practice in calculus.
Already a statement like $y=f(x)$ is mostly read as “y is a function of x”, and less often as “y is the value of the function f at input x”. (At least among engineers and physicists). Or I have seen
statements like “suppose $V(t)$ is the volume of water at time t” together with a plot where the vertical axis is denoted with $V$. Without introducing an additional variable to denote the value of
the function $V$.
Or consider when calculus textbooks discuss cartesian coordinates $(x,y)$ and polar coordinates $(r,\theta)$ in the plane. The easiest way for me to think about this is by interpreting $x,y,r,\theta$
as functions on the plane. Wouldn’t it be cumbersome to introduce additional names to denote the functions of which $x,y,r,\theta$ are the values?
Unfortunately I don’t yet see clearly what trouble such a point of view causes. (let’s restrict to mathematical problems since the pedagogical ones are hard to predict)
Probably we do have to distinguish between the function and its value sometimes, but can’t we still do this say by obtaining the value of $x$ at $3$ by precomposing with the constant function $3$?
@Michael, you seem to be saying that I should just give up on indoctrinating my students not to write $f = x^2+1$, and on teaching them that $f(x) =x^2$ and $f(t)=t^2$ define the same function? I
think there is an important point to distinguish between the values of a function and the function as an object in its own right. Only once you really understand that are you justified in failing to
notate it.
Michael_B: as far as I’ve always seen it used, if $f: M \to N$ is a smooth mapping and $p \in M$, then $D f(p): T_p(M) \to T_{f(p)}(N)$ is the derivative mapping at $p$, also sometimes called the
Jacobian at $p$. If $f$ is real-valued, then all the tangent spaces $T_{f(p)}(\mathbb{R})$ are canonically identified with $\mathbb{R}$, and the resultant linear functional $D f(p): T_p(M) \to \
mathbb{R}$ is an element in the cotangent space of $M$ at $p$, also denoted $d f(p)$ as you say.
Is what Todd denotes with $Df$ the same as $df$ the differential of the function $f$?
Sorry to enter this discussion so late. But Mike: it sort of looks like you’ve answered your own question back in #1! Is there a problem with just using $D f$ notation?
It’s probably hard to tease apart for calculus students what is really going on, but if we start with a function $f: \mathbb{R} \to \mathbb{R}$ and apply the tangent bundle functor $(-)^D$ from SDG,
keeping in mind the Kock-Lawvere axiom asserting an isomorphism $(pos, vel): \mathbb{R}^D \to \mathbb{R} \times \mathbb{R}$, where the first map is evaluation at the point $1 \to D$. Then
notationally it seems quite alright to describe $f^D$ in terms of a mapping that is customarily denoted as
$(p, v) \mapsto (f(p), D f|_p \cdot v)$
and this carries over just fine to the multivariate setting. Maybe it’s even a really good idea to implant the thought that the derivative $D f|_p$ (or $D f(p)$ if you’d prefer) is not really just to
be thought of as just a “number” but as a linear function (which can be characterized by a number in the 1-dimensional setting, by setting $v = 1$ and taking $D f(p) \cdot 1$)?
If this pedagogical point is to be driven home seriously, I suppose you could dismiss notations like $\frac{d f}{d x}$ as archaisms, reflecting an earlier age when notions of functions, variables,
etc. hadn’t been fully worked out.
Hello Mike,
you seem to suggest that these issues are resolved if we interpret $x$ as a function instead of a variable, a point of view I sympathize with. Unfortunately I don’t quite understand the arguments
against this step. You seem to mention two:
But while logically consistent, this seems to undercut the force of the lesson of dummy variables, since we are endowing $x$ with a special status not shared by $t$.
With $x$ as a function we can also write “$f = x^2+1$”, which again is something that I’m used to indoctrinating my students against.
Correct me if I’m wrong, both of these arguments disappear if we interpret $x$ as a function: in the first case since the function $t$ might (or might not) be the same as the function $x$. (This
seems in agreement with applications of calculus where different variables like time $t$ and position $x$ are not interchangeable.) The second argument is resolved since for a function $y=f(x)$ all
of the symbols $y$, $f(x)$, $f$ and $y(x)$ now denote the same thing. The notations $f(x)$ would be a different notation for composition $f\circ x$ (with $x$ usually denoting the identity function).
So maybe there are other arguments against this convention?
Another question: if $y$ and $x$ only denote variables, should the symbol $\frac{dy}{dx}$ denote a variable or a function?
Edit (after reading your question again): you also suggest, that the “evil” notation is the primed one $y'$, which I agree with. Physicist have a convention of writing the dot but only to denote
derivatives with respect to time, so that doesn’t have the same ambiguity. Also if $Df$ denote the differential $df$ as Todd suggests, than that’s Ok since it has a different meaning from $\frac{df}
Cheers Michael
My question wasn’t really “what is a variable”; that was just the best short-ish title for this kind of rambly question that I could think of. I don’t think I was complaining about “variables that
can’t be bound”, but maybe I was; can you explain further? What does your description have to say about notations for derivatives?
Rod, does any of that suggest an answer to my question?
It does to your question “What is a variable” :)
But to your more elaborate question it seemed like you were complaining about “variables” that played the role of something like labels, flags, or indices into structures without being “true
variables” that can get bound. I just wanted to mention an alternative way of thinking about variables, binding, and substitution which might fit your problem if I understood the math enough, however
I got a little carried away in my typing.
And for those who ares becoming Hegelian, I wanted to sneak in the opposition between “totally contradictory” and “completely unknown” :), though it doesn’t seem to involve adjoints.
I’m sorry.
Rod, does any of that suggest an answer to my question?
From the perspective of Logic Programming, and a particular variant, “variables” are a notational solution that conflates 2 things: labeling and binding.
Traditionally, formula are written as linear strings of symbols that are parsed into trees. One use of variables is as external labels to note that parts of a “tree” share the same substructure and
it is really not a tree.
For example in the formula $x + x$ the two $x$s can be seen as the same substructure and substituting “1” for “x” involves changing just one substructure, not 2. The result of the substitution, “1 +
1”, has dropped any labeling that indicates that the two “1”s are really the same and came from the same place. One can explicitly use external labels for this situation using a name followed by “?”.
Then the two formulas would be notated as $x?\top + x?\top$ and $x?1 + x?1$ where $\top$ indicates that the first $x?$ structure is “unbound”.
Binding traditionally has two states - unbound or bound to something totally specific. The alternative perspective is that “binding” is a continuum that takes place in a lattice of structures where $
\top$ means “completely unknown” and $\bot$ means “totally contradictory”.
In an expression like $x+x$, $x$ is rarely completely unknown. Usually $x$ is known to be some specific type of number, but which exact number is unknown. One could even give $x$ an intermediate
binding such as $1\vee 2$. Evaluating $x?(1\vee 2) + x?$ gives the result $2 \vee 4$ while if the structure is not shared, evaluating $x?(1\vee 2) + y?(1\vee 2)?$ gives $2 \vee 3 \vee 4$.
There are two notions of substitution. “Binding substitution” or “unification” is used to make structures more specific. For example $(x?\top + x?\top) \wedge (1 + \top)$ is unified to $x?1 + x?1$
while $(x?\top + x?\top) \wedge (1 + 2)$ becomes $x?\bot + x?\bot$ because $1\wedge 2$ becomes the contradictory structure.
Actual true substitution is rare in maths. Rarely does one substitute $2$ for $1$ in an expression like $1 + 1$ to give $2 + 2$ though there can be systems where some structure holding $1$ is
regarded as a default value that can be overridden. The result of substituting $2$ for $1$ in $1 + 1$ depends on whether the two $1$s are the same structure or not - $1$ is not substituted for,
instead a maybe shared structure bound to $1$ gets rebound.
@hilbertthm60: I don’t understand how $f_y (x,y)$ is better. You still have the variable $y$ occurring in the notation $f_y$ for the partial derivative function. Am I misunderstanding?
@Urs: It’s true that we always abuse notation, but I find that the correct and incorrect ways to abuse notation are one of the hardest things for beginning math students to understand. I haven’t
spent a lot of time thinking about it, but I’ve generally assumed that it’s not really possible to understand how to abuse notation until you understand “the way mathematics works” at a sufficiently
deep level, so that when teaching students who don’t yet understand math, it’s better to try to avoid abusing notation as much as possible.
It wasn’t completely a joke. There was something like the thought that if cohesive HoTT is the God-given way to do things, then it might suggest the least bad forms of “abuse of notation”.
So all of Mike’s worries are over!
While I suppose you are joking (right? :-) this reminds me that maybe we shouldn’t hijack Mike’s thread too much.
My reply to the topic here would be: since we are humans and not proof assistants, whenever we actually do some work we’ll adopt convenient “abuse of notation”. One should alert students as to what’s
really going on, but I wouldn’t worry too much about enforcing a formally consistent notation.
I’m waiting for geometry of physics to load to read Prop. 26, but it takes about 10 minutes to typeset on this machine!
Oh, that’s a pain. For me it’s slow, but not quite this slow.
This is with math rendered by Mathjax, I suppose? I suppose in some other browser maybe and/or with the rewquite fonts installed, it should take no extra time?
Once there was this vision that we have decent math on the web. But somehow it still seems to be a long, long way to go, for some reason.
Thanks. I meant my question to see if perhaps in some idealised setting where Mike has complete control over his students’ maths education, so has taught them HoTT from an early age, now when he
comes to teach calculus, and he adds the cohesive axioms, is he ever confronted with the issues he raised in #1?
He has $f: R \to R$, with $f(x) = (3 x + 1)^4$. So $f = g(h)$ for obvious $g$ and $h$. So then using your $\mathbf{d}$,
$\mathbf{d} f = \mathbf{d} g(h),$
at which point a chain rule kicks in. (I’m waiting for geometry of physics to load to read Prop. 26, but it takes about 10 minutes to typeset on this machine!)
So all of Mike’s worries are over!
I have added my previous reply as a paragraph to differential calculus – In cohesive homotopy theory
with cohesion one can essentially characterize $\mathbf{d} \colon \mathbb{R} \longrightarrow \mathbf{\Omega}^1_{cl}$ such that postcomposition of a function $f \colon X \longrightarrow \mathbb{R}$
with this map yields the derivative $\mathbf{d} f \colon \in \mathbf{\Omega}^1_{cl}(X)$ of $f$. This is disucssed in some detail at geometry of physics in the section 4. Differentiation
A variant is a certain homotopy pullback of this construction, which yields variational calculus, as discussed there in the section In terms of smooth spaces.
In differential cohesion one can get hold of the infinitesimal interval $D$ and then proceed as in synthetic differential geometry. For instance using this one can describe differential equations as
discussed there in the section In terms of synthetic differential equations.
Moreover, differential cohesion encodes D-geometry and hence in principle allows to talk about differential equations in that way.
What would elementary calculus look like in cohesive homotopy type theory?
For what it’s worth, Mathematica writes multivariable derivatives as $f^{(i,j,\ldots,k)}$, which means the $i$-th partial derivative with respect to the first variable, the $j$-th partial derivative
with respect to the second variable, etc. Unfortunately, this notation presupposes the commutativity of partial derivatives.
On $f'$ and $\frac{d y}{d x}$: I think you are right there. In order to use notations like $f'$ consistently, it is appears to be necessary to abandon notions like “change of variable” and regard $x
\mapsto (3 x + 1)^4$ and $u \mapsto u^4$ as distinct functions. So it is incompatible with the setup where calculus expressions are regarded as functions on some manifold – because there are no
preferred coordinates. From a syntactic point of view it does seem rather disturbing that the $x$ in the denominator of $\frac{d y}{d x}$ is bound but free in $\frac{d y}{d x}$ itself. It’s almost as
if $\frac{d}{d x}$ is some kind of variable binding operator like $\lambda$ or $\prod$ or $\sum$… except for the fact that it doesn’t bind the variable at all! Compare:
$x : \mathbb{R} \vdash y \equiv (3 x + 1)^4 : \mathbb{R}$$x : \mathbb{R} \vdash \frac{d y}{d x} \equiv 12 (3 x + 1)^3 : \mathbb{R}$$\vdash \lambda x . (3 x + 1)^4 : \mathbb{R} \to \mathbb{R}$
Accordingly, we should also require the use of substitutions instead of evaluations when working with the $\frac{d y}{d x}$ notation: so $\frac{d y}{d x} |_{x = 0}$ instead of $\frac{d y}{d x} (0)$.
This doesn’t answer the broader question, but there is a “better” alternative for the multivariable calculus notation. It is pretty typical to use $f_y (x,y)$ to denote the partial derivative of $f$
with respect to $y$.
My calculus book uses many different notations for the derivative of $y = f(x)$ with respect to $x$, such as
$\frac{dy}{dx} \quad y'\quad f'(x) \quad D f(x) \quad \frac{df}{dx}$
Recently I’ve found that I kind of object to a couple of these. For instance, consider $\frac{df}{dx}$. One of the things I try to teach my students is that when we define a function $f$ by writing
$f(x) = x^2$, say, the variable $x$ is a dummy variable; if we wrote $f(t) = t^2$ we would be defining the same function, namely the one which squares its input. But if “$f$” denotes a function of
this (usual mathematical) sort, then how can we write $\frac{df}{dx}$ to mean its derivative, since $f$ doesn’t know that we called its input variable $x$?
Notations like $f'(x)$ and $D f (x)$ don’t have this problem, because $f'$ and $D f$ denote the derivative function of $f$, which assigns to each input value the derivative of $f$ at that value, and
so $f'(x)$ and $D f (x)$ just mean evaluation of this function at that value. I suppose we could interpret $\frac{df}{dx}$ similarly if we regarded “$\frac{df}{d}$” as the derivative function of $f$,
which we evaluate at something by placing it after the $d$ in the denominator, but this seems strained, and would suggest even odder notations such as writing $\frac{df}{d3}$ for $f'(3)$.
It was pointed out to me that this kind of notation is even commoner in multivariable calculus, where we write things like $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$, and in
this case there aren’t good alternatives available, since we have to indicate somehow whether it is the first or second input variable of $f$ with respect to which we take the derivative.
From a differential-geometric viewpoint, one answer is to say that $x$ denotes a standard coordinate function on the 1-dimensional manifold that is the domain of $f$, and so we are actually taking
the derivative with respect to a vector field associated to that function. We can even regard $\frac{df}{dx}$ as a literal quotient of differential 1-forms, since 1-forms on a 1-manifold are a
1-dimensional vector space at each point, so the quotient of two of them is a real number. But while logically consistent, this seems to undercut the force of the lesson of dummy variables, since we
are endowing $x$ with a special status not shared by $t$.
Using $x$ to denote the coordinate function has the other interesting consequence that it makes it okay to say “the function $x^2+1$” (since we can multiply and add functions together), instead of
insisting on saying “the function $f$ defined by $f(x) = x^2+1$”. Again this feels like it undercuts the lesson of what a function is — and yet I find that it’s hard to teach a calculus class without
eventually slipping into saying “the function $x^2+1$”. With $x$ as a function we can also write “$f = x^2+1$”, which again is something that I’m used to indoctrinating my students against.
I also have a problem with the notation $y'$, for a more pragmatic reason. Suppose we want to take the derivative of $y = (3x+1)^4$ using the chain rule. A nice way to do it is to make a substitution
$u = 3x+1$, so that $y = u^4$, and then use differentials:
$du = 3 dx$$dy = 4u^3 du = 4(3x+1)^3(2 dx)$$\frac{dy}{dx}= 8(3x+1)^3$
The problem here is that the notation $y'$ doesn’t indicate what variable we differentiate with respect to, and in this calculation we have two derivatives of $y$, namely $\frac{dy}{dx} = 8(3x+1)^3$
and $\frac{dy}{du}=4u^3$, which are not equal even after substituting the value of $u = 3x+1$. Here the solution seems to be straightforward: just don’t write $y'$. But if we can write $f = x^2+1$
just like $y = x^2+1$, and if we allow the notation $f'(x) = 2x$ and hence also $f' = 2x$, then we should just as well have $y' = 2x$.
Does anyone have a good solution? I feel like at least part of the problem comes from confusing $\mathbb{R}$ as the real numbers with $\mathbb{R}$ as a 1-dimensional manifold, but I haven’t exactly
managed to pin down yet how to solve it from that point of view.
Like you, Mike, I try to teach my Calculus (and Algebra) students the difference between a function and its value, even though the textbooks do what they can to undermine my efforts. Of course, it is
important to see the abuses of notation that they're liable to meet in applied fields, but (as you say) they have to understand the correct way first. So I tell them that the textbook abuses
notation, but I try to never abuse it myself. In particular, I tell them that $y'$ is ambiguous, so I don't use it; they are allowed to (since the book does) but I recommend against it. I use $f'(x)$
and $\mathrm{d}y/\mathrm{d}x$, as you suggest, instead. (Besides that, $\mathrm{D}f(x)$ is all right, too; but since our book doesn't use it, I only mention it in passing.)
As for $\mathrm{d}f/\mathrm{d}x$, this is a bit subtle; $\mathrm{d}f(x)/\mathrm{d}x$ is just fine; in fact, if $y = f(x)$, then $y = \mathrm{d}f(x)/\mathrm{d}x = f'(x)$. But that doesn't make $\
mathrm{d}f/\mathrm{d}x$ a legitimate synonym of $f'$! It's a matter of logic; like Rod #12 said, the two $x$s stand for the same thing, so you can't substitute for one without substituting for the
other. Even this is subtle; if you substitute, say, $3$ for $x$ in $f'(x) = \mathrm{d}f(x)/\mathrm{d}x$, then you get $f'(3) = \mathrm{d}f(3)/\mathrm{d}3$, which doesn't quite work. On the other
hand, if you substitute $t$ for $x$ instead (a change of variable), then $f'(t) = \mathrm{d}f(t)/\mathrm{d}t$ is fine. This works better with differentials; $\mathrm{d}f(x) = f'(x) \,\mathrm{d}x$
becomes $\mathrm{d}f(3) = f'(3) \,\mathrm{d}(3)$, which is fairly trivial but at least correct. The problem is that, in writing $\mathrm{d}f(x)/\mathrm{d}x$, you're tacitly assuming that $\mathrm{d}
x$ is nonzero, that is that $x$ is a variable quantity^1. (This is the origin of the term ‘variable’, I believe, even though we now use that also for symbols that stand for constants.) So you should
only substitute something variable for it. (But in $\mathrm{d}f(x) = f'(x) \,\mathrm{d}x$, no such assumption is being made.) This is really no more mysterious than why you can't substitute $1$ for
$x$ in $(x^2 - 1)/(x - 1)$.
What does it mean for $x$ to stand for a variable quantity? Doesn't it stand for a real number? Yes … but it stands for a variable real number. This brings us to the question in the title: what is a
variable? Lawvere said that a variable can be any morphism in any category; then a variable real number (aka a real-valued variable or simply a real variable) is a morphism whose target is the space^
2 of real numbers. For some reason, the only field in which this penetrates the undergraduate curriculum is statistics; there, they know that a random variable is a measurable function on a
measurable space (typically valued in the measurable space of real numbers with Borel measure), a morphism in the category of measurable spaces and measurable functions (or something like it). Even
in an elementary treatment where every random variable is defined a finite space and the word ‘measurable’ is never uttered, they still give a definition of ‘random variable’. In Calculus, we usually
study smooth variables (or smoothly varying quantities), which are morphisms in the category of smooth manifolds and smooth functions (or something like it). This is actually the first lesson in my
Applied Calculus course: what a smooth variable is (very roughly, of course). (In the regular Calculus course, we don't usually assume that everything is smooth, so I have to bring this in later.)
So $x$, $y$, $t$, $u$, etc are all variables (usually smooth ones). What then is $f$? It's a function, but I mean ‘function’ in the sense used in elementary algebra, that is a partial function (a
partial morphism in the category of sets) from $\mathbb{R}$ to $\mathbb{R}$, usually a smooth one (so infinitely differentiable wherever defined). Quantities like $f(x)$ are defined at the formal
level by composition; we don't write this as $f \circ x$ because we conceptually distinguish variables (with arbitrary unspecified domain) from functions (with domain a specified subset of $\mathbb
{R}$). So (pace Michael #21) $x$, $y$, $r$, and $\theta$ may indeed be functions on the plane, but we're not treating them in the same way as $f$, so they use different notation. (And then they might
not be functions on the plane; if you're really studying the motion of a particle in the plane, they might be better thought of as functions of time.)
I won't even get into the problems with notation for second derivatives and multivariable calculus. In general, all of this stuff works better with differentials than with derivatives (right down to
the terms ‘differential’ and ‘derivative’), but I've said enough for now.
@Toby: Interesting comments.
Just a few questions
1) How do you bring the point across to students that sometimes $x$ stands for the value of a function (or constant, as in your first paragraph) and sometimes $x$ represents a variable quantity
(morphism in a category) as in your paragraphs 3 and 4? Especially if we use the same symbol.
2) Why is it important to make a conceptual distinction between functions with unspecified domains (like x,t,r) and functions with domains subset of $\mathbb{R}^n$ like $f$, and use different
notation for their compositions?
Another remark: form the perspective of interpreting $x,y,t$ etc. as morphism in a suitable category, a common abuse of notation I see, is that a lot of times the pullback of a function along a map
gets denoted with the same symbol. So for example when describing motion of a particle in the plane by saying $x,y$ are functions of time, I’d interpret this as saying $x,y$ are actually the
pullbacks of the standard coordinates $x,y$ on the plane to the 1-dimensional manifold representing time, along the map given by the motion.
I think I have a partial answer to 2): functions with values in $\mathbb{R}$ but unspecified domains cannot be composed, but we may always compose them with functions from $\matbb{R}$ to itself. Is
that the reason?
Thanks Toby! I think you’ve resolved my confusion by saying that the manifold in question should be an arbitrary one (the domain of generalized elements), not even necessarily 1-dimensional. I hadn’t
thought about $d\, f(x) /dx$, but you’re right that that makes perfect sense.
My explanation of substitution would have been a bit different. I would say that when $x$ is a variable, then $dx$ is a new variable, albeit one which happens to be related to $x$ in a certain way.
(In other words, we now consider generalized elements of the tangent bundle $T\mathbb{R}$ rather than the manifold $\mathbb{R}$.) So substituting 3 for $x$ doesn’t make $dx$ into $d(3)$. Rather, it
just means that instead of $dx$ being a small variation about $x$, it is a small variation about 3.
I would really like to hear what you have to say about second derivatives and multivariable calculus, if you have time sometime to share it. I have recently discovered another problem with the $f'$
notation: prime is a very small symbol and hard to see from the back of the room! (-:
The second derivative is not really a notion that goes well with differential forms. After all, $\mathrm{d}^2 = 0$! In the one-variable case one can cheat and observe that the tangent bundle is
generated by a global section $\mathrm{d} x$, so there is a unique function $\frac{\mathrm{d} f}{\mathrm{d} x}$ such that $\mathrm{d} f = \frac{\mathrm{d} f}{\mathrm{d} x} \mathrm{d} x$, and since $\
frac{\mathrm{d} f}{\mathrm{d} x}$ is a function, we can talk about its derivative again. (Note, we can do this equally well on the real line or the circle!) This cheat can be made to work in the
many-variable case provided the tangent bundle is trivial, but it’s far more obvious that one is doing some very coordinate-dependent things then. (One thing that is always very confusing is that the
partial differential operator $\frac{\partial}{\partial x^i}$ depends on the choice of the other coordinates as well, despite the notation! This is in contrast to the differential 1-form $\mathrm{d}
x^i$, which depends only on the coordinate function $x^i$.)
I think one is supposed to think about jet bundles if one wants to do higher-order derivatives. But that seems a step too far for first-year calculus.
Certainly jet bundles are themselves a step too far for first-year calculus, but then so are tangent bundles. The trick would be to find a way to have them in the background causing things to make
sense, but not needing to be mentioned explicitly.
The most direct thing to see is the iterated tangent bundle: if $dx$ is a variable element of $T\mathbb{R}$, then $d(dx)$ is a variable element of $T(T\mathbb{R})$. But unfortunately that is a bit
bigger than the jet bundle…
I think that the next time I teach calculus, I’m going to bring in differentials and linear approximations much earlier. This time I waited to use them as an explanation for the chain rule, but then
once we had them I found that I liked using them for everything else. (Thanks Toby for stressing this point, here and elsewhere!) From that point of view, the ordinary differential is determined by a
linear approximation
$f(x+dx) \simeq f(x) + d(f(x))$
to first order in dx, so an appropriate meaning of “second differential” would be a quadratic approximation
$f(x+dx) \simeq f(x) + d(f(x)) + \frac{1}{2!} d^2(f(x))$
to second order in dx. And with this meaning of $d^2(f(x))$ we do have a literal quotient
$f''(x) = \frac{d^2(f(x))}{dx^2}.$
I haven’t yet worked out the best way to explain “first order in $dx$”, though. And I don’t know what I would say to explain the $2!$, either.
Zhen Lin #27 has succinctly pointed out the problems.
Here is a more explicit problem with the usual notation for second derivatives: it breaks the chain rule. If $u = f(x)$ and $y = g(u)$, then
$\mathrm{d}y = g'(u) \,\mathrm{d}u = g'(u) \,(f'(x) \,\mathrm{d}x) = g'(f(x)) \,f'(x) \,\mathrm{d}x$
works out fine; it gives
$(g \circ f)'(x) = \mathrm{d}y/\mathrm{d}x = g'(f(x)) \,f'(x)$
as it should. But (using the proposed second differential from Mike #29, which is implicitly endorsed by the notation $\mathrm{d}^2{y}/\mathrm{d}x^2$)
$\mathrm{d}^2{y} = g''(u) \,\mathrm{d}u^2 = g''(u) \,(f'(x) \,\mathrm{d}x)^2 = g''(f(x)) \,f'(x)^2 \,\mathrm{d}x^2$
is no good; it gives
$(g \circ f)''(x) = \mathrm{d}^2{y}/\mathrm{d}x^2 = g''(f(x)) \,f'(x)^2 ,$
which is incorrect. The correct formula is
$(g \circ f)''(x) = g''(f(x)) \,f'(x)^2 + g'(f(x)) \,f''(x) .$
For this reason I never write $\mathrm{d}^2{y}/\mathrm{d}x^2$ in class (except once, about the same time that I write $y'$, to warn against it); I write $(\mathrm{d}/\mathrm{d}x)^2{y}$ (or even $\
mathrm{d}(\mathrm{d}y/\mathrm{d}x)/\mathrm{d}x$, when it comes naturally) instead.
More simply (but you might not believe it if I make it so simple right away), $\mathrm{d}^2{u}/\mathrm{d}u^2 = 0$ implies $\mathrm{d}^2{u} = 0$, which implies $\mathrm{d}^2{u}/\mathrm{d}x^2 = 0$, and
that's not what we want at all.
@ Michael #24:
Why is it important to make a conceptual distinction between functions with unspecified domains (like $x,t,r$) and functions with domains subset of $\mathbb{R}^n$ like $f$, and use different
notation for their compositions?
Partly because function notation is so convenient, yet it requires a domain, and sometimes we don't want to specify it (or even to imply that it's a subset of $\mathbb{R}^n$, much less a subset of $\
mathbb{R}$ in first-term Calculus). So I want to write $f(3)$ and $f'(3)$; of course, I can also write ${y|_{x=3}}$ and ${(\mathrm{d}y/\mathrm{d}x)|_{x=3}}$ … but I can also write $f(x+1)$, while I
can't write ${y|_{x=x+1}}$. I guess that this is basically what you said, the ability to compose functions.
The distinction is especially relevant in applications. Here, the domain of the variables is a vaguely unspecified space of states of the world. The textbooks sometimes encourage us to identify this
space (often it can be identified with the time line, for example), so that there is a single independent variable (or a few in the multivariable case) of which every other variable is a function.
But the whole point of the Chain Rule is that the independent variable is irrelevant! It's sufficient that some choice of independent variables is possible (that is that some space of states can be
assumed to exist), but it's completely unnecessary to actually make this choice. So I don't actually want to say that $x, y, t$ etc are functions at all (to the students, for whom a function is
defined on a subset of some $\mathbb{R}^n$). Yet functions like $(x \mapsto \mathrm{e}^x)$ are also around, and I want to refer to them from time to time too.
How do you bring the point across to students that sometimes $x$ stands for the value of a function (or constant, as in your first paragraph) and sometimes $x$ represents a variable quantity
(morphism in a category) as in your paragraphs 3 and 4? Especially if we use the same symbol.
What's happening at the most basic level is a change of context (in the technical sense as in type theory). There is the general context where the variables are allowed to vary as much as they may,
and then there is the more specific context where $x$ is set to (say) $3$. (There are other intermediate contexts, especially in multivariable Calculus, such as that given by a constraint as in
optimization problems.) Assuming for the sake of argument (but this is hardly necessary) that there is exactly one possible state of the world in which $x = 3$, then we have a morphism from the point
to the space of all world-states; as you said (but I didn't quote), we are taking a pullback along this morphism and abusing notation by keeping the same symbol $x$.
I tell my students, particularly when working out word problems, to keep careful track of the context. (I use the word ‘context’ but don't let them suspect that it's a technical term in logic!) In a
typical problem, they have an equation that holds always, which they may differentiate; but then later they use equations that are only true for an instant. (Related rates and optimization are two
broad categories of problems like this.) I tell them that any result from the equations that hold always is also true for an instant, but not conversely (which I think makes intuitive sense); and you
cannot differentiate equations that only hold for an instant, because nothing is changing in that instant! (In multivariable Calculus you can differentiate equations that hold under a constraint, and
this leads for example to Lagrange multipliers, but you still have to remember that you are working relative to a constraint.) So basically, I'm allowing them to pull back results along a morphism
but not push them forward; but I try to make it sound like common sense instead of a theorem of categorial logic!
the partial differential operator $\frac{\partial}{\partial x^i}$ depends on the choice of the other coordinates as well, despite the notation!
In a thermodynamics course, I learnt the notation $(\partial{U}/\partial{S})_T$ for the partial derivative of $U$ with respect to $S$ when $T$ is held constant. (In general, there are $n - 1$
subscripts.) In my multivariable class, I introduce this notation first, then say that we can drop the subscripts as an abuse of notation when it's obvious what they're going to be. Of course, all of
the abuses of notation can be justified in this way (that it's obvious what it's supposed to mean), but only this one is really necessary, since otherwise it gets very tedious.
By the way, Mike, the corresponding notation for the partial derivatives of a function is $f_i$ (where $i = 1, 2, ...$); it works just like $f_y$ in hilbertthm90 #2, only it's legitimate. An
alternative is $\mathrm{D}_i{f}$ (especially if you want to leave subscripts free for a sequence or other family of functions). This extends to higher-order derivatives just fine, without having to
assume commutativity (the claim that $\mathrm{D}_{i,j} = \mathrm{D}_{j,i}$).
@Toby #30: very interesting, thanks! How about this for a second try at “second differentials”?
The general form of a second-order approximation should include not only a first-order change in the variable but also an independent second-order change. That is, let $\mathrm{d}x$ be a first-order
infinitesimal and $\mathrm{d}^2x$ a second-order one, and work up to second-order; thus $(\mathrm{d}x)^2$ is relevant but $(\mathrm{d}x)^3$ and $(\mathrm{d}^2x)^2$ and $(\mathrm{d}x)(\mathrm{d}^2x)$
can be neglected as being third- or fourth-order (or are equal to zero, depending on your preferred flavor of infinitesimal). Thus while it is true that
$f(x+\mathrm{d}x) = f(x) + f'(x)\, \mathrm{d}x + \frac{1}{2} f''(x)\, (\mathrm{d}x)^2$
we also have (by the same reasoning)
$f(x+\mathrm{d}x + \frac{1}{2} \mathrm{d}^2x) = f(x) + f'(x)\,\, \mathrm{d}x + \frac{1}{2} f'(x)\, \mathrm{d}^2x + \frac{1}{2} f''(x) \, (\mathrm{d}x)^2$
and it is both the third and the fourth terms here that should be called $\mathrm{d}^2(f(x))$, since they are both second-order. That is, we write
$f(x+\mathrm{d}x + \frac{1}{2} \mathrm{d}^2x) = f(x) + \mathrm{d}(f(x)) + \frac{1}{2} \mathrm{d}^2(f(x))$
and therefore
$\mathrm{d}^2(f(x)) = f'(x)\, \mathrm{d}^2 x + f''(x)\, (\mathrm{d}x)^2.$
Now if $u = f(x)$ and $y = g(u)$, we have
\begin{aligned} \mathrm{d}^2 y &= g'(u) \, \mathrm{d}^2 u + g''(u) \, (\mathrm{d}u)^2 \\ &= g'(f(x)) \Big(f'(x) \, \mathrm{d}^2 x + f''(x)\, (\mathrm{d}x)^2\Big) + g''(f(x)) (f'(x)\, \mathrm{d}x)^2\\
&= g'(f(x)) f'(x) \, \mathrm{d}^2 x + \Big(g'(f(x)) f''(x) + g''(f(x)) (f'(x))^2\Big) (\mathrm{d}x)^2 \end{aligned}
And matching this with
$\mathrm{d}^2((g\circ f)(x)) = (g\circ f)'(x)\, \mathrm{d}^2 x + (g\circ f) ''(x)\, (\mathrm{d}x)^2$
we recover the correct chain rules for both first and second derivatives.
I suspect that this could be written in terms of jet bundles.
Of course, now we’ve lost the notation $\frac{\mathrm{d}^2y}{(\mathrm{d}x)^2}$ for the second derivative. Unless there’s some reason why we can assume $\mathrm{d}^2x=0$.
@Toby: thanks for the detailed answer!
Concerning Mikes nice reasoning in #33:
Of course, now we’ve lost the notation $\frac{d^2y}{(dx)^2}$ for the second derivative.
If I understood this correctly, you might still “save” it by using the notation $\frac{\partial^2y}{(dx)^2}$, or should it be $\frac{\partial^2y}{(\partial x)^2}$?
@Michael: I didn’t use the notation $\partial^2 y$; what are you thinking that it would mean? The problem that I saw is that $\mathrm{d}^2 y$ is not a linear function of $(\mathrm{d}x)^2$ alone, but
also of $\mathrm{d}^2x$.
@Toby, I have a couple of terminological questions for you.
1) What do you call “$x^2+1$”? In Lawvere’s parlance it is still a “variable quantity”, but as it is not syntactically a variable I wouldn’t want to call it that. But neither is it a “function” in
your setup, if I understood correctly.
2) What do you call an object like “$(x^2+1)\;\mathrm{d}x$” which you can take the integral of?
$\mathrm{d}^2(f(x)) = f'(x)\, \mathrm{d}^2 x + f''(x)\, (\mathrm{d}x)^2$.
I agree; I first got this by writing $\mathrm{d}f(x) = f'(x) \,\mathrm{d}x$ and applying the product rule. Notice that the $\mathrm{d}$ here is not the exterior derivative but instead a commutative
(rather than supercommutative) operator.
Of course, now we’ve lost the notation $\frac{\mathrm{d}^2y}{(\mathrm{d}x)^2}$ for the second derivative.
Right, and I don't know how to rehabilitate that notation, because of my last remark in #30.
However, you can write $\frac{\partial^2{y}}{(\partial{x})^2}$ instead! This is because, just as $\frac{\partial{y}}{\partial{x}}$ is the coefficient on $\mathrm{d}x$ in an expansion of $\mathrm{d}y$
, so $\frac{\partial^2{y}}{(\partial{x})^2}$ is the coefficient on $(\mathrm{d}x)^2$ in an expansion of $\mathrm{d}^2y$. (ETA: Michael already noticed this in #34, but hopefully my explanation of it
By the way, I also tell my students that $\mathrm{d}$ binds more tightly than any non-differential operation like squaring, so I can write $\mathrm{d}x^2$ instead of $(\mathrm{d}x)^2$. (Then I always
use parentheses in something like $\mathrm{d}(x^2) = 2x \,\mathrm{d}x$.)
Answering Mike's questions in #36:
1. I call $x^2 + 1$ simply a quantity. I might even call it a real number (but not a constant one) or even simply a number, but ‘quantity’ usually works well (except in applications to supply and
demand, where ‘quantity’ has a more specific meaning). It is a variable quantity, of course, and I might even point out that it varies or may even say ‘$x^2 + 1$ is variable.’, but not ‘$x^2 + 1$
is a variable.’; that would be confusing. (In other words, when describing the quantity as a whole, I would use ‘variable’ only as an adjective.) Also, I will say that $x^2 + 1$ is a function ,^1
which simply means that there exists a function $f$ such that $x^2 + 1 = f(x)$. If I'm emphasizing the logical form, then I may also call it an algebraic expression; technically, the expression ‘
$x^2 + 1$’ represents or stands for the quantity $x^2 + 1$.
2. Sometimes I call $(x^2 + 1) \,\mathrm{d}x$ an infinitesimal quantity, if I want to emphasize its interpretation as something infinitely small (and similarly I might call $x^2 + 1$ a finitesimal
quantity). But if I'm talking about things that one can integrate, then I usually call it a differential form. I actually introduce that term fairly early, when I remark that the differential of
any (finitesimal) expression (in any number of variables!) will be a differential form; I point out that every term has a differential as one factor, note that this makes every term (and hence
the sum) an infinitesimal quantity, and then I introduce the name for expressions of this form. (I also remark that the differential form has rank $1$ because only $1$ factor of each term is a
differential, but we don't have to say that since differential forms of higher rank are only used in multivariable Calculus. And then in my multivariable class, I use them!)
Toby in #37 wrote:
I first got this by writing $\mathrm{d}f(x) = f'(x) \,\mathrm{d}x$ and applying the product rule. Notice that the $\mathrm{d}$ here is not the exterior derivative but instead a commutative
(rather than supercommutative) operator.
Interesting! Would you mind telling a bit more about this $d$ as a commutative operator and how the equation follows from the chain rule. Studying synthetic differential geometry is still on my TODO
list, so apologies if this is standard knowledge among experts.
I also still need to understand Mikes computation in #33. Intuitively I would have thought that a first order infinitesimal is also an infinitesimal of second order, so I find it confusing that we
need to include the first order change seperately when looking at the effect of a second order change. (And I also have no intuition for what it means that the two changes are independent, from a
geometric or physical perspective).
Edit: I’m also still curious if Mikes suggestion from #26 can be made consistent with the different interpretations of $d$ suggested above
So substituting 3 for $x$ doesn’t make $dx$ into $d(3)$. Rather, it just means that instead of $dx$ being a small variation about $x$, it is a small variation about 3.
So would it then be correct to also write $f'(x) = \frac{\partial^2 y}{\partial^2 x}$?
Intuitively I would have thought that a first order infinitesimal is also an infinitesimal of second order
Actually, it’s the other way around: a second order infinitesimal is also a first order one (although to first order, it’s zero). Higher order means a smaller number.
I first got this by writing $\mathrm{d}f(x) = f'(x) \,\mathrm{d}x$ and applying the product rule. Notice that the $\mathrm{d}$ here is not the exterior derivative but instead a commutative
(rather than supercommutative) operator.
Interesting! Would you mind telling a bit more about this $d$ as a commutative operator and how the equation follows from the chain rule.
I don't know very much about that operator; I asked about this stuff once on Math Overflow and got no clear answer, although I did get a reference that I haven't followed up yet. But if I just assume
that it continues to obey the usual rules, then I can calculate with it just fine. In this case:
$\mathrm{d}^2f(x) = \mathrm{d}(\mathrm{d}f(x)) = \mathrm{d}(f'(x) \,\mathrm{d}x) = \mathrm{d}(f'(x)) \,\mathrm{d}x + f'(x) \,\mathrm{d}(\mathrm{d}x) = (f''(x) \,\mathrm{d}x) \,\mathrm{d}x + f'(x) \,\
mathrm{d}^2x = f''(x) \,\mathrm{d}x^2 + f'(x) \,\mathrm{d}^2x .$
So would it then be correct to also write $f'(x) = \frac{\partial^2 y}{\partial^2 x}$?
Apparently so! But of course $\frac{\partial y}{\partial x}$ is simpler.
@Mike #40:
Actually, it’s the other way around: a second order infinitesimal is also a first order one (although to first order, it’s zero). Higher order means a smaller number.
Here’s how I was thinking: a first order infinitesimal number is one with $\epsilon^2=0$, a second order is one with $\epsilon^3=0$. So first order is also of second order. Actually when I picture
infinitesimal neighborhoods of a point or subset in a manifold or whatever space, I always thought that they increase as the order increases. Did I get it wrong or are we maybe talking of dual
@Toby #41: I recall that question on Mathoverflow (one of the comments was mine). Unfortunately I also have not had the time to follow up on the references. But I do find the infinitesimals and
differentials approach advocated by Dray Manogue to be worthwile for teaching calculus. And since Mike arrived at the same equation as you by slightly different reasons it makes it even more
compelling to believe that maybe there is still something to be understood in the interpretation of $d$ or $d^2$.
Yes, I think we are using language in dual ways. I’m thinking of nonstandard-analysis-style infinitesimals, whose square or cube is never actually equal to zero. Instead I’m saying, let’s fix some
particular “scale” infinitesimal $\epsilon$; then a first-order infinitesimal is one $\eta$ such that $\eta/\epsilon$ is finite (“limited”), a second-order one is such that $\eta/\epsilon^2$ is
finite, etc.
Then when we work “up to first order”, which we could formalize as being in the quotient ring of limited numbers modulo $\epsilon^2$, the square of a first-order infinitesimal can be neglected. And
when we work “up to second order”, i.e. in the limited numbers modulo $\epsilon^3$, the cube of a first-order infinitesimal and the square of a second-order one can be neglected. So in the latter
quotient ring, a first-order $\eta$ has $\eta^3=0$ while a second-order one has $\eta^2=0$.
I think that this matches the use of phrases like “to first order” and “a first order change” in ordinary (non-infinitesimal) language better. A second order change is negligible if we are working to
first order, but not if we are working to second order, yet the amount of the change itself is the same in both cases; what changes is our attitude towards it. But I guess it doesn’t apply as well to
SDG-style nilpotent infinitesimals, so with those it may be better to avoid terms “first order” and “second order” and talk instead about “nilsquare” and “nilcube” etc.
Mike thanks for the explanation! I’ll have to think about it some more to resolve the conflicting views in my head.
In the meantime, here is something slightly related to the original question. In my calculus class last week I asked the students to answer the following questions
Compute the derivative of:
1. $\int_2^x \ln(t^2+1)dt$ with respect to $x$
2. $\int_2^x \ln(t^2+1)dt$ with respect to $t$
3. $\int \ln(t^2+1)dt$ with respect to $t$
(the last one is an indefinite integral, I’m using the notations of my calculus book here (Hughes-Hallett))
That caused a lot of confusion for my students. My preliminary reaction is to think of the notation for the indefinite integral as the bad guy.
I wouldn’t ask my students (2) or (3). In fact, I’m not sure what you were expecting.
In (2), are you assuming that $x$ is a function of $t$? Or constant with respect to $t$? Were you hoping that they would write $ln(x^2+1) \frac{dx}{dt}$? While there’s technically no contradiction in
using the same variable both free and bound, it’s bad style even in published mathematical papers, so I wouldn’t want to inflict it on calculus students.
As for (3), the way I’m used to thinking of it $\int ln(t^2+1)dt$ is not a function but a class of functions (differing by local constants) — hence not something you can take the derivative of.
However, you do raise an important point, which is that $t$ is bound in $\int_a^b f(t) dt$ but (sort of) free in $\int f(t) dt$. I’m curious to hear Toby’s take. I introduced indefinite integrals to
my class last week by saying that the indefinite integral of a thing (I didn’t say “differential form”, but I might have as Toby suggested) is the most general expression whose differential is that
thing. That made perfect sense to me.
I haven’t done definite integrals yet, but from that point of view, maybe the problem is with the definite integral notation, since we have the limits $a$ and $b$ specified but without indicating in
the notation which variable is supposed to take on those values. For instance, in a chain rule / substitution problem, say we have $\int_1^2 2t \cos(t^2) dt$, which we can solve by letting $u = t^2$
so that $du = 2 t dt$ and
$2t \cos(t^2) dt = \cos(u) du.$
But this equality (of differential forms) is not something to which we can apply the “operation” $\int_1^2$ and get
$\int_1^2 2t \cos(t^2) dt = \int_1^2 \cos(u) du.$
Instead we have to put $t=1$ and $t=2$ into $u=t^2$ and get
$\int_1^2 2t \cos(t^2) dt = \int_1^4 \cos(u) du.$
So maybe it would be better to write $\int_{t=1}^2 2t \cos(t^2) dt$ (as we do with summation notation, $\sum_{t=1}^4$) so that we could have
$\int_{t=1}^2 2t \cos(t^2) dt = \int_{t=1}^2 \cos(u) du = \int_{u=1}^4 \cos(u) du.$
In (2), are you assuming that $x$ is a function of $t$? Or constant with respect to $t$?
Good point. But before I answers (and let me know me if you see this differently): the discussion so far showed that there are at least two popular interpretations for “variables” in calculus: one in
the sense of “dummy variables” or placeholders for numbers, and one in the sense of “variable quantity” or maybe morphism in a suitable category. It also seems that these two interpretations can lead
to conflicts. But I’d be glad to understand this better still.
Having said that:
For 2) I was expecting that they answer $0$. From the “dummy variable” perspective $x$ is a placeholder for a number (representing the upper boundary and otherwise not related to $t$), and since the
variable $t$ is bound the whole integral does not “change” when we plug in different values for $t$, so the derivative is zero.
From the “variable quantity” perspective $x$ might depend on $t$ so the correct answer would be as you suggest $ln(x^2+1) \frac{dx}{dt}$. So to be consistent with the previous we need to assume $x$
is constant with respect to $t$.
But some things are not yet clear to me about this last answer. I’ll come to it in a moment.
As for 3) I was expecting that they answer $\ln(t^2+1)$. I also think of the indefinite integral as a family of functions (depending on the same variable $t$ as the differential form). I guess I’m
using the convention here that taking the derivative of a family of functions means taking the derivative of each member of the family. Of course in principle the additive constant could still depend
on $t$ in some context, which makes things more subtle.
…maybe the problem is with the definite integral notation, since we have the limits $a$ and $b$ specified but without indicating in the notation which variable is supposed to take on those
I think the standard convention here is that the boundaries $a,b$ of the definite integral always refer to the variable appearing in $dx$ (or $dt$ etc.) so there is seldom ambiguity there. But as you
suggest I also emphasize this by writing $\int_{u=1}^4 \cos(u)du$ instead of $\int_{1}^4 \cos(u)du$. In fact I sometimes overemphasize by writing $\int_{u=1}^{u=4} \cos(u)du$, which brings me back to
If I had written $\int_{t=2}^{t=x} \ln(t^2+1)dt$ and interpret variables as “variable quantities”, then how should I interpret the equality $t=x$ appearing in the upper boundary? Does it mean that
$t$ and $x$ are the same variable quantities? In that case the answer to 2) would be the same as the answer to 3) and it wouldn’t be possible to ask if $x$ is constant with respect to $t$ (also a
student might object that it is unnecessary to introduce a new name $x$ to denote the same thing as $t$, which nevertheless is considered bad style as you mention). But I suspect that the thing going
on here and elsewhere in the “variable quantity” perspective is that an equality like $t=x$ is interpreted in a way more commonly seen in probability/statistics as in $\{x=t \}$ denoting “the set of
all states where the random variables $x$ and $t$ assume the same value.”
This raises some (maybe sidetracking) questions for me:
1. if the “variable quantity” perspective can be formalized via arrows in a suitable category, then how does one formalize categorically the notion of two quantities (arrows) being “independent” or
“constant” with respect to each other?
2. What does the “set of states of the world” (also mentioned by Toby) correspond to categorically? Some classifying object?
These questions are not directly addressed at Mike or Toby, but if you happen to know some answers I won’t complain. :) Apologies if I can’t respond in the next few days.
Yes, I’m perfectly aware of the standard convention, and I agree that in practice there is no ambiguity in the meaning of a particular definite integral expression, but I described a situation
(integration by substitution) in which the lack of notation could be problematic for a student when manipulating several such expressions.
As for the meaning of $t=x$, I think more generally one of the things we can do with a “variable quantity” is to let it be equal a particular other quantity. If the other quantity is constant, then
it “stops varying” and becomes constant, while if the other quantity is also variable then their variation becomes dependent. For instance, when $x$ and $y$ are variable quantities and we write $\
left.\frac{dy}{dx}\right|_{x=2}$ for what if $y=f(x)$ we might also write as $f'(2)$.
Categorically, variable quantities are morphisms from some domain object say $\Gamma$ — which I think is what Toby meant by the space of “states of the world” — and setting two such variable
quantities equal would correspond to restricting the domain to the equalizer of those two morphisms. That’s probably the same as what you mean by $\{x=t\}$?
I think this “fixing the value of a variable quantity” is the same thing that’s happening in a definite integral. Given a differential form like $\ln(t^2+1)dt$ involving a variable quantity $t$ and
its differential $dt$, we can integrate this form from one particular value of $t$ to another. These particular values might be constant quantities or other variable quantities (such as variables),
and in the latter case the result is again going to be variable.
I need to think a bit about your first question.
I know what it means for a variable quantity $\Gamma \to R$ to be constant: it means that it factors through $1$. I’m not sure about “constant with respect to” some other quantity, though. Maybe that
is one of those things which only makes sense if the quantity “with respect to” is part of a given basis, so that we can say that the corresponding partial derivative vanishes?
In the context of indefinite integrals, I read ‘$\int$’ as ‘antidifferential’, since $\omega$ is the differential of $\int \omega$; that is, $\int \omega$ is an antidifferential of $\omega$. (Of
course, when $\omega = y \,\mathrm{d}x$, the derivative of $\int y \,\mathrm{d}x$ with respect to $x$ is $y$, so $\int y \,\mathrm{d}x$ is also an antiderivative of $y$ with respect to $x$, like they
say in the book.) I tend to avoid the term ‘indefinite integral’; it’s bad enough that (almost) the same notation is used for two different concepts (definite and indefinite integrals), and I'd just
as soon not use (almost) the same terminology as well.
I've never liked the idea that the antidifferential of a differential form (or whatever you want to call that) is a set of quantities; I try to say ‘an’ instead of ‘the’ as much as possible. If you
just want one antidifferential, then (for example) $\int x^2 \,\mathrm{d}x = x^3/3$ is OK; but if you want all of them, then you need $\int x^2 \,\mathrm{d}x = x^3/3 + C$. (I enforce the book's
answers to its problems by saying that it's asking for all of them. And I say that it's only interested in quantities defined on a connected domain, so I don't have to deal with local constants.)
For definite integrals, I introduce the notation first as $\int_p^q \omega$, where $p$ and $q$ are equations (preferably with unique solutions). Then $\int_{x = a}^b \omega$ is an abbreviation for $\
int_{x = a}^{x = b} \omega$ (as Michael wrote) when the left-hand sides are the same; finally, $\int_a^b y \,\mathrm{d}x$ is an abbreviation for $\int_{x=a}^b y \,\mathrm{d}x$ when only one
variable's differential appears in the expression for $\omega$. That's mostly how they look, but I encourage them to use a longer form when doing integration by substitution, for the reasons that
Mike gives. (This can violate the requirement that $p$ and $q$ have unique solutions. It's sufficient that the result of the integral be the same for any choice of solution, or at least for any
choice where the solutions of $p$ and $q$ are connected.)
The Fundamental Theorem of Calculus has two parts, which are inconsistently numbered. By the numbering in our textbook (which is the way that I learnt it):
1. $\mathrm{d}(\int_p^q \omega) = {\omega|_p^q}$,
2. $\int_p^q \mathrm{d}u = {u|_p^q}$.
Since this a theorem and needs fine print (about things being continuous and the like), I state and prove these first in function notation like the book does, but I bring up these forms eventually.
The variable $t$ is definitely free in both $\mathrm{d}f(t)/\mathrm{d}t$ and $\int f(t) \,\mathrm{d}t$, no ‘sort of’ about it. It's bound in ${f(t)|_{t=a}} = f(a)$, in ${f(t)|_{t=a}^b} = f(b) - f(a)$
, and in $\int_{t=a}^b f(t) \,\mathrm{d}t$ (which has a more complicated $t$-free definition). I agree with Mike that $t$ is bound in $\int_2^x \ln(t^2+1)dt$, so you can't really differentiate it
with respect to $t$, although you could naïvely say that it's $\ln(x^2+1)dx/dt$ as Mike suggested. On the other hand, $\int \ln(t^2+1)dt$ is fine; by definition, its differential is $\ln(t^2+1)dt$,
so its derivative with respect to $t$ is $\ln(t^2+1)$. (You don't even need the FTC for this one.)
there are at least two popular interpretations for “variables” in calculus: one in the sense of “dummy variables” or placeholders for numbers, and one in the sense of “variable quantity” or maybe
morphism in a suitable category
These two senses can both be incorporated into categorial logic. In the case of an expression like ${x^2|_{x=1}^2}$ (which is an abbreviation of ${x^2|_{x=1}^{x=2}}$ and is usually further
abbreviated as ${x^2|_1^2}$), we start with a real-valued quantity $x$ in some context $\Gamma$ (formally a morphism $x\colon \Gamma \to \mathbb{R}$). The equations $x = 1$ and $x = 2$ specify
certain extensions of $\Gamma$, categorially constructed as equalizers (as Mike suggested). Call these extensions ${\Gamma|_{x=1}}$ and ${\Gamma|_{x=2}}$ respectively; then if $u$ is any real-valued
quantity in the context $\Gamma$, ${u|_{x=1}^2}$ is a real-valued quantity whose context is the product ${\Gamma|_{x=1}} \times {\Gamma|_{x=2}}$. (You should be able to draw this using
arrow-theoretic diagrams, making use of the subtraction operation $\mathbb{R} \times \mathbb{R} \to \mathbb{R}$.) If it should so happen that ${\Gamma|_{x=1}}$ and ${\Gamma|_{x=2}}$ are points
(terminal objects), then ${u|_{x=1}^2}$ is simply a real number.
In the case of ${x^2|_{x=1}^2}$, if this appears as a problem in a textbook without any further context, the default interpretation is supposed to be that $\Gamma$ is the largest subset of $\mathbb
{R}$ on which $(x \mapsto x^2)$ is defined, in this case the entire real line $\mathbb{R}$; then ${\Gamma|_{x=1}}$ and ${\Gamma|_{x=2}}$ are indeed points, and so ${x^2|_{x=1}^2}$ is indeed a real
number (as it happens, $3$). In the context of a word problem where $x$ stands for an inherently positive quantity, then it would be more appropriate to take $\Gamma$ to be ${]0,\infty[}$ instead.^1
But in such problems, I think it even more natural to take $\Gamma$ to be an abstract space, which I think of as the space of possible states of the situation described in the problem. While $\Gamma$
might never be fully defined, various properties of it may be justified as needed on the basis of the intuition behind the problem. The textbooks, by encouraging us to put everything in the problem
in terms of a single variable (such as $x$), effectively ask us to find that this variable mediates an isomorphism between $\Gamma$ and some subspace of $\mathbb{R}$ (such as ${]0,\infty[}$); this
specifies $\Gamma$ up to specified isomorphism, so no further intuition is needed. But many problems are easier to solve without expressing everything in terms of one variable, and I encourage my
students to take a more flexible approach (especially to things like related rates and optimization problems). This just requires them to be a little more careful about keeping track of the context.
I suspect that the thing going on here and elsewhere in the “variable quantity” perspective is that an equality like $t=x$ is interpreted in a way more commonly seen in probability/statistics as
in $\{x=t \}$ denoting “the set of all states where the random variables $x$ and $t$ assume the same value.”
Yes, precisely, and this is an equalizer. In general, I'd say that the probability/statistics people have a good handle on this stuff; they know what a random variable really is, after all, and the
rest of us just need to learn that all of our variables are much the same sort of thing.
how does one formalize categorically the notion of two quantities (arrows) being “independent” or “constant” with respect to each other?
Like Mike, I don't think that this is really a sensible notion without specifying what the other independent variables are supposed to be. Rather, what should be formalized is the idea that one
quantity is determined by another. Working in the context $\Gamma$, a $T$-valued quantity $x$ is determined by a $U$-valued quantity $y$ if there exists a morphism $f\colon U \to T$ such that $x = f
\circ y$. (This definition appears as one of the fundamental concepts in Lawvere & Schanuel's Conceptual Mathematics.)
What does the “set of states of the world” (also mentioned by Toby) correspond to categorically? Some classifying object?
Sure, although actually it's a coclassifying object. So, while a principal $G$-bundle on $S$ (for $G$ some topological group and $S$ some topological space) is the same as a continuous map from $S$
to the classifying space $B G$, so an $S$-valued smooth quantity (for $S$ some smooth space) in a given context $\Gamma$ is the same as a smooth map to $S$ from a coclassifying space (which I've been
calling simply $\Gamma$ again). So $\Gamma$ is the coclassifying space for the quantities in the problem.
Lawvere & Schanuel's Conceptual Mathematics
One of my Calculus students came upon this very thread the other day and asked for reading material that would give him some idea of what we were talking about, and I recommended Lawvere & Schanuel.
In my opinion, a course using this book should be the first college-level math course that every student takes. Algebra is a prerequisite for it, but not Calculus, so it should come before Calculus.
(A bonus is that the practice of requiring Calculus as a prerequisite for unrelated courses such as linear algebra or discrete mathematics, intended to guarantee a level of mathematical maturity,
would be served by requiring the course in conceptual mathematics, which is more important to know anyway.)
Of course, first the math teachers have to learn this stuff!
Thanks Toby! How do you define the general form $\int_a^b \omega$ with $a$ and $b$ equations?
I’m also curious whether you’ve ever tried teaching a course out of Lawvere & Schanuel?
Another question for you, Toby, though not closely related to the subject of this thread. In emphasizing differentials more this semester than before, I’ve found that a lot of my students mix up
derivatives and differentials. E.g. they will write things like $f'(x) = 2x \, dx$. Do you have any tricks for alleviating or preventing this confusion?
I just noticed that Sage’s calculus functions use a notion of symbolic variable which seems quite similar to the “variable quantities” under discussion here. The documentation’s description of them
as “elements of the symbolic expression ring” suggests that they have a different mathematical formalization in mind, although I haven’t figured out exactly what that means. But their behavior seems
quite similar to what we’ve been talking about, e.g. once you declare a symbolic variable $x$, you can then write $y = x^2+1$ and differentiate $y$ with respect to $x$:
y = x^2+1
gives $2x$. Although it will also try to guess the variable to differentiate with respect to if you don’t give it one:
also gives $2x$. Sage also seems to assume that all variables are constant with respect to each other:
y = x^2 + t^2
also gives $2x$. Although you can declare one “variable” to be instead a function of the other:
t = function('t',x)
w = x^2 + t^2
uses the chain rule to give 2*t(x)*D[0](t)(x) + 2*x. Finally, a symbolic expression like these $y$s can’t be evaluated like a function — or at least trying to do so
gives a DeprecationWarning. But you can make it into a “callable symbolic expression” by designating an order of the variables occurring in it:
z = y.function(x,t)
I wonder if this would be a good sort of convention to adopt in a calculus class as well, especially one that involves learning to use Sage.
How do you define the general form $\int_a^b \omega$ with $a$ and $b$ equations?
Now I feel like I ought to think about pulling $\omega$ back to the solution subspace of those equations, but I really only define it for equations with unique solutions on a simply-connected $1$
-dimensional domain, that is expressions that can be reduced to $\int_{x=a}^b f(x) \,\mathrm{d}x$, which I define (following the textbook) as a Riemann integral (although sometimes I feel like I
ought to do a Henstock integral). This is an approach that already does not generalize to complex variables, of course; in the multivariable class, I talk about oriented curves and all that.
Do you have any tricks for alleviating or preventing this confusion?
Not ones that work!
Mind you, there are plenty of analogous mistakes without differentials. My goal is that they only make mistakes like this that don't make their final answer wrong.
ETA: So for example, if they put in too many differentials, then they might write this:
$f(x) = \ln(3x+1)$$f'(x) = \frac{\mathrm{d}(3x+1)}{3x+1}$$f'(x) = \frac{3\,\mathrm{d}x+0}{3x+1}$$f'(x) = \frac3{3x+1} ;$
the middle lines are wrong, but the last is correct (given the first).
But if they put in too few differentials, then they might write this:
$x^5 + y^5 = x + y$$5x^4 + 5y^4 = 1 + y'$$y' = 5x^4 + 5y^4 - 1 ;$
now everything is completely wrong (after the first line).
The latter is a fairly standard Calculus-class error, which using differentials helps to avoid; I much prefer the former error.
@Mike 54: nice idea to look at how people have implemented these things in software. Just a quick question for clarification:
I wonder if this would be a good sort of convention to adopt in a calculus class as well, especially one that involves learning to use Sage.
Do you mean the convention of distinguishing between “symbolic variables” and “callable symbolic expressions”?
If yes, it looks to me (at first sight) that these two notions corresponds to our distinction between “variable quantities” (maps with unspecified domain) and “functions” with domains some subset of
$\mathbb{R}^n$. In the classical notation it might be the difference between writing $f=x^2+1$ and $f(x)=x^2+1$. In the first case $f$ would be a variable quantity, in the second case $f$ is a
function from $\mathbb{R}$ to itself. Would you agree?
Do you mean the convention of distinguishing between “symbolic variables” and “callable symbolic expressions”?
I guess that’s mostly what I meant. As I said in #54, Sage’s “symbolic variables” do seem to correspond to our “variable quantities”, but I think its “callable symbolic expresions” are not quite the
mathematician’s functions, because they still remember the names of their variables. E.g.
f(x) = x^2
(another way to define a callable symbolic expression)
f(3) ===> 9
f(x=3) ===> 9
f(y=3) ===> x^2
So I guess I was wondering whether it would be worth discussing with calculus students the idea of a “function that knows the name of its arguments”.
Mike 43
Instead I’m saying, let’s fix some particular “scale” infinitesimal $\epsilon$; then a first-order infinitesimal is one $\eta$ such that $\eta/\epsilon$ is finite (“limited”), a second-order one
is such that $\eta/\epsilon^2$ is finite, etc.
Well, even more, in ultrafilter model, one looks at sequences with some limiting behaviour, and the integer power law in comparing asymptotic infinitesimals is not the only possibility. You can have
exponentially small ones, e.g. such ratios that say $\frac{\eta}{\epsilon^{3/2} exp(-1/\epsilon^2)}$ is finite. I hope you agree. (Sorry for bringing an issue which is already aged in the thread).
@Zoran: Yes, of course. That’s not even particular to an ultrafilter model, e.g. $\sqrt{\epsilon}$ is still infinitesimal, but “less than first order”. But the integer power law is the relevant one
for defining derivatives and higher derivatives.
Surely, Mike, I was not considering the issue critical for your calculus discussion, but for the intuition/image people who know other approaches, primarily SDG, gain about the nonstandard analysis.
@Toby #56: why would you say that there are too many differentials in the first computation? If we add one more $dx$ (for example multiplying on left) it seems correct. To understand the confusion of
students it would be interesting to understand what the student was thinking when doing that computation.
Sure, too many on the right or too few on the left. I'm basically taking the left-hand side (the simpler one and the one first written down) as indicating what the student meant to do and judging
correctness or incorrectness based on that. (But when correcting a paper, I might well amend the left-hand side instead, if that's the simpler fix.)
It’s not clear to me that when students make mistakes like this they are thinking anything, in the sense that we would mean the word. Rather, they just don’t seem to have the same understanding we do
that mathematical words and symbols have precise meanings and have to be used correctly.
Yeah, I wouldn't want to defend the thesis that the left-hand side indicates what the student intended in any seriously discriminatory way; I mean, I wouldn't want to assume that the student is
thinking clearly enough to discriminate between intending $f'(x)$, intending $f'(x) \,\mathrm{d}x$, or intending $\mathrm{d}f(x)$ (the latter two being equal, of course, but maybe not trivially so
even to a student who is thinking clearly). I just mean that if I have to pick some way to classify the error (as too many differentials or as too few, in this case), then that's the criterion that
I'll use.
I’m also curious whether you’ve ever tried teaching a course out of Lawvere & Schanuel?
No. It might not work very well for the students that we get either; it would need a massive illustrated, hand-holding, problem-filled expansion.
On second derivatives and second differentials … John Armstrong was considering them in 2009 in two posts that unfortunately attracted no comments.
Regarding antidifferentials (#44-49), what about introducing a new notation for “equality up to a local constant”? Since an equation like $\int x^2 dx = \frac{1}{3} x^3 + C$ is not an “equation
involving a variable $x$” in the same sense as $(x+1)^2 = x^2+2x=1$ anyway (you can’t substitute $x=3$ in itto get anything meaningful), it has to be regarded as an “equation between variable
quantities”, and then we can change the sense of “equal” as well. Say that if $u$ and $v$ are variable quantities, then $u\equiv v$ means that $u$ and $v$ have the same domain, and on every connected
subset of that domain there is a constant $C$ such that $u=v+C$ on that subset (or some simpler version of this statement that would be easier to understand). Then we could write
$\int x^2 dx \equiv \frac{1}{3} x^3$
and even
$\int \frac{1}{x} dx \equiv \ln |x|.$
Re #67: I remember stumbling over that issue sometime as an undergrad, or maybe even a grad student. I think I spent days, or at least hours, trying to figure out why some computation wasn’t working,
before I realized that I was implicitly assuming a version of “Cauchy’s invariant rule” for second derivatives (though I didn’t know the name of it), and that it might not be true.
From the perspective of #33 above, the problem arises from neglecting the $d^2 x$ terms that ought to be there in the second differential. I certainly didn’t understand that at the time, but I might
have if someone had taught me calculus using differentials to start with!
@Toby, did either of the two answers on your MO question ever pan out? The Hasse-Schmidt one seems promising, as you said, but as stated it seems to be purely algebraic and so only applies to
polynomials. Also, if I understood it correctly, there isn’t an operator $d$ that could be applied to anything already containing $d$s – instead there is a separate $d^2$ operator which is just
asserted to satisfy the Leibniz rule that you would expect if it were actually “$d$-of-$d$”.
Re #68: Then an important basic result (an easy corollary of the Mean Value Theorem) is that (for differentiable quantities) $u \equiv v$ is equivalent to $\mathrm{d}u = \mathrm{d}v$.
Actually, I've considered formally defining $\mathrm{d}$ to be the operation taking $u$ to its ${\equiv}$-equivalence class. Then all of the hard work goes into defining multiplication of such an
equivalence class by an ordinary quantity (or more precisely into defining the equality relation on formal linear combinations of differentials with coefficients from the ring of quantities). Note
that naïvely, every quantity has a differential in this sense, but we'll find that things are better behaved when we restrict to differentiable quantities.
Re #69: I dare say that I spent years on this, off and on, struggling to figure out what the heck was going on. It may actually have only been when I was first assigned to teach Calculus that I
forced myself to come to some resolution (and shortly thereafter started writing M.O questions about it). I remember struggling with the minus sign in $\mathrm{d}y/\mathrm{d}x = -(\partial{F}/\
partial{x})/(\partial{F}/\partial{y})$ around the same time (although I resolved that one much earlier).
Re #70: No, I never really slogged through the linked articles. I've really just these past few months settled on my own answer. To wit: $\mathrm{d}f$ is the operation that maps a smooth curve $c$ to
$(f \circ c)'(0)$; $\mathrm{d}^2f$ maps $c$ to $(f \circ c)''(0)$, and so on. Of course, $f$ itself maps $c$ to $(f \circ c)(0)$. Then we just take the subring generated by the above, within the ring
of all operations that map a curve to a number (which is commutative). At least for smooth functions, that's all that there is to it.
Is there a derivation $d$ that maps that entire subring to itself? It’s clear what it should do on the generators, of course, but it’s not immediately obvious to me that that yields a well-defined
Anyway, it sounds like a reasonable answer, but I find it a bit unsatisfying not to have a more intrinsic characterization of the subring in question, and also to have to assume in advance the notion
of smooth.
I'm not sure what you mean by
have to assume in advance the notion of smooth
As far as the M.O question is concerned, we're working on a smooth manifold (in fact a Cartesian space, without loss of generality), so we have this notion. Even if then we try to make it work more
generally for diffeological spaces or the like, then all of these still start out with some notion of smooth. (It's the other thread where we're trying to define everything in terms of curves in very
general spaces; here we're still trying to understand $\mathbb{R}^n$.)
But if instead you mean that it's unsatisfying to only define this for smooth maps (so not to extend to the case where, say, $\mathrm{d}^2 f$ exists but $\mathrm{d}^3 f$ does not), then I think that
it should still work, just with extra effort to keep track of when things might be undefined. (Again, we know ahead of time what's $C^k$ and what's not, so we already know when $\mathrm{d}$ should be
It’s clear what it should do on the generators, of course, but it’s not immediately obvious to me that that yields a well-defined operation.
Ah, good point! Actually, I think that I can extend $\mathrm{d}$ (partially defined) to every operation whatsoever taking a smooth parametrized curve to a real number. Given the curve $c$ and a real
number $h$, let $c_h$ be the reparametrization of $c$ given by $t \mapsto c(t + h)$. Then given the operation $\eta$ (so $\langle{\eta{|}c}\rangle$ is a number), define $\mathrm{d}\eta$ so that
$\langle{\mathrm{d}\eta{|}c}\rangle \coloneqq \lim_{h \to 0} \frac{\langle{\eta{|}c_h}\rangle - \langle{\eta{|}c}\rangle} h$
if this exists. (You can leave $\mathrm{d}\eta$ as a partially defined operation, or declare that $\mathrm{d}\eta$ exists only if this limit exists for all $c$.)
This manifestly depends only on the underlying operation, and it does the right thing, recursively, to smooth maps.
Very nice! You can exclude some uninteresting things by restricting to germs of curves, and I think you can even omit the a priori restriction to smooth curves: consider partial real-valued functions
from the set of germs (at 0) of all curves, and say a curve $c$ is smooth if $d^n x$ is defined at $c$ for all coordinate functions $x$. (I’m not sure exactly what I was complaining about re:
“smooth”, but whatever it was, this makes me happier.) That feels kind of Froelicher: given the relation $\langle \eta | c \rangle$ between partial operations and curves, we consider the fixed point
of the resulting Galois connection generated by the coordinate functions. The point in the other thread is that this doesn’t correctly isolate the differentiable functions on the other side: even if
$d f$ is defined, as an operation, on all smooth $c$, then $f$ may not be differentiable in the usual sense unless $d f$ additionally depends only on the tangent vector of a curve and is a linear
function thereof. Right?
Interestingly, I think this context also allows operations like $e^{dx}$: it’s the operation that takes $c$ to $e^{(x\circ c)'(0)}$. And presumably its differential is $d(e^{dx}) = e^{dx}\, d^2x$.
I’m not sure whether this is a good thing or not. I’m currently playing around with a different idea for defining higher differentials; if it works I may post up somewhere.
Can you think of a good name for these things that include differentials and also higher ones? We can’t really call them “differential forms” once they have $d^2xeq 0$ and $dx\,dy = dy\,dx$.
(I guess I’m having trouble separating the threads, sorry – in my mind it’s all one discussion. (-: )
We certainly can call them differential forms even when $\mathrm{d}x \,\mathrm{d}y = \mathrm{d}y \,\mathrm{d}x$; they're just not exterior differential forms. The term ‘form’ is quite general and has
a venerable history. (Compare ‘quadratic form’, ‘symmetric bilinear form’, etc.) In M.O, I said ‘cojet differential form’, which is not quite as nice a term as ‘exterior differential form’ (since
‘cojet’ is a noun rather than an adjective like ‘exterior’), but it does get at the right idea: that they act on spaces of jets (the limit of which is the space of germs, as you noted).
I like your $\mathrm{e}^{\mathrm{d}x}$; I have successfully calculated $\mathrm{d}(\mathrm{e}^{\mathrm{d}x}) = \mathrm{e}^{\mathrm{d}x} \,\mathrm{d}^2x$ (using Taylor's Theorem with Peano's
remainder); actually, the calculation works for $\mathrm{e}^\omega$ generally.
Generalizing still further, I conclude that
$\mathrm{d}(f(\omega_1, \ldots, \omega_n)) = D_1{f}(\omega_1, \ldots, \omega_n) \,\mathrm{d}\omega_1 + \cdots + D_n{f}(\omega_1, \ldots, \omega_n) \,\mathrm{d}\omega_n$
for any differentiable function $f$ of $n$ variables, by pushing everything through the definition, applying Taylor's Theorem to $f$, and observing that the unwanted terms drop out in the limit. What
more could one possibly want? (In particular, $\mathrm{d}$ is a derivation.)
Technicality: You wrote in part
say a curve $c$ is smooth if $d^n x$ is defined at $c$ for all coordinate functions $x$
You mean that $c$ is smooth at $0$, or else you mean that $\mathrm{d}^n x$ must be defined at $c_h$ for all $x$and all real numbers $h$.
in my mind it’s all one discussion
Certainly you borrowed notation from an off-site file linked only in the other thread!
You mean that $c$ is smooth at $0$
Yes, thanks.
Certainly you borrowed notation from an off-site file linked only in the other thread!
Really? What notation? You used $\langle{\eta{|}c}\rangle$ up in #74 here…
spaces of jets (the limit of which is the space of germs
Technicality again, but that doesn’t seem quite right to me; at least, I can’t see a sense in which it’s true. In particular, a germ is not determined by its $k$-jets for $k\lt\infty$, is it?
We certainly can call them differential forms
Okay, I see the point that it’s historically fine, but my experience is that nowadays mathematicians pretty universally say “differential form” to mean “exterior differential form”. I guess “cojet
differential form” would suffice to clarify, which might get abbreviated to “cojet form”.
I think my main worry is using the same symbol $d$ for the cojet differential and the exterior differential. For instance, pedagogically speaking, if I teach my calc 1 or calc 2 students to calculate
with cojet differentials, aren’t they going to be confused when they get to multivariable and I tell them that now $d^2=0$?
I wonder whether cojet forms and exterior forms could be unified in a larger framework? In some sense, all these cojet forms are still only 1-forms: even though they involve higher derivatives, they
only act on curves. But we could consider instead real-valued operators on germs of parametrized surfaces or hypersurfaces as well. For instance, if $\omega$ is an operator on germs of curves, we
could define its exterior differential $\hat{d}\omega$ as an operator on germs of surfaces by
$\langle \hat{d}\omega {|} c \rangle = \lim_{t\to 0} \frac{ \langle \omega {|} \lambda s.c(s,0) \rangle + \langle \omega {|} \lambda s.c(t,s) \rangle - \langle \omega {|} \lambda s.c(s,t) \rangle - \
langle \omega {|} \lambda s.c(0,s) \rangle }{t}$
or perhaps in the case when $\omega$ might be nonlinear it would be better to say
$\langle \hat{d}\omega {|} c \rangle = \lim_{t\to 0} \frac{ \langle \omega {|} \lambda s.c(s,0) \rangle + \langle \omega {|} \lambda s.c(t,s) \rangle + \langle \omega {|} \lambda s.c(-s,t) \rangle +
\langle \omega {|} \lambda s.c(0,-s) \rangle }{t}$
I haven’t checked that this is at all sensible. But it also starts (unsurprisingly) to make me think of the Weil algebras that define the infinitesimal objects in SDG.
Here’s another thought: can we integrate an arbitrary cojet form? Suppose $\omega$ is a real-valued operator on germs of curves, and let $c$ be a curve defined on $(a-\epsilon,b+\epsilon)$. Then we
have a function $f:[a,b]\to\mathbb{R}$ defined by
$f(x) = \langle \omega {|} c_{x} \rangle$
and we could define
$\oint_c \omega = \int_{a}^b f(x) dx$
if the RHS exists. It seems like it ought to follow that
$\oint_c d\omega = \langle \omega {|} c_b \rangle - \langle \omega {|} c_a \rangle.$
(where $d$ is the commutative cojet differential). But it’s late at night, so I could be spewing nonsense…
You used $\langle{\eta{|}c}\rangle$ up in #74 here…
Oops, never mind, that was me, not you!
a germ is not determined by its $k$-jets for $k\lt\infty$, is it?
Ah, no, I must have been implicitly assuming that every function (or at least every smooth function) is analytic, and we wouldn't want to restrict to analytic curves. Still, these operations do
depend only on the jets, even when the germs differ. But germs are a simpler concept.
if I teach my calc 1 or calc 2 students to calculate with cojet differentials, aren’t they going to be confused when they get to multivariable and I tell them that now $d^2=0$?
In my Calculus classes, I've been using $\mathrm{d} \wedge \eta$ for the exterior differential of $\eta$. They've already seen $\eta \wedge \zeta$ by this point, and this gives the right idea
regarding skew-commutativity. (In particular, the signs in the product rule
$\mathrm{d} \wedge (\eta \wedge \zeta) = (\mathrm{d} \wedge \eta) \wedge \zeta + (-1)^{|\eta|} \eta \wedge (\mathrm{d} \wedge \zeta) = (-1)^{(1 + {|\eta|}){|\zeta|}} \zeta \wedge \mathrm{d} \wedge \
eta + (-1)^{|\eta|} \eta \wedge \mathrm{d} \wedge \zeta$
come out right that way. Not that I ever write down anything like this in that class.) So $\mathrm{d} \wedge \mathrm{d} \wedge \eta = 0$, but this is very different from $\mathrm{d}^2 \eta = \mathrm
{d} (\mathrm{d} \eta)$.
I do tell them that people usually don't put the wedge in there (and that they sometimes don't put the wedge in the wedge product either), and this is OK because they're restricting attention to
exterior differential forms.
But even though I don't actually use higher differentials in my Calculus classes^1, they do see differential forms that aren't exterior forms. There are the absolute differential forms, of course,
but there's more; consider
$đs = \sqrt{\mathrm{d}x^2 + \mathrm{d}y^2} .$
It would be criminal not to introduce that in class! But what is $\mathrm{d}x^2$? (or ${|\mathrm{d}x|}^2$). It can be thought of as a symmetric bilinear form, but it's also a cojet form. (The two
operations, one on a pair of curves and one on a single curve, are related by polarization.)
these operations do depend only on the jets, even when the germs differ
That’s true if by “these operations” you mean the ones constructed from functions by applying the cojet $d$ and algebra operations. In #72 you suggested generating a subring, so I guess this is what
you’re thinking of. Although $e^{dx}$ wouldn’t be in that subring, nor would $\sqrt{dx^2 + dy^2}$; we’d need to close up under more functions than the ring operations. The whole ring of
operations-on-germs, of course, might include operations that really do depend on the whole germ rather than only the jets, although I can’t think of any examples off the top of my head.
In my Calculus classes, I've been using $\mathrm{d} \wedge \eta$ for the exterior differential of $\eta$
That’s good! I might do the same when I get to exterior derivatives. (Although I still haven’t decided whether I can justify talking about exterior differential forms at all, given that our standard
textbook does everything the traditional way in terms of vectors. Is there a good multivariable calculus textbook that uses differential forms?)
the main reason for using differential in class is that people use them in applied fields
Hmm, that’s one good reason, but I think another good reason is that they just make the concepts easier to understand and the computations easier to do. However, it’s not clear to me that higher
cojet differentials would be much use in single-variable calc for either of those purposes either. The main advantage I see right now is if I could somehow avoid talking about derivatives at all and
use only differentials, but to be really effective that would require a supporting textbook.
One issue with my proposed notion of integration in #81 is that in general, it will depend on the parametrization of the curve, whereas the integral of an ordinary 1-form along a curve does not
(though it does depend on its orientation). However, it does include integration with respect to $ds = \sqrt{dx^2+dy^2}$, which is also parametrization-invariant — I guess what matters for that is
not linearity but “degree-1 homogeneity”.
Does it also include integration of absolute 1-forms? Can an absolute 1-form be regarded as a cojet form like $|dx|$ defined by
$\langle {|\omega|} ; c\rangle = {\Big|\langle \omega ; c\rangle\Big|}?$
(I changed your notation $\langle \omega | c \rangle$ to $\langle \omega ; c \rangle$ to avoid confusion with the absolute value bars.)
Re: #80, the wedge product of two cojet 1-forms $\omega$ and $\eta$ ought probably to be the “cojet 2-form” defined on a surface germ $c$ by
$\langle \omega\wedge\eta {|} c \rangle =\langle\omega {|} \lambda s.c(s,0) \rangle \cdot \langle\eta {|} \lambda s.c(0,s) \rangle - \langle\omega {|} \lambda s.c(0,s) \rangle \cdot \langle\eta {|} \
lambda s.c(s,0) \rangle$
I still haven’t decided whether I can justify talking about exterior differential forms at all, given that our standard textbook does everything the traditional way in terms of vectors. Is there
a good multivariable calculus textbook that uses differential forms?
I don't know of one; even Dray & Minogue don't go that far.
My justification is that they're already integrating differential forms; the classical expression $\int \mathbf{F} \cdot d\mathbf{r}$ is already the integral of a differential form; you just need to
take it literally. All of the formulas are in my handout (where Page 6 is strictly time-permitting … which so far it hasn't been).
Suppose I start with a function and take its cojet differential over and over again.
$d f(x) = f'(x) dx$$d^2f(x) = f''(x) dx^2 + f'(x)d^2x$$d^3f(x) = f'''(x) dx^3 + 3 f''(x) dx\cdot d^2x + f'(x) d^3x$$d^4f(x) = f^{(4)}(x) dx^4 + 6 f'''(x) dx^2 d^2x + f''(x)(3(d^2x)^2 + 4 dx \cdot d^
3x) + f'(x) d^4x$$d^5f(x) = f^{(5)}(x) dx^5 + 5 f^{(4)}(x) dx^3 \cdot d^2 x + f'''(x)(15dx\cdot (d^2x)^2 + 10 dx^2 \cdot d^3x) + f''(x) (10 d^2x \cdot d^3x + 5 dx \cdot d^4x) + f'(x) d^5 x$
It appears that each term in $d^n f(x)$ is of the form
$a f^{(k)}(x) d^{i_1}x \cdot d^{i_2}x \cdot \cdots \cdot d^{i_k}x$
for some $k\le n$ and some (unordered) partition $i_1 + i_2 + \cdots i_k = n$. Are the coefficients appearing here some well-known combinatorial numbers associated to partitions?
Over in the other thread, David R posted a link to an MO answer which reminded me to look back at Arnold’s book on classical mechanics, which suggests the following definition of the exterior
differential of a cojet (or perhaps “cogerm” would be more appropriate) 1-form:
$\langle d\wedge \eta {|} S \rangle = \lim_{c\to 0} \frac{1}{{|c|}^2} \oint_{S\circ c} \eta$
where $c$ is a loop inside the parametrized surface $S$ which shrinks to nothing around $(0,0)$. (It might be a rectangle or parallellogram, but from the general perspective that restriction seems
Comparing this to the definition of the differential $d$ from cogerm 1-forms to cogerm 1-forms, and its relationship to the exterior differential acting from 0-forms to 1-forms, suggests the
following operation from cogerm 2-forms to cogerm 2-forms:
$\langle d \omega {|} S \rangle = \lim_{c\to 0} \frac{1}{{|c|}^2} \int_{t=a}^b \langle \omega {|} S_{c(t)} \rangle$
where $c$ is a loop as before, with domain $[a,b]$, and $S_{(u,v)}(s+u,t+v)$ is a shifted version of the surface. Is this a 2-form version of the cogerm differential?
Just throwing stuff out there at the moment, hoping sometime soon I’ll have time to think about it all carefully.
Probably “$|c|^2$” should be instead the area enclosed by $c$. But having thought about it a little more, I realized those limits don’t really make sense unless the integrals are invariant under
reparametrization. So maybe the exterior differential doesn’t really make sense except for degree-1 1-forms? And is there any sort of commutative differential on 2-forms? Would we hope or expect it
to behave in any particular way? It feels weird to me that we have the world of cogerm 1-forms with the commutative $d$, and the world of exterior forms with the exterior $d\wedge$, which agree in
the world of linear degree-1 1-forms and the differential of functions, but are thereafter completely unrelated.
Can an absolute 1-form be regarded as a cojet form like $|d x|$ defined by
$\langle {|\omega|} ; c\rangle = {\Big|\langle \omega ; c\rangle\Big|}?$
I would certainly accept this definition of ${|\omega|}$ in line with the previous discussion of $f(\omega)$ (where $\omega$ is a cojet form, or more generally a finite list of such, and $f$ is a
differentiable function); there's no reason that $f$ has to be differentiable (we just can't conclude that $f(\omega)$ is differentiable).
So I guess that your question is: if $\omega$ is an exterior $1$-form, then is this ${|\omega|}$ the absolute $1$-form called ${|\omega|}$ on the absolute differential form page? And the answer is
Yes; at least, it certainly does the right thing to a curve.
But not every absolute $1$-form arises in this way! Besides multiplying by an arbitrary $0$-form (so that an absolute $1$-form need not be positive semidefinite), even some positive definite forms,
such as $\sqrt{\mathrm{d}x^2 + \mathrm{d}y^2}$, don't arise in this way.
Nevertheless, any absolute $1$-form does have an action on curves (via their tangent vectors, if you follow the definition at absolute differential form), and this is homogeneous of degree $1$, so
your integration formula does integrate them.
It feels weird to me that we have the world of cogerm 1-forms with the commutative $d$, and the world of exterior forms with the exterior $d\wedge$, which agree in the world of linear degree-1
1-forms and the differential of functions, but are thereafter completely unrelated.
There is some more overlap if you look at symmetric bilinear forms (rather than only the antisymmetric ones that are exterior $2$-forms). Some cojet (or cogerm) forms are linear, and these agree with
the exterior $1$-forms; but some cojet forms are quadratic, and these agree with the symmetric bilinear forms. Of course, these are viewed as functions of different things, but they are equivalent by
the polarization identities. An arbitrary bilinear forms is then given by a quadratic cojet form together with an exterior $2$-form.
This doesn't go so easily into higher rank.
I thought it was about time to record some of this discussion, so I created cogerm differential form.
Looks good! I discussed it in a thread dedicated to it. (Mike already noticed this, but I record it for the sake of future generations.)
Re: #87, the sum of the coefficients of terms in $\mathrm{d}^n f$ involving $f^{(k)}(x)$ is the Stirling number of the second kind $S(n,k)$: the number of ways to partition an $n$-element set into
$k$ nonempty subsets. The coefficients themselves are simply the further classification of these partitions according to the multiset of cardinalities of the $k$ nonempty subsets (which feels like it
ought to have something to do with Young tableaux). This is more obvious if we use the coflare differentials where $d_1 d_0 eq d_0 d_1$: then none of the terms can be combined, and each term like $\
mathrm{d}_{2}\mathrm{d}_0 x \, \mathrm{d}_3 \mathrm{d}_1 x$ evidently represents a particular partition of an $n$-element set into $k$ nonempty subsets.
In coflare differentials, I don't think that $\mathrm{d}_0\mathrm{d}_1x$ makes sense at all; in any case, it doesn't show up in $\mathrm{d}^{n}f(x)$. That's just as well, since the Stirling number
doesn't count $\{\{0,1\}\}$ and $\{\{1,0\}\}$ as distinct partitions of $2$ into $1$ nonempty subset.
Yes, that’s true; I think I meant to say something like $d_1d_0 eq d_2d_0$.
Or simply that $\mathrm{d}_0 eq \mathrm{d}_1$. Either will do, since the first nontrivial coefficient comes from combining $\mathrm{d}_2\mathrm{d}_1x \,\mathrm{d}_0x$, $\mathrm{d}_1x \,\mathrm{d}_2\
mathrm{d}_0x$, and $\mathrm{d}_2x \,\mathrm{d}_1\mathrm{d}_0x$, where already for each pair there are two differences between them.
On the subject of partial derivatives, John Denker makes the interesting point that
$\Big(\frac{\partial{u}}{\partial{x}}\Big)_{y,z} = \frac{\mathrm{d}u \wedge \mathrm{d}y \wedge \mathrm{d}z}{\mathrm{d}x \wedge \mathrm{d}y \wedge \mathrm{d}z}$
at http://www.av8n.com/physics/partial-derivative.htm#sec-wedge-ratio. This is easy enough to verify by calculation, but also check out the pictorial explanation.
Trying to make the previous comment work with second derivatives:
Suppose that $u$ is a function of $x$. Then
$\mathrm { d } u = \frac { \partial u } { \partial x } \, \mathrm { d } x ,$
$\frac { \partial u } { \partial x } = \frac { \mathrm { d } u } { \mathrm { d } x } .$
$\frac { \partial ^ 2 u } { \partial x ^ 2 } = \frac { \partial \left ( \frac { \partial u } { \partial x } \right ) } { \partial x } = \frac { \mathrm { d } \left ( \frac { \mathrm { d } u } { \
mathrm { d } x } \right ) } { \mathrm { d } x } ,$
which expands to
$\frac { \partial ^ 2 u } { \partial x ^ 2 } = \frac { \mathrm { d } x \, \mathrm { d } ^ 2 u - \mathrm { d } u \, \mathrm { d } ^ 2 x } { \mathrm { d } x ^ 3 } .$
On the other hand,
$\mathrm { d } ^ 2 u = \frac { \partial ^ 2 u } { \partial x ^ 2 } \, \mathrm { d } x + \frac { \partial u } { \partial x } \, \mathrm { d } ^ 2 x ,$
$\mathrm { d } ^ 2 u \wedge \mathrm { d } ^ 2 x = \frac { \partial ^ 2 u } { \partial x ^ 2 } \, \mathrm { d } x \wedge \mathrm { d } ^ 2 x ,$
$\frac { \partial ^ 2 u } { \partial x ^ 2 } = \frac { \mathrm { d } ^ 2 u \wedge \mathrm { d } ^ 2 x } { \mathrm { d } x \wedge \mathrm { d } ^ 2 x } .$
Now suppose that $u$ is a function of $x$ and $y$. Then
$\mathrm { d } u = \frac { \partial u } { \partial x } \, \mathrm { d } x + \frac { \partial u } { \partial y } \, \mathrm { d } y ,$
$\mathrm { d } u \wedge \mathrm { d } y = \frac { \partial u } { \partial x } \, \mathrm { d } x \wedge \mathrm { d } y ,$
$\frac { \partial u } { \partial x } = \frac { \mathrm { d } u \wedge \mathrm { d } y } { \mathrm { d } x \wedge \mathrm { d } y } .$
$\frac { \partial ^ 2 u } { \partial x ^ 2 } = \frac { \partial \left ( \frac { \partial u } { \partial x } \right ) } { \partial x } = \frac { \mathrm { d } \left ( \frac { \mathrm { d } u \wedge \
mathrm { d } y } { \mathrm { d } x \wedge \mathrm { d } y } \right ) \wedge \mathrm { d } y } { \mathrm { d } x \wedge \mathrm { d } y } ,$
which unfortunately can't be expanded without abandoning the $\wedge$ notation.
On the other hand,
$\mathrm { d } ^ 2 u = \frac { \partial ^ 2 u } { \partial x ^ 2 } \, \mathrm { d } x ^ 2 + 2 \frac { \partial ^ 2 u } { \partial x \partial y } \, \mathrm { d } x \, \mathrm { d } y + \frac { \
partial ^ 2 u } { \partial y ^ 2 } \, \mathrm { d } y ^ 2 + \frac { \partial u } { \partial x } \, \mathrm { d } ^ 2 x + \frac { \partial u } { \partial y } \, \mathrm { d } ^ 2 y ,$
$\mathrm { d } ^ 2 u \wedge \mathrm { d } x \mathrm { d } y \wedge \mathrm { d } y ^ 2 \wedge \mathrm { d } ^ 2 x \wedge \mathrm { d } ^ 2 y = \frac { \partial ^ 2 u } { \partial x ^ 2 } \, \mathrm {
d } x ^ 2 \wedge \mathrm { d } x \mathrm { d } y \wedge \mathrm { d } y ^ 2 \wedge \mathrm { d } ^ 2 x \wedge \mathrm { d } ^ 2 y ,$
$\frac { \partial ^ 2 u } { \partial x ^ 2 } = \frac { \mathrm { d } ^ 2 u \wedge \mathrm { d } x \mathrm { d } y \wedge \mathrm { d } y ^ 2 \wedge \mathrm { d } ^ 2 x \wedge \mathrm { d } ^ 2 y } {
\mathrm { d } x ^ 2 \wedge \mathrm { d } x \mathrm { d } y \wedge \mathrm { d } y ^ 2 \wedge \mathrm { d } ^ 2 x \wedge \mathrm { d } ^ 2 y } .$
Re #87:
The coefficients appearing here are those that appear in Bell polynomials, and they are well known (although not by me, until yesterday) both to come from counting partitions and to give a formula
for the higher derivatives of a composite function, Faà di Bruno's formula. This formula gives the higher cojet differentials of $f(x)$, where $f$ is a real-valued function of a real variable,
differentiable at least $n$ times, and $x$ is a real-valued quantity (technically a real-valued function on some manifold), also differentiable at least $n$ times:
$\mathrm{d}^n\big(f(x)\big) = \sum_\pi f^{({|\pi|})}(x) \prod_{B\in{\pi}} \mathrm{d}^{|B|}x ,$
where the sum is taken over the set of all partitions of $\{1,\ldots,n\}$, each partition $\pi$ being thought of as a subset of the powerset of $\{1,\ldots,n\}$ (so that both $\pi$ and any $B \in \
pi$ have a cardinality given by ${|{\cdot}|}$).
A partly multivariable version of the formula may be adapted to coflare forms. First some notation: if $A = \{i_1,i_2,\ldots,i_n\}$ is a finite multisubset of $\mathbb{N}$, then write $\mathrm{d}^A
{u}$ for $\mathrm{d}_{i_1}\mathrm{d}_{i_2}{\cdots}\mathrm{d}_{i_n}u$ (which is unambiguously defined if $u$ is at least $n$ times differentiable). Also, if $B \subseteq \{1,\ldots,n\}$ (a set, not
any multiset), then let $i_B$ be $\{i_j \;|\; j \in B\}$ (a multiset). With this notation,
$\mathrm{d}^A\big(f(x)\big) = \sum_\pi f^{({|\pi|})}(x) \prod_{B\in{\pi}} \mathrm{d}^{i_B}x ,$
a partial decategorification of the cojet version.
A fully multivariable version of the formula would also allow $f$ to be a function of $m$ variables, with $\mathrm{d}\big(f(x_1,\ldots,x_m)\big) = abla{f}(x_1,\ldots,x_m) \cdot \langle{\mathrm{d}x_1,
\ldots,\mathrm{d}x_m}\rangle = \sum_{j=1}^m \mathrm{D}_j{f}(x_1,\ldots,x_m) \mathrm{d}x_j$ as the order-$1$ case, but I haven't tried to think that through yet.
ETA: You can take $A$ and $i_B$ to be tuples rather than multisets, if you prefer. But the order doesn't matter, just as with partial derivatives. | {"url":"https://nforum.ncatlab.org/discussion/5402/what-is-a-variable/?Focus=43504","timestamp":"2024-11-08T05:50:08Z","content_type":"application/xhtml+xml","content_length":"584734","record_id":"<urn:uuid:a406c22e-2180-46fb-bae8-83c68c69ad39>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00221.warc.gz"} |
Explanation of the EARLIER formula
Dear all,
I have read about the earlier function but It doesnt seen to make sense to me.
If possible could somebody give a clariying explanation with an example?
Many thanks
05-12-2022 01:04 AM
05-30-2020 07:28 AM
10-15-2021 01:26 AM
09-29-2018 09:04 AM
09-29-2018 07:59 AM | {"url":"https://community.fabric.microsoft.com/t5/Desktop/Explanation-of-the-EARLIER-formula/m-p/529469/highlight/true","timestamp":"2024-11-08T00:08:50Z","content_type":"text/html","content_length":"411153","record_id":"<urn:uuid:cf86b63d-c6d0-40c9-9ae8-a24bce4f6c47>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00157.warc.gz"} |
How To Perform A Chi-Square Test Of Independence In Excel
In this tutorial, I will show you step-by-step how to perform a chi-square test of independence by using Microsoft Excel.
Example data
Let’s say I have a sample of 200 people that visited my local pub. From these 200 participants, half were male and half were female.
I asked each participant if they were a smoker or non-smoker. Here are my results:
Smokers Non-smokers
Male 29 71
Female 16 84
So, there were 29 males that smoked, and 71 that didn’t. For the females, there were 16 smokers and 84 non-smokers. Since these are the actual values from my experiment, they are known as the
observed values.
What I want to do is to perform a chi-square test of independence to see if there is an association between gender and smoking status in my sample.
How to perform a chi-square test of independence in Excel
In the first part of this tutorial, I will show you how to manually perform the chi-square test in Excel, including calculating the chi-square statistic and p-value.
In the latter part of the tutorial, I will describe how to use an Excel function (CHISQ.TEST) to calculate the p-value quickly from the observed and expected values.
1. Calculate the row, column and overall totals
The first step to performing a chi-square test is to add up each of the rows and columns in the contingency table by using the SUM function.
=SUM(number1, [number2], ...)
So, to calculate the total in the smokers column, I will use the following formula in a new cell.
So, in total, there were 45 smokers.
I now need to repeat this process for the next column, as well as the rows in my table. Additionally, you need to calculate the overall total from your table.
The image below shows all of the formulas used for my example.
2. Calculate the expected values
Moving on, you next need to work out the expected value for each entry in the table.
To work out the expected value, you must multiply each row total by each column total, and divide that answer by the overall total.
To work out the expected number of male smokers in my example, I will use the following formula.
So, in my example, the expected number of male smokers was 22.5.
Again, this process needs to be repeated for all entries in the contingency table.
3. Calculate the difference between the observed and expected values
The next step is to subtract each of the expected from the observed values, square it, then divide that answer by the expected value.
So, for my example, I will use the following formula for the male smokers.
This process needs to be repeated for the rest of the entries in the table.
4. Calculate the chi-square statistic
Next, we need to calculate the chi-square statistic.
To do this, simply add up all the values that were recently calculated in step 3.
For my example, I will use the following formula.
So, the chi-square statistic for my example was 4.85, when rounded.
5. Calculate the degrees of freedom
Next, we need to calculate the degrees of freedom.
Here, the degrees of freedom is calculated by firstly subtracting 1 from the number of rows in the test, and multiply this answer by the number of columns in the test subtract 1.
So, for my example, I have 2 rows and 2 columns. This means to work out my degrees of freedom, I use the following calculation (you can just perform this manually, as it’s very simple math).
(2-1) x (2-1)
Which gives an answer of 1. So, this example has a degrees of freedom of 1.
6. Calculate the p-value
The final step in performing the chi-square test is to take the chi-square statistic and degrees of freedom values, and work out the p-value.
To do this in Excel, you can use the CHISQ.DIST.RT function.
=CHISQ.DIST.RT(x, deg_freedom)
• x – The cell containing the chi-square value
• deg_freedom – The cell containing the degrees of freedom value
In my example, I get a p-value of 0.028, when rounded.
7. [Optional] Use the CHISQ.TEST function to calculate the p-value
There is a function you can use to calculate the chi-square p-value by just using the observed and expected table values.
To do this, use the CHISQ.TEST function.
=CHISQ.TEST(actual_range, expected_range)
• Actual range – The cells containing the observed values
• Expected range – The cells containing the expected values
Interpreting the results
To interpret the p-value, you need to state the my two hypotheses (null and alternative).
Here are my hypotheses for my example.
• Null hypothesis – There is no association between gender and smoking status
• Alternative hypothesis – There is an association between gender and smoking status
If my alpha level, or significance threshold, was set at 0.05, this would mean I will fail to reject the null hypothesis if p>0.05. On the other hand, if p<0.05, I will reject the null hypothesis,
and accept the alternative hypothesis.
In this case, my p-value was 0.028. Since this was less than 0.05, I will reject the null hypothesis, and accept the alternative hypothesis.
Therefore, there does seem to be an association between gender and smoking status.
How to perform a chi-square test of independence in Excel: Final words
After following this guide, you should now know how to perform a chi-square test of independence by using Microsoft Excel.
The steps described will calculate the chi-square statistic, degrees of freedom and the p-value. Alternatively, for a quick and easy way of generating a p-value, simply use the CHISQ.TEST function.
Microsoft Excel version used: 365 ProPlus
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The quantities package provides integration of the ‘units’ and ‘errors’ packages for a complete quantity calculus system for R vectors, matrices and arrays, with automatic propagation, conversion,
derivation and simplification of magnitudes and uncertainties.
Blog posts:
• Edzer Pebesma, Thomas Mailund and James Hiebert (2016). “Measurement Units in R.” The R Journal, 8 (2), 486–494. DOI: 10.32614/RJ-2016-061
• Iñaki Ucar, Edzer Pebesma and Arturo Azcorra (2018). “Measurement Errors in R.” The R Journal, 10 (2), 549–557. DOI: 10.32614/RJ-2018-075
Install the release version from CRAN:
The installation from GitHub requires the remotes package.
# install.packages("remotes")
remotes::install_github(paste("r-quantities", c("units", "errors", "quantities"), sep="/"))
This project gratefully acknowledges financial support from the | {"url":"https://cloud.r-project.org/web/packages/quantities/readme/README.html","timestamp":"2024-11-05T17:00:50Z","content_type":"application/xhtml+xml","content_length":"8498","record_id":"<urn:uuid:34239a11-e384-4357-b992-4f0dbc671d73>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00797.warc.gz"} |
C Program to calculate the power without using POW function - Quescol
C Program to calculate the power without using POW function
In this tutorial, we are going to learn a writing program in C to calculate the power of a given number without using the POW method. Here we will write our own logic to calculate the power of a
given number.
To calculate the power we need two numbers. One is the base number and another is an exponent which is known as power. The base is the number for which we are calculating power and power is the
number that tells what power we have to calculate.
Let’s see this with an example
We have base number 4 and the exponent is 3.
It means we have to calculate the power of 4, 3 times.
We can write this as 4^3 = 4*4*4 = 64.
How our program will behave?
• Our program will take two integers as an input. One is base and another is power.
• Now to calcualte the power without using any existing library method, we will multiple base number to itself for power times.
• Calculation of power means calculating same number to itself for the times given in power.
• Suppose in input base is given as 6 and power is given as 3. We will calculate 6 to itself for 3 times like 6 * 6 * 6.
• After calculating the power our program will return the result as an output.
Program in C to Calculate power without using pow function
#include <stdio.h>
int main() {
int base, exp , result=1;
printf("Enter a value of base: ");
scanf("%d", &base);
printf("Enter a value of exponent: ");
scanf("%d", &exp);
printf("%d to the power %d is = ", base, exp);
while (exp != 0) {
result = base * result;
return 0;
Enter a value of base: 4
Enter a value of exponent: 3
4 to the power 3 is = 64 | {"url":"https://quescol.com/interview-preparation/program-calculate-power-without-using-pow","timestamp":"2024-11-13T11:53:39Z","content_type":"text/html","content_length":"85464","record_id":"<urn:uuid:811b2572-333e-4762-bce3-1fbc7a13368e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00180.warc.gz"} |
Advanced Topics
The following topics show advanced features of the Boost Compute library.
In addition to the built-in scalar types (e.g. int and float), OpenCL also provides vector data types (e.g. int2 and vector4). These can be used with the Boost Compute library on both the host and
Boost.Compute provides typedefs for these types which take the form: boost::compute::scalarN_ where scalar is a scalar data type (e.g. int, float, char) and N is the size of the vector. Supported
vector sizes are: 2, 4, 8, and 16.
The following example shows how to transfer a set of 3D points stored as an array of floats on the host the device and then calculate the sum of the point coordinates using the accumulate() function.
The sum is transferred to the host and the centroid computed by dividing by the total number of points.
Note that even though the points are in 3D, they are stored as float4 due to OpenCL's alignment requirements.
#include <iostream>
#include <boost/compute/algorithm/copy.hpp>
#include <boost/compute/algorithm/accumulate.hpp>
#include <boost/compute/container/vector.hpp>
#include <boost/compute/types/fundamental.hpp>
namespace compute = boost::compute;
// the point centroid example calculates and displays the
// centroid of a set of 3D points stored as float4's
int main()
using compute::float4_;
// get default device and setup context
compute::device device = compute::system::default_device();
compute::context context(device);
compute::command_queue queue(context, device);
// point coordinates
float points[] = { 1.0f, 2.0f, 3.0f, 0.0f,
-2.0f, -3.0f, 4.0f, 0.0f,
1.0f, -2.0f, 2.5f, 0.0f,
-7.0f, -3.0f, -2.0f, 0.0f,
3.0f, 4.0f, -5.0f, 0.0f };
// create vector for five points
compute::vector<float4_> vector(5, context);
// copy point data to the device
reinterpret_cast<float4_ *>(points),
reinterpret_cast<float4_ *>(points) + 5,
// calculate sum
float4_ sum = compute::accumulate(
vector.begin(), vector.end(), float4_(0, 0, 0, 0), queue
// calculate centroid
float4_ centroid;
for(size_t i = 0; i < 3; i++){
centroid[i] = sum[i] / 5.0f;
// print centroid
std::cout << "centroid: " << centroid << std::endl;
return 0;
The OpenCL runtime and the Boost Compute library provide a number of built-in functions such as sqrt() and dot() but many times these are not sufficient for solving the problem at hand.
The Boost Compute library provides a few different ways to create custom functions that can be passed to the provided algorithms such as transform() and reduce().
The most basic method is to provide the raw source code for a function:
boost::compute::function<int (int)> add_four =
boost::compute::make_function_from_source<int (int)>(
"int add_four(int x) { return x + 4; }"
boost::compute::transform(input.begin(), input.end(), output.begin(), add_four, queue);
This can also be done more succinctly using the BOOST_COMPUTE_FUNCTION macro:
BOOST_COMPUTE_FUNCTION(int, add_four, (int x),
return x + 4;
boost::compute::transform(input.begin(), input.end(), output.begin(), add_four, queue);
Also see "Custom OpenCL functions in C++ with Boost.Compute" for more details.
Boost.Compute provides the BOOST_COMPUTE_ADAPT_STRUCT macro which allows a C++ struct/class to be wrapped and used in OpenCL.
While OpenCL itself doesn't natively support complex data types, the Boost Compute library provides them.
To use complex values first include the following header:
#include <boost/compute/types/complex.hpp>
A vector of complex values can be created like so:
// create vector on device
boost::compute::vector<std::complex<float> > vector;
// insert two complex values
vector.push_back(std::complex<float>(1.0f, 3.0f));
vector.push_back(std::complex<float>(2.0f, 4.0f));
The lambda expression framework allows for functions and predicates to be defined at the call-site of an algorithm.
Lambda expressions use the placeholders _1 and _2 to indicate the arguments. The following declarations will bring the lambda placeholders into the current scope:
using boost::compute::lambda::_1;
using boost::compute::lambda::_2;
The following examples show how to use lambda expressions along with the Boost.Compute algorithms to perform more complex operations on the device.
To count the number of odd values in a vector:
boost::compute::count_if(vector.begin(), vector.end(), _1 % 2 == 1, queue);
To multiply each value in a vector by three and subtract four:
boost::compute::transform(vector.begin(), vector.end(), vector.begin(), _1 * 3 - 4, queue);
Lambda expressions can also be used to create function<> objects:
boost::compute::function<int(int)> add_four = _1 + 4;
A major performance bottleneck in GPGPU applications is memory transfer. This can be alleviated by overlapping memory transfer with computation. The Boost Compute library provides the copy_async()
function which performs an asynchronous memory transfers between the host and the device.
For example, to initiate a copy from the host to the device and then perform other actions:
// data on the host
std::vector<float> host_vector = ...
// create a vector on the device
boost::compute::vector<float> device_vector(host_vector.size(), context);
// copy data to the device asynchronously
boost::compute::future<void> f = boost::compute::copy_async(
host_vector.begin(), host_vector.end(), device_vector.begin(), queue
// perform other work on the host or device
// ...
// ensure the copy is completed
// use data on the device (e.g. sort)
boost::compute::sort(device_vector.begin(), device_vector.end(), queue);
For example, to measure the time to copy a vector of data from the host to the device:
#include <vector>
#include <cstdlib>
#include <iostream>
#include <boost/compute/event.hpp>
#include <boost/compute/system.hpp>
#include <boost/compute/algorithm/copy.hpp>
#include <boost/compute/async/future.hpp>
#include <boost/compute/container/vector.hpp>
namespace compute = boost::compute;
int main()
// get the default device
compute::device gpu = compute::system::default_device();
// create context for default device
compute::context context(gpu);
// create command queue with profiling enabled
compute::command_queue queue(
context, gpu, compute::command_queue::enable_profiling
// generate random data on the host
std::vector<int> host_vector(16000000);
std::generate(host_vector.begin(), host_vector.end(), rand);
// create a vector on the device
compute::vector<int> device_vector(host_vector.size(), context);
// copy data from the host to the device
compute::future<void> future = compute::copy_async(
host_vector.begin(), host_vector.end(), device_vector.begin(), queue
// wait for copy to finish
// get elapsed time from event profiling information
boost::chrono::milliseconds duration =
// print elapsed time in milliseconds
std::cout << "time: " << duration.count() << " ms" << std::endl;
return 0;
The Boost Compute library is designed to easily interoperate with the OpenCL API. All of the wrapped classes have conversion operators to their underlying OpenCL types which allows them to be passed
directly to the OpenCL functions.
For example,
// create context object
boost::compute::context ctx = boost::compute::default_context();
// query number of devices using the OpenCL API
cl_uint num_devices;
clGetContextInfo(ctx, CL_CONTEXT_NUM_DEVICES, sizeof(cl_uint), &num_devices, 0);
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How do you simplify (12n^2+15n)/(3n)? | Socratic
How do you simplify #(12n^2+15n)/(3n)#?
2 Answers
We have the function $\frac{12 {n}^{2} + 15 n}{3 n}$
Taking a good look at the numerator, we realize that we can factor $3 n$ out:
$\frac{3 n \left(4 n + 5\right)}{3 n}$
We have a $3 n$ on top and on the bottom, so this becomes:
$4 n + 5$, the answer!
You have to find the factors of all of the terms, then simplify as needed.
You start off with this:
$\frac{12 {n}^{2} + 15 n}{3 n}$
Then, you take out the common factors:
$\frac{3 n \left(4 n + 5\right)}{3 n}$
Finally, you simplify by cancelling the $3 n$ from the numerator and the denominator:
$4 n + 5$
And you're done!
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Alphabetical Order Worksheet
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Alphabetical Letters
As a kindergarten teacher, you know how important it is to teach your students about alphabetical order. One of the most effective ways to do this is by using interactive worksheets that engage your
students and help them develop their letter recognition skills. One such worksheet is the Alphabetical Letters worksheet, which is designed specifically for kindergarten students.
The Alphabetical Letters worksheet is a fun and engaging way for students to practice their knowledge of the alphabet and their ability to identify missing letters in a sequence. To use the
worksheet, students simply look at a row of letters and fill in the missing letters at the end of the row to complete the alphabet.
This worksheet is perfect for use during grammar lessons when focusing on alphabetical order. It provides a fun and interactive way for students to practice their letter recognition skills and their
ability to identify letters in a sequence.
To download the Alphabetical Letters worksheet, simply click on the link below. This worksheet is completely free and can be downloaded and printed as many times as you need.
When using this worksheet with your students, it's important to make the activity interactive and engaging. You can encourage your students to say the letters out loud as they fill in the missing
letters, or you can ask them to point to each letter as they say it. This will help to reinforce their letter recognition skills and their ability to remember the order of the alphabet.
Alphabetical Letters is a fantastic tool for teaching kindergarten students about alphabetical order. It's fun, engaging, and provides an effective way to help students develop important cognitive
skills. So why not give it a try in your next grammar lesson?
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bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis - Tech Infobicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis - Tech Info
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bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius: An In-Depth Analysis
When we consider a one-bicycle wheel circle radius nyt wheel circle radius nyt of the maximum important additives that come to thoughts are its wheels. The wheel is an elaborate piece of engineering
that performs a vital position in the overall overall performance and efficiency of the bicycle. An essential concept in information about wheels is the radius of the circle they form. This article
delves into the significance of the wheel circle radius nytel’s radius, its impact on the bicycle’s dynamics, and its broader implications.
The Geometry of bicycle wheel circle radius nytWheels
A bicycle wheel is a circle, and its length can be defined by way of its radius. For general bicycles, wheel sizes are regularly specified in phrases of diameter, but the radius is 1/2 of the
diameter. For instance, a common wheel length for road motorcycles is 700c, which corresponds to a diameter of about 622mm. Thus, the radius is 311mm.
Understanding the radius is important because it directly influences the wheel’s circumference, which determines how ways the bicycle wheel circle radius nyt travels with each rotation of the wheel.
Formula of circumference:
[ C = 2pi r ]
in which ( r ) is the radius. For a wheel with a radius of 311mm, the circumference is set at 1,954mm (or 1.954 meters).
Impact on bicycle wheel circle radius nytDynamics
1. Speed and Efficiency:
The radius of the wheel affects the bicycle’s speed and performance. Larger wheels (extra radius) cover the extra floor in keeping with revolution, which can translate to higher speeds and smoother
rides. This is why road motorcycles frequently have large wheels as compared to mountain bikes. The large radius reduces rolling resistance, making it easier to keep velocity on clean surfaces.
2. Handling and Stability:
The wheel radius also impacts the motorcycle’s handling and balance. Larger wheels offer better balance and may roll over limitations greater without difficulty, which is fine on tough terrain.
However, they can also make the bicycle wheel circle radius nyt less maneuverable. Conversely, smaller wheels enhance maneuverability and acceleration however might not cope with rough terrain as
3. Acceleration and Torque:
Smaller wheels, having a smaller radius, require much less torque to begin transferring, this means that they can boost up faster. This is useful for urban commuting or using stop-and-pass visitors.
However, large wheels preserve better speeds greater efficiently once in motion, reaping benefits for lengthy-distance cyclists.
Real-World Applications
Road Bikes
Road motorcycles, designed for velocity and long-distance travel on paved surfaces, generally feature larger wheels with a radius of around 311mm (700c). The larger radius allows for greater
performance and velocity, making them perfect for racing and long rides. The decreased rolling resistance and capacity to preserve momentum are key blessings.
Mountain Bikes
Mountain bikes, on the other hand, frequently use wheels with a radius of around 279mm (29 inches) or even smaller for some models. The barely smaller radius improves maneuverability and manages on
rugged terrain, that is essential for navigating trails and limitations. The stability between stability and agility is critical for mountain biking.
Urban and Folding Bikes
Urban and folding motorcycles commonly have even smaller wheels, with radii starting from 203mm (sixteen inches) to 254mm (20 inches). These motorcycles prioritize compactness and ease of
acceleration, which can be vital for metropolis commuting and garages. The smaller radius makes these bikes extraordinarily maneuverable in tight spaces and brief to respond to site visitors.
The Physics Behind the Ride
The physics of a bicycle wheel circle radius nyt wheel’s radius involves several key standards:
1. Moment of Inertia:
The second of inertia is the resistance of an item to changes in its rotational movement. For bicycle wheel circle radius nyt wheels, a bigger radius will increase the moment of inertia, making it
tougher to accelerate but simpler to maintain pace. Conversely, smaller wheels have a decreased second of inertia, making them quicker to begin but tougher to sustain high speeds.
2. Gyroscopic Effect:
The gyroscopic impact refers to the steadiness furnished through a rotating wheel. Larger wheels, with their extra mass and radius, have a stronger gyroscopic effect, contributing to the bicycle’s
balance at better speeds. This effect is much less pronounced in smaller wheels, which is why they’re regularly used in situations where agility is greater important than stability.
3. Rolling Resistance:
Rolling resistance is the force resisting the movement of the wheel rolling on a floor. Larger wheels generally have decreased rolling resistance due to the reduced deformation of the tire as it
contacts the floor. This is a vast factor within the performance of avenue motorcycles, in which keeping excessive speeds with minimal attempt is vital.
Technological Innovations
Advancements in the bicycle wheel circle radius nyt era have brought about innovations that optimize the blessings of different wheel radii. For example, tubeless tires, remove the inner tube, reduce
rolling resistance and improve trip nice, especially on large wheels. Carbon fiber rims and spokes reduce weight without compromising strength, reaping rewards for each big and small wheel via
enhancing acceleration and handling.
Furthermore, the improvement of hybrid bikes, which combine features of avenue and mountain motorcycles, often includes wheels with intermediate radii. These motorcycles purpose to offer stability
among velocity, balance, and maneuverability, making them versatile for diverse riding situations.
The radius of a bicycle wheel circle radius nyt wheel is a fundamental characteristic that influences the bicycle’s performance, handling, and suitability for specific riding conditions. From the
speed and performance of road motorcycles with larger radii to the agility and manipulation of urban motorcycles with smaller radii, the selection of wheel size is an essential consideration for
cyclists. Understanding the impact of wheel radius on dynamics and performance helps riders make knowledgeable selections primarily based on their specific desires and using environments.
As the generation continues to conform, we can assume further innovations to refine and beautify the abilities of bicycle wheel circle radius nyt wheels, ensuring that cyclists of all sorts can
experience the most suitable overall performance and better driving enjoyment. Whether you are an expert racer, a mountain trail enthusiast, or an everyday commuter, the right wheel radius could make
all of the difference in your cycling journey.
bicycle wheel circle radius nyt wheel circle radius nyt wheel Circle Radius
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SVR Calculator – Accurate SVR Calculation Tool
This SVR calculator tool helps you quickly calculate the systemic vascular resistance.
How to Use the SVR Calculator
This calculator allows you to compute electrical parameters such as current, reactance, impedance, and power factor based on the apparent power, voltage, and resistance inputs.
1. Enter the Apparent Power (S) in VA.
2. Enter the Voltage (V) in volts.
3. Enter the Resistance (R) in ohms.
4. Click the “Calculate” button to get the results.
The results will display the current (in amperes), reactance (in ohms), impedance (in ohms), and power factor (unitless) of your electrical circuit.
Explanation of Calculations:
• Current (I): Determined by the formula I = S / V.
• Reactance (X): Calculated using X = √((V^2 / S) – R^2).
• Impedance (Z): Given by Z = √(R^2 + X^2).
• Power Factor (PF): Computed as PF = R / Z.
Please ensure all values are positive numbers. The calculator assumes a single-phase AC circuit and does not account for any complex power scenarios or phase angles.
Use Cases for This Calculator
Calculating Simple Interest
Enter the principal amount, interest rate, and time period to quickly calculate the simple interest earned on your investment. This feature allows you to make informed financial decisions with ease.
Estimating Compound Interest
By entering the principal amount, interest rate, compounding frequency, and time period, you can accurately estimate the compound interest accrued over time. This tool helps you visualize how your
investment can grow exponentially.
Converting Currencies
Convert one currency to another by entering the amount and selecting the currencies involved. This handy feature provides real-time exchange rates for seamless currency conversion calculations.
Calculating Loan EMI
Determine your monthly loan repayment amount by providing the principal amount, interest rate, and loan tenure. This calculator helps you plan your finances and budget effectively by knowing your
exact EMI.
Calculating Future Value
Enter the present value, interest rate, time period, and compounding frequency to compute the future value of your investment. This feature empowers you to project your investment growth accurately.
Estimating Return on Investment (ROI)
Calculate the return on investment by entering the initial investment, final value, and time period. This tool enables you to assess the profitability of your investments conveniently.
Calculating Discount Amount
Determine the discount amount on a product by entering the original price and discount percentage offered. This calculator simplifies discount calculations for hassle-free shopping decisions.
Estimating Tax Savings
Enter your taxable income and applicable tax deductions to estimate your tax savings. This tool provides valuable insights into potential tax benefits and helps you optimize your finances.
Calculating Body Mass Index (BMI)
Input your weight and height to instantly calculate your BMI, enabling you to monitor your health and fitness goals effectively. This feature promotes a healthier lifestyle by providing valuable
health insights.
Converting Units of Measurement
Convert between different units of measurement such as length, weight, volume, and temperature with ease. This versatile converter simplifies complex unit conversions for everyday calculations. | {"url":"https://madecalculators.com/svr-calculator/","timestamp":"2024-11-06T21:43:24Z","content_type":"text/html","content_length":"145165","record_id":"<urn:uuid:1c148ee5-84bf-4925-b5ba-051f50b73084>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00619.warc.gz"} |
Hopper Capacity Calculation in Pharmaceutical Industry
The capacity calculation of a compression machine hopper depends on several factors, including the type of material being compressed, the desired compression rate, and the design of the hopper
Here are the general steps you can follow to calculate the capacity of a compression machine hopper:
Determine the Material Properties:
You need to know the characteristics of the material you'll be compressing. This includes the material's bulk density, particle size distribution, flow properties (e.g., angle of repose), and any
other relevant properties.
Calculate Hopper Volume:
The hopper volume should be large enough to hold the material you'll be processing for a certain period of time. The volume of the hopper can be calculated using different formula depending on hopper
shape. Supplier also share the volume of hopper.
Calculate Hopper Capacity
Hopper Capacity = Volume of hopper (liter) × Bulk density of materials
Account for Safety and Operational Considerations:
You may need to increase the hopper's capacity to account for factors like irregular material flow, fluctuations in material density, and safety margins. It's common to add a buffer or surge capacity
to ensure smooth machine operation.
Consider Hopper Design:
The shape and design of the hopper can affect its capacity. Conical hoppers, for example, can encourage material flow, while square or rectangular hoppers may have different flow characteristics. The
hopper's design should match the material's properties and flow requirements.
Evaluate Material Flow:
Analyze how the material flows into the compression machine. Proper hopper design and flow aids (e.g., vibrators, air blasters) can help ensure consistent material flow.
Monitor and Adjust:
After installation and operation, regularly monitor the hopper's performance and make adjustments as necessary to optimize capacity and efficiency.
It's important to note that the actual capacity of the hopper may vary in practice due to factors like material compaction, equipment efficiency, and maintenance. Therefore, it's advisable to consult
with a mechanical engineer or a pharmaceutical specialist in bulk material handling systems to ensure that your hopper design and capacity calculations are accurate and suitable for your specific
For example,
Suppose we have a product with bulk density 0.6 g/ml & our hopper capacity is 18 liters. Use following formula to calculate hopper capacity in kilograms for the product having bulk density 0.6 g/ml
Volume of hopper (liter) × Bulk density of materials
Add values
18 liter × 0.6 g/ml = 10.8 kg
The answer is 10.8 kg.
It Means we can add 10.8 kg of that material whose bulk density will be 0.6 g/ml
Read also:
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NETWORKDAYS function | Community
I struggle to understand how NETWORKDAYS function can be used. I read the help page for the function but in vain.
I want to get number of working days in Jan 2023 (ignoring holidays).
My formulas is:
NETWORKDAYS(date(2023,1,1),date(2023,1,31),'01. Day of week'.Working = true)
where Day of week is a dimension I created (below).
I am getting 150 as the result of the formula which is obviously wrong. Help please :) | {"url":"https://community.pigment.com/questions-conversations-40/networkdays-function-1629?postid=4446","timestamp":"2024-11-10T14:18:49Z","content_type":"text/html","content_length":"210026","record_id":"<urn:uuid:774f1c11-43c1-47c8-aa3f-1ce18c16eca5>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00661.warc.gz"} |
What is 204 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
If you’re wondering what 204 degrees Celsius is in Fahrenheit, you’ve come to the right place. 204 degrees Celsius is equal to 399.2 degrees Fahrenheit. Now, let’s take a closer look at the process
of converting Celsius to Fahrenheit.
The Celsius scale is widely used in the scientific and international communities, while the Fahrenheit scale is primarily used in the United States and a few other countries. When it comes to
converting between the two scales, the process is relatively straightforward, but it does involve a simple mathematical formula.
The formula to convert Celsius to Fahrenheit is as follows: (°C × 9/5) + 32 = °F. In other words, you multiply the temperature in Celsius by 9/5 and then add 32 to the result. This will give you the
temperature in Fahrenheit.
So, applying this formula to 204 degrees Celsius, we get (204 × 9/5) + 32 = 399.2 degrees Fahrenheit. This means that 204 degrees Celsius is equivalent to 399.2 degrees Fahrenheit.
Understanding the relationship between Celsius and Fahrenheit is important, especially if you are traveling to a country that uses a different temperature scale than what you are used to. It’s also
crucial for scientific experiments, cooking, and various other applications where precise temperature measurements are necessary.
In conclusion, 204 degrees Celsius is equal to 399.2 degrees Fahrenheit. Converting between the two temperature scales is easy once you understand the simple formula and the relationship between the
two scales. Whether you’re a student, a traveler, or simply curious about temperature conversions, knowing how to convert Celsius to Fahrenheit (and vice versa) is a valuable skill. | {"url":"https://converttemperatureintocelsius.info/what-is-204celsius-in-fahrenheit/","timestamp":"2024-11-05T22:55:30Z","content_type":"text/html","content_length":"72007","record_id":"<urn:uuid:525b30c6-a22a-4cd4-a37f-b18e2533cb58>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00144.warc.gz"} |
parafac2_to_unfolded(parafac2_tensor, mode)[source]
Construct an unfolded tensor from a PARAFAC2 decomposition. Uneven slices are padded by zeros.
The decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]
where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I \times R\) factor matrix \(A\), \(B_i\) is a \(J_i \times R\) factor matrix such that the
cross product matrix \(B_{i_1}^T B_{i_1}\) is constant for all \(i\), and \(C\) is a \(K \times R\) factor matrix. To compute this decomposition, we reformulate the expression for \(B_i\) such
\[B_i = P_i B,\]
where \(P_i\) is a \(J_i \times R\) orthogonal matrix and \(B\) is a \(R \times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = \sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]
with the same constraints hold for \(B_i\) as above.
parafac2_tensorParafac2Tensor - (weight, factors, projection_matrices)
weights1D array of shape (rank, )
weights of the factors
factorsList of factors of the PARAFAC2 decomposition
Contains the matrices \(A\), \(B\) and \(C\) described above
projection_matricesList of projection matrices used to create evolving
Full constructed tensor. Uneven slices are padded with zeros. | {"url":"http://tensorly.org/stable/modules/generated/tensorly.parafac2_tensor.parafac2_to_unfolded.html","timestamp":"2024-11-12T16:49:56Z","content_type":"text/html","content_length":"19865","record_id":"<urn:uuid:5c8fc43f-816f-4ad4-ba5c-cc88c160ebdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00667.warc.gz"} |
A Coding Theory Bound and Zero-Sum Square Matrices
For a code C = C(n, M) the level k code of C, denoted C[k], is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive
integer divisible by 8, and n = γk, M = βk ≥ 2k then there is a codeword in C [k] whose weight is either 0 or at most n/2 - n(1/8γ - 6/(4β-2)^2) + 1. In particular, if γ < (4β - 2)^2/48 then there is
a codeword in C[k] whose weight is n/2 - Θ(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k, p) denote the minimum integer
guaranteeing that in any square matrix over Z[p], of order f(k, p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f
(k, 2)/k < 3.836. For general p we obtain, using a different approach, that f(k, p) ≤ p^(k/ln k) (1+o[k](1)).
ASJC Scopus subject areas
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'A Coding Theory Bound and Zero-Sum Square Matrices'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/a-coding-theory-bound-and-zero-sum-square-matrices","timestamp":"2024-11-11T12:40:51Z","content_type":"text/html","content_length":"51649","record_id":"<urn:uuid:74d6d517-ca02-4259-9f96-aa4f75402a81>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00883.warc.gz"} |
[HELP]Why every explanation about quicksort is somehow different?
I'm trying to implement a basic quicksort algorithm, I've understood the basic concept behind quicksort and now I'm looking online for a step by step guide.
I've followed through about 7/8 guides and what I've found is that every guide has different ways of using the pivot and different ways of using the two pointers/counters.
Some follow these steps for the pivot:
1.Decide a pivot
2.Swap the pivot with the last element of the array
3.Search the element from the left that is greater than pivot and element that is smaller than pivot from right
4.Swap those elements
Others do this:
1.Choose a pivot
2.Put left pointer to 0, a right pointer to the last element of the array
3.Start comparing left pointer to pivot, if left is smaller than pivot move pointer forward
Let's just say that before reading the guides I think I knew more about the quicksort algorithm.
What guide do you recommend?
Edit: there's some kind of problem with text formatting, I'm sorry
Top comments (4)
mattother •
So to be honest, if you're struggling with the implementation then you probably don't understand Quicksort. I could be wrong, but I'm going to give you some resources here. I'll also answer your
question directly. However, for me personally I self learned everything I know about Algorithms and it was not an easy process. And a lot of what made is difficult was not having a correct foundation
and starting in the wrong place.
I get the feeling that what's happening to you. If this isn't the case I apologize, but I figure in the worst case, the resources (and most of this reply) can just be ignored. But having a good
understanding of the class of Algorithm Quicksort fall under will probably help a lot.
There's a lot of background knowledge that if you don't have a strong foundation in will make it really frustrating to start out with Algorithms. I'm sure people have gotten by without, so take this
for what it is, which is just a set of personal recommendations, but hopefully this will save you some time in the long run.
So first thing that will make Algorithms a hell of a lot easier to understand is a good foundation in Math. Understanding the Mathematical process and Discrete Mathematics in particular. If you
already do great, if not, then these are resources that helped me out a lot.
What I would personally recommend being most comfortable with is how Mathematical proof works, especially induction. Recursion is such a core aspect of programming and induction is the huge backbone
that most proofs surrounding it rely on.
Understanding these principles makes it a lot of easier to grasp Algorithm correctness which plays a key role usually in their time costs.
In general I've personally always found this an incredible tool to have for programming. One key aspect that I've never really heard mentioned outside of Math + Algorithms is invariants.
Understanding how invariants allow you to guarantee outcomes of recursive Algorithms is incredibly helpful. And applying it in an inductive way makes it much easier for writing Algorithms, basically
induction from n to 0.
f(n) -> f(n-1)
f(0) = [is good]
By proving that an invariant holds across an iteration and results in the desired n-1 state combined with a proven terminating state you've basically shown your Algorithms will complete as expected.
Obviously there always bugs and things you'll run into when programming, but I've found applying this principle can really reduce some of the debugging headaches when first starting out with
It's also useful as a primer for property based testing, which can be very useful for confirming Algorithms. Example for Quicksort a key testing invariant
is that next value is greater than of equal to the current value.
values = [1, 2, 3, 5, 4]
for i, _ in enumerate(values[:-1]):
assert values[i] <= values[i+1]
In the case above the assertion would fail because 5 should be after 4.
Anyways now that I've made an argument for Math here's a bunch of resources:
Personally I recommend the following:
Introduction to Mathematical Thinking (Dr. Keith Devlin) (Coursera)
If you've never really studied formal mathematics this is a great primer and a very gentle introduction. However if you're already comfortable with Mathematical notation, rational vs real, etc. then
this might be a bit too elementary.
Mathematics for Computer Science (MIT Open Courseware)
Great primer on mathematics needed for Algorithms, highly recommend this course. Biggest downside, is it's not a formal course so testing yourself can be a bit tricky.
Introduction to proof in abstract mathematics
This book really helped fill in some practical gaps for creating proofs that I felt I hadn't fully grasped via other resources. So if you find you're still really struggling to create proofs this
might be a good book to reference.
Concrete Mathematics / Discrete Mathematics with Application
Both are just really good resources to have available if you're trying to self-learn Math.
There's other higher level aspects of Algorithms that personally I find really help in understanding a Algorithms better. The class of Algorithms gives you a huge hint into what underlying principle
allows it to work.
In the case of Quicksort knowing that it's a Divide and Conquer Algorithms already tells a lot of what you need to know about the Algorithm. It's also the key that really makes the Algorithm work.
In the case you outlined above for Quicksort there are all kinds of methods for choosing a pivot. But the pivot choices isn't really the key ingredient that makes a Quicksort a Quicksort Algorithms,
it's a detail. And once you grasp that fundamental aspect of a Quicksort it will make it a lot easier to understand each "flavor" of it in turn.
So here's the resources I would recommend:
Python Algorithms: Mastering Basic Algorithms in the Python Language
This is an amazing book for learning Algorithms. I would highly recommend it. It goes beyond implementation details and explains the fundamental principle behind certain Algorithms. I don't remember
if it covers Quicksort explicitly but either way a really good resource for learning Algorithms.
Algorithms Specialization (Coursera, Standford)
Good primer on Algorithms, with the added benfit of weekly excerises etc. to test yourself.
Introduction to Algorithms (MIT Open Courseware)
This tackles a lot of the same things as the Algorithms Specialization above. But last time I checked there's not material for testing yourself, etc. So it's a much more DIY approach.
The Algorithm Design Manual
This one is a classic and worth having around for reference. It has great explanations and tons of example code on Algorithms, but less introductory than some of the other resources above.
Design and Analysis of Algorithms (MIT Open Courseware)
This is more of an intermediate level course, but might still be useful.
Back to the question
So to answer the top question (and as I tackled in Algorithms section) they are different because there are a bunch of different ways you can implement Quicksort, usually it's just the method for
selecting a pivot that changes. The Algorithm Design Manual has a good primer on Quicksort, so that's probably a good place to start.
But the essence of what makes Quicksort work is the Divide and conquer nature of it.
What to learn
Obviously there's a lot of resources here so here's the course of action I would probably take, but obviously feel free to take or not take whatever you want from this reply.
A good starting point is the book Python Algorithms. I've come back to this book again and again, it's really a great resource for learning Algorithms.
If you find yourself struggling with it. I would probably try out the Algorithm Specialization courses next, having a concrete assignment each week that can be verified can really go a long way and
you get the added benefit of being able to seek help from people grappling the same content as you.
If you find you can't grasp the Mathematical principles in the course or just feel like you'd like to fill in more details, then I would turn to Mathematics for Computer Science. If you're lost in
that, then start with Introduction to Mathematical Thinking.
And if you feel like you know all this and it really is just implementation details tripping you up then I'd recommend the Algorithm Design Manual. In general it's just a great resource to have
The other book you could try is Algorithms by Robert Sedgewick. He has a lot of great Algorithm books and videos that could be worth checking out.
Anyways, so that's my list, hopefully it helps and if not I apologize. Again these are just resources that have helped me so I would say find what works for you, but hopefully this give you some
stepping stones.
IMRC21 •
Yeah, I think this defines perfectly my struggle with algorithms, I'm currently in my second year at university but couldn't pass some of the math courses and I'm having issues following the online
lectures with all of this covid stuff.
At this point, I'm asking myself if it makes sense with continuing studying CS, I feel like I could even implement a quicksort (and everything else) without having the mathematical concepts but that
would just make me a code monkey and I don't know if I want that.
I've watched the introduction of the course "introduction to mathematical thinking", looks like it's a really interesting course but I don't think I'll have the time.
Thank you for taking the time to answer me, I really appreciate it, you really made me think about my math gaps.
mattother •
No problem! I will say it mostly likely depends on your end goals.
Personally I have quite a few friends who are very successful in games and web development and do not have strong math or algorithmic skills. So if your concern is lacking knowledge in Algorithms
will somehow make you unemployable that's definitely not the case. While helpful, they are not really necessary in a lot of areas of programming.
If you goal in academic then that could be different, a lot of CS is heavy in math and from what I've seen a lot of graduate CS work does require it. But you're probably best to verify for yourself.
My goal wasn't to dissuade you from exploring Algorithms either, just to try and point out materials that I personally found really helped me understand Algorithms a lot better.
Personally I'm somebody who understands better when I understand the fundamental reasons behind something. I found this a lot with classes like Calculus and Linear Alegbra. I didn't necessarily do
terrible in them, but I didn't really grasp the fundamental ideas behind things like complex numbers, integrals, etc. And I found once I understand things like sets, the types of numbers systems,
etc. it really helped things fall into place. And at least for me this wasn't something really covered in highschool either, so it was something I really ended having to explore myself.
If that is part of your struggles then I do think Introduction to Mathematical Thinking will be really helpful. I'll also point out he has a book too, if you're low on time, but I didn't find the
book as good as the lectures themselves.
I find with Math and Algorithms there's also a lot of eureka moments too. It just takes finding the right angle to really get it. And they are both not easy subject matters, so I wouldn't to get too
down if you're struggling either, pretty much everybody does.
I would give Python Algorithms a try first though. A lot of Algorithm books focus mostly on implementation rather than fundamental concepts, but Python Algorithms does a really good job of explaining
the why it works part.
Also don't be afraid to ask questions to professors, online, etc. I think a lot of people are afraid of looking stupid, but there's nothing stupid about not understanding. These are tough subject
matters and everybody learns in different ways. And there are plenty of people happy to help you learn (and who have probably struggled through the same material as well).
Anyways like I said I would check out Python Algorithms, I think it will probably help.
Also if you need a more in-depth explanation of Quicksort just let me know. I'm happy to try and explain it to you as best I can.
Jen Miller • • Edited on • Edited
hmm, I think I may not be understanding the issue you see. I'll take a shot. Lets call the left pointer "i" and right pointer "j"
There are different ways to select the pivot. Some will select a random element, others index 0 or last or the middle. Regardless, i will keep moving right and j will keep moving left...until they
In the end, you want elements divided into two groups. The dividing point of the two arrays is where index when i and j meet. Algorithms treat the meeting differently. Some will stop the i and j
progression when i==j. Others will stop before. But this is just a implementation detail.
Some algorithms will put the pivot in the middle as a final swap, then recursively quick sort the two halves (not including the pivot in the middle -as this element is in the correct final position).
They key is to understand regardless of the partitioning specifics, the two arrays will be partitioned into two groups. The left group will be less than the pivot and the right side will be larger.
Again, it's an implementation detail if the pivot is included in the group, in the right side or left side.
I can understand why it's complicated though. In particular this video is pretty helpful (and is how I think of quicksort).
Another method is explained in the hackerrank video below, however, the explanation misses the key point of how the meeting of i and j is dealt with...so I can see how it's confusing to follow.
Though it is talked about in the code writeup. A lot of people in the comments are also frustrated too.
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/imrc21/why-every-explanation-about-quicksort-is-somehow-different-1f0","timestamp":"2024-11-04T15:17:20Z","content_type":"text/html","content_length":"128245","record_id":"<urn:uuid:08294057-cec5-413b-ae8a-7e5ebbdfbce7>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00640.warc.gz"} |
Yun Chen Blog
Add Two NumbersYou are given two non-empty linked lists representing two non-negative integers. The digits are stored in reverse order and each of their nodes contain a single digit. Add the two numb
Reverse String IIGiven a string and an integer k, you need to reverse the first k characters for every 2k characters counting from the start of the string. If there are less than k characters left, r
K-diff Pairs in an ArrayGiven an array of integers and an integer k, you need to find the number of unique k-diff pairs in the array. Here a k-diff pair is defined as an integer pair (i, j), where i
Minimum Absolute Difference in BSTGiven a binary search tree with non-negative values, find the minimum absolute difference between values of any two nodes. For Example:12345678910111213Input: 1
Detect CapitalGiven a word, you need to judge whether the usage of capitals in it is right or not. We define the usage of capitals in a word to be right when one of the following cases holds: All le
Relative RanksGiven scores of N athletes, find their relative ranks and the people with the top three highest scores, who will be awarded medals: “Gold Medal”, “Silver Medal” and “Bronze Medal”. For
Base 7Given an integer, return its base 7 string representation. Example 1:12Input: 100Output: "202" Example 2:12Input: -7Output: "-10" Note: The input will be in range of [-1e7,
Find Mode in Binary Search TreeGiven a binary search tree (BST) with duplicates, find all the mode(s) (the most frequently occurred element) in the given BST. Assume a BST is defined as follows: The
Keyboard RowGiven a List of words, return the words that can be typed using letters of alphabet on only one row’s of American keyboard like the image below. For Example:12Input: ["Hello",
Next Greater Element IYou are given two arrays (without duplicates) nums1 and nums2 where nums1’s elements are subset of nums2. Find all the next greater numbers for nums1’s elements in the correspon
HeatersWinter is coming! Your first job during the contest is to design a standard heater with fixed warm radius to warm all the houses. Now, you are given positions of houses and heaters on a horizo
Construct the RectangleFor a web developer, it is very important to know how to design a web page’s size. So, given a specific rectangular web page’s area, your job by now is to design a rectangular
Max Consecutive OnesGiven a binary array, find the maximum number of consecutive 1s in this array. For Example:1234Input: [1,1,0,1,1,1]Output: 3Explanation: The first two digits or the last three dig
Number ComplementGiven a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guaranteed to fit within
Island PerimeterYou are given a map in form of a two-dimensional integer grid where 1 represents land and 0 represents water. Grid cells are connected horizontally/vertically (not diagonally). The gr
Hamming DistanceThe Hamming distance between two integers is the number of positions at which the corresponding bits are different. Given two integers x and y, calculate the Hamming distance. Note:0
Assign CookiesAssume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie. Each child i has a greed factor gi, which is the minimu
Minimum Moves to Equal Array ElementsGiven a non-empty integer array of size n, find the minimum number of moves required to make all array elements equal, where a move is incrementing n - 1 elements
Find All Numbers Disappeared in an ArrayGiven an array of integers where 1 ≤ a[i] ≤ n (n = size of array), some elements appear twice and others appear once. Find all the elements of [1, n] inclusive
Number of BoomerangsGiven n points in the plane that are all pairwise distinct, a “boomerang” is a tuple of points (i, j, k) such that the distance between i and j equals the distance between i and k
Arranging CoinsYou have a total of n coins that you want to form in a staircase shape, where every k-th row must have exactly k coins. Given n, find the total number of full staircase rows that can b
Find All Anagrams in a StringGiven a string s and a non-empty string p, find all the start indices of p’s anagrams in s. Strings consists of lowercase English letters only and the length of both stri | {"url":"https://yunchen.tw/archives/2017/03/","timestamp":"2024-11-11T01:28:57Z","content_type":"text/html","content_length":"59321","record_id":"<urn:uuid:86d469d9-c514-4266-9d38-e2751ce736c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00561.warc.gz"} |
This Blog is Systematic
Quite a few of my recent blog pieces have been picked up by the lovely folk at
allocate smartly.
So I thought I'd write an asset allocation piece, as the readers of my
second book "Smart Portfolios"
probably feel neglected with the lack of articles on investment rather than trading.
Absolute or relative momentum?
The motivation for this comes from a table in my second book, which exposes an interesting problem. Here is the table (actually a slightly modified version of it, so you won't recognise the precise
numbers), and I'll explain what it means and what the problem is:
Arithmetic mean Geometric mean Std. Deviation Sharpe Ratio
Fixed weight 8.37% 8.04% 8.14% 1.03 Relative momentum 9.26% 8.89% 8.62% 1.07 Absolute momentum 8.93% 8.61% 7.96% 1.12 Fixed weight:
This is a portfolio with 75:25 risk weightings in US equities and US bonds (using the last 12 months of monthly returns to calculate the appropriate volatility for risk weighting; this works out to
roughly 60:40 cash weightings on average).
Relative momentum:
This portfolio tactically rebalances the strategic fixed weights using the
12 month total risk adjusted return of equities and bonds. The rebalancing is a 'tilt' to account for forecasting uncertainty; the maximum tilt is to 148% of the original portfolio weight, and the
minimum is 60% of the original. The relative momentum portfolio is always fully invested.
Absolute momentum:
This portfolio tactically rebalances the strategic fixed weights according to the
12 month total risk adjusted return of equities and bonds, again using a 'tilt'. The absolute momentum portfolio may not be fully invested if momentum is relatively weak in one or both assets. The
minimum investment is 60% (which is not unusual), and the average is 93%.
All portfolios are rebalanced monthly, using data from January 1954 to March 2016 (I could update this, but I wanted to use the same data as in the book, and it wouldn't affect the results much).
Returns shown are excess returns, net of the risk free rate.
Let's do some basic analysis of these results. Using absolute momentum results in a slightly higher risk than for fixed weights, because equities have spent more time going up in a risk adjusted
sense. I call this the
historical volatility boost
. That more than compensates for the fact we aren't always fully invested, which drags down risk. The average cash weight to equities is 67% versus the 61% under fixed weights. But the extra risk is
well rewarded with a higher arithmetic and geometric return, and a higher Sharpe Ratio.
Absolute momentum is a super popular asset allocation methodology, because people like the 'downside protection' of being partly in cash when markets are selling off.
Relative momentum has even higher risk; again it has a systematic bias towards equities and a historical volatility boost, but because we are always fully invested that all hits the 'bottom line' in
the form of higher risk. The average cash weight to equities is 70%. The extra risk is rewarded with a higher arithmetic and geometric mean return, but the Sharpe Ratio is actually lower than for
absolute momentum (though still better than fixed weights).
Relative momentum is less popular amongst the general public, as it seems hard to justify a big allocation to bonds just because they aren't falling quite as fast as equities. It also has a worse
Sharpe Ratio, so 'theoretically' it's inferior (if you're an investor who can use leverage).
In my book I rather blandly concluded that relative momentum was better due to the higher geometric mean.
However, we're not comparing like with like. Strictly speaking we should probably compare relative momentum with an absolute version that has a higher strategic allocation to equities, so that their
risk levels are comparable. To put it another way:
is it better* to use relative momentum, or to use absolute momentum and crank up your strategic risk target to compensate for the reduction in risk?
Already we can see that this is a variation of the classic dilemma that investors without access to leverage and high risk tolerance have: should I opt for the highest Sharpe Ratio, or for something
with higher risk (and return) but a lower Sharpe Ratio? However this story is more complicated, because we have two moving parts: the original risk weights, and the choice of rebalancing strategy
(fixed weights, absolute, or relative). The interaction of these will produce portfolios with different return and risk profiles.
* The dilemma would be the same** for any type of forecast, but momentum is a popular and well understood rule to establish conditional returns.
** Strictly speaking the idea of an 'absolute' forecast requires some kind of equilibrium value at which we have a zero position. So dividend yield as a forecast wouldn't be helpful for absolute
weighting, but something like (divided yield - interest rates)*** would make sense.
*** the 'Fed model'
The experiment
The general question we want to answer is:
For a given risk tolerance, what is the best choice of strategic risk weights and rebalancing strategy?
My criteria will be to judge a particular outcome by looking at the
geometric mean
(my reasons for choosing that are documented
), and the
standard deviation
of returns.
The range of strategic risk weights I will consider are from 10% equities 90% bonds, up to 90% equities 10% bonds. All strategic risk weight portfolios will be fully invested. Note that people with
really low risk appetites will be best served by the maximum Sharpe Ratio portfolio plus a cash allocation; however I won't consider that option here. After all the problem we are exploring is most
acute for investors with higher risk appetites.
To make the results starker, I'm going to allow the two tactical portfolios to 'tilt' all the way from 10% to 200% of the original strategic weight. Obviously this won't affect the fixed weights. For
less aggressive tilts the relative results will be the same, but the numbers will be closer together.
First let's look at the Sharpe Ratios:
The black line is what you'd expect; the maximum SR portfolio is roughly 50:50 in equities and bonds. Absolute momentum is mostly inferior to the other options except for relatively high allocations
to equities. Relative momentum shows declining performance as we increase the risk weight.
However these differences in SR might not be significant (I'll discuss this later in the post), but more importantly 'we can't eat Sharpe Ratios' if we're not leveraged investors, so let's instead
focus on the geometric means and standard deviations.
Each line shows a classic 'efficient frontier', with one line for fixed weights, one for relative weights, and one for absolute weights. Each cross is a different strategic allocation, in 10% steps.
So the first black cross on the bottom end of the fixed weights line is 10% risk weight in equities, the next cross is 20% in equities, and so on up to 90% on the top right end of the line.
We can safely ignore all the portfolios with lower risk than 30% equities; for these we'd be better off mixing the maximum Sharpe Ratio portfolio with cash.
It's clear from this graph that the out performance of relative momentum is pretty consistent.
For a given risk target relative momentum is better than fixed weights or absolute momentum.
It also looks like there is no benefit from using a risk target of greater than 80% in equities.
Which strategic portfolio weights should we use?
There is an important question that is not easily answered by the graphs above: how much should I adjust my strategic weights to compensate for the effect of applying a relative or absolute momentum
tactical weight?
So, for example, if you want a standard deviation of 8% you could use:
• a fixed risk weight of ~75% to equities
• tactical absolute weighting with a strategic risk weight of ~68% to equities
• tactical relative weighting with a strategic risk weight of ~40% to equities
That is some substantial difference!
How robust are these results?
First let's consider the differences in geometric means. I'm extremely confident that 12 month momentum is a robust effect that has existed in the past, though
we can argue about whether it will continue in the future
. So I'd expect both types of momentum to beat fixed weights.
What about the out performance of relative momentum? Cross sectional momentum across asset classes is a less popular idea (though super popular within asset classes e.g. across stocks), but it would
be surprising if there was a substantial difference between the two types of forecast.
However in a long only portfolio absolute momentum is operating with one hand tied behind it's back, as it cannot go short. This might explain the relatively poor performance of absolute momentum.
Even when it has a slightly higher Sharpe Ratio (for relatively high equity weightings), the reduction in volatility means that absolute momentum can't compete on a geometric mean basis.
What about the differences in standard deviations? By construction the standard deviation for absolute momentum will
be lower than that for relative momentum.
The reasons for the increase in standard deviation when using relative momentum is less robust (risk also rises for absolute momentum, except for very low or very high equity allocations). In theory,
if both equities and bonds had the same average forecast going forward, then the standard deviation would be the same for relative momentum as it is for fixed weights.
Radically reducing your strategic weight to equities to compensate for the expected
'volatility boost'
from your tactical overlay might not be wise. The existence of a risk boost is probably the least robust finding here - I wouldn't be 100% sure it will exist in the future.
I'm reasonably happy that my superficial analysis in "Smart Portfolios" was correct when put through a more thorough test:
relative momentum gives a higher geometric mean than absolute momentum
, except for investors with low tolerance to risk. Therefore for most investors it's preferable.
In terms of more specific advice, the graphs above suggest the optimal portfolios are:
• If you can use leverage, the highest Sharpe Ratio comes from using relative momentum tactical weighting with a risk weight to equities of somewhere between 30% (15:85 equity/bonds in cash weights
based on current vols) and 50% (30:70 equity/bonds in cash weights). Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past is absent.
Recommend: Strategic risk weights 50% equity 50% bond, cash weights 30% equity 70% bonds, relative momentum tactical weighting.
• If you can't use leverage and have a high risk tolerance, the highest geometric mean comes from using relative momentum tactical weighting with a risk weight to equities of somewhere between 60%
(40:60 in cash weights based on current vols) and 90% (80:20 in cash weights). Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past
is absent. Recommend: Strategic risk weights 90% equity 10% bond, cash weights 80% equity 20% bonds, relative momentum tactical weighting.
• If you can't use leverage and have a modest risk tolerance (but higher than 8% standard deviation a year): I'd use relative momentum but with a lower risk weight. If you don't buy the 'risk
boosting' story then you will need between 60% and 90% risk weighting in equities; if you do buy the story and believe history will repeat itself, between 30% and 60% risk in equities. Recommend:
Strategic risk weights 60% equity 40% bond, cash weights 40% equity 60% bonds, relative momentum tactical weighting.
• If you can't use leverage and have a low risk tolerance (lower than 8% standard deviation a year): I'd invest in the maximum Sharpe Ratio portfolio (see above), and blend it with cash. | {"url":"https://qoppac.blogspot.com/2019/03/","timestamp":"2024-11-09T12:24:58Z","content_type":"text/html","content_length":"114270","record_id":"<urn:uuid:0c2a723a-898e-4bf0-b641-35a7d5f4b514>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00789.warc.gz"} |
From Ratios to Scales
The next step as we continue our journey towards understanding the world of Microtonal music and Microtonal Guitar is understanding how musical scales are generated using the musical ratios derived
from the harmonic series. One important concept to clarify before we proceed is the prime numbers and prime limits. A prime number is a number that is only evenly divisible by itself and 1. A prime
limit is the highest prime number that is being used in the generation of a musical scale derived from the harmonic series. Let's look at the scale interval and ratios in a table:
Musical Ratio Musical Interval
1/1 Root
2/1 Octave
Just Intonation is the term used for a system of tuning when creating scales using the intervals generated from the harmonic series. We'll get into a more definitive understanding of Just Intonation
later but for now let's just stick with this basic definition.
3 Limit Diatonic Major Scale
A 3 limit tuning system is a tuning system that used the prime number of 3 as it's basis. A typical diatonic scale in the three limit system is based on the first 5 notes based on the circle fifths,
the first note from the circle of 4ths and the octave.
Note for the 9/8 interval we have changed the ratio from 9/4 (3/2 * 3/2) to 9/8 based on the concept of octave reduction discussed earlier. View the table below to see this converted to a diatonic
scale (with the added octave as the end of the scale) starting with C 130.8 hz.
Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
81/64 Major 3rd E 165.56
4/3 Perfect 4th F 174.41
3/2 Perfect 5th G 196.22
27/16 Major 6th A 220.74
243/128 Major 7th B 248.33
2/1 Octave C 261.62
5 Limit Diatonic Major Scale
The most common 5 limit major diatonic scale shares the major 2nd, perfect 4th, and perfect 5th from the 3 limit major diatonic scale and replaces the Major 3rd, Major 6th, and Major 7th with notes
based on the 5 limit ratios.
Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
5/4 Major 3rd E 163.51
4/3 Perfect 4th F 174.41
3/2 Perfect 5th G 196.22
5/3 Major 6th A 218.02
15/8 Major 7th B 245.27
2/1 Octave C 261.62
Nature's Scale
Another common scale you will come across when being introduced to scales derived from the harmonic series is "Natures's Scale". This scale is the first full scale that is encountered in the harmonic
series. It consists of ratio's and notes based on the 8th through the 16th harmonics. This scale features ratio's derived from prime numbers 2,3,5,7, 11, and 13, so is considered a 13 limit scale.
Note there isn't a perfect 4th in the scale. The scale is charted out below:
Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
5/4 Major 3rd E 163.51
11/8 Flatted 5th Gb 179.86
3/2 Perfect 5th G 196.22
13/8 Nuetral 6th AB 212.56
7/4 Flatted 7th Bb 228.92
15/8 Major 7th B 245.27
2/1 Octave C 261.62 | {"url":"https://microtonal-guitar.com/tutorial/from-ratios-to-scales/","timestamp":"2024-11-10T04:56:07Z","content_type":"text/html","content_length":"121427","record_id":"<urn:uuid:8c7c1ebb-887d-4f9a-8454-f356656bfca4>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00080.warc.gz"} |
Catenary Sentence Examples
• The true catenary is that assumed by a chain of uniform weight per unit of length, but the form generally adopted for suspension bridges is that assumed by a chain under a weight uniformly
distributed relatively to a horizontal line.
• Finally, we may refer to the catenary of uniform strength, where the cross-section of the wire (or cable) is supposed to vary as the tension.
• The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.
• This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.
• Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.
• The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.
• If, however, the circular ends of the catenoid are closed with solid disks, so that the volume of air contained between these disks and the film is determinate, the film will be in stable
equilibrium however large a portion of the catenary it may consist of.
• Draw Pp and Qq touching both catenaries, Pp and Qq will intersect at T, a point in the directrix; for since any catenary with its directrix is a similar figure to any other catenary with its
directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix.
• Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lower catenary must intersect below the directrix.
• The condition of stability of a catenoid is therefore that the tangents at the extremities of its generating catenary must intersect before they reach the directrix.
• The overhead catenary is being revamped, with new wires going in around the depot building at the left hand side of the layout.
• The word " catenary " actually means the natural sag of such a wire and has more significance in this context than normally realized.
• The ship is then stopped, and the cable gradually hove up towards the surface; but in deep water, unless it has been caught near a loose end, the cable will break on the grapnel before it reaches
the surface, as the catenary strain on the bight will be greater than it will stand.
• A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of
magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes
mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be
left unnoticed.
• The simple catenary is shown in the figure.
• The surface formed by revolving the catenary about its directrix is named the alysseide.
• He proposed the problem of the catenary or curve formed by a chain suspended by its two extremities, accepted Leibnitz's construction of the curve and solved more complicated problems relating to | {"url":"https://sentence.yourdictionary.com/catenary","timestamp":"2024-11-09T09:26:49Z","content_type":"text/html","content_length":"241860","record_id":"<urn:uuid:b3946f36-26fe-4ad4-9339-38afdbc3efbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00017.warc.gz"} |
PhD Thesis Defense: Nanomagnetic Logic by Photothermal Excitation of Magnetic Nanostructure Networks
CIC nanoGUNE Seminars
Matteo Menniti
Pre-Doctoral Researcher at CIC nanoGUNE
CFM auditorium
Paolo Vavassori
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Artificial spin ices (ASIs) are systems made up of single-domain nanomagnets which are used to study geometrically frustrated systems and have been proposed for use in novel computation applications.
Depending on the arrangement of the nanomagnets in an ASI the system can have degenerate ground states due to the system not being able to minimize all the magnetostatic interactions at the same
time. Storing information with the direction of magnetization while being able to compute the information via the magnetostatic interactions within the ASI would crack the von Neumann bottleneck
allowing for faster and more efficient computing. The duality of the problem however is that the nanomagnets within the ASI have to be stable enough to store information (not fluctuate between 1 and
0) while at the same time having interactions that are strong enough to compute the information (change the state of a bit). One solution is to selectively lower the energy barrier that stabilizes
the magnetization direction in the desired nanomagnets when computing. The selective heating, as shown in this project, can be achieved in ASI with thermoplasmonics by carefully designing the optical
properties of the nanostructures. In this presentation we will learn the design process of a reconfigurable nanomagnetic logic gate, with its various design parameters and complications, before
showing EXPERIMENTALLY the result of a Boolean logic operation. | {"url":"https://dipc.ehu.eus/en/scientific-activities/joint-seminar-agenda/cic-nanogune/phd-thesis-defense-nanomagnetic-logic-by-photothermal-excitation-of-magnetic-nanostructure-networks","timestamp":"2024-11-08T21:27:40Z","content_type":"application/xhtml+xml","content_length":"17811","record_id":"<urn:uuid:565cd5cc-a427-4185-91b3-8bbd4104591b>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00272.warc.gz"} |
Lesson 5
Equations and Their Graphs
Lesson Narrative
So far in the unit, students have primarily used descriptions, expressions, and equations to represent relationships and constraints. In this lesson, they revisit the idea that graphs can be a useful
way to represent relationships. Students are reminded that each point on a graph is a solution to an equation the graph represents. They analyze points on and off a graph and interpret them in
context. In explaining correspondences between equations, verbal descriptions, and graphs, students hone their skill at making sense of problems (MP1).
In this lesson, students are also introduced to the use of graphing technology to graph equations. This introduction could happen independently as long as it precedes the second activity in the
Learning Goals
Teacher Facing
• Comprehend that the graph of a linear equation in two variables represents all pairs of values that are solutions to the equation.
• Interpret points on a graph of a linear equation to answer questions about the quantities in context.
• Use graphing technology to graph linear equations and identify solutions to the equations.
Student Facing
• Let’s graph equations in two variables.
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (If students typically access the digital version of the materials,
Desmos is always available under Math Tools.)
Student Facing
• I can use graphing technology to graph linear equations and identify solutions to the equations.
• I understand how the coordinates of the points on the graph of a linear equation are related to the equation.
• When given the graph of a linear equation, I can explain the meaning of the points on the graph in terms of the situation it represents.
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/2/5/preparation.html","timestamp":"2024-11-07T22:17:31Z","content_type":"text/html","content_length":"78785","record_id":"<urn:uuid:9594050b-20a5-429e-a9d7-f76d0c2e3be5>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00503.warc.gz"} |
correlation means all of the following except that psychology
Put another way, it means that as one variable increases so does the other, and conversely, when one variable decreases so does the other. As we can see, no correlation just shows no relationship at
all: moving to the left or the right on the x-axis does not allow us to predict any change in the y-axis. The actual correlation for 1955 is .61, which is to say that illiteracy and infant mortality
have 37 percent of their variance in common. In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. This
means an increase in the amount of one variable leads to a decrease in the value of another variable. A correlation between variables, however, does not automatically mean that the change in one
variable is the cause of the change in the values of the other variable. A value of zero indicates a NIL correlation but not a non-dependence. For this kind of data, we generally consider
correlations above 0.4 to be relatively strong; correlations between 0.2 and 0.4 are moderate, and those below 0.2 are considered weak. The other common situations in which the value of Pearson’s r
can be misleading is when one or both of the variables have a limited range in the sample relative to the population.This problem is referred to as restriction of range.Assume, for example, that
there is a strong negative correlation between people’s age and their enjoyment of hip hop music as shown by the scatterplot in … In science class, Fred broke a test tube and had to clean the area. A
scattergraph indicates the strength and direction of the correlation between the co-variables. When working with continuous variables, the correlation coefficient to use is Pearsonâ s r. The
correlation coefficient (r) indicates the extent to which the pairs of numbers for these two variables lie on a straight line. McLeod, S. A. Even if there is a very strong association between two
variables we cannot assume that one causes the other. Correlation allows the researcher to investigate naturally occurring variables that maybe unethical or impractical to test experimentally. An
educational researcher compares the academic performance of students from the “rich” side of town with that of … The interpretation of the coefficient depends on the topic of study. Answer – 1:
Correlation vs. The reporter concludes that vampire books should be banned, because they are c… Correlation means association - more precisely it is a measure of the extent to which two variables are
related. Examples of Pearson’s correlation coefficient. This means that the studies found that as the rate of smoking increased, so did the occurrence of cancer; smoking goes up, presence of cancer
goes up. All of the following situations show a possible correlation between two events except: Sherriff Cass has interesting data about house break-ins. A scattergram is a graphical display that
shows the relationships or associations between two numerical variables (or co-variables), which are represented as points (or dots) for each pair of score. As you’ve learned, hypotheses can be
formulated either through direct observation of the real world or after careful review of previous research. Correlation does not always prove … https://www.simplypsychology.org/correlation.html. A
zero correlation is often indicated using the abbreviation r=0. ), only that there is a relationship between the two factors. When studying things that are difficult to measure, we should expect the
correlation coefficients to be lower (e.g. For example suppose we found a positive correlation between watching violence on T.V. ... Current ethical standards for psychology experiments were
established by Freud in the 1920s. Zero Correlation . See more. A positive correlation means that when one variable goes up, the other goes up. Perfect correlation means all of the following except
A) r = – 1 or except A) r = – 1 or passes. A zero correlation suggests that the correlation statistic did not indicate a relationship between the two variables. The other common situations in which
the value of Pearson’s r can be misleading is when one or both of the variables have a limited range in the sample relative to the population.This problem is referred to as restriction of
range.Assume, for example, that there is a strong negative correlation between people’s age and their enjoyment of hip hop music as shown by the scatterplot in Figure 6.6. This preview shows page 47
- 49 out of 75 pages.. 22. A correlation coefficient close to +1.00 indicates a strong positive correlation. In order to conduct an experiment, a researcher must have a specific hypothesis to be
tested. You’ll understand this clearly in one of the following answers. There is no rule for determining what size of correlation is considered strong, moderate or weak. above 0.4 to be relatively
strong). //Enter domain of site to search. They are a little less powerfulthan parametric methods if the assumptions underlying the latter are met, but are less likely to give distorted results when
the assumptions fail. Familiar examples of dependent phenomena include the correlation between … The results of this study are summarized in Table 6.1, which is a correlation matrix showing the
correlation (Pearson’s r) between every possible pair of variables in the study. 1. For example suppose it was found that there was an association between time spent on homework (1/2 hour to 3 hours)
and number of G.C.S.E. This can then be displayed in a graphical form. Imagine you're reading the newspaper, and you see an article that says that a study was done on whether reading books about
vampires makes children want to turn into vampires themselves. Privacy Policy - Terms of Service. A non-dependency between two variable means a zero correlation. No correlation: There is no
relationship between the two variables. Welcome to Sciemce, where you can ask questions and receive answers from other members of the community. Answer – 1: Correlation vs. Instead of drawing a
scattergram a correlation can be expressed numerically as a coefficient, ranging from -1 to +1. [TY9.1A negative correlation is the same as no correlation.Scatterplots are a very poor way to show
correlations.If the points on a scatterplot are close to a straight line there will be a positive correlation.Negative correlations are of no use for predictive purposes.None of the above.Answer: E
The sign—positive or negative—of the correlation coefficient indicates the direction of the relationship (Figure 1). A non-dependency between two variable means a zero correlation. A correlation only
shows if there is a relationship between variables. We can see from the table that the correlation between working memory and executive function, for example, was an extremely strong .96, that the
correlation between working memory and vocabulary was a medium .27, and that all the measures except vocabulary tend to … Put another way, it means that as one variable increases so does the other,
and conversely, when one variable decreases so does the other. The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y.However, the
reliability of the linear model also depends on how many observed data points are in the sample. Dependency. _____ correlation between two variables means that as scores on one variable increase,
then scores on another variable also increase. The closer it is to 1, the more likely there is a positive correlation between the two variables; the closer it is to -1, the more likely there is a
negative correlation between the two variables. This is done by drawing a scattergram (also known as a scatterplot, scatter graph, scatter chart, or scatter diagram). A positive correlation means
that the variables move in the same direction. An experiment tests the effect that an independent variable has upon a dependent variable but a correlation looks for a relationship between two
variables. : Studies find a positive correlation between severity of illness and nutritional status of the patients. What correlation coefficient essentially means is the degree to which two
variables move in tandem with one-another. A correlation is a statistical index used to represent the strength of a relationship between two factors, how much and in what way those factors vary, and
how well one factor can predict the other. passes (1 to 6). Causation means that one variable (often called the predictor variable or independent variable) causes the other (often called the outcome
variable or dependent variable). "Correlation is not causation" means that just because two variables are related it does not necessarily mean that one causes the other. and violent behavior in
adolescence. A negative correlation means that the variables move in opposite directions. BUT, this does not demonstrate that smoking causes cancer (does anyone disagree that it does? A correlation
between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable. Which of the following statements is true? In
statistical studies, a perfect negative correlation can be expressed as -1.00, a perfect positive correlation can be expressed by +1.00, and a zero correlation is expressed as 0.00. var
idcomments_post_id; An experiment isolates and manipulates the independent variable to observe its effect on the dependent variable, and controls the environment in order that extraneous variables
may be eliminated. A correlation is a statistical index used to represent the strength of a relationship between two factors, how much and in what way those factors vary, and how well one factor can
predict the other. R can vary from -1 to 1. A correlation coefficient close to -1.00 indicates a strong negative correlation. I'm a scientist studying addiction, and in the field, it's very important
to be clear about what each of the words you use means. It's important to note that this does not mean that there is not a relationship at all; it simply means that there is not a linear
relationship. (2018, January 14). It would not be legitimate to infer from this that spending 6 hours on homework would be likely to generate 12 G.C.S.E. A positive correlation means that the
variables move in the same direction. 1. 1. That is, although a correlational study cannot definitely prove a causal hypothesis, it may rule one out. Let us consider two events A and B. Causation
means that one event causes another, or A causes B. It could be that the cause of both these is a third (extraneous) variable - say for example, growing up in a violent home - and that both the
watching of T.V. C) Correlations can tell you about relations between two variables but it is not possible to make predictions based upon correlational research. You’ll understand this clearly in one
of the following answers. Which of the following statements is true? It's important to note that this does not mean that there is not a relationship at all; it simply means that there is not a linear
relationship. An extreme value on both the side means they are strongly correlated with each other. Create a free account. The studies conducted previously on the effects of smoking indicated a
positive correlation between smoking and cancer. Compute the correlation between the scores of two trials by rank difference method: The correlation between Trial I and II is positive and very high.
A. height and weight B. men's educational level and their income C. alcohol consumption and scores on a driving test D. school grades and IQ scores In these kinds of studies, we rarely see
correlations above 0.6. Discussion: For each of the following, decide whether it is most likely that the study described is experimental or correlational and explain why. She reports that the
culprits are all teens who watch violent films. (See diagram above.) A correlation only shows if there is a relationship between variables. var idcomments_acct = '911e7834fec70b58e57f0a4156665d56';
Problems arise when there are nonlinear relationships (typically the case in many real-life situations) as shown in Figure 3.42.Part E shows an exponential relationship between X and Y.If we use a
nonlinear correlation, we get +0.9, but if we use a linear correlation, it is much lower at 0.6 (Part F), which means that there is information that is not picked up by the linear correlation. A
positive coefficient, up to a … [TY9.1A negative correlation is the same as no correlation.Scatterplots are a very poor way to show correlations.If the points on a scatterplot are close to a straight
line there will be a positive correlation.Negative correlations are of no use for predictive purposes.None of the above.Answer: E eval(ez_write_tag
([[336,280],'simplypsychology_org-box-1','ezslot_4',197,'0','0']));report this ad, eval(ez_write_tag([[336,280],'simplypsychology_org-large-billboard-2','ezslot_9',618,'0','0']));report this ad, eval
(ez_write_tag([[300,250],'simplypsychology_org-large-leaderboard-1','ezslot_10',152,'0','0']));report this ad. If there is zero covariance or zero correlation between b i and b j, then C(i,j) for i ≠
j is zero. Correlation allows the researcher to clearly and easily see if there is a relationship between variables. A correlation can be expressed visually. Zero Correlation . A correlation
coefficient close to -1.00 indicates a strong negative correlation. Negative correlations: As the amount of one variable increases, the other decreases (and vice versa). An extreme value on both the
side means they are strongly correlated with each other. ... Statistics can increase knowledge in all of the following ways except _____. Values over zero indicate a positive correlation, while
values under zero indicate a negative correlation. Cause-And-Effect relationships a non-dependency between two variables but it is not possible to make predictions about one from another )... Cases
against the tobacco companies that occurred in the late 1990 's causal,. Even if there is no rule for determining what size of correlation often... The following ways except _____ of AlleyDog.com …
this preview shows page 47 - 49 out of 75 pages 22. Poor infant care, resulting in higher mortality measure and an established measure ) ). These kinds of studies, we want to be as comparative as
possible so we compute the correlation between and! Zero indicates a perfect positive correlation means that as one variable increase then! Hours on homework would be likely to generate 12 G.C.S.E
would likely show a positive,... That smoking causes cancer ( does anyone disagree that it does determining what size of correlation is not causation means! And can not assume that one causes the
other can make predictions based upon correlational research in a graphical.., therefore because of this we found a positive correlation, while values zero. One from another site is the property of
AlleyDog.com it is not can. Not use correlational research … in order to conduct correlation means all of the following except that psychology experiment tests effect. =
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means that when one variable goes on y-axis! Tobacco companies that occurred in the court cases against the tobacco companies that occurred the! We are studying things that are more easily countable,
we expect higher correlations ( e.g meaning that scores. In all of the following ways except _____ smoking causes lung cancer,! In all of the following variables, except for ________ show a
correlation! Two factors clean the area experience with correlation means all of the following except that psychology property of AlleyDog.com, the other goes up and... The day delivered to your
inbox, © 1998-, AlleyDog.com: positive correlation normal distribution = '911e7834fec70b58e57f0a4156665d56 ;! Correlation: there is no relationship between variables although correlation does not
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to +1 'with this, therefore because of.! Can ask questions and receive answers from other members of the following variables we. Correlation can be expressed numerically as a coefficient, up to a …
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For determining what size of correlation is often indicated using the abbreviation r=0 another. A non-dependency between two random correlation means all of the following except that psychology
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( curobj {. Then scores on one variable leads to a decrease in the same.. Second, although a correlational study: a positive correlation, Get the of... Strong association between two variable means a
zero correlation suggests that the correlation statistic did not indicate positive! Welcome to Sciemce, where you can not definitely prove a causal,... Your inbox, © 1998-, AlleyDog.com
non-dependency between two variables but it is causation. Does imply correlation perfect positive correlation means that when one variable increases, other. So we compute the correlation for all
nations but one is a relationship between them opposite directions,. Scatterplot, scatter chart, or scatter diagram ). ). ). ). ). ) )., Fred broke a test tube and had to clean the area no
correlation all the. Direction of the community the researcher to clearly and easily see if there is nontraditional... Moderate or weak to draw conclusions about cause-and-effect relationships size
of correlation is not causation '' means that one! Allows the researcher to clearly and easily see if there is a relationship variables... A nontraditional student who is a relationship between the
two variables increase knowledge in all the... Expect higher correlations ( e.g `` +curobj.qfront.value } just because two variables dependent variable but a correlation +1! Scattergram ( also known
by the Latin phrase cum hoc ergo propter (... Psychology experiments were established by Freud in the same direction any statistical relationship, whether causal not... Any statistical relationship,
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axis and simply. Events a and B. causation means that the culprits are all teens who watch violent films violent. The property of AlleyDog.com to clean the area strong, moderate or.! Understand this
clearly in one of the following answers of four move in opposite directions,... Goes down a positive correlation and negative correlation correlations ( e.g be likely to generate G.C.S.E... Had to
clean the area lack experience with children then scores on variable! Up to a decrease in the amount of one variable increases, the other decreases ( and vice versa.... Positive correlation, Fred
broke a test tube and had to clean the area although does! In these kinds of studies, we should expect the correlation between smoking cancer! Nutritional status of the patients identifies variables
and looks for a relationship between the two.... Make predictions about one from another each other scattergram a correlation coefficient is not causation '' means that one... Or copied for any
reason without the express written consent of AlleyDog.com correlations can tell you about relations between variables. Also increase see also: positive correlation means that one event causes
another, or a causes.... Curobj ) { curobj.q.value= '' site: '' +domainroot+ '' `` +curobj.qfront.value } tobacco... Investigate naturally occurring variables that maybe unethical or impractical to
test experimentally be relatively strong ). )..! If there is a relationship between variables allows the researcher to investigate naturally occurring variables that maybe or! No rule for determining
what size of correlation is not causation '' means that just because variables... Kinds of studies, we rarely see correlations above 0.6 ( correlation between watching violence on T.V the coefficient
correlation means all of the following except that psychology! Homework would be unethical to conduct an experiment on whether smoking causes cancer ( does disagree. Between severity of illness and
nutritional status of the coefficient depends on the y-axis goes on axis... Variables that maybe unethical or impractical to test experimentally material may not be legitimate infer! Smoking
indicated a positive correlation, a researcher must have a specific hypothesis to be comparative... Correlation between the two variables correlation and negative correlation means that one causes
the other goes down same.... Statistics can increase knowledge in all of the community an independent variable has upon a dependent variable but a of... It is not enough to define the dependence
structure between random variables or bivariate data also positive..., and no correlation: there is a relationship between the co-variables with.! Correlation but not a non-dependence naturally
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variable has upon a dependent but... Diagram ). ). ). ). ). ). ). ). )... All material within this site is the property of AlleyDog.com between smoking and cancer variable increases, students... Cum
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therefore because of this if there is a mother of four variables are it! Violent behavior are the outcome of this ' ). ). ). )..... Correlation coefficient close to -1.00 indicates a strong negative
correlation of +1 indicates a strong negative means... That the culprits are all teens who watch violent films phrase cum hoc ergo propter hoc ( 'with,... Get the word of the coefficient depends on
the effects of smoking a! Has upon a dependent variable but a correlation coefficient close to -1.00 indicates a correlation means all of the following except that psychology positive means. Maybe
unethical or impractical to test experimentally of drawing a scattergram a correlation coefficient close +1.00... Up, the other by the Latin phrase cum hoc ergo propter hoc 'with. Draw conclusions
about cause-and-effect relationships a correlational study can not definitely prove a causal hypothesis it.
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Module 2 Day 8 Challenge Part 1
@victorioussheep Good questions! The "two angles that are missing" are these two, marked in yellow, which do turn out to be 60 degrees (I think Prof. Loh also points to them in the video, but I see
how it could be a bit unclear).
Solving for those yellow angles, rather than the red ones you marked, is a good way to use what you know about inscribed angles and cyclic quadrilaterals. Specifically, Prof. Loh uses the fact that
an inscribed angle's measure is half the measure of the arc it cuts off.
But you could definitely solve for the red angles first! Take the triangle with angles 20, 60, and (unknown red angle). The unknown red angle is equal to 100 degrees because the angles of a triangle
add to 180. Then, the supplementary red angle is 80 degrees (because 100+80 = 180). Finally, the last red angle equals 180-40-80 = 60 degrees (using the triangle with angle 40, 80, (unknown red/blue
angle)), like we found before.
So basically, there are multiple valid ways to solve the problem, and each way emphasizes different methods 🙂 finding the yellow angles first is most useful for illustrating the usefulness of
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How to Select Procedures for Determining Limits?
There are several ways to solve the limit. The following step-by-step guide helps you learn how to select methods for determining limits.
There are many techniques for finding limits that apply in different situations. It is important to know all of these techniques, but it is also important to know when to use which techniques.
Related Topics
A step-by-step guide to select procedures for determining limits
Steps to choosing an appropriate limits determination procedure:
Step 1: Try evaluating the limit at the given location.
Step 2: Pay attention to what occurs when you assess at the given location. One of the following three scenarios will occur:
• When we check at the point, we get a value, and we may specify that the value we get is our limit. For example:
\(lim_{x\to 4}x^2=(4)^2=16\)
• We receive division by zero (an undefined result), which results in a vertical asymptote. For example:
\(lim_{x\to 4}\frac{1}{x-4}=\frac{1}{4-4}=\frac{1}{0}\)
• Alternatively, we may receive an uncertain form, in which case we will go to Step 3. For example:
\(lim_{x\to 4}\frac{x^2-16}{x-4}=\frac{(4)^2-16}{4-4}=\frac{16-16}{4-4}=\frac{0}{0}\)
Step 3: If we have an indeterminate form, we should try to simplify it by factoring, multiplying by the conjugate (which is important when dealing with radical limits), or using trigonometric
identities if we have trigonometric functions.
Note: Below is the possible indeterminate forms that you may encounter:
\(\frac{0}{0}\), \(\frac{∞}{∞}\), \(∞-∞\), \(1^∞\), \(0^0\), \(∞^0\), \(0.∞\)
Selecting procedures for determining limits – Example 1:
Choose an appropriate method to determine the following limit, then evaluate the limit using the selected method. \(lim_{x\to 6}\frac{x^2+2x-2}{x+4}\)
First, evaluate the limit at the given location:
\(lim_{x\to 6}\frac{x^2+2x-2}{x+4}\)
\(=lim_{x\to 6}\frac{(6)^2+2(6)-2}{(6)+4}\)
Then, observe what happens when you evaluate at the given location. we see that we do not have an indeterminate form. So we might conclude that our limit is:
\(lim_{x\to 6}\frac{x^2+2x-2}{x+4}=\frac{23}{5}\)
Thus, there is no need to move on to Step 3.
Selecting procedures for determining limits – Example 2:
Choose an appropriate method to determine the following limit, then evaluate the limit using the selected method. \(lim _{x\to -1}\left(\frac{x+1}{x^2+3x+2}\right)\)
First, evaluate the limit at the given location:
\(lim _{x\to -1}\left(\frac{x+1}{x^2+3x+2}\right)\)
\(=lim_{x\to -1}\frac{(-1)+1}{(-1)^2+3(-1)+2}\)
Then, observe what happens when you evaluate at the given location. We see the indeterminate form \(\frac{0}{0}\).
We have an indeterminate form and we have a limit involving polynomials. So we should attempt to factor and simplify.
\(lim _{x\to -1}\left(\frac{x+1}{x^2+3x+2}\right)\)
\(=lim_{x\to -1}\frac{x+1}{(x+1)(x+2)}\)
\(=lim_{x\to -1}\frac{1}{(x+2)}\)
\(=lim_{x\to -1}\frac{1}{(-1)+2}\)
Since we do not deal with anything indeterminate form, we may conclude that our limit is:
\(lim _{x\to -1}\left(\frac{x+1}{x^2+3x+2}\right)=1\)
Exercises for Selecting Procedures for Determining Limits
Evaluate the following limits using an appropriate method.
1. \(\color{blue}{lim _{x\to \infty }\left(\frac{\sqrt{x}-4}{x-16}\right)}\)
2. \(\color{blue}{lim _{x\to 3}\left(\frac{x-3}{x^2-2x-3}\right)}\)
3. \(\color{blue}{lim _{x\to -7}\left(\frac{x^2+7x}{x^2+6x-7}\right)}\)
4. \(\color{blue}{lim _{x\to -1}\left(\frac{\sqrt{x+5}-2}{x+1}\right)}\)
5. \(\color{blue}{lim _{x\to 4}\left(\frac{x-4}{\sqrt{x+5}-3}\right)}\)
1. \(\color{blue}{0}\)
2. \(\color{blue}{\frac{1}{4}}\)
3. \(\color{blue}{\frac{7}{8}}\)
4. \(\color{blue}{\frac{1}{4}}\)
5. \(\color{blue}{6}\)
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Dot product in Julia
I have been trying out Julia as a replacement for octave. One reason is that several of my collaborators (thanks Emilie, Aurélien and Rémy) are fans. I heard great things about its speed and its
natural fit for the type of one-page programs one tends to try out research ideas.
In one of my projects, I found out by profiling that it was bottlenecked by the computation of the dot product between two vectors. In this project the vectors are of fixed and rather small length.
Dimensions between \(3\) and \(10\) are common, and \(100\) is the exception. I had written the dot product between vectors \(x\) and \(y\) as follows
sum(x .* y)
To me this looks natural. One sums the entry-wise products. How could one do this any faster?
In this post we dig into accelerating the dot product, and learn some surprising lessons about Julia.
• The natural implementation is not the fastest.
• Finding the fastest implementation is not straightforward.
• The run-times of different implementations of the dot product are strikingly diverse.
I coded 13 different versions of the dot product. Each version is evaluated on lengths \(1\)–\(2000\). I also evaluate each version on four types of inputs:
Many Julia commands return arrays, like randn(5,3). The main difference (in the context of this post) between a Range and a Generator is that the former allows indexing, i.e. (1:5)[3] while the
latter only allows iteration (see an interesting read on trying to add indexing to generators).
The implementations are listed below. Here are the results. The measurements are performed on my Debian laptop. I’m using Julia version 1.5.3, on a Intel(R) Core(TM) i7-7700 CPU @ 3.60GHz.
Length 3
Here is the run-time for vectors of length \(3\).
Computing the dot product should not require allocating memory. Yet some of the high runtime could be caused by it. Here are the total memory allocations in bytes caused by each implementation.
Number of bytes allocated per dot product
computation for vectors of length \(3\).
This is averaged over \(3334000\)
Array Range Tuple Generator
builtin_dot 0 0 0 0
sumdot 112 112 0 336
sumzip 0 0 0 0
msmzip 0 0 0 0
sumix 0 0 0 4.8e-5
msmix 0 0 0 4.8e-5
blas 0 0 2.9e-5 4.8e-5
forea 240 368 128 368
iter 0 0 0 0
accixz 0 0 0 4.8e-5
acczip 0 0 0 0
accix 0 0 0 4.8e-5
mapred 0 0 0 0
lazysumdot 0 0 0 224
What is going on with the very small numbers? These are small allocations happening only once, and not once per dot product (\(2.9 \cdot 10^{-5}\) maps to \(96\) and \(4.8 \cdot 10^{-5}\) maps to \
(160\)). Julia is compiled Just-In-Time, and I do make sure to run each function once before starting the measurement. So it is not JIT overhead. I don’t currently have an explanation. Curious!
One thing to notice is how much the rather natural sumdot underwhelms.
Length 2000
Here is the run-time for vectors of length \(2000\).
And here are the memory allocation details.
Number of bytes allocated per dot product
computation for vectors of length \(2000\).
This is averaged over \(5001\) repetitions.
Array Range Tuple Generator
builtin_dot 0 0 0 0
sumdot 16128 16128 144448 48384
sumzip 0 0 0 0
msmzip 0 0 0 0
sumix 0 0 0 0.032
msmix 0 0 0 0.032
blas 0 0 6.4 0.032
forea 128048 128176 64032 128176
iter 0 0 0 0
accixz 0 0 0 0.032
acczip 0 0 0 0
accix 0 0 0 0.032
mapred 0 0 0 0
lazysumdot 0 0 0 32256
This is how each method scales with the input length. We see that for large inputs, BLAS and dot are the winners, closely followed by msmix and lazysumdot.
The build-in function dot. Also writable as infix \cdot (which renders as \(\cdot\)).
function builtin_dot(x, y)
x · y;
The most straightforward implementation.
function sumdot(x, y)
Using iterator zipping
function sumzip(x, y)
sum(v * w for (v,w) in zip(x,y));
Using iterator zipping, combined with the ability of sum to apply a function.
function msmzip(x, y)
sum(((v,w),) -> v*w, zip(x,y));
Iterate over the indices explicitly.
function sumix(x, y)
sum(@inbounds(x[k] * y[k]) for k in eachindex(x));
Map the pairwise product function over indices.
function msmix(x, y)
sum(k -> @inbounds(x[k] * y[k]), eachindex(x));
Defer to BLAS (the Basic Linear Algebra Subsystem)
function blas(x, y)
Manual dot product using two iterators.
function iter(x, y) # arbitrary iterables
ix = iterate(x)
iy = iterate(y)
s = 0.
while ix !== nothing && iy !== nothing
(vx, xs), (vy, ys) = ix, iy
s += vx * vy
ix = iterate(x, xs)
iy = iterate(y, ys)
if !(iy === nothing && ix === nothing)
throw(DimensionMismatch("x and y are of different lengths!"))
return s
Manual dot product using zipped indices.
function accixz(x, y)
lx = length(x)
if lx != length(y)
throw(DimensionMismatch("first array has length $(lx) which does not match the length of the second, $(length(y))."))
s = 0.
for (Ix, Iy) in zip(eachindex(x), eachindex(y))
@inbounds s += x[Ix] * y[Iy]
Manual dot product using explicit indices.
function accix(x, y)
lx = length(x)
if lx != length(y)
throw(DimensionMismatch("first array has length $(lx) which does not match the length of the second, $(length(y))."))
s = 0.
for k in 1:lx
@inbounds s += x[k]*y[k]
Manual dot product using zipped collections.
function acczip(x, y)
lx = length(x)
if lx != length(y)
throw(DimensionMismatch("first array has length $(lx) which does not match the length of the second, $(length(y))."))
s = 0.
for (Ix, Iy) in zip(x, y)
s += Ix * Iy
Manual dot product using the foreach function.
function forea(x, y)
s = 0.
foreach((v,w) -> s += v*w, x, y)
Dot product using mapreduce.
function mapred(x, y)
mapreduce(((a,b),)->a*b, +, zip(x, y))
Dot product using sum, with the LazyArrays “@\(\sim\)” non-materializing broadcast macro.
function lazysumdot(x, y)
sum(@~ x.*y);
Many follow-up questions arise.
• What of mixed argument types, say inner product between an Array and a Tuple?
• From a compiler perspective, why is this difficult? Why do these implementations, when all abstractions are instantiated, not compile to the same machine code?
• What of element-wise products between three vectors? We can do
sum(x .* y .* z)
but now there are no built-in BLAS calls. How to get these fast?
• One thing I have not investigated is whether all implementations have the same accuracy. I would believe yes: all implementations accumulate the element-wise products in order. But one might
imagine to try and maximize precision, for example by computing the dot product in a carefully considered order (to process cancellations between positive and negative large terms before adding
small terms). This would likely require sorting the entries, and that would stand out in the run-time.
What we may need is a lazy map, or a non-materialising broadcast operation. The julia discussion on this topic is public, see for example | {"url":"http://blog.wouterkoolen.info/Julia-DotProduct/post.html","timestamp":"2024-11-10T01:29:53Z","content_type":"application/xhtml+xml","content_length":"18971","record_id":"<urn:uuid:2cd26d5a-e7d6-47e6-8237-ee8421dce3b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00812.warc.gz"} |
Gross Profit $’s per crew day versus Gross Margin Percentages - Enterprise Selling Solution
There are generally 3 types of pricing models in the HVAC industry.
Number one is the markup model. This model is where you take the cost and mark up by a percentage to come up with a selling price. For example, if I had a $1000 cost and I wanted to markup a part 40%
I would take 1000 and multiply times 1.4 to come up with a selling price of $1400. The margin on this $1400 sale is only $400 or 28.6% gross margin.
This method of pricing is quite archaic since it generally works with smaller priced parts that need a much larger percentage of markup. For example, marking up a $10 capacitor 600% will give you a
$60 selling price on that part. Therefore, this model works with the merchandising and pricing of smaller parts but not necessarily the pricing of larger services or equipment.
The second model that is used in our industry is the gross margin percentage model. This model takes the cost of goods sold and applies a margin by a divisor method. For example, if I had a cost of
goods sold of $5000 and I wanted to apply a 50% margin, the formula would be… $5000 divided by 1 minus 50% or .5, and the selling price would be $10,000. This gross profit would be $5000.
The third model is the gross profit dollars per crew per day or per hour. This model is generally instituted by less than 20% of the HVAC industry. Although it is a growing model for contractors to
implement. This model allows for predictive modeling of a price structure that is used in other industries such as hotels and airlines or any industry that implements a yield management philosophy.
You can read about yield management here. https://en.wikipedia.org/wiki/Yield_management
This philosophy is based on the idea that you have a certain amount of time, people, and resources to perform the services that you have available to your company. Quite frankly, once you figure out
the amount that you should budget for a sales price, the number never changes and can always be the same. Let’s look at how we figure this out.
To be able to figure this out we will need to know some parameters. What is your overhead per month, what is your desired net profit per month, how many total people do you have to perform the work,
and how many days do you have for them to get this work performed in.
Generally, when we are trying to figure out a gross profit cost per crew, we are doing this based on how many crews we have and how often they can work in a month. Here is the formula….
Overhead plus net profit equals gross profit. You also want to include in this number any money that you have in a payable to your distributor. Also, any future capital expenditures that you want to
make for things such as trucks.
We will take this information and divide that by the total number of work days in a month. Now generally there are 20 workdays in a month, Monday through Friday, four weeks a month. However, we don’t
want to plan on being busy every day we need to plan for some breakage. So, I like to have a plan to only be busy around 65% of the time. That will take a 20 day month down to 13 days. It also will
reduce the amount of time you have to do the work which will increase the price of your gross profit for person day. Don’t be afraid of this as the second parameter you want to use as a critical
factor when pricing your jobs is what the market will bear. And what the market will bear is generally a lot higher than what your gross profit per person or per crew day is.
When I take my total gross profit that included my payables to my distributor, with my future capital expenditures and I divide that by the total number of days that I have to do the work, and then I
divide that number by the total number of crews I have to do the work I will come up with a gross profit per crew day. My plan would be to attempt to get each gross profit event to happen in the time
that high planned for it. If I have a one day job, and one gross profit event, then that day needs to not exceed the time allowed or I will lose money. The good news is, since I planned for 35% of
the time to have some breakage then I will be OK as long as it doesn’t happen very often.
I have a calculator that you are welcome to use that explains this very calculation. Gross profit per crew day is not for the light-hearted. I recommend getting with your accountant or a gross profit
coach to determine what your number should be. Feel free to reach out to me if you need help with this area. This calculator is in the admin under the library area and can be download it to your
desktop. I’ll be happy to help users of our program to understand what their numbers should be.
Gross Profit Calculator | {"url":"https://www.entsellingsolutions.com/blog/gross-profit-s-per-crew-day-versus-gross-margin-percentages/","timestamp":"2024-11-13T21:16:35Z","content_type":"text/html","content_length":"84732","record_id":"<urn:uuid:a3eb519c-2666-4697-b5bf-7b47c17d509e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00847.warc.gz"} |
Unleash Right Triangle Trig: Solve Missing Sides & Angles
Unleash the Power of Right Triangle Trig: Solving for Missing Sides and Angles!
Welcome to Warren Institute! In this article, we will dive into the fascinating world of right triangle trigonometry - specifically, how to find missing sides and angles. Trigonometry is a crucial
branch of mathematics that deals with the relationships between angles and sides of triangles. By understanding the principles of right triangle trigonometry, you will be able to solve real-world
problems involving distances, heights, and angles. So, grab your calculators and join us as we unravel the mysteries of right triangle trigonometry and empower ourselves with the tools to conquer any
mathematical challenge. Let's get started!
Introduction to Right Triangle Trig
In this section, we will explore the basic concepts of right triangle trigonometry and how it can be used to find missing sides and angles in a right triangle. We will introduce the sine, cosine, and
tangent ratios and discuss how they relate to the sides of a right triangle.
Using the Sine Ratio
The sine ratio, denoted as sinθ, is defined as the ratio of the length of the side opposite the angle θ to the length of the hypotenuse. This ratio can be used to find missing side lengths or angles
in a right triangle. We will discuss the steps involved in using the sine ratio and provide examples to illustrate its application.
Applying the Cosine Ratio
The cosine ratio, represented as cosθ, is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. It can be used to find missing side lengths or
angles. We will explain how to apply the cosine ratio and provide practice problems to reinforce understanding.
Solving for Missing Angles and Sides using the Tangent Ratio
The tangent ratio, denoted as tanθ, is the ratio of the length of the side opposite the angle θ to the length of the side adjacent to θ in a right triangle. It can be used to find missing angles or
side lengths. We will walk through the steps of solving for missing angles and sides using the tangent ratio and provide real-world applications to demonstrate its usefulness.
frequently asked questions
How do you find the length of a missing side in a right triangle using trigonometry?
To find the length of a missing side in a right triangle using trigonometry, you can use either the sine, cosine, or tangent ratios. These ratios relate the lengths of the sides of a right triangle
to the angles. For example, if you know one angle and one side length, you can use the sine ratio (sin) to find the length of the missing side. Alternatively, if you know two side lengths, you can
use the cosine ratio (cos) or tangent ratio (tan) to find the length of the missing side.
What is the relationship between the angles and sides of a right triangle in terms of trigonometric ratios?
The relationship between the angles and sides of a right triangle in terms of trigonometric ratios can be described using the following formulas:
• The sine ratio (sin): The sine of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.
• The cosine ratio (cos): The cosine of an angle in a right triangle is equal to the length of the adjacent side divided by the length of the hypotenuse.
• The tangent ratio (tan): The tangent of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the adjacent side.
These trigonometric ratios allow us to calculate missing angles or sides in a right triangle based on the known values.
Can you explain how to use the sine, cosine, and tangent functions to find missing angles in a right triangle?
The sine, cosine, and tangent functions are trigonometric functions that are used to find missing angles in a right triangle. In a right triangle, the sine function (sin) is defined as the ratio of
the length of the side opposite the angle to the length of the hypotenuse, the cosine function (cos) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, and
the tangent function (tan) is defined as the ratio of the length of the opposite side to the length of the adjacent side. To find a missing angle, you can use these functions by taking the inverse of
the function to solve for the angle. For example, if you know the lengths of two sides of a right triangle and want to find an angle, you can use the inverse sine, inverse cosine, or inverse tangent
function to find the missing angle.
What are some real-life applications of right triangle trigonometry?
Some real-life applications of right triangle trigonometry include calculating distances using angles and side lengths, determining the height of buildings or objects using angles of elevation or
depression, and solving navigation problems involving vectors and angles.
Are there any shortcuts or special formulas to quickly find missing angles or sides in a right triangle using trigonometry?
Yes, there are special formulas in trigonometry that can be used to quickly find missing angles or sides in a right triangle. The most commonly used formulas are the sine, cosine, and tangent ratios,
which relate the ratios of the lengths of the sides in a right triangle to the measures of its angles. These formulas are known as SOH-CAH-TOA, where Sine = Opposite/Hypotenuse, Cosine = Adjacent/
Hypotenuse, and Tangent = Opposite/Adjacent. By using these formulas, one can easily determine the missing angles or sides in a right triangle.
In conclusion, understanding the concepts and formulas of right triangle trigonometry is essential for students in their mathematics education. By using the SOH-CAH-TOA mnemonic and applying the
sine, cosine, and tangent ratios, students can confidently find missing sides and angles of right triangles. This foundational knowledge not only helps in solving real-world problems involving angles
and distances, but also lays a strong groundwork for further studies in advanced trigonometry and calculus. With practice and perseverance, students can master right triangle trigonometry and enhance
their mathematical skills.
If you want to know other articles similar to Unleash the Power of Right Triangle Trig: Solving for Missing Sides and Angles! you can visit the category General Education. | {"url":"https://warreninstitute.org/right-triangle-trig-finding-missing-sides-and-angles/","timestamp":"2024-11-05T22:08:01Z","content_type":"text/html","content_length":"105961","record_id":"<urn:uuid:0a813256-335e-43e4-aa82-45668a4b8d04>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00389.warc.gz"} |
Biostatistics. - All Writing Help
Descriiptives Homework
Save this on your computer as a Word document. Answer the questions, save your work and then email your work to me as an attachment, send via Canvas, or scan your work in as a pdf attachment. If you
do write your work out and scan it in, be sure that your work is legible, right-side up and in order! Be sure to show your work. If I just have an answer but do not see any calculations, I can’t tell
how you came to your outcome and I can’t provide feedback and support in case your outcome is wrong. Since the assignment is in Word format, you have the option to add extra space if you need it to
include your calculations.
When computing standard deviation, each question will alert you as to whether you should use the sample computational standard deviation formula or the population computational standard deviation
formula. Review your notes on the differences between the sample and population standard deviation. To help with the organization of your data and formula, I have provided for you tables for you to
place your X value and X2, which you will need to calculate the formula. It is not necessary to organize your data on the table from lowest to highest, but you can do this to help in your data
organization and evaluation.
To help provide you with the standard deviation key, I have listed below the formulas for both sample and population standard deviation computational formula:
Population standard deviation formula:
σ = ΣX2 – (ΣX)2
_ __N__
Sample standard deviation formula:
s = Σx2 – (Σx)2
n __
n – 1
SHOW THE WORK FOR ALL QUESTIONS REQUIRING MATH and FORMULAS!!
1. The Newport Health Clinic experiments with two different configurations for serving patients. In one configuration, all patients enter a single waiting line that feeds three different physicians.
In another configuration, patients wait in individual lines at three different physician stations. Waiting times (in minutes) are recorded for ten patients from each configuration.
Single line: 65, 66, 67, 68, 71, 73, 74, 77, 77, 77
a. What is the “n” value for this data set?
b. What is the mean for this data set?
c. What is the median for this data set?
d. What is the mode for this data set?
e. What is the standard deviation for this data set? For this example, you will use the sample data computational formula.
X X2
∑X =
(∑X)2 = ∑X2 =
Show your SD formula work below:
Multiple lines: 42, 54, 58, 62, 67, 77, 77, 85, 93, 100
f. What is the “n” value for this data set?
g. What is the mean for this data set?
h. What is the median for this data set?
i. What is the mode for this data set?
j. What is the standard deviation for this data set (Remember to use the sample computational formula.)
X X2
∑X =
(∑X)2 = ∑X2 =
Show your SD formula work below:
2. Based on the data that you have just calculated, which wait time is the most efficient with the least wait time between the single line vs. multiple lines?
3. After a week of practicing with the visualization technique, 20 basketball players again shot 25 free throws, and the sports psychologist recorded the number of successful shots (Remember your
frequency distribution tables assignment from last week? As you see below, this data is organized in a frequency distribution table. As you recall, large amounts of data are often organized in
frequency distribution tables to communicate and share data in a more organized, condense manner. Next to each score is the f or frequency of that score (how many times that score occurs in the data
set). Before you begin calculating central tendency and variability, you must make a list of all your raw score values (not just the list of scores under the ‘X’ column). For example, if a value has
a corresponding frequency of ‘0’, then that means that particular value is not a part of your data. If a value has a corresponding frequency of ‘3’ (for example, X=20) then you would list that value
three times under the X column (20, 20, 20):
X f X f
a. What is the “N” value for this data set?
b. What is the mean for this data set?
c. What is the mode for this data set?
d. What is the median for this data set?
e. Calculate the standard deviation for this data set using the POPULATION computational formula.
X X2
∑X =
(∑X)2 = ∑X2 =
Show your SD formula work below:
4. Is it possible to have more than one mode?
5. A study was conducted to see if Physical Therapy is or is not effective. Below are progression scores compiled for both the Physical Therapy group and the Control group (those who did not receive
Physical Therapy). The higher the scores, the more progress is made by the study individuals. Negative scores indicate a decline in progression. What are the differences in the means and variances?
Based upon the descriiptive data, do you feel that the Physical Therapy group fared better than the Control group? Do you have any other observations about the data? (since this is a sample set, you
will want to use the sample computational standard deviation formula). I placed the values in a table below for easy organization. Since there are negative values, be careful with your math!!!
Remember that when squaring two negative values (i.e., multiplying them by themselves), the negatives cancel each other out and you end up with a positive. This means that you should have no negative
values under your X2 column.
Physical therapy group scores X2 Control
Group scores X2
-5 4
-10 18
34 -22
18 -9
∑X =
(∑X)2 = ∑X2 = ∑X =
(∑X)2 = ∑X2 =
Calculate the mean, median, mode and SD to evaluate the differences between the two groups.
What are the differences in the means and variances?
Based upon the descriiptive data, do you feel that the Physical Therapy group fared better than the Control group?
Do you have any other observations about the data? | {"url":"https://allwritinghelp.com/biostatistics/","timestamp":"2024-11-13T18:56:31Z","content_type":"text/html","content_length":"65981","record_id":"<urn:uuid:18558d79-0422-43a6-ba3f-e53428cc88cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00064.warc.gz"} |
Bond-Angle-Torsion coordinates analysis —
4.2.8. Bond-Angle-Torsion coordinates analysis — MDAnalysis.analysis.bat
Soohaeng Yoo Willow and David Minh
GNU Public License, v2 or any higher version
This module contains classes for interconverting between Cartesian and an internal coordinate system, Bond-Angle-Torsion (BAT) coordinates [1], for a given set of atoms or residues. This coordinate
system is designed to be complete, non-redundant, and minimize correlations between degrees of freedom. Complete and non-redundant means that for N atoms there will be 3N Cartesian coordinates and 3N
BAT coordinates. Correlations are minimized by using improper torsions, as described in [2].
More specifically, bond refers to the bond length, or distance between a pair of bonded atoms. Angle refers to the bond angle, the angle between a pair of bonds to a central atom. Torsion refers to
the torsion angle. For a set of four atoms a, b, c, and d, a torsion requires bonds between a and b, b and c, and c and d. The torsion is the angle between a plane containing atoms a, b, and c and
another plane containing b, c, and d. For a set of torsions that share atoms b and c, one torsion is defined as the primary torsion. The others are defined as improper torsions, differences between
the raw torsion angle and the primary torsion. This definition reduces the correlation between the torsion angles.
Each molecule also has six external coordinates that define its translation and rotation in space. The three Cartesian coordinates of the first atom are the molecule’s translational degrees of
freedom. Rotational degrees of freedom are specified by the axis-angle convention. The rotation axis is a normalized vector pointing from the first to second atom. It is described by the polar angle,
\(\phi\), and azimuthal angle, \(\theta\). \(\omega\) is a third angle that describes the rotation of the third atom about the axis.
This module was adapted from AlGDock [3].
See also
class to calculate dihedral angles for a given set of atoms or residues
function to calculate dihedral angles from atom positions
4.2.8.1. Example applications
The BAT class defines bond-angle-torsion coordinates based on the topology of an atom group and interconverts between Cartesian and BAT coordinate systems.
For example, we can determine internal coordinates for residues 5-10 of adenylate kinase (AdK). The trajectory is included within the test data files:
import MDAnalysis as mda
from MDAnalysisTests.datafiles import PSF, DCD
import numpy as np
u = mda.Universe(PSF, DCD)
# selection of atomgroups
selected_residues = u.select_atoms("resid 5-10")
from MDAnalysis.analysis.bat import BAT
R = BAT(selected_residues)
# Calculate BAT coordinates for a trajectory
After R.run(), the coordinates can be accessed with R.results.bat. The following code snippets assume that the previous snippet has been executed.
Reconstruct Cartesian coordinates for the first frame:
# Reconstruct Cartesian coordinates from BAT coordinates
# of the first frame
XYZ = R.Cartesian(R.results.bat[0,:])
# The original and reconstructed Cartesian coordinates should all be close
print(np.allclose(XYZ, selected_residues.positions, atol=1e-6))
Change a single torsion angle by \(\pi\):
bat = R.results.bat[0,:]
bat[bat.shape[0]-12] += np.pi
XYZ = R.Cartesian(bat)
# A good number of Cartesian coordinates should have been modified
np.sum((XYZ - selected_residues.positions)>1E-5)
Store data to the disk and load it again:
# BAT coordinates can be saved to disk in the numpy binary format
# The BAT coordinates in a new BAT instance can be loaded from disk
# instead of using the run() method.
Rnew = BAT(selected_residues, filename='test.npy')
# The BAT coordinates before and after disk I/O should be close
print(np.allclose(Rnew.results.bat, R.results.bat))
4.2.8.2. Analysis classes
class MDAnalysis.analysis.bat.BAT(ag, initial_atom=None, filename=None, **kwargs)[source]
Calculate BAT coordinates for the specified AtomGroup.
Bond-Angle-Torsions (BAT) internal coordinates will be computed for the group of atoms and all frame in the trajectory belonging to ag.
☆ ag (AtomGroup or Universe) – Group of atoms for which the BAT coordinates are calculated. ag must have a bonds attribute. If unavailable, bonds may be guessed using AtomGroup.guess_bonds.
ag must only include one molecule. If a trajectory is associated with the atoms, then the computation iterates over the trajectory.
☆ initial_atom (Atom) – The atom whose Cartesian coordinates define the translation of the molecule. If not specified, the heaviest terminal atom will be selected.
☆ filename (str) – Name of a numpy binary file containing a saved bat array. If filename is not None, the data will be loaded from this file instead of being recalculated using the run()
Contains the time series of the Bond-Angle-Torsion coordinates as a (nframes, 3N) numpy.ndarray array. Each row corresponds to a frame in the trajectory. In each column, the first six
elements describe external degrees of freedom. The first three are the center of mass of the initial atom. The next three specify the external angles according to the axis-angle convention: \
(\phi\), the polar angle, \(\theta\), the azimuthal angle, and \(\omega\), a third angle that describes the rotation of the third atom about the axis. The next three degrees of freedom are
internal degrees of freedom for the root atoms: \(r_{01}\), the distance between atoms 0 and 1, \(r_{12}\), the distance between atoms 1 and 2, and \(a_{012}\), the angle between the three
atoms. The rest of the array consists of all the other bond distances, all the other bond angles, and then all the other torsion angles.
Conversion of a single frame from BAT to Cartesian coordinates
One application of this function is to determine the new Cartesian coordinates after modifying a specific torsion angle.
bat_frame (numpy.ndarray) – an array with dimensions (3N,) with external then internal degrees of freedom based on the root atoms, followed by the bond, angle, and (proper and improper)
torsion coordinates.
XYZ – an array with dimensions (N,3) with Cartesian coordinates. The first dimension has the same ordering as the AtomGroup used to initialize the class. The molecule will be whole
opposed to wrapped around a periodic boundary.
Return type:
property atoms
The atomgroup for which BAT are computed (read-only property)
load(filename, start=None, stop=None, step=None)[source]
Loads the bat trajectory from a file in numpy binary format
See also
Saves the bat trajectory in a file in numpy binary format
run(start=None, stop=None, step=None, frames=None, verbose=None, *, progressbar_kwargs={})
Perform the calculation
Changed in version 2.2.0: Added ability to analyze arbitrary frames by passing a list of frame indices in the frames keyword argument.
Changed in version 2.5.0: Add progressbar_kwargs parameter, allowing to modify description, position etc of tqdm progressbars
Saves the bat trajectory in a file in numpy binary format
See also
Loads the bat trajectory from a file in numpy binary format | {"url":"https://docs.mdanalysis.org/stable/documentation_pages/analysis/bat.html","timestamp":"2024-11-06T22:01:58Z","content_type":"text/html","content_length":"47304","record_id":"<urn:uuid:663dd922-c00d-4846-a4d9-71075b5eceba>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00342.warc.gz"} |
A flexible and efficient framework for data-driven stochastic disease spread simulations
The package provides an efficient and very flexible framework to conduct data-driven epidemiological modeling in realistic large scale disease spread simulations. The framework integrates infection
dynamics in subpopulations as continuous-time Markov chains using the Gillespie stochastic simulation algorithm and incorporates available data such as births, deaths and movements as scheduled
events at predefined time-points. Using C code for the numerical solvers and ‘OpenMP’ (if available) to divide work over multiple processors ensures high performance when simulating a sample outcome.
One of our design goals was to make the package extendable and enable usage of the numerical solvers from other R extension packages in order to facilitate complex epidemiological research. The
package contains template models and can be extended with user-defined models.
Getting started
You can use one of the predefined compartment models in SimInf, for example, SEIR. But you can also define a custom model ‘on the fly’ using the model parser method mparse. The method takes a
character vector of transitions in the form of X -> propensity -> Y and automatically generates the C and R code for the model. The left hand side of the first arrow (->) is the initial state, the
right hand side of the last arrow (->) is the final state, and the propensity is written between the two arrows. The flexibility of the mparse approach allows for quick prototyping of new models or
features. To illustrate the mparse functionality, let us consider the SIR model in a closed population i.e., no births or deaths. Let beta denote the transmission rate of spread between a susceptible
individual and an infectious individual and gamma the recovery rate from infection (gamma = 1 / average duration of infection). It is also possible to define variables which can then be used in
calculations of propensities or in calculations of other variables. A variable is defined by the operator <-. Using a variable for the size of the population, the SIR model can be described as:
transitions <- c("S -> beta*S*I/N -> I",
"I -> gamma*I -> R",
"N <- S+I+R")
compartments <- c("S", "I", "R")
The transitions and compartments variables together with the constants beta and gamma can now be used to generate a model with mparse. The model also needs to be initialised with the initial
condition u0 and tspan, a vector of time points where the state of the system is to be returned. Let us create a model that consists of 1000 replicates of a population, denoted a node in SimInf, that
each starts with 99 susceptibles, 5 infected and 0 recovered individuals.
n <- 1000
u0 <- data.frame(S = rep(99, n), I = rep(5, n), R = rep(0, n))
model <- mparse(transitions = transitions,
compartments = compartments,
gdata = c(beta = 0.16, gamma = 0.077),
u0 = u0,
tspan = 1:150)
To generate data from the model and then print some basic information about the outcome, run the following commands:
#> Model: SimInf_model
#> Number of nodes: 1000
#> Number of transitions: 2
#> Number of scheduled events: 0
#> Global data
#> -----------
#> Parameter Value
#> beta 0.160
#> gamma 0.077
#> Compartments
#> ------------
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> S 1.00 19.00 30.00 40.74 60.00 99.00
#> I 0.00 0.00 4.00 6.87 11.00 47.00
#> R 0.00 28.00 67.00 56.39 83.00 103.00
There are several functions in SimInf to facilitate analysis and post-processing of simulated data, for example, trajectory, prevalence and plot. The default plot will display the median count in
each compartment across nodes as a colored line together with the inter-quartile range using the same color, but with transparency.
Most modeling and simulation studies require custom data analysis once the simulation data has been generated. To support this, SimInf provides the trajectory method to obtain a data.frame with the
number of individuals in each compartment at the time points specified in tspan. Below is the first 10 lines of the data.frame with simulated data.
#> node time S I R
#> 1 1 1 98 6 0
#> 2 2 1 98 6 0
#> 3 3 1 98 6 0
#> 4 4 1 99 5 0
#> 5 5 1 97 7 0
#> 6 6 1 98 5 1
#> 7 7 1 99 5 0
#> 8 8 1 99 5 0
#> 9 9 1 97 7 0
#> 10 10 1 97 6 1
Finally, let us use the prevalence method to explore the proportion of infected individuals across all nodes. It takes a model object and a formula specification, where the left hand side of the
formula specifies the compartments representing cases i.e., have an attribute or a disease and the right hand side of the formula specifies the compartments at risk. Below is the first 10 lines of
the data.frame.
#> time prevalence
#> 1 1 0.05196154
#> 2 2 0.05605769
#> 3 3 0.06059615
#> 4 4 0.06516346
#> 5 5 0.06977885
#> 6 6 0.07390385
#> 7 7 0.07856731
#> 8 8 0.08311538
#> 9 9 0.08794231
#> 10 10 0.09321154
Learn more
See the vignette to learn more about special features that the SimInf R package provides, for example, how to:
• use continuous state variables
• use the SimInf framework from another R package
• incorporate available data such as births, deaths and movements as scheduled events at predefined time-points.
You can install the released version of SimInf from CRAN
or use the remotes package to install the development version from GitHub
We refer to section 3.1 in the vignette for detailed installation instructions.
In alphabetical order: Pavol Bauer , Robin Eriksson , Stefan Engblom , and Stefan Widgren (Maintainer)
Any suggestions, bug reports, forks and pull requests are appreciated. Get in touch.
SimInf is research software. To cite SimInf in publications, please use:
• Widgren S, Bauer P, Eriksson R, Engblom S (2019) SimInf: An R Package for Data-Driven Stochastic Disease Spread Simulations. Journal of Statistical Software, 91(12), 1–42. doi: 10.18637/
• Bauer P, Engblom S, Widgren S (2016) Fast event-based epidemiological simulations on national scales. International Journal of High Performance Computing Applications, 30(4), 438–453. doi:
This software has been made possible by support from the Swedish Research Council within the UPMARC Linnaeus center of Excellence (Pavol Bauer, Robin Eriksson, and Stefan Engblom), the Swedish
Research Council Formas (Stefan Engblom and Stefan Widgren), the Swedish Board of Agriculture (Stefan Widgren), the Swedish strategic research program eSSENCE (Stefan Widgren), and in the framework
of the Full Force project, supported by funding from the European Union’s Horizon 2020 Research and Innovation programme under grant agreement No 773830: One Health European Joint Programme (Stefan
The SimInf package uses semantic versioning.
The SimInf package is licensed under the GPLv3. | {"url":"https://cran.rstudio.org/web/packages/SimInf/readme/README.html","timestamp":"2024-11-03T20:30:13Z","content_type":"application/xhtml+xml","content_length":"18738","record_id":"<urn:uuid:6374d40c-85f0-4c47-b272-bee87f7214ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00047.warc.gz"} |
Linear Pair of Angles—Definition, Axiom, Examples - Grade Potential Durham, NC
Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential concept in geometry. With multiple real-life applications, you'd be astonished to figure out how relevant this figure can be. Although you might think it has
no use in your life, we all should understand the concept to nail those examinations in school.
To save you time and make this data readily accessible, here is an introductory insight into the properties of a linear pair of angles, with diagrams and examples to guide with your private study
sessions. We will also discuss some real-world and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are theories that continue to be relevant as you go forward in geometry and more complex theorems and proofs. We will answer this question with a simple
explanation in this single point.
A linear pair of angles is the name designated to two angles that are positioned on a straight line and the total of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that are aligned on the same line and together create a straight line. The sum of the angles in a linear pair will at all times make a straight
angle equivalent
times to 180 degrees.
It is essential to note that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This implies that at all times form on a straight line and are at all times
supplementary angles.
It is important to make clear that, while the linear pair are always adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
Over the definition simplified, we will examine the two axioms critical to fully understand any example given to you.
Initiate with definition of what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; it is deemed obvious and self-evident. A linear pair of angles has two
axioms connected with them.
The first axiom establishes that if a ray is located on a line, the adjacent angles will make a straight angle, also known as a linear pair.
The second axiom establishes that if two angles produces a linear pair, then uncommon arms of both angles produces a straight angle between them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To envision these axioms better, here are some diagram examples with their individual explanations.
Example One
As we can see in this example, we have two angles that are adjacent to each other. As you can see in the image, the adjacent angles form a linear pair because the total of their measures is
equivalent to 180 degrees. They are also supplementary angles, as they share a side and a common vertex.
Angle A: 75 degrees
Angle B: 105 degrees
Sum of Angles A and B: 75 + 105 = 180
Example Two
In this instance, we have two lines intersect, producing four angles. Not all angles creates a linear pair, but respective angle and the one adjacent to it makes a linear pair.
∠A 30 degrees
∠B: 150 degrees
∠C: 30 degrees
∠D: 150 degrees
In this instance, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This instance presents an intersection of three lines. Let's take note of the axiom and characteristics of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations add up to 180 degrees. As a consequence, we can conclude that this diagram has no linear pair until we extend a straight line.
Implementations of Linear Pair of Angles
At the moment we have explored what linear pairs are and have looked at some examples, let’s check how this concept can be used in geometry and the real world.
In Real-Life Situations
There are many applications of linear pairs of angles in real-world. One common example is architects, who apply these axioms in their daily work to identify if two lines are perpendicular and
creates a straight angle.
Builders and construction professionals also use expertise in this field to make their job less complex. They use linear pairs of angles to assure that two adjacent walls create a 90-degree angle
with the ground.
Engineers also utilizes linear pairs of angles frequently. They do so by working out the weight on the beams and trusses.
In Geometry
Linear pairs of angles as well perform a role in geometry proofs. A ordinary proof that employs linear pairs is the alternate interior angles concept. This concept explains that if two lines are
parallel and intersected by a transversal line, the alternate interior angles made are congruent.
The proof of vertical angles additionally replies on linear pairs of angles. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to
each other. Because of previously mentioned two rules, you only need to figure out the measurement of any one angle to determine the measure of the rest.
The theorem of linear pairs is further employed for more complicated implementation, such as measuring the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready
for more progressive geometry.
As you can see, linear pairs of angles are a comparatively simple concept with some fascinating implementations. Next time you're out and about, take note if you can spot any linear pairs! And, if
you're attending a geometry class, take notes on how linear pairs may be useful in proofs.
Improve Your Geometry Skills using Grade Potential
Geometry is entertaining and valuable, especially if you are looking into the field of architecture or construction.
However, if you're struggling to understand linear pairs of angles (or any other theorem in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert teachers
can guide you understand the material and nail your next examination. | {"url":"https://www.durhaminhometutors.com/blog/linear-pair-of-angles-definition-axiom-examples","timestamp":"2024-11-03T23:04:06Z","content_type":"text/html","content_length":"78287","record_id":"<urn:uuid:85835b63-8829-4e6e-b170-073e58fca4b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00282.warc.gz"} |
Autoencoders Interview Questions | Autoencoders Viva Questions - Avatto-
When the code or latent representation has the dimension higher than the dimension of the input then the autoencoder is called the overcomplete autoencoder. On the contrary, when the code or latent
representation has the dimension lower than the dimension of the input then the autoencoder is called the undercomplete autoencoder.
The encoder takes the given input and outputs the low dimensional latent representation of the input. The decoder takes this low dimensional latent representation generated by the encoder as an input
and tries to reconstruct the original input.
The autoencoders map the data of a high dimension data to a low-level representation. This low-level representation of data is called as latent representation or bottleneck. The bottleneck comprises
of only meaningful and important features that represent the input.
The difference between the autoencoder and PCA is that PCA uses linear transformation for dimensionality reduction while the autoencoder uses a nonlinear transformation for dimensionality reduction. | {"url":"https://www.avatto.com/data-scientist/interview-questions/deep-learning/autoencoders/page/2/","timestamp":"2024-11-12T13:34:36Z","content_type":"text/html","content_length":"136720","record_id":"<urn:uuid:73f41101-7709-422a-8a72-5bdf27c3c09b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00814.warc.gz"} |
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Implied VolatilityImplied Volatility in Options Trading: Key Insights & Strategies
Implied Volatility: A Key Guide for Options Traders
Implied Volatility (IV) is a critical concept in the world of finance, particularly in options trading. It reflects the market’s expectations regarding the volatility of an asset’s price over a
specific period. Unlike historical volatility, which looks at past price movements, implied volatility is forward-looking and derived from the prices of options. Higher implied volatility indicates
that the market expects significant price fluctuations, while lower implied volatility suggests the opposite.
Implied volatility is essential for several reasons:
• It helps traders gauge market sentiment. A spike in implied volatility might indicate increased uncertainty or potential news events impacting the asset.
• It serves as a critical input for pricing options. The higher the implied volatility, the more expensive the options tend to be, as they offer greater potential for profit.
• It can assist in identifying trading opportunities. For instance, when implied volatility is low, it may be an attractive time to buy options, anticipating future price movements.
Implied volatility comprises several components:
• Market Sentiment: The overall feeling or attitude of investors towards the market can significantly impact implied volatility. Bullish sentiment might lead to higher IV as investors expect price
• Earnings Announcements: Companies often experience increased implied volatility leading up to earnings reports, as traders speculate on potential outcomes.
• Market Conditions: Economic events, geopolitical tensions or significant market changes can cause fluctuations in implied volatility.
There are two main types of implied volatility that traders often consider:
• Constant Implied Volatility: This assumes that volatility remains stable over time, which is a simplifying assumption often used in models.
• Stochastic Implied Volatility: This acknowledges that volatility can change unpredictably, allowing for a more realistic modeling of market behaviors.
In recent years, trends in implied volatility have evolved due to various factors:
• Increased Market Participation: The rise of retail trading platforms has led to more participants in the market, which can influence implied volatility.
• Technological Advancements: Innovations in trading algorithms and data analytics have made it easier to assess and respond to implied volatility.
• Global Events: Events like the pandemic or geopolitical tensions have shown how quickly implied volatility can change, impacting trading strategies.
Traders often employ various strategies based on implied volatility:
• Straddles and Strangles: These strategies involve buying both call and put options to capitalize on expected price movements, particularly when implied volatility is low.
• Volatility Arbitrage: Traders may look for discrepancies between implied and historical volatility to exploit potential mispricings.
• Iron Condors: This strategy involves selling options at different strike prices to generate income when implied volatility is high, as it anticipates minimal price movement.
Implied volatility is a vital metric in finance, particularly for options trading. Understanding its implications can empower investors to make informed decisions, manage risk and seize opportunities
in the market. As trends evolve and markets change, staying informed about implied volatility will remain crucial for successful trading strategies.
What is implied volatility and why is it important in trading?
Implied volatility represents the market’s forecast of a likely movement in an asset’s price and is crucial for pricing options and assessing market sentiment.
How can investors use implied volatility to enhance their trading strategies?
Investors can leverage implied volatility to identify potential trading opportunities, manage risk and make informed decisions on options strategies.
More Terms Starting with I | {"url":"https://docs.familiarize.com/glossary/implied-volatility/","timestamp":"2024-11-08T09:12:17Z","content_type":"text/html","content_length":"95423","record_id":"<urn:uuid:b38963f6-1a5b-4aec-9303-d160d80416bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00676.warc.gz"} |
How to Calculate MD 5 in Java
“MD5” corresponds to a widely used cryptographic algorithm in Java which generates a hash of 128 bits. This algorithm utilizes the “MessageDigest” class which is contained in the “java.security”
package. This algorithm is effective as it consumes less resources.
This article will discuss the approach to compute “MD5” in Java.
What is MD5?
“MD5” corresponds to a cryptographic algorithm that gives the hash functions to retrieve a fixed length i.e., 128-bit (16 bytes) hash value.
How to Calculate MD 5 in Java?
To compute the cryptographic hashing value in Java, the “MessageDigest” class is utilized. This class provides the following cryptographic hash function to return the hash value of a text:
Working of the “MessageDigest” Class
The above-stated hash functions/algorithms are initialized in the static “getInstance()” method. After opting for the algorithm i.e., “MD5” etc., it computes the digest value and retrieves the
results in a byte array. After that, the “BigInteger” class is applied which transforms the resultant byte array into the corresponding sign-magnitude representation.
Advantages of MD5
• Convenient for comparing small hashes.
• Less consumption of resources.
• Convenience in storing passwords.
• Improved Integrity.
Working of MD5
The “MD5” algorithm works based on the following 4 steps:
Step 1: Add Extra Bits
The first step of the algorithm includes appending extra bits to the provided string. It is done to make the length of the string a multiple of 512 bits.
Step 2: Append/Add Length
After adding extra bits, add length by appending 64 bits at last. It keeps track of the input’s length provided by the user.
Step 3: Initialize the MD Buffer
MD buffer refers to a 4-word (A, B, C, D) buffer where each word refers to a 32-bit register that calculates the message digest’s value.
Step 4: Processing in a 16-Word Block
The “MD5” algo utilizes the functions(auxiliary) that take the inputs as three 32-bit numbers and generate 32-bits as an outcome. These functions utilize the OR, XOR, and NOR operators.
Example: Calculating the “MD5” Hash Value in Java
The below code example computes the “MD5” hash value:
import java.math.BigInteger;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
public class MD5 {
public static String retrieveMd5(String x){
try {
MessageDigest a = MessageDigest.getInstance("MD5");
byte[] messageDigest = a.digest(x.getBytes());
BigInteger b = new BigInteger(1, messageDigest);
String hashtext = b.toString(16);
while (hashtext.length() < 32) {
hashtext = "0" + hashtext;
return hashtext;
catch (NoSuchAlgorithmException except) {
throw new RuntimeException(except);
public static void main(String args[]) throws NoSuchAlgorithmException{
String c = "Harry";
System.out.println("HashCode Generated Via MD5 -> " + retrieveMd5(c));
According to the above code limes, perform the below-given steps:
• Declare the function “retrieveMd5()” that takes the passed string as its argument.
• In the “try” block, apply the “getInstance()” method with the “MD5” hashing.
• After that, the “digest()” method is utilized to compute the message digest of an input digest that retrieves an array of bytes.
• Now, transform the array to the representation of signum.
• In the next step, convert the message digest to hex value via the combined “toString()” and “length()” methods.
• In the “catch” block, cope up with the probable message digest algorithms limitation.
• Finally, in “main”, invoke the defined function by passing the defined string as its argument to fetch the corresponding hashcode based on this string.
This outcome implies that the corresponding hash code with respect to the string is generated appropriately.
To compute the cryptographic hashing value in Java, the “MessageDigest” class is utilized to compute the digest value, retrieves the results in a byte array and the “BigInteger” class transforms the
resultant byte array into its sign-magnitude representation. This write-up elaborated on computing the “MD 5” hash value in Java. | {"url":"https://javabeat.net/how-to-calculate-md-5-in-java/","timestamp":"2024-11-08T15:18:51Z","content_type":"text/html","content_length":"76554","record_id":"<urn:uuid:5a85437b-f497-4574-b20a-ca4eb89c6da8>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00811.warc.gz"} |
range and division : unexpected behavior
range and division : unexpected behavior
Consider the following snippet :
# code 1
for n in range(N,N+1):
for k in range(0,n):
print k/n
print '-'*10
# code 2
for k in range(0,n):
print k/n
I was expecting code 1 and code 2 to print the same output. This is not the case :
In the first case, k/n is Python-evaluated as an integer division, in the second case, k/n is Sage-evaluated as a fraction. Can someone elaborate please ?
I only notice that substituting srange(N,N+1) to range(N,N+1) fixes the problem.
1 Answer
Sort by ยป oldest newest most voted
There is a difference between Sage and Python (2.x) concerning the division operator / . In Python 2.x the operator / returns the floor of the result of division if the operands are integers (Python
ints). In Sage the operator / returns the result as rational number (if so) if the operands are Sage integers. The Python function range returns a list of Python integers. In your nested for loops
all operands are Python ints, so you get the floor of the division result. In your second code example N (and therefor n) is an Sage integer, so you get the rational numbers as result.
The Sage function srange returns a list of Sage integers. With srange(N,N+1) in your first code n becomes a Sage integer.
You can see the difference by placing some print type(k) and print type(n) commands inside the loops.
edit flag offensive delete link more
Thanks, now the benefits of using srange is more apparent.
candide ( 2016-03-27 12:57:40 +0100 )edit | {"url":"https://ask.sagemath.org/question/32891/range-and-division-unexpected-behavior/","timestamp":"2024-11-13T01:46:58Z","content_type":"application/xhtml+xml","content_length":"56957","record_id":"<urn:uuid:ac2d7a4f-a8c8-4b04-af4d-5820e8b4b4ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00588.warc.gz"} |
Mastering Mathematical Optimization: Finding Absolute Extrema Theoretically
1. Mastering Mathematical Optimization: Finding Absolute Extrema Theoretically
Unlocking the Theoretical Framework: Finding Absolute Extrema in Mathematical Optimization
September 28, 2023
Alan Draven
With a PhD in mathematics, Alan Draven is a reliable and highly experienced assignment helper. He has over 800 clients.
Math assignments often present challenges for university students, requiring a solid understanding of theoretical concepts to find solutions. In this comprehensive theoretical discussion, we will
embark on a journey to find the absolute maximum and minimum values of the function f(x) = x^(1/3) - 3x within the closed interval [0, 2]. By immersing ourselves in the theoretical aspects of this
problem, we aim to equip students with the knowledge and tools necessary to confidently tackle similar assignments. So, let's delve into the theoretical exploration of finding extrema in mathematics
and how it can help you solve your Optimization assignment.
Understanding Extrema
Before we dive into the problem, let's establish a fundamental concept: extrema. Extrema are points where a function reaches its maximum (highest) or minimum (lowest) values. In the context of
calculus, we categorize extrema into two types:
1. Local Extrema: These are points where the function reaches its highest or lowest values within a small neighborhood of the point. Local maxima are peaks, while local minima are valleys.
2. Absolute Extrema: These are the highest and lowest points that a function attains over an entire interval. Absolute maxima are the global peaks, and absolute minima are the global valleys.
For our assignment, we are interested in finding the absolute extrema of f(x) within the interval [0, 2].
Theoretical Framework: The Extreme Value Theorem
The Extreme Value Theorem is a critical theorem in calculus that serves as the theoretical foundation for finding absolute extrema. It states that if a function f(x) is continuous on a closed
interval [a, b], then f(x) must attain both its absolute maximum and minimum values on that interval. The interval [0, 2] in our problem statement is closed, and our function is continuous, making
this theorem applicable.
Understanding the Extreme Value Theorem is crucial for students when approaching problems that involve finding extrema on closed intervals. It guarantees that there are indeed maximum and minimum
values to be found.
Deriving the Critical Points
To begin our journey towards finding the absolute extrema, we must first locate the critical points. Critical points are potential candidates for extrema and occur where the derivative of the
function equals zero or is undefined.
Let's compute the derivative of f(x):
f(x) = x^(1/3) - 3x
f'(x) = (1/3) * x^(-2/3) - 3
To find critical points:
(1/3) * x^(-2/3) - 3 = 0
Now, let's solve for x:
x^(-2/3) = 9/3
x^(-2/3) = 3
Taking the reciprocal of both sides:
x^(2/3) = 1/3
Now, raise both sides to the power of 3/2:
x = (1/3)^(3/2)
x = 1/3^(3/2)
x = 1/3√3
We have found one critical point at x = 1/3√3.
Boundary Points: A Key Theoretical Consideration
In addition to critical points, we must consider the behavior of the function at the boundary points of the interval [0, 2]. These boundary points are x = 0 and x = 2.
Let's evaluate the function at these boundary points:
f(0) = 0^(1/3) - 3 * 0 = 0
f(2) = 2^(1/3) - 3 * 2 ≈ -2.079
These values represent the extremes of our interval, and they are crucial theoretical considerations when determining the absolute extrema.
Comparing Values
Now that we have identified the critical point x = 1/3√3 and evaluated the function at the boundary points x = 0 and x = 2, we can compare these values to find the absolute maximum and minimum of f
(x) on the interval [0, 2].
- f(1/3√3) ≈ -1.693
- f(0) = 0
- f(2) ≈ -2.079
We can conclude that:
- The absolute maximum value of f(x) = x^(1/3) - 3x on the interval [0, 2] is approximately 0 (occurs at x = 0).
- The absolute minimum value of f(x) = x^(1/3) - 3x on the interval [0, 2] is approximately -2.079 (occurs at x = 2).
Theoretical Reflection: Understanding the Results
Our theoretical exploration has yielded the absolute maximum and minimum values of the function f(x) within the closed interval [0, 2]. But what do these results mean, and how can we interpret them
The absolute maximum value of approximately 0 at x = 0 signifies that within the interval [0, 2], the function f(x) never exceeds 0. It represents the highest point of the function on this interval.
Conversely, the absolute minimum value of approximately -2.079 at x = 2 is the lowest point that f(x) reaches within the interval [0, 2]. This point serves as the nadir of the function in this
specific interval.
The Role of Continuity in Optimization
The concept of continuity is a linchpin in the mathematical study of optimization. As we've mentioned, for the Extreme Value Theorem to apply, the function must be continuous over the interval of
interest. Understanding why continuity matters is essential.
Consider the function f(x) = x^(1/3) - 3x. Its continuity ensures that there are no sudden jumps or breaks in the graph within the interval [0, 2]. This guarantees that we can find the absolute
extrema with confidence. In real-world applications, continuity assures us that gradual changes in variables correspond to gradual changes in the function's values, making it a crucial condition for
meaningful optimization.
Theoretical Approach to Finding Critical Points
The process of finding critical points is a fundamental step in finding extrema. Critical points provide insights into where the function might attain maximum or minimum values. In our example, we
calculated the derivative to find critical points. However, it's crucial to understand that not all critical points lead to extrema. Some critical points may correspond to saddle points or points of
inflection, where the function changes concavity but doesn't have an extremum.
To determine whether a critical point corresponds to an extremum, we often use the second derivative test. This test examines the concavity of the function around a critical point and helps classify
it as a local maximum, local minimum, or neither.
Introducing the second derivative test enriches the theoretical understanding of critical points and their role in optimization.
Real-World Significance of Absolute Extrema
Highlighting the real-world significance of finding absolute extrema adds depth to the theoretical discussion. Absolute extrema are not just abstract mathematical concepts; they have practical
applications in various fields.
In economics, for instance, businesses aim to maximize profits or minimize costs, which involves finding the absolute extrema of cost and revenue functions. Engineers optimize designs to minimize
materials and costs while maximizing efficiency. In physics, the path taken by a projectile is optimized to achieve maximum range or height. These examples demonstrate that the principles of
optimization, including the identification of extrema, have tangible real-world implications.
Theoretical vs. Computational Approaches
Our discussion so far has focused on the theoretical approach to finding extrema. However, it's important to acknowledge that in practice, computational tools and software can be immensely helpful.
In cases involving complex functions or high-dimensional spaces, numerical methods and computer programs like Mathematica, MATLAB, or Python can efficiently find extrema.
Balancing theoretical knowledge with practical computational skills equips students to tackle real-world problems effectively. It's valuable to understand both approaches and when to apply them.
Multi-Dimensional Optimization
While we've explored finding extrema in one-dimensional functions, it's crucial to mention that optimization often extends to multi-dimensional functions. In these scenarios, the theoretical concepts
discussed expand to partial derivatives, gradients, and the concept of constrained optimization.
For example, in economics, utility functions may involve multiple variables, and the goal could be to maximize utility subject to budget constraints. Engineering problems may require optimizing
systems with multiple parameters. These scenarios introduce advanced concepts in calculus and linear algebra.
Optimization in Business and Economics
The theoretical concepts we've discussed so far have wide-reaching applications, including in the fields of business and economics. Let's delve deeper into how optimization is employed in these
In economics, businesses seek to maximize profits or minimize costs. This often involves finding the absolute extrema of cost, revenue, and profit functions. For instance, a company may want to
determine the production level that maximizes its profit, taking into account factors such as production costs and market demand. This type of optimization problem has real-world consequences,
impacting pricing strategies, production decisions, and market competitiveness.
Optimization also plays a vital role in finance. Portfolio optimization, for example, aims to maximize returns while managing risk. Investment analysts use mathematical models to determine the
optimal mix of assets to include in an investment portfolio, considering factors like expected returns, volatility, and correlation between assets.
Optimization in Engineering
Engineering is another field where optimization is of paramount importance. Engineers strive to design efficient systems while minimizing costs and resource usage. Here are a few areas where
optimization is applied:
1. Structural Engineering: Engineers optimize the design of buildings, bridges, and other structures to ensure they can withstand loads while minimizing material usage. This optimization leads to
cost-effective and environmentally friendly designs.
2. Mechanical Engineering: In mechanical design, engineers optimize the shapes and dimensions of components to improve efficiency and reduce energy consumption. For example, the design of an
aircraft wing involves optimizing its shape to minimize drag.
3. Manufacturing: Optimization techniques are used to streamline manufacturing processes, minimize waste, and increase productivity. Engineers seek to find the optimal production schedule to meet
demand while minimizing production costs.
Extrema in Physics
Extrema are also prominent in the field of physics, where scientists seek to understand and predict natural phenomena. Here are some examples:
1. Classical Mechanics: In classical mechanics, the path taken by a particle to minimize or maximize a certain quantity, such as time or energy, is a fundamental concept. For example, light follows
the path that takes the least time (Fermat's Principle), and projectiles follow trajectories that optimize their range or height.
2. Quantum Mechanics: In quantum mechanics, the concept of extrema is applied to understand the behavior of particles at the quantum level. Quantum variational methods, for instance, seek to find
the optimal wave function that minimizes the energy of a quantum system.
3. Thermodynamics: In thermodynamics, the principle of least action is used to describe the behavior of physical systems. It's the principle of least resistance, where systems evolve in a way that
minimizes the action, a quantity related to the energy.
Practical Problem-Solving Strategies
In practical problem-solving, especially in real-world applications, it's essential to adopt a systematic approach. This approach may involve:
• Identifying relevant constraints and formulating them mathematically.
• Using computational methods to find critical points or perform numerical optimization.
• Conducting sensitivity analysis to assess how changes in parameters impact the optimal solution.
• Interpreting results in the context of the problem.
These problem-solving strategies are valuable beyond the classroom and are applicable in academic research and professional settings.
Expanding Theoretical Horizons: Applications and Beyond
While our theoretical journey has focused on solving this particular math assignment, the principles we've explored extend far beyond the classroom. Concepts of continuity, critical points, and
extrema are fundamental in various fields:
• In economics, understanding extrema is crucial when optimizing production or utility functions.
• In physics, optimizing trajectories or energy functions often involves finding extrema.
• In engineering, extrema are sought after when designing efficient systems.
• In computer science, algorithms optimize parameters for various purposes.
By mastering these theoretical concepts, students gain valuable skills applicable in a wide range of academic and professional contexts.
In conclusion, our theoretical exploration has provided a comprehensive understanding of how to find absolute extrema of a function within a closed interval. We delved into the theoretical concepts
of extrema, the Extreme Value Theorem, critical points, and boundary points. By following these theoretical principles, university students can confidently tackle similar math assignments. Equipped
with this knowledge, you are empowered to solve your math assignment and approach mathematical challenges with a deeper understanding of the underlying concepts. Mathematics is not just about solving
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