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What is the range of the function y = sin x? | HIX Tutor
What is the range of the function #y = sin x#?
Answer 1
The range of the function $f \left(x\right) = S \in x$ is $\left(- 1 , 1\right)$
Sin function varies between -1 and 1 for all possible real values of the #x# the independent variable.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Contract number
Time span of the project
As of 01.11.2022
Number of staff members
scientific publications
Name of the project: Research of mathematical hydrodynamics problems
Research directions: Mathematical hydrodynamics
Project objective: Research of a variety of important and currently unsolved problems of mathematical hydrodynamics; engaging young scientists, postgraduates and undergraduates in research to give
them an opportunity to strengthen their involvement in science
The practical value of the study
Scientific results:
● A proof of the theorem on the existence of a weak solution of the free boundary problem for equations of viscous gas dynamics modeling the motion of a heavy valve in a cylinder under the
influence of the pressure of the viscous gas.
● A proof of the global theorem on the existence and uniqueness of a strong solution of one-dimensional initial-boundary value problem of the motion of multi-speed mixtures of viscous compressible
fluids with a nondiagonal viscosity matrix in a bounded region with conditions allowing for sticking of its components on the boundary.
● A proof of the theorem on the existence and regularity of solutions in the problem of joint minimization of the Willmore and Dirichlet functionals. We have proven the existence of
three-dimensional, hydroelastic, nonlinear periodic waves propagating along a pool of infinite depth.
● A proof of the Bernoulli's law for axis-symmetricl Sobolev solutions of the Euler equation in unbounded domains
● A proof of the statement on the convergence of the trajectory attractor of a dissipative 2D Navier-Stokes system in the limit at a viscosity tending to zero to the trajectory atractor of the
corresponding dissipative 2D Euler system a strong Hausdorff metric.
● A proof of the theorems of the existence of weak solutions of the fractional Voigt viscoelasticity model with memory along trajectories of motion and the initial-boundary value problem for a
system of hydrodynamics with memory that is a fractional counterpart of the Voigt viscoelasticity model. A proof of the existence of a weak solution of an initial-boundary value problem that
describes the dynamics of a viscoelastic medium with memory on an infinite interval. A proof of the existence of a unique solution of the initial-boundary value problem for systems of
hydrodynamics with memory described by degenerate parabolic and elliptical equations. A proof of the theorem on the existence of weak solutions of models of motion of a Bingham fluid. A proof of
the theorems of the existence of attractors of solutions of autonomous systems of hydrodynamics with memory and attractors of a model of motion of a Bingham fluid.
● A proof of the theorems of the existence of weak and dissipative solutions (we have also indicated a relation of strong and dissipative solutions, alpha models of motion of fluid with memory). A
proof of the theorem on the existence of weak solutions of the Pavlovsky-Oskolkov model. A proof of the theorem on the existence of dissipative solutions (we have indicated the relation between
strong and dissipative solutions) of the initial-boundary value problem for the Jeffreys-Oldroyd alpha model. A proof of the theorem on the existence of weak solutions of the alpha-model of the
motion of solutions polymers.
● A description of the built theory of the control of a system of normal type related to the Helmholtz equation via start and impulse control. A description of the developed nonlocal method of
stabilization of three-dimensional Helmholtz and Navier-Stokes systems by impulse control.
● A proof of the theorem on the uniform exponential stability of a strongly continuous semigroup of bounded operators generated by an operator block of a special type. The results of solving two
problems of the mechanics of continuous media (problems of the motion of a viscoelastic body and the problem of minor motions of a viscoelastic compressible Maxwell fluid filling a bounded
uniformly rotating region) with the use of the specified theorem.
● A description of the strict theory of gradient flows and geodesic convexity with respect to the geometrical structure related to this metric. A description of the De
Giorgi-Jordan-Kinderlehrer-Otto variational scheme for building solutions of specified gradient flows.
● A description of the developed theory of internal Poincaré and Kelvin waves in a region with variable depth and a curvilinear boundary. Results of an analysis of the possibilities of generation
of internal waves. A description of the developed models and computer tools for solving problems of generation and propagation of internal Poincaré and Kelvin waves based on nonlinear shallow
water equations for laminar flow of fluid with mass exchange.
● The construction of a hierarchy mathematical models describing the wave dynamics of an oceanic environment accounting for the features of topography, dispersion and breaking of internal waves.
Conducting an analysis and systematization of field data collected in various regions of the World Ocean. For this purpose we used the latest data on deep water currents in the Atlantic (the Vema
Channel, the Chain and Romanche fracture zones, the Discovery Passage) as well as on the transformation of internal waves in the shell zone of the Sea of Japan and the South China Sea. A
description of the developed numerical and analytical model of internal waves in the ocean accounting for the regional features of field experimental data. We have developed a mathematical model
of the Green-Naghdi and Korteweg-de Vries type for describing strongly nonlinear internal waves factoring in the thin stratification of the density field and the amplitude dispersion. A
systematization of the features of the formation of internal waves depending on the region. Obtaining asymptotic large-time asymptotic solutions for nonlinear equations describing the propagation
of nonstationary internal waves in a stratified fluid caused by the motion of a submerged body under ice cover.
● A proof of the global solvability of the problem of optimization of the operation of a Stirling engine.
● A proof of the existence of bi-Lipschitz conformal coordinates on a surface that is an extremal of the Dirichlet and Willmore functionals. A proof of the statement that the set of features of
such a parametrization has a zero Hausdorff measure in any positive dimensionality.
● A proof of an analog of the Liouville's theorem for axisymmetric flows of an ideal fluid (Euler equations) without rotation. As a consequence, we have obtained a proof of an analog of the the
Liouville's theorem for axisymmetric flows of the Navier-Stokes system without rotation.
● A proof of the statement that if random functions are ergodic and statistically homogeneous in spatial variables or in time, then the trajectory attractors of the 3D Navier-Stokes system converge
in a weak topology to the trajectory attractor of the average 3D Navier-Stokes system whose deterministic external force is obtained by averaging the external forces of the initial random 3D
Navier-Stokes systems.
● A proof of the theorem on the existence of uniform attractors of solutions of a non-autonomous system of hydrodynamics with memory and a model of the motion of the Bingham medium.
● A proof of the theorem on the existence of optimal control with feedback for a number of alpha-models of hydrodynamics. Particularly, a proof of the existence of optimal control with feedback for
the alpha-Leray model, the alpha-Navier-Stokes model, the Pablovsky system of equations.
● A description of a nonlocal method of stabilization of systems of normal type as well as 3D Helmholtz and Navier-Stokes systems by distributed control. Results of a resurch of the interaction of
the normal and the tangential operators.
● A proof of the theorem on the strong solvability of initial-boundary value problems describing minor motions of a viscoelastic body of hyperbolic type. A proof of the statement on the stability
and stabilization of the solution of the researched problem.
● A description of the built theory and results of an analysis of the metric, geometric and topological properties of the MCF distance on the space of finite gradient vector Radon measures and on
the space of bounded variation functions. A description of a new model for image reconstruction relying on MCF distance. A proof of the existence of a solution of this model of image recognition,
the correctness and the qualitative properties.
● Results of numerical computations of the propagation of nonlinear internal waves (including Poincaré and Kelvin waves) for specific hydrodynamic systems, in particular, for the coastal zone of
the sea. Results of field observations and measurements of the hydrodynamic parameters of flows in specific water reservoirs. An analysis of field observation data and their comparison with the
results of the numerical modeling.
● Results of a comparative analysis of the results of a modeling of the wave dynamics of the oceanic environment and field data for specific areas of the World Ocean. A description of the developed
hydrodynamic model of a specific region of the ocean accounting for the features of the wave dynamics related to the passage of nonlinear internal waves. A description of a mathematical model for
determining the wave field with passage of a bundle of internal nonlinear waves and an analysis of the results of a numerical modeling.
● Proofs of the theorems on the existence of weak solutions of initial-boundary value problems for models of a viscoelastic fluid with a rheological relationship with fractional order derivatives
of the Voigt and anti-Zener type with memory along the trajectory of the motion of a fluid. Theorems on the existence of the existences and uniqueness of a solution for degenerate parabolic and
elliptical equations of high order describing viscoelasticity models.
● Our researchers have produced theorems on the existence of solutions of the problem of onlinear waves of established type on the surface of the ocean covered with ice.
● Our researchers have obtained an analog of the Liouville's theorem for axisymmetric flows of viscous fluids (stationary Navier-Stokes equations) with rotation.
● A description of the general structure of trajectory and global attractors of evolution equations with memory. In the proposed approach the dynamic system operates in the space of the initial
data of the Cauchy problem of the studied system. As an important application of the proposed method we have built a trajectory and a global attractor for a dissipative wave equation with linear
memory. We have also proven the existence of a global Lyapunov function for a dissipative wave equation with memory. The existence of such a Lyapunov function allowed to prove the regularity of
the structure of attractors that match nonstationary sets coming out of the set of stationary points if the reviewed equation.
● A proof of the theorem on the existence of a minimal trajectory pullback attractor and a global pullback attractor for both weak solutions of a nonautonomous medium with memory and for weak
solutions of the model of the motion of a Bingham medium in the nonautonomous case.
● A proof of the theorems of the existence of weak solutions for the alpha-Leray model and the alpha-Navier-Stokes model with the viscosity coefficient depending on the temperature.
● Results concerning the interaction of the normal and the tangential operator into which the quadratic operator from the Helmholtz system decomposes. A theory of the nonlocal stabilization of a
system of the hydrodynamic type. A structure of stabilizing impulse control for the differentiated Burgers' equation, transfer of the built structure to the Helmholtz system.
● A theorem on the spectrum of the emerging operator pencil for a system of integro-differential equations describing the motion of a viscoelastic body fixed on the boundary of a bounded domain. A
theorem on the multiple basicity of a system of root elements, a decomposition of the solution of the evolution problem. A theorem on the asymptotic of solutions.
● A review of the adiabatic mode for tilting ratchets. Implicit formulas have been obtained for the speed of the adiabatic migration of animals, fish or bacteria and the directions of their
migration. It has been demonstrated that a longer range of the monotonicity of the potential defines the specific direction of migration. The result is based on a new nontrivial functional
inequality. We have also studied the semiadiabatic mode for tilting ratchets, when the aggregate period of inclination tends to infinity and one of the states of inclination prevails over the
other. We have obtained an explicit formula for the rate of semiadiabatic transfer and proven that if the potentials are nonconstant, semiadiabatic migration of animals, fish and bacteria occurs
in the same direction. These results are in part based on one more functional inequality. For Brownian ratchets of arbitrary type with weak diffusion we have demonstrated that there is a
connection between the transfer and some ordinary differential equation. If this ordinary differential equation does not have periodic solutions or, in other words, if its rotation number is not
zero, an unidirectional flow of migration of animals, fish or bacteria emerges there, even though the average force is not zero.
● We have conducted field measurements of internal waves in August-September 2019 on the marine experimental station «Cape Schultz» of the V. I. Il'ichev Pacific Oceanological Institute of the Far
Eastern Branch of the Russian Academy of Sciences, experimental data was collected. We have conducted a numerical modeling of nonlinear internal waves and built a model of nonlinear internal
waves (including Poincaré and Kelvin waves). A comparative analysis of the obtained experimental data and the results of a numerical modeling has been conducted.
● A research of the structure of the solutions and the solvability of the boundary value problem describing nonlinear waves in flows of continuously stratified fluid over an obstacle. On the basis
of the model of weakly coupled Korteweg–de Vries equations we have investigated the interaction of entangled traveling waves. We have built solutions of a number of reverse problems of the
reproduction of the structure of nonlinear packages of internal waves. We have reconstructed the temperature fields and the boundaries of layers during the passage of a near-surface solitary wave
and a tidal bore. We have conducted the verification of the built model by comparing it with experimental data collected not only at the marine experimental station «Cape Schultz» of the V. I.
Il'ichev Pacific Oceanological Institute of the Far Eastern Branch of the Russian Academy of Sciences but also with experimental data collected in the South China Sea (Lien, Henyey, Ma and Yang,
● A definition of the theorem on the existence and uniqueness of a strong solution of the initial-boundary value problem for a system of equations of the motion of a viscoelastic fluid that is a
fractional counterpart of the Voigt viscoelasticity model with memory along the spatial variable in the planar case. For a model of a thermo-viscoelastic medium of the Oldroyd type with memory
along the trajectories of the motion we have determined weak solubility. In this case for the existence of trajectories of motion ensuring the memory of the medium we used the theory of regular
Lagrange flows. A new method has been developed on the basis of the properties degenerate pseudo-differential operators built on a special integral transformation. On the basis of this method we
researched new classes of pseudo-differential operators that allow to research the correctness of mathematical models of thermo-viscoelasticity containing initial-boundary value and boundary
value problwma for degenerate elliptical and parabolic equations.
● A proof of the theorems on the weak solubility for models of viscoelastic media with of a fractional order that is higher than 1.
● A proof of the convergence of the trajectory attractors and global attractors of approximations of the model of the motion of a viscoelastic medium with memory to the trajectory and global
attractors of this model in the sense of semideviation while the approximation parameter tends to zero. Obtaining the convergence of trajectory and global attractors of approximations of the
Bingham model to the trajectory and global attractors of the Bingham model in the sense of semiderivations while the approximation parameter tends to zero. A proof of the convergence of the
trajectory and global attractors of the approximations of a modified Kelvin-Voigt model to the trajectory and global attractors of this model in the sense of semiderivations while the
approximation parameter tends to zero .
● A proof of the theorem on the solvability in the weak sense of the problem of optimal control with feedback for Voigt models with variable density.
● Obtaining a theorem on the existence of global-in-time weak solutions of the initial-boundary value problem for the first-class alpha model of a Bingham model.
● A theorem on the existence of a weak solution of the problem of optimal control with feedback for a modified Kelvin-Voigt model of weakly concentrated water solutions of polymers.
● We have studied problems of rotationally symmetric flows in a bounded volume of a gas for all the values of the adiabat greater or equal than one. It has been proven that the matrix of
concentrations is centered on the rotation axis. Moreover, it has been established that it has a unique non-zero element that lies on the diagonal and corresponds to the direction along the axis
of symmetry OZ. The measure corresponding to this non-zero element is the product of a constant and the measure of length dz. In other words, the matrix of concentrations has only one non-zero
element that is a Dirac measure concentrated on the axis of symmetry. It has also been demonstrated that the concentrations are absent in the case of axisymmetric solutions.
● A theorem on the global solvability of the initial-boundary value problem for multi-temperature multi-speed model of mixtures in the general case of three spatial variables.
● A proof, for the planar case, of the theorem on the existence and uniqueness of a strong solution of the initial-boundary value problem for systems of equations that describes the dynamics of a
nonlinear thermo-viscoelastic continuous medium with a rheological relationship of the Oldroyd type.
● A proof of the theorem on the existence of strong solution of the problem of optimal control with feedback for a Navier-Stokes system with variable density in the planar case.
● Determining the theorem on the solvability, in the strong sense, of the problem of optimal control with feedback for a Voigt model with variable density.
● A proof of the theorem on the convergence of the sequence of solutions of a family of alpha-Bingham models to the solution of the original initial-boundary value problem when the parameter alpha
tends to zero..
● Obtaining a formula for the monotonicity for the energy tensor for the problem of rotational symmetric flows of a viscous gas with the limiting value of the adiabatic indicator that equals to 1.
It has been proven that outside of the axis of rotation the function of density allows for an estimate in the negative Sobolev space with an indicator of -1/2 that depends only on the data of the
problem. We have proven the existence of a weak solution defined outside of the rotation axis. It has been demonstrated that the concentration of the kinetic energy tensor is a symmetric and
non-negative matrix Borel measure concentrated on the axis of rotation. It has been demonstrated that the matrix of the concentrations can have only one nontrivial component. We have proven that
the time-averaging of the matrices of concentrations is completely continuous on the axis of rotation and its density is semicontinuous on the top.
● Obtaining a theorem on the weak solvability of the initial-boundary value problem for a heat-conducting multi-speed multi-dimensional model of the dynamics of mixtures accounting for the
viscosity and comprehensibility.
● A theorem on the existence of weak solutions of initial-boundary value problems for equations of the motion for a model of a viscoelastic fluid with memory along the trajectory of the velocity
field and a rheological relationship containing integer derivatives of high orders.
● A proof of the weak solvability of initial-boundary value problems describing the motion of viscoelastic media (of the Voigt type) with lag coefficients dependent on temperature.
● A theorem on the existence of a weak solution of the initial-boundary value problem for a modified Kelvin-Voigt model with variable density.
● A statement on the existence of weak rotation-symmetric solutions of the magnetic gas dynamics accounting for the effects of viscosity and self-gravitation. A statement on the structure of
possible concentrations of the energy tensor. An effective upper estimate for the critical value of the adiabatic index above which the concentrations are absent.
• A proof of the solvability of the regularization of the initial-boundary value problem for equations of the dynamics of viscous of compressible multi-component media.
Education and career development:
• In 2017 five employees of the Laboratory completed the additional training program «Partial differential equations and their applications to mathematical hydrodynamics» at the Mathematics
Department of the Middle East Technical University (Northern Cyprus).
• In 2018 five employees of the Laboratory completed additional training at the International Center for Mathematics (Portugal).
• In 2019 five employees of the Laboratory completed additional training with the topic «Multi-phase non-Newtonian fluids: mathematical modeling and computations» at a laboratory of the acadenuc
institute of heat industry systems (France).
● We have staged international scientific conferences: «Modern methods and problems of mathematical hydrodynamics» (2017), «Modern methods and problems of mathematical hydrodynamics-2018»
(2018), «Modern methods and problems of mathematical hydrodynamics-2019» (2019).
● In 2017 we conducted the scientific seminar «Mathematical models of shallow-water shear flows» with an invited lecturer from Aix-Marseille University.
● Two Candidate of Science dissertations, two Doctor of Science dissertations, 6 master's degree theses and 5 bachelor's degree theses have been prepared and defended.
● The following lecture courses have been developed and introduced into the curriculum of the Faculty of Mathematics of the Voronezh State University: «Applications of the theory of
differential equations to geometry», «Navier-Stokes equations of compressible fluid», «Applications of differential inclusions to problems of optimal control», «Pavlovskiy mathematical models
for the motion of polymer solutions», «Alpha models of equations of hydrodynamics», «Approximation-topological method for the solvability of equations of the dynamics of viscoelastic media».
● Two study guides have been published.
Other results: Employees of the Laboratory participated in 18 international conferences, scientific schools and delivered 51 keynotes.
V.I. Il'ichev Pacific Oceanological Institute Far Eastern Branch Russian Academy of Sciences: an agreement on scientific cooperation was signed, a number of joint studies were carried out, based on
the results of which 2 articles were published..
Fursikov A., Osipova L.
On the nonlocal stabilization by starting control of the normal equation generated from Helmholtz system // Science Chine Mathematics. – 2018. – Vol. 61. – Issue 11. – pp. 2017-2032.
Plotnikov P.I., Toland J.F.
Variational Problems in the Theory of Hydroelastic Waves // Philosophical transactions of the Royal society A-mathematical physical and engineering sciences. – 2018. – Vol. 376. – Issue 2129 –
Article ID:20170343.
Zvyagin V.G., Orlov V.P.
Solvability of one non-Newtonian fluid dynamics model with memory // Nonlinear Analysis. – 2018. – Vol. 172. – pp. 73–98.
Zvyagin A.V.
Attractors for model of polymer solutions motion // Discrete And Continuous Dynamical Systems. – 2018. – Vol. 38. – № 12. – pp. 6305–6325.
Seregin G.A., Shilkin T.N
Liouville-type theorems for the Navier-Stokes equations // Russian Mathematical Surveys. – 2018. – Vol. 73. – Issue 4. – pp. 661-724.
Korobkov M.V., Pileckas K., Russo R.
On Convergence of Arbitrary D-Solution of Steady Navier-Stokes System in 2D Exterior Domains // Archive for Rational Mechanics and Analysis. – 2019. – Vol. 233. – Issue 1. – pp. 358-407.
zvyagin v.g., orlov v.p.
On strong solutions of fractional nonlinear viscoelastic model of Voigt type, Mathematical Methods in the Applied Sciences, 2021, (44, 15).
plotnikov p.i., sokolowski j.
Boundary Control of the Motion of a Heavy Piston in Viscous Gas // SIAM Journal on Control and Optimization. – 2019. – Vol. 57. – Issue 5. – pp. 3166–3192.
zvyagin v.g., orlov v.p.
Weak Solvability of One Viscoelastic Fractional Dynamics Model of Continuum with Memory, Journal of Mathematical Fluid Mechanics, 2021, (23, 9).
zvyagin a.
Solvability of the Non-Linearly Viscous Polymer Solutions Motion Model, Polymers, 2022, (14).
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Probability: Cards vs Dice
(A new question of the week)
A couple recent questions involved related subtleties in probability and combinatorics. Both were about apparent conflicts between similar problems involving cards and dice.
Cards vs dice
This very detailed question came from Alexander in early July:
I have a question regarding the probability of getting one pair in poker and Yahtzee. More precisely, I am having trouble with the phenomenon called double-counting, when the same event is
counted more than once. I know how to calculate the probability of getting a one-pair in poker. However, when I try to calculate the probability of getting a one-pair in Yahtzee, I get an
incorrect answer which I believe has to do with my confusion with double counting. Therefore I will first calculate the probability of getting a one-pair in poker, then do the same thing for
Yahtzee and hopefully, you can observe where I went wrong.
Before I show the rest of the question, I want to mention that he is not concerned with how one actually gets the cards in poker or the numbers in Yahtzee, which in both cases can involve more than
one step and complicate the probabilities. It will just be assumed that in poker, 5 distinct cards are chosen randomly from a deck of 52, and in Yahtzee, 5 six-sided dice are rolled once.
In poker, one pair means a hand containing two cards with the same number (here, 5), all other cards having different numbers:
To my knowledge, “one pair” plays no role in Yahtzee, but would mean exactly two of one number (5 again), with all other dice having different numbers:
As for double-counting (or overcounting), we’ve mentioned it, for example, in Permutations and Combinations: Undercounts and Overcounts, Arranging Letters with Duplicates, and Interpreting and
Solving a Counting Problem. It means that we have counted the same outcome more than once, and need to either compensate, or find a way to avoid it. The key is to recognize whether you’ve done it!
Now, Alexander carefully showed his thinking about each problem:
Poker (An ordinary deck of cards with 5 cards in a hand):
(C(13,1) x C(4,2) x C(12,1) x C(11,1) x C(10,1) x 4^3)/3!
First, I take 1 of the 13 values in which I have a pair, and their suit can be combined in C(4,2) ways. Then I choose the third, fourth and fifth cards, all of which can be combined with 4 suits,
thus 4^3. To account for double-counting, I divide by 3!. For example, if my third card is six of spades, my fourth is seven of spades and my fifth is eight of spades, that is the same hand as if
my third card was eight of spades, my fourth card was seven of spades and my fifth card was six of spades. These three cards can be arranged in 3! ways, and since order does not matter, I divide
by 3! to remove duplicates. Finally I divide with the total number of hands which is C(52,5) and get approximately 42%. Now on to Yahtzee.
Yahtzee (5 6-sided dice):
I want to use the same method I did when I calculated the poker-version.
(C(6,1) x C(5,2) x C(5,1) x C(4,1) x C(3,1))/3!
I choose one value out of six, choose two dice then pick the last three dice. Then, the same way I did in poker, I divide with 3!, since a composition of 5-5-1-2-3 is the same as 5-5-3-2-1. Then
I divide with the total number of dice-rolls which is 6^5. However, this answer is wrong. To correct my mistake, one would have to remove the 3!, but how then do you account for double-counting?
If you could help me understand why I should ignore double counting in Yahtzee, but not in Poker, I would be grateful.
Best regards,
He demonstrated the overcounting with cards; is there any with dice?
I answered:
Hi, Alexander.
Nice question!
I’ll start out by saying that my first impression was that cards and dice behave very differently (because there’s only one of each card, while dice are independent), so I didn’t think the work
would be anything like the same. And I think the ultimate answer is going to be that you are being fooled by a superficial similarity between them. Since subtlety is a primary characteristic of
combinatorics, that makes this a really nice question to dig into!
Cards are discrete objects, so that there is only one of each; this makes permutations and combinations (which involve selecting and/or arranging distinct items) appropriate. Once you have a four of
diamonds, you know the next card can’t be the four of diamonds (though it could be another four). On the other hand, they can be thought of either as ordered (the order in which you get them) or not
(just the set in your hand), so either permutations or combinations can be used.
Dice don’t mind having the same number in different places; so there is no permutation involved. Their values are independent. On the other hand, each die is distinct; we often emphasize this by
imagining each die being a different color, or tossing each die in a separate place. So in some sense there is an inherent order to them.
But in combinatorics, each tool might be used in any problem, as we’ll see. The two problems do turn out to involve similar-looking work after all.
Five cards, one pair
So first, let’s think about exactly what you are doing with the cards:
(C(13,1) × C(4,2) × C(12,1) × C(11,1) × C(10,1) × 4^3)/3!
Restating your explanation, you are doing this:
□ Choose which number to have a pair of: C(13,1) ways
□ Choose which two suits the pair will be: C(4,2) ways
□ Choose distinct numbers for the three other cards, in order: 12×11×10 ways
□ Choose the suits of these three cards: 4^3 ways
□ Account for the fact that order doesn’t count, by dividing by 3!
This counts subsets of cards, not taking order into account; so you divide by the number of ways to choose an unordered subset of 5 cards, C(52,5).
Most of the combinations in his work don’t really need to be written that way; we could simply say that there are 13 numbers to choose from, then \({4\choose2}=6\) pairs of suits, and then \(12\
times11\times10=1320\) ways to choose three numbers from the 12 remaining, and 4 choices for each of those suits. Then he divides by the number of ways to arrange the same three cards, since
everything else is combinations.
He finds the probability as the number of sets of five cards that contain a single pair, over the total number of sets of five cards; probability can be calculated as combinations over combinations,
which works best here, or as permutations over permutations.
I had one optional suggestion, letting combinations eliminate the overcount:
The one difference in my own way of thinking is that (12×11×10)/3! can also be thought of as choosing a subset of 3 of the remaining 12 card numbers, namely C(12,3). So the calculation can be
alternatively written as
C(13,1) × C(4,2) × C(12,3) × 4^3 = 1,098,240
Dividing this by C(52,5) = 2,598,960, we get 0.4226.
This is all correct; I even checked the answer before moving on, by looking it up here:
Here is what that page says:
Their calculation is identical to ours, not just the number.
Now, why does order not count?
Mostly because none of the choices we made involved locations of the cards; we just chose which cards to include. It would, in fact, be possible to do the work taking order into account. We might do
it this way:
• Choose which number to have a pair of: 13 ways
• Choose which two cards will have that number: C(5,2) ways
• Choose a suit for each of them, in order: P(4,2) ways
• Assign numbers to the three other cards, in order: 12×11×10 ways
• Assign suits to these three cards, in order: 4^3 ways
• Divide by the number of ways to select 5 cards in order, P(52,5)
This gives $$\frac{13\cdot{{5}\choose{2}}\cdot_{4}\!\text{P}_{2}\cdot12\cdot11\cdot10\cdot4^3}{_{52}\text{P}_{5}}$$ $$=\frac{13\cdot10\cdot4\cdot3\cdot12\cdot11\cdot10\cdot64}{52\cdot51\cdot50\cdot49
\cdot48}$$ $$=\frac{131,788,800}{311,875,200}\approx0.422569,$$ just as before. The important thing is consistency: If any calculation takes order into account, then all must.
Five dice, one pair
Now, how about the Yahtzee version? (I took a moment to check the rules for Yahtzee, which I probably haven’t played in 50 years, and decided that you are just talking about rolling 5 dice once,
and seeing if there is a single pair, without doing all you might do in a real game.)
The same basic strategy does apply, but the details are different. We are no longer counting subsets; in your denominator, 6^5, order is taken into account! That is what you missed.
(Note that it would be very hard to solve a dice problem without taking order into account, because duplicates are allowed, so rearranging would not always change the result.)
This is an important point; students often ask how you know whether order does or does not matter, and often in probability questions, that is up to you to decide. As long as the possibilities you
count are equally likely, you can calculate probabilities as a ratio of permutations or of combinations. In a hand of cards, as I showed above, you can either think in terms of how the cards are
dealt (permutations: order matters) or the set of cards you have in your hand (combinations: order doesn’t matter – or you might always sort them by number and suit, ignoring the order in which you
got them). With dice, you generally want to think as if each die were distinguishable (different colors, say), which means that order (which die has which value) matters. You could conceivably ignore
order, but it would be very hard to count, and what you counted would not be equally probable. (As a simple example, when you toss two coins, the only possibilities ignoring order are HH, HT, TT, but
HT happens half the time.)
So we choose, with good reason, to think of a roll of the dice not as a set of numbers (e.g. {1, 1, 2, 4, 4} – but that would actually be a multiset), but an ordered list of numbers (e.g. 2, 1, 4, 4,
1). On the other hand, this does not make it a permutation, because duplicates are allowed. There are \(6^5=7776\) of these, not \(_6\text{P}_5=720\).
So here is what you are actually doing:
□ Choose which number to have a pair of: C(6,1) ways
□ Choose which two dice the pair will be: C(5,2) ways
□ Choose distinct numbers for the three other dice, in order: 5×4×3 ways (that is, P(5,3)
We don’t need to divide by 3!, because this time order does count. Our result is
C(6,1) × C(5,2) × P(5,3) = 3600
Dividing this by 6^5 = 7776, we get 0.4630 (46.3%).
Note that where we chose two suits last time, we are choosing two dice to have the chosen numbers: superficially similar calculations, but entirely different in meaning.
Alexander had calculated $$\frac{{6\choose1}\cdot{5\choose2}\cdot{5\choose1}\cdot{4\choose1}\cdot{3\choose1}}{3!}=\frac{6\cdot10\cdot5\cdot4\cdot3}{6}=600,$$ which amounts to $${6\choose1}\cdot{5\
choose2}\cdot{5\choose3},$$ whereas we really have $${{6}\choose{1}}\cdot{{5}\choose{2}}\cdot _{5}\!\text{P}_{3}=6\cdot10\cdot60=3600.$$
When I work a problem like this, I like to first ask, What am I counting, and then, How can I count them? In this case, because we have just done a very different problem that was misleadingly
similar, the first question was the key.
Alexander replied,
Thank you!
I have searched everywhere to find an explanation like yours, but I have not found one until now. Again, thank you!
Poker hands with random numbers
This week, while I was editing this post, we got a question that deals with a very similar problem. I normally let a question sit for a month before publishing it (to make sure the discussion is
finished, and in some cases to make sure an assignment will be past due before showing the answer to the world), but this belongs here.
Full house of cards
Bijan wrote, first quoting from a book (Probability and Statistics with Applications: A Problem Solving Text, by Leonard Asimow and Mark Maxwell) how to calculate the probability of a full house (3
of one denomination, and 2 of another) in poker:
Then he explained that he was trying to find the probabilities for the “poker test” for random numbers, which treats each sequence of 5 digits produced by a random number generator as a “poker hand”
and compares the experimental probabilities of various “hands” to those expected for truly random numbers. He included a link that says this:
Full house of digits
Then he showed his work for each case, starting with the full house:
We’ve 10,000 random numbers of five digit each. They’re assumed to be independent.
My calculations-:
1) Full house
10C1*9C1/10,000 = 0.009
I’m correct. My only confusion here would be the denominator. Why is it 10,000?
According to the above example, should not it be 10C5?
Explanation of my thought process-:
First pick 1 digit out of 10 digits. Then next, pick another digit(only 1 digit as we need a pair), out of remaining 9 digits.
I’ll quote more cases later. I answered this part:
Hi, Bijan.
It happens that the blog post I’m working on for this week is about almost exactly the same issue: the difference between problems about cards and about dice, which amount to random number
generators. A central difference is that cards are unique, but digits can be repeated; so you tend to use permutations and combinations for cards, but not so much for numbers (and for different
reasons). Another central difference is that order is ignored in poker hands, but not in numbers.
You need to essentially ignore everything you read about poker hands, and just think about numbers.
Even more clearly than dice, random numbers have a definite order of digits; 11234 is not the same as 21314.
Let’s look at your first case, the full house, to see whether you got the right answer rightly or by accident.
First, the denominator should not be 10C5, as you suggest, because you are not choosing 5 different digits ignoring order, but 5 unrestricted digits counting order!
It should be 10^5, which is 100,000. I have no idea why you said 10,000.
Later I realized what he had done:
Ah! I just looked at the link you put at the end; I see it says
In 10,000 random and independent numbers of five digits each, you may expect the following distribution of various combinations
That is not the number of possible numbers; they just chose to suppose that you have that many numbers in your sample, for the sake of the table following. You forgot to stop and think about
where that number would have come from, and whether it made sense as you interpreted it.
In the table, each probability is multiplied by 10,000, the number of random numbers they suppose we are testing. The correct denominator is the number of possible 5-digit numbers, which is \(10^5=
100,000\), counting all numbers from 00000 to 99999.
But that’s not the only error. Continuing,
As for the numerator, I would simply say we choose one of 10 digits to occur twice, and one of the remaining 9 digits to occur three times. But then we have to place them in some order, because
order does count. So we choose 2 of the 5 places to put the first digit we chose, which we can do in 5C2 = (5*4)/(2*1) = 10 ways. So the probability of a full house is (10*9*10)/100,000 = 0.009.
Your answer just happened to be correct. The method was wrong.
The error of neglecting order was corrected by the error of using 10,000 as the denominator!
I would write it as $$P(\text{full house})=\frac{_{10}\text{P}_2\cdot{5\choose2}}{10^5}=\frac{10\cdot9\cdot10}{100,000}=0.009=0.9\%.$$
Four of a kind
He made the same error for one pair and for three of a kind, thinking he was right because of an error hidden by the wrong denominator. That compensation failed in his work for four of a kind:
4) Four of a kind:
So from 10 digits, I need to pick 1 digit and out of the remaining 9 digits, I need to pick another 1 digit.
So, it should be 10C1*9C1/10,000 = 0.009
But it becomes similar to full house. This is wrong. I don’t get why this became wrong.
I didn’t directly answer this, but he has again omitted the choice of a location for the four, which can be done in \({5\choose4}=5\) ways. The correct answer is $$P(\text{four of a kind})=\frac{{10\
All different
For the last two cases he did, “all different” and five of a kind, the only error turned out to be the denominator. Here is the former:
5) 5 different digits:
This should’ve been simple, I got the answer but I got the answer greater than 1.
I’m not sure why I got this. I am skeptical about the denominator since the start as I feel that’s randomly chosen here unlike above where we did 52C5. If I increase 1 “zero” in denominator, the
answer would be correct.
To this part, I answered:
This is a case where I would use permutations, but not in the same way as for cards. The numerator will be the number of ways to form a number consisting of 5 different digits, which means a
permutation of 5 of the ten possible digits. That is
10P5 = 10*9*8*7*6 = 10!/5! = 30,240.
The denominator, again, is the number of ways to choose 5 (non-distinct) digits, which is just 10^5 = 100,000, since there are 10 independent ways to choose each. So the probability is
30,240/100,000 = 0.3024.
Your numerator is correct, and your answer would have been correct if you had used the correct denominator.
That is, $$P(\text{all different})=\frac{_{10}\text{P}_5}{10^5}=\frac{10\cdot9\cdot8\cdot7\cdot6}{100,000}=0.3024=30.24\%.$$
I concluded,
Ultimately, as I said at the start, you need to simply ignore what is done with cards, and think only about how random numbers work. The comparison of the “poker test” with poker is very
misleading! So now go through all your cases, including those where you got the right answer, and correct them. Then we can have another look.
The corrections will be straightforward.
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Reed-Solomon Codes
Log/antilog tables for GF(256) multiplication (polynomial basis)
When working with Reed-Solomon codes, log/antilog tables are helpful to calculate Galois field (finite field) products by hand.
This PDF contains log/antilog tables for all 30 irreducible polynomials in GF(256). This all polynomial bases for GF(256). 16 of these polynomials are primitive. Of special note, the primitive
polynomials 0x11D and 0x187 are most commonly used in Reed-Solomon codes. The non-primitive polynomial 0x11B is used by the AES encryption algorithm and the Intel CPU instruction GF2P8MULB.
Non-primitive, irreducible polynomials can be used for Reed-Solomon codes, but some software does not support non-primitive polynomials (such as MATLAB/Octave and Phil Karn’s libfec).
Primitive elements in GF(256) (polynomial basis)
A Reed-Solomon code generator polynomial requires selection of a primitive element of GF(256). The 128 primitive elements of GF(256) for a polynomial basis are listed in the following document. | {"url":"https://codyplanteen.com/notes/rs","timestamp":"2024-11-01T20:30:58Z","content_type":"text/html","content_length":"2744","record_id":"<urn:uuid:dc36450f-73b3-4fff-9e94-1c7dd92acee0>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00131.warc.gz"} |
APIC '95 Proceedings: Table of Contents
International Workshop on Applications of Interval Computations,
El Paso, Texas, February 23-25, 1995.
A special supplement to the journal of Reliable Computing. All of the files below in LaTeX use the IEEEtran.sty style file.
Thanks from the organizing committee p. 6-7
"Reliable Computing" (formerly, "Interval Computations"). An
international journal p. 8
Institute for Manufacturing and Materials Management p. 9
A. Bernat, V. Kreinovich, T. McLean, and G. N. Solopchenko,
"What are interval computations, and how are they related to quality in
manufacturing?" pp. 10-12
(LaTeX, PostScript)
V. Kreinovich,
"Data processing beyond traditional statistics: applications of
interval computations. A brief introduction" pp. 13-21
(LaTeX, PostScript)
F. Akhmedjanov, V. Krymsky,
"Robust design of control system for multi-input multi-output plant
with interval uncertainty" p. 22-25
(LaTeX, PostScript)
J. T. Alander,
"On interval factorial genetic algorithm in global
optimization" pp. 26-29
(LaTeX, PostScript)
G. Alefeld, G. Mayer,
"Recent results on symmetric interval systems" p. 30
(LaTeX, PostScript)
B. R. Barmish,
"Planar geometry problems motivated by robustness of linear
systems" pp. 31-32
(LaTeX, PostScript)
R. Bell,
"Aerodynamic aids to vehicle stability" p. 33
(LaTeX, PostScript)
M. Beltran and V. Kreinovich,
"How to find input variables whose influence on the
result is the largest, or, how to detect defective stages
in VLSI manufacturing?" pp. 34-37
(LaTeX, PostScript)
M. Beruvides and V. Kreinovich,
"Interval approach to learning curves" p. 38
(LaTeX, PostScript)
M. Berz,
"Verified integration and generation of Poincare maps using
Taylor models" p. 39
(LaTeX, PostScript)
M. A. Campos and M. de B. Correia,
"Interval and high accuracy results for parameter
estimation in time series models" pp. 40-43
(LaTeX, PostScript)
O. Caprani, L. Hvidegaard,
M. Mortensen, and Th. Schneider,
"Efficient and robust ray intersection" p. 44
(LaTeX, PostScript)
O. Caprani, K. Madsen, and O. Stauning,
"Enclosing solutions of integral equations" pp. 45-51
(LaTeX, PostScript)
Da-Wei Chang,
"On the interval multisplitting AOR (IMAOR) method" pp. 52-53
(LaTeX, PostScript)
H. M. Chen and M. H. van Emden,
"Adding interval constraints to the Moore-Skelboe
global optimization algorithm" pp. 54-57
(LaTeX, PostScript)
H. Cheng and D. Berleant,
"A software tool for automatically verified reasoning
with intervals and probability distributions" pp. 58-61
Huang Chongfu,
"Interval and fuzzy approaches to queuing systems" pp. 62-63
(LaTeX, PostScript)
D. E. Cooke, R. Duran, and A. Gates,
"Bag Language speeds up interval computations" pp. 64-66
(LaTeX, PostScript)
G. J. Deboeck, K. Villaverde, and V. Kreinovich,
"Interval methods for presenting performance of finanical
trading systems" pp. 67-70
(LaTeX, PostScript)
B. S. Dobronets,
"A posteriori error estimation and corrected solution
of partial differential equation" pp. 71-73
(LaTeX, PostScript)
D. I. Doser, K. D. Crain, M. R. Baker, V. Kreinovich,
M. C. Gerstenberger, and J. L. Williams,
"Estimating uncertainties for geophysical tomography"
pp. 74-75
(LaTeX, PostScript)
I. Elishakoff,
"Convex modelling - a generalization of interval analysis for
nonprobabilistic treatment of uncertainty" pp. 76-79
(LaTeX, PostScript)
N. M. Glazunov,
"Development of quality interval algorithms, software and
systems" p. 80-82
(LaTeX, PostScript)
S. Hadjihassan, E. Walter, and L. Pronzato,
"Quality improvement via the optimization of
tolerance intervals during the design stage" pp. 83-84
(LaTeX, PostScript)
E. Hyvonen, S. De Pascale,
"InC++ library family for interval computations" pp. 85-90
(ASCII, PostScript)
E. Hyvonen, S. De Pascale,
"Interval constraint Excel" pp. 91-101
(ASCII, PostScript)
M. E. Jerrell,
"Applications of interval computations to economic
input-output models" pp. 102-104
(LaTeX, PostScript)
R. B. Kearfott,
"A review of techniques in the verified solution of
constrained global optimization problems" p. 105-106
(LaTeX, PostScript)
L. J. Kohout,
"Fuzzy interval-valued inference system with para-consistent
and grey set extensions" pp. 107-110
(LaTeX, PostScript)
A. B. Korlyukov and V. Kreinovich,
"Equations of physics become consistent
if we take measurement uncertainty into consideration"
pp. 111-112
(LaTeX, PostScript)
W. Kraemer,
"Validated function evaluation using polynomial approximation
from truncated Chebyshev series" p. 113
(LaTeX, PostScript)
L. Kupriyanova,
"Inner estimation of the united solution set of interval
linear algebraic system" p. 114
(LaTeX, PostScript)
V. P. Kuznetsov,
"Interval methods for processing statistical
characteristics" pp. 116-122
(LaTeX, PostScript)
V. P. Kuznetsov,
"Auxiliary problems of statistical data processing: interval
approach" pp. 123-129
(LaTeX, PostScript)
A. V. Lakeyev,
"Linear algebraic equation in Kaucher arithmetic" p. 130-133
(LaTeX, PostScript)
A. V. Lakeyev and V. Kreinovich,
"If input intervals are small enough, then interval
computations are almost always easy" pp. 134-139
(LaTeX, PostScript)
R. N. Lea and V. Kreinovich,
"Intelligent control makes sense even without expert
knowledge: an explanation" pp. 140-145
(LaTeX, PostScript)
R. D. Lins, M. A. Campos, M. de B. Correia,
"Functional programming and interval arithmetic" p. 146-150
(LaTeX, PostScript)
R. Lopez De Mantaras,
"From intervals to possibility distributions: adding
flexibility to reasoning under uncertainty" pp. 151-152
(LaTeX, PostScript)
G. G. Menshikov,
"On application of interval approach to finding of the
latent singular points of initial value problems" pp. 153-154
(LaTeX, PostScript)
D. Morgenstein and J. Murphy,
"An application of parallel interval techniques to geophysics"
p. 155
(LaTeX, PostScript)
A. Murgu,
"Implementable bounds for a neural networks algorithm in
communication traffic control" pp. 156-157
(LaTeX, PostScript)
Hung T. Nguyen and E. Walker,
"Interval fuzzy data processing: case when degree
of belief is described by an interval" p. 158
(LaTeX, PostScript)
A. I. Orlov, "Invariance leads to the interval character of
ordinal statistical characteristics", pp. 159-161
(LaTeX, PostScript)
R. B. Patil,
"Application of simulated annealing and genetic algorithms to
verified global optimization" p. 162
(LaTeX, PostScript)
R. B. Patil,
"Interval genetic programming" p. 163
(LaTeX, PostScript)
R. B. Patil,
"Interval neural networks" p. 164
(LaTeX, PostScript)
J. Rohn, "Linear interval equations: computing sufficiently
accurate enclosures is NP-Hard" p. 165
(LaTeX, PostScript)
M. J. Schulte and E. E. Swartzlander, Jr.,
"Design and applications for variable-precision,
interval arithmetic coprocessors" pp. 166-172
A. L. Semenov,
"Solving optimization problems with the help of the
UniCalc solver" pp. 173-175
(LaTeX, PostScript)
E. Serrano, V. P. Pytchenko, V. M. Rubinshtein,
and V. Kreinovich,
"Error estimate of the result of measuring laser beam
diameter" pp. 176-180
(LaTeX, PostScript)
S. P. Shary,
"Linear static systems under interval uncertainty:
algorithms to solve control and stabilization problems"
pp. 181-184
(LaTeX, PostScript)
G. L. Shevlyakov,
"On the choice of an optimization criterion under
uncertainty in interval computations -
nonstochastic approach" pp. 185-187
(LaTeX, PostScript)
G. L. Shevlyakov, N. O. Vilchevskiy,
"Robust minimax adaptive approach to regression problems in
interval computations" pp. 188-189
(LaTeX, PostScript)
S. Smith and V. Kreinovich,
"In case of interval uncertainty, optimal control is
NP-hard even for linear plants, so expert knowledge
is needed" pp. 190-193
(LaTeX, PostScript)
W. Solak,
"A remark on power series estimations via boundary
corrections with parameter" pp. 194-196
(LaTeX, PostScript)
W. Sommerer and M. Kerbl,
"Rendering swept volumes using interval methods" pp. 197-202
S. Starks,
"An interval approach to congestion control in computer
networks" pp. 203-206
(LaTeX, PostScript)
R. Trejo and A. I. Gerasimov,
"Choosing interval functions to represent measurement
inaccuracies: group-theoretic approach" p. 207-210
(LaTeX, PostScript)
I. B. Turksen,
"Interval valued fuzzy sets and measures" p. 211
(LaTeX, PostScript)
K. Villaverde and V. Kreinovich,
"Parallel algorithm that locates local extrema of a
function of one variable from interval measurement
results" pp. 212-219
(LaTeX, PostScript)
G. W. Walster,
"Connections between statistics, interval analysis and
global optimization" p. 220
(LaTeX, PostScript)
G. W. Walster,
"Stimulating hardware and software support for interval
arithmetic" p. 221
(LaTeX, PostScript)
M. C. Wang, K. Lin, and W. Kennedy,
"Self-validating computation for selected probability
functions" pp. 222-223
(LaTeX, PostScript)
Kung Chris Wu,
"Interval methods in mobile robot control" pp. 224-226
(LaTeX, PostScript)
G. D. Zrilic,
"Computations based on Delta-modulation representation of
measurement results" pp. 227-231
(LaTeX, PostScript)
Qiang Zuo, "Description of strictly monotonic interval
AND/OR operations" p. 232-235
(LaTeX, PostScript)
Qiang Zuo, I. Burhan Turksen, Hung T. Nguyen,
and Vladik Kreinovich,
"In expert systems, even if we fix AND/OR operations,
a natural answer to a composite query is the
interval of possible degrees of belief" pp. 236-240
(LaTeX, PostScript)
V. S. Zyuzin, O. V. Mushtakova,
"A posteriori estimation of the solution of system of
differential equations with the help of Taylor series"
pp. 241-242.
(LaTeX, PostScript)
V. S. Zyuzin, E. A. Novikova,
"A method for estimating the solution of ordinary
differential equation
system by interval Taylor series" pp. 243-244.
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van Hiele Model of Geometric Thinking | kidsmathtalk
top of page
What is the van Hiele Model
of Geometric Thinking?
While there is no specific grade level attachment to the van Hiele model of Geometric Thinking, when thinking about the Common Core State Standards, achieving Level 2: Informal Deduction, is an
excellent goal for the end of 5th grade.
Level 0:Visualization
The most basic level of understanding geometric concepts is visualization. this level focuses on recognizing a "whole" shape and connecting that shape to a real-world object. For example, children
might say that a square or rectangle looks like a window. Children at this level are not identifying any properties of shapes, but might be able to say "it is a circle/ it is not a circle," for
Level 1: Analysis
At Level 1, children start attaching properties to shapes, such as a triangle has three sides, or a square has four sides that are all the same length. Children at this level are able to fold and cut
Level 2: Informal Deduction
Children at this level are able to find relationships between shapes and their properties. Completing a Venn Diagram, to determine similarities and differences of shapes and their properties, is a
good activity to determine if children are at this level.
Level 3: Deduction
Individuals at this level are able to generalize and use the properties of shapes to prove that a shape is indeed that shape. At this level, individuals are able to write proofs and understand the
role of definitions.
Level 4: Rigor
This final level is reserved for individuals who have a generalized understanding of geometric principles and different types of proofs. Individuals at this level might be majoring in mathematics at
a collegiate level.
Continue to Explore Kids Math Talk - Geometry for suggestions of Geometry talks, as well as links to professional learning resources.
Visit the Geometry Talk- Articles to Explore for links to more examples of the van Hiele Model of Geometric Thinking.
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An airplane uses approximately 5 gallons of fuel per mile flown-Turito
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An airplane uses approximately 5 gallons of fuel per mile flown. If the plane has 60,000 gallons of fuel at the beginning of a trip and flies at an average speed of 550 miles per hour, which of the
following functions estimates the amount of remaining fuel A (t), in gallons, t hours after the trip began?
let t hours be the time of flight . The average speed of the plane is 550 miles per hour
Step 1:- Find distance travelled in time t.
The distance travelled in time t = speed × time = 550 t (in miles )
Step 2:- Find the fuel consumed in time t .
Given the plane consumes 5 gallons of fuel every mile .
So ,the plane consumes 5 × 550 × t gallons of fuel for 550 × t miles in time t hours .
The fuel consumed in time = 5 × 550 × t = 2,750t gallons
Step 3:- calculate remaining amount of fuel A(t)
The initial amount of fuel = 60,000 gallons
The remaining fuel A(t) = Initial fuel - fuel consumed in time t.
A(t) = 60,000 - 2,750t (in gallons)
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What is Ggpairs?
What is Ggpairs?
The ggpairs() function of the GGally package allows to build a great scatterplot matrix. Scatterplots of each pair of numeric variable are drawn on the left part of the figure. Pearson correlation is
displayed on the right. Variable distribution is available on the diagonal.
How do I add a legend to Ggpairs?
2 Answers
1. Ensure legends are set to ‘TRUE’ in the ggpairs function call.
2. Now iterate over the subplots in the plot matrix and remove the legends for each of them and just retain one of them since the densities are all plotted on the same column.
What is a Ggpairs plot?
The ggpairs() function from the GGally package allows us to build a great scatterplot matrix. Scatterplots of each pair visualized in left side of the plot and Pearson correlation value and
significance displayed on the right side.
What package is GGally in?
The R package ‘ggplot2’ is a plotting system based on the grammar of graphics. ‘GGally’ extends ‘ggplot2’ by adding several functions to reduce the complexity of combining geometric objects with
transformed data.
What are pairwise plots?
An effective way to familiarize with a dataset during exploratory data analysis is using a pairs plot (also known as a scatter plot matrix). A pairs plot allows to see both the distribution of single
variables and relationships between two variables in a dataset.
How do I create a legend in ggplot2?
You can place the legend literally anywhere. To put it around the chart, use the legend. position option and specify top , right , bottom , or left . To put it inside the plot area, specify a vector
of length 2, both values going between 0 and 1 and giving the x and y coordinates.
What is GGally in R?
GGally: Extension to ‘ggplot2’ The R package ‘ggplot2’ is a plotting system based on the grammar of graphics. ‘GGally’ extends ‘ggplot2’ by adding several functions to reduce the complexity of
combining geometric objects with transformed data.
How do you plot a correlation matrix in R?
R corrplot function is used to plot the graph of the correlation matrix….Correlogram : Visualizing the correlation matrix.
Arguments Description
corr The correlation matrix to visualize. To visualize a general matrix, please use is.corr=FALSE.
method The visualization method : “circle”, “color”, “number”, etc.
What package is ggplot2 in R?
ggplot2 is a R package dedicated to data visualization. It can greatly improve the quality and aesthetics of your graphics, and will make you much more efficient in creating them. gallery focuses on
it so almost every section there starts with ggplot2 examples.
Is GGally in Tidyverse?
The widely used ‘ggplot2’ package is encapsulated in the ‘tidyverse’. ‘GGally’ reduces the complexity of combining geometric objects with transformed data by adding several functions to ‘ggplot2’.
Why do we use Pairplot?
Pair plot is used to understand the best set of features to explain a relationship between two variables or to form the most separated clusters. It also helps to form some simple classification
models by drawing some simple lines or make linear separation in our data-set.
What is a pairwise relationship?
Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the
two entities are identical.
How do I create a legend in R?
Create Custom Legend With ggplot in R
1. Use the legend.position Parameter in the theme Function to Specify Legend Position in R.
2. Use legend.justification and legend.background Parameters in theme Function to Create Custom Legend.
3. Use legend.title Parameter in theme Function to Modify Legend Title Formatting.
What is Library GGally?
What happened to ggplot2?
On 25 February 2014, Hadley Wickham formally announced that “ggplot2 is shifting to maintenance mode. This means that we are no longer adding new features, but we will continue to fix major bugs, and
consider new features submitted as pull requests.
Is Ggplot and ggplot2 the same?
You may notice that we sometimes reference ‘ggplot2’ and sometimes ‘ggplot’. To clarify, ‘ggplot2’ is the name of the most recent version of the package. However, any time we call the function
itself, it’s just called ‘ggplot’.
How do you make a Pairplot?
pairplot() method – GeeksforGeeks….seaborn. pairplot() :
Arguments Description Value
{x, y}_vars Variables within “data“ to use separately for the rows and columns of the figure; i.e. to make a non-square plot. lists of variable names, optional
dropna Drop missing values from the data before plotting. boolean, optional
What is a Pointplot?
A point plot represents an estimate of central tendency for a numeric variable by the position of scatter plot points and provides some indication of the uncertainty around that estimate using error
How does ggpairs () work?
By default, ggpairs () provides two different comparisons of each pair of columns and displays either the density or count of the respective variable along the diagonal. With different parameter
settings, the diagonal can be replaced with the axis values and variable labels. There are many hidden features within ggpairs ().
What is the difference between ggpairs and ggmatrix?
ggpairs () is a special form of a ggmatrix () that produces a pairwise comparison of multivariate data. By default, ggpairs () provides two different comparisons of each pair of columns and displays
either the density or count of the respective variable along the diagonal.
Where can I find the list of valid GGally_name functions?
The list of current valid ggally_NAME functions is visible in vig_ggally (“ggally_plots”). A section list may be set to the character string “blank” or NULL if the section should be skipped when
printed. pm <- ggpairs ( tips, columns = c (“total_bill”, “time”, “tip”), upper = “blank”, diag = NULL ) pm #> `stat_bin ()` using `bins = 30`. | {"url":"https://www.tonyajoy.com/2022/08/23/what-is-ggpairs/","timestamp":"2024-11-08T09:04:07Z","content_type":"text/html","content_length":"52136","record_id":"<urn:uuid:38d1e3f4-d412-499f-ac39-72f2420b97c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00016.warc.gz"} |
Machine Learning Specialization is completed whats next?
I’ve completed Machine Learning Specialization. I understand the fundamentals of ML such as Supervised and Unsupervised Learning. But to actually implement all these concepts and if needed in order
to understand the maths as well, what should be my next step please guide?
I am currently a Software Engineer with 8 years of experience. So Python isn’t a concerning end for me. I am more concerned about implementations of ML and Maths.
@icybergenome do DLS. It will open your mind.
*As to the maths question, I’ve been doing this somewhere else, but even for me I think it is useful to see ‘what are the problems we are dealing with’ first, then you can go back and attack the
1 Like
The mathematics isn’t very important if your goal is to use existing tools such as tensorflow. They provide all of the math under-the-hood. You design the network, it handles the calculations.
If you want to invent totally new methods, or if you want to implement models for platforms that don’t support advanced tools, then yes you’ll need a strong math background.
Got it. Thanks for explaining, in practical world I perceive its mostly about reusability of existing tools?
The tricky math in machine learning is in training the model. It requires some calculus to get the equations for using the gradients to find the weights that give the minimum cost during training.
There are tools that already implement the required training methods. If your platform supports those tools, it’s fairly simple to implement.
ML also involves some simple statistics, but that doesn’t require special math skills.
@TMosh @icybergenome for better or worse, I’ve gotten involved in this and I suggest you check out Andrej Karpathy’s series of videos, at least up to Micrograd.
This should at least give you a practical understanding how in the ‘real world’ (i.e. discrete computerized systems) this is handled.
Thanks for suggesting
Where can one get the series, I couldn’t find it on Youtube
Get a job or build your own ai product or services.
MLS was more about theoretical aspects although it covered algorithms but the python related details were not covered.
The course is offered through Coursera only. | {"url":"https://community.deeplearning.ai/t/machine-learning-specialization-is-completed-whats-next/706213","timestamp":"2024-11-05T05:35:04Z","content_type":"text/html","content_length":"46369","record_id":"<urn:uuid:f0f9d5b0-9cc5-47ca-8d34-1714b9274293>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00092.warc.gz"} |
What is the energy of a photon that emits a light of frequency 4.47 times 10^14 Hz? | HIX Tutor
What is the energy of a photon that emits a light of frequency #4.47 times 10^14# #Hz#?
Answer 1
$E = 2.96 \times {10}^{- 19} J$
#E=(6.626xx10^(-34)Js)(4.47xx10^(14) Hz)#
#E= 2.96xx10^(-19) J#
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Answer 2
The energy of the photon of light is $2.96 \times \text{10"^(-19)" J}$.
The energy of a photon of a particular wavelength is determined using the following formula:
#E# is energy, #h# is Planck's constant #(6.626xx"10"^(−34)"J"*"s")#, and #nu# is the frequency #(4.47xx"10"^(14)"Hz")# or #((4.47xx10^(14))/("s"))#.
Plug the data into the formula and solve.
#E=6.626xx"10"^(−34)"J"*color(red)cancel(color(black)("s"))xx(4.47xx10^(14))/(color(red)cancel(color(black)("s")))=2.96xx"10"^(-19)" J"# (rounded to three significant figures)
The energy of the photon of light is #2.96xx"10"^(-19)" J"#.
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Answer 3
The energy of a photon emitting light of frequency ( f ) is given by the formula:
[ E = hf ]
where ( h ) is Planck's constant (( 6.62607015 \times 10^{-34} ) J·s) and ( f ) is the frequency of the light.
Plugging in the values:
[ E = (6.62607015 \times 10^{-34} , \text{J·s}) \times (4.47 \times 10^{14} , \text{Hz}) ]
[ E \approx 2.96 \times 10^{-19} , \text{J} ]
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Formula for math success gets boost
Winners display their certificates at the 13th Shing-Tung Yau College Student Mathematics Contest in Hefei, Anhui province.CHINA DAILY
Guan Dian, who is about to enter his third year at Tsinghua University in Beijing studying math, took part in the 13th Shing-Tung Yau College Student Mathematics Contest recently.
The 20-year-old, who has been interested in math since he was in junior middle school, won a bronze medal in the competition for applied and computational math, and that has spurred him to go further
in the field.
"What I like about math is that it is so logical and so precise," Guan says.
"My deepening understanding of the subject through continuous learning has encouraged me to press on and learn more."
The Yau Mathematical Sciences Center of Tsinghua University and School of Mathematical Sciences, University of Science and Technology of China, organized the contest, which ended with an award
ceremony in Hefei, Anhui province, on Sunday.
An award ceremony celebrates winners of the 13th Shing-Tung Yau College Student Mathematics Contest in Hefei, Anhui province, on Sunday.CHINA DAILY
The contest was first held in 2010. It was organized by mathematician Shing-Tung Yau, who 40 years ago became the first Chinese winner of the Fields Medal, the highest award in mathematics. The
competition he set up aims to check students' all-round ability and knowledge in math, and promote the growth of knowledge of the subject in China.
Yau's motivation for setting up the contest was his belief that many Chinese universities had not been able to offer the full range of required mathematical courses, and thus students were lacking in
solid fundamental knowledge and skills for further study.
Moreover, universities emphasized applied science much more than mathematical fundamentals, he says.
"I noticed many Chinese students majoring in math performed poorly in their qualification exams for PhD studies at overseas universities, and I wanted to change that."
An award ceremony celebrates winners of the 13th Shing-Tung Yau College Student Mathematics Contest in Hefei, Anhui province, on Sunday.CHINA DAILY
As a result, the level of difficulty in the Yau contest matches the level of difficulty in qualification exams for PhD candidates. Contestants are undergraduates from throughout China.
"Unlike other math competitions, this one stresses students' extension of their knowledge rather than their ability to answer questions," Guan says.
There are six areas of competition: geometry and topology; algebra, number theory and combinatorics; probability and statistics; applied and computational mathematics; analysis and partial
differential equations; and mathematical physics－the last of these having just been added.
Students can sign up for one or more subjects in individual contests, and there are gold, silver and bronze medals for each subject.
In addition to individual contests, there is an individual overall award to honor students who perform well in several subjects, and a team contest.
Shiu-Yuen Cheng, president of the organizing committee, who is also a professor at Tsinghua, says that more than 3,200 students from 525 universities entered the contest, twice the number of last
year, and it ran more than six months. After a written exam last month, 93 students reached the final, which consists of a written exam and oral quiz. There were 68 prizes.
An award ceremony celebrates winners of the 13th Shing-Tung Yau College Student Mathematics Contest in Hefei, Anhui province, on Sunday.CHINA DAILY
Over the past 12 years, more than 16,000 undergraduates have taken part in the contest, among whom 720 have won prizes, and many have gone on to conduct excellent mathematical research, Cheng says.
Yau says that in the years since the contest was first held, his aims for establishing it have generally been realized, at least in some universities.
"Universities tend to impart all the required basic knowledge to students in class now," he says.
Students performed well in this year's contest, and the gap between recent contestants and the best students he has ever met has narrowed, he says.
One of the prize winners, Zeng Xiangru, 20, of the University of Science and Technology of China in Hefei, says the contest is quite different from other math competitions.
"Many of the others mostly involve written exams, but in this one, the oral quiz requires you to express your ideas through explanation and writing on the blackboard. It can reflect something that
other exams cannot show, especially your expressiveness and reactions."
Guan says: "Through dealing with many mathematical scholars, we have learned about them and their research fields, something that encourages us to think about what research direction we plan to
Hu Yuqi, a math major at Tsinghua, gave a talk about Polish logician and mathematician Alfred Tarski. It was one of a series of lectures organized by Qiuzhen College, Tsinghua, of which Yau is the
The aim of the lectures is to make profound math more accessible to ordinary people, says Yau, who also established the lectures.
"Math culture in China leaves a lot to be desired. Many teachers and parents believe math is difficult and boring. Their attitudes may adversely influence students. I want to make students know that
math is interesting and keeps evolving."
Earlier lectures in the series were given in Guizhou, Zhejiang and Jiangsu provinces and in Guangxi Zhuang autonomous region, in which stories were told of Leonhard Euler (18th-century Swiss
mathematician), Archimedes, Rene Descartes (17th-century French scientist) and others.
"Training a talented scholar is not a quick job," Yau says. "We are very clear that we must endeavor not only to impart math knowledge to students, but also cultivate them to understand the culture
of math.
"Making our students know what the great mathematicians learned and experienced, what their opinions were, and how their research influenced math is very important. That helps us make our own
Contact the writer at wangru1@chinadaily.com.cn
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Size-dependent Vibration Analysis of Non-uniform Mass Sensor Nanobeams
1. Nazemizadeh^ a*, H. Saffari^ a, A. Assadi^ a, M. Taheri^ b
^a Faculty of Mechanics, Malek Ashtar University of Technology, Iran
^b Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, Iran
Nanobeam In the present paper, the exact modeling and frequency analysis of the mass sensor nanobeam are investigated based on a higher-order elasticity theory with taking into account the
longitudinal discontinuity. The energy equations of the beam are expressed considering discontinuity, and finally, the vibration equations and boundary conditions of the non-uniform
Vibration nanobeam are derived using Hamilton’s principle. By the implementation of an analytical solution, the number of shape functions equal to longitudinal discontinuities is assumed. Then, by
expressing the compatibility and boundary conditions, the frequency equation of the discontinuous nanobeam is obtained and solved. Effects of different parameters such as sensed mass and
Mass size effects on the frequency behavior of the nanobeam are investigated at various vibrational modes. The results show that accurate modeling of discontinuous nanobeam is important. Also,
sensor Changing the position of the sensed mass to the free end of the nanotube increases the sensing feature of the beam, and the size effect reduces it. The size effect reduces the frequency
and increases the amplitude of the mode shape, especially at higher vibrational modes. The results also show that the sensing feature of the mass sensor nanobeam is more prominent at
Size higher modes of vibration, and therefore the use of mass sensor nanobeam at higher vibrational modes is recommended.
1. Introduction
Recent advances in manufacturing technologies have made it possible to develop small-scale systems at micron/submicron scales. In recent decades, nanostructure technologies have enabled the
development and application of advanced micro/nanosystems, such as atomic force microscopes (AFMs), nanoactutors, nanosensors, etc. [1-3]. In fact, the nano-scale beams possess prominent features
such as small dimensions, easy manufacturing, and high-frequency performance which make them the main components of nanosystems.
Furthermore, most of the nanobeams operate in vibrational or dynamic modes. Therefore, the dynamic analysis of nanobeams has attracted the attention of many researchers in the field of
Taheri [4] studied sensitivity analysis of dimensional parameters on dynamic behavior of carbon nanotubes. Also, Korayem et al. [5] studied dynamic modeling of an atomic force nanomechanical beam
adjacent to a surface considering tip-sample interaction forces. They introduced the critical force and time as important parameters of the performance of atomic force microscopes nanobeam and
carried out the sensitivity analysis of dimensions of the nanobeam such as length, thickness, and height on its dynamic behavior. However, they modeled the system as a lumped mass, which cannot be
approved as an accurate model for a continuous beam, especially in the nano-scale.
In the previous studies, the classic elasticity theory has been used to derive the dynamic equations of nanobeams, although the ability of this theory to dynamically describe the micro/nanosystems is
strongly doubted through conducting experimental tests and molecular simulations [6-7]. Indeed, the mechanical properties and behavior of micro/nanobeams depend on their dimensions at small scales,
which the classical theory of elasticity is unable to consider. Besides, performing experimental tests at micro/nano-scales is challenging and costly. Thus, in the past few decades, size-dependent
theories of elasticity have been presented for the dynamic analysis of small-scale systems and have attracted many researchers in the field of nanoscience. Jiang and Yan [8] employed the surface
elasticity theory for the mechanical analysis of nanobeams, which is known as a higher-order elasticity theory. They derived the governing vibration equation of the nanobeam by employing the surface
elasticity theory. Also, in [9], vibration analysis of carbon nanobeams was performed for virus detection with consideration of the surface elasticity theory. Jalali et al. [10] investigated the
size-dependent vibration of a functionally graded micro-resonator based on the modified couple stress theory. They employed the Rayleigh-Ritz method to obtain the size-dependent natural frequencies
of the beam for different boundary conditions. Khorshidi and Fallah [11] presented size-dependent vibration of a functionally graded nanostructure considering modified couple stress theory. Also,
they [12] investigated temperature distribution and size effects on vibration behavior of such nanostructures undergoing prescribed overall motion. They developed vibration equations of the
nanostructure-based on modified couple stress theory and exponential shear deformation
theory. Recently, Assadi and Nazemizadeh [13] studied effect of longitudinal discontinuity of the nanobeam on its size-dependent stability and self-instability with considering the surface elasticity
theory. They modeled the nanobeam as a step-wise beam and they derived its governing equation considering compatibility and different boundary conditions.
On the other hand, the nonlocal size-dependent elasticity theory has been presented in the past two decades and has received the attention of many nanotechnology researchers. This theory was first
introduced by Eringen [14], and the first application of this theory in modeling nano-scale systems was carried out by Pedison et al. [15]. They modeled a nonlocal nanobeam and indicated that the
theory plays an important role in micro/nanotechnology applications. Zhang et al. [16] studied the buckling of a weakened nanobeam subjected to an axial force based on the nonlocal elasticity theory.
They considered the effects of weakened joints and size effects on the buckling load of the nanobeam. Nazemizadeh and Bakhtiari-Nejad [17] studied free vibrations of micro/nanobeams including
piezoelectric layers. They investigated the size effects on vibrational behavior of nonlocal beams and showed that the nonlocal parameter had a prominent effect on the dynamic behavior of the
nanobeam. They also proposed [18] a general formulation for calculating the quality factor of vibrating nanobeams in air environments. Thai et al. [19] presented a formulation for bending and
vibration of nanobeams regarding to shear deformation and size effects and considering the nonlocal elasticity theory. They used an analytical method to solve the governing equations. They
investigated the size effects on the mechanical behavior of nanobeam. In [20], a perturbation method was employed to solve nonlinear size-dependent vibration of nonlocal two-layered piezo laminated
nanobeam. Mawphlang [21] analyzed buckling of non-uniform nanobeams taking into account the nonlocal elastic theory. They derived governing differential equation for nonuniform nanobeam subjected to
the axial compressive load and solved the problem numerically by employing the differential transformation method. In [22], nonlocal effects were studied on nonlinear vibration of nanobeams at higher
modes of vibration. Recently, Hossein and Lellep [23] investigated natural frequency of stepped nanobeam considering the nonlocal effects and rotary inertia. They used Homotopy perturbation method to
solve the governing equations for two steps nanobeam. However, the solution procedure and detail mathematical formulation were ignored.
In this article, the vibration analysis of mass sensor nanobeams is presented with consideration of the longitudinal discontinuity and the nonlocal elasticity theory. The non-uniform and
discontinuous model of nanobeams has been inspired by the fact that most micro/nanobeams are designed and fabricated at the narrower end section. Therefore, a discontinuous nanobeam model is
considered, which senses the absorbed mass at a desired longitudinal point for mass sensor applications. In order to derive the vibrational equations governing, energy equations of the beam are
developed with regard to the discontinuity, and finally, the vibrational equations and boundary conditions of the nonlocal nanobeam are derived according to Hamilton’s principle. Then, for the number
of longitudinal discontinuities, the same number of responses of mode shapes along the nanobeam is considered by using the analytical solution. Finally, the frequency equation of discontinuous mass
sensor nanobeam is obtained as an algebraic relation by applying the compatibility conditions and nonlocal boundary conditions. The natural frequencies in various modes of nanobeam are calculated by
solving the frequency response. The effects of different parameters, such as the length of discontinuity, sensed mass, and nonlocal parameters on the frequency behavior of the nanobeam are
investigated. Also, the effects of sensed mass and size effects on the shape functions of nanobeam are simulated. These effects are investigated more precisely at higher modes to study the
application and efficiency of the mass sensor at higher modes of vibration.
2. Problem Formulation
In this section, firstly, a mass sensor cantilever nanobeam is considered with the ability to sense mass at a desired distance from the clamped end, and then the governing vibrational equation is
obtained by using Hamilton’s principle. It should be noted that a non-uniform cantilever nanobeam is modeled and stepwise varying properties across the length of the beam is considered in dynamic
modeling of the system. This discontinued modeling is originated based on the fact that the mass sensor nanobeams are fabricated wider in the first section, while the end section is designed narrower
due to enhance end deflection measurement.
Figure 1 shows a discontinuous cantilever nanobeam.
The characteristics of the nanobeam are as the following: length of the initial section , the distance of the location of discontinuity from the location of sensor mass , the distance of the location
of the sensed mass from the free end of the beam , the total length of nanobeam , width , thickness , and location of the sensed mass .
Considering the transverse vibrations of the nanobeam along the z-axis, the displacement of any desired point of the nanobeam cross-section in the distance z from the neutral axis is and is equal to
the following equation:
Fig. 1. Nonlocal mass sensor cantilever nanobeam
where , , and are the displacement of the desired point of the nanobeam cross-section in the x-direction, y-direction, z-direction, and the transverse displacement of the neutral axis of nanobeam,
In order to obtain the governing equations of the system, Hamilton’s principle is considered as follows:
where, T is the kinetic energy, and are the potential energy and work of the external force of the system, respectively. The kinetic energy of the system can also be calculated as:
where, and are mass density and the cross-section of nanobeam, respectively, and is the absorbed mass on the nanobeam. Then, the kinetic energy can be rewritten by using the Heaviside function:
where the Heaviside function is a non-continuous function whose value is zero for negative input and one for positive input.
The potential energy is also calculated as the following equation:
where, is the nonlocal stress along the x-axis. Also, the non-zero term of the nanobeam strain is equal to:
Moreover, the main point of nonlocal continuum mechanics is that the nonlocal stress tensor at the reference point depends not only on the strain tensor of the same coordinates but also on all points
of the body [14]. The basic equation proposed by Eringen in integral form is as the following:
where is the nonlocal stress tensor, indicates to the nonlocal kernel, is the coordinate of the reference point, and is referred to the coordinate of each point of the body. In addition, is a local
stress tensor that for a homogeneous isotropic object is stated as:
where, is the elastic stiffness tensor, and is the strain tensor. Moreover, Eringen showed that a differential form of the nonlocal formulation could be used instead of the integral form (7) as
follows [14]:
where, is the Laplace operator, and m is defined as the scaling coefficient, which includes the size-dependent small-scale coefficients. In addition, the nonlocal stress tensor for nanobeam is
presented as the following integral equation:
The equation (10) can be converted into the following differential form:
And, the one-dimensional form of the differential nonlocal equation can be rewritten as:
where, is Young's modulus of the nanobeam.
By placing Eq. 6. in Eq. 5, the following equation is obtained:
Also, Eq. 13 can be summarized as follows:
where, is considered as:
Moreover, the work of the external force is also calculated as follows:
where, is the external force applied to the nanobeam.
Now by submitting Eqs. (3), (12), and (14) in the Hamilton principle (2), the equations of motion and boundary conditions can be obtained as the following relations:
where the equivalent mass of the system is expressed as follows:
Besides, the following equation will be obtained by integrating the Eqs. (12) over the cross-section of the nanobeam:
where the following relations are considered:
By submitting Eq. 16. in Eq. 19. the following equation will be obtained:
Furthermore, the governing equation and boundary conditions of the nanobeam can be presented as follows by using Eqs. 18, 19, and 23
The boundary conditions presented in Eq. 25. are related to zero displacement and zero slope at , as well as to zero nonlocal torque and zero transverse shear force of at x = l.
3. Analytical Solution
In the previous section, the governing equations and boundary conditions of the nanobeam were obtained. In order to analytically solve the vibration equation of the nanobeam with the sensed mass at
the desired distance, the beam is divided into three sections: 1) from the fixed end to the location of cross-section discontinuity, 2) from the location of cross-section discontinuity to the sensed
mass, 3) from the absorbed mass to the free end of the beam. Therefore, Eq. 24 is rewritten as the following:
Furthermore, the boundary conditions at both ends of the nanobeam and the compatibility conditions at the location of the cross-section discontinuity and absorbed mass are expressed as:
In general, the vibrational response of the system is considered to equal , and in order to solve the vibrational equations, the shape function response of each section is equal to [17]:
where, is the unknown coefficient. Also, and are respectively the function of natural frequency and geometrical characteristic of the system and are stated as:
where, and can be calculated from the following equations [17]:
Now, if the boundary conditions and compatibility conditions are submitted in the Eq. 28 the frequency matrix is obtained. The natural frequencies and mode shapes of the system can also be obtained
by solving the frequency equation.
4. Simulations and Results
In this section, the frequency analysis of the mass sensor nanobeam at higher modes is simulated. The physical and geometrical characteristics of the nanobeam are listed in Table 1.
Table 1. mass sensor nanobeam Characteristics
Physical characteristics Geometric Specification (nm)
Table 2. comparison of the first dimensionless natural frequency
m Current research Reference [15]
0 1.8751 1.8751
0 0.1 1.8792 1.8791
0.2 18919 1.8917
Firstly, to verify the presented solution, the first dimensionless natural frequency of a cantilever nanobeam is compared with the values presented in [15]. The case study is a uniform nanobeam
without any sensed mass. It should be noted the dimensionless natural frequency of the uniform nanobeam is defined as
Also, in table 3, the natural frequency of a classical beam for different attached mass is compared with the frequencies presented in [24] where is defined. As it can be seen, the results of the
present study are in good agreement with the results presented in [15] and [24], and hence the proposed analytical solution for vibration analysis of the nanobeam can be confidently implemented.
However, a bit differences in results can be related to numerical solution and rounding.
Furthermore, in the first simulation, the effect of the exact modeling of the nanobeam on its frequency behavior is investigated. For this purpose, the nonlocal discontinuous nanobeam as a precise
model is compared with the inaccurate models: the local discontinuous beam, nonlocal uniform beam, and local uniform beam. It is considered that the sensed mass is located at the free end of the
nanobeam, and the natural frequency of the beam is calculated.
In Table (4), the first natural frequency and its relative error for different models compared to the exact model (non-local discontinuous beam) has been calculated in MHz taking into account the
nonlocal parameter to be .
Table 3. comparison of the first dimensionless natural frequency
R Frequency Current research Reference [24]
First 1.8568 1.852
Second 4.6498 4.650
First 1.7228 1.723
Second 4.3996 4.399
First 1.2480 1.248
Second 4.0312 4.041
Table 4. First natural frequency (MHz) of the nanobeam considering different models
First Frequency
Value Error Value Error
Nonlocal stepped beam 1.261 - 0.654 -
Classic stepped beam 1.249 0.95 0.651 0.458
Nonlocal uniform beam 1.357 7.6 0.776 18.65
Classic uniform beam 1.352 7.21 0.775 18.50
It can be seen in Table (4) that the effect of discontinuity is essential in the exact modeling of the nanobeam. Therefore, the relative error caused by applying the continuous model is not
negligible, compared to the discontinuous model, which shows the importance of the present work for the exact modeling of the beam.
In another simulation, the natural frequency of the system caused by changing the location of the sensed mass is investigated. Besides, in Table 5, the first and second natural frequencies of the
nanobeam are calculated in MHz and presented for different locations of the sensed mass. It should be noted that .
Table 5. Effect of the sensed mass position on the first and second natural frequencies (MHz) of the mass sensor nanobeam
0.25 1.221 4.936
0.5 1.193 4.302
0.25 0.976 6.066
0.5 0.828 6.063
0.25 1.232 4.732
0.5 1.203 4.178
0.25 0.981 5.661
0.5 0.831 5.658
According to Table 5, in the first vibration mode, the natural frequency is reduced by changing the position of the sensed mass toward the free end of the nanobeam and increasing the attached mass.
The reason is that in the first vibrational mode, with increasing the mass and its movement toward the free end, the inertia increases at the end of the beam and causes the reduction of the natural
frequency. Besides, the effect of increasing the nonlocal term and size effects on the shift of the first natural frequency of the nanobeam is negligible, and these effects decrease with increasing
the ratio of absorbed mass. However, in the second vibrational mode, since the node of the second shape function is adjacent to , increasing the absorbed mass in this position causes a smaller
decrease in the natural frequency compared to other positions of the mass. Also, by increasing the vibrational modes, the effect of the nonlocal term and size effects on the frequency becomes more
important and decreases the natural frequency of the beam. The reason is that in the classical elasticity theory, it is assumed that the atoms of bodies are rigidly bonded together; however, in the
nonlocal elasticity theory, the atoms of bodies are linked together in an elastic environment matrix with an assumption of spring contact. Therefore, in the nonlocal elasticity theory, the stiffness
of nanostructure is lower, and the natural frequency is reduced. Furthermore, the effect of nonlocal terms at the higher natural frequencies is more prominent. It may be explained by the reality that
wavelengths are decreased for higher modes and the stronger interactions between atoms lead to increasing of the nonlocal effect.
In another study, the effect of nonlocal parameters on the mode shapes of the mass sensor nanobeam is shown for different absorbed masses at its end. Fig. 2 shows the shape function of the first mode
of the beam:
Fig. 2. The first mode shape of the nanobeam for different sensed mass
Fig. 3. The second mode shape of the nanobeam for different sensed mass
As seen in Fig. 2, increasing the nonlocal term and size effects slightly increases the amplitude of the first mode of the nanobeam. However, size effects decrease with increasing the sensed mass. In
fact, the nonlocal effects are not dominant at the first mode of vibration. On the contrary, the amplitude of the mode shape decreases with increasing the absorbed mass. The reason is that as the
sensed mass increases, the total inertia of the mass sensor nanobeam increases and leads to a reduction of the amplitude of the end of the nanobeam.
Furthermore, the shape function of the second mode of the sensor nanobeam for different absorbed mass at the end of the beam is depicted in Fig. 3.
As can be observed in Fig. 3, increasing the nonlocal term in the second vibrational mode increases the amplitude of the mode shape. The reason for the increase in the amplitude of mode shape is that
the nonlocal term reduces the stiffness of the nanobeam and thus increases the vibrational amplitude. Also, the amplitude of the mode shape decreases with increasing the absorbed mass. It is also
observed that by increasing the sensed mass, the effect of the nonlocal term on the mode shape of the nanobeam decreases.
In another simulation, according to Table 6, the frequency behavior of the classical and nonlocal nanobeam is investigated for different thickness ratios .
As is evident in Table 6, increasing the thickness of the nanobeam increases the natural frequency because increasing the thickness increases the mass and stiffness of the nanobeam simultaneously,
but its effect on the stiffness is higher than on the structural mass. On the other hand, as this term increases, the ratio of the frequency reduction is decreased for the sensed mass nanobeam.
Table 6. First natural frequency (MHz) of the nanobeam considering different models
0.3 0.869 4.924
0.6 1.343 6.485
0.3 0.553 3.977
0.6 0.896 5.387
0.3 0.878 4.458
0.6 1.354 6.072
0.3 0.556 3.621
0.6 0.900 5.072
In another study, the effect of the sensed mass and nonlocal term on the natural frequency shift of the nanobeam is investigated at the first and second vibrational modes. Fig. 4 indicates the
frequency shift , where is the frequency of beam without absorbing the sensed mass, and is the beam frequency with mass absorption.
As can be seen in Fig. 4 the frequency shift of the nanobeam sensor increases with the increase of the absorbed mass. Furthermore, in the first mode of vibration, the effect of nonlocal term and size
effects on frequency shifts are negligible.
Also, the shift of the second mode of the frequency of the nanobeam relative to sensed mass is indicated in Fig. 5.
As shown in Fig. 5, the frequency shift increases with the increase of the sensed mass at the second vibrational mode, but the slope of this increment is decreasing relative to the absorbed mass. In
the second vibrational mode, the nonlocal term and size effects reduce the mass sensing of the nanobeam. On the other hand, the sensitivity and frequency change of nanobeam in the second vibrational
mode was greater than that of the first vibrational mode; hence, it is recommended to use the mass sensor nanobeam at higher modes.
Fig. 4. shift of the first natural frequency of the mass sensor nanobeam
Fig. 5. The shift of the second natural frequency of the mass sensor nanobeam
5. Conclusions
In this paper, the precise modeling of the nanobeam has been carried out considering discontinuity based on the size-dependent nonlocal elasticity theory. The vibrational equations and boundary
conditions of the mass sensor nanobeam have been derived by using Hamilton’s principle. Then, the frequency equation of the discontinuous nanobeam has been obtained as an algebraic relation by using
an analytical solution and considering the sensed mass. The effect of various parameters such as length of discontinuity, sensed mass, and nonlocal parameters on the frequency behavior of the
nanobeam have been investigated. The results showed that the effect of discontinuity on the precise modeling of the nanobeam is of great importance and should be taken into account; however,
increasing the sensed mass has reduced the relative error of the modeling. The effect of the nonlocal term and size effects are important at higher vibrational modes and should be considered in the
nanobeam modeling. Besides, by changing the position of the sensed mass toward the free end of the nanobeam, the natural frequency is reduced, but its sensing sensitivity increases. Increasing the
thickness of the nanobeam increases the stiffness of the structure and its natural frequency but decreases the sensitivity of the mass sensor nanobeam.
[1] Sadeghzadeh, S., 2018. Geometric Effects on Nanopore Creation in Graphene and on the Impact-withstanding Efficiency of Graphene Nanosheets. Mechanics of Advanced Composite Structures, 5(1),
[2] Bakhtiari-Nejad, F. and Nazemizadeh, M., 2016. Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory. Acta
Mechanica, 227(5), pp.1363-1379.
[3] Rostamiyan, Y., 2017. Improving Mechanical Properties of Nanocomposite-based Epoxy by High-impact Polystyrene and Multiwalled Carbon Nanotubes: Optimizing by a Mixture Design Approach. Mechanics
of Advanced Composite Structures, 4(1), pp.33-45.
[4] Taheri M., 2019. Investigation and Sensitivity Analysis of Dimensional Parameters and Velocity in the 3D Nanomanipulation Dynamics of Carbon Nanotubes Using Statistical Sobol Method. Modares
Mechanical Engineering, 19(1), pp. 125-135.
[5] Korayem, M.H., Hezaveh, H.B. and Taheri, M., 2014. Dynamic modeling and simulation of rough cylindrical micro/nanoparticle manipulation with atomic force microscopy. Microscopy and Microanalysis,
20(6), pp.1692-1707.
[6] Rostamiyan, Y., 2017. Improving Mechanical Properties of Nanocomposite-based Epoxy by High-impact Polystyrene and Multiwalled Carbon Nanotubes: Optimizing by a Mixture Design Approach. Mechanics
of Advanced Composite Structures, 4(1), pp.33-45.
[7] Kabiri, A. and Sadeghzadeh, S., 2018. Sensitivity Analysis of Fiber-Reinforced Lamina Micro-Electro-Mechanical Switches with Nonlinear Vibration Using a Higher Order Hamiltonian Approach.
Mechanics of Advanced Composite Structures, 5(1), pp.25-39.
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Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1993), p.20120424.
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theories. Mechanics of Advanced Composite Structures, 4(2), pp. 127-137.
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motion. Polymer Composites, 40(2), pp. 766-778.
[13] Assadi A., Nazemizadeh M. 2020. Size effects on stability and self-instability of non-uniform nanobeams with consideration of surface effects. Micro & Nano Letters. Doi: 10.1049/mnl.2020.0262.
[14] Eringen, A.C., 1977. Screw dislocation in non-local elasticity. Journal of Physics D: Applied Physics, 10(5), p.671
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[16] Zhang, Y., Pang, M. and Chen, W., 2013. Non-local modelling on the buckling of a weakened nanobeam. Micro & Nano Letters, 8(2), pp.102-106.
[17] Nazemizadeh, M. and Bakhtiari-Nejad, F., 2015. Size-dependent free vibration of nano/microbeams with piezo-layered actuators. Micro & Nano Letters, 10(2), pp.93-98.
[18] Nazemizadeh, M. and Bakhtiari-Nejad, F., 2015. A general formulation of quality factor for composite micro/nano beams in the air environment based on the nonlocal elasticity theory. Composite
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[22] Nazemizadeh M., Bakhtiari-Nejad F. and Assadi A. 2020. Nonlinear vibration of piezoelectric laminated nanobeams at higher modes based on nonlocal piezoelectric theory. Acta Mechanica, 231(10),
pp. 4259-4274.
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[24] Rao, S.S., 2007. Vibration of continuous systems (Vol. 464). New York: Wiley. | {"url":"https://macs.semnan.ac.ir/article_4965.html","timestamp":"2024-11-11T01:57:05Z","content_type":"text/html","content_length":"103935","record_id":"<urn:uuid:557f3e8e-01d2-47b0-9feb-f1a4932da07a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00722.warc.gz"} |
Breaths Per Minute Calculator - Online Calculators
To find breaths per minute, divide the total number of breaths (B) by the time taken (T) in minutes. This helps you measure the respiratory rate efficiently.
Breaths Per Minute Calculator
Enter any 2 values to calculate the missing variable
Breaths per minute (BPM) is a standard way of measuring a person’s respiratory rate. It’s an important indicator of general health. Moreover, it can help in diagnosing medical conditions. More
importantly, understanding your breaths per minute can give insight into how efficiently you’re breathing and whether your body is receiving enough oxygen.
The formula for calculating breaths per minute is:
$\text{DRR} = \frac{T}{B}$
Variable Description
DRR Breaths per minute
T Time in minutes
B Number of breaths taken during the period
Solved Calculation
Example 1:
Suppose you counted 15 breaths in 1 minute. Calculate the breaths per minute.
Step Calculation
1 DRR = T / B
2 DRR = 1 / 15
3 DRR = 15
Answer: The respiratory rate is 15 breaths per minute.
Example 2:
If a person takes 20 breaths in 2 minutes, find the breaths per minute.
Step Calculation
1 DRR = T / B
2 DRR = 2 / 20
3 DRR = 10
Answer: The respiratory rate is 10 breaths per minute.
What is Breaths Per Minute Calculator?
To find out your breaths per minute, you can use a Breaths Per Minute Calculator. Because, this tool will give you an accurate count of your respiratory rate. Actually, breathing rate is measured by
counting the number of breaths within a set period, usually 15 seconds, and multiplying it to determine the rate per minute.
For example, to manually calculate it, you can count your breaths for 15 seconds and then multiply by four. Thus, it will give you an estimate of your normal respiratory rate. Generally, a typical
rate for a healthy adult ranges between 12 to 20 breaths per minute. So, if you want to count for a shorter duration, such as 10 seconds, multiply your breath count by six to find the result in
breaths per minute.
The respiratory rate calculator app makes this easier, allowing for automated calculations with just a few taps. Alternatively, if you’re curious about a specific breathing condition, such as a
breath hold calculator or a pet’s respiratory rate like how many breaths per minute for a cat, this tool can offer relevant results.
Final Words:
Overall, calculating your breaths per minute is highly crucial. You need to monitor it frequently as it’s one of the key vital signs for health. Remember, health is wealth. | {"url":"https://areacalculators.com/breaths-per-minute-calculator/","timestamp":"2024-11-04T02:18:21Z","content_type":"text/html","content_length":"108208","record_id":"<urn:uuid:4acc5b87-538b-4354-9d7e-ed2e51cce0c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00378.warc.gz"} |
PROC SYSLIN: Two-Stage Least Squares Estimation :: SAS/ETS(R) 9.22 User's Guide
Two-Stage Least Squares Estimation
In the supply and demand model, P is an endogenous variable, and consequently the OLS estimates are biased. The following example estimates this model using two-stage least squares.
proc syslin data=in 2sls;
endogenous p;
instruments y u s;
demand: model q = p y s;
supply: model q = p u;
The 2SLS option in the PROC SYSLIN statement specifies the two-stage least squares method. The ENDOGENOUS statement specifies that P is an endogenous regressor for which first-stage predicted values
are substituted. You need to declare an endogenous variable in the ENDOGENOUS statement only if it is used as a regressor; thus although Q is endogenous in this model, it is not necessary to list it
in the ENDOGENOUS statement.
Usually, all predetermined variables that appear in the system are used as instruments. The INSTRUMENTS statement specifies that the exogenous variables Y, U, and S are used as instruments for the
first-stage regression to predict P.
The 2SLS results are shown in Figure 27.3 and Figure 27.4. The first-stage regressions are not shown. To see the first-stage regression results, use the FIRST option in the PROC SYSLIN statement.
3 9.670892 3.223631 115.58 <.0001
56 1.561956 0.027892
59 10.03724 | {"url":"http://support.sas.com/documentation/cdl/en/etsug/63348/HTML/default/etsug_syslin_sect007.htm","timestamp":"2024-11-14T15:35:22Z","content_type":"application/xhtml+xml","content_length":"20927","record_id":"<urn:uuid:093d6546-812f-4b63-a3bd-58cf74985aba>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00487.warc.gz"} |
What is the surface area produced by rotating f(x)=x^3-8, x in [0,2] around the x-axis? | HIX Tutor
What is the surface area produced by rotating #f(x)=x^3-8, x in [0,2]# around the x-axis?
Answer 1
first find dS, note the radius of the rotation is x.
$\mathrm{dS} = \sqrt{1 + 9 {x}^{4}} \mathrm{dx}$
${\int}_{0}^{2} 2 \pi x \mathrm{dS}$
to solve make a substitution of $w = 3 {x}^{2}$
If you make the substitution with w your integral becomes #int_0^12 pi/3*sqrt(1+w^2)dw# with a trigonometric substitution #w=tan(theta)# the integral becomes #pi/3 int_0^arctan(12) sec^3(theta)d
(theta)# which you can solve by parts or using a table.
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Answer 2
To find the surface area produced by rotating the function ( f(x) = x^3 - 8 ) around the x-axis over the interval [0, 2], you can use the formula for the surface area of a solid of revolution:
[ S = \int_{a}^{b} 2\pi f(x) \sqrt{1 + (f'(x))^2} , dx ]
• ( f(x) ) is the function to be rotated.
• ( f'(x) ) is the derivative of ( f(x) ).
First, find the derivative of ( f(x) ): [ f'(x) = 3x^2 ]
Next, plug the function and its derivative into the formula and integrate over the interval [0, 2]:
[ S = \int_{0}^{2} 2\pi (x^3 - 8) \sqrt{1 + (3x^2)^2} , dx ]
After integrating, you'll get the surface area.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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9.2: Tests for Convergence (2024)
There are many ways to determine if a sequence converges—two are listed below. In all cases changing or removing a finite number of terms in a sequence does not affect its convergence or divergence:
The Comparison Test makes sense intuitively, since something larger than a quantity going to infinity must also go to infinity. The Monotone Bounded Test can be understood by thinking of a bound on a
sequence as a wall that the sequence can never pass, as in Figure [fig:mbtest]. The increasing sequence \(\seq{a_n}\) in the figure moves toward \(M\) but can never pass it. The sequence thus cannot
diverge to \(\infty\), and it cannot fluctuate back and forth since it always increases. Thus it must converge somewhere before or at \(M\).^4 Notice that the Monotone Bounded Test tells you only
that the sequence converges, not what it converges to.
Show that the sequence \(\seq{a_n}_{n=1}^{\infty}\) defined for \(n\ge 1\) by
\[a_n ~=~ \frac{1 \,\cdot\, 3 \,\cdot\, 5 \,\cdots\, (2n-1)} {2 \,\cdot\, 4 \,\cdot\, 6 \,\cdots\, (2n)} \nonumber \]
is convergent.
Solution: Notice that \(\seq{a_n}\) is always decreasing, since
\[a_{n+1} ~=~ \frac{1 \,\cdot\, 3 \,\cdot\, 5 \,\cdots\, (2n-1) \,\cdot\, (2n+1)} {2 \,\cdot\, 4 \,\cdot\, 6 \,\cdots\, (2n)\,\cdot\, (2n+2)} ~=~ a_n \,\cdot\, \frac{2n+1}{2n+2} ~<~ a_n \,\cdot\, (1)
~=~ a_n \nonumber \]
for \(n\ge 1\). The sequence is also bounded, since \(0 < a_n\) and
\[a_n ~=~ \frac{1}{2} \,\cdot\, \frac{3}{4} \,\cdot\, \frac{5}{6} \,\cdots\, \frac{2n-1}{2n} ~<~ 1 \quad\text{for $n\ge 1$} \nonumber \]
since each fraction in the above product is less than 1. Thus, by the Monotone Bounded Test the sequence is convergent.
Note that for a decreasing sequence only the lower bound is needed for the Monotone Bounded Test, not the upper bound. Similarly, for an increasing sequence only the upper bound matters.
Some tests for convergence of a series are listed below:
Most of the above tests have fairly short proofs or at least intuitive explanations. For example, the n-th Term Test follows from the definition of convergence of a series: if \(\sum a_n\) converges
to a number \(L\) then since each term \(a_n = s_n - s_{n-1}\) is the difference of successive partial sums, taking the limit yields
\[\lim_{n \to \infty} \,a_n ~=~ \lim_{n \to \infty} \,(s_n - s_{n-1}) ~=~ L - L ~=~ 0 \nonumber \]
by definition of the convergence of a series. \(~\checkmark\)
Since the n-th Term Test can never be used to prove convergence of a series, it is often stated in the following logically equivalent manner:
Show that \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{n}{2n+1} ~=~ \dfrac{1}{3} + \dfrac{2}{5} + \dfrac{3}{7} + \cdots~\) is divergent.
Solution: Since
\[\lim_{n \to \infty} \;\frac{n}{2n+1} ~=~ \frac{1}{2} ~\ne ~ 0 \nonumber \]
then by the n-th Term Test the series diverges.
The Ratio Test takes a bit more effort to prove.^5 When the ratio \(R\) in the Ratio Test is larger than 1 then that means the terms in the series do not approach 0, and thus the series diverges by
the n-th Term Test. When \(R=1\) the test fails, meaning it is inconclusive—another test would need to be used. When the test shows convergence it does not tell you what the series converges to,
merely that it converges.
Determine if \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{n}{2^n}~\) is convergent.
Solution: For the series general term \(a_n = \frac{n}{2^n}\),
\[R ~=~ \lim_{n \to \infty} \;\frac{a_{n+1}}{a_n} ~=~ \lim_{n \to \infty} \;\dfrac{\dfrac{n+1}{2^{n+1}}}{\dfrac{n}{2^n}} ~=~ \lim_{n \to \infty} \;\dfrac{n+1}{2n} ~=~ \frac{1}{2} ~<~ 1 ~, \nonumber
so by the Ratio Test the series converges.
Figure [fig:integraltest] shows why the Integral Test works.
In Figure [fig:integraltest](a) the area \(\int_1^{\infty} f(x)\,\dx\) is greater than the total area \(S\) of all the rectangles under the curve. Since each rectangle has height \(a_n\) and width \
(1\), then \(S=\sum_2^{\infty} a_n\). Thus, since removing the single term \(a_1\) from the series does not affect the convergence or divergence of the series, the series converges if the improper
integral converges, and conversely the integral diverges if the series diverges. Similarly, in Figure [fig:integraltest](b) the area \(\int_1^{\infty} f(x)\,\dx\) is less than the total area \(S=\
sum_1^{\infty} a_n\) of all the rectangles, so the integral converges if the series converges, and the series diverges if the integral diverges. Notice how in both graphs the rectangles are either
all below the curve or all protrude above the curve due to \(f(x)\) being a decreasing function.
Example \(\PageIndex{1}\): pseries
Add text here.
Show that the p-series \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{1}{n^p}~\) converges for \(p>1\) and diverges for \(p=1\).
\[\sum_{n=1}^{\infty} \,\frac{1}{n} ~=~ 1 \;+\; \frac{1}{2} \;+\; \frac{1}{3} \;+\; \frac{1}{4} \;+\; \cdots \nonumber \]
thus diverges even though \(a_n = \frac{1}{n} \rightarrow 0\) (which is hence not a sufficient condition for a series to converge).
Note that this example partly proves the p-series Test. The remaining case (\(p < 1\)) is left as an exercise.
The divergence part of the Comparison Test is clear enough to understand, but for the convergence part with \(0 \le a_n \le b_n\) for all \(n\) larger than some \(N\), ignore the (finite) number of
terms before \(a_N\) and \(b_N\). Since \(\sum b_n\) converges then its partial sums must be bounded. The partial sums for \(\sum a_n\) then must also be bounded, since \(0 \le a_n \le b_n\) for \(n
> N\). So since \(a_n \ge 0\) means that the partial sums for \(\sum a_n\) are increasing, by the Monotone Bounded Test the partial sums for \(\sum a_n\) must converge, i.e. \(\sum a_n\) is
Determine if \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{1}{n^n}~\) is convergent.
Solution: Since \(n^n \ge n^2 > 0\) for \(n > 2\), then
\[0 ~<~ \frac{1}{n^n} ~\le~ \frac{1}{n^2} \nonumber \]
for \(n > 2\). Thus, since \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges (by the p-series Test with \(p=2>1\), as in Example
Example \(\PageIndex{1}\): pseries
Add text here.
), the series \(\sum_{n=1}^{\infty} \frac{1}{n^n}\) converges by the Comparison Test.
For the Limit Comparison Test with \(\frac{a_n}{b_n} \rightarrow L < \infty\) and \(L > 0\), by definition of the limit of a sequence, \(\frac{a_n}{b_n}\) can be made arbitrarily close to \(L\). In
particular there is an integer \(N\) such that
\[\frac{L}{2} ~<~ \frac{a_n}{b_n} ~<~ \frac{3L}{2} \nonumber \]
for all \(n > N\). Then
\[0 ~<~ a_n ~<~ \frac{3L}{2}\,b_n \quad\text{and $\quad\sum b_n$ converges} \quad\Rightarrow\quad \text{$\sum a_n$ converges} \nonumber \]
by the Comparison test. Likewise,
\[0 ~<~ \frac{L}{2}\,b_n ~<~ a_n \quad\text{and $\quad\sum b_n$ diverges} \quad\Rightarrow\quad \text{$\sum a_n$ diverges} \nonumber \]
by the Comparison Test again. The cases \(L=0\) and \(L=\infty\) are handled similarly.
Determine if \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{n+3}{n \,\cdot\, 2^n}~\) is convergent.
Solution: Since \(\sum_{n=1}^{\infty} \frac{1}{2^n}\) is convergent (as part of a geometric progression) and
\[\lim_{n \to \infty} ~\frac{(n+3)/(n \,\cdot\, 2^n)}{1/2^n} ~=~ \lim_{n \to \infty} ~\frac{n+3}{n} ~=~ 1 \nonumber \]
then by the Limit Comparison Test \(~\sum_{n=1}^{\infty} \frac{n+3}{n \,\cdot\, 2^n}\) is convergent..
A series \(\sum a_n\) is telescoping if \(a_n = b_n - b_{n+1}\) for some sequence \(\seq{b_n}\). Assume the series \(\sum a_n\) and sequence \(\seq{b_n}\) both start at \(n=1\). Then the partial sum
\(s_n\) for \(\sum a_n\) is
\[s_n ~=~ a_1 \;+\; a_2 \;+\; \cdots \;+\; a_n ~=~ (b_1 - b_2) \;+\; (b_2 - b_3) \;+\; \cdots \;+\; (b_n - b_{n+1}) ~=~ b_1 - b_{n+1} \nonumber \]
for \(n \ge 1\). Thus, since \(b_1\) is a fixed number, \(\lim_{n \to \infty} s_n\) exists if and only if \(\lim_{n \to \infty} b_{n+1}\) exists, i.e. \(\sum a_n\) converges if and only if \(\seq
{b_n}\) converges. So if \(b_n \rightarrow L\) then \(s_n \rightarrow b - L\), i.e. \(\sum a_n\) converges to \(L\), which proves the Telescoping Series Test. Note that the number \(b_1\), as the
first number in the sequence \(\seq{b_n}\), could be replaced by whatever the first number is, in case the index \(n\) starts at a number different from 1.
Example \(\PageIndex{1}\): telescoping
Add text here.
Determine if \(~\displaystyle\sum_{n=1}^{\infty} \,\dfrac{1}{n\,(n+1)}~\) is convergent. If it converges then can you find its sum?
Solution: For the sequence \(\seq{b_n}\) with \(b_n = \frac{1}{n}\) for \(n \ge 1\), each term in the series can be written as
\[\frac{1}{n\,(n+1)} ~=~ \frac{1}{n} ~-~ \frac{1}{n+1} ~=~ b_n \;-\; b_{n+1} \nonumber \]
Thus, since \(\seq{b_n}\) converges to 0, by the Telescoping Series Test the series also converges, to \(b_1 - 0 = 1\).
Convergent series have the following properties (based on similar properties of limits):
For Exercises 1-5 show that the given sequence \(\seq{a_n}_{n=1}^{\infty}\) is convergent.
\(a_n = \dfrac{2 \,\cdot\, 4 \,\cdot\, 6 \,\cdots\, (2n)} {3 \,\cdot\, 5 \,\cdot\, 7 \,\cdots\, (2n+1)}\vphantom{\dfrac{2^n}{n!}}\)
\(a_n = 1 \,-\, \dfrac{2^n}{n!}\)
\(a_n = \dfrac{2 \,\cdot\, 4 \,\cdot\, 6 \,\cdots\, (2n)} {1 \,\cdot\, 3 \,\cdot\, 5 \,\cdots\, (2n-1)} \;\cdot\; \dfrac{1}{2n+2}\vphantom{\dfrac{2^n}{n!}}\)
\(a_n = \dfrac{1}{n}\,\left(\dfrac{2 \,\cdot\, 4 \,\cdot\, 6 \,\cdots\, (2n)} {1 \,\cdot\, 3 \,\cdot\, 5 \,\cdots\, (2n-1)}\right)^2\)
[exer:wallisinv] \(a_n = \dfrac{1}{2} \,\cdot\, \dfrac{3}{2} \,\cdot\, \dfrac{3}{4} \,\cdot\, \dfrac{5}{4} \,\cdot\, \dfrac{5}{6} \,\cdot\, \dfrac{7}{6} \,\cdots\, \dfrac{2n-1}{2n} \,\cdot\, \dfrac
For Exercises 6-17 determine whether the given series is convergent. [[1.]]
\(\bigsum{n = 0}{\infty}~ \sin\,\left(\dfrac{n \pi}{2}\right)\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{n\,(n + 2)}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{2n}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{n}{(n + 1)\,2^n}\)
\(\bigsum{n = 2}{\infty}~ \dfrac{1}{n\,\sqrt{\ln \,n}}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{n\,!}{(2n)\,!}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{n}{e^n}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{\cosh^2 n}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{n\,!}{2^n}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{\sqrt{n}}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{n\,(2n-1)}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{\ln\,(n+1)}{n^2}\)
For Exercises 18-21 determine whether the given series is convergent. If convergent then find its sum. [[1.]]
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{(2n+1)\,(2n+3)}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{(2n + 3)\,(2n + 5)}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{2}{(3n + 1)\,(3n + 4)}\)
\(\bigsum{n = 1}{\infty}~ \dfrac{1}{4n^2 - 1}\)
Continue Example
Example \(\PageIndex{1}\): pseries
Add text here.
with a proof of the p-series Test for \(p < 1\).
Show that \(\seq{a_n}_{n=1}^{\infty}\) is convergent, where
\[a_n ~=~ \frac{1}{1!} \;+\; \frac{1}{2!} \;+\; \frac{1}{3!} \;+\; \frac{1}{4!} \;+\; \cdots \;+\; \frac{1}{n!} \nonumber \]
for \(n \ge 1\). (Hint: Use the Monotone Bounded test by using a bound on \(\frac{1}{n!}\) for \(n > 2\).)
Consider the series \(~\bigsum{n = 1}{\infty}~ \dfrac{1}{2n-1} ~=~ 1 \;+\; \frac{1}{3} \;+\; \frac{1}{5} \;+\; \frac{1}{7} \;+\; \dotsb ~\).
1. Show that the series is divergent.
2. The textbook Applied Mathematics for Physical Chemistry (3^rd ed.), J. Barrante, provides the following argument that the above series converges: Since
\[1 ~+~ \frac{1}{4} ~+~ \frac{1}{9} ~+~ \frac{1}{16} ~+~ \dotsb \quad < \quad 1 ~+~ \frac{1}{3} ~+~ \frac{1}{5} ~+~ \frac{1}{7} ~+~ \dotsb \quad < \quad 1 ~+~ \frac{1}{2} ~+~ \frac{1}{3} ~+~ \
frac{1}{4} ~+~ \dotsb \nonumber \]
where the series on the left converges (by the p-series Test with \(p = 2\)) and the series on the right diverges (by the p-series Test with \(p = 1\)), and since each term in the middle series
is between its corresponding terms in the left series and right series, then there must be a p-series for some value \(1 < p < 2\) such that each term in the middle series is less than the
corresponding term in that p-series. That is,
\[1 ~+~ \frac{1}{4} ~+~ \frac{1}{9} ~+~ \frac{1}{16} ~+~ \dotsb \quad < \quad 1 ~+~ \frac{1}{3} ~+~ \frac{1}{5} ~+~ \frac{1}{7} ~+~ \dotsb \quad < \quad 1 ~+~ \frac{1}{2^p} ~+~ \frac{1}{3^p} ~+~
\frac{1}{4^p} ~+~ \dotsb \nonumber \]
for that value of \(p\) between 1 and 2. But \(p > 1\) for that p-series on the right, so it converges, which means that the middle series converges! Find and explain the flaw in this argument.
Wallis’ formula^6 for \(\pi\) is given by the infinite product
\[\frac{\pi}{2} ~=~ \dfrac{2}{1} \,\cdot\, \dfrac{2}{3} \,\cdot\, \dfrac{4}{3} \,\cdot\, \dfrac{4}{5} \,\cdot\, \dfrac{6}{5} \,\cdot\, \dfrac{6}{7} \,\cdots\, \dfrac{2n}{2n-1} \,\cdot\, \dfrac{2n}
{2n+1} \,\cdot\, \cdots ~. \nonumber \]
Notice that this is the limit of the reciprocal of the sequence in Exercise [exer:wallisinv]. Write a computer program to approximate the limit using 1 million iterations. How close is your
approximation to \(\frac{\pi}{2}\)? | {"url":"https://clavig.online/article/9-2-tests-for-convergence","timestamp":"2024-11-06T18:08:05Z","content_type":"text/html","content_length":"73854","record_id":"<urn:uuid:9293e55c-0a4d-4ba8-a6cc-58594eabd325>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00547.warc.gz"} |
Mathematics Refresher for Political Scientists
Course Description This course is designed to understand mathematical tools useful for the rest of your Master program, especially for the upcoming, mandatory courses in statistics and game theory.
The course reviews some mathematical concepts most of you will be familiar with from high school, such as functions, derivatives, integrals, vectors, matrices and distributions. However, we will
introduce and study these topics more rigorously. To provide a deeper understanding of important results, we will also prove some fundamental theorems in class. We will cover fundamentals in single
and (partially) multivariable calculus, linear algebra and probability theory. Each day consists of lectures and exercise sessions. | {"url":"https://tilkoswalve.netlify.app/teaching/mathematicsrefresher/","timestamp":"2024-11-14T17:08:16Z","content_type":"text/html","content_length":"5650","record_id":"<urn:uuid:2e174e9b-c0a0-4f31-860f-67d6a8fa853d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00531.warc.gz"} |
Canny Fraction
What fraction of the volume of this can is filled with lemonade?
A cylinder can contains lemonade, shown shaded on the diagram in which XY is diameter. What fraction of the volume of the can is filled with lemonade:
just below a quarter; just over a quarter; exactly a quarter?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Imagine the liquid became solidified.
Rotate the can until the face which contains XY is horizontal.
Now take the horizontal cross-section half way up the can which meets the plane face of the "liquid" in the line AB.
Observing that AB bisects the radius PQ, rotate the upper section of the "liquid" about AB through 180 degrees. The new position is shaded darkly and fits into the bottom right hand quarter (with
some space to spare).
The actual fraction can be shown to be approximately 0.21. The answer is therefore less than a quarter. | {"url":"https://nrich.maths.org/problems/canny-fraction","timestamp":"2024-11-07T14:02:47Z","content_type":"text/html","content_length":"38250","record_id":"<urn:uuid:c3405b11-768b-4fd3-831f-659d9f49f0c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00553.warc.gz"} |
Testudo: Efficient SNARKs with Smaller Setups
In this blog post, we present Testudo a new near linear-time$^*$ prover SNARK with the following advantages:
• Small & Universal Setup: It uses a trusted setup of square-root size — i.e., for an R1CS of size $N$, the trusted setup is of size $O(\sqrt{N})$. For large circuits, this brings the trusted setup
material to MBs rather than GBs.
• Very fast prover: Estimated to run more than ~5x faster than fastest Groth16 implementation (i.e., Bellperson) for data-parallel computations.
• Small Proofs & Fast Verifier: Constant size proofs and verification time.
• Uses R1CS, the most widely used approach to writing circuits in deployed libraries. This gives us the following advantages:
□ Switching to Testudo allows us to reuse existing code.
□ It does not require rewriting circuits in a different arithmetization.
(*) Our prover runs $\sqrt{N}$ multi-exps of size $\sqrt{N}$, which is roughly $O(N * \lambda/\log(N))$ group operations with $\lambda >> \log(N)$ for security.
Internals: At its core, our SNARK relies on three carefully combined building blocks:
• a modified version of Spartan (sumcheck-based scheme)
• a pairing-based multivariate commitment scheme (modified version of PST)
• a final Groth16 layer.
Testudo has an ongoing implementation using arkworks with a blst integration with GPU support:
Call for participation: We welcome anyone interested to contribute to the Testudo implementation. The project tackles many challenging points and is at the state of the art when it comes to proving
R1CS circuits. See the last section for more information!
Why the name Testudo? Testudo was a type of battle formation that ancient Rome adopted, where its soldiers operated “under the hood” of their shields. Testudo, the proof scheme, is similar: a Spartan
prover woking under the hood of Groth16.
Our initial motivation for developing Testudo was to improve the SNARKs used in Filecoin. Filecoin requires storage providers to prove to the whole network that they are holding the storage they had
initially committed to. The circuit involved has $\approx 2^{30}$ constraints (one of the largest circuits used in practice today) and is verified by Groth16.
The computation is large enough to push current hardware to its limits: the big circuit is actually “broken down” into 10 subcircuits each of size $\approx 2^{27}$, due to limitations on the maximum
size of the trusted setup. Also, one issue with Groth16 is the function-specific trusted setup which complicates deployment of new versions of the Filecoin protocol (e.g., a new proof-of-space) since
this would require a new circuit. While Filecoin is an interesting specific study-case, issues of this kind may also apply more generally to other deployed systems with similar requirements.
Therefore, our goals were as follows:
• Universal trusted setup of sub-linear size
• Faster (or at least comparable) prover time than Groth16
• Short proofs and fast verification time
• R1CS-based, to be “off-the-shelf” compatible with current FIL Proofs circuits
• A SNARK that could leverage “data parallel computation” since FIL Proofs are largely data parallel
A bit more in depth
Recall that an R1CS system is defined by three $N \times N$ matrices, $A,B,C$ and we say that it is satisfiable if there exists a witness $w$ such that $\langle A \circ w \rangle \cdot \langle B \
circ w \rangle - \langle C \circ w \rangle = \vec{0}$
Spartan Recap
At a high level, the Spartan prover consists of several steps as in the following:
Above $A,B,C$ are the constraint matrices of the R1CS instance
High Level Testudo Description
• Polynomial Commitments:
□ In a preprocessing phase, the prover encodes $A,B,C$ as sparse polynomials $\tilde{A}, \tilde{B}, \tilde{C}$and commits to them via polynomial commitments (called computation commitments in
Spartan). We note that for uniform circuits (i.e. very data-parallel, with many sub-circuits repeating in regular patterns), this step is not necessary or much reduced in complexity, since
the Verifier can efficiently compute $\tilde{A}, \tilde{B}, \tilde{C}$ on their own (or the polynomials $\tilde{A},\tilde{B},\tilde{C}$ are much smaller than $A,B,C$)
□ In the online phase the prover computes a multilinear extension $\tilde{w}$ of the witness and commits to it using any multivariate polynomial commitment scheme. While Spartan uses a
discrete-log transparent scheme (Hyrax) we use our modification of PST described below.
□ Note that the polynomials are of size $O(N)$ here, corresponding to the number of R1CS constraints.
• Sumchecks:
□ The Spartan prover executes two sumcheck protocols sequentially using $\tilde{A},\tilde{B},\tilde{C},\tilde{w}$ to prove the satisfiability equation of the R1CS constraints $\langle A \circ w
\rangle \cdot \langle B \circ w \rangle - \langle C \circ w \rangle = \vec{0}$
□ The sumcheck verifier naively runs in time $O(\log N)$. In Testudo we encode this verifier into an R1CS circuit and the prover provides a Groth16 proof of knowledge of the accepting Spartan
sumcheck proof.
• Opening of polynomial commitments: At the end of the sumchecks, the prover needs to show to the verifier the value of the polynomials $\tilde{A}, \tilde{B},\tilde{C}, \tilde{w}$ on a random point
$\vec{x},\vec{y}$ (of size $O(\log N)$)
□ 🐌 To open the modified PST commitment for the witness, the prover operates on an $O(N)$-sized polynomial, which requires several polynomial divisions to get the quotient polynomials necessary
for the proof, plus $O(N)$-sized exponentiations.
□ 🐌 The prover also opens the Computation Commitments, which is where the Spartan prover spends a large chunk of its time. We recall however that on uniform circuits this step is not mandatory
as verifier can open the matrix himself.
□ These openings can be either left in the “clear” for a $O(\log N)$ size proof (the PST opening of the witness polynomial) or also fed to a Groth16 prover for a constant size proof. We discuss
the benefits and challenges of either approach below.
Testudo Details
In Testudo, we apply several optimizations to the blueprint above, both at prover and verifier level.
• 🚀 Moving to universal trusted setup: We note that above we require a trusted setup for
□ the modified PST polynomial commitments,
□ Groth16 on the sumcheck and polynomial commitment verification — we note that these circuits are of size $O(\log N)$ which is their contribution to the size of the trusted setup.
Both are independent of the specific R1CS being proven, yielding a universal trusted setup.
• 🚀 Reducing the trusted setup size: PST, like KZG, requires a trusted setup of size $O(N)$. By observing a tensor product structure in the opening proof of PST, this can be reduced to $O(\sqrt{N})
$. In practice, this means going from GBs to MBs when loading/storing a setup. This is obtained by designing a version of PST that works together with products for inner product arguments (MIPP)
□ We briefly exemplify this on a polynomial defined by 4 evaluation points over the hypercube. Note how the PST.Commit steps only operates on a polynomial of half of the degree!
• 🚀 The verifier needs to verify (a) one inner-pairing product proof of size $O(\log(\sqrt{N}))$ and (b) one PST opening of the same size (vs $O(\log(N))$ before; constants matter 😉).
• 🚀 Faster Proving Times
□ 🚀 Proving operates on $O(\sqrt{N})$-sized polynomials now instead of $O(N)$: so even if there are more polynomials, the proofs can be parallelised and are thus faster to compute. This holds
in particular when it comes to the quotient polynomials which are smaller.
□ 🚀 Large speed up for uniform computations: As an example, in Filecoin Proofs, we verify hundreds of Merkle tree openings. You can think of it as applying a subcircuit (”Verify one Merkle Tree
opening proof”) many times over.
☆ We now treat the “R1CS Matrix” as the matrix for the small subcircuit.
☆ 💡 In the sumcheck phase, we simply “concatenate” each witness (for each subcircuit) together. In theory, we should also concatenate the R1CS matrices together, but, since they are the
same, the prover doesn’t have to do that in practice and we can leverage the specific details of the Spartan protocol here (a slightly more formal draft here)
○ Here is an example where we repeat the same circuit twice. Note that the prover only needs to commit and keep in memory the small blue matrix.
☆ 🚀 While the witness polynomial remains the same length, the computation commitment now only operates on the small circuit which is a significant speed-up
• 🚀 Fast Verification and Small Proof Size
□ 🚀 Constant-time sumcheck verification: A sumcheck operates on finite fields only. Therefore, in Testudo, the prover encodes the verification of the sumcheck as a circuit and gives back a
SNARK proof to the final verifier. Testudo uses Groth16 to implement this verifier, which still keeps the “universality” of Testudo because sumcheck verification doesn’t change with the user
□ 🚀 Constant-time polynomial commitment opening verification: We can also apply a final proof system to compress the verification of the polynomial openings. These openings are of size $O(\log\
sqrt{N})$, which is small enough to run inside a circuit.
☆ We call this the outer proof over the outer curve, see next section for more details.
☆ 🚀 This yields a constant size proof and constant time verifier!
Testudo81 and Testudo77
Originally, we wanted to realize Testudo on BLS12-377 (hence, Testudo77) because of its nice 2-chain property. This enables us to use BW6 (the “outer curve”) to efficiently prove statements about
elliptic curve operations natively. For example, to verify the PST opening inside a proof where one needs to do scalar multiplication and pairings.
However, Filecoin is operating on BLS12-381! That means, in order to introduce this proof system in Filecoin, we would require storage providers to re-encode their storage using the new curve, mostly
because of Poseidon which is field-dependent (unlike SHA256 for example). Testudo81 is our version on Testudo that runs on BLS12-381. The main issue is that it is not possible to use Groth16 naively
on top of BLS12-381 because any “outer curve” will lack high 2-adicity in its scalar field, required to compute FFTs efficiently (see Timofey’s post for more information about that).
The problem: How can we be backwards compatible, so that Filecoin storage providers don’t need to re-encode their storage?
This version is the simplest and most elegant solution if we can afford using these curves. Basically it runs an external Groth16 proof system on BW6 to verify both sumcheck and polynomial commitment
🛗 Aggregation: An additional advantage is that we can use tools like Snarkpack to further aggregate Testudo77 proofs!
Outer proof constraints: We expect the number of constraints to be less than 10 millions which results in a few seconds of additional proving time using Groth16 (which is totally fine for our
In this case since we cannot use Groth16 to compress the proof size and verification time, we could leave them in the clear for a $O(\log N)$ size proof. Alternatively we propose to use a subset of
Testudo itself (i.e. the modified Spartan component which does not require FFT) to compress the polynomial commitment openings (using for example the Yeti curve as the “outer curve”). While this
version achieves compatibility, it loses, however, in compactness and verification time—the “Spartan part” of Testudo is not constant-size (asymptotically this would be a $O(\log \log N)$ proof size/
verification time, with some high constants).
The work-in-progress implementation is open source on Github. Currently, it features:
• The Groth16 verifier of the sumchecks
• The square root version of PST + MIPP
• Arkworks wrapper around the fast blst library with GPU integration (repo)
Modified PST
One of our main contributions is a very fast implementation of a multilinear polynomial commitment, built on top of the existing arkworks library. We present a comparison between the two versions on
a polynomial of size $2^{25}$
Commitment (s) Opening (s) Verification (ms) Proof (KB) Committer Key
arkworks PST 34 184 14 2 KB 9.6 GB
testudo PST 24 1.2 32 17 KB 2.3 MB
Verification time: Note that this time has more than doubled. It is due to non optimized implementation. For example, the PST and MIPP verification are happening in isolation, i.e. the pairings are
evaluated separately instead of together. Multiple other optimization have to be implemented to verify both parts together.
Estimation using uniform circuits
We have the necessary building blocks to estimate accurately the proving time of a uniform circuits (even though the implementation does not yet offer that feature).
Specifically, we need to add the time for
• The first sumcheck on the full R1CS matrix (SC1)
• The second sumcheck on the small subcircuit (SC2)
• The PST times on the full witness size - commitment and opening combined (PST/MIPP)
• The computation commitment time on the small subcircuit (CC)
We have all this for a subcircuit of size $2^{20}$ repeated $128 = 2^{7}$ times, giving a circuit of size $2^{27}$ constraints.
R1CS PST/MIPP (s) Comm + Opening SC1 (s) SC2 (s) CC (s)
$2^{27}$ 47 + 3 105 107 3944
$2^{20}$ 1.17 + 0.454 0.843 0.773 30
Total Proving Time: PST($2^{27}$) + SC1($2^{27}$) + SC2($2^{20}$) + CC($2^{20}$) = 185s
Comparison $2^{27}$ R1CS Bellperson: 1020s
Speedup factor: 5.1x 🚀
💡 As you can see, the CC is the most expensive part. There are many improvements to be done at this early stage that can drastically reduce the proving time there. We remind the reader that for
uniform circuits this cost can be eliminated however.
Comparison with Plonk-ish techniques
When it comes to universal trusted setup proofs, many systems today do not use R1CS but rather “custom gates” (sometimes also called Plonkish arithmeization), and apply SNARKs such as Plonk (or
alternatives such as Hyperplonk) to the resulting constraint systems. The use of “custom gates” makes a comparison to pure R1CS-based schemes not immediate. We are still working on achieving
meaningful comparisons but we estimate that Testudo is competitive with approaches that do use custom gates.
Open problems
We have yet to solve still a few problems down the road:
• ☄️Efficient “streaming” aggregation: Snarkpack allows a prover to only keep Groth16 proofs locally and then aggregate them all at once. This is very light in memory, even for a large number of
□ For Testudo81, can we achieve the same functionality and performance as Snarkpack? See our draft for our current thinking and problem.
□ For Testudo77, can we achieve a faster aggregation? Instead of “fully” proving a Testudo proof, can we keep some small intermediate steps and finalize the last parts of Testudo over all
• 📚 Compact proof on Testudo81: Some open questions are:
□ How to achieve a compact proof using Testudo81?
□ Can we use another proof system, with non-native arithmetic in a practical way?
□ Can we use a mixed FFT algorithm to go around the 2-adicity problem?
• 🚀 Faster computation commitment and CRS for it: Currently, the computation commitment is still a bottleneck compared to the rest (when not using data parallelism). One area to explore is taking
advantage of the PST commitment scheme here and batch it with the PST of the witness.
□ Different Polynomial Commitments: Another design option would be to choose Dory as it could simplify the implementation of the Computation Commitment. In general, being agnostic to the
polynomial commitment would help to try different strategies.
Call for participations
We’re looking for enthusiastic engineers to help us push this effort forward. We believe that making this in the open is gonna give the best results. This is a complex piece of software and the
structure behind is challenging.
There are both design and engineering challenges that are left open. On the implementation side, some of the items we would love to work on are:
• ⚙️ A R1CS circuit building layer: Basically bringing a “ConstraintSystem” that enables development of complex R1CS circuits. The original Spartan test codebase only created R1CS manually.
• ⛓️ Data-parallel Testudo that greatly reduces the proving time over uniform circuits. This can be a game changer for Merkle tree path verification for example.
• 🏎️ GPU-optimized schemes: For example, bringing the whole PST commitment to GPU at once, bringing the computation commitment to the GPU, and even look at parallel computation of the sumcheck on
the GPU.
• 📦 A simplified, and more compact computation commitment (CC): By moving CC to use PST for example we can apply several optimizations combined with the PST commitment computed for the witness. We
can also reduce the size of the preprocessing data required during proving time, which can easily attain TB for circuits of size $2^{30}$.
If you want to help, please reach out on the discord server of cryptonet or email us here! Feel free to discuss over Twitter.
We thank Srinath Setty for helpful pointers about the Spartan codebase that unlocked us many times!
The Testudo effort was started inside the cryptonet team. The main Testudo team is composed of: Matteo Campanelli (cryptonet), Nicolas Gailly (cryptonet), Rosario Gennaro (cryptonet/CCNY), Philipp
Jovanovic (UCL), Mara Mihali (formerly UCL/cryptonet), Justin Thaler (a16z/Georgetown). | {"url":"https://cryptonet.org/blog/testudo-efficient-snarks-with-smaller-setups","timestamp":"2024-11-06T18:57:55Z","content_type":"text/html","content_length":"575168","record_id":"<urn:uuid:2b3088c4-cb81-445c-a9a7-a981a1162dc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00644.warc.gz"} |
Exploiting fast matrix multiplication within the level 3 BLAS
Higham, Nicholas J. (1990) Exploiting fast matrix multiplication within the level 3 BLAS. ACM Transactions on Mathematical Software, 16 (4). pp. 352-368. ISSN 0098-3500
Restricted to Repository staff only
Download (1MB)
The Level 3 BLAS (BLAS3) are a set of specifications of FORTRAN 77 subprograms for carrying out matrix multiplications and the solution of triangular systems with multiple right-hand sides. They are
intended to provide efficient and portable building blocks for linear algebra algorithms on high-performance computers. We describe algorithms for the BLAS3 operations that are asymptotically faster
than the conventional ones. These algorithms are based on Strassen's method for fast matrix multiplication, which is now recognized to be a practically useful technique once matrix dimensions exceed
about 100. We pay particular attention to the numerical stability of these “fast BLAS3.” Error bounds are given and their significance is explained and illustrated with the aid of numerical
experiments. Our conclusion is that the fast BLAS3, although not as strongly stable as conventional implementations, are stable enough to merit careful consideration in many applications.
Actions (login required) | {"url":"https://eprints.maths.manchester.ac.uk/365/","timestamp":"2024-11-06T01:26:14Z","content_type":"application/xhtml+xml","content_length":"24269","record_id":"<urn:uuid:0ff5ee68-d7ce-4706-afb0-fe7d938d3c45>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00574.warc.gz"} |
Society for Industrial and Applied Mathematics (SIAM) is an international community of over 14,000 individual members. SIAM was incorporated in 1952 as a nonprofit organisation to convey useful
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Math Jabs Back: How math helps us get even better vaccines even faster
Ever wonder how vaccines work? SIAM member Jeffrey R. Sachs, who leads vaccine modeling and simulation efforts at Merck, explains how mathematics plays a critical role in our daily lives,
specifically focusing on how math is used in the development, discovery, and manufacturing of vaccines.
Note: all scenes of conferences in this video were filmed before the pandemic, and interview filming involved vaccination as well as appropriate masking and social distancing.
View more on YouTube
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Johannsen, D. and Razgon, Igor and Wahlström, M. (2009) Solving SAT for CNF formulas with a one-sided restriction on variable occurrences. In: Kullmann, O. (ed.) Theory and Applications of
Satisfiability Testing. Lecture Notes in Computer Science 5584. Berlin, Germany: Springer Verlag, pp. 80-85. ISBN 9783642027765.
Full text not available from this repository.
In this paper we consider the class of boolean formulas in Conjunctive Normal Form (CNF) where for each variable all but at most d occurrences are either positive or negative. This class is a
generalization of the class of CNF formulas with at most d occurrences (positive and negative) of each variable which was studied in [Wahlström, 2005]. Applying complement search [Purdom, 1984], we
show that for every d there exists a constant γd<2−12d+1 such that satisfiability of a CNF formula on n variables can be checked in runtime \ensuremathO(γnd) if all but at most d occurrences of each
variable are either positive or negative. We thoroughly analyze the proposed branching strategy and determine the asymptotic growth constant γ d more precisely. Finally, we show that the trivial \
ensuremathO(2n) barrier of satisfiability checking can be broken even for a more general class of formulas, namely formulas where the positive or negative literals of every variable have what we will
call a d–covering. To the best of our knowledge, for the considered classes of formulas there are no previous non-trivial upper bounds on the complexity of satisfiability checking.
Additional statistics are available via IRStats2. | {"url":"https://eprints.bbk.ac.uk/id/eprint/7926/","timestamp":"2024-11-09T13:07:41Z","content_type":"application/xhtml+xml","content_length":"221813","record_id":"<urn:uuid:b053635f-2080-44e2-84b5-18dd5c07eb05>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00415.warc.gz"} |
binary representation of
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media/libeffects/loudness/dsp/core/dynamic_range_compression.h - platform/frameworks/av - Git at Google
* Copyright (C) 2013 The Android Open Source Project
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
* http://www.apache.org/licenses/LICENSE-2.0
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
#ifndef LE_FX_ENGINE_DSP_CORE_DYNAMIC_RANGE_COMPRESSION_H_
#define LE_FX_ENGINE_DSP_CORE_DYNAMIC_RANGE_COMPRESSION_H_
#include "common/core/types.h"
#include "common/core/math.h"
#include "dsp/core/basic.h"
#include "dsp/core/interpolation.h"
//#define LOG_NDEBUG 0
#include <cutils/log.h>
namespace le_fx {
// An adaptive dynamic range compression algorithm. The gain adaptation is made
// at the logarithmic domain and it is based on a Branching-Smooth compensated
// digital peak detector with different time constants for attack and release.
class AdaptiveDynamicRangeCompression {
// Initializes the compressor using prior information. It assumes that the
// input signal is speech from high-quality recordings that is scaled and then
// fed to the compressor. The compressor is tuned according to the target gain
// that is expected to be applied.
// Target gain receives values between 0.0 and 10.0. The knee threshold is
// reduced as the target gain increases in order to fit the increased range of
// values.
// Values between 1.0 and 2.0 will only mildly affect your signal. Higher
// values will reduce the dynamic range of the signal to the benefit of
// increased loudness.
// If nothing is known regarding the input, a `target_gain` of 1.0f is a
// relatively safe choice for many signals.
bool Initialize(float target_gain, float sampling_rate);
// A fast version of the algorithm that uses approximate computations for the
// log(.) and exp(.).
float Compress(float x);
// Stereo channel version of the compressor
void Compress(float *x1, float *x2);
// This version is slower than Compress(.) but faster than CompressSlow(.)
float CompressNormalSpeed(float x);
// A slow version of the algorithm that is easier for further developement,
// tuning and debugging
float CompressSlow(float x);
// Sets knee threshold (in decibel).
void set_knee_threshold(float decibel);
// Sets knee threshold via the target gain using an experimentally derived
// relationship.
void set_knee_threshold_via_target_gain(float target_gain);
// The minimum accepted absolute input value and it's natural logarithm. This
// is to prevent numerical issues when the input is close to zero
static const float kMinAbsValue;
static const float kMinLogAbsValue;
// Fixed-point arithmetic limits
static const float kFixedPointLimit;
static const float kInverseFixedPointLimit;
// The default knee threshold in decibel. The knee threshold defines when the
// compressor is actually starting to compress the value of the input samples
static const float kDefaultKneeThresholdInDecibel;
// The compression ratio is the reciprocal of the slope of the line segment
// above the threshold (in the log-domain). The ratio controls the
// effectiveness of the compression.
static const float kCompressionRatio;
// The attack time of the envelope detector
static const float kTauAttack;
// The release time of the envelope detector
static const float kTauRelease;
float sampling_rate_;
// the internal state of the envelope detector
float state_;
// the latest gain factor that was applied to the input signal
float compressor_gain_;
// attack constant for exponential dumping
float alpha_attack_;
// release constant for exponential dumping
float alpha_release_;
float slope_;
// The knee threshold
float knee_threshold_;
float knee_threshold_in_decibel_;
// This interpolator provides the function that relates target gain to knee
// threshold.
sigmod::InterpolatorLinear<float> target_gain_to_knee_threshold_;
} // namespace le_fx
#include "dsp/core/dynamic_range_compression-inl.h"
#endif // LE_FX_ENGINE_DSP_CORE_DYNAMIC_RANGE_COMPRESSION_H_ | {"url":"https://android.googlesource.com/platform/frameworks/av/+/3371ce02b725abbd5304933862fc8e02821197c4/media/libeffects/loudness/dsp/core/dynamic_range_compression.h","timestamp":"2024-11-02T21:22:20Z","content_type":"text/html","content_length":"39003","record_id":"<urn:uuid:18a9fa38-6534-4b22-966e-c10a70577c7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00104.warc.gz"} |
This Little-Known Quantum Rule Makes Our Existence Possible - Science and Nonduality (SAND)
From macroscopic scales down to subatomic ones, the sizes of the fundamental particles play only a small role in determining the sizes of composite structures. Whether the building blocks are truly
fundamental and/or point-like particles is still not known, but we do understand the Universe from large, cosmic scales down to tiny, subatomic ones. MAGDALENA KOWALSKA / CERN / ISOLDE TEAM
Take a look around you at everything on Earth. If you were to investigate what any object is made out of, you could subdivide it into progressively smaller and smaller chunks. All living creatures
are made up of cells, which in turn are composed of a complex array of molecules, which themselves are stitched together out of atoms. Atoms themselves can be broken down further: into atomic nuclei
and electrons. These are the constituent components of all matter on Earth and, for that matter, all the normal matter we know of in the Universe.
It might make you wonder how this occurs. How do atoms, made of atomic nuclei and electrons, which come in less than 100 varieties, give rise to the enormous diversity of molecules, objects,
creatures and everything else we find? We owe the answer to one underappreciated quantum rule: the Pauli Exclusion Principle.
The atomic orbitals in their ground state (top left), along with the next-lowest energy states as you progress rightwards and then down. These fundamental configurations govern how atoms behave and
exert inter-atomic forces.
When most of us think of quantum mechanics, we think of the bizarre and counterintuitive features of our Universe on the smallest scales. We think about Heisenberg uncertainty, and the fact that it's
impossible to simultaneously know pairs of physical properties (like position and momentum, energy and time, or angular momentum in two perpendicular directions) beyond a limited mutual precision.
We think about the wave-particle nature of matter, and how even single particles (like electrons or photons) can behave as though they interfere with themselves. And we often think about
Schrödinger's cat, and how quantum systems can exist in a combination of multiple possible outcomes simultaneously, only to reduce to one specific outcome when we make a critical, decisive
Schrodinger's cat is a thought experiment designed to illustrate the bizarre and counterintuitive nature of quantum mechanics. A quantum system can be in a superposition of multiple states until a
critical measurement/observation is made, at which point there is only one measurable outcome.
Most of us barely give a second thought to the Pauli Exclusion Principle, which simply states that no two identical fermions can occupy the same exact quantum state in the same system.
Big deal, right?
Actually, it's not only a big deal; it's the biggest deal of all. When Niels Bohr first put out his model of the atom, it was simple but extremely effective. By viewing the electrons as planet-like
entities that orbited the nucleus, but only at explicit energy levels that were governed by straightforward mathematical rules, his model reproduced the coarse structure of matter. As electrons
transitioned between the energy levels, they emitted or absorbed photons, which in turn described the spectrum of each individual element.
When free electrons recombine with hydrogen nuclei, the electrons cascade down the energy levels, emitting photons as they go. In order for stable, neutral atoms to form in the early Universe, they
have to reach the ground state without producing a potentially ionizing, ultraviolet photon. The Bohr model of the atom provides the course (or rough, or gross) structure of the energy levels, but
this already was insufficient to describe what had been seen decades prior. BRIGHTERORANGE & ENOCH LAU/WIKIMEDIA COMMONS
If it weren't for the Pauli Exclusion Principle, the matter we have in our Universe would behave in an extraordinarily different fashion. The electrons, you see, are examples of fermions. Every
electron is fundamentally identical to every other electron in the Universe, with the same charge, mass, lepton number, lepton family number, and intrinsic angular momentum (or spin).
If there were no Pauli Exclusion Principle, there would be no limit to the number of electrons that could fill the ground (lowest-energy) state of an atom. Over time, and at cool enough temperatures,
that's the state that every single electron in the Universe would eventually sink to. The lowest energy orbital — the 1st orbital in each atom — would be the only orbital to contain electrons, and it
would contain the electrons inherent to every atom.
This artist's illustration shows an electron orbiting an atomic nucleus, where the electron is a fundamental particle but the nucleus can be broken up into still smaller, more fundamental
constituents. NICOLLE RAGER FULLER, NSF
Of course, this is not the way our Universe works, and that's an extremely good thing. The Pauli Exclusion Principle is exactly what prevents this from occurring by that simple rule: you cannot put
more than one identical fermion in the same quantum state.
Sure, the first electron can slide into the lowest-energy state: the 1s orbital. If you take a second electron and try to put it in there, however, it cannot have the same quantum numbers as the
previous electron. Electrons, in addition to the quantum properties inherent to themselves (like mass, charge, lepton number, etc.) also have quantum properties that are specific to the bound state
they're in. When they're bound to an atomic nucleus, that includes energy level, angular momentum, magnetic quantum number, and spin quantum number.
The electron energy states for the lowest possible energy configuration of a neutral oxygen atom. Because electrons are fermions, not bosons, they cannot all exist in the ground (1s) state, even at
arbitrarily low temperatures. This is the physics that prevents any two fermions from occupying the same quantum state, and holds most objects up against gravitational collapse. CK-12 FOUNDATION AND
The lowest-energy electron in an atom will occupy the lowest (n = 1) energy level, and will have no angular momentum (l = 0) and therefore a magnetic quantum number of 0 as well. The electron's spin,
though, offers a second possibility. Every electron has a spin of ½, and so will the electron in the lowest-energy (1s) state in an atom.
When you add a second electron, it can have the same spin but be oriented in the opposite direction, for an effective spin of -½. This way, you can fit two electrons into the 1s orbital. After that,
it's full, and you have to go to the next energy level (n = 2) to start adding a third electron. The 2s orbital (where l = 0, also) can hold an additional two electrons, and then you have to go to
the 2p orbital, where l = 1 and you can have three magnetic quantum numbers: -1, 0, or +1, and each of those can hold electrons with spin of +½ or -½.
Each s orbital (red), p orbital (yellow), d orbital (blue) and f orbital (green) can contain only two electrons apiece: one spin up and one spin down in each one. LIBRETEXTS LIBRARY / NSF / UC DAVIS
The Pauli Exclusion Principle — and the fact that we have the quantum numbers that we do in the Universe — is what gives each individual atom their own unique structure. As we add greater numbers of
electrons to our atoms, we have to go to higher energy levels, greater angular momenta, and increasingly more complex orbitals to find homes for all of them. The energy levels work as follows:
• The lowest (n = 1) energy level has an s-orbital only, as it has no angular momentum (l = 0) and can hold just two (spin +½ and -½) electrons.
• The second (n = 2) energy level has s-orbitals and p-orbitals, as it can have an angular momentum of 0 (l = 0) or 1 (l = 1), which means you can have the 2s orbital (where you have spin +½ and -½
electrons) holding two electrons and the 2p orbital (with magnetic numbers -1, 0, and +1, each of which holds spin +½ and -½ electrons) holding six electrons.
• The third (n = 3) energy level has s, p, and d-orbitals, where the d-orbital has an angular momentum of 2 (l = 2), and therefore can have five possibilities for magnetic numbers (-2, -1, 0, +1,
+2), and can therefore hold a total of ten electrons, in addition to the 3s (which holds two electrons) and 3p (which holds six electrons) orbitals.
The energy levels and electron wavefunctions that correspond to different states within a hydrogen atom, although the configurations are extremely similar for all atoms. The energy levels are
quantized in multiples of Planck's constant, but the sizes of the orbitals and atoms are determined by the ground-state energy and the electron's mass. Additional effects may be subtle, but shift the
energy levels in measurable, quantifiable fashions. POORLENO OF WIKIMEDIA COMMONS
Each individual atom on the periodic table, under this vital quantum rule, will have a different electron configuration than every other element. Because it's the properties of the electrons in the
outermost shells that determine the physical and chemical properties of the element it's a part of, each individual atom has its own unique sets of atomic, ionic, and molecular bonds that it's
capable of forming.
No two elements, no matter how similar, will be the same in terms of the structures they form. This is the root of why we have so many possibilities for how many different types of molecules and
complex structures that we can form with just a few simple raw ingredients. Each new electron that we add has to have different quantum numbers than all the electrons before it, which alters how that
atom will interact with everything else.
The way that atoms link up to form molecules, including organic molecules and biological processes, is only possible because of the Pauli exclusion rule that governs electrons.
The net result is that each individual atom offers a myriad of possibilities when combining with any other atom to form a chemical or biological compound. There is no limit to the possible
combinations that atoms can come together in; while certain configurations are certainly more energetically favorable than others, a variety of energy conditions exist in nature, paving the way to
form compounds that even the cleverest of humans would have difficulty imagining.
But the only reason that atoms behave this way, and that there are so many wondrous compounds that we can form by combining them, is that we cannot put an arbitrary number of electrons into the same
quantum state. Electrons are fermions, and Pauli's underappreciated quantum rule prevents any two identical fermions from having the same exact quantum numbers.
A white dwarf, a neutron star or even a strange quark star are all still made of fermions. The Pauli degeneracy pressure helps hold up all stellar remnants against gravitational collapse, preventing
a black hole from forming. CXC/M. WEISS
If we didn't have the Pauli Exclusion Principle to prevent multiple fermions from having the same quantum state, our Universe would be extremely different. Every atom would have almost identical
properties to hydrogen, making the possible structures we could form extremely simplistic. White dwarf stars and neutron stars, held up in our Universe by the degeneracy pressure provided by the
Pauli Exclusion Principle, would collapse into black holes. And, most horrifically, carbon-based organic compounds — the building blocks of all life as we know it — would be an impossibility for us.
The Pauli Exclusion Principle isn't the first thing we think of when we think of the quantum rules that govern reality, but it should be. Without quantum uncertainty or wave-particle duality, our
Universe would be different, but life could still exist. Without Pauli's vital rule, however, hydrogen-like bonds would be as complex as it could get.
Astrophysicist and author Ethan Siegel is the founder and primary writer of Starts With A Bang! His books, Treknology and Beyond The Galaxy, are available wherever books are sold.
This article was originally published in Forbes Magazine | {"url":"https://scienceandnonduality.com/article/this-little-known-quantum-rule-makes-our-existence-possible/","timestamp":"2024-11-09T00:40:54Z","content_type":"text/html","content_length":"245410","record_id":"<urn:uuid:89d6f982-40f5-42e0-881c-c7ef4de58453>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00722.warc.gz"} |
I think it's grossly unfair and saddening that people are judged differently based on their looks, talent, education, when many of these factors are out of their own control. Not just that they are
judged differently, but a person's fate can be largely dependent on factors outside their own control. For example, I can never become Einstein or Bach, but I am fortunate to live much more
comfortable life from someone born in an area plagued by war, even though I do not think I'm more entitled to such life than they are. However, I understand absolute equality can also be appalling as
depicted in Vonnegut's "Harrison Bergeron". Where do we draw the line? What do philosophers think?
Read another response about | {"url":"https://askphilosophers.org/comment/37210","timestamp":"2024-11-12T02:09:24Z","content_type":"text/html","content_length":"27373","record_id":"<urn:uuid:175493bf-1924-46bc-932f-a1fa3e824947>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00799.warc.gz"} |
Some of my talks and talks of my students are here. Some of these are crosslisted under “Research interests, and selected publications”. That page also contains some older talks.
• How do exponential size solutions arise in semidefinite programming? Foundations of Computational Mathematics, June 17, 2023
• How do exponential size solutions arise in semidefinite programming? Poster talk at the MIP 2021 conference, May 24, 2021 2nd prize to Alex Touzov
• On positive duality gaps in semidefinite programming
Johann Radon Institute, Workshop on Conic and Copositive Optimization, December 2020.
• Characterizing Bad Semidefinite Programs: Normal Forms and Short Proofs
Egon Balas Memorial Conference, Carnegie Mellon University, October 2019.
• Combinatorial characterizations in semidefinite programming: how elementary row operations help
CAM colloquium, University of Chicago, October 2019.
• Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs
Poster talk of Yuzixuan Zhu at the 2018 Princeton Optimization Day, sept 2018. First poster prize in the Algorithms category.
• Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs
Talk of Yuzixuan Zhu at ISMP 2018,July 2018
• Bad semidefinite programs, now with short proofs
Talk at Fields Institute, Workshop on Modern Convex Optimization and Applications July 2017
• Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming
Shinji Mizuno birthday conference, Aug 2016
• Combinatorial certificates in semidefinite programming
Danish Technical University, June 2016. This talk roughly combines the two talks below.
• Exact duality in semidefinite programming based on elementary reformulations
Tamas Terlaky’s birthday conference June 2015
• Bad semidefinite programs: they all look the same UC San Diego, May 2014 | {"url":"https://gaborpataki.web.unc.edu/talks/","timestamp":"2024-11-05T02:50:08Z","content_type":"text/html","content_length":"30752","record_id":"<urn:uuid:a4820923-bb68-4f31-aa6e-f34b59057ac9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00430.warc.gz"} |
Yukitaka ISHIMOTO | Professor | PhD | Saga University, Saga | Faculty of Sciences & Engineering | Research profile
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Learn more
Yukitaka Ishimoto has just moved to Faculty of Science and Engineering, Saga Univ. from Dept. of Mechanical Engineering, Akita Prefectural Univ. Yukitaka does research in Applied Mathematics,
Mathematical Physics, Fluid Science, Bioinformatics, Biophysics, and Bioengineering by using mathematical/physical/biological/engineering techniques. One of the lab's current projects is 'Tissue
mechanics by bubbly vertex dynamics model and other means'.
• Fluid dynamics, theoretical biophysics, and bioengineering
• Fluid dynamics, theoretical biophysics, and bioengineering | {"url":"https://www.researchgate.net/profile/Yukitaka-Ishimoto","timestamp":"2024-11-11T00:27:29Z","content_type":"text/html","content_length":"962753","record_id":"<urn:uuid:731973ff-348a-49ce-a3fd-f311e3919999>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00531.warc.gz"} |
Value of Options – Premium - Long Nifty Short
Value of Options – Premium
What is value of options? Why is it important?
Options are the derivatives of the Stocks or Index such as NIFTY or NIFTY BANK which are traded by the Stock Exchanges NSE and BSE. So, the value of the options is derived from the value of the
underlying asset. The value of an option is the current market price of that option at which is being traded in the stock market. Its value depends on various factors, but more importantly there are
two major factors which govern the value of an option, they are –
• Intrinsic Value
• Time Value
To understand the two terms in detail, we need to take an example in real scenario. Let us take the NIFTY 50 Index which is trading at 14,677.80 as on May 14, 2021. It is the benchmark index of NSE
having the market value of the top 50 companies having highest market capitalization. Market cap means the total number of shares of the company multiplied by the share price.
Strike Prices 14000 CE 14600 CE 15000 CE
Intrinsic Value 677.80 77.80 0
Time Value 74.2 199.20 81.50
Total Premium (May 27 2021 expiry) 752 277 81.50
The data in the green color can be found from the NSE Option Chain website (https://www.nseindia.com/option-chain) or from any brokers’ trading platform.
Intrinsic Value is the difference between the Current Value of the Underlying Stock/Index (NIFTY 50 index here) and the strike price of that option. It can never be in negative, either it is positive
or zero.
Time Value is the difference between the total premium value and the intrinsic value of the option.
From the table, we can summarize,
• The intrinsic value of the option decreases as we move above the current price of the underlying asset and increases as we move below the current price.
• The time value is maximum at the current market price and it decreases on the either side.
The Time value signifies that how much time is left from the date of expiry in terms of value. As the time will pass, the time value will go on decreasing and at the end of expiry, the time value
will decay down to 0 and only the intrinsic value will remain as the total premium value.
Classification of Options based on Intrinsic Value –
• ITM Option:- ITM option stands for In the Money Option. It means the strike price of the option is less than the Current Price of the underlying stock/index for a Call Option and it is more than
the current price for a Put Option or in other words, the intrinsic value is not equal to zero. For Ex – If Nifty is trading at 14600, So all the call options with Strike Prices lower than 14600
are ITM options and all the Put Options of strikes higher than 14600 are ITM options.
• ATM Option:- ATM option stands for At the Money Option. It means the strike price of the option is equal to very close to the current price of the underlying stock/index for both the PE and CE
options. In above example, 14600 CE and PE option will be ATM options.
• OTM Option:- OTM option stands for Out of the Money Option. It means the strike price of the option is more then the current price for a Call option and lower than the current price for a Put
Option. Again, for above example, all call options with strike prices more than 14600 are OTM and all put options with strike prices lesser than 14600 are OTM.
Nothing in the world comes for free. It has some premium!! | {"url":"https://longniftyshort.com/value-of-options-premium/","timestamp":"2024-11-02T21:16:55Z","content_type":"text/html","content_length":"59455","record_id":"<urn:uuid:36b328b7-b52f-4f35-b99c-abbd5714556c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00445.warc.gz"} |
CDLXXV in Hindu Arabic Numerals
CDLXXV = 475
M C X I
MM CC XX II
MMM CCC XXX III
CD XL IV
D L V
DC LX VI
DCC LXX VII
DCCC LXXX VIII
CM XC IX
CDLXXV is valid Roman numeral. Here we will explain how to read, write and convert the Roman numeral CDLXXV into the correct Arabic numeral format. Please have a look over the Roman numeral table
given below for better understanding of Roman numeral system. As you can see, each letter is associated with specific value.
Symbol Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
How to write Roman Numeral CDLXXV in Arabic Numeral?
The Arabic numeral representation of Roman numeral CDLXXV is 475.
How to convert Roman numeral CDLXXV to Arabic numeral?
If you are aware of Roman numeral system, then converting CDLXXV Roman numeral to Arabic numeral is very easy. Converting CDLXXV to Arabic numeral representation involves splitting up the numeral
into place values as shown below.
CD + L + X + X + V
500 - 100 + 50 + 10 + 10 + 5
400 + 50 + 10 + 10 + 5
As per the rule highest numeral should always precede the lowest numeral to get correct representation. We need to add all converted roman numerals values to get our correct Arabic numeral. The Roman
numeral CDLXXV should be used when you are representing an ordinal value. In any other case, you can use 475 instead of CDLXXV. For any numeral conversion, you can also use our roman to number
converter tool given above.
Current Date and Time in Roman Numerals
The current date and time written in roman numerals is given below. Romans used the word nulla to denote zero because the roman number system did not have a zero, so there is a possibility that you
might see nulla or nothing when the value is zero. | {"url":"https://romantonumber.com/cdlxxv-in-arabic-numerals","timestamp":"2024-11-12T04:06:19Z","content_type":"text/html","content_length":"89744","record_id":"<urn:uuid:05987d66-77a1-46b9-860c-5c6aed6c30e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00857.warc.gz"} |
How to write ratios in a word problem
Read,3 minutes
In mathematics, a ratio is defined as the comparison between two numbers. This is generally done to find out how big or small a number or a quantity is with respect to another. So, what method do we
use to find these ratios? Well, we use the division method. In a ratio, two numbers are divided. The dividend part is known as the ‘antecedent’, whereas the divisor is known as the ‘consequent.’
So, let’s look at an example. Suppose, in a market of \(40\) vendors, \(23\) of them sell apples, and the rest \(17\) sell bananas. So, the ratio of the vendors selling apples to those selling
bananas would be \(23 \ : \ 17\). Moreover, in mathematical terms, this is read as “\(23\) is to \(17\).”
Types of Ratios
Now, there are \(2\) types of ratios. They are called “part-to-part” and “part-to-whole.” So, when we say that in a group of \(17\) teachers, \(9\) teach math and \(8\) teach physics, then the
part-to-part ratio is \(9 \ : \ 8\). Basically, a part-to-part ratio denotes the clear distinction between two entities.
Now, if we say that out of \(10\) books in a library, \(7\) are of computer science. So, here part-to-whole ratio is \(7 \ : \ 10\). This means that out of \(10\) books in the library, every \(7\)
are of computer science. So, basically, a part-to-whole ratio denotes the ratio between the whole group to a specific entity.
Calculation of Ratios
To understand the exact process of calculating ratios, let’s take up the following problem.
Question: Suppose, \(24\) tigers and \(17\) lions make up for an entire zoo. So, what is the ratio of tigers and lions in that zoo?
• Firstly, identify the unique entities. In this case, \(24\) tigers and \(17\) lions are unique entities.
• Next, write these in a fraction form. So, we write it as \(\frac{24}{17}\)
• Now, we need to check if this fraction can be further simplified or not. Here, it can’t be simplified further.
• So, the ratio of tigers and lions in the zoo is \(24 \ : \ 17\).
Simplifying Ratios
Check the entities in the given problem for writing ratios. Then divide both sides of the ratio to simplify it further. Suppose there are \(14\) apples and \(7\) oranges. So, the ratio of apples to
oranges would be \(\frac{14}{7} \ = \ 14 \ : \ 7 \ = \ 2 \ : \ 1\).
Ratio and Rates Word Problems
Now suppose, a problem like this is given, where \(x \ : \ 4 \ = \ 4 \ : \ 16\) and we are required to find \(x\).
So, we can write, \(\frac{x}{4} \ = \ \frac{4}{16}\) and by cross multiplication and simplifying the fractions, we can get \(x=1\).
Ratio and Rates Word Problems Quiz | {"url":"https://testinar.com/article/proportions_and_ratios__ratio_and_rates_word_problems_course","timestamp":"2024-11-03T08:46:03Z","content_type":"text/html","content_length":"56407","record_id":"<urn:uuid:9d28a60e-ceaf-406e-aefb-4a04468a99e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00152.warc.gz"} |
Sudoku Puzzle! Help
Table of Contents
The selection of a Difficulty level indicates how hard it will be to solve the Sudoku puzzle. The more difficult the level, the harder it will be to figure out which numbers go into the empty blocks.
Note: More difficult levels do not necessarily mean less blocks will be pre-filled in the initial puzzle.
Puzzle ID
Often you may want to share a specific puzzle or print a puzzle solution some time after the original puzzle was created. This can be done by using a puzzle ID. A puzzle ID is a six (6) character
identifier that represents a specific Sudoku puzzle that was previously created. You can obtain a unique puzzle ID by viewing your original puzzle; it is located in the bottom-left corner of the
puzzle (e.g. MWYJKK). When you input a puzzle ID, select the Puzzle or Solution option, then click the Generate button to create the solution puzzle. Note: When a puzzle ID is used, the Difficulty
level is ignored as your puzzle ID already includes this information.
The Generate button allows you to create your Sudoku puzzle. Once you have selected your Difficulty level or Puzzle ID, you may select one of the following choices below and then click the Generate
• Puzzle - This choice will create a Sudoku puzzle. You can then print the puzzle to complete it immediately, save it for a later activity, or print it to share with someone else.
• Solution - This choice will create the solution to the Sudoku puzzle. The puzzle's original empty blocks will be highlighted and filled in. You can print the solution puzzle or view it online to
determine if your own solution is correct.
Each time you click the Generate button you will get the same Sudoku puzzle. If you would like to create other unique puzzles, then you can click the Reshuffle button. Once you click Reshuffle, you
click the Generate button once more to generate another unique puzzle. There is no limit to the number of times you can reshuffle to create additional unique puzzles.
How to play Sudoku
A standard Sudoku puzzle consists of a grid of 9 blocks. Each block contains 9 boxes arranged in 3 rows and 3 columns.
• When you start a game of Sudoku, some blocks will be pre-filled for you. You cannot change or move these numbers in the course of the game.
• Each column must contain all of the numbers 1 through 9 and no two numbers in the same column of a Sudoku puzzle can be the same.
• Each row must contain all of the numbers 1 through 9 and no two numbers in the same row of a Sudoku puzzle can be the same.
• Each block must contain all of the numbers 1 through 9 and no two numbers in a single block of a Sudoku puzzle can be the same.
• There is only one valid solution to each Sudoku puzzle. The only way the puzzle can be considered solved correctly is when all 81 boxes contain numbers and the other Sudoku rules have been
You can view a detailed explanation on how to play Sudoku on
When a puzzle is complete, the solved Sudoku puzzle appears as shown (white blocks are the ones filled in to solve the puzzle): | {"url":"https://www.mathfactcafe.com/Games/Sudoku/Help","timestamp":"2024-11-09T14:09:36Z","content_type":"text/html","content_length":"11291","record_id":"<urn:uuid:89ba15fa-b86e-4996-a193-d133f07ba71c>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00103.warc.gz"} |
Linear Regression basics - course note | Synesthesia
This notebook is based on the exercises in the Anaconda training Getting started with AI and ML.
I’m copying my notes here as a more in-depth test of the ability to publish directly from Jupyter notebooks, and also to put my notes somewhere I can access them later!
Linear Regression
The most commonly used supervised machine learning algorithm.
This module covered:
• Fit a line to data
• Measure loss with residuals and sum of squares
• Use `scikit-learn`` to fit a linear regression
• Evaluate a linear regression using R2 and train-test splits
• simple to understand and interpret
• doesn’t over-fit
When is Linear Regression suitable?
1. variables are continuous, not binary or categorical (use logistic regression for the latter)
2. input variables follow a Gaussian (bell curve) distribution
3. input variables are relevant to the output variables and not highly correlated with each other (collinearity)
Simple Linear Regression
ML often splits into two tasks - regression (predict quantity) and classification (predict a category)
E.g $y = mx+b$
Challenge is to define m and b for “best fit”
Multiple linear regression
With multiple independent variables
e.g. $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon$
$\epsilon $ is error due to noise
Multiple variables can get complex, important to use tools to help select only input variables correlated with the output variables.
• Pearson correlation and $R^2$
• adjusted $R^2$
• Akakike Information Criterion
• Ridge and lasso regression
Residuals are the difference between the data points and the equivalent regression. Linear Regression models aim to minimise the regressions by optimising a loss function such as Sum of Squares.
When ML model works well with training data but fails to predict correctly with new data. Linear regression tends to show low variance and high bias, so less likely to be overfitted. (define terms
variance and bias)
Train/Test Splits
Common technique to mitigate overfitting is the use of train/test splits. Training data is used to fit the model, then test data is used to test it with previously-unseen data, if necessary the model
can then be tweaked.
Evaluating the model with $R^2$
$R^2$ (the coefficient of determination) ratios the average y-value to the average of the residuals.
It measures how well the independent variables explain a dependent variable, with 0.0 meaning no connection and 1.0 meaning a perfect explanation.
Example using scikit-learn
The package scikit-learn contains many tools to support Machine LEarning techniquies such as Linear Regression.
This worked example demonstrates some of them.
First we import the packages we are going to use, making use of two key utilities from scikit-learn:
• train_test_split makes it easy to split a set of data into training and test subsets.
• LinearRegression fits a linear model with coefficients w = (w1, …, wp) to minimize the residual sum of squares between the observed targets in the dataset, and the targets predicted by the linear
import pandas as pd
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
We need to import our data into a pandas DataFrame. For convenience I am using one of the datasets provided by the course author
# Load the data
df = pd.read_csv('https://bit.ly/3pBKSuN', delimiter=",")
│ │x │ y │
│0 │1 │-13.115843 │
│1 │2 │25.806547 │
│2 │3 │-5.017285 │
│3 │4 │20.256415 │
│4 │5 │4.075003 │
│5 │6 │-3.530260 │
│6 │7 │24.045999 │
│7 │8 │22.112566 │
│8 │9 │5.968591 │
│9 │10│43.392339 │
│10│11│32.224643 │
│11│12│14.666142 │
│12│13│17.966141 │
│13│14│-2.754718 │
│14│15│25.156840 │
│15│16│20.182870 │
│16│17│22.281929 │
│17│18│16.757447 │
│18│19│54.219575 │
│19│20│60.564151 │
We need to split our data into inputs and the associated outputs
# Extract input variables (all rows, all columns but last column)
X = df.values[:, :-1]
# Extract output column (all rows, last column)
Y = df.values[:, -1]
We then need to create separate training and testing data to evaluate performance and reduce overfitting. Her ewe make use of the train_test_split utility.
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=1.0/3.0, random_state=10)
• we train the standard LinearRegression model provided by scikit-learn against our training data
• then we use the trained model to fit a regression to our test data
The utility allows us to easily score the model using $R^2$.
model = LinearRegression()
model.fit(X_train, Y_train)
result = model.score(X_test, Y_test)
print("R^2: %.3f" % result)
R^2: 0.182
Using matplotlib we can visualise the model output against the whole input data set
import matplotlib.pyplot as plt
plt.plot(X, Y, 'o') # scatterplot
plt.plot(X, model.coef_.flatten()*X+model.intercept_.flatten()) # line | {"url":"https://www.synesthesia.co.uk/note/2023/11/07/linear-regression/","timestamp":"2024-11-10T21:09:22Z","content_type":"text/html","content_length":"33583","record_id":"<urn:uuid:a96e6c14-4f47-4055-af00-dbccc60dece3>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00370.warc.gz"} |
What shape is a pentagonal prism?
What shape is a pentagonal prism?
A pentagonal prism has 15 edges, 7 faces, and 10 vertices. The base of a pentagonal prism is in the shape of a pentagon. The sides of a pentagonal prism are shaped like a rectangle. A pentagonal
prism is a type of heptahedron, which is a polyhedron with seven plane faces.
What is the angle of pentagonal prism?
Prisms & Antiprisms
Vertices: 10 (10[3])
Faces: 7 (5 squares + 2 regular pentagons)
Edges: 15
Symmetry: 5-fold Prismatic (D5h)
Pentagon-Square Angle: acos(0) 90 degrees
Is a pentagonal prism a polyhedron?
As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms
formed by square sides and two regular polygon caps.
How many sides does a 3d pentagon have?
A pentagon is a two-dimensional shape that has five straight sides. It is a polygon, so it has the same number of angles (five) as it has sides. Other examples of polygons include: squares, hexagons,
and octagons.
What do you call a prism with a pentagonal base?
All cross-sections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of
the prismatoids .
What is the shape of a prism in maths?
To recall, the prism shape in maths is a three-dimensional box i.e. a solid figure with a uniform cross-section and has two common bases. Edge: Intersection of two faces on a solid object. This is a
line. Vertex: Joining point of two edge sides.
What are the lateral sides of a Prism called?
The lateral side faces are called lateral edges. A prism is a right pentagonal prism when it has two congruent and parallel pentagonal faces and five rectangular faces that are perpendicular to the
triangular ones. The two important measures made on a pentagonal prism is to find its volume and surface area.
When are the rectangular faces of a pentagonal prism congruent?
All rectangular faces of a regular pentagonal prism are congruent. When the pentagonal faces of a regular pentagonal prism are the bases, the rectangular faces of the prism are said to be lateral. | {"url":"https://www.handlebar-online.com/writing-tips/what-shape-is-a-pentagonal-prism/","timestamp":"2024-11-10T01:53:25Z","content_type":"text/html","content_length":"43048","record_id":"<urn:uuid:7c6696a3-012f-4919-b13f-adcc88eb6fef>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00680.warc.gz"} |
Lesson 14
Comparing Mean and Median
Problem 1
Here is a dot plot that shows the ages of teachers at a school.
Which of these statements is true of the data set shown in the dot plot?
The mean is less than the median.
The mean is approximately equal to the median.
The mean is greater than the median.
The mean cannot be determined.
Problem 2
Priya asked each of five friends to attempt to throw a ball in a trash can until they succeeded. She recorded the number of unsuccessful attempts made by each friend as: 1, 8, 6, 2, 4. Priya made a
mistake: The 8 in the data set should have been 18.
How would changing the 8 to 18 affect the mean and median of the data set?
The mean would decrease and the median would not change.
The mean would increase and the median would not change.
The mean would decrease and the median would increase.
The mean would increase and the median would increase.
Problem 3
In his history class, Han's homework scores are:
The history teacher uses the mean to calculate the grade for homework. Write an argument for Han to explain why median would be a better measure to use for his homework grades.
Problem 4
The dot plots show how much time, in minutes, students in a class took to complete each of five different tasks. Select all the dot plots of tasks for which the mean time is approximately equal to
the median time.
Problem 5
Zookeepers recorded the ages, weights, genders, and heights of the 10 pandas at their zoo. Write two statistical questions that could be answered using these data sets.
Problem 6
Here is a set of coordinates. Draw and label an appropriate pair of axes and plot the points. \(A = (1, 0)\), \(B = (0, 0.5)\), \(C= (4, 3.5)\), \(D = (1.5, 0.5)\) | {"url":"https://im.kendallhunt.com/MS/teachers/1/8/14/practice.html","timestamp":"2024-11-05T22:53:00Z","content_type":"text/html","content_length":"77164","record_id":"<urn:uuid:2fa1f1f3-63af-4788-96c4-5e6476abb019>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00098.warc.gz"} |
3 easy hypothesis tests for the mean value | Your Data Teacher
3 easy hypothesis tests for the mean value
Data scientists and analysts often have to work with mean values and need to compare the mean value of a sample with a known expected value or the mean value of another sample. Statistics helps us
with a powerful set of hypothesis tests we can perform for such tasks.
The problem
Let’s say that we measure something like the height of Mount Everest. We know that it’s 8848 meters. After we measure it, we get 8840 meters with a standard error of 20 meters. Is our mean value
statistically different from the known height?
Let’s say that another research group has performed the same measure with different tools and has obtained 8850 meters with a standard error equal to 10 meters. Is their result statistically
different than ours?
These are two common problems when it comes to working with mean values. We often have to compare our mean value with a known expected value or with the mean value obtained by somebody else.
Comparing things with each other is a common problem in statistics and we can benefit from the theory of hypothesis tests in order to assess the statistical significance of such a comparison. In this
article, I’m going to talk about the Student’s t-tests for the mean value. They are the most common types of hypothesis tests that can be performed and are very useful.
They all rely on a basic assumption: that the samples have been generated from a normal distribution.
Here follow some examples using Python programming language. All these examples are based on NumPy, so we first have to import it and set the seed of the random number generator. Let’s import
matplotlib as well for histogram calculations.
import numpy as np
import matplotlib.pyplot as plt
Comparing the mean value with a known, expected value
The simplest form of a t-test is the one that compares the mean value of a sample with a known, expected value.
Let’s create a sample of 30 normally-distributed random numbers in Python using NumPy with a mean equal to 100 and standard deviation equal to 10.
x = np.random.normal(size=30,loc=100,scale=10)
The mean value of this sample is 104.43. We’d like to tell whether it’s statistically different from 100.
The null hypothesis for such a t-test on the mean value is: the mean value is equal to 100. So, we are working with a 2-tailed test.
In order to use it, we have to import the proper function from Scipy.
from scipy.stats import ttest_1samp
Then, we can use the “ttest_1samp” function, which gives us the p-value of such a 2-tailed test:
# Ttest_1sampResult(statistic=2.2044551627605693, pvalue=0.035580270712695275)
If we wanted to use a 1 tailed test, the null hypothesis would be one of these:
• The mean value is greater than 100
• The mean value is less than 100
By stating the alternative hypothesis, we can calculate the p-values of these two tests.
# Null hypothesis: x.mean() is greater than 100
# (alternative hypothesis, it's less than 100)
# Ttest_1sampResult(statistic=2.2044551627605693, pvalue=0.9822098646436523)
# Null hypothesis: x.mean() is less than 100
# (alternative hypothesis, it's greater than 100)
# Ttest_1sampResult(statistic=2.2044551627605693, pvalue=0.017790135356347637)
This is an example of a Student’s t-test on a single sample. Remember that the sample must have been created from a normal distribution.
Comparing the mean values of two samples with the same variance
Let’s now create a new sample starting from the same normal distribution of the previous one. We can change the size from 30 to 50.
y = np.random.normal(size=50,loc=100,scale=10)
Its mean value is 96.85.
Let’s say that we already know that the two samples come from two normal distributions with the same variance. We can use a 2-sample t-test to compare their mean values. We have to use the
“ttest_ind” function.
from scipy.stats import ttest_ind
We can now calculate the p-value of all the tests we need (two tails or one tail).
# Null hypothesis: x.mean() is equal to y.mean()
# Ttest_indResult(statistic=3.4565852447894163, pvalue=0.0008885072426696321)
# Null hypothesis: x.mean() is greater than y.mean()
# (alternative hypothesis: it's less than y.mean())
# Ttest_indResult(statistic=3.4565852447894163, pvalue=0.9995557463786652)
# Null hypothesis: x.mean() is less than y.mean()
# (alternative hypothesis: it's greater than y.mean())
# Ttest_indResult(statistic=3.4565852447894163, pvalue=0.00044425362133481604)
Comparing the mean values of two samples with different variances
Finally, the most common case: two samples that come from normal distributions with different variances.
Let’s create a new variable that comes from a normal distribution with a standard deviation equal to 5:
z = np.random.normal(size=50,loc=100,scale=5)
The function that performs the so-called Welch’s test (i.e. this particular case of Student’s t-test) is “ttest_ind” again, but we have to set the equal_var argument equal to False.
# Null hypothesis: x.mean() is equal to z.mean()
# Ttest_indResult(statistic=1.094819002420836, pvalue=0.2807390405295771)
# Null hypothesis: x.mean() is greater than z.mean()
# (alternative hypothesis: it's less than z.mean())
# Ttest_indResult(statistic=1.094819002420836, pvalue=0.8596304797352115)
# Null hypothesis: x.mean() is less than z.mean()
# (alternative hypothesis: it's greater than z.mean())
# Ttest_indResult(statistic=1.094819002420836, pvalue=0.14036952026478855)
In this article, I’ve explained 3 types of hypothesis tests based on the mean value: the one-sample t-test, which compares the mean value of a sample with a known expected value, the two-sample
t-test with equal variances and the two-sample t-test with different variances (also called Welch’s test). The last two tests compare the mean values of two samples. Although such tests are pretty
powerful, it’s always necessary to remember that they strongly require that the samples are generated from gaussian distributions. If this requirement is satisfied, these tests can be performed | {"url":"https://www.yourdatateacher.com/2022/10/25/3-easy-hypothesis-tests-for-the-mean-value/","timestamp":"2024-11-10T04:14:46Z","content_type":"text/html","content_length":"80138","record_id":"<urn:uuid:0f88f62e-c31f-4619-aa5d-60b41eb49f74>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00555.warc.gz"} |
Find the derivative of the following function and simplify the answer
Find the derivative of the following function and simplify the answers as much as possible: f(x)=\frac{2^{x^{2}}}{3^{x^{3}}} \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=\frac{x 2^{x^{2}}}{\left(3^{x^
{3}}\right)^{2}}\left(\ln 4-x^{2} \ln 27\right)\right) \frac{\mathrm{d} f}{\mathrm{~d} x}=(2 x)\left(\frac{2}{3}\right)^{x^{2}-1} \text { c)
} \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=\frac{x 2^{2}}{3^{x^{3}}}(\ln 4-x \ln 27)\right) \text { d) } \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=\frac{x 2^{x^{2}}}{\left(3^{3^{3}}\right)^{2}}(\ln
4-x \ln 27)\right) \text { e) } \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=x 2^{x^{2}} 3^{-x^{3}}(2 \ln 2-3 \ln 3)\right) \text { f) } \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=\frac{x 2^{x^{2}}}{3^{x^
{3}}}\left(\ln 4-x^{2} \ln 27\right)\right) \left.\frac{\mathrm{d} f}{\mathrm{~d} x}=\frac{2^{x^{2}}}{3^{3^{3}}}(2 \ln 2-3 x \ln 3)\right) \text { h) None of the answers shown here. }
Fig: 1
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Fig: 10 | {"url":"https://tutorbin.com/questions-and-answers/find-the-derivative-of-the-following-function-and-simplify-the-ans6672","timestamp":"2024-11-11T10:57:25Z","content_type":"text/html","content_length":"80204","record_id":"<urn:uuid:179a1b4e-f3f8-4d90-aa79-161164777a79>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00443.warc.gz"} |
How To Find Diameter Of A Tree? (Detailed Guide)
Measuring a tree without having felled it yet is at first sight a difficult task. However, there are various techniques that can be used to quickly and easily determine the height and circumference
of a tree.
Height of a tree
Let’s first look at the forestry square technique, an interesting alternative to measuring the tree by guesswork. It is a 45° square allowing the aiming. The positioning is done with a plumb bob. Be
careful, the plumb line must be parallel to the edge of the square!
Then, you just have to move backwards until the line of sight passes by the top of the tree. The height to be measured is simply the distance from the base of the tree, plus the height of your eye.
Circumference of a tree
The circumference of a tree is measured on the bark at 4 ft. above the ground using a forestry compass or a forestry tape. A reduction is then made to account for metric decay based on species
averages. Remember: the larger the tree, the greater the reduction.
How to measure a tree ?
There are 2 methods to easily measure a tree:
Measuring the height of a tree with the logger’s cross is the most used because the technique is simple and simply requires the use of 2 sticks of the same length. By aiming the cross at the foot and
the top of the tree, the distance between you and the tree gives you its height.
The second method is based on measuring the shadows of the tree and a person whose height you know.
These two methods are derived from the application of Thales’ theorem. There are simpler graduated sights to use for measuring small or curtained plants.
Note that the dimensions of a deciduous tree in nurseries are based on the circumference of the trunk at 1.30 m height and not on its height. The size of the trunk (diameter or circumference) gives a
more accurate idea of the age of the tree.
Here are 2 methods to measure a tree.
Method 1: Make a logger’s cross
Take 2 wooden sticks of the same length (l), 8 or 12 inches long, and hold them perpendicularly in front of your eye to form a cross.
Move away from the tree so that the base of the vertical stick coincides with the base of the tree and the end of the stick with the top of the tree.
The distance to the tree (L) is the height of the tree (H).
Count the number of steps (1 m long) from the tree or use a tape measure to make a more accurate measurement.
Method 2: Measure the tree’s shadows to determine its size
The shadows should be long enough to facilitate measurement. Avoid the middle of the day.
Stand with your back to the tree and align the shadow of the top of your head with the top of the tree. This point is called O.
Then measure:
The distance OB from the tree to point O, by counting the number of 1 m steps or using a decameter.
The height of your shadow OA.
Then do the following calculation: height of the tree = t × OB/OA, where t is your height in meters.
How to determine the age of a tree?
Accurate or crude, dating a tree can be done in several ways. How to determine the age of a tree?
The coring method
The scientific method consists of coring the tree with a tool called the Pressler auger. The core sample (piece of wood extracted from the trunk) is analyzed in the laboratory for dating thanks to
the measurement (width) of the rings represented graphically. This work is done by computer and is called dendrochronology.
Beyond the age, this method allows to measure the average annual growth and the growth speed which corresponds to the time necessary for the radius of the tree to increase by 1 inch.
It should be noted that a tree grows in height by an average of 10 inches per year with periods of rest, especially in winter, when the sap does not flow or only a little. This change is visualized
by a darker coloration of the wood visible on the rings present in the trunk. It is these rings that allow us to roughly identify the years when the tree is cut.
The calculation method
• A less invasive method allows dating by means of the tree’s circumference. However, the precise variety of the tree must be known, as it determines the multiplication factor by :
• 1.5 for silver maple, elm, poplar ;
• 2 for birch, white pine, red pine, Austrian pine, ash, red maple, oak and larch; by 2.5 for balsam fir, beech and ash;
• 3 for very slow growing trees such as red oak and walnut.
In practice
Measure the circumference of the tree at a height of about 1.40 m from the ground, then determine the diameter of the trunk by dividing the circumference by Pie (3.1416). Finally, multiply the trunk
diameter by the multiplication factor corresponding to your type of tree and you will have an approximate age of the tree. | {"url":"https://green-shack.com/how-to-find-diameter-of-a-tree/","timestamp":"2024-11-07T13:48:53Z","content_type":"text/html","content_length":"139243","record_id":"<urn:uuid:18344deb-2716-45dd-a4a1-7bdfb9aabdc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00106.warc.gz"} |
Z-Test Definition: Its Uses In Statistics Simply Explained With...
Home>Finance>Z-Test Definition: Its Uses In Statistics Simply Explained With Example
Z-Test Definition: Its Uses In Statistics Simply Explained With Example
Published: February 20, 2024
Learn the meaning of a Z-Test in finance, its applications in statistics, and understand how it works with a simple example. Discover the power of statistical analysis!
(Many of the links in this article redirect to a specific reviewed product. Your purchase of these products through affiliate links helps to generate commission for LiveWell, at no extra cost. Learn
The Z-Test: A Powerful Tool in Statistical Analysis
When it comes to understanding and analyzing data, statistics plays a crucial role. One of the most commonly used statistical tests is the Z-test. In this blog post, we will delve into the definition
of the Z-test, explore its uses in finance, and provide a simple example to help you better understand its application.
Key Takeaways:
• The Z-test is a statistical test used to determine if means or proportions from two different populations are significantly different.
• It is often employed in finance to compare investment returns, evaluate asset performance, or test hypotheses related to market trends.
Understanding the Z-Test
The Z-test is a hypothesis test that helps us determine if two means or proportions from different populations are significantly different from each other. It is based on the Z-statistic, which
measures how far a data point is from the mean, expressed in terms of standard deviations. By comparing the Z-statistic to a critical value, we can assess the statistical significance of our
In finance, the Z-test is a valuable tool for multiple purposes:
1. Comparing investment returns: Financial analysts and investors often use the Z-test to compare the performance of different investment portfolios. By applying the Z-test to the returns of two
investment strategies, they can determine if the difference in returns is statistically significant.
2. Evaluating asset performance: The Z-test can also be utilized to assess the performance of individual assets within a portfolio. For example, if an investor wants to compare the returns of a
specific stock to the average stock returns in the market, they can employ the Z-test to determine if the stock’s performance is significantly different.
3. Testing market trends: The Z-test is often used to test hypotheses related to market trends. For instance, researchers may use this test to determine if the difference in average salaries between
two industries is statistically significant, providing insights into market dynamics.
An Example: Using the Z-Test in Finance
Let’s consider an example to illustrate the application of the Z-test in finance. Suppose we have two investment portfolios: Portfolio A and Portfolio B. We want to determine if there is a
significant difference in returns between the two portfolios over a certain period.
Here are the details of the portfolios:
• Portfolio A has an average annual return of 10%, with a standard deviation of 2%.
• Portfolio B has an average annual return of 12%, with a standard deviation of 3%.
To perform the Z-test, we calculate the Z-statistic using the formula:
Z = (mean difference – hypothesized difference)/(standard deviation of the difference).
In our case, the hypothesized difference is 0 (no difference), and the standard deviation of the difference can be calculated using the following formula:
sigma_difference = sqrt((sigma_A^2/n_A) + (sigma_B^2/n_B)).
Where sigma_A and sigma_B represent the standard deviations of Portfolio A and Portfolio B, respectively, and n_A and n_B are the sample sizes.
By calculating the Z-statistic, we achieve a value of -2.33. Comparing this value to the critical value of -1.96 (assuming a confidence level of 95%), we determine that the returns of Portfolio B are
significantly higher than those of Portfolio A.
This example demonstrates how the Z-test can be utilized in finance to evaluate investment performance and make informed decisions based on statistical significance.
In Conclusion
The Z-test is a powerful statistical test that helps us compare means or proportions from different populations. In finance, it finds application in various contexts, including comparing investment
returns, evaluating asset performance, and testing market trends. By understanding the principles of the Z-test and its calculations, financial professionals and researchers can make more informed
decisions based on statistical evidence. | {"url":"https://livewell.com/finance/z-test-definition-its-uses-in-statistics-simply-explained-with-example/","timestamp":"2024-11-11T04:37:11Z","content_type":"text/html","content_length":"288920","record_id":"<urn:uuid:dfea62b4-e013-4260-aeac-82874163af06>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00271.warc.gz"} |
fission reaction energy release calculator
Fission Reaction Energy Release Calculator
Calculating the energy release from a fission reaction is crucial in various fields, including nuclear physics and energy production. With the right formula and tools, understanding this energy
release becomes more manageable. In this article, we’ll provide a guide on how to use a fission reaction energy release calculator, along with the necessary formula and examples for better
How to Use
To utilize the fission reaction energy release calculator effectively, follow these steps:
1. Input the mass defect (Δm) of the nuclei undergoing fission.
2. Enter the speed of light (c).
3. Provide the conversion factor for mass-energy equivalence (c²).
Once these values are entered, click the “Calculate” button to obtain the energy release.
The formula used to calculate the energy release from a fission reaction is based on Einstein’s famous equation, E=mc², where:
• E represents energy,
• m is the mass defect, and
• c is the speed of light.
The formula for calculating energy release (E) from a fission reaction is:
Example Solve
Q: What is a fission reaction?
A: Fission is a nuclear reaction in which the nucleus of an atom splits into two or more smaller nuclei.
Q: Why is it important to calculate energy release from fission reactions?
A: Understanding the energy release from fission reactions is crucial for various applications, including nuclear power generation and nuclear weapons development.
Q: How accurate is the provided formula for calculating energy release?
A: The formula is derived from Einstein’s theory of relativity and provides accurate results for calculating energy release from fission reactions.
Q: Can this calculator be used for other types of reactions?
A: No, this calculator is specifically designed for calculating energy release from fission reactions.
In conclusion, the fission reaction energy release calculator offers a convenient way to determine the energy released during nuclear fission. By understanding the formula and following the provided
steps, users can accurately calculate the energy release for various fission reactions. | {"url":"https://calculatordoc.com/fission-reaction-energy-release-calculator/","timestamp":"2024-11-13T21:29:25Z","content_type":"text/html","content_length":"93394","record_id":"<urn:uuid:ed44d39a-7e64-4338-9873-3409e92890b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00280.warc.gz"} |
Maturity amount calculation for recurring deposit in MS Excel
Maturity amount calculation for recurring deposit where interest is compounded quarterly, using the function FV:
Maturity amount = FV( (Rate of interest)/4, 4 * (Period in years), - (Value of each instlament) * (3 + (Rate of interest) / 2))
For example, an RD of Rs. 5000 each month for a period of 1.5 years at interest 7.5% p.a. compounded quarterly yields on maturity and amount of:
FV(7.5%/4, 4 * 1.5, -5000 * (3 + (7.5%/2))) = Rs. 95,504.78
The above formula is a general formula and individual banks might use a slightly different formula. For example, the Indian Banks' Association uses the following formula for computing the maturity
value where interest is compounded quarterly (Source: http://www.iba.org.in/formula.asp):
Maturity amount = ((Value of each instlament) * ((1+i)^n-1))/(1-(1+i)^(-1/3)),
where i = (Rate of interest) /4 and n = number of quarters.
Hence, taking the same example of RD of Rs. 5000 each month for a period of 1.5 years at interest 7.5% p.a. compounded quarterly, the maturity value, according to this formula is:
((5000) * ((1+(7.5%/4))^6-1))/(1-(1+(7.5%/4))^(-1/3)) = Rs. 95502.35
Extending the formula to other compounding periods:
For Monthly compounding: Maturity amount = ((Value of each instlament) * ((1+i)^n-1))/(1-(1+i)^(-1))
where i = (Rate of interest)/12 and n = number of months.
For half-yearly compounding: Maturity amount = ((Value of each instlament) * ((1+i)^n-1))/(1-(1+i)^(-1/6))
where i = (Rate of interest)/2 and n = number of years * 2.
Download the following app on your android phone to compute the maturity amount for fixed and recurring deposits: http://goo.gl/olkbN. You can also do these calculations online at https://
13 comments :
I think you have to change your formula
Round(FV(7.5%/4, 4 * 1.5, -5000 * (3 + 7.5%/2)),-2)
answer is = 95,500..
Nikhil Shah
E-mail Id : [email protected]
Hi smartmoneyindia,
THe only difference I see in the formula you suggested is the addition of the round function. Am I right?
Keshavaprasad B S
thanks man...
Will you pl explain why you used 3+ in the formula
[email protected]
What change is required in the formula if we need to calculate compound half yearly
The formula would be:
Maturity amount = FV( (Rate of interest)/2, 2 * (Period in years), - (Value of each instlament) * (6 + (Rate of interest) / 2))
to calculate compound half yearly.
We are calculating the maturity amount compounded quarterly in that formula. Each quarter has three months. The third argument in the formula needs the payment done per compounding period. Hence
using 3+ there.
What would be the formula for yearly calculation
I am getting some mistake
Please rectify
Is this correct??
Maturity amount = ((Value of each instalment) * ((1+i)^n-1))/(1-(1+i)^(-1/12))
yes, according to the IBA formula, it is:
((Value of each instlament) * ((1+i)^n-1))/(1-(1+i)^(-1/12)),
where n is the number of years,
and i is the rate of interest.
For example, for an amount of Rs. 10,000, for a period of 5 years at a rate of 10 percent interest, the maturity amount would be:
((10000) * ((1+10%)^5-1))/(1-(1+10%)^(-1/12)) = Rs. 7,71,717.40
Can You please tell me the formula to calculate Rate of interest compounded quarterly for Recurring deposit if all other parameters are known.
As mentioned in the blog, it is:
((Value of each installment) * ((1+i)^n-1))/(1-(1+i)^(-1/3))
where i is (rate of interest / 4), and n is number of quarters (number of years * 4) for which the deposit is made.
For example, for maturity amount for RD of Rs. 5000 per installment for a period of 1.5 years at interest 7.5% p.a. compounded quarterly,
i = 7.5%/4 = 1.875%
n = 1.5 * 4 = 6
Maturity amount = ((Value of each installment) * ((1+i)^n-1))/(1-(1+i)^(-1/3))
= ((Rs. 5000) * ((1+1.875%)^6-1))/(1-(1+1.875%)^(-1/3))
= Rs. 95502.3508376
This formula can be used to calculate maturity value but I want the formula(to be used in Excel) to calculate the rate of interest if maturity value is known in case of RD.
THANK A LOT
I AM SEARCHING IT, I GOT IT HERE.
THANKS AGAIN.
As this is useful to my Tally.ERP (Accounting Software) customization tdl programme.
-Hasmukh Patel | {"url":"http://blog.kbsbng.com/2011/01/maturity-amount-calculation-for.html","timestamp":"2024-11-10T14:36:40Z","content_type":"application/xhtml+xml","content_length":"165278","record_id":"<urn:uuid:c963a22f-1db4-4e82-91ea-5353fe99af5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00584.warc.gz"} |
Alana 1.0 review
A highly responsive Turing machine simulator written in Tcl/Tk
License: GPL (GNU General Public License)
File size: 25K
Developer: Markus Triska
A highly responsive Turing machine simulator written in Tcl/Tk. Contains many examples (unary and binary addition, subtraction, multiplication, a 5-state busy beaver, 2 string parsing examples,
divisibility test, primality test), a theoretical introduction to Turing machines, a proof of the undecidability of the halting problem and pointers to further literature.
It requires Tcl/Tk 8.3 or later.
Alana 1.0 search tags | {"url":"https://nixbit.com/software/alana-review/","timestamp":"2024-11-05T02:28:16Z","content_type":"text/html","content_length":"10659","record_id":"<urn:uuid:736ba5bf-fa89-42d7-9b6b-4fbd6a57019c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00427.warc.gz"} |
Average of Levels in Binary Tree
View Average of Levels in Binary Tree on LeetCode
Time Spent Coding
15 minutes
Time Complexity
O(n) - Every node in the tree must be visited, resulting in the O(n) time complexity.
Space Complexity
O(l) - The queue will have at most l elements, where l is the number of nodes in the largest level, resulting in the O(l) space complexity.
Runtime Beats
76.22% of other submissions
Memory Beats
73.90% of other sumbissions
The algorithm follows the breadth-first search algorithm and creates a queue of all nodes at the same level, iterates through exactly the number of elements at that level, and increments the cur_sum
variable while removing each node from the queue. At the same time, it is appending children nodes not equal to None to the back of the queue to be iterated through on the next loop (the next level).
Once the current level is exhausted, the algorithm divides the cur_sum by the number of elements in that level to get the average and then adds that value to the result list. Finally, when no nodes
are left in the queue, the algorithm returns the result list.
Algorithm & Data Structure Used
Breadth-first search - A tree/matrix traversal algorithm that searches for elements with a given property. Create a queue of elements yet to be visited and then iterate through each level (similar
properties) until there are no more elements of the same level.
Binary Tree - A rooted tree where every node has at most two children, the left and right children.
1 # Definition for a binary tree node.
2 # class TreeNode:
3 # def __init__(self, val=0, left=None, right=None):
4 # self.val = val
5 # self.left = left
6 # self.right = right
7 class Solution:
8 def averageOfLevels(self, root: Optional[TreeNode]) -> List[float]:
9 q = deque()
10 q.append(root)
11 res = []
13 while q:
14 cur_sum = 0
15 len_q = len(q)
16 for _ in range(len_q):
17 n = q.popleft()
18 cur_sum += n.val
19 if n.left: q.append(n.left)
20 if n.right: q.append(n.right)
22 res.append(cur_sum/len_q)
24 return res | {"url":"https://douglastitze.com/posts/average-of-levels-in-binary-tree/","timestamp":"2024-11-13T16:21:46Z","content_type":"text/html","content_length":"26865","record_id":"<urn:uuid:19307808-2f1a-42b2-be82-468018b8c9ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00066.warc.gz"} |
Case Studies
Your contributions to this department are invited! We're looking for short descriptions of interesting examples and/or fruitful applications of the algorithms or other ideas in A=B.
• Proof of an identity of Feigin-Stoyanovsky shows the proof of a complicated q-identity, handled by q-EKHAD in routine fashion, in response to a request from those who conjectured it and needed it
for other purposes. (An Acrobat file)
• How to do MONTHLY problems with your computer (by Nemes, Petkovsek, Wilf and Zeilberger) appeared in the American Math. Monthly. It has a large number of examples of the methods in action. (An
Acrobat file)
• A Mathematica notebook that exhibits all of the examples that are in the MONTHLY paper above. If Mathematica 3.0 is available to you, then with this notebook you will be able to see and to follow
in detail exactly how each of the MONTHLY problems was done by computer. If also you have downloaded the four basic packages (qzeil.m, zb.m, gosper.m, hyper.m) then you will be able not only to
follow the solutions of these problems, but also to run them for yourself and see the answers happen.
• An identity of J. S. Lomont and John Brillhart has a proof of an identity that looked difficult because it has no less than 7 binomial coefficients in it, but turned out to be an exercise in
Gosper's algorithm.
• The Filbert Matrix has entries that are reciprocals of Fibonacci numbers. Many interesting properties of the matrix have been found by Tom Richardson, some with the aid of the methods from "A=B".
• Here is a paper of Ivan Selesnick in which the methods of "A=B" are used to find a recurrence that has applications to lowpass filters.
• This is a proof of an identity of Gasqui and Goldschmidt, which they encountered during a study of Radon transforms.
Return to the A=B home page | {"url":"https://www2.math.upenn.edu/~wilf/CaseStudies.html","timestamp":"2024-11-11T19:50:37Z","content_type":"text/html","content_length":"4856","record_id":"<urn:uuid:dc78357f-3f02-4b52-881c-6eca97d4d30e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00203.warc.gz"} |
Elementary Linear Algebra
Elementary Linear Algebra
Publisher Description
Elementary Linear Algebra, Sixth Edition provides a solid introduction to both the computational and theoretical aspects of linear algebra, covering many important real-world applications, including
graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and
quadratic forms. In addition, many computational techniques in linear algebra are presented, including iterative methods for solving linear systems, LDU Decomposition, the Power Method for finding
eigenvalues, QR Decomposition, and Singular Value Decomposition and its usefulness in digital imaging.
- Prepares students with a thorough coverage of the fundamentals of introductory linear algebra
- Presents each chapter as a coherent, organized theme, with clear explanations for each new concept
- Builds a foundation for math majors in the reading and writing of elementary mathematical proofs | {"url":"https://books.apple.com/us/book/elementary-linear-algebra/id1601556809","timestamp":"2024-11-01T19:17:39Z","content_type":"text/html","content_length":"344948","record_id":"<urn:uuid:53f9ed4f-85de-4cbe-b0a0-7e3958bc7677>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00085.warc.gz"} |
How do Airplanes Fly?
Most recent answer: 10/01/2013
If gravity makes things fall to the ground, then why do airplanes don't fall to the ground?
- Laura (age 16)
laredo, Texas
Hi Laura,
I've heard a lot of arguments about exactly how airplanes fly... you can find a lot of material online, but I'm not sure you should trust any of it. (Aerodynamics is a very complicated subject, and a
lot of the problems can't be solved exactly. Most engineers and physicists study these problems by making complicated computer models.)
However, there are some things which we can say for sure. For starters, we know that the airplane doesn't fall out of the sky, so there must be a force upwards on the plane to balance the force of
gravity. What is pushing up on the airplane? It must be air: there is nothing else around.
Specifically, the airplane's wings push the air downward, and (by Newton's second law) the air pushes back upward (the so-called reaction force) on the wing. It is this reaction force from pushing
air downwards that keeps the airplane up.
But airplanes are often very heavy... how can air keep them up? I've always wanted to calculate this, so your question motivated me to finally do an estimation. The exact numbers I used are rough,
but should give you some idea of how it works.
Let's consider a fully-loaded, 500 ton Boeing 747 at take-off (I got these numbers from wikipedia). The surface area of a Boeing 747 is about 550 square meters. Since the air hits the wings at an
angle, the effective area is less; let's estimate this as 200 square meters. The take-off speed is about 300 meters per second, so the wings intersect 1.7 * 10^4 cubic meters (over 20 tons!) of air
in one second. To lift the 500 ton jet, the air must be forced downwards so that the rate of change in momentum is equal to the force of gravity. This fairly simple calculation shows that the 20 tons
of air are forced downwards by the wings at over 200 meters per second, almost 2/3 the speed of sound!
The moral of the story? Don't stand underneath a 747's wings during takeoff!
David Schmid
(published on 10/01/2013) | {"url":"https://van.physics.illinois.edu/ask/listing/24487","timestamp":"2024-11-13T07:58:52Z","content_type":"text/html","content_length":"29376","record_id":"<urn:uuid:07315f26-09b9-4b61-9c5d-17795395cf9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00522.warc.gz"} |
What is a Pip? - Definition & Meaning | What Is Their Value?
What is a Pip?
In the world of forex trading, one term that you will hear frequently is “pip.” If you are new to forex trading, you may be wondering what a pip is and how it is used. In this article, we will define
what a pip is and provide examples to help you understand its importance in forex trading.
What is a Pip?
A pip is a unit of measurement used to express the change in value between two currencies in a forex trade. The term “pip” stands for “percentage in point” or “price interest point.” It represents
the smallest price movement that a currency pair can make.
Most forex currency pairs are quoted to four decimal places. For example, the EUR/USD currency pair might be quoted as 1.2350. In this case, the fourth decimal place (the zero) represents one pip. If
the price of the EUR/USD currency pair moves from 1.2350 to 1.2360, that represents a movement of ten pips.
It is worth noting that some currency pairs are quoted to five decimal places, in which case the fifth decimal place represents a fractional pip, also known as a “pipette.” However, most traders
simply refer to these fractional pips as pips.
Why are Pips Important in Forex Trading?
Pips are important in forex trading because they determine the profits or losses of a trade. The value of a pip will vary depending on the currency pair being traded and the size of the trade.
Let’s look at an example. Suppose you decide to buy 10,000 units of the EUR/USD currency pair at a price of 1.2350, and you later sell those units at a price of 1.2360. This would represent a profit
of ten pips, which may not seem like much. However, the actual profit will depend on the size of the trade.
In forex trading, trades are typically measured in lots. A standard lot represents 100,000 units of the base currency, while a mini lot represents 10,000 units and a micro lot represents 1,000 units.
So, if you were trading one standard lot of the EUR/USD currency pair, your profit would be $100 (assuming a pip value of $10). However, if you were trading one mini lot, your profit would be $10
(assuming a pip value of $1).
Different Types of Pips
There are two types of pips: pipettes and fractional pips.
1. Pipettes: Pipettes are also known as fractional pips. They represent a movement of 0.1 pip. For example, if the EUR/USD currency pair moves from 1.2350 to 1.2351, that represents a movement of
one pipette.
2. Fractional Pips: Fractional pips represent a movement of 0.001 pip. They are sometimes referred to as “pip fractions” or “pip decimals.” Fractional pips are used when currency pairs are quoted to
five decimal places. For example, if the USD/JPY currency pair is quoted as 105.235, and it moves to 105.236, that represents a movement of one fractional pip.
How to Calculate the Value of a Pip
The value of a pip depends on the currency pair being traded, the size of the trade, and the exchange rate of the currency pair at the time of the trade.
To calculate the value of a pip, you can use the following formula:
Pip value = (1 pip / exchange rate) x trade size
Let’s use an example to illustrate this formula. Suppose you are trading one mini lot of the EUR/USD currency pair, and the current exchange rate is 1.2350. You can calculate the pip value as
Pip value = (1 pip / 1.2350) x 10,000
Pip value = 8.10 USD
This means that for every pip the EUR/USD currency pair moves, your profit or loss will be 8.10 USD if you are trading one mini lot.
It is important to note that pip values can vary depending on the currency pair being traded. For example, the pip value for the USD/JPY currency pair may be different than the pip value for the EUR/
USD currency pair, even if the trade size and exchange rate are the same.
Examples of Pips in Forex Trading
Let’s look at some examples to help you understand how pips work in forex trading.
Example 1: Suppose you are trading the USD/JPY currency pair, and you buy one standard lot at a price of 108.50. You hold the position for a few days, and the price of the currency pair rises to
109.50. This represents a movement of 100 pips, and your profit would be 1,000 USD (assuming a pip value of 10 USD).
Example 2: Suppose you are trading the GBP/USD currency pair, and you sell one mini lot at a price of 1.3800. You close the position a few hours later at a price of 1.3750. This represents a movement
of 50 pips, and your profit would be 50 USD (assuming a pip value of 1 USD).
Example 3: Suppose you are trading the EUR/USD currency pair, and you buy one micro lot at a price of 1.2000. The next day, the price of the currency pair falls to 1.1950. This represents a movement
of 50 pips, and your loss would be 5 USD (assuming a pip value of 0.1 USD).
In conclusion, a pip is a unit of measurement used to express the change in value between two currencies in a forex trade. Pips are important in forex trading because they determine the profits or
losses of a trade. The value of a pip will vary depending on the currency pair being traded, the size of the trade, and the exchange rate of the currency pair at the time of the trade. By
understanding pips and how to calculate their value, you can better manage your trades and make more informed trading decisions.
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AuctionDB Market Value
This page attempts to explain exactly how AuctionDB calculates the market value of items. AuctionDB uses an algorithm with a bunch of steps to make it as accurate as possible while using as little
memory/disk space to store the data as possible.
Let's use the following data set as an example (assume these are gold values and each number represents the buyout of a single item):
{5, 13, 13, 15, 15, 15, 16, 17, 17, 19, 20, 20, 20, 20, 20, 20, 21, 21, 29, 45, 45, 46, 47, 100}
Performing a simple average of this data set would give you a market value of 25.79 which is obviously too high. AuctionDB calculates the market value in multiple steps which attempt to correct for
outliers, give a moving value over time, and give a much more accurate market value in general than a simple average.
Step 1:
It is easy to see that the value of the item depends on how many is typically bought at a time. If you bought the 5 cheapest auctions in this example, you'd pay 12.2 gold per item. If you bought the
15 cheapest, you'd pay 16.3 gold per item. This is an inherent weakness with most market value estimates. AuctionDB attempts to factor this into its market value.
It is assumed that the number of an item that is required on average is proportional to the number currently on the auction house. This assumption turns out to be pretty accurate. For example, there
is more hypnotic dust on the auction house than greater celestial essences as hypnotic dust is used in much greater quantities than greater celestial essences. The same can be said for cloth,
leather, ore, volatiles, and other items where the quantities required by trade skills are larger.
AuctionDB uses this to detect more subtle outliers. After it is through 15% of the auctions, any increase of 20% or more in price from one auction to the next will trigger the algorithm to throw out
that auction and any above it. It will consider at most the lowest 30% of the auctions. In the example data above, there are no large increases in price between 15 and 30 percent of the way through,
but it would totally ignore everything after (but not including) the 16 because that's the last number in the bottom 30% of the data. It would not ignore the 13 even though it's more than 20% greater
than 5 because it is not yet 15% of the way through the data.
After step 1, the data set looks like this:
{5, 13, 13, 15, 15, 15, 16}
Step 2:
We now simply take the average of the data that survived step 1. In this case, the average is 13.143. Now, we find the standard deviation for our data. Our data has a standard deviation of 3.761.
In this step, AuctionDB throws out any data points that are more than 1.5 times the standard deviation away from the average. In this example, it throws out any data points that are not between 7.502
and 18.785 which means the 5 will get thrown out.
After step 2, the data set looks like this:
{13, 13, 15, 15, 15, 16}
Step 3:
Finally, we calculate our current market value by simply taking the average of the remaining data points. In this example, our final market value is 14.5. This method ensures that no poisoning of our
market value can take place by those who post high volume items at astronomical prices. It also gets rid of more subtle outliers to determine the average.
Step 4:
I bet you thought we were done...but no, we are just getting started! So far we have determined the current market value of an item based on one scan. But we want a moving market value that adjusts
overtime to get rid of market fluctuation. At this point, AuctionDB throws out all the individual auction data and for the purposes of market value, only cares about two pieces of data: the day on
which this market value was determined and the current market value itself. Again, this "current market value" isn't what shows up as the final market value because we need to take into account
previous scans' market values.
Essentially, we are going to take a weighted average of the previous 14 days' market values plus today's. First, let me explain how these "daily market values" are calculated. AuctionDB will store
the market values from every scan you've done today, but once it becomes tomorrow (and you log in), it will combine the market values from all your scans into a single market value using a simple
average. For the purposes of calculating the market value at this instant, we also average today's scans but each individual scan is still stored separately in the saved variables file until the next
day. So, now we're left with a list of daily market values. Here is a graph of the weighting we are going to be using:
The x-axis is the number of days old the data is and the y-axis is the weight (on an arbitrary scale). As you can see, the recent data is very heavily weighted. This results in the data having a
"half-life" of a little over 2 days. | {"url":"https://support.tradeskillmaster.com/en_US/tsm-addon-documentation/auctiondb-market-value","timestamp":"2024-11-04T16:49:35Z","content_type":"text/html","content_length":"73492","record_id":"<urn:uuid:5ab47785-92e9-405f-85dc-208a151bae9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00378.warc.gz"} |
Stats Fodder
I haven't taught Algebra 1 in a few years. But I was motivated to write something down in response to this thread, and in particular, Benjamin Dickman's tweet:
I sense there is a disconnect among meanings of 'teach factoring trinomials,' but I'm confident there's a way to broach factoring trinomials consonant with [e.g. youcube's description of] number
I view solving [some] quadratic equations like computing 18x5: flexible, etc!
— Benjamin Dickman (@benjamindickman) May 24, 2019
In particular, this.
"but I'm confident there's a way to broach factoring trinomials consonant with [e.g. youcube's description of] number sense."
I think I had a way to broach factoring trinomials that promotes number sense.
Consider this in the context of teaching kids how to multiply binomials and trinomials and such. I would emphasize the distributive property in conjunction with the area model for students who needed
a bit of structure or organization. I also made a conscious effort to make connections to multiplying integers.
I also would introduce students to dividing a trinomial by a binomial. This was a natural extension to what we were doing, and the question "Can we only multiply, or can we divide, too?" seemed to
always come up.
This probably took the better part of three to four weeks (so about 15 instruction days.)
Every day, in anticipation of introducing the idea of factoring after working on the idea of multiplying and dividing polynomials, I would provide my students with a short number puzzle or two as a
starter problem or bell ringer or whatever you would call it. The puzzles would be along the lines of something like this:
Find two numbers whose product is 18 and whose sum is 11.
This is the same question students ask themselves when they are asked to factor $x^2+11x+18$.
I would often provide two puzzles that would complement each other. Possible second puzzles to accompany the one one above might be:
Find two numbers whose product is 18 and whose sum is -9.
Find two numbers whose product is -18 and whose sum is 7.
Again, these are the questions kids would ask themselves when factoring $x^2-9x+18$ and $x^2+7x-18$, respectively.
When I would introduce students to the patterns arising from multiplying squaring binomials my number puzzle would be:,
Find two numbers whose product is 25 and whose sum is 10.
When I would introduce students to the patterns arising from multiplying conjugate binomials such as $x^2-5$ and $x^2+5$, my number puzzle would be:
Find two numbers whose product is -25 and whose sum is 0.
Needless to say, the kids became incredibly proficient in solving these number puzzles and creating their own for others to try. Now is the time to throw them a challenge:
Find four numbers $a, b, c,$ and $d$ where $ac=3$ and $bd=5$ and $ad+bc=16$.
This might be the question kids would ask themselves when factoring $3x^2+16x+5=(3x+1)(x+5)$
One day, the kids would come in, and their starter puzzle would be:
Two binomials were multiplied and the resulting product was $x^2+8x+12$. What are the two binomials that were multiplied together?
So after a month of number puzzles each day to start class, factoring became just another number puzzle. As the solution to the puzzle would percolate through the room, the realization was that this
is just another form of the number puzzles we had been doing. I would spend less than two weeks on factoring, because the kids had been factoring through number puzzles for a month by then. | {"url":"https://www.statsfodder.com/2019/05/","timestamp":"2024-11-05T03:24:14Z","content_type":"application/xhtml+xml","content_length":"39128","record_id":"<urn:uuid:6f23253b-d0ad-4dd0-b3e6-547fd7e614b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00471.warc.gz"} |
Food Webs and Niche Space. (Mpb-11), Volume 11
Joel E Cohen
Princeton University Press
What is the minimum dimension of a niche space necessary to represent the overlaps among observed niches? This book presents a new technique for obtaining a partial answer to this elementary question
read more…
niche space. The author bases his technique on a relation between the combinatorial structure of food webs and the mathematical theory of interval graphs. Professor Cohen collects more than thirty
food webs from the ecological literature and analyzes their statistical and combinatorial properties in detail. As a result, he is able to generalize: within habitats of a certain limited physical
and temporal heterogeneity, the overlaps among niches, along their trophic (feeding) dimensions, can be represented in a one-dimensional niche space far more often than would be expected by chance
alone and perhaps always. This compatibility has not previously been noticed. It indicates that real food webs fall in a small subset of the mathematically possible food webs. Professor Cohen
discusses other apparently new features of real food webs, including the constant ratio of the number of kinds of prey to the number of kinds of predators in food webs that describe a community. In
conclusion he discusses possible extensions and limitations of his results and suggests directions for future research.
BOOKSTORE TOTAL
{{condition}} {{price}} + {{shipping}} s/h
This book is currently reported out of stock for sale, but WorldCat can help you find it in your local library: | {"url":"https://bookchecker.com/0691082022","timestamp":"2024-11-11T18:26:49Z","content_type":"text/html","content_length":"115041","record_id":"<urn:uuid:826a2e69-efae-481d-aedd-beb6759814f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00584.warc.gz"} |
Exploring Graph Partitioning Strategies with Apache AGE: A Detailed Examination
Distributing data across multiple machines to maximize efficiency and minimize redundancy is a common concern when dealing with large datasets. One such approach is graph partitioning, which is
critical for distributed graph computation and scalable graph database solutions. Today, we'll explore graph partitioning strategies with Apache AGE.
"Partitioning is not merely dividing; it is about intelligently categorizing for optimal utilization and access."
Understanding Graph Partitioning
Before we dive into Apache AGE, let's first understand graph partitioning. It is a method of splitting a large graph into smaller sub-graphs, known as partitions. The goal is to divide the graph so
that the number of edges crossing between partitions is minimized, thereby reducing inter-machine communication and improving performance.
"Graph partitioning is like dividing a large city into neighborhoods. The aim is to minimize traffic between neighborhoods while maximizing it within them."
Introduction to Apache AGE
Apache AGE (A Graph Extension) is an extension of PostgreSQL that provides graph database functionality. AGE combines the robustness and familiarity of SQL with the flexibility and power of graph
databases, making it a great tool for managing complex, interrelated data.
"Apache AGE is the bridge between the relational database world and the graph database universe, offering the best of both realms."
Apache AGE
Graph Partitioning in Apache AGE
In a large graph database, efficient partitioning is critical for performance. Apache AGE does not inherently provide automated graph partitioning but given its flexible architecture and being an
extension of PostgreSQL, it allows users to implement their partitioning strategies at the PostgreSQL level.
Strategies for Graph Partitioning with Apache AGE
Now let's delve into some strategies for graph partitioning that can be employed with Apache AGE.
1. Hash-based Partitioning: In hash-based partitioning, a hash function is applied to some attribute of the nodes, and the resulting hash value is used to determine the partition. This can be easily
implemented in AGE by applying a hash function to a chosen attribute at the PostgreSQL level.
CREATE TABLE age_graph_data PARTITION BY HASH (node_id);
Here, 'node_id' is the attribute we've chosen to hash. We then create partitions for each possible hash value.
1. Range-based Partitioning: Range-based partitioning involves dividing data based on a specified range of the partition key. For example, you could partition a graph of people based on age ranges.
CREATE TABLE age_graph_data PARTITION BY RANGE (age);
1. List-based Partitioning: List-based partitioning involves partitioning based on a list of values of the partition key. For example, if you're dealing with a graph of geographic data, you might
partition based on a list of countries or regions.
CREATE TABLE geo_graph_data PARTITION BY LIST (region);
1. Composite Partitioning: Composite partitioning involves combining two or more partitioning strategies. For instance, you might first partition a graph based on a list of regions, then partition
each of those partitions based on a range of ages.
CREATE TABLE comp_graph_data PARTITION BY LIST (region) SUBPARTITION BY RANGE (age);
"Composite partitioning in Apache AGE combines different partitioning strategies, enhancing the flexibility and efficiency of data management."
Final Thoughts
While Apache AGE doesn't provide built-in automatic graph partitioning, its integration with PostgreSQL allows users to implement various partitioning strategies at the PostgreSQL level. This
flexibility provides AGE users with a wide range of possibilities for graph partitioning.
"Apache AGE's marriage with PostgreSQL allows for customizable and efficient graph partitioning strategies, making it a preferred choice for scalable graph databases."
With the right partitioning strategy, Apache AGE can provide a powerful and scalable solution for managing large and complex graph databases. Happy partitioning!
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Printable Multiplication Chart 1 20 2024 - Multiplication Chart Printable
Printable Multiplication Chart 1 20
Printable Multiplication Chart 1 20 – If you are looking for a fun way to teach your child the multiplication facts, you can get a blank Multiplication Chart. This will likely let your child to
complete the important points by themselves. You will discover empty multiplication charts for different product varieties, which includes 1-9, 10-12, and 15 goods. You can add a Game to it if you
want to make your chart more exciting. Here are some suggestions to get your little one started off: Printable Multiplication Chart 1 20.
Multiplication Charts
You can use multiplication maps as part of your child’s student binder to enable them to commit to memory math details. While many children can commit to memory their math concepts facts normally, it
requires lots of others time to do so. Multiplication graphs are an effective way to strengthen their learning and boost their confidence. In addition to being educational, these maps might be
laminated for more toughness. Listed here are some useful ways to use multiplication maps. You may also look at websites like these for useful multiplication fact assets.
This lesson includes the essentials in the multiplication desk. Along with studying the rules for multiplying, pupils will comprehend the idea of variables and patterning. Students will be able to
recall basic facts like five times four, by understanding how the factors work. They can also be able to utilize the house of one and zero to fix more advanced goods. Students should be able to
recognize patterns in multiplication chart 1, by the end of the lesson.
As well as the common multiplication graph or chart, individuals may need to develop a graph or chart with increased factors or less aspects. To make a multiplication graph or chart with more
variables, students should generate 12 desks, each and every with a dozen series and three posts. All 12 tables have to fit on one sheet of papers. Collections needs to be drawn by using a ruler.
Graph papers is perfect for this task. If graph paper is not an option, students can use spreadsheet programs to make their own tables.
Game concepts
Whether you are instructing a beginner multiplication session or working on the competence of the multiplication table, you are able to think of exciting and engaging video game suggestions for
Multiplication Graph 1. Several exciting suggestions are highlighted below. This video game demands the students to stay in work and pairs about the same problem. Then, they will likely all endure
their credit cards and discuss the best solution for any min. If they get it right, they win!
When you’re training kids about multiplication, among the finest instruments it is possible to allow them to have is really a printable multiplication graph. These printable sheets can come in a
range of models and might be imprinted in one page or many. Kids can find out their multiplication specifics by copying them from the chart and memorizing them. A multiplication graph can be helpful
for a lot of reasons, from helping them learn their mathematics information to educating them the way you use a calculator.
Gallery of Printable Multiplication Chart 1 20
Printable Multiplication Table Chart 1 20 PrintableMultiplication
Multiplication Chart 1 20 PrintableMultiplication
Printable Multiplication Table 1 To 20 Chart Worksheet In PDF The
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Introduction to Implementing Neural Network
Introduction to Implementing Neural Network
What do you know about Artificial Neural Network?
A Neural Network is a chain of algorithms that attempts to recognize underlying relationships in an array of data by mimicking the way the human brain operates. Neural Networks can either be
biological or artificial, that is they can be formed by biological neurons or artificial neurons. Artificial Neurons can be used to solve AI challenges.
Biological neural networks have influenced the development of artificial neural networks, but they are rarely exact replicas of their biological counterparts. Let us understand the different
components of artificial neural networks.
Neuron (Perceptron model)
To start with, one must understand the perceptron model. A perceptron is a single neuron that takes input and produces output. Here is a simple linear perceptron model.
The Linear Unit
The perceptron model does the following steps.
The input is x. Its connection to the neuron has a weight which is w. Whenever a value flows through a connection, you multiply the value by the connection’s weight. For the input x, what reaches the
neuron is w * x. A neural network “learns” by modifying its weights.
The b is a special kind of weight we call the bias. The bias doesn’t have any input data associated with it; instead, we put a 1 in the diagram so that the value that reaches the neuron is just b
(since 1 * b = b). The bias enables the neuron to modify the output independently of its inputs.
But, data is not always linear!
We can show non-linear relationships in data using Activation functions. In simple words, in order to fit curves, we will need to use activation functions(mathematical functions). There are many
activation functions available such as Relu, Selu, elu, tanh, etc. here is the list of activation functions to experiment with https://en.wikipedia.org/wiki/Activation_function
The below figure shows the modified version of the perceptron model. The model below includes activation functions that are used to show the non-linearity in data.
The steps that are carried out in the modified model are as follows
1. Inputs (x1, x2..etc) along with random weights are fed to the neurons
2. Each neuron performs a weighted summation of input, weights, and bias (constants)
3. The final output goes to the activation function, the activation function outputs the final output.
In order to build a neural network, the layers of neurons have to be stacked one after the other. Which will eventually result in a multi-layer perceptron model (neural network).
The neural network consists of an input layer, hidden layers(one or more), and an output layer.
Every neuron in one layer is connected to every other layer of neurons, forming a dense neural network. The connections from layer to another layer is called synapse in the context of biological
neural network. As mentioned above, every input layer, hidden layer, and output layer will have activation functions.
The commonly used activation functions in the input layer and hidden layer are Relu and tanh . However, in the output layer, the sigmoid activation function is used for binary classification and
softmax for multiclass classification problems. But, for the regression analysis, the activation function changes for the output layer such as the Linear activation function.
A neural network requires resources, training data, ability, and time to construct. To create a neural network, most data mining platforms provide at least the neural network algorithm. In this
article, we’re going to work through a classification problem with the Keras framework. In other words, taking a set of inputs and predicting what class those sets of inputs belong to.
The steps required to build a neural network are as follows:
ANN Implementation in Python
• 1. Data Preprocessing
• 1.1 Import the Libraries
• 1.2 Load the Dataset
• 1.3 Split Dataset into X and Y
• 1.4 Split the X and Y Dataset into the Training set and Test set
• 1.6 Perform Feature Scaling
• 2. Build Artificial Neural Network
• 2.1 Import the Keras libraries and packages
• 2.2 Initialize the Artificial Neural Network
• 3. Train the ANN
• 3.1 Fit the ANN to the Training set
• 4 Predict the Test Set Results
• 5. Make the Confusion Matrix
For implementation, we use the Pima Indians Diabetes Dataset. You can download the dataset from https://www.kaggle.com/uciml/pima-indians-diabetes-database. Artificial Neural networks can be used for
both classification and regression. And here we are going to use ANN for classification.
The dataset has the following features
The datasets consist of several medical predictor variables and one target variable, Outcome. Predictor variables include the number of pregnancies the patient has had, their BMI, insulin level, age,
and so on.
1.1 Import the Libraries
NumPy is a Python package that may be used to execute a variety of mathematical and scientific activities. NumPy is a Python library for working with arrays. It also provides functions for working
with matrices, Fourier transforms, and linear algebra. Matplotlib is a plotting library that is used for creating a figure, plotting an area in a figure, plotting some lines in a plotting area,
decorating the plot with labels, etc. Pandas is a tool used for data wrangling and analysis.
So in step 1, we imported all required libraries. Now the next step is-
1.2 Load the Dataset
So, when you load the dataset after running this line of code, the output is shown like this
As you can see in the dataset, there are 8 independent variables and 1 dependent variable.
1.3 Split Dataset into X and Y
We will also split the independent variables in X and a dependent variable in Y.
1.4 Split the X and Y Dataset into the Training set and Test set
We will use sklearn train_test_split class, to split the data into 80% percent train data and 20% test data
1.6 Perform Feature Scaling
Since the neural network performs well when the data is scaled. We will use a standard scaler object to standardize the independent variables
2. Build Artificial Neural Network
2.1 Import the Keras libraries and packages
Now, it is time to build a sequential model using Keras’s sequential class. Keras sequential is so simple, that you keep adding layers of neurons to build the neural network.
2.2 Initialize the Artificial Neural Network
We initialize the sequential object and start adding the input layer, hidden layer, and output layer. We use the add function to add all these layers one after the other. The input layer has 8
neurons as the first parameter, relu as the activation function as the second parameter and 8 input dimensions as the third parameter, this is because we have 8 independent variables in our dataset.
The hidden layer has 10 neurons and relu as an activation function. The last layer i.e output layer has one neuron because we need to predict whether a person is diabetic or non-diabetic. The
activation function used in the last layer is the sigmoid activation function.
Finally, we need to compile the model, the compile function includes an adam(Adaptive Moment Estimation) optimizer function to tune the weights of the model, the loss function is binary cross-entropy
(normally used for binary classification problem) and the metric we want to monitor is accuracy. The aim is to reduce the loss and improve the accuracy of the neural network.
Note: this article will not discuss the different optimizers used in neural networks. However, optimizers are mathematical functions that are used to tune the weights of the neural network in order
to improve the model’s predictive capabilities
3. Train the ANN
3.1 Fit the ANN to the Training set
Once the model is compiled, we need to train the neural network. The fit function in Keras does that job. The first parameter in the fit function is the 80% training part, the second parameter is the
batch size, which specifies the number of training samples to be passed to the model for the training purpose. The last parameter is an epoch, epoch is the number of times you iterate through the
entire training set in order to train the model. As you can see, as the number of epochs increases the accuracy of the model. However, care should be taken not to overfit the model by keeping a large
number of epochs.
4. Predict the Test Set Results
y_pred > 0.5 means if y-pred is in between 0 to 0.5, then this new y_pred will become 0(False). And if y_pred is larger than 0.5, then the new y_pred will become 1(True).
After running this code, you will get y_pred something like the above.
It is easy to explain how predictions are predicted correctly and how many are predicted incorrectly when we have a small dataset. But when we have a large dataset, it’s quite impossible. And that’s
why we use a confusion matrix, to clear our confusion.
5. Make the Confusion Matrix
In the above snapshot, the accuracy we obtained is 77%. That’s a pretty good one. However, you can always tune your model to improve the accuracy. Here is some common tuning we normally do to improve
the model performance
Improving a model
Steps to improve your neural network model
1. Creating a model– where you might want to add more layers, increase the number of hidden units (also called neurons) within each layer, and change the activation functions of each layer.
2. Compiling a model– you might want to choose a different optimization function (such as the Adam optimizer, which is usually pretty good for many problems) or perhaps change the learning rate of
the optimization function.
3. Fitting a model– perhaps you could fit a model for more epochs (leave it training for longer).
There are many different ways to potentially improve a neural network. Some of the most common include: increasing the number of layers (making the network deeper), increasing the number of hidden
units (making the network wider), and changing the learning rate. Because these values are all human-changeable, they’re referred to as hyperparameters) and the practice of trying to find the best
hyperparameters is referred to as hyperparameter tuning.
In this article, we tried explaining neural networks and their implementation using the Keras framework. Hope you have understood.
Happy Learning!
Complete code:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
dataset = pd.read_csv(‘/content/diabetes.csv’)
X = dataset.iloc[:,:-1]
y = dataset.iloc[:,-1]
from sklearn.model_selection import train_test_split
X_train,X_test,y_train,y_test = train_test_split(X,y,test_size = 0.2,random_state = 0)
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
import keras
from keras.models import Sequential
from keras.layers import Dense
classifier = Sequential()
classifier.add(Dense(8,activation = “relu”, input_dim = 8,
classifier.add(Dense(10,activation = “relu” ))
classifier.add(Dense(1,activation = “sigmoid”))
classifier.compile(optimizer = ‘adam’ , loss = ‘binary_crossentropy’,
metrics = [‘accuracy’] )
classifier.fit(X_train,y_train,batch_size = 10,epochs= 100)
y_pred = classifier.predict(X_test)
y_pred = (y_pred > 0.5)
from sklearn.metrics import confusion_matrix, accuracy_score
cm = confusion_matrix(y_test, y_pred)
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[USACO16OPEN] Diamond Collector S
Bessie the cow, always a fan of shiny objects, has taken up a hobby of mining diamonds in her spare time! She has collected $N$ diamonds ($N \leq 50,000$) of varying sizes, and she wants to arrange
some of them in a pair of display cases in the barn. Since Bessie wants the diamonds in each of the two cases to be relatively similar in size, she decides that she will not include two diamonds in
the same case if their sizes differ by more than $K$ (two diamonds can be displayed together in the same case if their sizes differ by exactly $K$). Given $K$, please help Bessie determine the
maximum number of diamonds she can display in both cases together. 奶牛Bessie很喜欢闪亮亮的东西（Baling~Baling~），所以她喜欢在她的空余时间开采钻石！她现在已经收集了N颗不同大小的钻石（N<=50,000），现
在她想在谷仓的两个陈列架上摆放一些钻石。 Bessie想让这些陈列架上的钻石保持相似的大小，所以她不会把两个大小相差K以上的钻石同时放在一个陈列架上（如果两颗钻石的大小差值不大于K，那么它们可以同时放在一个陈
The first line of the input file contains $N$ and $K$ ($0 \leq K \leq 1,000,000,000$). The next $N$ lines each contain an integer giving the size of one of the diamonds. All sizes will be positive
and will not exceed $1,000,000,000$.
Output a single positive integer, telling the maximum number of diamonds that Bessie can showcase in total in both the cases.
输入样例 #1
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• Activity Feed for heba2000
I am not really familiar with data generation and when I just copied your code it was accepted but the data was not showed to me it is clear that i am missing one command to get the data produced.
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I mean I wrote proc iml; x=j(60,5,.); x=int(5*ranuni(0))+1; but only one observation is generated and I need 60 observations with 5 variables
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but in this case only the first two variables will be zero and 1 but the other 3 all zeros because the condition for the second variable would be =0 if var1=1 and =1 if var1=0 so I canot use this
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I need to generate 5 binary variables but each observation must take 1 in one variable and zero in the other 4 variables.I need the code for this case Can anyone help in this case Thanks in advance
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I face the same problem in my model and I wonder if you found a solution for this problem or not Thanks for your help
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The constraints in NLPCG are put in matrix form with the first row representing the lower limit so I put the matrix as con={0. 0. 0. 0. 0. 0. 0. 0. 0. 0. . . , . . . . . . . . . . . . , 40. 51. 60.
24. 53. 80. 16. 34. 52. 84. 0. 42.894, 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. 1. }; put still the resulted variables have negative values as -7.05E-18 so how can I solve this problem. The other thing is
that when I generated xs by this way and added to them the outliers(generated as from same distribution with larger value) the the new variables donot have the same correlation determined in the
begining so how can I solve this problem? I also need to know how to make a condition so that: if correlation between y and x1 greater than or equal 0.5 x1 belongs to matrix H if correlation between
y and x1 less than 0.5 x1 belongs to matrix K Thanks
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Thanks a lot for your effort but I still have problem in this part If I need the outliers in the independent variables x's I would follow the same procedure? and how to determine the correlation
between the produced x's if I produced each x separetly? The other problem I have is that I am using NLPCG model and I determined the first row in the blc matrix as zeros as I need my decision
variable to be positive but still the produced variables have negative values how can I solve this problem? and I have another model that is linear in both objective function and constraint what is
the suitable Call ? Thanks in advance
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What I need is generating the independent variables in a regression relationship and I need the generated independent variables to contain outliers. By outliers I only mean values that are far away
from the set of data generated (either outliers up or down)and not according to certain criteria and not something related to CI . and it is not for a business context it is just for applying .Thanks
for your effort
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Hi everyone I need to know how can I determine the number of outliers in my simulation? Can anyone help?
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The Stacks project
112.5.12 Group actions on stacks
Actions of groups on algebraic stacks naturally appear. For instance, symmetric group $S_ n$ acts on $\overline{\mathcal{M}}_{g, n}$ and for an action of a group $G$ on a scheme $X$, the normalizer
of $G$ in $\text{Aut}(X)$ acts on $[X/G]$. Furthermore, torus actions on stacks often appear in Gromov-Witten theory.
• Romagny: Group actions on stacks and applications [romagny_actions]
This paper makes precise what it means for a group to act on an algebraic stack and proves existence of fixed points as well as existence of quotients for actions of group schemes on
algebraic stacks. See also Romagny's earlier note [romagny_notes].
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Points A and B are at (4 ,3 ) and (1 ,4 ), respectively. Point A is rotated counterclockwise about the origin by pi/2 and dilated about point C by a factor of 2 . If point A is now at point B, what are the coordinates of point C? | HIX Tutor
Points A and B are at #(4 ,3 )# and #(1 ,4 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #2 #. If point A is now at point
B, what are the coordinates of point C?
Answer 1
$C = \left(- 7 , 4\right)$
#"under a clockwise rotation about the origin of " pi/2#
#• " a point " (x,y)to(-y,x)#
#rArrA(4,3)toA'(-3,4)" where A' is the image of A"#
#"under a dilatation about C of factor 2"#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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I got a comment the other day asking about whether it is possible to get the expected number of goals scored from over/under odds, similar to how you can do this for odds for win, draw or lose
outcomes. The over/under odds refer to the odds for the total score (the sum of the score for two opponents) being over or under a certain value, usually 2.5 in soccer.
It is possible, and rather easy even, to get the expected total score from the over/under odds, at least if you assume that the number of goals scored by the two teams follows a Poisson distribution.
This is the same assumption that makes the method for extracting the expected goals from HDW odds possible. The Poisson distribution is really convenient and reasonable realistic probability model
for different scorelines. It is controlled by a single parameter, called lambda, that is also the expected value (and the expected goals in this case). One convenient property of the Poisson is that
the sum of two Poisson distributed variables with parameters lambda1 and lambda2 is also Poisson distributed, with the lambda being the sum of the two lambdas, i.e. lambdasum = lambda1 + lambda2.
So how can you find the expected total number of goals based on the over/under odds? First you need to convert the odds for the under outcome to a proper probability. How you do this depends on the
format your odds come in, but in R you can use the odds.converter package to convert them to decimal format, and then use my own package called implied to convert them to proper probabilities.
After you have the probabilities for the under probability, you can use the Poisson formula to find the value of the parameter lambda that gives the probability for the under outcome that matched the
probability from the odds. In R you can use the built-in ppois function to compute the probabilities for there being scored less than 2.5 goals when the expected total goals is 3.1 like this:
under <- 2.5
ppois(floor(under), lambda=3.1)
This will give us that the probability is 40.1% of two or less goals being scored in total, when the expected total is 3.1. Now you can try to manually adjust the lambda parameter until the output
matches your probability from the odds. Another way is to automate this search using the built-in uniroot function. The uniroot function takes as input another function, and searches for the input
value that gives the result 0. We therefore have to write a function that takes as input the expected goals, the probability implied by the odds, and the over/under limit, and returns the difference
between the probability from the Poisson model and the odds probability. Here is one such function:
obj <- function(expg, under_prob, under){
(ppois(floor(under), lambda=expg) - under_prob)
Next we feed this to the uniroot function, and gives a realistic search interval for the expected goals, between 0.01 and 10 in this case, and the rest of the parameters. For this example I used 62%
chance of there being scored less than 2.5 goals.
uniroot(f = obj,
interval = c(0.01, 10),
under_prob = 0.62,
under = 2.5)
From this I get that the expected total goals is 2.21.
You might wonder if it is possible to get the separate expected goals for the two teams from over/under odds using this method. This is unfortunately not possible. The only thing you can hope for is
to get a range of possible values for the two expected goals that sums to the total expected goals. In our example with the expected total goals being 2.21, the range of possible values for the two
expected goals can be plotted as a line like this:
If course, you can judge some pairs of expected goals being more likely than others, but there is no information about this in over/under odds alone. It might be possible, I am not 100% sure, that
other non-Poisson models, which would involve more assumptions, could exploit the over/under odds to get expected goals for both teams. | {"url":"https://opisthokonta.net/?m=202009","timestamp":"2024-11-05T22:09:00Z","content_type":"text/html","content_length":"38505","record_id":"<urn:uuid:f9e85bf9-1bc4-4c79-b38a-2e2cc17df355>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00635.warc.gz"} |
orksheets for 1st Class
Recommended Topics for you
Comparing Two-Digit Numbers
Adding Several Two-Digit Numbers
Comparing two digit numbers
Comparing two-digit numbers
Adding with two digit numbers and regrouping
Adding 10 to two digit numbers
Addition of Two-Digit Numbers
Two Digit Numbers with One Digit Numbers
Ascending Order (Two digit numbers)
Addition of Two Digit Numbers with Regrouping
Descending Order (two digit numbers)
Subtracting whole tens from two digit numbers
Addition with One and Two Digit Numbers
Adding and Subtracting one and two digit numbers
subtracting two digit numbers by one digit numbers
Ch 4 Math Test Review - Subtract Two-Digit Numbers
Explore Two-Digit Numbers Worksheets by Grades
Explore Two-Digit Numbers Worksheets for class 1 by Topic
Explore Other Subject Worksheets for class 1
Explore printable Two-Digit Numbers worksheets for 1st Class
Two-Digit Numbers worksheets for Class 1 are an essential resource for teachers looking to enhance their students' understanding of math and number sense. These worksheets provide a variety of
engaging activities that help young learners develop a strong foundation in mathematics. With a focus on number recognition, counting, and basic operations, these worksheets are designed to challenge
and inspire Class 1 students. Teachers can utilize these resources to create lesson plans, homework assignments, and in-class activities that cater to different learning styles and abilities. By
incorporating Two-Digit Numbers worksheets for Class 1 into their curriculum, teachers can ensure that their students are well-prepared for more advanced mathematical concepts in the future.
Quizizz, a popular educational platform, offers a wide range of resources for teachers, including Two-Digit Numbers worksheets for Class 1. This platform allows educators to create interactive
quizzes, assignments, and games that can be easily integrated into their lesson plans. With Quizizz, teachers can monitor student progress, provide instant feedback, and identify areas where students
may need additional support. In addition to worksheets, Quizizz offers a vast library of pre-made quizzes and activities that cover various topics in math, number sense, and other subjects. By
incorporating Quizizz into their teaching strategies, educators can provide a fun and engaging learning experience for their Class 1 students while ensuring that they develop a strong foundation in
two-digit numbers and other essential math skills. | {"url":"https://quizizz.com/en/two-digit-numbers-worksheets-class-1","timestamp":"2024-11-04T07:14:22Z","content_type":"text/html","content_length":"150549","record_id":"<urn:uuid:5b33268f-055d-4ba3-91cb-41370e0b6730>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00062.warc.gz"} |
quantity library
A superset of all the other libraries available in the quantity package. Import this library to have access to all the quantity types, units and constants defined in this package. Alternatively,
import only the quantity_si library to stick to the types and units consistent with the International System of Units (SI).
The mean energy imparted to matter per unit mass by ionizing radiation. See the Wikipedia entry for Absorbed Dose for more information.
The rate of mean energy imparted to matter per unit mass by ionizing radiation. See the Wikipedia entry for Absorbed Dose for more information.
Units acceptable for use in describing AbsorbedDoseRate quantities.
Units acceptable for use in describing AbsorbedDose quantities.
The rate of change of speed of an object. See the Wikipedia entry for Acceleration for more information.
Units acceptable for use in describing Acceleration quantities.
A measure of the effective concentration of a species in a mixture. See the Wikipedia entry for Thermodynamic activity for more information.
Units acceptable for use in describing Activity quantities.
The size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. See the Wikipedia entry for Amount of substance for more information.
Units acceptable for use in describing AmountOfSubstance quantities.
A planar (2-dimensional) angle, which has dimensions of 1 and is a measure of the ratio of the length of a circular arc to its radius.
An immutable angle range with a start angle, an end angle and an implicit direction.
Units acceptable for use in describing Angle quantities.
The rate of change of angular speed. See the Wikipedia entry for Angular acceleration for more information.
Units acceptable for use in describing AngularAcceleration quantities.
A measure of the quantity of rotation of a system of matter, taking into account its mass, rotations, motions and shape. See the Wikipedia entry for Angular momentum for more information.
Units acceptable for use in describing AngularMomentum quantities.
The rate of change of an angle. See the Wikipedia entry for Angular_velocity for more information.
Units acceptable for use in describing AngularSpeed quantities.
The extent of a two-dimensional figure or shape. See the Wikipedia entry for Area for more information.
Units acceptable for use in describing Area quantities.
Represents an integer as a binary number.
The period of 365 days (or 366 days in leap years) starting from the first of January; used for reckoning time in ordinary affairs.
The ability of a body to store an electrical charge, See the Wikipedia entry for Capacitance for more information.
Units acceptable for use in describing Capacitance quantities.
The increase in rate of a chemical reaction caused by the presence of a catalyst. See the Wikipedia entry for Catalysis for more information.
Units acceptable for use in describing CatalyticActivity quantities.
The property of matter that causes it to experience a force when placed in an electromagnetic field See the Wikipedia entry for Electric charge for more information.
Electric charge per unit volume of space. See the Wikipedia entry for Charge density for more information.
Units acceptable for use in describing ChargeDensity quantities.
Units acceptable for use in describing Charge quantities.
Complex numbers have both a real and an imaginary part.
The abundance of a constituent divided by the total volume of a mixture. See the Wikipedia entry for Concentration for more information.
Units acceptable for use in describing Concentration quantities.
The ease with which an electric current passes through a conductor (the inverse of Resistance). See the Wikipedia entry for Electrical resistance and conductance for more information.
Units acceptable for use in describing Conductance quantities.
Money in any form when in actual use or circulation as a medium of exchange. See the Wikipedia entry for Currency for more information.
Units acceptable for use in describing Currency quantities.
The flow of electric charge. See the Wikipedia entry for Electric current for more information.
The electric current per unit area of cross section. See the Wikipedia entry for Current density for more information.
Units acceptable for use in describing CurrentDensity quantities.
Units acceptable for use in describing Current quantities.
Represents a digit in four bits of a single byte. This wastes four bits but that's a decent trade-off for simplicity and better anyway than the 4+ bytes allocated for a regular int.
The Dimensions class represents the dimensions of a physical quantity.
Represents the stochastic health effects (probability of cancer induction and genetic damage) of ionizing radiation on the human body. See the Wikipedia entry for Equivalent dose for more
Units acceptable for use in describing DoseEquivalent quantities.
Wraps Dart's core double type, so that it can share a common base type with other Numbers.
A measure of a fluid's resistance to gradual deformation by shear stress or tensile stress. See the Wikipedia entry for Viscosity for more information.
Units acceptable for use in describing DynamicViscosity quantities.
The magnitude of the force per unit charge that an electric field exerts. See the Wikipedia entry for Electric field for more information.
Units acceptable for use in describing ElectricFieldStrength quantities.
A measure of the intensity of an electric field generated by a free electric charge, corresponding to the number of electric field lines passing through a given area. See the Wikipedia entry for
Electric_flux for more information.
Units acceptable for use in describing ElectricFluxDensity quantities.
The difference in electric potential energy between two points per unit electric charge See the Wikipedia entry for Voltage for more information.
Units acceptable for use in describing ElectricPotentialDifference quantities.
The ability of a system to perform work; cannot be created or destroyed but can take many forms. See the Wikipedia entry for Energy for more information.
The amount of energy stored in a given system or region of space per unit volume. See the Wikipedia entry for Energy density for more information.
Units acceptable for use in describing EnergyDensity quantities.
The rate of transfer of energy through a surface. See the Wikipedia entry for Energy density for more information.
Units acceptable for use in describing EnergyFlux quantities.
Units acceptable for use in describing Energy quantities.
A version of scientific notation in which the exponent of ten must be divisible by three (e.g., 123.345 x 10^3).
The number of specific ways in which a thermodynamic system may be arranged, commonly understood as a measure of disorder. See the Wikipedia entry for Entropy for more information.
Units acceptable for use in describing Entropy quantities.
The radiant energy received by a surface per unit area. See the Wikipedia entry for Radiant exposure for more information.
Units acceptable for use in describing Exposure quantities.
Represents a level of a field quantity, a logarithmic quantity. Level of a field quantity is defined as ln(F/F0), where F/F0 is the ratio of two field quantities and F0 is a reference amplitude
of the appropriate type.
Constructs a FiscalYear time period.
Any interaction that, when unopposed, changes the motion of an object. See the Wikipedia entry for Force for more information.
Units acceptable for use in describing Force quantities.
A convenient way to represent fractional numbers.
The number of occurrences of a repeating event per unit time. See the Wikipedia entry for Frequency for more information.
Units acceptable for use in describing Frequency quantities.
Heat rate per unit area. See the Wikipedia entry for Heat flux for more information.
Units acceptable for use in describing HeatFluxDensity quantities.
Represents an integer as a hexadecimal number.
The total luminous flux incident on a surface, per unit area. See the Wikipedia entry for Illuminance for more information.
Units acceptable for use in describing Illuminance quantities.
Represents an imaginary number, defined as a number whose square is negative one.
The property of an electrical conductor by which a change in current flowing through it induces an electromotive force in both the conductor itself and in any nearby conductors by mutual
inductance. See the Wikipedia entry for Inductance for more information.
Units acceptable for use in describing Inductance quantities.
Amount of data. See the Wikipedia entry for Information for more information.
The flow of information, per unit time. See the Wikipedia entry for Information for more information.
Units acceptable for use in describing InformationRate quantities.
Units acceptable for use in describing Information quantities.
Wraps Dart's core int type, so that it can share a common base type with other Numbers.
The resistance to flow of a fluid, equal to its absolute viscosity divided by its density. See the Wikipedia entry for Viscosity for more information.
Units acceptable for use in describing KinematicViscosity quantities.
Represents the length physical quantity (one of the seven base SI quantities). See the Wikipedia entry for Length for more information.
Units acceptable for use in describing Length quantities.
Represents logarithmic physical quantities and has dimensions of 1 (Scalar). Level of a field quantity and level of a power quantity are two common logarithmic quantities.
Units acceptable for use in describing Level quantities.
The intensity of light emitted from a surface per unit area. See the Wikipedia entry for Luminance for more information.
Units acceptable for use in describing Luminance quantities.
The perceived power of light. It differs from radiant flux, the measure of the total power of electromagnetic radiation (including infrared, ultraviolet, and visible light), in that luminous flux
is adjusted to reflect the varying sensitivity of the human eye to different wavelengths of light See the Wikipedia entry for Luminance for more information.
Units acceptable for use in describing LuminousFlux quantities.
Represents the luminous intensity physical quantity (one of the seven base SI quantities), the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle.
See the Wikipedia entry for Luminous intensity for more information.
Units acceptable for use in describing LuminousIntensity quantities.
The intensity of a magnetic field. See the Wikipedia entry for Magnetic field for more information.
Units acceptable for use in describing MagneticFieldStrength quantities.
The magnetic flux density passing through a closed surface. See the Wikipedia entry for Magnetic flux for more information.
The amount of magnetic flux in an area taken perpendicular to a magnetic flux's direction See the Wikipedia entry for Magnetic flux for more information.
Units acceptable for use in describing MagneticFluxDensity quantities.
Units acceptable for use in describing MagneticFlux quantities.
Represents the mass physical quantity (one of the seven base SI quantities), that determines the strength of a body's mutual gravitational attraction to other bodies. See the Wikipedia entry for
Mass for more information.
Mass per unit volume. See the Wikipedia entry for Density for more information.
Units acceptable for use in describing MassDensity quantities.
The mass of a substance which passes per unit of time. See the Wikipedia entry for Mass flow rate for more information.
Units acceptable for use in describing MassFlowRate quantities.
The mass of a substance which passes per unit of time. See the Wikipedia entry for Mass flow rate for more information.
Units acceptable for use in describing MassFluxDensity quantities.
Units acceptable for use in describing Mass quantities.
A MiscQuantity is a general (miscellaneous) Quantity having arbitrary dimensions (including possibly the same dimensions as a named Quantity subclass). MiscQuantity objects may be used, for
example, in less common domains or as intermediate results in equations.
Energy per mole of a substance. See the Wikipedia entry for Specific energy for more information.
Units acceptable for use in describing MolarEnergy quantities.
Entropy content per mole of substance. See the Wikipedia entry for Standard molar entropy for more information.
Units acceptable for use in describing MolarEntropy quantities.
MutableQuantity supports updates to its value, dimensions and uncertainty. Changes are broadcast over the onChange stream.
The abstract base class for all Number types.
NumberFormatSI implements the International System of Units (SI) style conventions for displaying values of quantities. Specifically:
Represents an integer as an octal number.
The ability of a material to support the formation of a magnetic field within itself. See the Wikipedia entry for Permeability (electromagnetism) for more information.
Units acceptable for use in describing Permeability quantities.
The resistance that is encountered when forming an electric field in a medium. See the Wikipedia entry for Permittivity for more information.
Units acceptable for use in describing Permittivity quantities.
Amount of energy per unit time. See the Wikipedia entry for Power (physics) for more information.
Represents a level of a power quantity, a logarithmic quantity. Level of a power quantity is defined as 0.5*ln(P/P0), where P/P0 is the ratio of two powers and P0 is a reference power.
Units acceptable for use in describing Power quantities.
Precise represents an arbitrary precision number.
Force applied perpendicular to the surface of an object per unit area over which that force is distributed. See the Wikipedia entry for Pressure for more information.
Units acceptable for use in describing Pressure quantities.
The abstract base class for all quantities. The Quantity class represents the value of a physical quantity and its associated dimensions. It provides methods for constructing and getting the
quantity's value in arbitrary units, methods for mathematical manipulation and comparison and optional features such as arbitrary precision and uncertainty.
QuantityRange<Q extends Quantity>
Represents a range of quantity values.
The radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. See the Wikipedia entry for Radiance for more information.
Units acceptable for use in describing Radiance quantities.
Radiant flux is another way to express power.
The radiant flux emitted, reflected, transmitted or received, per unit solid angle. See the Wikipedia entry for Radiant intensity for more information.
Units acceptable for use in describing RadiantIntensity quantities.
Provides a common handle for all Real Numbers.
A measure of the difficulty passing an electric current through a conductor. See the Wikipedia entry for Electrical resistance and conductance for more information.
Units acceptable for use in describing Resistance quantities.
A one-dimensional physical quantity that can be described by a single real number. See the Wikipedia entry for Scalar (physics) for more information.
Units acceptable for use in describing Scalar quantities.
Formats a number as a single integer digit, followed by decimal digits and raised to a power of 10 (e.g., 1.2345 x 10^3).
A two-dimensional angle in three-dimensional space that an object subtends at a point. See the Wikipedia entry for Solid angle for more information.
Units acceptable for use in describing SolidAngle quantities.
Energy per unit mass. See the Wikipedia entry for Specific energy for more information.
Units acceptable for use in describing SpecificEnergy quantities.
The heat capacity per unit mass of a material. See the Wikipedia entry for Heat capacity for more information.
Units acceptable for use in describing SpecificHeatCapacity quantities.
The ratio of the substance's volume to its mass. See the Wikipedia entry for Specific volume for more information.
Units acceptable for use in describing SpecificVolume quantities.
Irradiance of a surface per unit frequency. See the Wikipedia entry for Radiometry for more information.
Units acceptable for use in describing SpectralIrradiance quantities.
The rate of change of position. See the Wikipedia entry for Speed for more information.
Units acceptable for use in describing Speed quantities.
The elastic tendency of liquids which makes them acquire the least surface area possible. See the Wikipedia entry for Surface tension for more information.
Units acceptable for use in describing SurfaceTension quantities.
An objective comparative measure of hot or cold. See the Wikipedia entry for Thermodynamic temperature for more information.
The difference between two temperatures, where temperature is an objective comparative measure of hot or cold. See the Wikipedia entry for Thermodynamic temperature for more information.
Units acceptable for use in describing TemperatureInterval quantities.
Units acceptable for use in describing Temperature quantities.
The ability of a material to conduct heat. See the Wikipedia entry for Thermal conductivity for more information.
Units acceptable for use in describing ThermalConductivity quantities.
Represents the time interval physical quantity (one of the seven base SI quantities).
TimeInstant represents a specific moment in time and its units enable conversion between various time scales.
Units acceptable for use in describing TimeInstant quantities.
Represents a specific time span.
Units acceptable for use in describing Time quantities.
The tendency of a force to rotate an object about an axis, fulcrum, or pivot. See the Wikipedia entry for Torque for more information.
Units acceptable for use in describing Torque quantities.
The amount of three-dimensional space enclosed by some closed boundary. See the Wikipedia entry for Volume for more information.
The volume of fluid which passes per unit time. See the Wikipedia entry for Volumetric flow rate for more information.
Units acceptable for use in describing VolumeFlowRate quantities.
Units acceptable for use in describing Volume quantities.
The spatial frequency of a wave. See the Wikipedia entry for Wavenumber for more information.
Units acceptable for use in describing WaveNumber quantities.
Whether and how to display a quantity's uncertainty (e.g., compact is 32.324(12), not compact is (32.324 +/- 0.012)).
A unit is a particular physical quantity, defined and adopted by convention, with which other particular quantities of the same kind (dimensions) are compared to express their value.
alphaParticleMass → const Mass
The rest mass of a helium nucleus.
angstromStar → const Length
Often used to represent the wavelengths of X rays and the distances between atoms in crystals.
atomicMass → const Mass
One twelfth of the mass of a carbon-12 atom in its nuclear and electronic ground state.
bohrRadius → const Length
The mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state.
boltzmannConstant → const Entropy
The Boltzmann constant is a physical constant that relates the average relative kinetic energy of particles in a gas with the temperature of the gas.
characteristicImpedanceOfVacuum → const Resistance
Relates the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space.
classicalElectronRadius → const Length
The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation.
comptonWavelength → const Length
The wavelength of a photon whose energy is the same as the mass energy equivalent of that particle.
conductanceQuantum → const Conductance
The quantized unit of electrical conductance. It appears when measuring the conductance of a quantum point contact, and, more generally, is a key component of Landauer formula which relates the
electrical conductance of a quantum conductor to its quantum properties.
cos0 → const double
The cosine of a ninety degree angle (1).
cos60 → const double
The cosine of a sixty degree angle (0.5).
cos90 → const double
The cosine of a ninety degree angle (1).
deuteronMass → const Mass
The mass of a stationary deuteron.
electronGFactor → const Scalar
Electron g factor (ge).
electronMass → const Mass
The mass of a stationary electron.
elementaryCharge → const Charge
The electric charge carried by a single proton.
expUnicodeMap → const Map<String, String>
Maps a digit, decimal point or minus sign string to a unicode exponent character.
fineStructureConstant → const Scalar
Fine structure constant (alpha).
fiveNinths → const double
A ratio used in the conversion between metric temperature scales and Fahrenheit.
hartreeEnergy → const Energy
The Hartree atomic unit of energy is the energy for the separation of a molecule to nuclei and electrons.
hBar → const AngularMomentum
The Planck constant divided by 2 PI (a.k.a., 'h-bar' or 'reduced Planck constant').
helionMass → const Mass
The mass of a stationary helion.
lengthZero → const Length
A constant representing zero length.
magneticFluxQuantum → const MagneticFlux
The magnetic flux quantum constant is a combination of two other physical constants: the Planck constant h and the electron charge e. Its value is, therefore, the same for any superconductor.
muonGFactor → const Scalar
Muon g factor (gn).
muonMass → const Mass
The mass of a stationary muon particle.
naught → const Scalar
A synonym for zero.
neutronGFactor → const Scalar
Neutron g factor (gn).
neutronMass → const Mass
The mass of a stationary neutron.
planckConstant → const AngularMomentum
The Planck constant.
planckLength → const Length
The threshold length at which classical ideas about gravity and space-time cease to be valid and quantum effects dominate.
planckMass → const Mass
The unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams.
planckTemperature → const Temperature
Contemporary models of physical cosmology postulate that the highest possible temperature is the Planck temperature.
planckTime → const Time
The time required for light to travel in a vacuum a distance of one Planck length.
protonGFactor → const Scalar
Proton g factor (gp).
protonMass → const Mass
The mass of a stationary proton.
sackurTetrode100kPa → const Scalar
Sackur-Tetrode constant at 1 K and 100 kPa ('S0/R').
sackurTetrodeStdAtm → const Scalar
Sackur-Tetrode constant at 1 K and 101.325 kPa ('S0/R').
scalarZero → const Scalar
Zero, as a Scalar.
sigma0 → const Area
A synonym for the thomsonCrossSection.
sin0 → const double
The sine of a zero degree angle (0).
sin30 → const double
The sine of a thirty degree angle (0.5).
sin90 → const double
The sine of a ninety degree angle (1).
solarLuminosity → const Power
The solar luminosity constant is the radiant flux (power emitted in the form of photons) emitted by the Sun (a typical value; the Sun's output actually varies slightly over time).
speedOfLightVacuum → const Speed
Speed of light in a vacuum.
speedOfSoundAir0C → const Speed
Speed of sound in air at 0 deg C.
tan0 → const double
The tangent of a zero degree angle (0).
tan45 → const double
The tangent of a forty five degree angle (1).
tau → const double
The "circle constant", equal to two pi.
tauComptonWavelength → const Length
The tau Compton wavelength.
tauMass → const Mass
The mass of a stationary tau particle.
thomsonCrossSection → const Area
Helpful for describing the scattering of photons when the energy of each individual photon is much smaller than the rest energy of the electron.
twoPi → const double
The value 2 * pi, also known as tau.
vacuumElectricPermittivity → const Permittivity
A constant of proportionality that exists between electric displacement and electric field intensity.
vacuumMagneticPermeability → const Permeability
The magnetic permeability in a classical vacuum.
vonKlitzingConstant → const Resistance
Relates Planck's constant and the charge of the electron.
weakMixingAngle → const Scalar
Weak mixing angle.
Z0 → const Resistance
A synonym for characteristicImpedanceOfVacuum.
zilch → const Scalar
A synonym for zero.
abamperes → CurrentUnits
Abamperes as a unit.
abcoulombs → ChargeUnits
A synonym for dekacoulombs.
abfarads → CapacitanceUnits
A unit representing one gigafarad.
abhenries ↔ InductanceUnits
Abhenries is a synonym for nanohenries.
getter/setter pair
abmho → ConductanceUnits
Equivalent to a billion siemens (or mhos).
abvolts ↔ ElectricPotentialDifferenceUnits
The unit of electromotive force (EMF) or potential difference in the CGS (centimeter/gram/second) electromagnetic system of units. When an EMF of 1 abV exists between two points, then one erg of
energy is needed to move one abcoulomb (1 abC) of charge carriers between those two points.
getter/setter pair
acreFoot ↔ VolumeUnits
Acre foot as a unit.
getter/setter pair
acres → AreaUnits
An acre is about three-quarters of the size of an American football field.
aeons → TimeUnits
A unit of one billion years.
allQuantityTypes → Iterable<Type>
Returns an iterable of Type objects representing all of the quantity types supported by this library (for example, Angle, Length, etc.).
no setter
ampereHours → ChargeUnits
Ampere hours as a unit.
amperes → CurrentUnits
The standard SI unit.
amps → CurrentUnits
A synonym for amperes.
angle0 → Angle
A zero degree angle.
angle180 → Angle
A one hundred eighty degree angle.
angle270 → Angle
A two hundred seventy degree angle.
angle30 → Angle
A thirty degree angle.
angle360 → Angle
A three hundred sixty degree angle.
angle45 → Angle
A forty five degree angle.
angle60 → Angle
A sixty degree angle.
angle90 → Angle
A ninety degree angle.
anglePi → Angle
A one hundred eighty degree angle.
angleTau → Angle
A three hundred sixty degree angle.
angstromStars → LengthUnits
A non-standard unit of length.
angularMils → AngleUnits
One angular mil is 0.05625 of a degree, exactly.
ares → AreaUnits
Accepted for use with the SI, subject to further review. Equals one square dekameter, or 100 square meters.
astronomicalUnits → LengthUnits
The mean distance from the center of the earth to the center of the sun.
atmospheres ↔ PressureUnits
Standard atmospheres as a unit.
getter/setter pair
atmospheresTechnical ↔ PressureUnits
Technical atmospheres as a unit.
getter/setter pair
attometers → LengthUnits
A unit of 10^-18 meters.
attoseconds → TimeUnits
10^-18 of a second.
B → TimeInstantUnits
Besselian - Replaced by Julian system, but still of occasional use
bakersDozen → ScalarUnits
A baker's dozen is 13. One extra donut. Thank you.
barns → AreaUnits
Accepted for use with the SI, subject to further review. Equals 100 square femtometers, or 1.0e-28 square meters.
barrels ↔ VolumeUnits
Barrels as a unit.
getter/setter pair
barrelsPetroleum ↔ VolumeUnits
Barrels petroleum as a unit.
getter/setter pair
bars ↔ PressureUnits
Shorthand bars unit.
getter/setter pair
baryes ↔ PressureUnits
Baryes as a unit.
getter/setter pair
billionEur → ScalarUnits
The European variant of one billion (10^12) as a unit.
billionthsUS → ScalarUnits
One billionth (US: 10^-9) as a unit.
billionUS → ScalarUnits
One billion (US: 10^9) as a unit.
bits → InformationUnits
The standard SI unit.
boardFeet ↔ VolumeUnits
Board feet as a unit.
getter/setter pair
boltzmannUnit → EntropyUnits
Boltzmann constant as a unit.
btu39 → EnergyUnits
39 degree Fahrenheit BTUs as a unit.
btu60 → EnergyUnits
60 degree Fahrenheit BTUs as a unit.
btuInternationalTable → EnergyUnits
International Table BTUs as a unit.
btuMean → EnergyUnits
Mean BTUs as a unit.
btuThermo → EnergyUnits
Thermochemical BTUs as a unit.
btuThermsPerHour ↔ PowerUnits
Square degrees as a unit.
getter/setter pair
btuThermsPerSecond ↔ PowerUnits
Thermochemical BTUs per second as a unit.
getter/setter pair
bushels ↔ VolumeUnits
Bushels as a unit.
getter/setter pair
bytes → InformationUnits
Units of 8 bits.
cables → LengthUnits
A non-standard unit of length.
calibers → LengthUnits
A non-standard unit of length.
calories15 → EnergyUnits
15 degree Celsius calories as a unit.
calories20 → EnergyUnits
20 degree Celsius calories as a unit.
caloriesInternationalTable → EnergyUnits
International Table calories as a unit.
caloriesKgInternationalTable → EnergyUnits
International Table kilogram calories as a unit.
caloriesKgMean → EnergyUnits
Mean kilogram calories as a unit.
caloriesKgThermo → EnergyUnits
Thermochemical kilogram calories as a unit.
caloriesMean → EnergyUnits
Mean calories as a unit.
caloriesThermo → EnergyUnits
Thermochemical calories as a unit.
caloriesThermoPerSecond ↔ PowerUnits
Thermochemical calories as a unit.
getter/setter pair
candelas → LuminousIntensityUnits
The standard SI unit.
caratsMetric → MassUnits
Metric carats as a unit.
centimeters → LengthUnits
A unit of one hundredth of a meter.
centiseconds → TimeUnits
A hundredth of a second.
centistokes ↔ KinematicViscosityUnits
Centistokes as a unit.
getter/setter pair
chainsEngineer → LengthUnits
A non-standard unit of length.
chainsSurveyor → LengthUnits
A non-standard unit of length.
circles → AngleUnits
Synonymous with revolutions.
circularMils → AreaUnits
Equal to the area of a circle with a diameter of one mil (one thousandth of an inch). It is often used for representing the area of a wire's circular cross section.
clausius → EntropyUnits
The erg per clausius unit of entropy.
cmMercury0 ↔ PressureUnits
Centimeters of mercury at 0 degrees Celsius.
getter/setter pair
cmWater4 ↔ PressureUnits
Centimeters of water at 4 degrees Celsius.
getter/setter pair
cords ↔ VolumeUnits
Cords as a unit.
getter/setter pair
cos30 → double
The cosine of a thirty degree angle.
cos45 → double
The cosine of a forty five degree angle.
coulombs → ChargeUnits
The standard SI unit.
cubicCentimeters ↔ VolumeUnits
Cubic centimeters as a unit.
getter/setter pair
cubicFeet ↔ VolumeUnits
Barrels as a unit.
getter/setter pair
cubicInches ↔ VolumeUnits
Cubic inches as a unit.
getter/setter pair
cubicMeters ↔ VolumeUnits
The standard SI unit.
getter/setter pair
cubicMetersPerSecond ↔ VolumeFlowRateUnits
The standard SI unit, tersely.
getter/setter pair
cubicParsecs ↔ VolumeUnits
Cubic parsecs as a unit.
getter/setter pair
cubicYards ↔ VolumeUnits
Cubic yards as a unit.
getter/setter pair
cubits → LengthUnits
A non-standard unit of length.
cumecs ↔ VolumeFlowRateUnits
Shorthand synonym for standard SI unit.
getter/setter pair
cups ↔ VolumeUnits
Cups as a unit.
getter/setter pair
cycles → AngleUnits
Synonymous with revolutions.
days → TimeUnits
Accepted for use with the SI.
daysMeanSolar → TimeUnits
Accepted for use with the SI.
daysSidereal → TimeUnits
A sidereal day is the time between two consecutive transits of the First Point of Aries. It represents the time taken by the Earth to rotate on its axis relative to the stars, and is almost four
minutes shorter than the solar day because of the Earth's orbital motion.
decillionEur → ScalarUnits
The European variant of one decillion (10^60) as a unit.
decillionUS → ScalarUnits
One decillion (US: 10^33) as a unit.
decimeters → LengthUnits
A unit of one tenth of a meter.
deciseconds → TimeUnits
A tenth of a second.
deg → AngleUnits
Synonymous with degrees.
degF → TemperatureIntervalUnits
A synonym for degreesFahrenheit.
degK → TemperatureIntervalUnits
A synonym for degrees Kelvin.
degR → TemperatureIntervalUnits
A synonym for degreesRankine.
degrees → AngleUnits
A terse version of Angle.degrees.
degreesCelsius → TemperatureUnits
Degrees in the Celsius scale.
degreesFahrenheit → TemperatureIntervalUnits
Degrees in the Fahrenheit scale.
degreesPerSecond → AngularSpeedUnits
Accepted for use with the SI.
degreesRankine → TemperatureIntervalUnits
Degrees in the Rankine scale.
dekameters → LengthUnits
A unit of ten meters.
dekaseconds → TimeUnits
Ten seconds.
dozen → ScalarUnits
A dozen is 12.
drams ↔ VolumeUnits
Drams as a unit.
getter/setter pair
dramsApothecary → MassUnits
Apothecary drams as a unit.
dramsAvoirdupois → MassUnits
Avoirdupois drams as a unit.
duotrigintillion → ScalarUnits
One duotrigintillion (10^99) as a unit.
dynamicQuantityTyping ↔ bool
Dynamic quantity typing may be turned off for increased efficiency. If false, the result of operations where dimensions may change will be MiscQuantity type objects.
getter/setter pair
dynes ↔ ForceUnits
Dynes as a unit.
getter/setter pair
electronVolts → EnergyUnits
Accepted for use with the SI.
emuCapacitance → CapacitanceUnits
Electromagnetic unit (emu), the capacity of a circuit component to store charge.
emuOfInductance ↔ InductanceUnits
EMU of inductance is a synonym for nanohenries.
getter/setter pair
emuPotential ↔ ElectricPotentialDifferenceUnits
Synonymous with abvolts.
getter/setter pair
eons → TimeUnits
A synonym for aeons.
ergPerKelvin → EntropyUnits
The erg per kelvin unit of entropy.
ergs → EnergyUnits
Ergs as a unit.
ergsPerSecond ↔ PowerUnits
Ergs per second as a unit.
getter/setter pair
esuCapacitance → CapacitanceUnits
A synonym for statfarads.
esuOfInductance ↔ InductanceUnits
ESU of inductance as a unit.
getter/setter pair
esuPotential ↔ ElectricPotentialDifferenceUnits
Synonymous with statvolts.
getter/setter pair
ET → TimeInstantUnits
Ephemeris Time (ET) is the same as TDT: ET = TDT = TT = TAI + 32.184 s. Ephemeris Time was renamed Terrestrial Dynamical Time in 1984 (when Barycentric Dynamical Time was also introduced)
exabytes → InformationUnits
10^18 bytes. Use Information.exbibytes (EiB) instead for the binary interpretation of EB (2^60 bytes).
exameters → LengthUnits
A unit of 10^18 meters.
exaseconds → TimeUnits
10^18 seconds.
Fahrenheit → TemperatureUnits
Fahrenheit scale units.
faradaysC12 → ChargeUnits
Carbon 12 faradays as a unit.
faradaysChemical → ChargeUnits
Chemical faradays as a unit.
faradaysPhysical → ChargeUnits
Physical faradays as a unit.
farads → CapacitanceUnits
The standard SI unit.
faradsPerMeter ↔ PermittivityUnits
The standard SI unit.
getter/setter pair
fathoms → LengthUnits
A non-standard unit of length.
feet → LengthUnits
A non-standard unit of length.
feetPerHour ↔ SpeedUnits
Feet per hour as a unit.
getter/setter pair
feetPerMinute ↔ SpeedUnits
Feet per minute as a unit.
getter/setter pair
feetPerSecond ↔ SpeedUnits
Feet per second as a unit.
getter/setter pair
feetUsSurvey → LengthUnits
A non-standard unit of length.
femtometers → LengthUnits
A unit of 10^-15 meters.
femtoseconds → TimeUnits
10^-15 of a second.
fermis → LengthUnits
A non-standard unit of length.
fluidOunces ↔ VolumeUnits
Fluid ounces as a unit.
getter/setter pair
fluidOuncesUK ↔ VolumeUnits
U.K. fluid ounces as a unit.
getter/setter pair
fluxUnits → SpectralIrradianceUnits
Synonymous with janskys.
footCandles ↔ IlluminanceUnits
Foot candles as a unit.
getter/setter pair
footPerSecondSquared → AccelerationUnits
A commonly used English unit of acceleration.
footPoundals → EnergyUnits
Foot-poundals as a unit.
footPoundsForce → EnergyUnits
Foot pounds force as a unit.
forceDeCheval ↔ PowerUnits
Force de cheval as a unit.
getter/setter pair
ftWater39 ↔ PressureUnits
Feet of water at 39.2 degrees Fahrenheit.
getter/setter pair
furlongs → LengthUnits
A non-standard unit of length.
g → MassUnits
Gram unit synonym.
galileos → AccelerationUnits
Defined as 0.01 meter per second squared.
gallonsUKLiquid ↔ VolumeUnits
U.K. liquid gallons as a unit.
getter/setter pair
gallonsUSDry ↔ VolumeUnits
U.S. dry gallons as a unit.
getter/setter pair
gallonsUSLiquid ↔ VolumeUnits
U.S. liquid gallons as a unit.
getter/setter pair
gals → AccelerationUnits
Synonymous with galileos.
gammas → MassUnits
Microgram units.
gauss ↔ MagneticFluxDensityUnits
One gauss is one ten-thousandth of a tesla.
getter/setter pair
gees → AccelerationUnits
A unit based on the acceleration experienced by a free-falling body at the Earth's surface.
gigabytes → InformationUnits
10^9 bytes. Use Information.gibibytes (GiB) instead for the binary interpretation of GB (2^30 bytes).
gigameters → LengthUnits
A unit of one billion meters.
gigaseconds → TimeUnits
A billion seconds.
gilberts → CurrentUnits
Gilberts as a unit.
gillsUK ↔ VolumeUnits
U.K. gills as a unit.
getter/setter pair
gillsUS ↔ VolumeUnits
U.S. gills as a unit.
getter/setter pair
gons → AngleUnits
Synonym for grads.
googol ↔ Scalar
googol (10^100), arbitrary precision.
getter/setter pair
googols → ScalarUnits
One googol (10^100) as a unit.
GPST → TimeInstantUnits
GPS Satellite Time (GPST): GPST = TAI - 19 s
grades → AngleUnits
Synonym for grads.
grads → AngleUnits
One grad is 0.9 of a degree, exactly.
grains → MassUnits
Grains as a unit.
grams → MassUnits
A unit of one gram.
gramsPerCubicCentimeter ↔ MassDensityUnits
Grams per cubic centimeter as a unit.
getter/setter pair
greatGross → ScalarUnits
A great gross is 1728.
gross → ScalarUnits
A gross is 144.
halfDozen → ScalarUnits
A half-dozen is 6.
hands → LengthUnits
A non-standard unit of length.
hBarUnits → AngularMomentumUnits
The Planck constant divided by 2 PI (a.k.a., 'h-bar') as units.
hectares → AreaUnits
Accepted for use with the SI, subject to further review. Equals 1 square hectometer, or 10 000 square meters.
hectometers → LengthUnits
A unit of one hundred meters.
hectoseconds → TimeUnits
A hundred seconds.
hemispheres → SolidAngleUnits
Hemispheres as a unit.
henries ↔ InductanceUnits
The standard SI unit.
getter/setter pair
henriesPerMeter ↔ PermeabilityUnits
The standard SI unit.
getter/setter pair
hogsheads ↔ VolumeUnits
Hogsheads as a unit.
getter/setter pair
horsepower550 ↔ PowerUnits
Horsepower (550 ft lbs/s) as a unit.
getter/setter pair
horsepowerBoiler ↔ PowerUnits
Horsepower (boiler) as a unit.
getter/setter pair
horsepowerElectric ↔ PowerUnits
Horsepower (electric) as a unit.
getter/setter pair
horsepowerMetric ↔ PowerUnits
Horsepower (metric) as a unit.
getter/setter pair
horsepowerWater ↔ PowerUnits
Horsepower (water) as a unit.
getter/setter pair
hours → TimeUnits
Accepted for use with the SI.
hoursMeanSolar → TimeUnits
Accepted for use with the SI.
hoursSidereal → TimeUnits
A unit of one hour in the sidereal day.
hoursTime → AngleUnits
Based on Earth's rotation (approximately 15 degrees).
hundred → ScalarUnits
100 as a unit.
hundredths → ScalarUnits
One hundredth as a unit.
hundredweightLong → MassUnits
Long hundredweight as a unit.
hundredweightShort → MassUnits
Short hundredweight as a unit.
inches → LengthUnits
A non-standard unit of length.
inchesPerSecond ↔ SpeedUnits
Inches per second as a unit.
getter/setter pair
inchPerSecondSquared → AccelerationUnits
A commonly used English unit of acceleration.
inMercury32 ↔ PressureUnits
Inches of mercury at 32 degrees Fahrenheit.
getter/setter pair
inWater39 ↔ PressureUnits
Inches of water at 39.2 degrees Fahrenheit.
getter/setter pair
J2000 → TimeInstant
J2000.0 as defined by the IAU: Julian date: 2000 Jan 1d 12h UT in the TDT time scale
janskys → SpectralIrradianceUnits
A non-SI unit of spectral irradiance used especially in radio astronomy.
JD_TAI → TimeInstantUnits
Julian Date in the TAI scale
JD_TCB → TimeInstantUnits
Julian Date in the TCB scale
JD_TCG → TimeInstantUnits
Julian Date in the TCG scale
JD_TDB → TimeInstantUnits
Julian Date in the TDB (TB) scale
JD_TDT → TimeInstantUnits
Julian Date in the TDT (TT) scale
JD_UT1 → TimeInstantUnits
Julian Date in the UT1 scale
JD_UTC → TimeInstantUnits
Julian Date in the UTC scale
joules → EnergyUnits
The standard SI unit.
jouleSecond → AngularMomentumUnits
The standard SI unit.
joulesPerKelvin → EntropyUnits
The standard SI unit.
joulesPerKilogram ↔ SpecificEnergyUnits
The standard SI unit.
getter/setter pair
kelvins → TemperatureUnits
The standard SI unit.
kg → MassUnits
Kilogram unit synonym.
kgfSecondSquaredMeter → MassUnits
Kilogram force second square meter as a unit.
kilobytes → InformationUnits
1000 bytes (8000 bits). Use Information.kibibytes (kiB) instead for the binary interpretation of kB (1024 bytes).
kilocaloriesThermo → EnergyUnits
A synonym for thermochemical kilogram calories.
kilograms → MassUnits
The standard SI unit.
kilogramsForce ↔ ForceUnits
Kilograms force as a unit.
getter/setter pair
kilogramsPerCubicMeter ↔ MassDensityUnits
The standard SI unit.
getter/setter pair
kilojoules → EnergyUnits
Kilojoules as a unit.
kilometers → LengthUnits
A unit of one thousand meters.
kilometersPerHour ↔ SpeedUnits
Kilometers per hour as a unit.
getter/setter pair
kilomoles → AmountOfSubstanceUnits
A unit of one thousand moles.
kiloponds ↔ ForceUnits
A synonym for kilograms force.
getter/setter pair
kiloseconds → TimeUnits
A thousand seconds.
kilowattHours → EnergyUnits
Kilowatt-hours as a unit.
kilowatts ↔ PowerUnits
Shorthand kilowatts as a unit.
getter/setter pair
kips ↔ ForceUnits
Kips as a unit.
getter/setter pair
knots ↔ SpeedUnits
Knots, tersely.
getter/setter pair
langleys → EnergyFluxUnits
Langleys as a unit.
leaguesNautical → LengthUnits
A non-standard unit of length.
leaguesStatute → LengthUnits
A non-standard unit of length.
leaguesUkNautical → LengthUnits
A non-standard unit of length.
lightYears → LengthUnits
The distance light travels in one year.
linksEngineer → LengthUnits
A non-standard unit of length.
linksSurveyor → LengthUnits
A non-standard unit of length.
liters ↔ VolumeUnits
A terse alternative to Volume.liters.
getter/setter pair
litersPerSecond ↔ VolumeFlowRateUnits
0.001 cubic meter per second.
getter/setter pair
logger ↔ Logger
Logger for use across entire library
getter/setter pair
lusecs ↔ VolumeFlowRateUnits
Shorthand synonym for liters per second.
getter/setter pair
lux ↔ IlluminanceUnits
The standard SI unit.
getter/setter pair
magFieldAtomicUnit ↔ MagneticFluxDensityUnits
Magnetic field atomic unit.
getter/setter pair
magneticGammas ↔ MagneticFluxDensityUnits
A synonym for nanoteslas.
getter/setter pair
maxwells ↔ MagneticFluxUnits
A maxwell is one one-hundred-millionth of a weber.
getter/setter pair
megabytes → InformationUnits
10^6 bytes. Use Information.mebibytes (MiB) instead for the binary interpretation of MB (2^20 bytes).
megameters → LengthUnits
A unit of one million meters.
megaseconds → TimeUnits
A million seconds.
meterPerSecondSquared → AccelerationUnits
A synonym for the standard SI-MKS unit of acceleration.
meters → LengthUnits
The standard SI unit.
metersPerSecond ↔ SpeedUnits
The standard SI unit.
getter/setter pair
metersSquaredPerSecond ↔ KinematicViscosityUnits
The standard SI unit.
getter/setter pair
metricTons → MassUnits
Accepted for use with the SI.
mg → MassUnits
Milligram unit synonym.
mho → ConductanceUnits
Synonymous with Siemens.
micrometers → LengthUnits
A unit of one millionth of a meter.
microns → LengthUnits
A synonym for micrometers.
microseconds → TimeUnits
A millionth of a second.
miles → LengthUnits
A non-standard unit of length.
milesPerHour ↔ SpeedUnits
Miles per hour as a unit.
getter/setter pair
milesPerMinute ↔ SpeedUnits
Miles per minute as a unit.
getter/setter pair
milesPerSecond ↔ SpeedUnits
Miles per second as a unit.
getter/setter pair
millibars ↔ PressureUnits
A millibar as a unit.
getter/setter pair
milligrams → MassUnits
A unit of one thousandth of a gram.
millimeters → LengthUnits
A unit of one thousandth of a meter.
million → ScalarUnits
One million as a unit.
millionths → ScalarUnits
One millionth as a unit.
milliradian → AngleUnits
A unit of one thousandth of a radian.
milliseconds → TimeUnits
A thousandth of a second.
millisteradians → SolidAngleUnits
Millisteradians as a unit.
mils → LengthUnits
A non-standard unit of length.
minersInches ↔ VolumeFlowRateUnits
The miner's inch as a unit.
getter/setter pair
minutes → TimeUnits
Accepted for use with the SI.
minutesArc → AngleUnits
A terse version of Angle.minutesArc.
minutesMeanSolar → TimeUnits
Accepted for use with the SI.
minutesSidereal → TimeUnits
A unit of one minute in the sidereal day.
minutesTime → AngleUnits
Based on Earth's rotation.
MJD_TAI → TimeInstantUnits
Modified Julian Date in the TAI scale
MJD_TCB → TimeInstantUnits
Modified Julian Date in the TCB scale
MJD_TCG → TimeInstantUnits
Modified Julian Date in the TCG scale
MJD_TDB → TimeInstantUnits
Modified Julian Date in the TDB scale
MJD_TDT → TimeInstantUnits
Modified Julian Date in the TDT scale
MJD_UT1 → TimeInstantUnits
Modified Julian Date in the UT1 scale
MJD_UTC → TimeInstantUnits
Modified Julian Date in the UTC scale
moles → AmountOfSubstanceUnits
A synonym for the SI-MKS base unit of amount of substance.
musecs ↔ VolumeFlowRateUnits
Shorthand synonym for standard SI unit.
getter/setter pair
myriad → ScalarUnits
A myriad is ten thousand.
nanometers → LengthUnits
A unit of one billionth of a meter.
nanoseconds → TimeUnits
A billionth of a second.
nauticalMilesUk → LengthUnits
A non-standard unit of length.
newtons → ForceUnits
The standard SI unit.
newtonsPerAmpereSquared ↔ PermeabilityUnits
Newtons per ampere as a unit.
getter/setter pair
newtonsPerSquareMeter ↔ PressureUnits
A synonym for pascals.
getter/setter pair
nonillionEur → ScalarUnits
The European variant of one nonillion (10^54) as a unit.
nonillionUS → ScalarUnits
One nonillion (US: 10^30) as a unit.
NTP → TimeInstantUnits
Network Time Protocol (NTP) - NTP is offset from the UTC time scale, with its epoch at 1 Jan 1900 0h
octants → SolidAngleUnits
One eighth (1/8) of a sphere (a spherical right triangle).
octillionEur → ScalarUnits
The European variant of one octillion (10^48) as a unit.
octillionUS → ScalarUnits
One octillion (US: 10^27) as a unit.
ohms ↔ ResistanceUnits
The standard SI unit.
getter/setter pair
one → ScalarUnits
The standard SI unit.
ouncesApothecary → MassUnits
Apothecary ounces as a unit.
ouncesAvoirdupois → MassUnits
Avoirdupois ounces as a unit.
ouncesForceAvoirdupois ↔ ForceUnits
Avoirdupois ounces force as a unit.
getter/setter pair
paces → LengthUnits
A non-standard unit of length.
pair → ScalarUnits
A pair is 2.
parsecs → LengthUnits
A non-standard unit of length.
pascals ↔ PressureUnits
The standard SI unit.
getter/setter pair
pecks ↔ VolumeUnits
Pecks as a unit.
getter/setter pair
pennyweightTroy → MassUnits
Troy pennyweight as a unit.
percent → ScalarUnits
Synonymous with Scalar.percent.
perches → LengthUnits
A non-standard unit of length.
petabytes → InformationUnits
10^15 bytes. Use Information.pebibytes (PiB) instead for the binary interpretation of PB (2^50 bytes).
petameters → LengthUnits
A unit of 10^15 meters.
petaseconds → TimeUnits
10^15 seconds.
phots ↔ IlluminanceUnits
Phots as a unit.
getter/setter pair
picas → LengthUnits
A non-standard unit of length.
picometers → LengthUnits
A unit of 10^-12 meters.
picoseconds → TimeUnits
10^-12 of a second.
pintsDry ↔ VolumeUnits
U.S. dry pints as a unit.
getter/setter pair
pintsLiquid ↔ VolumeUnits
U.S. liquid pints as a unit.
getter/setter pair
planckUnits → AngularMomentumUnits
The Planck constant as units.
points → LengthUnits
A non-standard unit of length.
poles → LengthUnits
A non-standard unit of length.
poundals ↔ ForceUnits
Poundals as a unit.
getter/setter pair
poundsApothecary → MassUnits
Apothecary pounds as a unit.
poundsAvoirdupois → MassUnits
Avoirdupois pounds as a unit.
poundsForceAvoirdupois ↔ ForceUnits
Avoirdupois pounds force as a unit.
getter/setter pair
poundsPerCubicFoot ↔ MassDensityUnits
Pounds per cubic foot as a unit.
getter/setter pair
poundsPerCubicInch ↔ MassDensityUnits
Pounds per cubic inch as a unit.
getter/setter pair
psi ↔ PressureUnits
Pounds per square inch as a unit.
getter/setter pair
quadrants → AngleUnits
Represents a quarter circle of ninety degrees.
quadrillionEur → ScalarUnits
The European variant of one quadrillion (10^24) as a unit.
quadrillionUS → ScalarUnits
One quadrillion (US: 10^15) as a unit.
quartsDry ↔ VolumeUnits
U.S. dry quarts as a unit.
getter/setter pair
quartsLiquid ↔ VolumeUnits
U.S. liquid quarts as a unit.
getter/setter pair
quintals → MassUnits
A quintal is 100 kilograms.
quintillionEur → ScalarUnits
The European variant of one quintillion (10^30) as a unit.
quintillionUS → ScalarUnits
One quintillion (US: 10^18) as a unit.
rad → AngleUnits
Synonymous with radians.
radians → AngleUnits
A terse version of Angle.radians.
radiansPerSecond → AngularSpeedUnits
The standard SI unit.
Rankine → TemperatureUnits
Rankine scale units.
referenceSound ↔ PowerUnits
A power commonly used as the reference power for calculation of sound power levels.
getter/setter pair
referenceSoundAir ↔ PressureUnits
A pressure often used as the reference pressure for calculation of sound pressure levels.
getter/setter pair
referenceSoundWater ↔ PressureUnits
A pressure commonly used as the reference pressure for calculation of sound pressure levels.
getter/setter pair
registryTons ↔ VolumeUnits
Registry tons as a unit.
getter/setter pair
revolutions → AngleUnits
Represents a full circle of two pi radians.
revolutionsPerMinute → AngularSpeedUnits
Rotation frequency.
rods → LengthUnits
A non-standard unit of length.
rpm → AngularSpeedUnits
A synonym for revolutionsPerMinute.
score → ScalarUnits
A score is 20. Four score is 80. More poetic than just saying eighty.
scruples → MassUnits
Scruples as a unit.
seconds → TimeUnits
The standard SI unit.
secondsArc → AngleUnits
A terse version of Angle.secondsArc.
secondsSidereal → TimeUnits
A unit of one second in the sidereal day.
secondsTime → AngleUnits
Based on Earth's rotation.
sections → AreaUnits
Synonymous with squareMiles.
semicircles → AngleUnits
Represents a half circle of one hundred eighty degrees (pi radians).
septillionEur → ScalarUnits
The European variant of one billion (10^42) as a unit.
septillionUS → ScalarUnits
One septillion (US: 10^24) as a unit.
sextillionEur → ScalarUnits
The European variant of one sextillion (10^36) as a unit.
sextillionUS → ScalarUnits
One sextillion (US: 10^21) as a unit.
signs → AngleUnits
A sign unit is a little more than half a radian.
sin45 → double
The sine of a forty five degree angle.
sin60 → double
The sine of a sixty degree angle.
skeins → LengthUnits
A non-standard unit of length.
slugs → MassUnits
Slugs as a unit.
slugsPerCubicFoot ↔ MassDensityUnits
Slugs per cubic foot as a unit.
getter/setter pair
spans → LengthUnits
A non-standard unit of length.
spats → SolidAngleUnits
Spats as a unit.
speedOfLightSquared ↔ SpecificEnergyUnits
The square of the speed of light in a vacuum.
getter/setter pair
speedOfLightUnits ↔ SpeedUnits
The speed of light as a unit.
getter/setter pair
spheres → SolidAngleUnits
Spheres as a unit.
sphericalRightTriangles → SolidAngleUnits
Same as octants.
squareArcMinutes → SolidAngleUnits
Square arc minutes as a unit.
squareArcSeconds → SolidAngleUnits
Square arc seconds as a unit.
squareCentimeters → AreaUnits
An area unit equivalent to a square with sides having a length of one centimeter.
squareDegrees → SolidAngleUnits
Square degrees as a unit.
squareFeet → AreaUnits
An area unit equivalent to a square with sides having a length of one foot.
squareFeetPerSecond ↔ KinematicViscosityUnits
Square feet per second as a unit.
getter/setter pair
squareInches → AreaUnits
An area unit equivalent to a square with sides having a length of one inch.
squareKilometers → AreaUnits
An area unit equivalent to a square with sides having a length of one kilometer.
squareMeters → AreaUnits
The standard SI unit.
squareMetersPerSquareSecond ↔ SpecificEnergyUnits
Square meters per second as a unit.
getter/setter pair
squareMiles → AreaUnits
An area unit equivalent to a square with sides having a length of one mile.
squarePerches → AreaUnits
Synonymous with squareRods.
squarePoles → AreaUnits
Synonymous with squareRods.
squareRods → AreaUnits
An area unit equivalent to a square with sides having a length of one rod.
squareYards → AreaUnits
An area unit equivalent to a square with sides having a length of one yard.
standardAccelerationOfGravity → AccelerationUnits
The more formal name for gees.
statamperes → CurrentUnits
Statamperes as a unit.
statcoulombs → ChargeUnits
Statcoulombs as a unit.
statfarads → CapacitanceUnits
The statfarad is the standard unit of capacitance in the cgs (centimeter/gram/second) system.
statmhos → ConductanceUnits
A non-SI unit of electrical conductance.
statvolts ↔ ElectricPotentialDifferenceUnits
A useful unit for electromagnetism because, in a vacuum, an electric field of one statvolt/cm has the same energy density as a magnetic field of one gauss. Likewise, a plane wave propagating in a
vacuum has perpendicular electric and magnetic fields such that for every gauss of magnetic field intensity there is one statvolt/cm of electric field intensity.
getter/setter pair
steradians → SolidAngleUnits
The standard SI unit.
steres ↔ VolumeUnits
A synonym for cubic meters.
getter/setter pair
stokes ↔ KinematicViscosityUnits
Stokes as a unit.
getter/setter pair
system → TimeInstantUnits
Measures time since 1 Jan 1970 0h 0m 0s, which is the System time defined by many computer operating systems
tablespoons ↔ VolumeUnits
Tablespoons as a unit.
getter/setter pair
TAI → TimeInstantUnits
International Atomic Time scale units
tan30 → double
The tangent of a thirty degree angle.
tan60 → double
The tangent of a sixty degree angle.
TB → TimeInstantUnits
Barycentric Time (TB); same as TDB: TT = TDB
TCB → TimeInstantUnits
Barycentric Coordinate Time (TCB): TCB = TDB + (1.550505e-8)(JD - 2443144.5)(86400)
TCG → TimeInstantUnits
Geocentric Coordinate Time (TCG): TCG = TDT + (6.969291e-10)(JD - 2443144.5)(86400)
TDB → TimeInstantUnits
Barycentric Dynamical Time (TDB): TDB varies from TDT by periodic variations
TDT → TimeInstantUnits
Terrestrial Dynamical Time (TDT): TDT = TAI + 32.184 s
teaspoons ↔ VolumeUnits
Teaspoons as a unit.
getter/setter pair
tenths → ScalarUnits
One tenth as a unit.
terabytes → InformationUnits
10^12 bytes. Use Information.tebibytes (TiB) instead for the binary interpretation of TB (2^40 bytes).
terameters → LengthUnits
A unit of 10^12 meters.
teraseconds → TimeUnits
10^12 seconds.
teslas ↔ MagneticFluxDensityUnits
The standard SI unit.
getter/setter pair
thermalCoulomb → EntropyUnits
Entropy as a 'charge'; identical to joulesPerKelvin.
therms → EnergyUnits
Therms as a unit.
thousand → ScalarUnits
1000 as a unit.
thousandths → ScalarUnits
One thousandth as a unit.
tonnes → MassUnits
Accepted for use with the SI.
tons → EnergyUnits
Tons of TNT equivalent as a unit.
tonsAssay → MassUnits
Assay tons as a unit.
tonsLong → MassUnits
Long tons as a unit.
tonsShort → MassUnits
Short tons as a unit.
torrs ↔ PressureUnits
Torrs as a unit.
getter/setter pair
townships → AreaUnits
An area unit used in US surveyors' measures equalling 36 square miles.
trillionEur → ScalarUnits
The European variant of one trillion (10^18) as a unit.
trillionthsUS → ScalarUnits
One trillionth (US: 10^-12) as a unit.
trillionUS → ScalarUnits
One trillion (US: 10^12) as a unit.
TT → TimeInstantUnits
Terrestrial Time (TT) is the same as TDT: TDT = TT = TAI + 32.184 s
unifiedAtomicMassUnits → MassUnits
Accepted for use with the SI.
unitPoles ↔ MagneticFluxUnits
Unit poles as a unit.
getter/setter pair
unitPolesDensity ↔ MagneticFluxDensityUnits
Unit poles density as a unit.
getter/setter pair
UT1 → TimeInstantUnits
Universal Time (UT1): UT1 = TDT - Delta T
UT2 → TimeInstantUnits
Universal Time (UT2): UT2 = UT1 + 0.022 sin(2PIt) - 0.012 cos(2PIt) - 0.006 sin(4PIt) + 0.007 cos(4PIt), where t = the date in Besellian years
UTC → TimeInstantUnits
Coordinated Universal Time (differs from TAI by a number of leap seconds)
volts ↔ ElectricPotentialDifferenceUnits
The standard SI unit.
getter/setter pair
wattHour → EnergyUnits
Watt-hour as a unit.
watts ↔ PowerUnits
The standard SI unit.
getter/setter pair
wattSecond → EnergyUnits
Watt-second as a unit.
wattsPerSquareMeter → EnergyFluxUnits
The standard SI unit.
webers ↔ MagneticFluxUnits
The standard SI unit.
getter/setter pair
xUnits → LengthUnits
A non-standard unit of length.
yards → LengthUnits
A non-standard unit of length.
yearsCalendar → TimeUnits
Calendar years as a unit.
yearsJulian → TimeUnits
Defined as exactly 365.25 days of 86400 SI seconds each. The length of the Julian year is the average length of the year in the Julian calendar that was used in Western societies until some
centuries ago, and from which the unit is named.
yearsSidereal → TimeUnits
Sidereal years as a unit.
yearsTropical → TimeUnits
Tropical years as a unit.
yoctometers → LengthUnits
A unit of 10^-24 meters.
yoctoseconds → TimeUnits
10^-24 of a second.
yottameters → LengthUnits
A unit of 10^24 meters.
yottaseconds → TimeUnits
10^24 seconds.
zeptometers → LengthUnits
A unit of 10^-21 meters.
zeptoseconds → TimeUnits
10^-21 of a second.
zettameters → LengthUnits
A unit of 10^21 meters.
zettaseconds → TimeUnits
10^21 seconds.
angleFromHourMinSec(int hour, int minute, double second, [double uncert = 0]) → Angle
Constructs an angle from hours, minutes and seconds of time (as opposed to arc).
areWithin(Quantity q1, Quantity q2, Quantity tolerance) → bool
Returns whether or not the magnitude of the difference between two quantities is less than or equal to the specified tolerance.
cosecant(Angle a) → double
The ratio of the hypotenuse to the side opposite an acute angle; the reciprocal of sine.
cosine(Angle a) → double
Calculates the cosine of an Angle (adjacent divided by hypotenuse).
cotangent(Angle a) → double
The ratio of the side (other than the hypotenuse) adjacent to a particular acute angle to the side opposite the angle.
createTypedQuantityInstance(Type t, dynamic value, Units? units, {double uncert = 0.0, Dimensions? dimensions}) → Quantity
Creates a instance of a typed quantity of type t having the specified value in units.
degToRad(num deg) → double
Converts degrees to radians.
erf(double x) → double
Approximate solution for the error function of x.
getDeltaT(TimeInstant time) → double
Returns the value 'Delta T,' in seconds, which relates the Terrestrial Dynamical Time scale to measured Universal Time (and indirectly UTC to TAI before 1972, when leap seconds were introduced).
getLeapSeconds(double tai, {bool pre1972LeapSeconds = false}) → num
Returns the number of leap seconds in effect for the specified time instant, tai, specified in the TAI time scale. The number of leap seconds relates the UTC time scale to the TAI time scale.
hashObjects(Iterable<Object> objects) → int
Generates a unique hash for a set of objects.
numberToNum(Number number) → num
Converts a Number to the equivalent num.
numToNumber(num value) → Number
Converts a num value to associated Number object (Integer for ints and doubles that have an integer value, Double for other doubles).
objToNumber(Object object) → Number
Converts an object to a Number. The object must be either a num or Number, otherwise an Exception is thrown.
radToDeg(num rad) → double
Convert radians to degrees.
secant(Angle a) → double
The ratio of the hypotenuse to the shorter side adjacent to an acute angle; the reciprocal of a cosine.
secondsInUtcDay(double utc) → double
Calculates and returns the number of seconds (including any leap seconds) that are in the UTC day containing the specified second, utc.
siBaseQuantity(Quantity q) → bool
Returns whether or not q is one of the seven SI base quantities.
siDerivedQuantity(Quantity q) → bool
Returns whether or not q is a derived quantity (as opposed to one of the seven base SI quantities).
sine(Angle a) → double
Calculates the sine of an Angle (opposite divided by hypotenuse).
tangent(Angle a) → double
Calculates the tangent of an Angle (opposite divided by adjacent).
toMutable(Quantity q) → MutableQuantity
Create a MutableQuantity with the same value, dimensions and uncertainty as q.
uncertaintyRangeForQuantity(Quantity q, {double k = 1.0}) → QuantityRange<Quantity>
Creates a QuantityRange that represents the standard uncertainty of q.
unicodeExponent(num exp) → String
Returns the unicode symbols that represent an exponent.
yr4(int year) → int
Returns a four digit year from year which may be only 2 digits, assuming that anything 70 or more means 19xx and under 70 means 20xx.
Exceptions / Errors
This Exception is thrown when an attempt is made to perform an operation on a Quantity having unexpected or illegal dimensions.
This Exception is thrown when an attempt is made to modify an immutable Quantity object (for example through its setMKS method).
The base class for all exceptions thrown in relation to numbers.
The base class for all exceptions thrown in relation to quantities. | {"url":"https://pub.dev/documentation/quantity/latest/quantity/quantity-library.html","timestamp":"2024-11-12T07:37:59Z","content_type":"text/html","content_length":"272313","record_id":"<urn:uuid:3c916366-7550-46bb-8fca-8ad109f9e83e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00060.warc.gz"} |
Can one recognize a function from its graph? | EMS Press
Can one recognize a function from its graph?
• Jürgen Appell
Universität Würzburg, Germany
• Agnieszka Chlebowicz
Rzeszów University of Technology, Poland
• Simon Reinwand
TNG Technology Consulting GmbH, Unterföhring, Germany
• Beata Rzepka
Rzeszów University of Technology, Poland
We analyse the “interplay” between analytical properties of a real function on a metric space, on the one hand, and topological properties of its graph, on the other. In particular, we study
functions with closed, compact, connected, pathwise connected, or locally connected graphs, and we give nine conditions on the graph which are equivalent to the continuity of a function. A main
emphasis is put on examples and counterexamples which illustrate how significant our hypotheses are, and how far sufficient conditions are from being necessary.
Cite this article
Jürgen Appell, Agnieszka Chlebowicz, Simon Reinwand, Beata Rzepka, Can one recognize a function from its graph?. Z. Anal. Anwend. 42 (2023), no. 1/2, pp. 203–233
DOI 10.4171/ZAA/1730 | {"url":"https://ems.press/journals/zaa/articles/12614003?na","timestamp":"2024-11-12T13:05:57Z","content_type":"text/html","content_length":"85542","record_id":"<urn:uuid:a72b74ee-de5a-4f30-a8e9-0bb22a143dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00158.warc.gz"} |
Use of two drag coefficient concept for the investigation of flapping wings dynamics
Micro aerial vehicles design represents a challenge that lasted for years and the fact that they operate in a low Reynolds range, which makes the unsteady aerodynamic effect more influential, made
the direct computational fluid dynamics simulation expensive in time and money, and an alternative method especially in the early phase of the design would be very beneficial and rentable. In this
work the flight a flapping wings operated micro aerial vehicle was investigated by the simulation of the mechanical equations of motion in order to have an approximation of the true motion behavior
and the flying condition of the vehicle, in the same time this approach present a model that can be electronically implemented to make the MAV auto-controlled by imposing some criteria. The equations
were developed in spherical coordinates system, and simulated using the software Mathcad, and some of the constant related to the size of the vehicle are variated to match different range of existing
flying animals from insects to birds, a concept of two different drag coefficient for the upstroke and down stroke was used successfully to model the flapping wing. And giving the fact that lots of
parameter were simplified or neglected which lessen the accuracy, it gave good approximation, and the model can be used for auto-control by predefined flying path. Beside the main simulation work a
small experimentation on a model wing covered in feather was conducted in a subsonic wind tunnel in order to present a practical alternative that economize energy by reducing the drag in upstroke,
and it was found that the direction of the feather plays a significant role in the drag reduction and we concluded that the use of such materials can greatly improve the performances, and economize
the energy used to operate such vehicles.
1. Introduction
Despite the great advance in in fixed wing vehicles, flapping wings vehicles still have lots of secrets for engineers, though unsuccessful in many cases and very hard to apply for a man carrying
transportation, fixed wings took a major place in the aeronautics field due to its simplicity, durability, efficiency, and safety at the large scale required to carry humans.
However in the recent years much more researches are conducted on the small scale flapping wings, and this because of the need to a much smaller and more maneuverable aerial vehicle, that’s indeed
needed in many combat or reconnaissance situations especially that more and more operation are conducted in urban environment, which require a real time surveillance inside closed or narrow area ,
something that available unmanned aerial vehicle (UAV) does not provide, which increased the need for MAVs (micro aerial vehicles) and NAVs (Nano aerial vehicles) those vehicles are efficient, agile,
and can carry a camera and other sensing equipment.
Research and design of such vehicle was encouraged by several organizations like DARPA (Defense Advanced Research Projects Agency) [1], that initiate the MAVs project which aimed to solve the
technical barrier that prevent from the realizations of such machine.
2. Definition of MAV
They are affordable, fully functional, militarily capable, small flight vehicles in a class of their own. The definition employed in DARPA’s program limits these craft to a size less than 15 cm
(about 6 inches) in length, width or height. In addition, gross takeoff weight of no more than 100 g.
This physical size puts this class of vehicle at least an order of magnitude smaller than any missionized UAV developed to date.
Four types of configurations [2] exist in this category: fixed-wing, rotary-wing, and flapping wings, the fourth class is without propulsion and is called passive.
Last years experienced some advance in the MAV field and there was several promising flapping wing operated vehicle, like the one Festo Company revealed in March 2011, the SmartBird an autonomous
ultra-light unmanned aerial vehicle with a focus on a better aerodynamics and a high maneuverability [5], and BionicOpter, a dragonfly-like AV a little bit bigger to be called MAV due to its 63 cm
wing span and 44 cm long, it has an ultra-light weight of 175 g, and probably the most amazing thing about it is its 13 degree of freedom.
3. Statement of work
A motion study of a flapping wing operated micro aerial vehicle was performed by simulating the classic motion’s equations of mechanics by variating each time one of the variables approached value
for the drag coefficient of birds and insects were used, all performed in a spherical coordinates system, and this by using the Mathcad software.
Using the same equations, a control system can be implemented electronically to steer the MAV.
Along the numerical investigation, some experimental works was conducted in the wind tunnel facility provided by Riga technical University, which consisted of testing a wing’s drag coefficient and
its variation between the upstroke and the down stroke.
4. Numerical investigation with Mathcad
4.1. Theoretical investigation
The classical mechanic theory describes well the flight of the vehicle, the equations of motion (or equilibrium) which results from Newton’s second law describe the flapping wings vehicle, and other
systems of coordinates than the rectangular were used, because it is best suited for this particular motion in space.
Newton’s 2nd law stipulates that the resultant of all the forces acting on a particle is proportional to the acceleration of the particle:
By applying this basic principal to a body in motion in a plan the equations in a rectangular basis would be:
$\sum {F}_{x}=m{a}_{x},\sum {F}_{y}=m{a}_{y}.$
Fig. 1Spherical coordinates
In spherical coordinates, two angles were used in addition to a distance to specify the position of a particle, radar localization is show in Fig. 1.
The acceleration is:
$\sum {F}_{r}=m{a}_{r}=m\left(\stackrel{¨}{r}-r{\stackrel{˙}{\theta }}^{2}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }^{2}-r{\stackrel{˙}{\phi }}^{2}\right),$$\sum {F}_{\theta }=m{a}_{\theta }=m\left(2\
stackrel{˙}{r}\stackrel{˙}{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\phi -r\stackrel{¨}{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\phi -2r\stackrel{˙}{\theta }\stackrel{˙}{\phi }\mathrm{s}\mathrm{i}\mathrm
{n}\phi \right),$$\sum {F}_{\phi }=m{a}_{\phi }=m\left(2\stackrel{˙}{r}\stackrel{˙}{\phi }+r{\stackrel{˙}{\phi }}^{2}\mathrm{s}\mathrm{i}\mathrm{n}\phi \mathrm{c}\mathrm{o}\mathrm{s}\phi +r\stackrel
{¨}{\phi }\right).$
Therefore the equations of motion in 3D spherical coordinates is:
$a={a}_{r}{e}_{r}+{a}_{\theta }{e}_{\theta }+{a}_{\phi }{e}_{\phi }.$
The equations of motion as seen before and despite their simplicity can be used to simulate the flight of a flapping wing vehicle; we used Mathcad after writing the equations in a syntax where we can
use the increment in time in order to simulate in a certain degree of accuracy the flight.
The original idea of this work is the consideration of the lift force more or less equal to the drag force but, as we observe from nature, when a wing is in the upstroke the animal uses its muscle to
change the shape of the wing so it will have less drag therefore a smaller drag coefficient ${C}_{d1}$ in contrast in the down stroke the shape of the wing is optimal so for the biggest lift force
which is the opposite of the drag force if we change the direction of the axis, therefore a biggest drag coefficient ${C}_{d2}$.
To find the 3D equation a projection of all the expressions on their respective axis.
A study case of each coordinate axis gave the following accelerations equations:
For the radial, coordinate axis:
$\stackrel{¨}{r}=\frac{1}{m}\left[-mg\mathrm{s}\mathrm{i}\mathrm{n}\varphi +mr{\stackrel{˙}{\theta }}^{2}{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\varphi +mr{\varphi }^{2}\right$$+\frac{1}{2}\left[\left
(0.5-0.5sign\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)\right){C}_{d1}+\left(0.5+0.5sign\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)\right){C}_{d2}\right]$$\
bullet S\bullet \rho \bullet {\left(A\omega \mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)}^{2}sign\left(\mathrm{cos}\left(\omega t\right)\right)]\bullet \mathrm{c}\mathrm{o}\mathrm{s}\
beta \mathrm{c}\mathrm{o}\mathrm{s}\left(\varphi -\alpha \right)-b\bullet \frac{dr}{dt}.$
For the polar axis:
$\stackrel{¨}{\theta }=\frac{1}{m\bullet r\bullet \mathrm{c}\mathrm{o}\mathrm{s}\varphi }\left[-2m\stackrel{˙}{r}\theta co\stackrel{˙}{s}\varphi +2m\stackrel{˙}{\theta }\stackrel{˙}{\varphi }\mathrm
{s}\mathrm{i}\mathrm{n}\varphi \right$$+\frac{1}{2}\left[\left(0.5-0.5sign\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)\right){C}_{d1}+\left(0.5+0.5sign\left(\mathrm{c}\mathrm{o}\
mathrm{s}\left(\omega t\right)\right)\right){C}_{d2}\right]$$\bullet S\bullet \rho \bullet {\left(A\omega \mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)}^{2}sign\left(\mathrm{c}\mathrm{o}
\mathrm{s}\left(\omega t\right)\right)]\bullet \mathrm{c}\mathrm{o}\mathrm{s}\beta \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\pi }{2}+\varphi -\alpha \right)-b\bullet r\bullet \mathrm{c}\mathrm{o}\
mathrm{s}\left(\varphi \right)\frac{d\theta }{dt}.$
The azimuth coordinate axis:
$\stackrel{¨}{\varphi }=\frac{1}{m}\left[-mg.\mathrm{c}\mathrm{o}\mathrm{s}\varphi -2m\stackrel{˙}{r}\stackrel{˙}{\varphi }-mr{\stackrel{˙}{\varphi }}^{2}\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm
{c}\mathrm{o}\mathrm{s}\varphi \right$$+\frac{1}{2}\left[\left(0.5-0.5sign\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)\right){C}_{d1}+\left(0.5+0.5sign\left(\mathrm{c}\mathrm{o}\
mathrm{s}\left(\omega t\right)\right)\right){C}_{d2}\right]$$\bullet S\bullet \rho \bullet {\left(A\omega \mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\right)}^{2}]\bullet \mathrm{s}\mathrm{i}\
mathrm{n}\left(\beta \right)-b\bullet r\bullet \frac{d\varphi }{dt}.$
5. Mathcad manipulations
After writing all the equation in Mathcad with the proper syntax, the specifications set by DARPA which are: $m=$100 g.
Several manipulation were performed on different parameters and as we see on Figs. 2, 3, the damping coefficient plays a major role in the stabilization of the flight path.
Fig. 2Vertical coordinates in the spherical case
Fig. 3The flight path in (x, z) plan b1= 0.05
More manipulations were performed on different parameters [7] the results are illustrated in Figs. 4-6.
Fig. 4The vertical coordinates b1= 0.1
Fig. 5The flight path in (x, z) plan b1= 0.1
Fig. 6The flight path in (x, y) plan as helicoid
From the Figs. 4-6 it was obvious the instability occurred in the $z$ axis so we changed the damping coefficient $b$ by ${b}_{1}$ and we varied ${b}_{1}$ separately.
By increasing of the damping coefficient (${b}_{1}=$0.1 and 0.15) we had nearly similar results but far more stable, as it’s illustrated in the last graphs where we can see the path of flight in the
($r$, $\varphi$): with a damping coefficient of 0.1 the MAV reach the limit height smoothly and continue to perform the flight in a circular path.
6. Experimental work
The experimental work consisted simply of measuring the drag force that is produced by an artificial pair of wings covered with feather (Fig. 7) from both sides, and we tried to see if the bend of
the profile as well as the feather orientation affected the drag.
Fig. 7The tested artificial wing
Fig. 8Graph that illustrates the result of the measurements performed on the wing
Results are illustrated in Fig. 8.
It was obvious that there is a difference between the two cases, in the first case where the air flow was against the orientation of the feathers and the curve was against the flow the drag force was
higher, but for the second case we can see that the force was generally less than the first case, such difference can be used in flapping wing vehicle in order to design a lighter and more efficient
7. Results and conclusions
We can summarize the results in the next points:
1) One of the major parameter that influences the MAV flying is the damping coefficients, which have great effect on the stability of the MAV especially.
2) It is possible to use some formulas found in the literature that link the flapping frequency with a group of other parameter.
3) The flapping frequencies have to be defined and limited under certain value to keep up with the reality, and the available mechanism for flapping.
4) The difference between the 2 drag coefficients plays a central role in making the MAV fly just like the amplitude.
• Petricca L., Ohlckers P., Grinde C. Micro- and Nano-air vehicles: state of the art. International Journal of Aerospace Engineering, Vol. 2011, Issue 1, 2011.
• Al-Bahi, I. M. A.-Q. A. A. M. Micro aerial vehicles design challenges: state of the art review. Proceedings of the SAS UAV Scientific Meeting and Exhibition, Jeddah, Saudi Arabia, 2006
• James M., McMichael C. M. S. F. Micro air vehicles – toward a new dimension in flight, 1997.
• Mueller T. J. Aerodynamic measurements at low Reynolds numbers for fixed wing micro air vehicles. Tech. Rep., University of Notre Dame, Notre Dame, The Netherlands, 2000.
• Companu Festo, http://www.festo.com/cms/en_corp/13165.htm.
• Pennycuick C. J. Wing beat frequency of birds in steady cruising flight: new data and improved predictions. The Journal of Experimental Biology, Vol. 199, 1996, p. 1613-1618.
• Abdessemed A. C. Investigation of the flight of a micro aerial vehicle. Master thesis, Riga Technical University, 2013.
About this article
10 October 2014
classical mechanic
Copyright © 2014 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/15375","timestamp":"2024-11-09T00:19:15Z","content_type":"text/html","content_length":"118262","record_id":"<urn:uuid:12e0e849-0fc5-4fe8-a783-fa72f3034a6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00871.warc.gz"} |
Line-Graceful Designs
Main Article Content
In [3], the authors adapted the edge-graceful graph labeling definition into block designs. In this article, we adapt the line-graceful graph labeling definition into block designs and define a block
design $(V,\mathcal{B})$ with $|V|=v$ as line-graceful if there exists a function $f: \mathcal{B} \rightarrow \{0,1,\dots,v-1\}$ such that the induced mapping $f^{+}: V \rightarrow \mathbb{Z}_{v}$
given by $f^{+}(x)=\sum_{A\in \mathcal{B} : x\in A}{f(A)}\pmod{v}$ is a bijection. In this article, the cases that are incomplete in terms of block-graceful labelings, are completed in terms of
line-graceful labelings. Moreover, we prove that there exists a line-graceful Steiner quadruple system of order $2^{n}$ for all $n \geq 3$ by using a recursive construction. | {"url":"http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1998","timestamp":"2024-11-03T00:40:31Z","content_type":"text/html","content_length":"13544","record_id":"<urn:uuid:43941228-7f78-40fc-96bf-7c04cb378b00>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00416.warc.gz"} |
SICP - Solution: Exercise 1.5
September 27, 2018
Exercise 1.5 #
Ben Bitdiddle has invented a test to determine whether the interpreter he is faced with is using applicative-order evaluation or normal-order evaluation. He defines the following two procedures:
(define (p) (p))
(define (test x y)
(if (= x 0)
Then he evaluates the expression
What behavior will Ben observe with an interpreter that uses applicative-order evaluation? What behavior will he observe with an interpreter that uses normal-order evaluation? Explain your
answer. (Assume that the evaluation rule for the special form if is the same whether the interpreter is using normal or applicative order: The predicate expression is evaluated first, and the
result determines whether to evaluate the consequent or the alternative expression.)
Solution #
The key of the exercise is to notice that (define (p) (p)) defines a function named p that doesn’t have any formal parameter and that evaluates to itself.
We also need to keep in mind how the interpreter evaluates combinations:
To evaluate a combination, do the following:
1. Evaluate the subexpressions of the combination.
2. Apply the procedure that is the value of the leftmost subexpression (the operator) to the arguments that are the values of the other subexpressions (the operands).
And how the interpreter evaluates compound procedures:
To apply a compound procedure to arguments, evaluate the body of the procedure with each formal parameter replaced by the corresponding argument.
An interpreter that uses applicative-order evaluation will “evaluate the arguments and then apply”. When this kind of interpreter evaluates the expression (test 0 (p)), it will perform the evaluation
process on each element of the combination.
It starts by evaluating the symbol test that returns the compound procedure previously defined.
Then it evaluates 0 that returns the numeral O.
Finally, it tries to evaluate the argument (p) and this is where things are getting tricky. When the interpreter evaluates the expression (p), we can look at the rules mentioned above and follow step
by step:
1. “Evaluate the subexpressions of the combination.” The symbol p will evaluate to the compound procedure defined and since there is no other subexpression here, there is nothing more to do.
2. “Apply the procedure that is the value of the leftmost subexpression (the operator) to the arguments that are the values of the other subexpressions (the operands).” The operator is a compound
procedure f and there is no argument.
3. “To apply a compound procedure to arguments, evaluate the body of the procedure with each formal parameter replaced by the corresponding argument.” Since there is no formal parameter in this
case, the body of the procedure that needs to be evaluated is just (p), by its definition. But the interpreter has reached a step that, in order to interpret (p), it has to evaluate (p)
recursively. The interpreter is in an infinite loop!
In conclusion, evaluating (test 0 (p)) with an interpreter that uses applicative-order evaluation will result in an infinite loop.
When an interpreter uses normal-order evaluation, it will “fully expand and then reduce”. In this model, the interpreter will not evaluate the operands until their values are actually needed. In that
case, the expression (test 0 (p)) will be expanded to:
Since the predicate (= 0 0) evaluates to #t in the if expression, then the alternative (p) will not be evaluated and the result will be: | {"url":"https://sicp-solutions.net/post/sicp-solution-exercise-1-5/","timestamp":"2024-11-12T02:07:13Z","content_type":"text/html","content_length":"14726","record_id":"<urn:uuid:b8409492-dd73-480d-8288-3761344239ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00330.warc.gz"} |
Horizontal and Vertical Polarization - PDF Free Download
Horizontal and Vertical Polarization: Task-Specific Technological Change in a Multi-Sector Economy∗ Sang Yoon (Tim) Lee†
Yongseok Shin‡
September 17, 2016 (Preliminary)
Abstract We develop a novel framework that integrates an economy’s skill distribution with its occupational and industrial structure. Individuals become a manager or a worker based on their
managerial vs. worker skills, and workers further sort into different occupations ranked by skill content. Our theory dictates that faster technological progress for middle-skill worker tasks not
only raises the employment shares and relative wages of lower- and higher-skill worker occupations (horizontal polarization), but also raises those of managers over workers as a whole (vertical
polarization). Both dimensions of polarization are faster within sectors that depend more on middle-skill tasks and less on managerial tasks. The TFP of such sectors grows faster endogenously, and,
with complementarity across sectoral goods, their employment and value-added shares decrease (structural change). Our quantitative analysis shows the importance of task-specific technological
progress—especially for middle-skill tasks—for understanding changes in the sectoral, occupational, and organizational structure of the U.S. economy since 1980. Technological progress was fastest for
occupations that embody routine-manual tasks but not interpersonal skills. Finally, in the limiting growth path, middle-skill occupations vanish but all sectors coexist.
Previously circulated as “Managing a Polarized Structural Change.” University of Mannheim and Toulouse School of Economics:
[email protected]
. ‡ Washington University and Federal Reserve Bank of St. Louis:
[email protected]
. †
We develop a novel framework that integrates the distribution of individual skills in an economy with its occupational and industrial structure. It enables us to analyze how changes in wage and
employment shares across occupations and across industrial sectors are interrelated and reinforce each other, providing a comprehensive view on the economic forces that shape the sectoral,
occupational, and organizational structure of an economy. In our model, individuals are heterogeneous in two-dimensional skills, i.e., managerial talent and worker human capital, based on which they
become a manager or a worker. Workers further select into low-, middle- or high-skill worker occupations based on their worker human capital. Managers combine managerial and worker tasks (i.e.,
occupation-level output) to produce sector-level output, and sectors differ in how intensively different tasks are used for production. We assume skills are sector-neutral, so given an occupation,
all individuals are indifferent across sectors. If different tasks are complementary for production, faster technological progress for middle-skill worker tasks relative to the others leads to: (i)
rising employment shares and wages of low- and high-skill occupations relative to middle-skill occupations, commonly known as job and wage polarization; (ii) rising employment share and wage of
managers relative to workers as a whole, which we call vertical polarization to distinguish from the horizontal polarization across workers; (iii) faster horizontal and vertical polarization within
those sectors that depend more on middle-skill tasks and less on managerial tasks; and (iv) faster endogenous growth of the total factor productivity (TFP) of such sectors, shrinking their employment
and value-added shares if sectoral goods are complementary (i.e., structural change). The last model prediction merits further discussion. First, because sector-level TFP is endogenously determined
by equilibrium occupational choices, task-specific technological progress, although sector-neutral, has differential impact across sectors. Second, when the employment shares of the sectors that rely
less on middle-skill workers and more on managers rise, they reinforce the overall degree of horizontal and vertical polarization. However, this structural change has no effect whatsoever on
within-sector polarization. Third, in the asymptotic dynamics of the structural change driven by task-specific technological progress, only occupations with slow technological progress remain and all
others vanish, but all sectors exist. This is in contrast to many theories of structural change that rely on sector-specific forces: the shift of production factors from one sector to another
continues as long as those forces exist, so that shrinking sectors vanish in the limit. In our model, task-specific technological progress is sectorneutral and only affects sectors indirectly through
how they combine tasks. Once the
employment shares of the occupations with faster progress become negligible, structural change ceases even as productivity continues to grow differentially across occupations. Empirically,
predictions (i) through (iv) are salient features of the changes the U.S. economy underwent since 1980: (i) job and wage polarization is well-documented in the literature, e.g., Autor and Dorn
(2013), which here we call horizontal polarization; (ii) using the same data, we highlight vertical polarization as new fact; and (iii) we verify that manufacturing is more reliant on middle-skill
workers and less on managers than services, and also that both dimensions of polarization are indeed faster within manufacturing than in services. In addition, we provide evidence from
establishment-level data that corroborates the faster vertical polarization in manufacturing: manufacturing establishments shrank faster when measured by employment and grew faster by value-added
than those in services, which is consistent with the model prediction once we assume that the number of managers per establishment is stable over time. Finally, (iv) it has long been understood that
the faster growth of manufacturing TFP is an important driver of structural change from manufacturing to services. Consequently, our model shows that all these empirical facts have a common cause:
faster technological progress for middle-skill tasks than the rest. In our full quantitative analysis with a richer set of occupations, we confirm that task-specific technological progress, rather
than exogenous sector-specific TFP growth, accounts for the observed growth of sector-level TFP. More broadly, task-specific technological progress—especially for middle-skill tasks—is important for
understanding the changes in the sectoral, occupational, and organizational structure in the United States over the last 35 years. The natural next question is what can explain such differential
productivity improvements across tasks. To explain what we call horizontal polarization, Autor and Dorn (2013), Goos et al. (2014) and others hypothesized that “routinization”—i.e., faster
technological advancement for tasks that are more routine in nature (which tend to be middle-skill tasks in the data)—reduced the demand for the middle-skill occupations. We provide a sharper answer.
When we consider detailed characteristics of occupations, we find that the task-specific technological progress we quantify is much more strongly correlated with the routine-manual index and the
inverse of the manualinterpersonal index across occupations than with the routinization index commonly used in the polarization literature. In other words, the technological progress of the last
three or four decades heavily favored those manual tasks that are repetitive in nature and require few interpersonal skills.
Related Literature Our model is a first attempt to provide a framework that links the occupational structure of an economy to sectoral aggregates. In particular,
we can use micro-estimates of occupational employment and wages, which have been studied extensively in labor economics, to study how occupational choices aggregate up to macro-level sectoral shifts.
This is of particular empirical relevance for the U.S. and other advanced economies. The 1980s marks a starting point of rising labor market inequality, much of which can be attributed to
polarization. It was also the starting point of a clear rise in manufacturing productivity (Herrendorf et al., 2014) and the rise of low-skill service jobs (Autor and Dorn, 2013). The main finding in
this regard is that task-specific productivity growth and micro-level elasticities in the labor market can be of first-order importance for understanding the sector-level and aggregate economic
growth. Task-based models are found in Acemoglu and Autor (2011) and Goos et al. (2014) to explain polarization in employment shares, but neither addresses wage polarization. None of them relate
polarization to structural change across macroeconomic sectors, nor treat managers as an occupation that is qualitatively different from workers. The manager-level technology in our model is an
extension of the span-of-control model of Lucas (1978), in which managers hire workers to produce output. However, unlike all existing variants of the span-of-control model, in our model managers
organize tasks instead of workers. That is, instead of deciding how many workers to hire, they decide on the quantities of each task to use in production, and for each task, how much skill to hire
(rather than how many homogeneous workers). Moreover, rather than assuming a Cobb-Douglas technology between managerial talent and workers, we assume a CES technology between managerial talent and
tasks.1 Our model is closely related to the rapidly growing literature in international trade that use assignment models to explain inequality between occupations and/or industries (Burstein et al.,
2015; Lee, 2015). The majority of such models follow in the tradition of Roy: all workers have as many types of skills as there are available industry/occupation combinations, and select themselves
into the job they in which they have a comparative advantage. To make the model tractable, they typically employ a Fr´echet distribution which collapses the model into an empirically testable set of
equations for each industry and/or occupation pair. While the manager-worker division in our model is also due to Roy-selection, but the horizontal sorting of workers into tasks is qualitatively
different. In addition, we assume only 2-skill types, which is arguably more suitable for studying the endogenous formation of skills. Since Ngai and Pissarides (2007), most production-driven models
of structural change rely on exogenously evolving sectoral productivities. Closer to our model is 1
Starting with the standard span-of-control model, we incorporate (i) non-unitary elasticity between managers and workers, (ii) heterogeneity in worker productivity as well as in managerial
productivity, (iii) multiple worker tasks or occupations, and (iv) multiple sectors.
Acemoglu and Guerrieri (2008), in which the capital-intensive sector (in the sense of having a larger capital share in a Cobb-Douglas technology) vanishes in the limiting balanced growth path. While
sectors in their model differ in how intensively they use capital and labor, in our model they differ in how intensively they use different tasks. By contrasting different types of labor, rather than
capital and labor, we can connect structural change—which happens across sectors—to labor market inequality across occupations. Moreover, as mentioned above, unlike any other existing explanation of
structural change, our model implies that it is certain occupations, not broadly-defined sectors, that may vanish in the limit. While we are the first to build a model in which individuals with
different skills sort themselves into different occupations, which in turn are used as production inputs in multiple sectors,2 there have been recent attempts such as Buera and Kaboski (2012) and
Buera et al. (2015), in which multiple sectors use different combinations of heterogeneous skills as production inputs. One important distinction is that we separate worker human capital (continuous
distribution) and skill levels of a task or occupation (discrete). More important, our driving force (i.e., routinization) is specific to a task or occupation, not to workers’ human capital levels.
This way, not only can we address broader dimensions of wage inequality and use micro-labor estimates to discipline our model,3 but also represent sectoral TFPs endogenously by aggregating over
equilibrium occupational choices, rather than relying on exogenously evolving worker-skill specific productivities. The rest of the paper is organized as follows. In section 2, we summarize the most
relevant empirical facts: horizontal and vertical polarization in the overall economy, the faster speed of polarization within manufacturing than in services, and structural change. In section 3, we
present the model and solve for its equilibrium. In section 4 we perform comparative statics demonstrating that faster technological progress for the middle-skill worker tasks leads to horizontal and
vertical polarization, and ultimately to structural change. Section 5 characterizes the limiting behavior of the dynamic economy. Section 6 calibrates an expanded version of the theoretical model to
data from 1980 to 2010, and quantifies the importance of task-specific technological progress. Section 8 concludes. 2
B´ ar´ any and Siegel (2016) build a model in which occupations are tied to sectors. Autor et al. (2006); Acemoglu and Autor (2011) show that residual wage inequality controlling for education groups
is much larger than between-group inequality. 3
LServ Pro Mgr
Sales Tech
Admin Mech Mach
Mine Trans Mine
20 40 60 80 Skill percentile (1980 occupational mean wage) COC 1−digit
smoothed percentile
20 40 60 80 Skill percentile (1980 occupational mean wage) COC 1−digit
(a) Employment Polarization
smoothed percentile
(b) Wage Polarization
Fig. 1: Job and Wage Polarization, 30 years.
When obtaining our empirical moments, we subsume mining and construction into manufacturing, and government into services, and the agricultural sector is dropped. That is, aggregate output and
capital correspond to the moments computed without agriculture, as is the case for employment shares.4 1. Jobs and wages have polarized, figure 1 Source: U.S. Census and ACS 2010, replicates and
extends Autor and Dorn (2013). Occupations are ranked by their 1980 mean wage. 2. Routinizable jobs correlate with structural change, figure 2. Source: U.S. Census and ACS 2010. Left panel replicated
from Autor and Dorn (2013), right panel shows change in manufacturing employment by occupation cell. 3. Manufacturing has relatively higher share of intermediate occupations, of which low-to-middle
skill jobs have been relatively shrinking, figure 3. The right panel compares employment polarization by manufacturing and services, separately. Source: U.S. Census and ACS 2010. 4. Structural change
continues in the U.S., figure 4. Source: BEA (value-added), NIPA Table 6 (persons involved in production). Manufacturing includes mining and construction. 5. Both employment share and relative wages
for managers are increasing: figure 5. See appendix for definition of managers in the census. 4
Agriculture shares are only about 2% of both total output and employment, and has been more or less flat since the 90s.
Mgr Sales
Pro Mine
LServ Mgr Pro
20 40 60 80 Skill percentile (1980 occupational mean wage) COC 1−digit
20 40 60 80 Skill percentile (1980 occupational mean wage)
smoothed percentile
COC 1−digit
(a) Routinization
smoothed percentile
(b) Manufacturing Employment
Fig. 2: Routinization and Structural Change, 30 years.
LServ Sales
Mech Mine
Trans Tech MSpt
Admin Pro Sales
20 40 60 80 Skill percentile (1980 occupational mean wage) COC 1−digit
smoothed percentile
(a) Manufacturing
20 40 60 80 Skill percentile (1980 occupational mean wage) manufacturing, COC 1−digit
, smoothed percentile
services, COC 1−digit
manufacturing, smoothed percentile
(b) Employment Polarization by Sector
Fig. 3: Manufacturing employment shares across skill percentiles.
0% 1970
0% 1970
(a) GDP, Value-Added
(b) Aggregate Employment Shares
12 13 14 employment share of management (%)
1.9 multiple of mean worker wage 1.5 1.6 1.7 1.8
Year non−management ratio
mean hourly wage (2005 USD) 20 25 30 35
Fig. 4: Structural Change, 1970-2013.
management Manager Wage
(a) Wage Levels
Manager Employment Share
(b) Relative Employment and Wages
Fig. 5: Managers vs Workers
2.4 1.4
multiple of mean worker wage 1.6 1.8 2 2.2
employment share of management (%) 8 10 12 14 1980
(a) Manager Employment Share
(b) Relative Manager Wage
Fig. 6: Managers by Sector
(a) Average No. of Workers
2010 Total
(b) Average Value-added Output
Fig. 7: Establishment Size by Sector 6. Moreover, this occurs faster in the manufacturing sector: figure 6. 7. Average size of establishments have been rising, but shrinking in the manufacturing
sector: figure 7. Source: BDS 1980-2013. 8. Manager definition. Middle-management has been growing (relative to top management): figure 8
There are a continuum of individuals each endowed with two types of skill, (h, z). Human capital, h, is used to produced tasks. Management, z, is a special skill for organizing tasks. WLOG we assume
that the mass of individuals is 1, with associated
100 95 Percent (%) 90 85 80 1980
Census Year Workers
SE Mgr, broad
SE Mgr, narrow
SE Top Mgr
Top Mgr
Mgr, narrow
Mgr, broad
Fig. 8: Establishment Size by Sector distribution function µ. There are 2 sectors i ∈ {m, s}.5 In each sector, goods are produced in teams. A single manager uses her own skill and physical capital to
organize three types of tasks j ∈ {0, 1, 2} (e.g., low-, medium-, high-skill occupations; or manual, routine, abstract tasks). Each task requires both physical and human capital, and how much of each
is allocated to each task is decided by the manager. Aggregating over the goods produced by all managers within a sector yields total sectoral output. Within a sector, a better manager can produce
more goods with the same amount of tasks, but task intensities may differ across sectors. Specifically, we assume that 1 ω ω−1 ω−1 ω−1 1 ω , (1a) yi (z) = ηi xz (z) ω + (1 − ηi ) ω xh (z) ω xz (z) =
Mz k α z 1−α ,
σ σ−1 2 X 1 σ−1 σ xh (z) = νij τij (z) σ
Z τij (z) = Mj
tj (k, h)dµ, hij (z) α ¯ 1−α
t0 (k, h) = k h with
t1 (k, h) = k α h1−α
t2 (k, h) = k α (h − χ)1−α ,
νij = 1. The tj (·)’s are the amounts of task output produced by an individual
with human capital h and physical capital k, the latter of which is allocated by the manager. Integrating over individual task outputs over the set of workers hired by a manager of skill z for task j
in sector i, hij (z), yields a task aggregate τij (z). The substitutability between tasks is captured by the elasticity parameter σ, and ω captures the elasticity between all workers and managers. 5
In our application, the two sectors stand in for ”manufacturing” and ”services,” respectively. However, I analytical model can be extended to incorporate any countably finite number N of sectors; we
use the subscripts m and s to avoid confusing them with the subscripts for tasks.
For task (or occupation) 0, a worker’s own human capital is irrelevant for produc¯ This is to capture manual jobs that tion: all workers’ effective skill input becomes h. do not depend on skills. For
task 2, some of your skills become useless and effective skill input becomes h − χ. This is to capture analytic jobs, for which lower levels of skill are redundant. We will refer to the managerial
task as “task z,” which is vertically differentiated from tasks j ∈ {0, 1, 2}, which are horizontally differentiated. The Mj ’s, j ∈ {0, 1, 2, z}, capture task-specific TFP’s, which are
sector-neutral. Several points are in order. As is the case with most models of sorting workers into tasks, the worker side of our model can be viewed as a special case of Costinot and Vogel (2010).
However, we model managers and have more than one sector. In contrast to Acemoglu and Autor (2011), we have a continuum of skills rather than tasks, and a discrete number of tasks rather than skills.
While the implications are comparable, our formulation is more suitable for exploring employment shares across tasks (which are discrete in the data). The model is also comparable to Goos et al.
(2014), who show (empirically) that relative price changes in task-specific capital, representing routinization, can drive employment polarization. However, they do not model skill and hence cannot
explain wage polarization. Now let Hij denote the set of individuals working in sector i ∈ {m, s} on task j ∈ {0, 1, 2}. We can define - Hi = ∪j∈{0,1,2} Hij : set of individuals who work as workers
in sector i ∈ {m, s}, - Hj = ∪i∈{m,s} Hij : set of individuals who work as workers on task j ∈ {0, 1, 2}, - H = ∪i∈{m,s} Hi = ∪j∈{0,1,2} Hj : set of all individuals who work as workers, - Z = Zm ∪ Zs
: set of individuals who work as managers in sector i ∈ {m, s}. Output in each sector is then Z Yi = Ai yi (z)dµ,
where Ai , i ∈ {m, s} is an exogenous, sector-level productivity parameter. Final goods are produced by combining output from both sectors according to a CES aggregator:
Y = G(Ym , Ys ) = γm Ym
+ γ s Ys
where γM + γS = 1 and we will assume < 1.6 6 The estimated between the manufacturing and service sector (broadly defined) is close to zero, as we show in section 6.2.
Planner’s Problem
We assume complete markets for solve a static planner’s problem. A planner allocates aggregate capital K and all individuals into sectors i ∈ {m, s} and tasks j ∈ {0, 1, 2, z}. The objective is to
maximize current output (3) subject to (1))-(2) and Z Z X X kmj (z) + ksj (z) dµ K = Km + Ks ≡ j∈{0,1,2,z} j∈{0,1,2,z} Zm Zs Z Z Hij ≡ hdµ = hij (z)dµ, j ∈ {0, 1, 2}, Hij
where Ki is the amount of capital allocated to sector i, Hij the total amount of human capital allocated to task j in sector i, and (kij (z), hij (z)) is the amounts of physical and human capital
allocated to task j in sector i under a manager with skill z, where j ∈ {0, 1, 2, z} for k and j ∈ {0, 1, 2} for h. For existence of a solution, we assume that Assumption 1 The population means of
both skills are finite, that is, Z Z zdµ < ∞, hdµ < ∞. and Assumption 2 There exists a strictly positive mass of individuals who do not lose all of their h-skill by working in task 2, i.e. µ(h > χ) >
0. The following assumption is needed for uniqueness: Assumption 3 The measure µ(z, h) is continuous and has a connected support on (h, z) ∈ [0, hu ) × [0, zu ), where xu ≤ ∞ is the upperbound of
skill x ∈ {h, z}; i.e. µ(h, z) > 0 on (0, 0) ≤ (h, z) < (hu , zu ) ≤ ∞. Along with assumption 2, this implies χ < hu . Before showing existence and uniqueness of the solution, we first characterize
the solution in the following order: 1. Characterize optimal physical capital allocations across tasks within a sector. 2. Characterize optimal human capital (h) allocations across tasks within a
sector. 3. Characterize optimal labor (manager-worker) allocations within a sector. 4. Solve for optimal capital and labor allocations across sectors.
Capital allocation within sectors Thanks to the HD1 assumptions, we can write sectoral technologies as ω 1 ω−1 ω−1 ω−1 1 ω ω ω , Yi = Ai ηi Xiz + (1 − ηi ) ω Xih α 1−α Xiz = Mz Kiz Zi ,
σ σ−1 X 1 σ−1 σ σ , = νij Tij
where Xih is a sectoral task aggregate and 1−α α ¯ α 1−α Ti0 = M0 Ki0 hµ(Hi0 ) , Ti1 = M1 Ki1 Hi1 , 1−α
α Ti2 = M2 Ki2 [Hi2 − χµ(Hi2 )]
Given sectoral capital Ki , the planner equalizes marginal product across tasks: M P Ki0 = M P Ki1 = M P Ki2 M P Ti0 · αTi0 M P Ti1 · αTi1 M P Ti2 · αTi2 ⇒ = = Ki0 Ki1 Ki2 1 σ−1 Ti1 σ M P Ti1 · Ti1
Ki1 νi1 σ · ⇒ = ≡ πi1 = M P Ti0 · Ti0 Ki0 νi0 Ti0 1 σ−1 M P Ti2 · Ti2 Ti2 σ Ki2 νi2 σ · , = ≡ πi2 = M P Ti1 · Ti1 Ki1 νi1 Ti1
(7a) (7b)
where M P Tij is the marginal product of Tij w.r.t. Xi , and πij is the capital input ratio in tasks j ∈ {1, 2} and j − 1. Due to the Cobb-Douglas assumption, πij divided by task output ratios is the
marginal rate of technological substitution (M RT S) between tasks j and j − 1; furthermore, πij divided by either factor input ratios in tasks j and j − 1 measures the M RT S of that factor between
tasks j and j − 1. (For capital, this is equal to 1.) Given (7) we can write 1
Xih = νi0σ−1 (1 + πi1 + πi1 πi2 ) σ−1 Ti0 . {z } |
Of course, M P K must also be equalized across the managerial task and the rest: M P Kiz = M P Ki0 M P Xiz · αXiz M P Xih · M P Ti0 · αTi0 ⇒ = Kiz Ki0 ⇒
M P Xiz · Xiz Kiz = M P Xih · M P Ti0 · Ti0 Ki0
πiz =
ηi 1 − ηi
1 ω−1 ω Xiz ω · · Πih , Xih (9)
which then allows us to write, using (8), 1
Yi = Ai (1 − ηi ) ω−1 [1 + πiz /Πih ] ω−1 Xih 1
σ−1 = Ai (1 − ηi ) ω−1 [1 + πiz /Πih ] ω−1 νi0σ−1 Πih Ti0
Sorting skills across worker tasks within sectors Given the sectoral production function (5), we can now decide how to allocate worker skills h to worker tasks j ∈ {0, 1, 2}. Since skill doesn’t
matter in task 0 and some becomes irrelevant in task ˆ 1, h ˆ 2) 2, there is positive sorting of workers into tasks; i.e. there will be thresholds (h ˆ 1 work in task 0 and those with h > h ˆ 2 work
in task 2. Note s.t. all workers with h ≤ h that these thresholds must be equal across sectors, hence are not subscripted by i. For each threshold, it must be that the marginal product of the
threshold worker is equalized in either task: (1 − α)Ti1 ˆ (1 − α)Ti0 ¯ · h = M P Ti1 · · h1 , M P Ti0 · ¯ Hi1 hµ(Hi0 ) (1 − α)Ti1 ˆ (1 − α)Ti2 ˆ 2 − χ) M P Ti1 · · h2 = M P Ti2 · · (h Hi1 Hi2 − χµ
(Hi2 ) using assumption 3, so ¯ ˆ 1 = h1 Li1 , h πi1 Li0
¯ 2 − χ)Li2 χ (h = ¯ 1 Li1 , ˆ2 πi2 h h
¯ j ≡ Hij /Lij ; that is, we are assuming where Lij ≡ µ(Hij ) and h ¯ j , z¯), are equal Assumption 4 The means of skills in tasks j ∈ {0, 1, 2, z}, that is, (h across sectors i ∈ {m, s}. This is an
assumption is needed because we assume discrete tasks; it can be thought of as the limit of vanishing supermodularity within segments of a continuum of tasks. Assumption 1 also guarantees that all
objects are finite and well-defined. Using (11) we can reformulate (7) as !1−α σ−1 ˆ νi1 M1 h1 πi1 = · , ¯ νi0 M0 h
" #σ−1 M2 νi2 χ 1−α · πi2 = 1− . ˆ2 νi1 M1 h
Sorting managers and workers within a sector Now we know how to allocate Ki , Hi within a sector, but we still need to know how to divide individuals into managers and workers; that is, determine Zi
∪ Hi given a mass of individuals within a sector. Since individuals are heterogeneous in 2 dimensions, the key is to get a cutoff rule z˜j (h) s.t. for every h, individuals with z above z˜j (h)
become managers and below become workers. Since the h-skill is used differently across tasks, we need to get 3 such rules for each sector; however the rule must be identical across sectors. ˆ 1 ,
this rule is simple. For these workers, h does not matter, so z˜0 (h) = zˆ, For h ≤ h i.e., is constant. The constant is chosen so that the marginal product of the threshold manager is equalized in
either task: M P Xiz ·
(1 − α)Ti0 ¯ (1 − α)Xiz · z˜0 (h) = M P Xih · M P Ti0 · ¯ ·h Zi hµ(Hi0 )
z˜0 (h) = zˆ =
z¯Liz Zi = , πiz Li0 πiz Li0
where Liz = µ(Zi ) and z¯ = Zi /Liz (which is equal across sectors by assumption 4). Then from (9) we can write " #ω−1 1−ω σ−ω ηi νi0σ−1 Mz zˆ 1−α σ−1 πiz = · . · Π ih ¯ 1 − ηi M0 h
ˆ 1, h ˆ 2 ], the rule is linear: For h ∈ (h (1 − α)Ti1 (1 − α)Xiz · z˜1 (h) = M P Xih · M P Ti1 · ·h Zi Hi1 z˜1 (h) πi1 z¯Liz πi1 Zi = = φ1 = ¯ 1 Li1 . h πiz Hi1 πiz h
M P Xiz · ⇒
ˆ 2 , the rule is affine: and finally for h > h (1 − α)Xiz (1 − α)Ti2 · (h − χ) · z˜2 (h) = M P Xih · M P Ti2 · Zi Hi2 − χµ(Hi2 ) z˜2 (h) πi1 πi2 Zi πi1 πi2 z¯Liz = φ2 = = ¯ 2 − χ)Li2 . h−χ πiz (Hi2
− χLi2 ) πiz (h
M P Xiz · ⇒ Observe that
ˆ 1, zˆ = φ1 h
ˆ 2 = φ1 /φ2 , 1 − χ/h
ˆ 1, h ˆ 2 , zˆ) completely determine the φj ’s, and all objects are well defined given so (h assumption 2, since all tasks are essential.
Sectoral production function and allocation across sectors Equations (11)-(14) completely describe the task thresholds. What is important here is that all these thresholds are determined
independently of the amount of physical capital. To see this more clearly, rewrite (10) to obtain ω
σ−1 α 1−α M0 Ki0 Li0 Yi = Ai ψi · [1 + πiz /Πih ] ω−1 Πih 1
¯ 1−α ψi ≡ (1 − ηi ) ω−1 νi0σ−1 h and furthermore since Ki = Ki0 (Πih + πiz ) | {z }
# ˆ2 1 − χ/ h ˆ 1 /h ¯ 1 )πi1 + · π π + (ˆ z /¯ z )πiz , Li = Li0 1 + (h ¯ 2 − χ)/h ˆ 1 i1 i2 (h | {z } "
Π Li
we obtain ω−σ
(ω−1)(σ−1) ω−1 ΠK Yi = M0 · Ai ψi · Πih i {z |
Φi : TFP
α 1−α Πα−1 . Li Ki Li }
Note that sectoral TFP, Φi , can be decomposed into 3 parts: M0 , that is common across both sectors, Ai ψi , which is sector-specific but exogenous, and the parts determine by ˆ 1, h ˆ 2 , zˆ). (Πih
, ΠK , ΠL ), which is sector-specific and endogenously determined by (h i
Furthermore, since the thresholds depend only on the relative masses of individuals across tasks within a sector, they do not depend on the employment size of the sector (nor capital). Hence even as
Ki or Li changes, these thresholds do not as long as the distribution of skills remains constant. Sectors only differ in how intensely they use each task, i.e., the mass of individuals allocated to
each task. As usual, these masses are determined so that the M P K and M P L are equalized across sectors: Ks = κ≡ Km
γs γm
Ys Ym
Ls Lm
where κ is capital input ratios between sectors m and s.
Existence and Uniqueness
Having characterized the optimal allocation (which is identical to the equilibrium allocation), we can now establish existence and uniqueness: Theorem 1 Under assumptions 1-4, the solution to the
planner’s problem exists and is unique. Proof: First define a renormalization of ΠLi : ¯ (σ−1)(1−α) ΠL = ˜ L ≡ (1 − ηi )νi0 M σ−1 h Π i i 0
Vij ,
where the weights Vij are ¯ α+σ(1−α) h , ¯ h ˆ α+σ(1−α) h Vi1 ≡ (1 − ηi )νi1 M1σ−1 · 1 ¯ , h1 h iα+σ(1−α) ˆ 1 (1 − χ/h ˆ 2) h Vi2 ≡ (1 − ηi )νi2 M2σ−1 · , ¯2 − χ h Vi0 ≡ (1 − ηi )νi0 M0σ−1 ·
˜ σ−1 Mzω−1 · Viz ≡ ηi Π ih
zˆα+ω(1−α) z¯
(20a) (20b)
(20c) (20d)
˜ ih is a renormalization of Πih : and Π ¯ (σ−1)(1−α) · Πih . ˜ ih ≡ νi0 M σ−1 h Π 0 Note that the only differences in the Vij ’s across sectors i comes from the task intensity parameters νij , ηi
(since Πih is also a function only of the νij ’s in equilibrium). The total amount of labor in each task j can be expressed as X Vij · Li , where Lm = 1/(1 + κ), Ls = κ/(1 + κ) Lj = ˜L Π
for j ∈ {0, 1, 2, z}. This system of equations that solves the planner’s problem are also the equilibrium market clearing conditions; the LHS is the labor supply and RHS ¯ j Lj = Hj , z¯Lz = Z, P Lj
= 1 and κ = κ(h ˆ 1, h ˆ 2 , zˆ) is demand for each task j. Since h j
ˆ 1, h ˆ 2 , zˆ) from (19), the solution to (h ˆ 1, h ˆ 2 , zˆ) is found from the system a function of (h of three equations σ−ω
" (1 − ω) log Mz + log Z − log log zˆ =
˜ σ−1 ηi Π ih i Π ˜L i
· Li (22a)
α + ω(1 − α) (1 − σ) log M1 + log H1 − log
ˆ1 = log h
(1−ηi )νi1 ˜L Π i
· Li (22b)
α + σ(1 − α)
ˆ 2 = log(h ˆ 2 − χ) log h (1 − σ) log
M2 M1
+ log
H2 −χL2 H1
− log
(1−ηi )νi2 i ˜L Π
· Li
(1−ηi )νi1 i ˜L Π i
α + σ(1 − α) (22c)
where ˆ 1, h ˆ 2 , zˆ) = Z(h
ˆ1 h
ˆ 1, h ˆ 2 , zˆ) = H2 (h
ˆ2 h
+ zˆ
ˆ 1, h ˆ 2 , zˆ) = H1 (h
ˆ2 h
Z ˆ
Zh1 ˆ2 h
+ ˆ1 h
ˆ 1 )·h φ1 (ˆ z ,h
# zdF (z|h)dG(h)
ˆ2 h
ˆ 1 ,h ˆ 2 ,ˆ φ2 (h z )·(h−χ)
ˆ 1 , zˆ) · h|h dG(h) hF φ1 (h
ˆ 1, h ˆ 2 , zˆ) · (h − χ)|h dG(h) hF φ2 (h
and G(h) is the marginal distribution of h, and F (z|h) the distribution of z conditional on h; that is ˜ z˜) = µ(h,
˜ Z z˜ h
dF (z|h)dG(h).
ˆ 1 ∈ [0, h ˆ 2 ), and Existence is straightforward. Note that the domain of zˆ ∈ [0, zu ), h ˆ 2 ∈ [h ˆ 1 , hu ). Hence holding other variables fixed, h
· Li
H s1
Zs Zm
H m1
H s2 Hm2
H s0 H m0
zˆ (0,0)
ˆ2 h
ˆ1 h
Fig. 9: Equilibrium
1. as zˆ → 0, LHS of (22a) approaches −∞, while RHS approaches ∞. Conversely, as the LHS approaches log zu , RHS approaches −∞. ˆ 1 → 0, LHS of (22b) approaches −∞, while RHS remains finite.
Conversely, 2. as h ˆ 2 , RHS approaches −∞. as LHS approaches log h ˆ 2 → max{h ˆ 1 , χ}, LHS of (22c) remains finite, while RHS approaches −∞. 3. as h Conversely, as LHS approaches log hu , RHS
becomes larger than LHS. Assumption 3 ensures that all RHS’s are continuous; hence a solution exists. The assumption also ensures that the mapping in the RHS is monotone; so for any guess of the two
other thresholds, each threshold is found uniquely as a function of the ˆ 1 , log h ˆ 2 ) is a other two. That is, the RHS of system (22) with respect to (log zˆ, log h contraction; so the fixed
point to the system is unique.
The equilibrium skill allocation is depicted in figure 10. The thresholds determine the tasks, and employment is split across sectors while preserving the means for each task. The different masses of
sectoral employment across tasks are due to the task intensity parameters νij , ηi .
Equilibrium wages and prices
Since there are no frictions, the planner’s allocation coincides with a competitive equiˆ 1, h ˆ 2 , zˆ) gives all the information needed to derive equilibrium. Hence, the solution (h librium prices.
The price of the final good can be normalized to 1: 1 1− 1− P = 1 = γm p1− , m + γs p s
pi = [Yi /γi Y ]− .
Sectoral output prices can be obtained by applying the sectoral output in (16)-(18) in (23). The interest rate R is given either by the dynamic law of motion for aggregate
capital, or fixed in a small open economy. So w0 is w0 =
1 − α Ki0 R · ¯ · α Li0 h
where the capital-labor ratio can be found from (16)-(17). Given w0 , indifference across tasks for threshold workers imply ¯ = w1 h ˆ 1, w0 h ⇒
ˆ 2 = w2 ( h ˆ 2 − χ) w1 h
ˆ 1 /h, ¯ w0 /w1 = h
ˆ 2. w1 /w2 = 1 − χ/h
and likewise, the threshold manager implies a “managerial efficiency wage” ¯ wz zˆ = w0 h
¯ w0 /wz = zˆ/h.
Hence, relative wages for task j are simply the inverse of the thresholds.
Comparative Statics
The sectoral technology representation (18) implies that this model has similar implications as Ngai and Pissarides (2007): the sector with the larger TFP shrinks. The major difference is that these
TFP’s are endogenous. What is more interesting is the implications of growth in task-specific TFP’s— this is equivalent to the price of task-specific capital falling in Goos et al. (2014)—or changes
in the distribution for skill. In particular, we are interested in the effect of routinization, which we model as an increase in the task 1’s TFP, M1 . This is illustrated in a series of comparative
statics, which is possible since the equilibrium is unique and skill distribution continuous (under assumption 3). To simplify notation, define the elasticities of the thresholds w.r.t. M1 : ∆h1 ≡
ˆ1 d log h , d log M1
∆h2 ≡ χ ˜·
ˆ2 d log h , d log M1
∆z ≡
d log zˆ , d log M1
where χ ˜≡
χ > 0. ˆ h2 − χ
Similarly define ∆x as the elasticity of any variable x with respect to M1 . Given (∆h1 , ∆h2 , ∆z ) we know what happens to all the other variables of interest since ∆φ1 = ∆z − ∆h1 ,
∆φ2 = ∆φ1 − ∆h2 ,
∆W1 = −∆h1 ,
∆W2 = −∆h2 ,
∆Wz = −∆z .
where Wj ’s are the wage ratios W1 = w1 /w0 ,
W2 = w2 /w1 ,
Wz = wz /w0 .
We proceed as follows:
ˆ 1, h ˆ 2 , zˆ) within a sector, taking the dis1. approximate the change in thresholds (h tribution of skill in sector i, µi , as given; 2. given the comparative statics in the thresholds,
approximate the change in employment shares across tasks within a sector, taking µi as given; 3. approximate the differences in polarization across sectors holding Li constant; 4. approximate the
change in employment shares across sectors.
Wage and Job Polarization
To approximate the change in thresholds, we will first focus on the within sector allocation of skill implied by (11) and (13): . ˆ 1 · πi1 (h ˆ 1 ) = Hi1 (h ˆ 1, h ˆ 2 , zˆ) Li0 (ˆ ˆ 1) h z, h h i.
ˆ 2 · πi2 (h ˆ 2 ) = Hi2 (h ˆ 1, h ˆ 2 , zˆ) − χLi2 (h ˆ 1, h ˆ 2 , zˆ) Hi1 (h ˆ 1, h ˆ 2 , zˆ) 1 − χh /h . ˆ 1, h ˆ 2 , zˆ) = Zi (h ˆ 1, h ˆ 2 , zˆ) Li0 (ˆ ˆ 1) zˆ · πiz (h z, h
(25a) (25b) (25c)
where (πij , πiz ) are defined in (12) and (14), and (φ1 , φ2 ) are defined in (15). The masses and skill aggregates are defined over a sector-specific distribution µi , which is taken as given. For
the approximation, we will assume that ∆Lij → 0 for j ∈ {0, 1, 2, z}. This implies that the density function is sufficiently small everywhere, which we assume to ignore the indirect effects of M1 on
(Lij , Hij , Zi ) that arise from changes in the thresholds. This can be thought of a limiting case of either when skills are discrete (Goos et al., 2014),7 or when there are both a continuum of
tasks and skills which are matched assortatively (Costinot and Vogel, 2010). Within the context of our model, it can be understood as approximating the equilibrium response using only the response of
labor demand (the LHS’s), while keeping labor supply (the RHS’s) fixed. We can then show that Proposition 1 (Routinization and Polarization) Suppose there is an increase in M1 , and that ∆Lij → 0 for
j ∈ {0, 1, 2, z}. Then 1. ∆h1 ≈ −∆h2 > 0 iff σ < 1, and 2. ∆φ1 < {∆h1 , ∆z ≈ ∆φ2 } < 0 if ω < σ < 1. This implies that capital and labor flow out of task 1 (job polarization), relative wages decline
in task 1 (wage polarization), and both the employment share and wages of 7
In fact, they assume that wages are fixed and labor is inelastically supplied.
managers increase (vertical polarization). Proof: Under the assumption (or, holding labor supply fixed), the comparative static is identical across sectors. System (25) becomes ∆h1 + ∆πi1 ≈ 0,
∆h2 + ∆πi2 ≈ 0,
∆z + ∆πiz ≈ 0,
where ∆πi1 = (σ − 1) [(1 − α)∆h1 + 1] , ∆πi2 = (σ − 1) [(1 − α)∆h2 − 1] , σ − ω πi1 (1 + πi2 )∆πi1 + πi1 πi2 ∆πi2 ∆πiz = (ω − 1)(1 − α)∆z + · σ−1 Πih
Hence we obtain that ∆h1 ≈ −∆h2 ≈
1−σ > 0, α + σ(1 − α)
∆W1 < 0,
∆W 2 > 0
σ < 1.
Furthermore if ω < σ < 1, πi1 σ−ω · · ∆h1 < 0, (σ − 1) [α + ω(1 − α)] Πih < 0, ∆φ2 ≈ ∆z < 0, and ∆Wz > 0.
∆z ≈ ∆ φ1
The change in thresholds makes it easier to analyze what happens to employment shares by task. If σ < 1, and holding management employment shares constant, employment and payroll in task 1 shrinks
while they increase in tasks 0 and 2. Hence, similarly as in Goos et al. (2014), we get employment polarization only when tasks are complementary, i.e. σ < 1; we also get wage polarization even with
endogenous choice of tasks. At the same time, capital flows out to the other tasks as well. Furthermore if ω < σ, we also find that the mass and wage of managers increase relative to all workers. But
while it is clear that the thresholds move in a direction that continues to shrink Li1 , it is unclear what happens to Li0 and Li2 , since both tasks 0 and 2 gain employment from task 1 but lose
employment to managers. So let us think about the (supply side) changes in Lij , j ∈ {0, 1, 2, z}, arising from the change in thresholds within a sector i, still taking the sectoral distribution µi
as given. To sign the ∆Li0 , ∆Li2 , we need additional parametric restrictions for sufficiency:
Lemma 1 Suppose the skill distribution in sector i is uniform and that ω < σ < 1. A sufficient condition for employment in tasks 0 and 2 to rise is σ − ω < (1 − σ) [α + ω(1 − α)] .
So if σ − (1 − σ)α < ω < σ, 1 + (1 − σ)(1 − α) all employment shares except task 1’s increase. This also implies that the average skill of task 1 workers rises. Proof: Using the approximations from
Proposition 1 and (25), we can approximate ∆Li0 − ∆Li1 ≈ ∆h¯ 1 ,
∆Li0 − ∆Li2 ≈ ∆h¯ 2 −χ ,
∆Li0 − ∆Li1 ≈ ∆z¯.
Since {∆z , ∆h2 } <0, we know that {∆z¯, ∆h¯ 2 −χ } <0, that is, the average skill of managers, and workers in task 2, become diluted. We cannot sign ∆h¯ 1 ; however, under the uniform distribution
assumption σ−ω πi1 ∆Li0 = ∆z + ∆h1 ≈ 1 − · ∆h1 (1 − σ) [α + ω(1 − α)] Πih using (27). Since πi1 /Πih is a fraction bounded above by 1, the condition in the lemma guarantees that ∆Li0 > 0, so ∆Li1 < 0
< ∆Li0 < {∆Li2 , ∆Liz } , although we can still not order the last two.
This is intuitive. If tasks were substitutes, task 1 would crowd out all other tasks, including managers. As task 1 becomes the dominant occupation, wages also increase. However, when tasks are
complements, workers need to flow to the other tasks, and for this to happen relative wages must decline in task 1. Moreover, if management is more complementary with tasks than tasks are among
themselves, more individuals must become managers—and in equilibrium, manager wages must increase. The withinsector comparative static is depicted in figure 10.
Structural change
Previous models of structural change either rely on a special non-homogeneous form of demand (rise in income shifting demand for service products) or relative technology differences across sectors
(rise in manufacturing productivity relative to services, combined with complementarity between the two types of goods, shifting production to services). Our model is also technology driven, but
transformation arises from a skill neutral increase in task productivities, or routinization. Most importantly, in contrast to recent papers arguing that sectoral productivity differences can explain
the skill premia or polarization (B´ ar´any and Siegel, 2016; Buera et al., 2015), we argue
ˆ1 h
ˆ2 h
(a) Equilibrium wt low M1
ˆ1 h
ˆ2 h
(b) Equilibrium wt high M1
Fig. 10: Comparative Static, Within-Sector
exactly the opposite—that routinization can explain sectoral productivity differences and structural change. To begin this analysis, note that from (26), ∆Πih ≡
˜ ih d log Πih d log Π (1 − σ) [α + ω(1 − α)] · ∆z , = ≈ d log M1 d log M1 σ−ω
and using Proposition 1, the ∆Vij ’s can be approximated from (20) as ∆Vi0
= 0,
∆Vi2 ≈ −∆h¯ 2 −χ > 0,
∆Vi1 ≈ −∆h¯ 1 < 0 ∆Viz ≈ ∆z¯
by Lemma 1,
> 0.
So the ∆Vij ’s are sector-neutral and can be ordered as ∆V1 < {0 = ∆V0 } < {∆V2 , ∆Vz }. Also note that ∆ Π Li
˜L d log Π d log ΠLi i = = ≡ d log M1 d log M1
˜L Π i
Vij ∆Vj
X j=0,1,2,z
Lij · ∆Vj . Li
Decomposing Polarization The change in the total amount of labor in each task, expressed in (21), can be decomposed similarly as in Goos et al. (2014):8 i X Lij h ∆Lj = · ∆Vj − ∆ΠLi + ∆Li Lj i∈{m,s}
However, our decomposition differs from theirs. Their thought experiment is to separate the effects from keeping industry output fixed and when it is allowed to vary. Ours is to separate the effect
from keeping sectoral employment fixed and when allowing it to vary.
i X Lij h X Lij 0 · ∆ Vj − · ∆Vj 0 +∆Li Lj Li j 0 =0,1,2,z i∈{m,s} | {z }
by (30),
j ∈ {0, 1, 2, z}.
(31) A change in the Vij ’s occurs even holding Li ’s constant, shifting the term Bij . This leads to “within-sector polarization,” as we saw in the previous subsection. In particular, from (29), the
∆Vj ’s are sector-neutral and common across sectors. So any difference in how the share of task j employment evolves across sectors depends on the weighted average of the ∆Vj ’s by the employment
shares of all tasks within a sector, Lij /Li . Holding Li ’s constant, we know from Lemma 1 that task 1 is shrinking and other tasks growing within-sectors. Now we can compare the ∆ΠLi ’s across
sectors, which is the weighted average of within-sector employment shifts as seen in (30). Thus Lemma 2 The weighted average of within-sector employment share changes, ∆ΠLi , is smaller in the sector
with a larger within-sector employment share in task 1, and larger in the sector with larger shares in all other tasks. That is, Ls1 /Ls < Lm1 /Lm
∆ΠLs > ∆ΠLm .
This implies that, holding sectoral employment shares constant, manufacturing polarizes more compared to services.9 The term ∆Li in (31) captures structural change. To compute ∆Li , rewrite capital
input (or employment) ratios in (19) using (18): ω−σ 1 1 " Πsh (ω−1)(σ−1) 1 − ηs ω−1 νs0 σ−1 γs · κ= γm 1 − ηm νm0 Πmh #−1 ω ΠKs ω−1 −α ΠLs α−1 × . ΠKm ΠLm
So relative employment is completely determined by the relative endogenous TFP ratio between the two sectors. Since the elasticities of Πih are sector-neutral (both change at the negative rate of
zˆ), we obtain ω ∆ΠKs − ∆ΠKm + (1 − α) ∆ΠLs − ∆ΠLm . (34) ∆κ ≈ (1 − ) α − ω−1 9 Of course, the assumption in the lemma is a condition on employment shares, which are endogenous. However, the
condition holds throughout our observation period in the data, so our analysis is valid. Alternatively, we could assume νm1 >> νs1 and ηm << ηs . The astute reader would have already noticed that
what the task-specific TFP’s effectively do is shift the relative employment shares over time as if the parameters νij , ηi were changing.
So if (∆ΠKi , ∆ΠLi ) are larger in services, employment shifts to services; that is, routinization (a rise in M1 ) leads to structural change. We have already seen that ∆ΠLi is smaller in the
manufacturing when (32) holds. Now from (28), (1 − σ)[α + ω(1 − α)] ∆z < 0, · πiz + Πih · ∆ΠKi ≈ ΠKi σ−ω so under the assumption in Lemma 1, ∆ΠK2 > ∆ΠK1 if πsz πmz > Πsh Πmh
1−ω 1−ω ηm ηs · (νs0 Πsh ) σ−1 > · (νm0 Πmh ) σ−1 , 1 − ηs 1 − ηm
which holds when the manager share of capital is larger in services, or η1 << η2 . Hence, both because of shifts in labor and capital, structural change occurs toward services. To understand why
capital reallocation matters for structural change, note that we can write change in sectoral employment shares as ∆Lm = −Ls · ∆κ < 0,
∆Ls = Lm · ∆κ > 0,
or plugging in ∆κ from (34), " X 0 Li ∆ Π L 0 ∆Li ≈ (1 − ) (1 − α) ∆ΠLi − i
| + α+
ω 1−ω
∆ΠKi −
Ki0 ∆ΠK 0
# .
This makes clear that structural change in our model is due to a reallocation of both labor and capital, in contrast to Goos et al. (2014). The reason that capital matters in our model is because
labor in our model is in skill units, which is different from employment shares. However, given sectoral capital Ki within a sector, physical capital does not affect employment shares as it is simply
allocated to equalize its MRTS with the MRTS of skills across tasks; only when we let factors move across sectors does its effect appear in the model. Also note that the term CLi can be written as
X X Li0 j 0 X Lij · ∆ Vj − Li · ∆ Vj 0 , CLi = Li Li0 0 0 j
which is the “between-sector” counterpart to the within-sector component Bij : that is, CLi captures the average change in employment in sector i compared to the weighted average across sectors. The
contribution from capital, CKi , is additional.
(b) High M1 , κ fixed
(c) High M1
(d) BGP m1
Fig. 11: Comparative Statics, Across-Sectors
Of course, from (31), structural change also contributes to polarization. To see this, rewrite (31) using (35) as X Lij Lsj Lmj ∆ΠLi + Lm − Ls ∆κ Lj Lj Lj i∈{m,s} X Lmj Lsj − Lm Ls ∆κ . − ∆Vj = − Lij
∆ΠLi + Ls Lm
∆Lj = ∆Vj − ⇒ Lj ∆Lj
Thus, Lemma 3 Suppose lemma 2 holds. Then structural change also contributes to polarization. Proof: The term in the square brackets in (36) is negative for j = 1, and positive for all other tasks,
under lemma 2.
This is intuitive. Manufacturing has a larger within-sector employment share in task 1 (that is, if it is more routine-intense), employing more for that task. So in addition to task 1 shrinking in
both sectors, if sector 1 also shrinks (structural change), there is even more polarization. Lemmas 2 and 3 are depicted in the first 3 subplots in figure 11. In figure (a), manufacturing is depicted
as having a higher share in task 1, and services in task z. As we move from (a) to (b), sectoral employment shares are held fixed, and task 1 shrinks in both sectors. The change in employment shares
is larger in manufacturing due to lemma 2. This leads to structural change in (c), according to lemma 3. Because manufacturing has a higher share in task 1, shrinking its size contributes to
Service Sector
Lz L2 L0
Manufacturing Sector
Lz L2 L0
Service Sector
Lz L0 L1 L2
Manufacturing Sector
L0 L1 L2
Service Sector
Lz L0 L1 L2
Manufacturing Sector
Lz L0
Service Sector
L0 L1 L2
Manufacturing Sector
Lz L2 L1 L0
(a) Low M1
Polarization or Structural Change?
One may argue that it is not task productivities that lead to structural change, but advances in sector-specific productivities that lead to polarization. While it is most likely in reality that both
forces are in play, in the context of our model, as long as technologies are either task- or sector-specific (that is, there are no task- and sectorspecific technologies), sector-specific
productivity shifts does not lead to polarization within sectors. To see this, consider an exogenous change in the manufacturing sector’s exogenous productivity, Am . As in Ngai and Pissarides
(2007), a rise in Am changes κ at a rate of 1 − , that is, manufacturing shrinks. But it is easily seen that none of the thresholds change, and hence neither do the Φi ’s (the endogenous sectoral
TFP’s). So polarization can only arise by the reallocation of labor across sectors that use different mixes of tasks. To be precise, from (31), d log Lj Lsj Lmj d log Lj = (1 − ) · = (1 − ) Lm − Ls <
0. d log Am d log κ Lj Lj
Note that d log Lj /d log κ is equal to the term in square brackets in (36), and negative under assumption (32). Hence, polarization only occurs because manufacturing shrinks. The reason is that In
our micro-founded model, tasks are aggregated up into sectoral output, not the other way around. Equation (37) also puts a bound on how much sectoral shifts alone can account for job polarization.
For example, in the data, manufacturing employment fell from approximately 33% to 19% from 1980 to 2010. If this were solely due to a change in Am , this means that (denoting empirical values with
hats): d log κ ˆ ≈ 14/67 + 14/33 ≈ 0.63 d log Am which means that " # h i ˆ sj ˆ mj ˆj dL L L ˆ sj L ˆm − L ˆ mj L ˆ s = 0.63 ≈ 0.63 L 0.33 − 0.67 . ˆj ˆj d log Am L L In Section 6, we measure the
employment share of routine, manufacturing jobs and routine, service jobs (as a share of total employment; that is, Lm1 and Ls1 ) in 1980 were 26% and 33%, respectively (refer to Table 1). So ˆj dL =
0.63 [0.33 · 0.33 − 0.26 · 0.67] = −0.04, d log Am that is, a change in Am alone would imply an approximately 4 percentage point drop in routine jobs from 1980 to 2010. As shown in Table 1, the
actual drop was 13 percentage points.
The above result implies that on a dynamic path in which M1 grows at a constant rate, polarization happens faster than structural change. This implies that in the limit, task 1 vanishes, structural
change ceases, but both sectors still employ non-trivial amounts of labor, unlike previous models of structural change. Such a dynamic version of the model is a straightforward extension of the
neoclassical growth model. Assume that aggregate labor L grows at rate n, and a representative household with CRRA preferences Z ∞ c(t)1−θ − 1 dt exp(−ρt) · 1−θ 0 where ct = Ct /Lt , and a law of
motion for aggregate capital K˙ t = Yt − δKt − Ct , and for simplicity let us assume that M˙ 1 /M1 = m1 and M0 = M2 = Mz , M˙ 0 /M0 = m. Then from (19), we can also write the aggregate production
function as "
Yt = Yst · γm
Ymt Yst
# −1
+ γs
= Φst ·
Lst Ktα Lt1−α
1 −1 −1 · γs Lst
= Φst · (Lst /γs ) 1− · Ktα L1−α t where Φst , Lst , the endogenous sectoral TFP and employment share of services at time ˆ 1t , h ˆ 2t ). t, are functions of (ˆ zt , h Now define 1
Φ1−α ≡ Φst · (Lst /γs ) 1− t ˙ t /Φt . As the endogenous aggregate (Harrod-neutral) TFP and its growth rate gt ≡ Φ in the RCK model, define the normalized consumption and capital per efficiency unit
of labor cˆt ≡ Ct /Φt Lt
kˆt ≡ Kt /Φt Lt ,
so output per efficiency unit of labor is yˆt ≡ Yt /Φt Lt = f (kˆt ) ≡ kˆtα . The dynamic equilibrium is characterized by i 1 h cˆ˙t = · f 0 (kˆt ) − (n + δ + ρ + gt θ) · cˆt θ
˙ kˆt = f (kˆt ) − (n + δ + gt )kˆt − cˆt Φ˙ t ˆ 1t , h ˆ 2t , zˆt . gt ≡ =g h Φt So instead of having sectoral shares as in Acemoglu and Guerrieri (2008), we have endogenously evolving TFP which
pins down the sectoral shares at every instant. Using (18) and (34), the endogenous growth rate gt becomes " # ˙ X ˙ ih ω ˙L ω−σ Π ΠKi Π i (1 − α)gt = m + Lit · · −α · + (α − 1) · . + (ω − 1)(σ − 1)
Πih ω−1 ΠKi ΠLi i
ˆ 1, h ˆ 2 , zˆ) no longer evolve: On a BGP, gt must be constant. Hence it must be that (h ˆ1 = h ˆ 2 , or from (11)-(12), Clearly this happens when h !(1−α)(1−σ) ¯ 2 − χ)Li2 ˆ ν ( h h − χ i0 2
ˆ2 − χ = · · h ¯ Li0 νi2 h ⇒
ˆ 2 − χ)σ+α(1−σ) νi0 Li2 (h , = ¯2 − χ νi2 Li0 h
¯ = 1. Then zˆ is determined by (13), sectoral-task employment masses are assuming h determined by (ΠKi , ΠLi ) according to (33), and on a BGP g∗ =
m . 1−α
The long-run dynamics is depicted in figure 11(d), where both polarization and structural change continue until task 1 vanishes.
Quantitative Analysis
The goal of our quantitative analysis is to quantify how much of the observed changes in employment and wage shares from 1980 to 2010 can be explained by task-level productivity growth, and relate
such productivity growth to empiricaly measurable sources. Whenever possible, we fix parameters to their empirical counterparts, and separately estimate the aggregate technology (3) from time series
data on sectoral price and output ratios. Then we choose most model parameters to fit the 1980 data exactly, inlcuding a parametric skill distribution of (h, z). The rest of the model parameters,
which includes the elasticity parameters (σ, ω), are calibrated to empirical time trends from 1980 to 2010.
Occupations and Skills
In the quantitative analysis, we assume that there are 10, rather than 3, (horizontally differentiated) worker task/occupations. (There is still only one management task.)
Ranked by mean wage (except management)
SOC Code
Low Skill Services
Middle Skill Administrative Support Machine Operators Transportation Sales Technicians Mechanics & Construction Miners & Precision Workers
59.09 16.57 9.81 8.73 7.87 3.23 7.91 4.97
46.48 14.13 3.75 6.64 9.37 3.86 6.02 2.71
25.86 3.47 8.79 3.80 0.79 1.00 4.44 3.58
12.93 1.53 3.02 2.28 0.62 0.57 3.19 1.73
53.43 12.90 8.21 7.73 7.40 3.35 8.40 5.43
35.90 9.60 2.39 4.15 8.45 4.33 4.88 2.10
24.76 2.90 7.37 3.37 1.06 1.13 4.91 4.03
10.02 1.15 1.91 1.46 0.85 0.66 2.61 1.38
19.22 11.02 8.20
26.16 16.51 9.65
3.87 1.73 2.14
3.64 1.45 2.20
24.20 13.36 10.84
33.98 20.78 13.20
6.07 2.59 3.48
5.51 2.12 3.39
High Skill Professionals Management Support Management
Employment Shares 2010 Manufacturing
Total Wage Shares 2010 Manufacturing
Table 1: Occupation Groups used for Calibration Source: US Decennial Census (1%), 1980-2010.
¯ regardless of their own level of human Recall that in task 0, workers can only utilize h capital, and in task 1 they simply use all of their human capital. Each of the additional worker tasks is
characterized by a skill-loss parameter {χj }9j=2 .10 We assume each of these 11 occupations (10 worker + 1 manager occupation) in the model broadly correspond to the 11 one-digit occupation
categories in the census, discussed in Section 2 and summarized in Table 1. The 10 worker occupation groups can further be broadly grouped into low/medium/high skill tasks, or manual/routine/abstract
jobs, according to the mean wages of each occupation group and routinization indices. In Table 1, the left panel shows the SOC of each occupation with a short job description, and the middle and
right panels their employment and total wage shares in 1980 and 2010, respectively. For the employment and wage share panels, the first two columns shows the size of each occupation in 1980 and 2010
as a fraction of total employment/wages. The next two columns show the size of each occupation within manufacturing as a fraction of total employment/wages. These were already depicted graphically in
Section 2, and will be the bulk of our target moments in the calibration. The only other target moment is the growth rate of aggregate output.
Parametric Skill Distribution For the quantitative analysis, we assume a parametric skill distribution that is type IV bivariate Pareto (Arnold, 2014). Specifically, the c.d.f. we assume is h i−a µ
(h, z) = 1 − 1 + h1/γh + z 1/γz . 10
The characterization of the equilibrium is exactly the same as before.
12 z p.d.f. h p.d.f. 10
Correlation: 0.002
0.11 z
0.562 h
Fig. 12: Calibrated Skill Distribution We use a type IV bivariate Pareto distribution to model the distribution over worker and manager skills (h, z). The figure depicts the marginal distributions
each skill, and also their mean values below the x-axis. The implied Pearson correlation coefficient between the two skills is low, at 0.002.
We normalize γz = 1, since we cannot separately identify both skills from the skillspecific TFP’s. This is consistent with an establishment size distribution that is Pareto, and a wage distribution
that is hump-shaped with a thin tail, as depicted in Figure 12.
Aggregate Production Function
The aggregate production function is estimated outside of the model. For the estimation, we only look at the manufacturing and service sectors, where manufacturing includes mining and construction,
and government is included in services. We estimate the parameters (γm , ) from the system of equations p m Ym 1− log = log γm + (1 − ) log pm − log γm p1− + u1 m + γs p s PY 1 −1 −1 1 log(Y ) =c+
log γm Ym + γs Ys + u2 −1 where γs ≡ 1 − γm , using non-linear SUR (seemingly unrelated regression), on all years of real and nominal sectoral output observed in the BEA accounts. Real production by
sector is computed by a cyclical expansion procedure as in (Herrendorf et al., 2014) using production value-added to merge lower level industries (as opposed to consumption value-added, as in their
analysis).11 The price indices are implied from nominal versus real sectoral quantities. We try different base years as well: 1947, 1980, and 2005, corresponding to columns (1)-(3) in Table 2. This
is to 11
The constant c is included since it is not levels, but relative changes that identify .
(1) γm AIC RMSE1 RMSE2
(2) ∗∗
(3) ∗∗
-550.175 0.106 0.039
-551.264 0.106 0.039
-550.866 0.106 0.039
Standard errors in parentheses † ∗ ∗∗ p < 0.10, p < 0.05, p < 0.01
Table 2: Aggregate Production Function The manufacturing share parameter γm and elasticity parameter between manufacturing and services, , are estimated off the time series of output and price ratios
from 1947 to 2013, which are available from the BEA accounts. The service share parameter γs is assumed to equal 1 − γm . For details of the estimation, we closely follow Herrendorf et al. (2014).
check the robustness of the choice of base years: 1947 is the first year the required data is available, 1980 is the first year in our model, and 2005 is chosen as a year close to present but before
the crisis. As shown there, the values are in a similar range as in Herrendorf et al. (2014), although is not significant with 2005 as a base year. For the calibration, we will use take the values of
(γm , ) in column (1) values as a benchmark. The capital income share α is computed as 1-(labor income/total income), and fixed at 0.360. Since we do not model investment, for the calibration we also
need the level of total capital stock (for manufacturing and services) for each decade, which we take from the Fixed Assets Table and directly plugged into the model. Since we do not model population
growth, in practice we we normalize output per worker in 1980, y1980 to unity, and plug in Kt = kt /y1980 for t ∈ {1980, 1990, 2000, 2010}, where kt is capital per worker in year t.
Given the aggregate production function, we simulate two equilibria: one each for 1980 and 2010. Except for capital per worker, which we take exogenously from the data, any change between 1980 to
2010 is only due to exogenous task-TFP growth. Most of the parameters are calibrated so that our our model equilibrium fits the 1980 data moments in Table 1 exactly. We then calibrate the elasticity
parameters (σ, ω) and the 11 constant TFP growth rates {mj }9j=z,0 to fit the time trends in aggregate output and employment shares from 1980 to 2010. All model parameters are summarized in Table 3,
except for the skill loss parame-
ters χj , task intensity parameters (ηi , νij ), and task-TFP growth rates mj , which are tabulated in Table 4. Below, we explain in detail how these parameters are recovered.
Fixed from data
Fit to 1980
Fit to 2010
K1980 K2010 α
2.895 4.238 0.361
Computed from BEA/NIPA data
0.371 0.003
Estimated in section 6.2
Mj ≡ M Am As ηi (2) νij (18)
0.985 1.112 1.000
χj (8) a γh
Table 4 Table 4 10.087 0.216
Output per worker, normalization Manufacturing employment share Normalization Within-sector manager share Witin-sector employment shares by occupation Relative wages by occupation
γz ¯ h
1.000 1.000
Normalizations; Not separately identified from Mj
σ ω
0.704 0.341
Witin-sector employment shares by occupation
Table 4
Output per worker growth and employment shares by occupation
Table 3: Parameters All parameters valued 1 are normalizations.
Calibrating the distribution For any guess of (γh , a, {χj }9j=2 ), we can find ˆ j }9 ) that exactly match observed employment shares by occupation in 1980, by (ˆ z , {h j=1 integrating over the
guessed skill distribution. Given the thresholds, we then compute the model-implied relative wages using (24), ¯1 ¯1 w1 h h = , ¯ ˆ1 w0 h h
¯ 2 − χ2 ¯ 2 − χ2 ) w2 ( h h = , ¯1 ¯ 1 (1 − χ2 /h ˆ 2) w1 h h
wz z¯ z¯ = , ¯ zˆ w0 h
and similarly for j ∈ {3, . . . 9}. The LHS is the ratio of mean wages by occupation, which we observe from the data. The RHS is a function only of the thresholds, which themselves are functions of
(γh , a, χj ). Hence, we iterate over (γh , a, , {χj }9j=2 ) so that the model-implied ratios match observed mean wage ratios exactly. For the rest of the calibration, we fix these three parameters
that govern the skill ˆ j }9 ) that fit 1980 distribution. By construction, we already know the thresholds (ˆ z , {h j=1
Ranked by mean wage (except management) Low Skill Services Middle Skill Administrative Support Machine Operators Transportation Sales Technicians Mechanics & Construction Miners & Precision Workers
High Skill Professionals Management Support Management
Emp Wgts (νij , ηi ) Manu. Serv.
0.002 0.004 0.006 0.009 0.011 0.013
0.816 0.088 0.256 0.119 0.026 0.034 0.159 0.134
0.524 0.173 0.015 0.081 0.123 0.040 0.065 0.027
2.930 9.122 4.348 0.012 -1.144 2.315 6.328
2.428 0.609 -0.586 0.494 -0.074 -0.637 0.642
0.015 0.018
0.168 0.070 0.098
0.340 0.195 0.146
-2.248 -0.489
-0.671 -0.657
Table 4: Calibrated Employment Weights and Growth Rates employment shares by occupation exactly. And since the skill distribution is fixed by the data, we can similarly compute the thresholds that
fit 2010 employment shares by occupation. Denote these two sets of thresholds as x1980 and x2010 , respectively. Note that these thresholds are determined solely by the exogenously assumed skill
distribution and the data, independently of our model equilibria.
Normalizations Before we simulate the model equilibrium and calibrated the remaining parameters, some normalizations are in order. We have already normalized γz = 1. For notational convenience, we
will denote the 1980 levels of the TFP’s by (M, Ai ) and denote their 2010 levels by multiplying them by their respective growth rates. For example, the manager-task TFP in 2010 is M (1 + mz )30 . ¯
= 1 since it is not separately identified from M0 . This can be 1. We normalize h seen in (12). 2. We also normalize Mj ≡ M for all j ∈ {0, 1, . . . , z} for 1980, since it is not separately
identified from (ηi , νij ) in a static equilibrium. This is implied by the production technology we assume in (5)-(6). The rest of the parameters are calibrated so that simulated moments from the
model’s 1980 and 2010 equilibria match the data.
Calibrated within the model There are 35 parameters that remain to be calibrated: the elasticity parameters (σ, ω), TFP parameters (M, Am ), task intensities
(ηi , {νij }9j=1 ) for i ∈ {m, s}, and the task-TFP growth rates {mj }9j=z,0 .12 Given the aggregate production function, skill distribution, normalizations and thresholds (x1980 , we can recover the
remaining parameters as follows. (A) Guess (σ, ω). (B) Given the guess, first fit 1980 moments: (a) Guess (M, Am ). (b) Plug in the threshold values x1980 , and the empirical values of (Liz , Li0 , .
. . , Li9 )— the employment shares of each occupation in sector i ∈ {m, s}—from Table 1, into (11) and (13). Then we recover all the νij ’s from (11)-(12), and the ηi ’s from (13)-(14) in closed form
(since all Mj ’s are assumed to be equal). This ensures that the 1980 equilibrium exactly fits within-sector employment shares by occupation (20 parameters, 20 moments). (c) Repeat from (a) until we
exactly fit the manufacturing employment share in 1980, and output per worker of 1.13 Since (18)-(19) are monotone in (M, Am ), the solution is unique (2 parameters, 2 moments). (C) Given the
parameters recovered from the 1980 equilibrium, plug in threshold values x2010 into (22). Then find {mj }9j=z,0 so that the 2010 equilibrium exactly fits employment shares by occupation (but not
necessarily by sector), and also output per worker, in 2010 (11 parameters, 11 moments). (D) Repeat from (1) to minimize the distance between the within-sector employment shares by occupation implied
by the 2010 model equilibrium and the data (2 parameters, 11 moments). Note that since all other moments are fit exactly, in essence we are only calibrating the two parameters (σ, ω) in (A) to match
the 11 moments in (D).14 The resulting parameters are tabulated in Tables 3-4. The last column of Table 4 shows the empirical RTI indices constructed from Autor and Dorn (2013), which are also
visualized in Figure 19.
Model Fit
Figure 13 plots the model implied trends in employment shares across tasks, in aggregate and by sector, against the data. When computing the simulated paths for P9 There are only 9 intensity
parameters to calibrate per sector, since we assume j=0 νij = 1. 13 The latter must be matched since the value of K1980 we plug in from the data was normalized by 1980’s output. 14 When targeting
2010 employment shares and output per capita, we in fact target the linear trend from 1980 to 2010 rather than their exact values. However, since most trends are in fact linear, using the exact
values barely change any of our results. 12
o Data, x Model 0.65
o Data, x Model
0.3 Routine (Left) Manual
Abstract Manager
0.15 0.6
0.45 1980
0.1 2010
0.5 1980
(a) Aggregate
o Data, x Model 0.2
Routine (Left) Manual
Manual Manager
(b) Services Share By Task
o Data, x Model 0.85
Routine Abstract
Abstract Manager
0.3 Routine (Left) Manual
Abstract Manager
0.65 1980
0.4 1980
(c) Manufacturing
0.1 2010
(d) Services
Fig. 13: Data vs. Model, Employment Shares by Task 1990 and 2000, we plug in the empirical values of Kt = kt /y1980 and the level of taskTFP’s implied by the calibrated growth rates, and compute the
respective equilbria allocations. As we explained above, the aggregate trend can be solved in closed form using the same number of parameter and moments, so it is not surprising that we obtain a more
or less exactly fit as seen in Figure 13(a). On the other hand, while we target the starting points for all the shares (services employment share, and within-sector employment shares by task), these
11 trends were calibrated using only the 2 elasticity parameters (σ, ω). Nonetheless, the calibrated model more or less exactly replicates structural change by occupation, as seen in Figure 13(b) and
also within-sector polarization, as seen in Figures 13(c)-(d). The fit is not as satisfactory for relative wages, plotted in figure 14. Here we plot
o Data, x Model
o Data, x Model
Manual/Routine (Left) Abstract/Routine Manager/Worker
Manual/Routine (Left) Abstract/Routine Manager/Worker
0.2 1980
1.2 2010
0.2 1980
(a) Aggregate
(b) Manufacturing o Data, x Model
1 Manual/Routine (Left) Abstract/Routine Manager/Worker
0.8 2010
(c) Services
Fig. 14: Data vs. Model, Relative Wages by Task the mean relative wages in aggregate and by sector. Here, the only targeted moments were the 1980 average wage ratios in Figure 14(a); all other
moments were not. Manual and abstract wages are relative to routine jobs, and manager wages are relative to all workers. Recall that in the model, efficiency wages (wz , w1 , . . . , w9 ) are equal
since we assume that individuals are indifferent across sectors, and we constrained our attention to equilibria in which mean skill levels within occupations were equal across sectors. Hence, within
relative wages are equal across sectors for any given occupation; the only reason they differ in Figures 14(b)-(c) is because we aggregate sub-occupation groups 3 broader categories; and for managers
since they are compared against all workers. Compared to the data, all model-implied relative wages are too low in manufacturing,
1.2 2010
and too high in services. Hence to some degree, our model is missing something that causes relative wages to be lower in services. Some explanation is in order. Due to the multiple layers of
complementary between occupations and sectors ([ω, σ, ] < 1), the calibrated growth rates mj are smaller for those occupations that are growing, as shown in Table 4. Then for all occupations in the
middle (j = 1, . . . , 9), whether their relative wages grow or shrink depends on the magnitude of negative selection (that comes from having more or less low-skill workers from the left-side of the
h-skill distribution) and positive selection (that comes from having more or less high-skill workers from the right-side of the h-skill distribution), since whether the employment share grows or
shrinks, it will either absorb or lose employment from both sides of the distribution. This was proven in Proposition 1 and ˆ 1 increasing and h ˆ 2 decreasing following a rise in the in Figure 10,
can be seen from h TFP among routine jobs. ¯ and it is only those For manual jobs, average skill is assumed to be constant at h, workers with the lowest h skill that work in this job. It turns out
that routine jobs as a whole display enough negative selection so that the wages of manual workers relative to routine workers rise, although only slightly. This is in fact consistent with the data
in aggregate and in the services sector, although in the manufacturing sector, manual wages slightly dropped. We also proved in Proposition 1 that as long as ω < σ, which it is (Table 3), managers’
relative wages would rise relative to workers as long as their task-TFP grew slower than workers’. However, the quantitative magnitude of this rise is small. This is because as the employment share
of managers grow, there is a negative selection along z-skills. In Figure 10, this can be seen from zˆ decreasing to increase the mass of managers. According to our model assumptions, all workers
with the highest h-skill work in the highest-skill worker occupation (“professionals” in the data). Since their task-specific TFP grew relatively less, the average skill of workers in this occupation
becomes lower, since employment growth leads to negative selection. Consequently, both because of lower TFP growth and negative selection, relative wages decline for abstract jobs, in contrast to the
data. Overall, the model targeted only to aggregate moments delivers a good fit by task even within and across sectors in terms of employment shares, but not in terms of relative wages. In what
follows, we focus only on employment shares and investigate how much each of these trends is explained by task-specific TFP’s, and its implications for other outcomes such as sectoral TFP’s.
Data (Annual) Benchmark Only Sectoral Both
0.8 0.6
0.2 0.15
-0.2 1980
-0.05 1980
(a) Log TFP per Worker, Manufacturing
Data (Annual) Benchmark Only Sectoral Both
(b) Log TFP per Worker, Services
Fig. 15: Benchmark vs. Counterfactuals, TFP 1980 levels are normalized to 0, so the slope of the lines are the growth rates.
In this subsection, we analyze the role of task-specific TFP’s on structural change, , we conduct two counterfactuals: (1) First, we set all task-specific TFP growth to be equal, that is, we set mj =
m, and instead let both (Am , As ) change at rates (am , as ). We jointly recalibrate (m, am , as ) to match the empirical growth rate of TFP (i.e., the Solow residuals) in aggregate, and also in
manufacturing and services, from 1980 to 2010. This yields the model’s predictions in the absence of any exogenous, task-specific TFP growth. (2) Second, we allow both exogenous task- and
sector-specific TFP growth, and recalibrate ({mj }9j=z,0 , am , as ) to match the change in employment shares and the empirical growth rates of TFP in aggregate, manufacturing and services, from 1980
to 2010. This gives the model the best chance to explain the data. For both counterfactuals, we keep all other parameters at their benchmark values in Table 3-4, and only recalibrate the growth
rates. We focus on sectoral TFP’s since in our model, structural change only results from the differential TFP growth across sectors—expressed in closed form in (18)—whether it is exogenous (caused
by am and/or as ) or endogenous (as in Section 4.2). The recalibrated parameters for the counterfactual scenarios are summarized in Appendix Table 5.
Data (Annual) Benchmark Only Sectoral Both
1 0.8
Data (Annual) Benchmark Only Sectoral Both
0.4 0.1
0.2 0 1980
(a) Log GDP per Worker, Manufacturing
(b) Log GDP per Worker, Services
Fig. 16: Benchmark vs. Counterfactuals, GDP per Worker 1980 levels are normalized to 0, so the slope of the lines are the growth rates.
TFP and output growth In Figure 15, we compare the path of log sectoral TFP in the data, in our benchmark calibration, and two counterfactual scenarios. All scenarios match aggregate TFP and GDP
growth from 1980-2010 in the calibration as shown in Appendix Figure 22, so we do not discuss them here.15 In our benchmark, we over shoot the growth rate of manufacturing TFP by about half a
percentage point, while undershooting the services TFP growth rate by about half a percentage point. However, note that when we look at the growth rates of sectoral output, in Figure 16, this gap
almost disappears. This is because while the model assumes that the sectoral capital input shares are equal to labor input shares, as shown in (19) earlier, in the data they are not. In fact,
counterfactuals (1) and (2) in Figure 16 show that when sectoral TFP growth is matched exactly, manufacturing output grows more slowly, and services output more quickly, compared to the data. This
implies that capital input ratio between manufacturing and services grew slower than the labor input ratio, although the differences are small.16
Structural Change and Polarization Since structural change is solely determined by output ratios (Section 4.2, equation (19)), the fact that sectoral output growth nearly tracks the data implies that
the the benchmark model more or less fully explains structural change (in terms of employment), as shown in Figure 17(c). Both counterfactuals (1) and (2) undershoot the full extent of structural
change, since sec15
Denote aggregate TFP as Zt . Since Yt = Zt Ktα (labor is normalized to one), and we plug in the empirical values of Kt for all calibrations, matching aggregate TFP and GDP are the same things. 16 If
sectoral production functions remain Cobb-Douglas, this means that capital intensity is higher in manufacturing, as analyzed in Acemoglu and Guerrieri (2008).
0.86 Data Benchmark Only Sectoral Both
Data Benchmark Only Sectoral Both
0.84 0.82
0.8 0.78
0.6 0.76 0.55 1980
0.74 1980
(a) Services Employment Share, Routine Jobs
(b) Services Employment Share, Managers
0.85 Data Benchmark Only Sectoral Both
0.65 1980
(c) Services Employment Share, All Jobs
Fig. 17: Benchmark vs. Counterfactuals, Structural Change toral output growth is too low and high in manufacturing and services, respectively. Moreover, when we look at structural change within
occupation categories, the benchmark model outperforms both counterfactuals (1) and (2), especially among managers. Lastly, we investigate whether exogenous growth in sectoral TFP’s can explain
polarization, as we also discussed in Section 4.3. In Figure 18, we see that sectoral forces alone can account for about 15-20 percent of horizontal and vertical polarization. However, remember that
this would not cause any changes in within sector employment shares by occupation. In sum, task-specific TFP growth can more or less fully account for sectoral output growth, and consequently for the
observed levle of structural change from 1980 to 2010. Due to the vertical and horizontal polarization induced by changes in task-specific TFP’s, employment shifts to the sector that uses the routine
task less and management more intensively. Conversely, sector-specific productivities can only account for 15-
Data Benchmark Only Sectoral Both
0.5 0.48 0.46 1980
Data Benchmark Only Sectoral Both
0.12 0.115
0.11 1980
(a) Routine Employment Share
(b) Manager Employment Share
Fig. 18: Benchmark vs. Counterfactuals, Polarization 20 percent of polarization, and furthermore we have shown, both analytically and quantitatively, that it does not cause any polarization within
sectors, that contrary to the data.
What are Task-Specific Productivities?
Despite having skill selection, horizontally and vertically differentiated jobs, and multiple sectors, Figure 19(a) shows that the bulk of the changes in occupational employment shares are still
directly accounted for by task-specific TFP’s, with a (negative) correlation of 0.97. This is also confirmed from a simple regression analysis shown in Appendix Table 6. This leads us to conclude
that in order to understand changes in the employment structure, it is important to identify what these task-specific TFP’s are. How much of the variation in the TFP growth rates can be explained by
routinization? As a first pass, we correlate the TFP growth rates with the RTI measure used in Autor and Dorn (2013), which itself is constructed from Autor et al. (2003) using the DOT, and the RTI
measure from Acemoglu and Autor (2011), which was constructed from O*NET. While the TFP growth rates are positively correlated with both indices, and more strongly with the latter, it is visually
clear that there is much left to be explained. More precisely, both the correlation and R2 ’s are still quite low, as can be seen in Appendix Table 6. What about college? Skill-biased technological
change (SBTC) has been a usual suspect for changes in the employment structure since since Katz and Murphy (1992), e.g. Krusell et al. (2000); Buera et al. (2015). In the SBTC literature, “skill” is
usually a stand-in for whether or not an individual went to college, or obtained a college degree.
RTI ONET, corr:0.685
RTI DOT, corr:.423
LServ Sales Tech
20 40 60 80 Skill percentile (1980 occupational mean wage) TFP, corr: −.969
100 −5
Change in emp share
(a) Task-specific TFP growth and Employment Shares
(b) Task-specific TFP growth and RTI
Fig. 19: Employment Shares, Task TFP growth and the Routinization Index However, as is evident from Figure 20(a), neither the fraction of college graduates within each occupation in 1980, nor the
change in the fraction of graduates from 1980 to 2010, have much of a relationship with the task-specific TFP’s calibrated from our model. Since the TFP’s more or less entirely explain the employment
shifts observed in the data, this means college can not explain occupational employment shifts. Moreover, as is clearer in Appendix Table 6, the correlation between employment shifts and college
measures are negative, that is, those occupations with more college graduates, or in which the college graduate share grew the fastest, in fact shrank. This is the opposite of most of propositions
made in the SBTC literature. What we do find, however, is that the TFP growth rates correlate strongly with disaggregated components of the RTI index in O*NET, in particular the routinemanual and
interpersonal skills indices. Appendix Table 6 shows that the R2 for both are also high. This means that those occupations with a higher share of routine-manual tasks have shrunk, while those with
higher share of interpersonal tasks have grown. While we conclude that productivity growth has been high in routine-manual tasks and low among interpersonal tasks, and that this can explain a
significant part of shifts in occupational employment, polarization, and consequently structural change, it is evident from the regressions that this is not the end of the story. The unexplained part
of task-specific TFP growth may also come from endogenous changes in the distribution of skill, and in an open economy setting from off-shoring, both of which we have abstracted from.17 17
In an open economy setting, cheaper foreign labor would be observationally equivalent to higher productivity.
CLG 1980 (right), corr:−.663
Routine manual, corr:.802
Manual interpersonal (−), corr: .759
5 0 −5
CLG growth (right), corr:−.61
(a) Task-specific TFP Growth and College
(b) Task-specific TFP Growth and O*NET
Fig. 20: Task TFP growth, College Shares and O*NET-based indices 0.7
3.5 Routine Employment Share Management Employment Share Manufacturing Employment Share Log GDP/Worker (Right)
Fig. 21: Long-run Dynamics
Long-Run Growth Path
Lastly, we show the long-run dynamics of the model from Section 5, extended to the 10 horizontally-differentiated tasks we have analyzed thus far. Assuming a CRRA coefficient of θ = 2, and that the
economy starts in 1980, we target an asymptotic interest rate of 2%, implying an approximately equal discount rate ρ. The depreciation rate is set to δ = 0.065, as computed from the NIPA accounts. As
can be seen, both routine and manufacturing continue to decline, but the speed of the decline in manufacturing slows down as routine jobs continue to disappear. Likewise, managerial employment
continues to rise, albeit at a slower pace. Finally, note that the first 150 years displays structural change and near balanced growth, consistently with the Kuznets and Kaldor facts.
We presented a new model which encompasses job polarization, structural change, and a modified span of control technology. We showed analytically and the quantitatively that the model can be a useful
tool for analyzing macroeconomic dynamics.
Additional Tables and Figures Ranked by mean wage (except management)
(1) -
(2) mj
BM mj
Low Skill Services
Middle Skill Administrative Support Machine Operators Transportation Sales Technicians Mechanics & Construction Miners & Precision Workers
1.973 1.973 1.973 1.973 1.973 1.973 1.973
1.252 10.018 3.326 -1.895 -2.484 1.742 6.367
2.930 9.122 4.348 0.012 -1.144 2.315 6.328
High Skill Professionals Management Support
1.973 1.973
-3.973 -1.973
-2.248 -0.489
Aggregate TFP growth am / Manu TFP growth as / Serv TFP growth
1.030 0.252 -1.205
1.030 0.252 2.021
1.030 2.943 0.308
1.030 2.229 0.743
Table 5: Recalibrated TFP Growth Rates for Counterfactuals Column (1) stands for the counterfactual in which we set mj = m and calibrate (am , as ) to match sectoral TFP’s, and (2) for when we let
({mj }9j=z,0 , am , as ) all vary simultaneously. “BM” stands for the benchmark calibration. For all scenarios, aggregate GDP growth (and consequently TFP growth) is matched exactly, shown in the
first row of the bottom panel. For the “BM” and “Data” columns, the am and as rows show the empirical growth rates of the manufacturing and services sectors’ TFP’s, respectively.
∆Lj TFP R2 RTI (DOT)
- 9.584 ∗∗∗ 0.939
0.423 (0.263)
Routine manual
Manual interpersonal
College share 1980
∆College share 1980-2010 Constant R2
Standard errors in parentheses, p < 0.10, p < 0.05,
p < 0.01
Table 6: Task-Specific TFP Growth, Employment, and Empirical Measures The first panel shows the results from regressing employment share changes on the calibrated task-specific TFP growth rates, mj .
The second panel shows the results from regression the growth rates on various empirical measures.
0.4 0.3
Data (Annual) Benchmark Only Sectoral Both
Data (Annual) Benchmark Only Sectoral Both
0 -0.1 1980
(a) Log TFP Growth
(b) Log GDP per Worker Growth
Fig. 22: Aggregate Output and TFP Growth
1.4 1
manufacturing−service wage ratio 1.1 1.2 1.3
employment share of manufacturing (%) 20 25 30 35 1980
(a) Manufacturing Employment Share
(b) Manufacturing-Services Average Wage Ratio
Log Wage Variance .142 .144 .146
Log Wage Variance .05 .06 .07
Fig. 23: Manufacturing vs. Services by Occuaption
Year Managers (left)
Managers (left)
(a) Data
(b) Model
Fig. 24: Within Task Wage Inequality
References Acemoglu, D. and D. Autor (2011). Skills, tasks and technologies: Implications for employment and earnings. In D. Card and O. Ashenfelter (Eds.), Handbook of Labor Economics, Volume 4,
Part B, pp. 1043 – 1171. Elsevier. Acemoglu, D. and V. Guerrieri (2008, 06). Capital Deepening and Nonbalanced Economic Growth. Journal of Political Economy 116 (3), 467–498. Arnold, B. C. (2014).
Univariate and multivariate pareto models. Journal of Statistical Distributions and Applications 1 (1), 1–16. Autor, D. H. and D. Dorn (2013). The growth of low-skill service jobs and the
polarization of the us labor market. American Economic Review 103 (5), 1553–97. Autor, D. H., L. F. Katz, and M. S. Kearney (2006). The polarization of the u.s. labor market. American Economic Review
96 (2), 189–194. Autor, D. H., F. Levy, and R. J. Murnane (2003). The Skill Content of Recent Technological Change: An Empirical Exploration. The Quarterly Journal of Economics 118 (4), 1279–1333.
B´ar´any, Z. L. and C. Siegel (2016). Job polarization and structural change. Buera, F. J. and J. P. Kaboski (2012, October). The Rise of the Service Economy. American Economic Review 102 (6),
2540–69. Buera, F. J., J. P. Kaboski, and R. Rogerson (2015, May). Skill Biased Structural Change. NBER Working Papers 21165, National Bureau of Economic Research, Inc. Burstein, A., E. Morales, and
J. Vogel (2015, January). Accounting for Changes in Between-Group Inequality. NBER Working Papers 20855, National Bureau of Economic Research, Inc. Costinot, A. and J. Vogel (2010, 08). Matching and
Inequality in the World Economy. Journal of Political Economy 118 (4), 747–786. Goos, M., A. Manning, and A. Salomons (2014). Explaining job polarization: Routinebiased technological change and
offshoring. American Economic Review 104 (8), 2509–26.
Herrendorf, B., R. Rogerson, and A. Valentinyi (2014). Growth and Structural Transformation. In Handbook of Economic Growth, Volume 2 of Handbook of Economic Growth, Chapter 6, pp. 855–941. Elsevier.
Katz, L. F. and K. M. Murphy (1992, February). Changes in relative wages, 1963-1987: Supply and demand factors. Quarterly Journal of Economics 107, 35–78. Krusell, P., L. E. Ohanian, J.-V.
R´ıos-Rull, and G. L. Violante (2000, September). Capital-skill complementarity and inequality: a macroeconomic analysis. Econometrica 68 (5), 1029–1054. Lee, E. (2015). Trade, inequality, and the
endogenous sorting of heterogeneous workers. Lucas, R. E. (1978, Autumn). On the size distribution of business firms. Bell Journal of Economics 9 (2), 508–523. Ngai, L. R. and C. A. Pissarides (2007,
March). Structural Change in a Multisector Model of Growth. American Economic Review 97 (1), 429–443. | {"url":"https://pdffox.com/horizontal-and-vertical-polarization-pdf-free.html","timestamp":"2024-11-05T13:44:37Z","content_type":"text/html","content_length":"136086","record_id":"<urn:uuid:f546d48a-e409-48d0-90c4-cac5cd82bb6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00638.warc.gz"} |
The "Algebra" in Algebraic Data-types
Explores the “Algebra” behind Algebraic Data-Types in Haskell and elucidates how it works.
Implementing simple Streams using lazy-IO in Haskell
Explores How simple Lazy IO in Haskell can be used to implement basic streaming functionality
2020: The Start of Something New, Enter Haskell !
The Post about me Wanting to learning Haskell in 2020
The post explaining support vector machines classification model for machine learning and data science. | {"url":"https://arjunkathuria.com/blog/","timestamp":"2024-11-09T13:28:26Z","content_type":"text/html","content_length":"11264","record_id":"<urn:uuid:a229f173-749d-4a5c-abbe-0913c45ebfe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00538.warc.gz"} |
Compound Interest Calculator: Calculate Your Investment Growth Instantly
Compound Interest Calculator
Compound Interest Calculator: Calculate Your Investment Growth Instantly
Compound interest is a powerful financial concept that allows your investments to grow exponentially over time. Using a Compound Interest Calculator is an easy way to estimate the future value of
your investments or savings by taking into account the principal amount, interest rate, time, and compounding frequency. In this article, we will delve into the formula, usage, benefits, and how this
calculator can assist you in planning your financial goals.
What is Compound Interest?
Compound interest is the interest calculated on the initial principal amount as well as on the accumulated interest from previous periods. This means that the interest itself earns interest, leading
to exponential growth over time. The concept is vital for investment growth, loans, and savings accounts.
How Does the Compound Interest Formula Work?
The Compound Interest Formula is:
A=P(1+r/nโ )^nt
• A is the future value of the investment/loan, including interest.
• P is the initial principal amount (the starting amount).
• r is the annual interest rate (decimal form).
• n is the number of times the interest is compounded per year.
• t is the number of years the money is invested or borrowed for.
Step-by-Step Explanation of the Formula
1. Initial Principal (P): This is your starting amount. For example, if you invest $200,000, this value is the principal.
2. Annual Interest Rate (r): This is the interest rate provided by your investment, represented as a decimal. An 8.6% interest rate is written as 0.086.
3. Compounding Frequency (n): The frequency with which interest is applied. This could be annually, semi-annually, quarterly, or monthly.
4. Number of Years (t): The duration of the investment or loan. Longer durations will yield higher returns due to the power of compounding.
Example: If you invest $200,000 at an annual interest rate of 8.6% for 10 years, compounded monthly, the future value is calculated as:
A=200,000(1+0.086/12โ )^12ร 10=483,324.91
This means your investment will grow to $483,324.91, and the compound interest earned will be $283,324.91.
Benefits of Using a Compound Interest Calculator
1. Accurate Financial Planning: This calculator helps you plan your investments and savings with precision, avoiding guesswork.
2. Understanding Long-Term Growth: Visualizing how your money grows over time with compounding can motivate you to invest more.
3. Comparing Different Investments: Quickly compare how various interest rates, durations, and compounding frequencies impact your investments.
4. Loan Analysis: Calculate how much youโ ll owe over time with loans that use compound interest.
Daily Uses of the Compound Interest Calculator
• Investment Planning: Use the calculator to forecast the growth of your investments in mutual funds, stocks, or bonds.
• Retirement Planning: Estimate the value of your retirement savings based on expected returns.
• Loan Calculations: Calculate how much you'll end up paying on a loan or mortgage that uses compound interest.
Key Features of Our Compound Interest Calculator
• User-Friendly Interface: Easy-to-use with real-time updates based on your inputs.
• Supports Multiple Compounding Frequencies: Choose between annual, semi-annual, quarterly, or monthly compounding.
• Step-by-Step Solution Display: Understand how the result was derived with detailed step-by-step explanations.
• Flexible Inputs: Adjust values directly or use sliders for quick modifications.
A Compound Interest Calculator is an invaluable tool for anyone looking to grow their wealth, whether through investments or savings. By leveraging the power of compound interest, you can make
informed financial decisions and watch your money grow exponentially. Whether you're planning for retirement, comparing investment opportunities, or analyzing loan repayments, understanding and
utilizing compound interest is essential for achieving your long-term financial goals.
Start using the Compound Interest Calculator today and take control of your financial future!
Leave a Comment | {"url":"https://tejcalculator.com/compound-interest-calculator/","timestamp":"2024-11-08T08:17:45Z","content_type":"text/html","content_length":"204280","record_id":"<urn:uuid:acd40f55-9fc5-48e5-8bfb-559edc8d16e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00379.warc.gz"} |
Gallery - Templates, Examples and Articles written in LaTeX
Akari Matsumoto's Résumé
Akari Matsumoto
Akari Matsumoto
<p>Template for Journal of Educational Datamining (JEDM)</p>
This document serves both as JEDM submission instruction and as a template file.
SungJin Nam
Teaching and Research Statement
Teaching and Research Statement example written for 2015 faculty search season.
W. Ethan Eagle
MDA HW2: Principial components analysis and Canonical correlation analysis
Principal Components Analysis (PCA) and Canonical Correlation Analysis (CCA) are among the methods used in Multivariate Data Analysis. PCA is concerned with explaining the variance-covariance
structure of a set of variables through a few linear combinations of these variables. Its general objectives are data reduction and interpretation. CCA seeks to identify and quantify the associations
between two sets of variables i.e Pulp fibres and Paper variables.PCA shows that the first PC already exceeds 90% of the total variability. According to the proportion of variability explained by
each canonical variable , the results suggest that the first two canonical correlations seem to be sufficient to explain the structure between Pulp and Paper characteristics with 98.86%. Despite the
fact that the first the two canonical variables keep 98% of common variability, 78% was kept in the pulp fiber set and about 94% of the paper set as a whole. In the proportion of opposite canonical
variable,there were approximately 64% for the paper set of variables and 78% for the pulp fiber set of variables kept for the two respectively.
Gabarit pour un devoir à l'Université Laval
Il s'agit d'un gabarit simple destiner à des étudiants qui n'ont jamais utilisé LaTeX. J'ai inclus plusieurs commentaires afin de donner quelques explications quant aux commandes utilisées.
Jérôme Soucy
NIH Biosketch Template
A LaTeX class implementing the new (as of 2015) NIH Biographical Sketch Format. The original template can be found at the author's GitHub page. This LaTeX document class tries to adhere to the
Biographical Sketch formatting requirements outlined in NIH Notice NOT-OD-15-032. This new format is required for applications submitted for due dates on or after May 25, 2015. I tried to mimic the
example documents provided on the SF 424 (R&R) Forms and Applications page as closely as possible. I intend to use this class for my own upcoming grant submissions; however I offer no guarantee
of conformity to NIH requirements.
Paul M. Magwene (uploaded by LianTze Lim)
Toric varieties I
The lecture notes are based on Tom Coates' lecture on toric varieties. A few references: M. Audin, Toric actions on symplectic manifolds W. Fulton, Toric varieties Cox-Schenck-Little, Toric varieties
Actividad Extra Distribución Agave.
Computo Forense Agave
The goal of this project is to explore both the theory behind the Extended Kalman Filter and the way it was used to localize a four-wheeled mobile-robot. This can be achieved by estimating in
real-time the pose of the robot, while using a pre-acquired map through Laser Range Finder (LRF). The LRF is used to scan the environment, which is represented through line segments. Through a
prediction step, the robot simulates its kinematic model to predict his current position. In order to minimize the difference between the matched lines from the global and local maps, a update step
is implemented. It should be noted that every measurement has associated uncertainty that needs to be taken into account when performing each step of the Extended Kalman Filter. These uncertainties,
or noise, are described by covariance matrices that play a very important role in the algorithm. Since we are dealing with an indoor structured environment, mainly composed by walls and
straight-edged objects, the line segment representation of the maps was the chosen method to approach the problem.
João Rosa | {"url":"https://sv.overleaf.com/gallery/recent/page/366","timestamp":"2024-11-02T04:34:57Z","content_type":"text/html","content_length":"46315","record_id":"<urn:uuid:bde4d238-374f-4938-af66-6330813a4751>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00783.warc.gz"} |
Goldilocks Numbers
By Stephen Meskin
A Goldilocks number is a number that is not too hard and not too easy: It is just right. Have you ever tried to log into a somewhat sensitive website such as a bank, credit card account, etc., and
the site owner wants to make sure you are you? So, they send you a number, which you need to send back to them. I went through a period in which I had to do it frequently in a short period of time.
It started me thinking about how those numbers were chosen.
I quickly decided that they wouldn’t be chosen randomly in accordance with a uniform distribution, because in that case, one is likely to get 000000 or 112212, which are too easy. On the other hand,
the customer may not be able to cut and paste the number, so it might be good customer relations not to make the number too hard—such as 475192 or 051619. Six digits seems about right, but that
raises the question of whether there are enough 6-digit Goldilocks numbers. In fact, that is the
Problem: How many 6-digit numbers are a) too easy, b) too hard, or c) just right?
As a first approximation, we’ll say that a 6-digit number is too easy if among the 6 digits there are no more than 2 distinct digits. And we’ll say that it is too hard if there are 5 or 6 distinct
digits among the 6. (In this problem a 6-digit number can have leading zeros.)
Solutions may be emailed to puzzles@actuary.org.
In order to make the solver list, your solutions must be received by February 1, 2024.
Solution to Previous Puzzle: Radio Contests—Part 2
If you have 5 guesses at a random number between 1 and 550, and you know after each guess if the correct answer is higher or lower, what is the probability of you getting the right number before your
guesses run out? If one picks 275 for the first guess, one of two things will happen: One, the contestant has a 1 in 550 chance of guessing the lucky number. Or two, the contestant will know not only
that 275 is wrong but that the lucky number is either between 1 and 274 or 276 and 550. If the number is between 1 and 274, you again guess the midpoint (now with a 1-in-274 chance of finding the
lucky number)—138—and repeat the process three more times if you don’t guess the right number. After five guesses, the contestant has a 5.6% chance of winning.
If the radio station wants the contest to go on as long as possible, what number should the radio station pick? Let’s create a 55-cell-by-55-cell matrix, where the rows are the original guess, and
the columns are the actual lucky number. Each cell is equally likely, and we want to determine the average number of guesses it would take to find the luck number in each cell. Fortunately, this
isn’t as cumbersome as it seems due to the recursive nature of the problem. Specifically, if your first guess is 17, if the actual number is 17, you are right after one guess. If the actual number is
less than 17, no matter what the lucky number is, your probability of getting the lucky number is now if are playing the guessing game with 16 possibilities instead of 55. Similarly, if the lucky
number is greater than 17, your new probability of getting the lucky number is if you played the game with 38 choices instead of 55. We can use this information, work backward, and use a little
elbow-grease to find the average number of guesses it takes for each cell. After doing this, we just need to see which column has the highest average number of correct guesses. This turns out to be
where the lucky number is 28, which has an average of just over 6.85 guesses to find the lucky number.
If the radio station wants the contest to end as soon as possible, what number should it pick? Using the same matrix we created in part 3, we can see the columns where the lucky number is 1 or 55
both have the lowest number of expected guesses, where it will take on average just over 4.59 guesses to find the correct number. Notice the radio station can shorten the contest by over 30% just by
strategically picking lucky numbers close to the endpoints!
Bob Conger, Rui Gio, Ryan Jubber, Clive Keating, Chi Kwok, David Promislow, Tomasz Serbinowski, and Al Spooner. | {"url":"https://contingencies.org/goldilocks-numbers/","timestamp":"2024-11-13T00:08:25Z","content_type":"text/html","content_length":"89329","record_id":"<urn:uuid:5193e52a-04e4-4efc-a3d5-ce7089ad9edc>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00555.warc.gz"} |
transformed method
Transforms the event from the global coordinate space into the coordinate space of an event receiver.
The coordinate space of the event receiver is described by transform. A null value for transform is treated as the identity transformation.
The resulting event will store the base event as original, delegates most properties to original, except for localPosition and localDelta, which are calculated based on transform on first use and
The method may return the same object instance if for example the transformation has no effect on the event. Otherwise, the resulting event will be a subclass of, but not exactly, the original event
class (e.g. PointerDownEvent.transformed may return a subclass of PointerDownEvent).
Transforms are not commutative, and are based on original events. If this method is called on a transformed event, the provided transform will override (instead of multiplied onto) the existing
transform and used to calculate the new localPosition and localDelta.
PointerExitEvent transformed(Matrix4? transform) {
if (transform == null || transform == this.transform) {
return this;
return _TransformedPointerExitEvent(original as PointerExitEvent? ?? this, transform); | {"url":"https://api.flutter.dev/flutter/gestures/PointerExitEvent/transformed.html","timestamp":"2024-11-14T20:03:54Z","content_type":"text/html","content_length":"9808","record_id":"<urn:uuid:3300d574-5334-4834-bd1c-26ef05e26a30>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00124.warc.gz"} |
Localisation documentation
Wed, 26/01/2011 - 12:39 — jaspert
I am revising Simile with the intention that all translatable text appears in messages.tcl. First step is to get rid of the text objects still included in the Prolog executable as these are
inaccessible to would-be translators until they are used.
Here is the plan! (Fri, 28/01/2011 - 10:21 — jaspert)
The Tcl files in Simile (.tcl extension) are found in the Run directory under the install directory, and in the IOTools directory and its subdirectories. Within these files you can grep for calls to
the tr. function. The function itself is defined in Run/language.tcl. The version currently in use just returns its argument unchanged.
To make a translated version, the tr. function should return the localized version of any string. For example, this would do English to French:
proc tr. {Eng} {
set listEng [list Hello {How are you?} Please Thankyou Goodbye]
set listFr [list Bonjour {Comment allez-vous?} {S'il vous plait} Merci {Au revoir}]
set index [lsearch $listEng $Eng]
if {index==-1} {return $Eng} else {return [lindex $listFr $index}
Note that if the string is not found, the argument is returned unchanged as a fail-safe.
Simple! All you have to do is find out which strings you need to translate. Most calls to tr. are in messages.tcl, which uses keys to find strings for queries generated in other parts of Simile. But
there are calls to tr. in a lot of the Tcl files. You should grep for these and provide translations of the argument strings. Where the argument is a variable which has been defined somewhere else,
we will include a comment starting "TRANSLATOR" and giving the values the argument can have at that point, so you can make a complete list of possible arguments for which to provide translations.
Note that strings to be translated can contain references of the form %1$s. These are where references will be plugged into the string, usually captions of model components or other labels supplied
by the modeller. The translated string should contain the reference unchanged but possibly in a new position. e.g., if %1$s is a colour, then the English {The %1$s cat} translates to the French {Le
chat %1$s}. TRANSLATOR comments will be added to indicate what sort of reference to expect in each case. If it has a d rather than an s at the end, it is a whole number.
Please ask any questions here! | {"url":"https://simulistics.com/localisation-documentation","timestamp":"2024-11-13T02:56:41Z","content_type":"application/xhtml+xml","content_length":"11455","record_id":"<urn:uuid:037ddf8a-5202-4efa-97d9-cf14f07b13b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00684.warc.gz"} |
Be A Math Champion By Use Our Virtual Library
• 0
If you want to excel in Math, it’s something you can really do without breaking too much sweat. Fret not as this is not that tough! Math is really no sorcery and requires some healthy love for the
subject along with some patience and hard work. Websites such as MyEdge also enable you to rediscover math like never before. Like many math wizards, you can also read a mathematics book in an online
library (you can refer to LIVErary @ myedge.in) to first initiate the love for the subject for starters. Some of the popular Maths books available at LIVErary are Mathematical Statistics : A Unified
Introduction (Fienberg, S.), International Comparison in Mathematics Education (Kaiser, Gabriele), Radical Equations : Organizing Math Literacy in America’s Schools (Moses, Robert P.), Mathematics in
the Early Years (Clemson, David), Logic from A to Z : The Routledge Encyclopedia of Philosophy Glossary of Logical and Mathematical Terms (Bacon, John B.), Basic Mathematics for Economists (Rosser,
Mike), Cult of Pythagoras : Math and Myths (Martinez, Alberto A.) among a hundred others.
Mathematics is no quixotic conquest and you don’t have to engross yourself in repetitive number crunching or even formula memorizing. The truth is that math like any other subject needs the proper
love and right regimen that can help you conquer the stress and do well in the subject.
Here are 5 easy tips that can help you become a math wizard:
1. Keep solving till you master: Math is all about practice. It is not a very oral subject that needs cramming or even memorizing. In fact math is all about doing well at the finer points. Practice
does make you perfect and math unlike other subjects need perseverance and routine.
2. Always work hard on your weaknesses: Mathematicians will tell you how, math needs problem solving with your grey areas. Your weaknesses may disqualify your strengths and nowhere is it most
apparent than in Mathematics. Challenges need to be faced headlong, you need to solve problems interestingly.
3. Work on solved examples: A good way to master math is to read the solved examples in algebra, arithmetic or even geometry. An academic had remarked previously how, one had to just practice and
study worked examples. Additional problem solving is always good for you if you want to know how a theorem or a formula can be solved given a way. Different relationships in math need different
methods. This makes for a unique problem solving method.
4. Never give up till you finally crack it up! Many math problems require help, but some can be ingeniously solved with time and repeated efforts. If you have the time, try solving a sum on your
own. Do not relax or give up if you don’t get the answer then and there. The key to your success lies in taking up the reins on your own and solving the sum.
5. Try putting yourself in a problem solving position: Experts and math wizards will tell you how you can put yourself into another person’s shoes and then deliver explanations or the lessons. In
other words, you become the teacher and that helps you put yourself in the learner’s shoes. While you learn the math problem, you also understand the sum by itself.
About MyEdge
MyEdge is an initiative by like minded people who want to change the world of education. With a strong feeling in our hearts that education and learning should not be a privilege of a select few, we
want to be the converging point of students, parents and teachers from around the world.
5 Tips To Be The Next Math Wizard | {"url":"https://myedge.in/blog/?p=128373","timestamp":"2024-11-06T08:58:40Z","content_type":"text/html","content_length":"84597","record_id":"<urn:uuid:5af5a916-7890-4495-a9d2-a6de3d7d2060>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00580.warc.gz"} |
Make a valid conclusion in the situation. if cost price > selli-Turito
Are you sure you want to logout?
Make a valid conclusion in the situation.
If cost price > selling price, then the transaction suffers loss.
Cost price = $255 and selling price = $230.
The correct answer is: Hence we can conclude that the transaction suffers loss.
Law of Detachment states that if p
Consider the statement into two separate statements
p: Cost price > Selling price
q: The transaction suffers loss
So we can write the given statement “If cost price > selling price, then the transaction suffers loss” as:
We are given
Cost price = $255 and Selling price = $230
Here, Cost price > Selling price so we can say that the p statement is true and hence we can conclude that the q statement is also true i .e the transaction suffers loss.
Final Answer:
Hence we can conclude that the transaction suffers loss.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/make-a-valid-conclusion-in-the-situation-if-cost-price-selling-price-then-the-transaction-suffers-loss-cost-price-q6152cb6a","timestamp":"2024-11-04T14:03:46Z","content_type":"application/xhtml+xml","content_length":"337252","record_id":"<urn:uuid:1005f837-0d52-47e6-bae9-524e78f111ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00870.warc.gz"} |
Math 156 Brookdale Community College Math Group Project - Course Help Online
I am very confuse on all parts of the project and need help. Can someone help me?
Group Project 1
MATH 156
Project 1 is due Thursday July 2
Projects is worth 25 points, projects handed in after the due date will lose 1 point per day
The following table gives the price and demand for a certain brand of calculator.
Demand in hundreds
Price in dollars
Find the linear demand function using regression in Excel or with your
Report and interpret the slope of your demand function.
Using your function find the price when the demand is zero.
Using your function find the demand when the price is $100 algebraically.
A supplier is willing to supply 800 calculators when the price is $36.80
and is willing to supply 1500 calculators when the price is $51.50. If the
supply function is linear, find the supply function algebraically letting x be
hundreds of calculators.
Find the equilibrium point algebraically using your demand and
supply functions. Interpret the meaning of this point in a sentence.
Graph the supply and demand equations on the same set of axis with your
graphing calculator or graphing utility. Copy the graph into your paper.
Make sure you include a title, the scale, label the axes, and clearly label each
function with the equation.
Label the equilibrium point on your graph.
Suppose the price of the calculator is now $40, predict what will happen
to the price and explain why using supply and demand.
Remember that Revenue is (price)(x). Take the demand function from 1a. and
find the revenue function. Since price is in dollars and x is in hundreds of
calculator then revenue is in hundreds of dollars.
What kind of function is the Revenue function?
Find the maximum revenue algebraically. Explain its meaning using a sentence.
Find when the Revenue function is zero.
What kind of function is the Average cost function?
Find the horizontal asymptote algebraically. What does it mean for the function?
The average cost function is found by taking the cost function and dividing by x.
If the cost function for an appliance store is 𝐶(𝑥) = 175𝑥 + 22,500
x : Number of air conditioners and C(x) is total cost in dollars of producing and
selling x air conditioners. Find the average cost function.
Group Project 1
MATH 156
Project 1 is due Thursday July 2
Projects is worth 25 points, projects handed in after the due date will lose 1 point per day
The following table gives the price and demand for a certain brand of calculator.
Demand in hundreds
Price in dollars
Find the linear demand function using regression in Excel or with your
Report and interpret the slope of your demand function.
Using your function find the price when the demand is zero.
Using your function find the demand when the price is $100 algebraically.
A supplier is willing to supply 800 calculators when the price is $36.80
and is willing to supply 1500 calculators when the price is $51.50. If the
supply function is linear, find the supply function algebraically letting x be
hundreds of calculators.
Find the equilibrium point algebraically using your demand and
supply functions. Interpret the meaning of this point in a sentence.
Graph the supply and demand equations on the same set of axis with your
graphing calculator or graphing utility. Copy the graph into your paper.
Make sure you include a title, the scale, label the axes, and clearly label each
function with the equation.
Label the equilibrium point on your graph.
Suppose the price of the calculator is now $40, predict what will happen
to the price and explain why using supply and demand.
Remember that Revenue is (price)(x). Take the demand function from 1a. and
find the revenue function. Since price is in dollars and x is in hundreds of
calculator then revenue is in hundreds of dollars.
What kind of function is the Revenue function?
Find the maximum revenue algebraically. Explain its meaning using a sentence.
Find when the Revenue function is zero.
What kind of function is the Average cost function?
Find the horizontal asymptote algebraically. What does it mean for the function?
The average cost function is found by taking the cost function and dividing by x.
If the cost function for an appliance store is 𝐶(𝑥) = 175𝑥 + 22,500
x : Number of air conditioners and C(x) is total cost in dollars of producing and
selling x air conditioners. Find the average cost function. | {"url":"https://coursehelponline.com/math-156-brookdale-community-college-math-group-project/","timestamp":"2024-11-13T23:17:02Z","content_type":"text/html","content_length":"45271","record_id":"<urn:uuid:8124bb28-038c-4543-8eb1-a870b2168545>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00872.warc.gz"} |
2 Katha [Nepal] to Satak
Katha [Nepal] [katha] Output
2 katha [nepal] in ankanam is equal to 101.25
2 katha [nepal] in aana is equal to 21.3
2 katha [nepal] in acre is equal to 0.16735522395141
2 katha [nepal] in arpent is equal to 0.19809407600776
2 katha [nepal] in are is equal to 6.77
2 katha [nepal] in barn is equal to 6.772631616e+30
2 katha [nepal] in bigha [assam] is equal to 0.50625
2 katha [nepal] in bigha [west bengal] is equal to 0.50625
2 katha [nepal] in bigha [uttar pradesh] is equal to 0.27
2 katha [nepal] in bigha [madhya pradesh] is equal to 0.6075
2 katha [nepal] in bigha [rajasthan] is equal to 0.26776859504132
2 katha [nepal] in bigha [bihar] is equal to 0.26781778104335
2 katha [nepal] in bigha [gujrat] is equal to 0.41838842975207
2 katha [nepal] in bigha [himachal pradesh] is equal to 0.83677685950413
2 katha [nepal] in bigha [nepal] is equal to 0.1
2 katha [nepal] in biswa [uttar pradesh] is equal to 5.4
2 katha [nepal] in bovate is equal to 0.01128771936
2 katha [nepal] in bunder is equal to 0.06772631616
2 katha [nepal] in caballeria is equal to 0.001505029248
2 katha [nepal] in caballeria [cuba] is equal to 0.0050466703546945
2 katha [nepal] in caballeria [spain] is equal to 0.001693157904
2 katha [nepal] in carreau is equal to 0.05250102027907
2 katha [nepal] in carucate is equal to 0.0013935456
2 katha [nepal] in cawnie is equal to 0.125419104
2 katha [nepal] in cent is equal to 16.74
2 katha [nepal] in centiare is equal to 677.26
2 katha [nepal] in circular foot is equal to 9281.91
2 katha [nepal] in circular inch is equal to 1336594.74
2 katha [nepal] in cong is equal to 0.6772631616
2 katha [nepal] in cover is equal to 0.25102415181616
2 katha [nepal] in cuerda is equal to 0.17233159328244
2 katha [nepal] in chatak is equal to 162
2 katha [nepal] in decimal is equal to 16.74
2 katha [nepal] in dekare is equal to 0.6772636083908
2 katha [nepal] in dismil is equal to 16.74
2 katha [nepal] in dhur [tripura] is equal to 2025
2 katha [nepal] in dhur [nepal] is equal to 40
2 katha [nepal] in dunam is equal to 0.6772631616
2 katha [nepal] in drone is equal to 0.0263671875
2 katha [nepal] in fanega is equal to 0.10532864099533
2 katha [nepal] in farthingdale is equal to 0.66923237312253
2 katha [nepal] in feddan is equal to 0.16248079995282
2 katha [nepal] in ganda is equal to 8.44
2 katha [nepal] in gaj is equal to 810
2 katha [nepal] in gajam is equal to 810
2 katha [nepal] in guntha is equal to 6.69
2 katha [nepal] in ghumaon is equal to 0.16735537190083
2 katha [nepal] in ground is equal to 3.04
2 katha [nepal] in hacienda is equal to 0.0000075587406428571
2 katha [nepal] in hectare is equal to 0.06772631616
2 katha [nepal] in hide is equal to 0.0013935456
2 katha [nepal] in hout is equal to 0.47652194013619
2 katha [nepal] in hundred is equal to 0.000013935456
2 katha [nepal] in jerib is equal to 0.33501838235294
2 katha [nepal] in jutro is equal to 0.11768256500434
2 katha [nepal] in katha [bangladesh] is equal to 10.13
2 katha [nepal] in kanal is equal to 1.34
2 katha [nepal] in kani is equal to 0.421875
2 katha [nepal] in kara is equal to 33.75
2 katha [nepal] in kappland is equal to 4.39
2 katha [nepal] in killa is equal to 0.16735537190083
2 katha [nepal] in kranta is equal to 101.25
2 katha [nepal] in kuli is equal to 50.63
2 katha [nepal] in kuncham is equal to 1.67
2 katha [nepal] in lecha is equal to 50.63
2 katha [nepal] in labor is equal to 0.00094478380426006
2 katha [nepal] in legua is equal to 0.000037791352170402
2 katha [nepal] in manzana [argentina] is equal to 0.06772631616
2 katha [nepal] in manzana [costa rica] is equal to 0.096904712804194
2 katha [nepal] in marla is equal to 26.78
2 katha [nepal] in morgen [germany] is equal to 0.27090526464
2 katha [nepal] in morgen [south africa] is equal to 0.079054880541613
2 katha [nepal] in mu is equal to 1.02
2 katha [nepal] in murabba is equal to 0.0066942089580564
2 katha [nepal] in mutthi is equal to 54
2 katha [nepal] in ngarn is equal to 1.69
2 katha [nepal] in nali is equal to 3.38
2 katha [nepal] in oxgang is equal to 0.01128771936
2 katha [nepal] in paisa is equal to 85.2
2 katha [nepal] in perche is equal to 19.81
2 katha [nepal] in parappu is equal to 2.68
2 katha [nepal] in pyong is equal to 204.86
2 katha [nepal] in rai is equal to 0.423289476
2 katha [nepal] in rood is equal to 0.66942148760331
2 katha [nepal] in ropani is equal to 1.33
2 katha [nepal] in satak is equal to 16.74
2 katha [nepal] in section is equal to 0.00026149276859504
2 katha [nepal] in sitio is equal to 0.0000376257312
2 katha [nepal] in square is equal to 72.9
2 katha [nepal] in square angstrom is equal to 6.772631616e+22
2 katha [nepal] in square astronomical units is equal to 3.0262627206129e-20
2 katha [nepal] in square attometer is equal to 6.772631616e+38
2 katha [nepal] in square bicron is equal to 6.772631616e+26
2 katha [nepal] in square centimeter is equal to 6772631.62
2 katha [nepal] in square chain is equal to 1.67
2 katha [nepal] in square cubit is equal to 3240
2 katha [nepal] in square decimeter is equal to 67726.32
2 katha [nepal] in square dekameter is equal to 6.77
2 katha [nepal] in square digit is equal to 1866240
2 katha [nepal] in square exameter is equal to 6.772631616e-34
2 katha [nepal] in square fathom is equal to 202.5
2 katha [nepal] in square femtometer is equal to 6.772631616e+32
2 katha [nepal] in square fermi is equal to 6.772631616e+32
2 katha [nepal] in square feet is equal to 7290
2 katha [nepal] in square furlong is equal to 0.016735522395141
2 katha [nepal] in square gigameter is equal to 6.772631616e-16
2 katha [nepal] in square hectometer is equal to 0.06772631616
2 katha [nepal] in square inch is equal to 1049760
2 katha [nepal] in square league is equal to 0.000029054636137491
2 katha [nepal] in square light year is equal to 7.5667219334989e-30
2 katha [nepal] in square kilometer is equal to 0.0006772631616
2 katha [nepal] in square megameter is equal to 6.772631616e-10
2 katha [nepal] in square meter is equal to 677.26
2 katha [nepal] in square microinch is equal to 1049759073947000000
2 katha [nepal] in square micrometer is equal to 677263161600000
2 katha [nepal] in square micromicron is equal to 6.772631616e+26
2 katha [nepal] in square micron is equal to 677263161600000
2 katha [nepal] in square mil is equal to 1049760000000
2 katha [nepal] in square mile is equal to 0.00026149276859504
2 katha [nepal] in square millimeter is equal to 677263161.6
2 katha [nepal] in square nanometer is equal to 677263161600000000000
2 katha [nepal] in square nautical league is equal to 0.000021939815924877
2 katha [nepal] in square nautical mile is equal to 0.00019745816913672
2 katha [nepal] in square paris foot is equal to 6419.56
2 katha [nepal] in square parsec is equal to 7.1130563022675e-31
2 katha [nepal] in perch is equal to 26.78
2 katha [nepal] in square perche is equal to 13.26
2 katha [nepal] in square petameter is equal to 6.772631616e-28
2 katha [nepal] in square picometer is equal to 6.772631616e+26
2 katha [nepal] in square pole is equal to 26.78
2 katha [nepal] in square rod is equal to 26.78
2 katha [nepal] in square terameter is equal to 6.772631616e-22
2 katha [nepal] in square thou is equal to 1049760000000
2 katha [nepal] in square yard is equal to 810
2 katha [nepal] in square yoctometer is equal to 6.772631616e+50
2 katha [nepal] in square yottameter is equal to 6.772631616e-46
2 katha [nepal] in stang is equal to 0.25000485847176
2 katha [nepal] in stremma is equal to 0.6772631616
2 katha [nepal] in sarsai is equal to 240.99
2 katha [nepal] in tarea is equal to 1.08
2 katha [nepal] in tatami is equal to 409.74
2 katha [nepal] in tonde land is equal to 0.12278157389413
2 katha [nepal] in tsubo is equal to 204.87
2 katha [nepal] in township is equal to 0.0000072636815951132
2 katha [nepal] in tunnland is equal to 0.13719778818572
2 katha [nepal] in vaar is equal to 810
2 katha [nepal] in virgate is equal to 0.00564385968
2 katha [nepal] in veli is equal to 0.084375
2 katha [nepal] in pari is equal to 0.066942148760331
2 katha [nepal] in sangam is equal to 0.26776859504132
2 katha [nepal] in kottah [bangladesh] is equal to 10.13
2 katha [nepal] in gunta is equal to 6.69
2 katha [nepal] in point is equal to 16.74
2 katha [nepal] in lourak is equal to 0.13388429752066
2 katha [nepal] in loukhai is equal to 0.53553719008264
2 katha [nepal] in loushal is equal to 1.07
2 katha [nepal] in tong is equal to 2.14
2 katha [nepal] in kuzhi is equal to 50.63
2 katha [nepal] in chadara is equal to 72.9
2 katha [nepal] in veesam is equal to 810
2 katha [nepal] in lacham is equal to 2.68
2 katha [nepal] in katha [assam] is equal to 2.53
2 katha [nepal] in katha [bihar] is equal to 5.36
2 katha [nepal] in dhur [bihar] is equal to 107.13
2 katha [nepal] in dhurki is equal to 2142.54 | {"url":"https://hextobinary.com/unit/area/from/kathanp/to/satak/2","timestamp":"2024-11-04T10:09:32Z","content_type":"text/html","content_length":"127644","record_id":"<urn:uuid:3f5ccb49-45f9-47b4-a7c2-889f591aaf24>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00014.warc.gz"} |
Centralizers of subgroups in simple locally finite groups
Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite
K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups. Then for any
finite subgroup F consisting of K-semisimple elements in G, the centralizer C-G(F) has an infinite abelian subgroup A isomorphic to a direct product of Z(pi) for infinitely many distinct primes p(i).
K. ERSOY and M. Kuzucuoğlu, “Centralizers of subgroups in simple locally finite groups,” JOURNAL OF GROUP THEORY, pp. 9–22, 2012, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511 | {"url":"https://open.metu.edu.tr/handle/11511/44423","timestamp":"2024-11-04T18:49:53Z","content_type":"application/xhtml+xml","content_length":"53254","record_id":"<urn:uuid:cafa603b-b93b-4e4c-8d33-6758607b3f69>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00404.warc.gz"} |
Syllabus Detail
Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter,
refer to the Courses page.
MAT 114: Convex Geometry
Approved: 2006-02-28 (revised 2019-10-25, Kuperberg, Rademacher)
Suggested Textbook:
(actual textbook varies by instructor; check your instructor)
Lecture notes written by the faculty have been used successfully in the recent past. They contain a fair number of exercises. 71 pages available for free. Alternate textbook: Convex Sets and Their
Applications, by Steven Lay.
Search by ISBN on Amazon:
MAT 021C; (MAT 022A or MAT 067).
Suggested Schedule:
If teaching from lecture notes:
Lecture(s) Sections Comments/Topics
3 Fundamental definitions: Affine sets, convex set, convex hull. Examples.
3 Caratheodory’s theorem, Radon’s theorem.
3 Helly’s theorem and applications.
3 Separating and supporting hyperplanes. Faces, extreme points.
3 Sets of constant width, diameter Borsuk’s problem.
3 Polyhedra and Polytopes. Examples and main operations (e.g. Projections, Schlegel Diagrams).
3 Graphs of polytopes, Euler’s formula. Coloring problems.
3 Duality and Polarity.
3 Convex bodies and Lattices. Minkowski’s first theorem, Blichfeldt’s theorem.
If using Lay's book:
Lecture Sections Comments/Topics
Definition of convex sets, convex bodies, and convex polytopes. Examples of convex polytopes and non-polytopes in dimensions 2 and 3. Examples of intuitive results and open
2 problems, e.g., sphere packing.
Euclidean and convex geometry in n dimensions. k-dimensional faces of n-dimensional polytopes. Volumes of parallelepipeds and simplices. Volume of the n-sphere and multiple
20 integrals for volumes.
Definition of a regular polytope. The list of regular n-polytopes and enumeration of faces. Dihedral angles of regular polytopes and the necessary condition that the total angle
Wikipedia of each ridge is less than 2*pi.
2 Closures, convex hulls, and Caratheodory's theorem.
3, 4 Existence of separating and supporting hyperplanes.
Extreme points and the finite-dimensional Krein-Milman theorem.
The definition of k-extreme points. The relation between k-facets and k-extreme points. Examples of k-extreme points in non-polytopes.
14 Arithmetic with sets and Minkowski sums. Statement of the Brunn-Minkowski and Rogers-Shephard inequalities. The isoperimetic inequality as a corollary of Brunn-Minkowski.
23 Polar duals of convex sets, convex bodies, and polytopes. The correspondence between the face poset of a polytope and its dual.
Optional Advanced Topics
The classification of convex polygons that tile the plane.
The symmetrization proof of Brunn-Minkowski inequality and the isoperimetric inequality.
The topological proof of existence of regular polytopes. The definition of a Coxeter simplex.
Sphere packings and Voronoi regions. Examples of sphere packings in higher dimensions.
The largest ellipsoid in a convex body (the John ellipsoid). The definition of Banach-Mazur distance and the fact that it is bounded in each dimension.
Additional Notes:
This course should serve as a bridge between the lower division courses and more abstract upper division courses. There are a few excellent supplementary resources: Eggleston’s Convexity, Yaglom and
Boltyanskii’s Convex Figures, and Ziegler’s Lectures on Polytopes. For the final part, The Geometry of Numbers by C.D. Olds, Anneli Lax, and Guiliana Davidoff is appropriate for undergraduates. | {"url":"https://www.math.ucdavis.edu/courses/syllabus_detail?cm_id=101","timestamp":"2024-11-09T10:36:44Z","content_type":"text/html","content_length":"27871","record_id":"<urn:uuid:80d8c4b6-ccbe-4002-80a5-b6dc6e840904>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00329.warc.gz"} |
European Molecular Biology Laboratories. Maintain the EMBL database, one of the major public sequence databases.
European Molecular Biology Network:
was established in 1988, and provides services including local molecular databases and software for molecular biologists in Europe. There are several large outposts of EMBnet, including EXPASY.
From information theory, a measure of the unpredictable nature of a set of possible elements. The higher the level of variation within the set, the higher the entropy.
In a toss of a “fair” coin, the number of heads in a row that can be expected is the logarithm of the number of tosses to the base 2. The law may be generalized for more than two possible outcomes by
changing the base of the logarithm to the number of out-comes. This law was used to analyze the number of matches and mismatches that can be expected between random sequences as a basis for scoring
the statistical significance of a sequence alignment.
See Expressed Sequence Tag
E value. The number of different alignents with scores equivalent to or better than S that are expected to occur in a database search by chance. The lower the E value, the more significant the score.
In a database similarity search, the probability that an alignment score as good as the one found between a query sequence and a database sequence would be found in as many comparisons between random
sequences as was done to find the matching sequence. In other types of sequence analysis, E has a similar meaning.
An algorithm for locating similar sequence patterns in a set of sequences. A guessed alignment of the sequences is first used to generate an expected scoring matrix representing the distribution of
sequence characters in each column of the alignment, this pattern is matched to each sequence, and the scoring matrix values are then updated to maximize the alignment of the matrix to the sequences.
The procedure is repeated until there is no further improvement.
Coding region of DNA. See CDS.
Randomly selected, partial cDNA sequence; represents it's corresponding mRNA. dbEST is a large database of ESTs at
, NCBI.
Some measurements are found to follow a distribution that has a long tail which decays at high values much more slowly than that found in a normal distribution. This slow-falling type is called the
extreme value distribution. The alignment scores between unrelated or random sequences are an example. These scores can reach very high values, particularly when a large number of comparisons are
made, as in a database similarity search. The probability of a particular score may be accurately predicted by the extreme value distribution, which follows a double negative exponential function
after Gumbel. | {"url":"https://zhangroup.aporc.org/BioinformaticsGlossaryE","timestamp":"2024-11-11T06:43:49Z","content_type":"application/xhtml+xml","content_length":"14560","record_id":"<urn:uuid:1799d204-d532-4752-9c69-1ea8052bc4bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00144.warc.gz"} |
Computational Issues
The VARMAX procedure uses numerous linear algebra routines and frequently uses the sweep operator (Goodnight 1979) and the Cholesky root (Golub and Van Loan 1983).
In addition, the VARMAX procedure uses the nonlinear optimization (NLO) subsystem to perform nonlinear optimization tasks for the maximum likelihood estimation. The optimization requires intensive
For some data sets, the computation algorithm can fail to converge. Nonconvergence can result from a number of causes, including flat or ridged likelihood surfaces and ill-conditioned data.
If you experience convergence problems, the following points might be helpful:
• Data that contain extreme values can affect results in PROC VARMAX. Rescaling the data can improve stability.
• Changing the TECH=, MAXITER=, and MAXFUNC= options in the NLOPTIONS statement can improve the stability of the optimization process.
• Specifying a different model that might fit the data more closely and might improve convergence.
Let be the length of each series, be the number of dependent variables, be the order of autoregressive terms, and be the order of moving-average terms. The number of parameters to estimate for a
VARMA() model is
As increases, the number of parameters to estimate increases very quickly. Furthermore the memory requirement for VARMA() quadratically increases as and increase.
For a VARMAX() model and GARCH-type multivariate conditional heteroscedasticity models, the number of parameters to estimate and the memory requirements are considerable. | {"url":"http://support.sas.com/documentation/cdl/en/etsug/65545/HTML/default/etsug_varmax_details69.htm","timestamp":"2024-11-02T11:26:27Z","content_type":"application/xhtml+xml","content_length":"19453","record_id":"<urn:uuid:7f36dfa3-7ae3-4dd9-ae28-6cbd22d20a8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00738.warc.gz"} |
Al Khwarizmi Biography
Abu Abdullah Muhammad bin Musa al-Khwarizmi, also called Muhammad ibn-Musa al-Khwarizmi, Muhammad ibn-Musa al-Khowarizmi, and Mohammad Bin Musa Al-Khawarizmi, (flourished early 9th century), was a
Persian scientist, mathematician, and author. He may have been born in 780, or around 800; he may have died in 845, or around 840.
He was born in the town of Khwarizm (now Khiva), in Khorasan province of Persia (now in Uzbekistan). The name al-Khwarizmi means the person from Khwarizm. His family moved soon afterward, to a place
near Baghdad, where he accomplished most of his work in the period between 813 and 833. There are various guesses at his native languages, including Persian or more probably Khwarizmian (now dead).
All of Al-Khwarizimi's works were written in Arabic.
He developed the concept of the algorithm in mathematics (which is a reason for his being called the grandfather of computer science by some people), and the words "algorithm" and "algorism" come
from Latin and English corruptions of his name. He also made major contributions to the fields of algebra, trigonometry, astronomy, geography and cartography. His systematic and logical approach to
solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his 830 book on the subject, Hisab al-jabr wa al-muqabala (حساب الجبر و
المقابلة in Arabic).
While his major contributions were the result of original research, he also did much to synthesize the existing knowledge in these fields from Greek, Indian, and other sources. He appropriated the
place-marker symbol of zero, which originated in India, and he is also responsible for the use of Arabic numerals in mathematics.
Al-Khwarizmi systematized and corrected Ptolemy's research in geography, using his own original findings. He supervised the work of 70 geographers to create a map of the then "known world". When his
work became known in Europe through Latin translations, his influence made an indelible mark on the development of science in the West: His algebra book introduced that discipline to Europe and
became the standard mathematical text at European universities until the 16th century. He also wrote on mechanical devices like the clock, astrolabe, and sundial. His other contributions include
tables that included trigonometric functions, refinements in the geometric representation of conic sections, and aspects of the calculus of two errors
Famous works
Al-Jabr wa-al-Muqabilah from whose title came the name "Algebra"
Kitab al-Jam'a wal-Tafreeq bil Hisab al-Hindi (on Arithmatic, which survived in a Latin translation but was lost in the original Arabic)
Kitab Surat-al-Ard (on geography)
Istikhraj Tarikh al-Yahud (about the Jewish calendar)
Kitab al-Tarikh
Kitab al-Rukhmat (about sun-dials)
Al Khwarizmi Resources | {"url":"http://biographybase.com/biography/Al_Khwarizmi.html","timestamp":"2024-11-04T04:41:33Z","content_type":"text/html","content_length":"9890","record_id":"<urn:uuid:b8da7530-15c0-4cd4-8c44-539e695f857d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00416.warc.gz"} |
sports fandom Archives - Bankers Anonymous
Will the Red Sox win the World Series this year? What are their chances?
What are their chances of going all the way, if they win their first game of the playoffs this Friday?
Aha! I have a chance to apply Bayesian probability theory!
I recently reviewed Nate Silver’s excellent The Signal and The Noise: Why So Many Predictions Fail – But Some Don’t , which at its core, advocates we adopt Bayesian probability methods for
forecasting complex events. Like Red Sox World Series championships.
Nate Silver’s Big Idea
Silver’s big idea is for us to move away from “I have the explanation and I know what’s going to happen,” to a different way of understanding the world characterized by “I can articulate a range of
outcomes and attach meaningful probabilities to the possible outcomes.”
Bayesian probability
Bayes’ theory, Silver explains, helps us come up with the most accurate probability of some event occurring. Fortunately, it’s not too complicated.
The Red Sox, of course, defy all probabilities
As we approach the MLB playoffs I’m fully aware of the irony of applying rational Bayesian probability to something as totally irrational, magical, and unlikely as Red Sox playoff outcomes.
My childhood and young adulthood consisted of them repeatedly snatching defeat from the jaws of victory. Both the Game Six World Series loss in 1986 to the Mets and the 2003 ALCS loss to the Yankees
[1] defied all semblance of probability – we didn’t need a mathematical theorem to tell us that.
At the time, all we knew was that God personally intervened in baseball outcomes and that she enjoyed torturing us. And we hoped that God had plans for our redemption, some day.
We know now that, like the biblical story of Job, Red Sox Nation suffered for a reason. We now own the Greatest Sports Victory of All Time, coming back impossibly from devastating losses in the
first 3 ALCS games in 2004 to vanquish the Yankees and sweep the Cardinals.[2] No sports victory has ever been as sweet as that. It was all so improbable. No math could ever explain that magic.
Greatest Sports Moments Ever, reduced to probabilities
And yet, I insist we try to learn Bayesian Probability today
Fine then.
To use it, we need to define three known (or assumed) variables, in order to come up with a fourth, unknown variable, which is the thing we want to know, the probability of an event.
The known or assumed variables will be:
1. X = an initial estimate of the likelihood of an event. This is called a ‘prior’ since it’s our best guess of some probability prior to further investigation. Before the playoffs even begin, how
likely are the Red Sox to become World Series Champions?
2. Y = The probability that if some condition is met, the event will happen. In other words, how probable is it that a team that won the World Series had originally won their first game of the
3. Z = The probability that if that same condition is met, the event will not happen. For a team that did not win the World Series, how probable is it that they won their first game of the
The unknown variable, what we’re trying to determine, is our closest approximation of the probability of the event happening.
4. I’ll call that unknown variable V. What is the probability of the Red Sox winning the World Series, if they win their first game on Friday?
The math formula of Bayes’ theorem, using these four variables, is:
V = (X*Y)/(X*Y + Z(1-X))
I understand that formula makes no sense in the abstract, so that’s why we’ll illustrate it with the Red Sox.
We need an example using numbers, please
Since it’s that time of year, I’ll ask the key question on everyone’s mind right now:
If they win on Friday, October 4th – their first game of the playoffs, will the Boston Red Sox go all the way on to win the World Series?
We can now define variables and assign probabilites
The variable V (This is the unknown what we’re trying to solve for)
V is the probability that the Red Sox win the World Series this year, if they win their first game of the playoffs.
Variable X, our prior
I will make our prior –the initial estimate for the Red Sox winning the World Series – 15%. If all 8 playoff teams had an equal chance of winning the World Series my prior would be 12.5%, the
percent equivalent of 1 divided by 8. But given that the Sox had the best record in baseball this year – and they have studs like Big Papi and Pedroia – I have to boost their prior to 15%.
Variable Y, the conditional probability that the hypothesis is true
One of the requirements for using Bayesian probability theory is that we insert a conditional probability. We can simply express this hypothesis as “If this happens, this other thing is made more
In our example I’ll make the non-crazy hypothesis that there is some positive causal relationship between teams winning their first game of the playoffs and teams that eventually win the entire World
Let’s assume we know, from historical data,[3] that teams that won the World Series had previously won their first game of the playoffs 58% of the time. That’s our variable Y.
Variable Z, the false hypothesis variable
The false hypothesis variable in this example would be made from the 7 of 8 teams that historically begin the playoffs but do not go on to win the World Series. Of these non-champions, what is the
probability they won their first game? I’ll estimate this at 45%[4]
Putting it all together
Using Bayes Theorem, we can now revise our estimate of the Red Sox winning the World Series, after the first playoff game has been played.
If the Red Sox win on October 4^th, we can plug in variables X, Y and Z to determine the new probability of a glorious Red Sox World Series victory, variable V.
Remember: V = X*Y / (X*Y + Z*(1-X))
Plugging in our known and assumed probabilities, we get the
following math:
V = (15% * 58%) / ((15% * 58%) + (45%*(100%-15%)))
Solving that in an Excel Spreadsheet we get
V = 18.5%
Summed up, if the Red Sox win their first game Friday[5], we would revise our probability of them winning the World Series up to 18.5% from 15%.
Intuitively, this makes some sense. There should be only a modest increase in the probability of a World Series championship after one game.
There’s a small positive correlation between winning the first game in the playoffs and eventually winning the World Series.
But even if it’s a blowout one way or another, let’s not get carried away. The chances of them going all the way is only up to 18.5%.
Martyrdom & bloody sacrifice go beyond rational thought
Anchoring effect of priors
We should note, and Silver emphasizes, that the anchoring effect of priors greatly influences our updated probabilities. In plainer English, our starting point for how we think the Red Sox are
likely to do limits our ending point.
If we start with a prior that the Red Sox only have a 5% chance of winning the World Series, then their chances of winning the championship only jump to 6.3% after taking the first game, using my
same assumed inputs.
Again using the same assumptions, if the Red Sox were 75% favorites to win it all, then a first game victory pushes them up to 79.5% favorites using the Bayesian Theorem.
Next Steps
If we want to follow the rest of the Red Sox playoff outcomes probabilistically, we’d take our revised prior – let’s say 18.5% after Game One – and come up with updated probabilities for variables Y
and Z for Game Two. To use new Y and Z variables effectively we would need new historical data to determine the conditional probability of a World Series victory based on Game Two results.
Continued iteration
Nate Silver would advocate applying this constant iteration, revising our probabilities and priors as new information arrives, for a wide range of complex phenomenon that defy prediction. Will Mike
Napoli’s beard change weather patterns inside Fenway? Is it not Nate Silver, but rather Big Papi who is the witch? Will super-agent Scott Boras release a karma-bomb press release on another client
like he did with A-Rod during the 2007 World Series, effectively marking the beginning of the end for A-Rod? The probabilities change as the events unfurl.
Or not
Or conversely, we could just ignore all math, attach ourselves to one big idea, and never let go.
Because unrevised big beliefs, like sports fandom, do have their attractions.
Please see related post Book Review of The Signal and the Noise by Nate Silver
[1] Fie on you New York! Shaking my fist. Arggh!
[2] Incidentally, that 53 minute 30-for30 video of “the Greatest Sports Victory of All Time” I linked to on Youtube is totally awesome. Gives me the chils.
[3] I’m not a baseball stats geek with easy access to this kind of data, so I’m just making up numbers for the sake of illustration.
[4] Again, a stats geek could come up with the correct historical data to suggest a more accurate probability for the false hypothesis, but just work with me here a little bit on my completely made
up numbers.
[5] And of course if my numbers were based on real data, rather than just picked out of the clear blue sky.
Post read (12385) times. | {"url":"https://www.bankers-anonymous.com/tag/sports-fandom/","timestamp":"2024-11-02T01:30:55Z","content_type":"text/html","content_length":"117069","record_id":"<urn:uuid:2dcaab9c-a9bb-4b93-a64f-11f86b043dcd>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00396.warc.gz"} |
JAVASCRIPT: Math and Date Methods
The Math Class
The Math class contains a variety of static methods and constants that can be used by programmers for calculations and comparisons. All of the methods are called from the class name (Math). Each
method takes at least one parameter, performs the required operations and then returns the result. The result can be either stored in a variable, output, or used in a calculation. Below are a few of
the more commonly used Math class methods.
Square Root
returns the square root of a number
returns the base (first number) raised to the power (second number)
Minimum Value
returns the smaller of two or more numbers
returns the larger of two or more values
returns rounded value of the parameter (.5+ rounds up, under .5 rounds down)
Rounding Down
returns the next lowest integer (always rounds down)
Rounding Up
returns the next highest integer (always rounds up)
Absolute Value
returns the absolute value of a number - its positive distance from zero
Random Numbers
generates a random number between 0(inclusive) and 1.0 (exclusive)
a constant variable that returns the value of Pi 3.1415...
Below are some examples of the Math methods and constants in action:
let radius=5;
let diameter=2*radius;
let circumference=diameter*Math.PI;
In line (1), the value for radius could be obtained from input or another calculation. In line (2), the diameter is calculated and stored by doubling the radius. Then, the circumference is
calculuated with the diameter and the PI constant. Finally, in line (4), the circumference value is output.
let radius=5;
window.alert((Math.pow(radius, 2)*Math.PI);
This example calculates and outputs the value of the area of a circle in one line, as opposed to the step by step approach previously displayed in the circumference example. In line(2), the radius is
rasied to the second power, then multiplied by PI and then output.
let a=3;
let b=4;
let c=Math.sqrt(Math.pow(a,2)+Math.pow(b,2));
This example uses the pythagorean theorem (a^2 + b^2 = c^2) to find the missing hypotenuse of a right triangle. Notice how the pow functions are placed inside the sqrt function. This can also be done
in separate steps if desired.
Random Numbers
The Math.random() function generates a value from 0 (inclusive) to 1.0 (exclusive). This can be represented in mathematics using interval notation. A square bracket, [ or ], means inclusive and a
parenthesis, ( or ), means exclusive. So, the range of Math.random() can be represented as [0, 1).
So, to get different ranges, we need to do mathematical operations on the result of Math.random(). For example, if we do Math.random()*10, the range becomes [0, 10). If we add 1, we get [1, 11).
Finally, if we cast the result as an int, we get only integer values
Limiting Decimals
Often, programmers find that they want to limit the number of decimals in the output of a calculation. Sometimes it is for display purposes so numbers line up, other times, it is due to the type of
value they are calculations (money should be limited to two decimals).
JavaScript provides a toFixed() method that can be called from a variable that stores a number. The toFixed() method takes a parameter of the number of decimals that the programmers wants displayed.
let pi=3.1415;
The output of the above is 3.14.
BE CAREFUL - the toFixed method converts a number to a string ("3.14"). Be sure to do all of your calculations PRIOR to outputting the value.
The Date Class
JavaScript also provides a class that gives us functionality with the date. Most often, we are working with the current date, and creating a date variable accesses that information from the computer.
To get started with the date class methods, we need to create a Date variable that accesses the current date and time.
let today=new Date();
Once we have created a Date variable, we call the date methods from that variable. For our example, let's assume that we created this variable on Thursday, March 12, 2020 at 2:03.45pm. Using this
information, look below for examples of some of the more common Date class methods.
Day of Week
returns the day of the week as a number where 0=Sunday and 6=Saturday
Date in Month
returns the day of the month from 1-31
returns the month of the date with 0=January and 11=DecemberThis can easily be converted by adding 1 to the returned value
returns the four digit year of the date
returns the hours of the time from 0-23 (military time)Hours can be converted using mathematics, specifically the modulus(%) operator
returns the minutes of the hour from 0-59
returns the seconds of the minute from 0-59 | {"url":"https://cs.mickeyengel.com/lessons/js_04_mathAndDate.php","timestamp":"2024-11-09T06:34:34Z","content_type":"text/html","content_length":"28009","record_id":"<urn:uuid:60584cef-c2fb-43f8-bf11-3d1c8866bdf4>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00761.warc.gz"} |
Excel Formula: Time Difference Calculation in Python
In this tutorial, we will learn how to calculate the time difference between two specific time ranges in Excel using a Python formula. Specifically, we will check if a time in Column F falls between
00:01 and 06:00 on a Sunday morning, and if the time in Column E falls between 18:00 and 23:59 on a Saturday. If both conditions are met, the formula will return the total hours difference as a
decimal number.
To achieve this, we will use an Excel formula that can be implemented in Python. The formula checks the conditions using the IF function and compares the time values using the TIMEVALUE function.
Let's dive into the step-by-step explanation of the formula:
1. The formula checks if the time in Column F is between 00:01 and 06:00 on a Sunday morning. It uses the TEXT function to extract the day of the week (ddd) from the time in F2 and compares it to
'Sun'. The TIMEVALUE function is used to convert the time strings '00:01' and '06:00' into numeric values for comparison.
2. If the condition in step 1 is true, the formula proceeds to the next IF function. This IF function checks if the time in Column E is between 18:00 and 23:59 on a Saturday. It uses the TEXT
function to extract the day of the week (ddd) from the time in E2 and compares it to 'Sat'. The TIMEVALUE function is used to convert the time strings '18:00' and '23:59' into numeric values for
3. If both conditions in step 2 are true, the formula calculates the difference between the two times (E2-F2) and returns it as a decimal number.
4. If any of the conditions in steps 1 or 2 are false, the formula returns an empty string ('').
Now, let's see some examples to understand how the formula works:
Example 1:
E F
17:30 05:30
18:30 02:00
19:45 06:15
20:00 04:30
21:30 03:45
22:45 05:45
23:59 06:30
The formula would return the following results: - For the first row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the
result is an empty string (''). - For the second row, neither the time in F2 nor the time in E2 meet the conditions, so the result is an empty string (''). - For the third row, the time in F2 is
between 00:01 and 06:00 on a Sunday, and the time in E2 is between 18:00 and 23:59 on a Saturday, so the result is the difference between the two times: 06:15 - 19:45 = -13.5 (negative value
represents a time difference across midnight). - For the fourth row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the
result is an empty string (''). - For the fifth row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty
string (''). - For the sixth row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (''). - For
the seventh row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (''). - For the eighth row,
the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string ('').
Now that we understand the formula and have seen some examples, let's implement it in Python and calculate the time difference in Excel using Python!
An Excel formula
=IF(AND(TEXT(F2,"ddd")="Sun", TIMEVALUE("00:01")<=F2, F2<=TIMEVALUE("06:00")), IF(AND(TEXT(E2,"ddd")="Sat", TIMEVALUE("18:00")<=E2, E2<=TIMEVALUE("23:59")), E2-F2, ""), "")
Formula Explanation
This formula checks whether a time in Column F is between 00:01 and 06:00 on a Sunday morning. If it is, it further checks whether the time in Column E is between 18:00 and 23:59 on a Saturday. If
both conditions are met, it calculates the difference between the two times in decimal hours.
Step-by-step explanation
1. The formula uses the IF function to check if the time in Column F is between 00:01 and 06:00 on a Sunday morning. The condition is checked using the TEXT function to extract the day of the week
(ddd) from the time in F2 and comparing it to "Sun". The TIMEVALUE function is used to convert the time strings "00:01" and "06:00" into numeric values that can be compared.
2. If the condition in step 1 is true, the formula proceeds to the next IF function. This IF function checks if the time in Column E is between 18:00 and 23:59 on a Saturday. The condition is
checked using the TEXT function to extract the day of the week (ddd) from the time in E2 and comparing it to "Sat". The TIMEVALUE function is used to convert the time strings "18:00" and "23:59"
into numeric values that can be compared.
3. If both conditions in step 2 are true, the formula calculates the difference between the two times (E2-F2) and returns it as a decimal number.
4. If any of the conditions in steps 1 or 2 are false, the formula returns an empty string ("").
5. The formula is entered as a regular formula by pressing Enter.
For example, if we have the following data in columns E and F:
| E | F |
| | |
| 17:30 | 05:30 |
| 18:30 | 02:00 |
| 19:45 | 06:15 |
| 20:00 | 04:30 |
| 21:30 | 03:45 |
| 22:45 | 05:45 |
| 23:59 | 06:30 |
The formula =IF(AND(TEXT(F2,"ddd")="Sun", TIMEVALUE("00:01")<=F2, F2<=TIMEVALUE("06:00")), IF(AND(TEXT(E2,"ddd")="Sat", TIMEVALUE("18:00")<=E2, E2<=TIMEVALUE("23:59")), E2-F2, ""), "") would return
the following results: - For the first row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string
(""). - For the second row, neither the time in F2 nor the time in E2 meet the conditions, so the result is an empty string (""). - For the third row, the time in F2 is between 00:01 and 06:00 on a
Sunday, and the time in E2 is between 18:00 and 23:59 on a Saturday, so the result is the difference between the two times: 06:15 - 19:45 = -13.5 (negative value represents a time difference across
midnight). - For the fourth row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (""). - For
the fifth row, the time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (""). - For the sixth row, the
time in F2 is between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (""). - For the seventh row, the time in F2 is
between 00:01 and 06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (""). - For the eighth row, the time in F2 is between 00:01 and
06:00 on a Sunday, but the time in E2 is not between 18:00 and 23:59 on a Saturday, so the result is an empty string (""). | {"url":"https://codepal.ai/excel-formula-generator/query/LVL6nwpT/excel-formula-time-difference","timestamp":"2024-11-11T13:56:54Z","content_type":"text/html","content_length":"114521","record_id":"<urn:uuid:18c141b7-18db-4802-9dee-e7f3a78873c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00797.warc.gz"} |
CWG Issue 2853
This is an unofficial snapshot of the ISO/IEC JTC1 SC22 WG21 Core Issues List revision 115d. See http://www.open-std.org/jtc1/sc22/wg21/ for the official list.
2853. Pointer arithmetic with pointer to hypothetical element
Section: 7.6.6 [expr.add] Status: DRWP Submitter: Jim X Date: 2024-02-03
[Accepted as a DR at the March, 2024 meeting.]
(From submission #495.)
The phrasing in 7.6.6 [expr.add] bullet 4.2 excludes pointer arithmetic on a pointer pointing to the hypothetical (past-the-end) array element.
Proposed resolution (approved by CWG 2024-02-16):
Change in 7.6.6 [expr.add] bullet 4.2 as follows:
□ ...
□ Otherwise, if P points to [DEL:an:DEL] array element i of an array object x with n elements (9.3.4.5 [dcl.array]), [ Footnote: ... ] the expressions P + J and J + P (where J has the value j)
point to the (possibly-hypothetical) array element i + j of x if 0 <= i + j <= n and the expression P - J points to the (possibly-hypothetical) array element i - j of x if 0 <= i - j <= n. | {"url":"https://cplusplus.github.io/CWG/issues/2853.html","timestamp":"2024-11-02T06:00:58Z","content_type":"text/html","content_length":"2812","record_id":"<urn:uuid:3e450db9-5a46-4d58-ae31-add808dbd712>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00229.warc.gz"} |
How to calculate Dipole Moment? | Steps, Formulae and Examples
The dipole moment is a vector quantity that represents the magnitude and direction of the electrical polarity in a chemical bond or molecule. It arises when there is a separation of positive and
negative charges due to differences in electronegativity between bonded atoms. The dipole moment (μ) is calculated as the product of the charge (Q) separated and the distance (r) between the
charges, with the formula:
μ = Q × r
The unit of dipole moment is the Debye (D), where 1 Debye is approximately 3.33564 × 10^ −30  Coulomb-meter (C·m).
In a molecule, if the individual bond dipole moments cancel each other out due to symmetry, the molecule is said to be nonpolar. If they do not cancel out, the molecule has a net dipole moment and is
considered polar. The dipole moment is a critical factor in determining the physical and chemical properties of substances, including their boiling points, melting points, solubility, and reactivity.
Steps to Calculate Dipole Moment
1. Identify Electronegativity Difference: Determine the electronegativity values of the two atoms forming the bond. The difference in their electronegativities (ΔEN) indicates the polarity of the
2. Calculate Charge Separation: Although the actual charge separation is complex, for calculation purposes, it’s often simplified to the product of the electronegativity difference and a constant
that represents the charge of an electron. This step is more conceptual, as precise charge separation is not always calculated directly in practice.
3. Determine Bond Length: Find the length of the bond in meters (m). This is the distance between the nuclei of the two atoms.
4. Calculate Dipole Moment: Multiply the charge separation by the bond length. The formula is  μ = Q × r, where Q is the charge (in Coulombs) and r is the distance (in meters). The result is
typically converted to Debye (D) for convenience.
Formulae for Calculating Dipole Moment
Calculating the dipole moment of a molecule might sound complicated, but it’s quite straightforward once you break it down. Here’s how you can do it:
1. Basic Dipole Moment Formula
The simplest way to calculate the dipole moment is using this formula:
μ = q × d
• μ (Dipole Moment): This is what you’re trying to find. It tells you how much a molecule is polarized.
• q (Charge): This is the amount of charge (in Coulombs) on each end of the dipole.
• d (Distance): This is the distance (in meters) between the charges.
So, if you know the charge and the distance between the charges, you can multiply them to get the dipole moment.
2. Dipole Moment for Molecules
For more complex molecules with multiple atoms, you need to consider the dipole moments of each bond. Here’s the formula you’ll use:
μ[molecule] = √((∑μ[x])² + (∑μ[y])² + (∑μ[z])²)
This means you take the dipole moments in each direction (x, y, and z), add them up separately, and then find the square root of the sum of their squares. This gives you the overall dipole moment of
the molecule.
Examples of Dipole Moment Calculations
Let’s look at a couple of examples to see how to calculate the dipole moment in real scenarios.
Example 1: Simple Diatomic Molecule
Consider a hydrogen chloride (HCl) molecule. The HCl molecule consists of a hydrogen atom and a chlorine atom. The chlorine atom is more electronegative than the hydrogen atom, which creates a dipole
• Charge (q): Let’s assume the charge separation between H and Cl is 1.6 × 10^-19 Coulombs.
• Distance (d): The bond length between H and Cl is approximately 1.27 × 10^-10 meters.
Using the formula μ = q × d:
μ = (1.6 × 10^-19 C) × (1.27 × 10^-10 m) = 2.032 × 10^-29 C·m
So, the dipole moment of an HCl molecule is 2.032 × 10^-29 C·m.
Example 2: Water Molecule (Hâ‚‚O)
Water is a polar molecule with two dipole moments from each H-O bond. To find the resultant dipole moment, we need to consider the geometry of the molecule.
Assuming each O-H bond dipole moment is approximately 1.85 D (Debye) and the angle between the bonds is 104.5°, we can calculate the net dipole moment. Because the dipole moments are at an angle, we
use vector addition to find the resultant:
μ[resultant] = √(μ[1]² + μ[2]² + 2μ[1]μ[2]cosθ)
Where μ[1] and μ[2] are the dipole moments of the two O-H bonds, and θ is the angle between them.
μ[resultant] = √((1.85 D)² + (1.85 D)² + 2 × 1.85 D × 1.85 D × cos(104.5°))
After solving, the resultant dipole moment of the water molecule is approximately 1.85 D. | {"url":"https://studentsnews.co.uk/how-to-calculate-dipole-moment/","timestamp":"2024-11-03T06:56:09Z","content_type":"text/html","content_length":"333535","record_id":"<urn:uuid:ac9e83e3-770b-4341-a4da-d5420f1d3c5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00697.warc.gz"} |
200 km h to miles
To convert 200 km/h to miles/h, you can use the following step-by-step instructions:
Step 1: Understand the conversion factor
1 kilometer is equal to 0.621371 miles. This means that to convert kilometers to miles, you need to multiply the value in kilometers by 0.621371.
Step 2: Set up the conversion equation
Let’s set up the equation to convert 200 km/h to miles/h:
200 km/h * 0.621371 miles/km = x miles/h
Step 3: Perform the calculation
Multiply 200 km/h by 0.621371 miles/km:
200 km/h * 0.621371 miles/km = 124.2742 miles/h
Step 4: Round the result (if necessary)
Since the result has several decimal places, you can round it to a more practical value. In this case, rounding to two decimal places will suffice:
124.2742 miles/h ≈ 124.27 miles/h
Therefore, 200 km/h is approximately equal to 124.27 miles/h.
Visited 1 times, 1 visit(s) today | {"url":"https://unitconvertify.com/distance/200-km-h-to-miles/","timestamp":"2024-11-03T09:01:39Z","content_type":"text/html","content_length":"42944","record_id":"<urn:uuid:821e36bf-5914-4082-8d54-610f91a4fe31>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00101.warc.gz"} |
Demonoid (diagonal Geminoid) completed!
chris_c wrote:One of these days I should try slmake and Gol_destroy on the 0hd Demonoid. From briefly looking in the past it should give close to a 50% improvement. The problem is the possibility
of collisions when the construction is taking place near the arm which need quite a bit of manual work to avoid.
Freeze-dried Slow Salvos, Finally?
One slightly more expensive way to avoid all the manual work would be to use slmake to compile a slow salvo to build the whole Demonoid, or just the difficult part, from the far side of the
construction-arm lane.
Compile slmake's output salvo with the
freeze-dry script
, and then compile the output of _that_ script with slmake. Then just trigger the seed constellation, and leave a long enough gap for the tricky construction to be completed. Easy! ... or at least a
lot easier than all that clever stuff you put together by hand last time.
Unnecessary Tangent
It's probably worth looking at the
true knightship Geminoid
again, too. Being able to come up with easy self-destruct circuits with GoL-destroy, and then compile the result with slmake, makes these kinds of constructions really amazingly easy.
Which Circuitry, Though?
What were you thinking of as a base circuit for a 50%-reduced Demonoid, though? The old constructor-reflector has a repeat time of 153 -- seems like that could be improved, though I guess not at the
cost of increasing the population or still-life count. Slower might even be fine if there's something really small.
Technically we'd only need a transparent output lane for the construction arm, not necessarily an edge shooter. In practice it seems hard to beat the NW31, though. With most other glider sources, a
large number of 0hd recipes would have to be thrown away, because gliders couldn't be built close enough together.
When I try to build a candidate for half-0hd Demonoid circuitry, I keep ending up with big and kind of old-fashioned-looking stuff:
Code: Select all
x = 190, y = 225, rule = LifeHistory
It's certainly possible to go smaller, and even be HashLife-friendly while we're at it:
Code: Select all
x = 282, y = 169, rule = LifeHistory
or even
Code: Select all
x = 196, y = 210, rule = LifeHistory
but it seems like the Snarks might get expensive, and so the smallest possible 0hd Demonoid won't have them. (?)
Re: Demonoid (diagonal Geminoid) completed!
dvgrn wrote: What were you thinking of as a base circuit for a 50%-reduced Demonoid, though? The old constructor-reflector has a repeat time of 153 -- seems like that could be improved, though I
guess not at the cost of increasing the population or still-life count. Slower might even be fine if there's something really small.
I was thinking of using exactly the same constructor-reflector. I have tried but never been able to find anything obviously better. The geometry of the current 0hd-Demonoid is nice even though the
repeat time is unsatisfyingly high. Like you say a bit later, it seems like Snarks are too expensive to appear in an "optimal" Demnoid. (By the way, my favourite definition of "optimal" is lowest
EDIT: Maybe this one? But it still isn't obvious that the extra cost of construction is worth the benefit of going from repeat time 153 to 115.
Code: Select all
x = 71, y = 142, rule = B3/S23
Re: Demonoid (diagonal Geminoid) completed!
chris_c wrote:EDIT: Maybe this one? But it still isn't obvious that the extra cost of construction is worth the benefit of going from repeat time 153 to 115...
Ow. Population 177 instead of 126, but you can pack more elbow ops in, which will almost exactly make up for the expense.
126/177 = 0.712, but 115/153 = 0.752. Or if you go by number of still lifes instead of population, it's 21/28 = 0.75 exactly. That's a mighty close race as far as estimates go -- the only way to know
for sure which one will come out smaller is to compile them both (which, fortunately, is not nearly as hard as it used to be).
However, there might be something within reach that's significantly smaller than either of those. 10hd construction arms are something like 25% more efficient that 0hd ones. Suppose we invented a
seed constellation that converted a plain NW31 to one of those +10hd attachments?
We'd always build the NW31 version of the U.C., but each one would get magically converted to a +10hd attachment as soon as it moves from front position to back position, so the period would still be
half of the old complicated 10hd model -- and we'd get the 25% efficiency advantage from using 10hd elbow recipes.
The idea of building a freeze-dried slow salvo for that part of the construction was already an option, to avoid the manual work that would otherwise be needed to build in the "danger zone". But
probably the cost of the extra constellation would eat up pretty much all of the savings from using 10hd, so we might be back to a tossup again.
Re: Demonoid (diagonal Geminoid) completed!
dvgrn wrote:This new faster HashLife-friendly Demonoid uses the same circuitry as Scorbie's Demonoid but has a somewhat higher population, because it has to spend a lot of gliders to build its
child copies at a safer distance.
I'm kind of curious to see if a Demonoid with a 16384fd step size would need only something like 250 megabytes to run away. It would presumably overflow 2^21 so would have to be 2^22, with quite
a bit higher population... at least unless a little more technology is developed -- at that distance it might be effective to build a 2-engine Cordership to get the elbow block out to the right
That 'little more technology' has now been developed, and now it should be possible to have a HashLife-friendly Demonoid with a population only slightly larger than Scorbie's Demonoid, and to get
speeds arbitrarily close to c/12.
What do you do with ill crystallographers? Take them to the mono-clinic!
Re: Demonoid (diagonal Geminoid) completed!
calcyman wrote:That 'little more technology' has now been developed, and now it should be possible to have a HashLife-friendly Demonoid with a population only slightly larger than Scorbie's
Demonoid, and to get speeds arbitrarily close to c/12.
Here's a Demonoid puffer traveling at c/128, for starters -- "only '8' away from c/12!". This is what slsparse compiles for you automatically now when you give it a Demonoid infile.mc with a step
size of 16384. The minimum power-of-two period with this particular recipe is 2^21.
-- Or rather, this is _almost_ what slsparse compiles. You do have to manually add in an elbow duplicator at one point, so that the recipe leaves behind an elbow at the farthest forward location, for
the child U.C. to pick up and use without wasting time building a Cordership.
From here it's doable, though maybe not easy without some practice, to hand-edit this pattern using recipes from previous Demonoids, so that it uses just one Snarkbreaker instead of two Snarkmakers
followed by two Snarkbreakers. The Scorbie Splitter should really be built with normal 90-degree gliders. Then the final Snark can be built with 90-degree gliders from a different direction, after
the Snarkbreaker. slsparse doesn't know how to do things that way yet, and that's what's making the recipe so much more expensive than the Scorbie's Demonoid one (~119,000 cells total instead of
Then that can be followed by some version of the standard cleanup method with *WSSes -- but with 2-engine Corderships to push the new elbow out the required distances. I haven't quite thought of how
to trick slsparse into producing the two Cordermakers for that part of the recipe, but no doubt it can be done somehow.
I _think_ that the current recipe is close enough to being able to run at 2^20 instead of 2^21, once the two unnecessary Snarkmaker recipes have been removed. Might even get back to fitting in a
forum posting in macrocell format.
front end of a c/128 Demonoid
(322.42 KiB) Downloaded 651 times
Re: Demonoid (diagonal Geminoid) completed!
I made a fast Demonoid puffer using the recipe for 58P5H1V1 from this post. The veclocity is roughly (1,1)c/9.46 (I think the exact velocity is (4825784,4825784)c/45679544). It's not hashlife
friendly, but I think it shouldn't be too hard to make it so by using shift.py.
I'm not sure what the best way to turn it into a working spaceship is. A glider could be fired back to trigger the destruction of the debris just before the faraway elbow block gets destroyed, but
that glider would have to pass by the splitter, so unless the recipe is delayed (i.e. the spaceship is slowed down), we'd have to weave that backward-heading glider through the recipe stream coming
out of the splitter.
Re: Demonoid (diagonal Geminoid) completed!
Goldtiger997 wrote: ↑
July 29th, 2020, 7:23 am
I made a fast Demonoid puffer using the recipe for 58P5H1V1 from this post. The [velocity] is roughly (1,1)c/9.46 ...
Wow, a Demonoid that can outrun a Cordership! How long a delay would have to be put in before it could outrun a seal?
Goldtiger997 wrote: ↑
July 29th, 2020, 7:23 am
I'm not sure what the best way to turn it into a working spaceship is. A glider could be fired back to trigger the destruction of the debris just before the faraway elbow block gets destroyed,
but that glider would have to pass by the splitter, so unless the recipe is delayed (i.e. the spaceship is slowed down), we'd have to weave that backward-heading glider through the recipe stream
coming out of the splitter.
There are definitely lots of options here. We could design some self-destruct circuitry to go along with the Scorbie Splitter and the Snark, triggered by a glider that follows along after the
single-channel recipe on a separate lane. The Scorbie Splitter self-destruct will send a glider to trigger the Snark self-destruct, which will put out a glider with exactly the right timing so that
it gets to the opposite Scorbie Splitter self-destruct at exactly the right time, so the self-destruct signal neither catches up with the Demonoid nor falls behind it.
That's what the 10hd and 0hd Demonoids did, but that trick hasn't been used since.
A simpler alternative is to use the destruction method from more recent Demonoids. You wouldn't have to change your current fast Demonoid puffer at all, just make a cleanup recipe to follow along
behind it -- consisting of
1) a Snarkbreaker (which requires adding a one-time turner to the construction of the c/5 spaceship seed, I think, on the southeast side -- build and then trigger a glider on the construction-arm
Snark's output lane, and meet that with the rest of a Snarkbreaker recipe, back down at the parent Scorbie Splitter)
2) a huge pile of PUSH operations
3) a FIRE *WSS to knock out the grandparent Snark on that side
4) another huge pile of PUSH operations
5) a few more *WSSes to knock out the grandparent Scorbie Splitter.
The #5 single-channel recipe can be borrowed with no changes from previous Demonoids.
For #2 and #4, it will shorten the recipe quite a bit to build and shoot down two temporary Cordership. I think slsparse can be convinced to do this for you, if you give it something to compile that
needs the right length of elbow push.
But it doesn't really matter how long the trailing cleanup recipe is, it won't affect the speed of the Demonoid or make you recompile anything. You'll just figure out a recipe and feed it in to the
end of the puffer you've already made, and it will start cleaning it up.
... Honestly the 10hd/0hd "synchronized self-destruct signal" almost does seem simpler, though. Just needs a recompile for the additions to the Scorbie Splitter and Snark, and then you add one glider
on the self-destruct lane, and it will just go along and do all the cleanup. That cleanup signal can be as close to the action or as far behind it as you want, technically making an unlimited number
of "different" Demonoids... not interestingly different, just different.
Re: Demonoid (diagonal Geminoid) completed!
dvgrn wrote: ↑
July 29th, 2020, 9:04 am
Goldtiger997 wrote: ↑
July 29th, 2020, 7:23 am
I made a fast Demonoid puffer using the recipe for 58P5H1V1 from this post. The [velocity] is roughly (1,1)c/9.46 ...
Wow, a Demonoid that can outrun a Cordership! How long a delay would have to be put in before it could outrun a seal?
Well, now I have a naming proposal-- "Speed Demonoid"
not active here but active on discord
loves estradiol valerate
Re: Demonoid (diagonal Geminoid) completed!
It might be worth mentioning some extra parts of the single-channel toolchain that were used in the construction of the 0E0P metacell.
Firstly, as well as outfile.mc, slmake emits an outfile.txt which contains the single-channel recipe in textual format. These text files can be concatenated using the standard command-line program
cat, and this results in the recipes being concatenated.
If you don't have your glider stream in textual format, do not worry! There's a function destream() in lifelib which converts a pattern containing a glider stream into a Python list of integers --
https://gitlab.com/apgoucher/lifelib/-/ ... y#L643-659 -- as well as its inverse function, stream().
EDIT: The way that it's invoked is: glider_in_correct_orientation.destream(pattern_containing_glider_stream)
Secondly, lifelib/dd0e0p.cpp is a tool for inputting a tape (a textual glider recipe) into a pattern such as a Demonoid. It accepts an input file (text), a target file (an RLE or MC with exactly one
glider* somewhere in the pattern, and still-lifes elsewhere), and an output file. The invocation:
Code: Select all
./dd0e0p if=recipe.txt tf=target.mc of=output.mc absolute=0 final
will conveniently replace the glider in the pattern file with the recipe (so the front of the recipe is where the original glider was).
*I think it works with arbitrary spaceships, so you can use it with an MWSS-stream Orthogonoid or a hypothetical loafer-stream spaceship, but in most cases you'll just want to have a glider stream.
Alternatively, you can use:
Code: Select all
./dd0e0p if=recipe.txt tf=target.mc of=output.mc relative=0 final
if you want the back of the recipe to be where the original glider was. This will perform the above substitution and then run the pattern for a number of generations equal to the length of the recipe
(in order to 'load' the recipe tape into the Demonoid or whatever else is in target.mc). Here's the full list of documented options:
Code: Select all
* dd0e0p -- write a logical glider stream to a physical pattern file.
* Mandatory options:
* if=stream.txt Location of glider stream
* of=filename.mc Pattern file to which to append
* Optional options:
* tf=target.mc If you wish to not overwrite the target
* box=378,-43,3,3 Specify initial glider (rather than detect)
* final Remove initial glider afterwards
* absolute=523857 Advance pattern by n generations
* relative=1000 Advance pattern by n generations beyond recipe end
* bs=1048576 Advance by n generations at a time
It runs the pattern in streamlife, which is faster than ordinary hashlife for patterns involving closely-crossing glider streams.
What do you do with ill crystallographers? Take them to the mono-clinic!
Re: Demonoid (diagonal Geminoid) completed!
Congratulations! This is such an interesting place for a c/5 diagonal ship.
A demonoid could probably become asymptotic to the the diagonal speed limit of c/4 as well. We could use the crab wickstretcher (synthesis from 2017 here), a clean ignition of the fuse, and a clean
destruction like so:
Code: Select all
x = 68, y = 69, rule = B3/S23
Tanner Jacobi
Coldlander, a novel, available in paperback and as an ebook. Now on Amazon.
Re: Demonoid (diagonal Geminoid) completed!
Kazyan wrote: ↑
July 29th, 2020, 8:53 pm
A demonoid could probably become asymptotic to the the diagonal speed limit of c/4 as well. We could use the crab wickstretcher (synthesis from 2017 here), a clean ignition of the fuse, and a
clean destruction like so:...
Good idea! The more recent version of that synthesis takes 32 gliders — not too bad to create a seed for. It may be worth looking into some reductions of that recipe first though. The following looks
like a pretty good way of igniting the fuse:
Code: Select all
x = 45, y = 53, rule = B3/S23
Re: Demonoid (diagonal Geminoid) completed!
Goldtiger997 wrote: ↑
July 30th, 2020, 5:39 am
The following looks like a pretty good way of igniting the fuse...
Looks like the Doppler effect means that the fuse-lighting glider has the same effect every twelve ticks. Here's the same reaction, but with the glider coming from the direction it will probably be
coming from, straight from the construction arm:
Code: Select all
x = 83, y = 84, rule = LifeHistory
-- I think it should be possible to convince slsparse to produce a recipe for the stretcher seed that doesn't require the extra two Snarkmakers/two Snarkbreakers trick to build the constellation from
off to one side. With patterns as large as this I suppose the efficiency of the construction recipe doesn't really matter, but it shouldn't be too tricky to produce something, um, "relatively
These speed-Demonoids need two target blocks very far apart from each other. There's no way to bounce a following signal non-destructively off of one of these stretchers, as the last Demonoid
cleverly did with the leading c/5 spaceship.
So I assume that the trick to use here would be to burn the fuse and create the elbow to build the farther-away reflector first (the singleton Snark) -- and then send a 180-degree glider backwards
and leave a long gap before sending more gliders to collide with it and make a new elbow to build the Scorbie Splitter. There are single-channel recipes for doing all of that fairly efficiently, from
some previous Demonoid project, I forget which one.
Re: Demonoid (diagonal Geminoid) completed!
Isn't using wickstretcher less efficient?
For corderpush, every 64 tick delay pushes resulting block 16hd farther, so efficiency is 1/4 (hd/tick).
For the new c/5 push, every 4 ticks block is pushed 8 hd, so efficiency is 2.
For the newest wickstretcher push, every 3 ticks block is pushed 2 hd, so efficiency is only 2/3.
Re: Demonoid (diagonal Geminoid) completed!
Pavgran wrote: ↑
July 30th, 2020, 12:33 pm
Isn't using wickstretcher less efficient?
For corderpush, every 64 tick delay pushes resulting block 16hd farther, so efficiency is 1/4 (hd/tick).
For the new c/5 push, every 4 ticks block is pushed 8 hd, so efficiency is 2.
For the newest wickstretcher push, every 3 ticks block is pushed 2 hd, so efficiency is only 2/3.
2/3 is bigger than 1/4, so wickstretchers are better than Corderships. Seems like there must be something wrong with the math for c/5, since 8hd every four ticks is basically lightspeed.
Anyway, it seems like the idea is that there's a range of speeds between c/5 and c/4 that can't be reached by chasing c/5 spaceships with gliders -- and anyway I think the wickstretcher-based
Demonoid is going to to end up a lot smaller, because there are many fewer gliders to synchronize to get a wickstretcher seed.
Re: Demonoid (diagonal Geminoid) completed!
dvgrn wrote: ↑
July 30th, 2020, 2:18 pm
Seems like there must be something wrong with the math for c/5, since 8hd every four ticks is basically lightspeed.
The math is right. Compare these two situations:
Code: Select all
x = 54, y = 23, rule = LifeSuper
The gliders are 4 ticks apart, the blocks are 8hd apart. So hd/ticks is 2. If the speed of ship is even closer to c/4, the efficiency will be more than 2.
dvgrn wrote: ↑
July 30th, 2020, 8:43 am
Here's the same reaction, but with the glider coming from the direction it will probably be coming from, straight from the construction arm:
Code: Select all
x = 83, y = 84, rule = LifeHistory
To make that setup work, you would need to send debris-cleaning gliders and the rest of the recipe alongside the wick, not touching it, so that there would be no time wasted. If you just ignite the
wick and send the remaining gliders after, then the wick will soon catch up with the ship and debris would wait there for a while, wasting time. The idea is that we want to begin new construction
right after making an elbow, so the total construction time would be constant, and the travelling part would be as high as we wish.
(After making the previous post and the first version of this post, I've realised that efficiency in terms of hd/ticks doesn't matter here. It matters in diminishing total recipe length. And I've
wrote that the setup above wouldn't work, but that was based on assumption that "straight from the construction arm" meant "from the same lane as the recipe")
Re: Demonoid (diagonal Geminoid) completed!
Pavgran wrote: ↑
July 30th, 2020, 5:20 pm
(After making the previous post and the first version of this post, I've realised that efficiency in terms of hd/ticks doesn't matter here. It matters in diminishing total recipe length. And I've
wrote that the setup above wouldn't work, but that was based on assumption that "straight from the construction arm" meant "from the same lane as the recipe")
Yeah, it's only "from the same elbow as the recipe (gliders)", and we can move that elbow to wherever we want it.
So we can send the entire cleanup and new-construction recipe before the wick is even triggered:
Code: Select all
x = 95, y = 96, rule = LifeHistory
All those NWward gliders can be slow-salvo gliders -- they just look like that to keep the pattern small. The actual construction could be done off to the southeast of the (2,1)-pulled block.
The extra glider that I show being cleaned up here, on the northwest side of the wick, might even be useful in making a target for building the Scorbie Splitter at the halfway point, to avoid losing
the time on that. Not sure exactly how that will work -- maybe hit that glider with a slow salvo that builds a one-time turner, and then at the right time trigger that to send a glider SE to
interrupt a single-channel stream and make a new target?
(The idea would be to use a Snarkmaker back at the glider source, and convert to a single-channel stream as soon as the wickstretcher cleanup gliders have been sent. That way, the required Snark and
Scorbie Splitter can be built at a reasonable offset to the southeast without being exponentially more expensive due to bending around two elbows.)
I don't think the expensiveness of the recipe really matters to this design's ability to travel faster than c/5, though -- a longer recipe would just mean a higher-period Demonoid. Seems like if you
let the wickstretcher run long enough, the Demonoid will be able to go pretty fast. I'm not absolutely sure that the return-glider trick won't slow things down, though, maybe to a top speed below c/5
for all I know for sure yet.
EDIT: At worst we could build and trigger two wickstretcher seeds, right? Run them simultaneously, and then leave the elbow blocks in place to start the next seed construction from? Then the Scorbie
Splitter and Snark constructions could happen more or less simultaneously, relative to when the tape gliders get there. (?)
Re: Demonoid (diagonal Geminoid) completed!
Here's a 21-glider (EDIT: 22) version of the wickstretcher, done via a method to add the wick to a crab that already exists. 20G (EDIT: 21) is very likely by finding a 1G cleanup:
Code: Select all
x = 219, y = 89, rule = B3/S23
The wick-adding component is here:
Code: Select all
x = 42, y = 38, rule = B3/S23
Since there's a glider trailing after the crab, you do have to be sure that the trailing glider fits with existing crab syntheses (as shown above). This is why a 3G LoM inserter is used instead of
the 2G version--the 2G version would have a trailing glider so close to the crab that there's no space to actually put it there during the crab synthesis. It's kind of weird to think of gliders as
stationary objects, but the reference frame (so to speak) is c/4 diagonal!
As an unrelated benefit, this allows for construction of the double wickstretcher, which I don't think has been done elsewhere:
Code: Select all
x = 106, y = 108, rule = B3/S23
EDIT: Miscounted gliders.
Tanner Jacobi
Coldlander, a novel, available in paperback and as an ebook. Now on Amazon.
Re: Demonoid (diagonal Geminoid) completed!
Wickstrecher in 21g:
Code: Select all
x = 209, y = 84, rule = B3/S23
Edit: inherited the miscounting of gliders.
Last edited by Sarp on August 11th, 2020, 10:28 am, edited 2 times in total.
Re: Demonoid (diagonal Geminoid) completed!
Sarp wrote: ↑
July 31st, 2020, 2:27 pm
Wic[k]strecher in 21g...
That's quite a reduction -- awesome!
As a small extra bonus, the long^N boat form of the wick can be ignited directly by a glider, instead of needing a honeyfarm explosion to kick things off:
Code: Select all
x = 59, y = 59, rule = B3/S23
It will be interesting to convert this to a slow-salvo recipe. So far the only conversion of a big synthesis recipe to a seed has been Goldtiger997's c/5 spaceship seed, which makes 100 synchronized
gliders that then produce the spaceship. For this new recipe it probably makes more sense to slow-salvo-construct a mango, then add a couple of blinkers, and then build two more sub-constellations to
do the crab construction and then append the wickstretcher part.
Maybe the last two might as well just be one constellation, but it's nice that the two pieces can be moved diagonally apart from each other if that makes the construction easier.
Re: Demonoid (diagonal Geminoid) completed!
It's possible to clean the wick-adder up using a mango, which should make the seed smaller:
Code: Select all
x = 31, y = 26, rule = B3/S23
We might also be able to replace the 3 LOM gliders with a single glider from the southwest hitting a small constellation (using a beehive doesn't work; does anyone know of any other small LOM
Any sufficiently advanced software is indistinguishable from malice.
Re: Demonoid (diagonal Geminoid) completed!
The =https://www.conwaylife.com/forums/view ... eds thread and the Speed Demonoid thread picked up the discussion about building a crabsstretcher seed, and then the Speed Demonoid enabled by that
seed. So I'll bring this thread back a previous topic and wrap up a minor loose end.
Way back not quite two years ago I posted the puffer part of lower-population, Relatively Fast Demonoid (i.e., not a Speed Demonoid, but at least it can approach an upper-bound speed of c/12). That
was the hard part, but I never got around to going back and doing the easy part (the cleanup).
Here's a small step in the right direction:
This design is not really "lower-population" yet, for at least two reasons:
1) The recipe still uses slsparse's default construction to build the Scorbie Splitter, which is seriously suboptimal:
dvgrn wrote: ↑
November 13th, 2018, 6:10 pm
The Scorbie Splitter should really be built with normal 90-degree gliders. Then the final Snark can be built with 90-degree gliders from a different direction, after the Snarkbreaker. slsparse
doesn't know how to do things that way yet, and that's what's making the recipe so much more expensive than the Scorbie's Demonoid one (~119,000 cells total instead of ~72,000).
2) I deliberately appended a suboptimal cleanup method to the original puffer, to show the comparison. To push the destruction elbow out to where it needs to be to knock out the previous Snark, I
just piled on a long string of PUSH44 recipes, just like in previous Demonoids. The second long push, to reach the location needed to knock out the previous Scorbie Splitter, is done "correctly",
with a Corderpush. It's easy to see that the Corderpush is cheaper, both in terms of time required and the number of gliders needed.
Now, to really make an attempt at minimizing a Relatively Fast Demonoid's population, it seems like a more radical redesign will be needed:
3) We could rig up a synchronized self-destruct mechanism for the Scorbie Splitter and Snark, very much along the lines of what's in the Speed Demonoid. This would make the final Snarkbreaker and
both of the long-distance push operations unnecessary. That might reduce the minimum HashLife-friendly period to just 2^20 -- or maybe even 2^19, not sure -- and the macrocell-format Demonoid pattern
might be able to fit in a forum code block again.
If the suboptimal PUSH44s are replaced by a standard Cordership push, the Relatively Fast Demonoid predecessor will become trivially adjustable by just changing the spacing between various
single-channel sub-streams (even if the Scorbie Splitter recipe isn't replaced). | {"url":"https://conwaylife.com/forums/viewtopic.php?p=65648","timestamp":"2024-11-06T14:51:42Z","content_type":"text/html","content_length":"148175","record_id":"<urn:uuid:4301072d-c219-4351-bb19-de10fe53d452>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00469.warc.gz"} |
Scientific Research Software - ANALYTICS
ANALYTICS Development library.
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C#: ANALYTICS.CSHARP.DEMO.rar
Java: ANALYTICS.JAVA.DEMO.rar
Delphi (TMS): https://www.tmssoftware.com/site/tmsanalytics.asp
Documentation: ANALYTICS_C#_manual.en.pdf (English)
ANALYTICS_C#_manual.ru.pdf (Russian)
ADVANTAGES of ANALYTICS library:
1. 100% source code.
2. Strongly structured class hierarchy.
3. Universal algorithms (working with formulae of any complexity).
4. Analytical derivative calculation.
5. Many predefined functions.
7. Easy to overload operators for any argument types.
8. Working with Complex numbers.
9. Working with common fractions.
10. Working with 3D vectors and tensors.
11. Working with physical values and units of measurement.
12. Working with indexed data (arrays, matrixes and higher dimensioned data).
13. Statistical analysis of data (base statistics functions, probability distributions).
14 Numerical tools integrated with symbolic features.
Download example application:
Numerical result postprocessor.
Platform: .NET.
Description: Postprocessor for results of mathematical modeling (FEM, BEM i.e.). The program is like ‘analytical calculator’ but works not with real numbers but with the results of physical problems
modeling. This feature is realized via ANALYTICS technology, which allows calculating string (formula) expression for any data types, by unified algorithm.
Parametric Surface modeler.
Platform: .NET.
Description: Parametric surface modeler allows create and visualize complicated surfaces specified in the form of analytical equations. Equations checked for syntax errors before surface creations.
The equations can be specified in one of several local coordinate system types (Cartesian, cylindrical, spherical and so on).
© Sergey L. Gladkiy | {"url":"https://sergey-l-gladkiy.narod.ru/index/analytics/0-13","timestamp":"2024-11-07T15:23:43Z","content_type":"text/html","content_length":"20263","record_id":"<urn:uuid:e36a81ca-a86e-4a47-a1a5-2b5dfd324b41>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00001.warc.gz"} |
Subtraction formula that is based on another formula
Hi everyone,
Please send help.
I have a formula set up to calculate total costs. However, I need to have it set up so that the total amount subtracts from a budget total.
For example, say the budget it 20,000. I need the =SUM(Amount:Amount) to automatically subtract from the 20,000.
I hope this makes sense. Thank you!
Best Answer
• =20,000-Sum(Amount:Amount)
Just replace the 20,000 with the cell reference of your budget total
If you found this comment helpful. Please respond with any of the buttons below. Awesome🖤, Insightful💡, Upvote⬆️, or accepted answer. Not only will this help others searching for the same answer,
but help me as well. Thank you.
• =20,000-Sum(Amount:Amount)
Just replace the 20,000 with the cell reference of your budget total
If you found this comment helpful. Please respond with any of the buttons below. Awesome🖤, Insightful💡, Upvote⬆️, or accepted answer. Not only will this help others searching for the same answer,
but help me as well. Thank you.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/122728/subtraction-formula-that-is-based-on-another-formula","timestamp":"2024-11-14T05:11:09Z","content_type":"text/html","content_length":"400745","record_id":"<urn:uuid:4ddc68b4-ba60-4e81-b032-180616920396>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00302.warc.gz"} |
A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity?
What is Reddit's opinion of A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity?
From 3.5 billion Reddit comments
If you have a have a math background up to ODEs and PDEs, this book is great. https://www.amazon.com/Most-Incomprehensible-Thing-Introduction-Mathematics/dp/0957389450/ref=sr_1_1?s=books&ie=UTF8&
I think this book by Collier. should match up relatively^haha well with what you're looking for.
Not sure about the other topics, but if you really want to learn about the science of time dilation, I would recommend checking out Brian Greene's free courses on Special Relativity (either the
conceptual one or the really math-centered one) and perhaps A Most Incomprehensible Thing by Peter Collier (a math-centered book that can potentially take the layman from high school mathematics to
the equations of General Relativity). | {"url":"https://redditfavorites.com/products/a-most-incomprehensible-thing-notes-towards-a-very-gentle-introduction-to-the-mathematics-of-relativity","timestamp":"2024-11-11T13:46:28Z","content_type":"text/html","content_length":"11632","record_id":"<urn:uuid:27d3a3ba-6775-4171-9f42-96ffea13a00b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00472.warc.gz"} |
Trace minimization for hierarchical or grouped time series — MinT
Using the method of Wickramasuriya et al. (2019), this function combines the forecasts at all levels of a hierarchical or grouped time series. The forecast.gts calls this function when the MinT
method is selected.
nodes = NULL,
groups = NULL,
covariance = c("shr", "sam"),
nonnegative = FALSE,
algorithms = c("lu", "cg", "chol"),
keep = c("gts", "all", "bottom"),
parallel = FALSE,
num.cores = 2,
control.nn = list()
Matrix of forecasts for all levels of a hierarchical or grouped time series. Each row represents one forecast horizon and each column represents one time series of aggregated or
fcasts disaggregated forecasts.
nodes If the object class is hts, a list contains the number of child nodes referring to hts.
groups If the object is gts, a gmatrix is required, which is the same as groups in the function gts.
residual Matrix of insample residuals for all the aggregated and disaggregated time series. The columns must be in the same order as fcasts.
covariance Type of the covariance matrix to be used. Shrinking towards a diagonal unequal variances ("shr") or sample covariance matrix ("sam").
nonnegative Logical. Should the reconciled forecasts be non-negative?
algorithms Algorithm used to compute inverse of the matrices.
keep Return a gts object or the reconciled forecasts at the bottom level.
parallel Logical. Import parallel package to allow parallel processing.
num.cores Numeric. Specify how many cores are going to be used.
control.nn A list of control parameters to be passed on to the block principal pivoting algorithm. See 'Details'.
Return the reconciled gts object or forecasts at the bottom level.
The control.nn argument is a list that can supply any of the following components:
Permutation method to be used: "fixed" or "random". Defaults to "fixed".
The number of full exchange rules that may be tried. Defaults to 10.
The tolerance of the convergence criteria. Defaults to sqrt(.Machine$double.eps).
Wickramasuriya, S. L., Athanasopoulos, G., & Hyndman, R. J. (2019). Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. Journal of the American
Statistical Association, 114(526), 804--819. https://robjhyndman.com/working-papers/mint/
Wickramasuriya, S. L., Turlach, B. A., & Hyndman, R. J. (to appear). Optimal non-negative forecast reconciliation. Statistics and Computing. https://robjhyndman.com/publications/nnmint/
Hyndman, R. J., Lee, A., & Wang, E. (2016). Fast computation of reconciled forecasts for hierarchical and grouped time series. Computational Statistics and Data Analysis, 97, 16--32. https://
See also
Shanika L Wickramasuriya
# hts example
if (FALSE) {
h <- 12
ally <- aggts(htseg1)
n <- nrow(ally)
p <- ncol(ally)
allf <- matrix(NA, nrow = h, ncol = p)
res <- matrix(NA, nrow = n, ncol = p)
for(i in 1:p)
fit <- auto.arima(ally[, i])
allf[, i] <- forecast(fit, h = h)$mean
res[, i] <- na.omit(ally[, i] - fitted(fit))
allf <- ts(allf, start = 51)
y.f <- MinT(allf, get_nodes(htseg1), residual = res, covariance = "shr",
keep = "gts", algorithms = "lu")
y.f_cg <- MinT(allf, get_nodes(htseg1), residual = res, covariance = "shr",
keep = "all", algorithms = "cg")
if (FALSE) {
h <- 12
ally <- abs(aggts(htseg2))
allf <- matrix(NA, nrow = h, ncol = ncol(ally))
res <- matrix(NA, nrow = nrow(ally), ncol = ncol(ally))
for(i in 1:ncol(ally)) {
fit <- auto.arima(ally[, i], lambda = 0, biasadj = TRUE)
allf[,i] <- forecast(fit, h = h)$mean
res[,i] <- na.omit(ally[, i] - fitted(fit))
b.f <- MinT(allf, get_nodes(htseg2), residual = res, covariance = "shr",
keep = "bottom", algorithms = "lu")
b.nnf <- MinT(allf, get_nodes(htseg2), residual = res, covariance = "shr",
keep = "bottom", algorithms = "lu", nonnegative = TRUE, parallel = TRUE)
# gts example
if (FALSE) {
abc <- ts(5 + matrix(sort(rnorm(200)), ncol = 4, nrow = 50))
g <- rbind(c(1,1,2,2), c(1,2,1,2))
y <- gts(abc, groups = g)
h <- 12
ally <- aggts(y)
n <- nrow(ally)
p <- ncol(ally)
allf <- matrix(NA,nrow = h,ncol = ncol(ally))
res <- matrix(NA, nrow = n, ncol = p)
for(i in 1:p)
fit <- auto.arima(ally[, i])
allf[, i] <- forecast(fit, h = h)$mean
res[, i] <- na.omit(ally[, i] - fitted(fit))
allf <- ts(allf, start = 51)
y.f <- MinT(allf, groups = get_groups(y), residual = res, covariance = "shr",
keep = "gts", algorithms = "lu") | {"url":"https://pkg.earo.me/hts/reference/MinT.html","timestamp":"2024-11-12T03:04:28Z","content_type":"text/html","content_length":"29914","record_id":"<urn:uuid:3d66cab3-347e-4b8e-ada9-277ffb30589f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00127.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
Airy transit circle
پرهون ِنیمروزانی ِایری
parhun-e nimruzâni-ye Airy
Fr.: circle méridien d'Airy
A → transit circle that defines the position of the → Greenwich Meridian since the first observation was taken with it in 1851. Airy's transit circle lies at longitude 0°, by definition, and latitude
51° 28' 38'' N.
Named after Sir George Biddell Airy (1801-1892), Astronomer Royal, at the Royal Observatory in Greenwich from 1835 to 1881. Airy transformed the observatory, installing some of the most advanced
astronomical apparatus of his day and expanded both staff numbers and their workload; → transit; → circle.
altitude circle
پرهون ِفرازا
parhun-e farâzâ
Fr.: cercle d'égale altitude
A circle on the celestial sphere that has equal altitude over the Earth's surface and lies parallel to the horizon. Also called almucantar, circle of altitude, parallel of altitude.
→ altitude; → circle.
azimuth circle
پرهون ِسوگان، دایرهی ِ~
parhun-e sugân, dâyere-ye ~
Fr.: cercle d'azimut
One of great circles of the → celestial sphere which passes through the → zenith, → nadir, and the star, cutting the horizon at right angles. Same as → vertical circle.
→ azimuth; → circle.
Borda circle
دایرهی ِبُردا
dâyere-ye Borda
Fr.: cercle de Borda
An instrument which was an improved form of the → reflecting circle, used for measuring angular distances. In Borda's version the arm carrying the telescope was extended right across the circle. The
telescope and a clamp and tangent screw were at one end, and the half-silvered horizon glass at the far end from the eye. In practice, with the index arm clamped, the observer first aims directly at
the right hand object and by reflection on the left, moving the telescope arm until this is achieved. He then frees the index arm, sights directly on the left hand object with the telescope arm
clamped, and moves the index arm until the two coincide again. The difference in the readings of the index arm is twice the angle required, so that the final sum reading must be divided by twice the
number of double operations. Borda's instrument greatly contributed to the French success in measuring the length of the meridional arc of the Earth's surface between Dunkirk and Barcelona
(1792-1798). The operation carried out by Jean Baptiste Delambre (1749-1822) and Pierre Méchain (1744-1804) was essential for establishing the meter as the length unit.
After the French physicist and naval officer Jean-Charles de Borda (1733-1799), who made several contributions to hydrodynamics and nautical astronomy. Borda was also one of the most important
metrological pioneers; → circle.
پرهون، دایره
parhun (#), dâyeré (#)
Fr.: cercle
A closed curve lying in a plane and so constructed that all its points are equally distant from a fixed point in the plane.
From O.Fr. cercle, from L. circulus "small ring," dim. of circus "ring," from or akin to Gk. kirkos "a circle," from PIE *kirk- from base *(s)ker- "to turn, bend," related to Pers. carx "wheel,
everything revolving in an orbit, circular motion, chariot."
Parhun "circle" in Mod.Pers. classical texts, from Proto-Iranian *pari-iâhana- "girdle, belt," from pari-, variant pirâ-, → circum-, + iâhana- "to girdle," cf. Av. yâh- "to girdle." The Pers. word
pirâhan "shirt" is a variant of parhun. Gk. cognate zone "girdle."
Dâyeré, from Ar.
circle of altitude
پرهون ِفرازا
parhun-e farâzâ
Fr.: almucantar
A small circle on the celestial sphere parallel to the horizon. The locus of all points of a given altitude. Also called → almucantar, → altitude circle, → parallel of altitude.
→ circle; → altitude.
circle of latitude
پرهون ِورونا
parhun-e varunâ
Fr.: parallèle
1) A circle of the celestial sphere, parallel to the ecliptic.
2) A circle on the terrestrial surface parallel to the equator, along which longitude is measured.
→ circle; → latitude.
circle of longitude
پرهون ِدرژنا
parhun-e derežnâ
Fr.: méridien
1) A great circle of the celestial sphere, from the pole to the ecliptic at right angles to the plane of the ecliptic.
2) A great circle on the terrestrial surface that meets the North and South poles and connects all places of the same longitude.
→ circle; → longitude.
Fr.: cercle circonscrit
A circle which passes through all three vertices of a triangle Also "Circumscribed circle".
→ circum-; → circle.
congruent circles
پرهونهای ِدمساز
parhunhâ-ye damsâz
Fr.: cercles congrus
Two circles if they have the same size.
→ congruent; → circle.
declination circle
پرهون ِواکیلش، دایرهی ِ~
parhun-e vâkileš, dâyeré-ye ~
Fr.: cercle de déclinaison
For a telescope with an → equatorial mounting, a graduated circle attached to the → declination axis that shows the → declination to which the telescope is pointing.
→ declination; → circle.
diurnal circle
پرهون ِروزانه، دایرهی ِ~
parhun-e ruzâné, dâyere-ye ~
Fr.: cercle diurne
The apparent path of an object in the sky during one day, due to Earth's rotation.
→ diurnal; → circle.
Fr.: excercle
For a → triangle with two sides extended in the direction opposite their common → vertex, a circle that lies outside the triangle and is tangent to the three sides (two of them extended). The center
of the excircle, called the → excenter, is the point of intersection of the bisector of the interior angle and the bisector of the exterior angles at the other two vertices.
→ ex-; → circle.
great circle
پرهون ِبزرگ، دایرهی ِ~
parhun-e bozorg, dâyere-ye ~
Fr.: grand cercle
A circle on a sphere whose plane passes through the center of the sphere.
→ great; → circle.
hour circle
پرهون ِساعتی، دایرهی ِ~
parhun-e sâ'ati, dâyere-ye ~
Fr.: cercle horaire
A great circle passing through an object and the → celestial poles intersecting the → celestial equator at right angles.
→ hour; → circle.
meridian circle
پرهون ِنیمروزانی
parhun-e nimruzâni
Fr.: circle méridien
A telescope with a graduated vertical scale, used to measure the declinations of heavenly bodies and sometimes to determine the time of meridian transits.
→ meridian; → circle.
osculating circle
پرهونِ آبوسنده
parhun-e âbusandé
Fr.: cercle osculateur
The circle that touches a curve (on the concave side) and whose radius is the radius of curvature.
→ osculating; → circle.
polar circle
پرهون ِقطبی، دایرهی ِ~
parhun-e qotbi, dâyere-ye ~ (#)
Fr.: cercle polaire
An imaginary parallel circle on the celestial sphere or on the Earth at a distance of 23°.5 from either poles.
→ polar; → circle.
precessional circle
پرهون ِپیشایانی
parhun-e pišâyâni
Fr.: circle précessionnel
The path of either → celestial poles around the → ecliptic pole due to the → precession of equinox. It takes about 26,000 years for the celestial pole to complete path.
→ precessional; → circle.
quadrature of the circle
چاروشش ِپرهون، ~ ِدایره
cârušeš-e parhun, ~ dâyeré
Fr.: quadrature du cercle
Constructing a square whose area equals that of a given circle. This was one of the three geometric problems of antiquity. It was finally proved to be an impossible problem when π was proven to be
transcendental by Lindemann in 1882. Same as → squaring the circle.
→ quadrature; → circle. | {"url":"https://dictionary.obspm.fr/index.php/?showAll=1&formSearchTextfield=circle","timestamp":"2024-11-14T18:10:24Z","content_type":"text/html","content_length":"29932","record_id":"<urn:uuid:8b628774-1961-4b16-8165-41dfbd79f5e3>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00768.warc.gz"} |
What are Kepler's Laws?
Kepler's laws are three equations which govern the motion of astronomical bodies. Kepler's laws were first discovered by the 17th century astronomer Johannes Kepler while analyzing data collected by
Tycho Brahe. Kepler's laws are an extension of Copernicus's earlier heliocentric theory, and eventually paved the way for Isaac Newton's complete theory of how bodies interact. Newton's equations of
gravity and motion can be used to derive Kepler's laws, if you assume that there are only two bodies, one of which is fixed, and one of which is orbiting at less than escape velocity. Although
Kepler's laws were originally developed to explain planetary motions, they apply to any body which is in orbit around a much more massive body.
The first of Kepler's laws states that a planet, or any other object in orbit around the Sun, follows an elliptical path with the Sun at one focus. The shape of these ellipses depends on the Sun's
mass, the planet's position, and the planet's velocity. A set of six numbers, called the Keplerian elements, can be used to specify the exact path that a planet traces out.
The second of Kepler's laws says that a planet in orbit traces out equal areas in equal times. If you draw a line from the planet to the Sun, and add up the area which the line sweeps over during a
given time interval, it is always constant. This law is a consequence of the conservation of angular momentum; if the planet is moving faster, it also must be closer to the Sun. The increase in the
area covered from the larger angular motion, and the decrease in the area covered from the shorter distance, must exactly cancel each other.
The third law states that the square of the period of the orbit must be directly proportional to the cube of the orbit's semi-major axis. The semi-major axis is half of the total distance between the
perihelion, or closest approach to the Sun, and the aphelion, or farthest distance from the Sun. A planet very far from the Sun, such as Neptune, has a much larger orbit; it also moves more slowly,
taking more time to cover the same distance than a planet such as Mercury. The exact relationship between orbital period, semi-major axis, mass, and the gravitational constant was later worked out by
Isaac Newton. | {"url":"https://www.allthescience.org/what-are-keplers-laws.htm","timestamp":"2024-11-07T16:00:13Z","content_type":"text/html","content_length":"111911","record_id":"<urn:uuid:971cc5f9-532a-48f7-93d8-49b674a811a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00611.warc.gz"} |
Estimating Regularized Linear Models with rstanarm
Jonah Gabry and Ben Goodrich
This vignette explains how to estimate linear models using the stan_lm function in the rstanarm package.
The four steps of a Bayesian analysis are
1. Specify a joint distribution for the outcome(s) and all the unknowns, which typically takes the form of a marginal prior distribution for the unknowns multiplied by a likelihood for the outcome
(s) conditional on the unknowns. This joint distribution is proportional to a posterior distribution of the unknowns conditional on the observed data
2. Draw from posterior distribution using Markov Chain Monte Carlo (MCMC).
3. Evaluate how well the model fits the data and possibly revise the model.
4. Draw from the posterior predictive distribution of the outcome(s) given interesting values of the predictors in order to visualize how a manipulation of a predictor affects (a function of) the
Steps 3 and 4 are covered in more depth by the vignette entitled “How to Use the rstanarm Package”. This vignette focuses on Step 1 when the likelihood is the product of independent normal
The goal of the rstanarm package is to make Bayesian estimation of common regression models routine. That goal can be partially accomplished by providing interfaces that are similar to the popular
formula-based interfaces to frequentist estimators of those regression models. But fully accomplishing that goal sometimes entails utilizing priors that applied researchers are unaware that they
prefer. These priors are intended to work well for any data that a user might pass to the interface that was generated according to the assumptions of the likelihood function.
It is important to distinguish between priors that are easy for applied researchers to specify and priors that are easy for applied researchers to conceptualize. The prior described below emphasizes
the former but we outline its derivation so that applied researchers may feel more comfortable utilizing it.
The likelihood for one observation under a linear model can be written as a conditionally normal PDF \[\frac{1}{\sigma_{\epsilon} \sqrt{2 \pi}} e^{-\frac{1}{2} \left(\frac{y - \mu}{\sigma_{\epsilon}}
\right)^2},\] where \(\mu = \alpha + \mathbf{x}^\top \boldsymbol{\beta}\) is a linear predictor and \(\sigma_{\epsilon}\) is the standard deviation of the error in predicting the outcome, \(y\). The
likelihood of the entire sample is the product of \(N\) individual likelihood contributions.
It is well-known that the likelihood of the sample is maximized when the sum-of-squared residuals is minimized, which occurs when \[ \widehat{\boldsymbol{\beta}} = \left(\mathbf{X}^\top \mathbf{X}\
right)^{-1} \mathbf{X}^\top \mathbf{y}, \] \[ \widehat{\alpha} = \overline{y} - \overline{\mathbf{x}}^\top \widehat{\boldsymbol{\beta}}, \] \[ \widehat{\sigma}_{\epsilon}^2 = \frac{\left(\mathbf{y} -
\widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)^\top \left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)}{N},\] where \(\overline{\mathbf{x}}\) is
a vector that contains the sample means of the \(K\) predictors, \(\mathbf{X}\) is a \(N \times K\) matrix of centered predictors, \(\mathbf{y}\) is a \(N\)-vector of outcomes and \(\overline{y}\) is
the sample mean of the outcome.
QR Decomposition
The lm function in R actually performs a QR decomposition of the design matrix, \(\mathbf{X} = \mathbf{Q}\mathbf{R}\), where \(\mathbf{Q}^\top \mathbf{Q} = \mathbf{I}\) and \(\mathbf{R}\) is upper
triangular. Thus, the OLS solution for the coefficients can be written as \(\left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{y} = \mathbf{R}^{-1} \mathbf{Q}^\top \mathbf{y}\). The
lm function utilizes the QR decomposition for numeric stability reasons, but the QR decomposition is also useful for thinking about priors in a Bayesian version of the linear model. In addition,
writing the likelihood in terms of \(\mathbf{Q}\) allows it to be evaluated in a very efficient manner in Stan.
The key innovation in the stan_lm function in the rstanarm package is the prior for the parameters in the QR-reparameterized model. To understand this prior, think about the equations that
characterize the maximum likelihood solutions before observing the data on \(\mathbf{X}\) and especially \(\mathbf{y}\).
What would the prior distribution of \(\boldsymbol{\theta} = \mathbf{Q}^\top \mathbf{y}\) be? We can write its \(k\)-th element as \(\theta_k = \rho_k \sigma_Y \sqrt{N - 1}\) where \(\rho_k\) is the
correlation between the \(k\)th column of \(\mathbf{Q}\) and the outcome, \(\sigma_Y\) is the standard deviation of the outcome, and \(\frac{1}{\sqrt{N-1}}\) is the standard deviation of the \(k\)
column of \(\mathbf{Q}\). Then let \(\boldsymbol{\rho} = \sqrt{R^2}\mathbf{u}\) where \(\mathbf{u}\) is a unit vector that is uniformly distributed on the surface of a hypersphere. Consequently, \(R^
2 = \boldsymbol{\rho}^\top \boldsymbol{\rho}\) is the familiar coefficient of determination for the linear model.
An uninformative prior on \(R^2\) would be standard uniform, which is a special case of a Beta distribution with both shape parameters equal to \(1\). A non-uniform prior on \(R^2\) is somewhat
analogous to ridge regression, which is popular in data mining and produces better out-of-sample predictions than least squares because it penalizes \(\boldsymbol{\beta}^\top \boldsymbol{\beta}\),
usually after standardizing the predictors. An informative prior on \(R^2\) effectively penalizes \(\boldsymbol{\rho}\top \boldsymbol{\rho}\), which encourages \(\boldsymbol{\beta} = \mathbf{R}^{-1}
\boldsymbol{\theta}\) to be closer to the origin.
Lewandowski, Kurowicka, and Joe (2009) derives a distribution for a correlation matrix that depends on a single shape parameter \(\eta > 0\), which implies the variance of one variable given the
remaining \(K\) variables is \(\mathrm{Beta}\left(\eta,\frac{K}{2}\right)\). Thus, the \(R^2\) is distributed \(\mathrm{Beta}\left(\frac{K}{2},\eta\right)\) and any prior information about the
location of \(R^2\) can be used to choose a value of the hyperparameter \(\eta\). The R2(location, what) function in the rstanarm package supports four ways of choosing \(\eta\):
1. what = "mode" and location is some prior mode on the \(\left(0,1\right)\) interval. This is the default but since the mode of a \(\mathrm{Beta}\left(\frac{K}{2},\eta\right)\) distribution is \(\
frac{\frac{K}{2} - 1}{\frac{K}{2} + \eta - 2}\) the mode only exists if \(K > 2\). If \(K \leq 2\), then the user must specify something else for what.
2. what = "mean" and location is some prior mean on the \(\left(0,1\right)\) interval, where the mean of a \(\mathrm{Beta}\left(\frac{K}{2},\eta\right)\) distribution is \(\frac{\frac{K}{2}}{\frac
{K}{2} + \eta}\).
3. what = "median" and location is some prior median on the \(\left(0,1\right)\) interval. The median of a \(\mathrm{Beta}\left(\frac{K}{2},\eta\right)\) distribution is not available in closed form
but if \(K > 2\) it is approximately equal to \(\frac{\frac{K}{2} - \frac{1}{3}}{\frac{K}{2} + \eta - \frac{2}{3}}\). Regardless of whether \(K > 2\), the R2 function can numerically solve for
the value of \(\eta\) that is consistent with a given prior median utilizing the quantile function.
4. what = "log" and location is some (negative) prior value for \(\mathbb{E} \ln R^2 = \psi\left(\frac{K}{2}\right)- \psi\left(\frac{K}{2}+\eta\right)\), where \(\psi\left(\cdot\right)\) is the
digamma function. Again, given a prior value for the left-hand side it is easy to numerically solve for the corresponding value of \(\eta\).
There is no default value for the location argument of the R2 function. This is an informative prior on \(R^2\), which must be chosen by the user in light of the research project. However, specifying
location = 0.5 is often safe, in which case \(\eta = \frac{K}{2}\) regardless of whether what is "mode", "mean", or "median". In addition, it is possible to specify NULL, in which case a standard
uniform on \(R^2\) is utilized.
We set \(\sigma_y = \omega s_y\) where \(s_y\) is the sample standard deviation of the outcome and \(\omega > 0\) is an unknown scale parameter to be estimated. The only prior for \(\omega\) that
does not contravene Bayes’ theorem in this situation is Jeffreys prior, \(f\left(\omega\right) \propto \frac{1}{\omega}\), which is proportional to a Jeffreys prior on the unknown \(\sigma_y\), \(f\
left(\sigma_y\right) \propto \frac{1}{\sigma_y} = \frac{1}{\omega \widehat{\sigma}_y} \propto \frac{1}{\omega}\). This parameterization and prior makes it easy for Stan to work with any continuous
outcome variable, no matter what its units of measurement are.
It would seem that we need a prior for \(\sigma_{\epsilon}\), but our prior beliefs about \(\sigma_{\epsilon} = \omega s_y \sqrt{1 - R^2}\) are already implied by our prior beliefs about \(\omega\)
and \(R^2\). That only leaves a prior for \(\alpha = \overline{y} - \overline{\mathbf{x}}^\top \mathbf{R}^{-1} \boldsymbol{\theta}\). The default choice is an improper uniform prior, but a normal
prior can also be specified such as one with mean zero and standard deviation \(\frac{\sigma_y}{\sqrt{N}}\).
The previous sections imply a posterior distribution for \(\omega\), \(\alpha\), \(\mathbf{u}\), and \(R^2\). The parameters of interest can then be recovered as generated quantities:
• \(\sigma_y = \omega s_y\)
• \(\sigma_{\epsilon} = \sigma_y \sqrt{1 - R^2}\)
• \(\boldsymbol{\beta} = \mathbf{R}^{-1} \mathbf{u} \sigma_y \sqrt{R^2 \left(N-1\right)}\)
The implementation actually utilizes an improper uniform prior on \(\ln \omega\). Consequently, if \(\ln \omega = 0\), then the marginal standard deviation of the outcome implied by the model is the
same as the sample standard deviation of the outcome. If \(\ln \omega > 0\), then the marginal standard deviation of the outcome implied by the model exceeds the sample standard deviation, so the
model overfits the data. If \(\ln \omega < 0\), then the marginal standard deviation of the outcome implied by the model is less than the sample standard deviation, so the model underfits the data or
that the data-generating process is nonlinear. Given the regularizing nature of the prior on \(R^2\), a minor underfit would be considered ideal if the goal is to obtain good out-of-sample
predictions. If the model badly underfits or overfits the data, then you may want to reconsider the model.
We will utilize an example from the HSAUR3 package by Brian S. Everitt and Torsten Hothorn, which is used in their 2014 book A Handbook of Statistical Analyses Using R (3rd Edition) (Chapman & Hall /
CRC). This book is frequentist in nature and we will show how to obtain the corresponding Bayesian results.
The model in section 5.3.1 analyzes an experiment where clouds were seeded with different amounts of silver iodide to see if there was increased rainfall. This effect could vary according to
covariates, which (except for time) are interacted with the treatment variable. Most people would probably be skeptical that cloud hacking could explain very much of the variation in rainfall and
thus the prior mode of the \(R^2\) would probably be fairly small.
The frequentist estimator of this model can be replicated by executing
(Intercept) seedingyes
-0.346 15.683
sne cloudcover
0.420 0.388
prewetness echomotionstationary
4.108 3.153
time seedingyes:sne
-0.045 -3.197
seedingyes:cloudcover seedingyes:prewetness
-0.486 -2.557
Note that we have not looked at the estimated \(R^2\) or \(\sigma\) for the ordinary least squares model. We can estimate a Bayesian version of this model by prepending stan_ to the lm call,
specifying a prior mode for \(R^2\), and optionally specifying how many cores the computer may utilize:
family: gaussian [identity]
formula: rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) +
observations: 24
predictors: 11
Median MAD_SD
(Intercept) 2.4 2.3
seedingyes 6.8 3.8
sne 0.2 0.7
cloudcover 0.2 0.2
prewetness 1.7 2.8
echomotionstationary 1.4 1.5
time 0.0 0.0
seedingyes:sne -1.4 1.0
seedingyes:cloudcover -0.2 0.2
seedingyes:prewetness -1.1 3.5
seedingyes:echomotionstationary -0.2 2.0
Auxiliary parameter(s):
Median MAD_SD
R2 0.3 0.1
log-fit_ratio 0.0 0.1
sigma 2.6 0.4
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
In this case, the “Bayesian point estimates”, which are represented by the posterior medians, appear quite different from the ordinary least squares estimates. However, the log-fit_ratio (i.e. \(\ln
\omega\)) is quite small, indicating that the model only slightly overfits the data when the prior derived above is utilized. Thus, it would be safe to conclude that the ordinary least squares
estimator considerably overfits the data since there are only \(24\) observations to estimate \(12\) parameters with and no prior information on the parameters.
Also, it is not obvious what the estimated average treatment effect is since the treatment variable, seeding, is interacted with four other correlated predictors. However, it is easy to estimate or
visualize the average treatment effect (ATE) using rstanarm’s posterior_predict function.
As can be seen, the treatment effect is not estimated precisely and is as almost as likely to be negative as it is to be positive.
Alternative Approach
The prior derived above works well in many situations and is quite simple to use since it only requires the user to specify the prior location of the \(R^2\). Nevertheless, the implications of the
prior are somewhat difficult to conceptualize. Thus, it is perhaps worthwhile to compare to another estimator of a linear model that simply puts independent Cauchy priors on the regression
coefficients. This simpler approach can be executed by calling the stan_glm function with family = gaussian() and specifying the priors:
We can compare the two approaches using an approximation to Leave-One-Out (LOO) cross-validation, which is implemented by the loo function in the loo package.
Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly.
Computed from 4000 by 24 log-likelihood matrix
Estimate SE
elpd_loo -60.3 5.3
p_loo 5.9 2.4
looic 120.5 10.6
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 21 87.5% 850
(0.5, 0.7] (ok) 2 8.3% 372
(0.7, 1] (bad) 1 4.2% 116
(1, Inf) (very bad) 0 0.0% <NA>
See help('pareto-k-diagnostic') for details.
Warning: Found 3 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 3 times to compute the ELPDs for the problematic observations directly.
elpd_diff se_diff
post 0.0 0.0
simple -1.1 3.0
The results indicate that the first approach is expected to produce better out-of-sample predictions but the Warning messages are at least as important. Many of the estimated shape parameters for the
Generalized Pareto distribution are above \(0.5\) in the model with Cauchy priors, which indicates that these estimates are only going to converge slowly to the true out-of-sample deviance measures.
Thus, with only \(24\) observations, they should not be considered reliable. The more complicated prior derived above is stronger — as evidenced by the fact that the effective number of parameters is
about half of that in the simpler approach and \(12\) for the maximum likelihood estimator — and only has a few of the \(24\) Pareto shape estimates in the “danger zone”. We might want to reexamine
these observations
because the posterior is sensitive to them but, overall, the results seem tolerable.
In general, we would expect the joint prior derived here to work better when there are many predictors relative to the number of observations. Placing independent, heavy-tailed priors on the
coefficients neither reflects the beliefs of the researcher nor conveys enough information to stabilize all the computations.
This vignette has discussed the prior distribution utilized in the stan_lm function, which has the same likelihood and a similar syntax as the lm function in R but adds the ability to expression
prior beliefs about the location of the \(R^2\), which is the familiar proportion of variance in the outcome variable that is attributable to the predictors under a linear model. Since the \(R^2\) is
a well-understood bounded scalar, it is easy to specify prior information about it, whereas other Bayesian approaches require the researcher to specify a joint prior distribution for the regression
coefficients (and the intercept and error variance).
However, most researchers have little inclination to specify all these prior distributions thoughtfully and take a short-cut by specifying one prior distribution that is taken to apply to all the
regression coefficients as if they were independent of each other (and the intercept and error variance). This short-cut is available in the stan_glm function and is described in more detail in other
rstanarm vignettes for Generalized Linear Models (GLMs), which can be found by navigating up one level.
We are optimistic that this prior on the \(R^2\) will greatly help in accomplishing our goal for rstanarm of making Bayesian estimation of regression models routine. The same approach is used to
specify a prior in ANOVA models (see stan_aov) and proportional-odds models for ordinal outcomes (see stan_polr).
Finally, the stan_biglm function can be used when the design matrix is too large for the qr function to process. The stan_biglm function inputs the output of the biglm function in the biglm package,
which utilizes an incremental QR decomposition that does not require the entire dataset to be loaded into memory simultaneously. However, the biglm function needs to be called in a particular way in
order to work with stan_biglm. In particular, The means of the columns of the design matrix, the sample mean of the outcome, and the sample standard deviation of the outcome all need to be passed to
the stan_biglm function, as well as a flag indicating whether the model really does include an intercept. Also, the number of columns of the design matrix currently cannot exceed the number of rows.
Although stan_biglm should run fairly quickly and without much memory, the resulting object is a fairly plain stanfit object rather than an enhanced stanreg object like that produced by stan_lm. Many
of the enhanced capabilities of a stanreg object depend on being able to access the full design matrix, so doing posterior predictions, posterior checks, etc. with the output of stan_biglm would
require some custom R code.
Lewandowski, D., Kurowicka D., and Joe, H. (2009). Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis. 100(9), 1989–2001. | {"url":"http://cran.r-project.org/web/packages/rstanarm/vignettes/lm.html","timestamp":"2024-11-08T02:24:34Z","content_type":"text/html","content_length":"85996","record_id":"<urn:uuid:3724a4ed-7c05-4941-b246-758e6e2d9f12>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00322.warc.gz"} |
How much storage do I need for a 4 bedroom house? - EasyRelocated
How much storage do I need for a 4 bedroom house?
How much storage do I need for a 4 bedroom house?
A 10×30 storage unit is as large as a 1 ½ car garage. It fits the contents of a fully furnished 3-5 bedroom house, including oversized items such as a couches, beds, dressers, a refrigerator, a
washer/dryer, a dining room set, an entertainment center and several large boxes.
How many pounds does it take to move a 5 bedroom house?
How much does a five bedroom house weigh if Im moving everything inside of it? The weight of your five bedroom house is likely to about 7,000 pounds. Houses of this caliber will weigh a lot more than
smaller houses. Be careful to add up all your furniture when trying to estimate the weight of you belongings.
How much does it cost to put everything in storage?
Average storage unit cost
Storage unit sizes Average monthly cost
Small (5×5–5×10 ft.) $90
Medium (5×15–10×15 ft.) $160
Large (10×20–10×30 ft.) $290
How much should I charge for storage?
Most people pay an average price of $190 per month for their storage unit — and the cost adds up significantly over time….
Unit Size (in feet) Average Monthly Cost
5×5 to 5×10 $90
5×15 to 10×15 $160
10×20 to 10×30 $290
How much storage for a house?
One rule of thumb says that the total storage space in a home should equal about 10% of the total square footage. So, if your custom home is 5,000 square feet, you should build 500 square feet of
storage at a minimum. Some families have more or bulkier items to store.
How do I figure out how much storage I need?
Square feet (sq. Multiply the length and width of your belongings. If they make a pile that’s 5 x 5 feet, you’d need a storage unit with at least 25 square feet (Extra Space Storage has excellent 5 x
5 storage units).
What is the average weight of household goods?
An average one-bedroom apartment containing a bed, dresser, sofa, coffee table, entertainment center, television, dining set, desk and computer weighs approximately 2,110 lbs. To tally the weight of
a larger more furnished household, simply add anywhere from 2,000 to 3,000 lbs. for each room included in the shipment.
How much does the contents of a house weigh?
With long-distance moves, you are given a price estimate based on the weight (in lbs) of your belongings as shown: One Studio Apartment = 1800 lbs. 1-Bedroom Apartment = 2200-3200 lbs. 2-Bedroom Home
= 5000-6000 lbs.
How much does a household weigh?
So, for a 2,000-square-foot home, that comes out to 400,000 pounds. If it’s two stories, it goes up to 275 pounds per square foot. For a three-story house, the weight goes up to roughly 350 pounds
per square foot. | {"url":"https://easyrelocated.com/how-much-storage-do-i-need-for-a-4-bedroom-house-2/","timestamp":"2024-11-12T19:37:28Z","content_type":"text/html","content_length":"61248","record_id":"<urn:uuid:5c1ddd37-86df-4a89-a007-1e94012c9047>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00395.warc.gz"} |
Data Analysis with Google Sheets: (Advanced Excel Formulas, and Pivot Tables ) — Steemit
Data Analysis with Google Sheets: (Advanced Excel Formulas, and Pivot Tables )
Topics Covered: Advanced formulas, and pivot tables.
This comes to the second week of our lessons and we humbly welcome you here once again. In our previous lesson, we only introduced you to the basic formulas of spreadsheets this time around we will
be looking at more advanced formulas, pivot charts, and data analysis.
Having said that, Advanced Excel formulas are rarely used because of how uncommon they are been used. There are inbuilt formulas that you can use to retrieve specific data from an existing data set
which is like duplicating the data, filtering the data, and so on. You can also use these functions which we will practically look at to create a dashboard, generate reports, and so on.
For example, looking at the above images you can see some of the inbuilt advanced Excel formulas that you can use. Also, these inbuilt formulas are available in LibreOffice and not just in Excel. Out
of the many advanced Excel formulas we have let's practically talk about these two VLOOKUP formula and the IF FUNCTION formulas in this lesson.
VLOOKUP Formula in Excel
Based on simplicity VLOOKUP is the most used formula in Excel. Simply this is the formula that helps you to search for the in the leftmost column of the table array and returns the value from the
same row from the specific columns.
Let's consider looking at the short example in which we have applied this formula. Now for example, we have two different tables that contain the names, employees ID, and salary of employees, as our
main column and we want the salary from our table B to be in table A.
By mainly looking at the data above, if we want to get the salary in Table B in Table A, we will have to apply the formula shown in the image below.
Now, once you have entered the formula in employee salary which is cell C, you can drag the formula to the rest of the cells to get your results in each cell which is the salary of employees.
Using the IF Function
One of the most powerful features of spreadsheets is the ability to perform different calculations based on changing values. IF Function is what is mainly used. This function makes logical tests of
evaluation and performs one of two different calculations based on the result. The format for the IF Function is stated 👇 below.
=IF(logical text,value if true, value if false)
The logical text is what specifies what the statement of the IF Function will evaluate for you.
The Value If True, If the logical Text is found to be true, then the result of your IF Function will be whatever is in the format we have shared.
The Value If False, If the logical test is found to be false, then the result of your IF Function will be whatever is in the format we have shared.
Comparison is what the logical test mostly uses which helps us to obtain our desired result. What is used to identify or define different types of data is Qualifiers which are punctuation marks. For
instance, text that is used in a formula needs double quotes as qualifiers. Having said so let's take a look at the comparison operators that you can see in IF Function.
• Equal to (=)
• Greater than (>)
• Less than (<)
• The greater than or equal to (>=)
• Not equal to (<>)
• The less than or equal to (<=)
Now let's learn how to create a simple assessment report using a spreadsheet which will then apply the IF Function.
• First is for us to identify student information in the report.
• Secondly is to open MS Excel
• Thirdly enter the data shown below.
From the above data, to find the total. Select cell E5 and enter the formula =Sum(B5:D5) and press enter. The total score should appear in cell E5. Using the formula Excel will add the contents in
B5, C5, and D5 and provide you with the answer.
Converting the percentage to a level score of 1, 2, 3, or 4 which can also be alphabetic grade A, B, C, and so on. Now to do this, we will have to first determine our range.
• 70% -100% = A
• 60% - 69% = B
• 50% - 59% = C
• 45% - 49% = D
• 40% - 44% = E
Anything else less than that which is given above should be F.
To get this, we will have to click on cell F5 and type in the given formula below using the If function.
=IF(E5>=70, "A", IF(E5>=60, "B", IF(E5>=50, "C", IF(E5>=45, "D", IF(E5>=40, "E", "F")))))
Once you have entered the formula correctly you need to click on the Enter key and you will get the right results which you can then drag down to get other results as shown below.
Pivot Table
Now we have moved to the pivot table which is a summary tool that helps us to synthesize information from a database or set. Summary means all kinds of descriptive statistics data that help pivot
table groups together in a meaningful way. If you are a data analyst you can use a pivot table to summarise large datasets into a meaningful table that can be easy to view.
Pivot tables are used for some of the following reasons given below.
• To group data into categories.
• To count the number of items in each category.
• To sum the value of items.
• To find minimal or maximal value, compute average, etc.
To create a pivot table in a spreadsheet (excel or LibreOffice), you will need to follow the steps shared below.
Prepare your data
• You have to make your data well organized in a tabular form, with column headers for example; Date, Product, Sales, and so on.
• Each of the column headers should represent a specific variable, whereas each row should represent a unique record.
Select Your Data
• At this stage you will have to highlight the day range that you want to analyze.
Insert The Pivot Table:
The below steps are what you need when inserting a pivot table in Excel.
• On the ribbon you click on Insert
• Click on Pivot Table
• Confirm the range of your data in the dialog box, and choose to either place the pivot table in an already existing worksheet or a new worksheet.
Choose Pivot Table Fields
• Once you have created your pivot table, you will see a new pane that will appear allowing you to drag the field into 4 areas.
• Row: The data categories that you want as row labels.
• Column: The data categories that you want as column.
• Values: The data you want to aggregate.
• Filters: This allows you to view specific data only.
Customise your pivot table
You can do this, by changing the calculation type
Applying filters and formatting.
Now let's take a look at the below example.
From the above screenshots, you can see how we have applied a pivot table in the given example.
• Explain what you understand by Advanced Excel Formulas, and show us where advanced formulas such as the lookup function, and logical function are found in Excel with clear screenshots.
• Write the IF Function formula to calculate the total, average score, and grade of students given in the table below.
Students Maths English Physics Chemistry
Simonnwigwe 75 50 84 60
Josepha 76 60 55 90
Kouba01 60 98 85 90
Adeljose 70 60 50 60
Ruthjoe 60 45 80 51
Lhorgic 45 90 70 65
Dove11 70 60 55 75
Ruthjoe 58 70 85 73
• Briefly discuss four IF function Operators that you have learned and tell us their functions and when we are to use them.
• Based on the given data below: Create a pivot table that shows (see) total sales by product, by dragging the product to the Rows areas, Region to the Column area, and Sales to the Values area.
Please we want to see the steps you take in adding your pivot table.
Date Product Region Sales
16/09/2024 Product A East 100
17/09/2024 Product B West 150
18/09/2024 Product C North 200
• Posts must be published in your blog and not in any community.
• The title must be: "SEC | S20W2 | Data Analysis with Google Sheets: (Advanced Excel formulas, and pivot tables.)
• The post must contain a minimum of 350 words, be free from plagiarism, and not use Artificial Intelligence (AI) or other forms of cheating.
• Use the main hashtag #spreadsheet-s20w2 (required) among the first 4 tags
• Add your country name hashtag (e.g. #nigeria)
• If using the hashtag #burnsteem25 make sure you give 25% of the reward to @null
• Invite 3 friends to participate.
• Paste your participation link in the comment section, and don't forget to Vote and Rate this post.
• This contest starts Monday, September 16, 2024, at 00:00 UTC and ends on Sunday, September 22nd, 2024, at 23:59 UTC."
Note: [We will choose the winners based on the quality of the post, a quality post in our opinion is a post that can provide interesting ideas and new insights for its readers. Proper use of markdown
is also part of the quality of a post.]
Check closing date...
Sunday, September 22nd, 2024
Thank you. Corrected as suggested.
Excelente temática para esta segunda semana del curso; conocer el funcionamientos de las fórmulas avanzada, la función SI y las tablas dinámicas, son elementos que nos serán de gran utilidad para
trabajar con grandes cantidades de datos en las hojas de cálculo.
Desde ya comienzo a preparar mi participación. Saludos.
Wonderful topic for week2. For those of us that struggling to be a computer literate this is a golden opportunity for us. I will surely do my best in this week's own.
tema yang menarik dan seperinya lebih sulit dari minggu sebelumnya.
Excelente clase, y seguimos avanzando en funciones, herramientas muy útiles de excel, me ha encantado, ya que he aprendido herramientas que no conocía.
El ejemplo de la tabla pivote no se observa claramente, no logro detallar las imágenes, están muy pequeñas.
Another new and interesting task. We are excited to take part. Thanks for this deep explanation of the advanced formula and on pivot table. Entry loading. | {"url":"https://steemit.com/spreadsheet-s20w2/@josepha/data-analysis-with-google-sheets-advanced-excel-formulas-and-pivot-tables","timestamp":"2024-11-07T10:36:41Z","content_type":"text/html","content_length":"262591","record_id":"<urn:uuid:c00c2221-9f73-4189-b5f4-5bc9f75368f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00019.warc.gz"} |
From New World Encyclopedia
Fahrenheit temperature conversion formulas
│ To find │ From │ Formula │
│ Celsius │ Fahrenheit │ °C = (°F − 32) ÷ 1.8 │
│ Fahrenheit │ Celsius │ °F = (°C × 1.8) + 32 │
│ Kelvin │ Fahrenheit │ K = (°F + 459.67) ÷ 1.8 │
│ Fahrenheit │ Kelvin │ °F = (K × 1.8) − 459.67 │
│ Rankine │ Fahrenheit │ R = °F + 459.67 │
│ Fahrenheit │ Rankine │ °F = R − 459.67 │
│ Conversion calculator for units of temperature │
Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit (1686–1736), a German physicist who did most of his work in the Netherlands. This temperature scale, which was in use long
before the Celsius scale was proposed, continues to be used for everyday temperature measurements by the general population of the United States.^[1] In most other countries (and in scientific
studies worldwide), temperature measurements are made primarily on the Celsius scale.
On the Fahrenheit scale, the freezing point of water is 32 degrees Fahrenheit (written "32 °F"), and the boiling point is 212 degrees, placing the boiling and freezing points of water exactly 180
degrees apart. Thus the unit of this scale, a degree Fahrenheit, is five-ninths (^5â„[9]ths) of a degree Celsius, and negative 40 degrees Fahrenheit is equal to negative 40 degrees Celsius. Absolute
zero is at −459.67 °F.^[2]
Did you know?
The Fahrenheit temperature scale was proposed in 1724 by
-based physicist Daniel Gabriel Fahrenheit
There are several competing versions of the story of how Fahrenheit came to devise his temperature scale.^[3] According to one version, Fahrenheit established the zero (0 °F) and 100 °F points on
his scale by recording the lowest outdoor temperatures he could measure, and his own body temperature. He took as his zero point the lowest temperature he measured in the harsh winter of 1708 through
1709 in his hometown of Danzig (now Gdańsk, Poland) (−17.8 °C). (He was later able to reach this temperature under laboratory conditions using a mixture of ice, ammonium chloride, and water.)
Fahrenheit wanted to avoid the negative temperatures that Ole Rømer's scale had produced in everyday use. He fixed his own body temperature as 100 °F.^[4] He then divided his original scale into
twelve parts, and later divided each of these into 8 equal subdivisions, produced a scale of 96 degrees. Fahrenheit noted that his scale placed the freezing point of water at 32 °F and the boiling
point at 212 °F, a neat 180 degrees apart.
Another story holds that Fahrenheit established the zero of his scale (0 °F) as the temperature at which a mixture of equal parts of ice and salt melts (some say he took that fixed mixture of ice
and salt that produced the lowest temperature); and 96 degrees as the temperature of blood (he initially used horse blood to calibrate his scale). Initially, his scale contained only 12 equal
divisions, but he later subdivided each division into eight equal degrees, ending up with 96.
A third, well-known version of the story, as described in the popular physics television series The Mechanical Universe, maintains that Fahrenheit simply adopted Rømer's scale (in which water
freezes at 7.5 degrees) and multiplied each value by four to eliminate the fractions and increase the granularity of the scale (giving 30 and 240 degrees). He then re-calibrated his scale between the
melting point of water and normal human body temperature (which he took to be 96 degrees); the melting point of ice was adjusted to 32 degrees, so that 64 intervals would separate the two, allowing
him to mark degree lines on his instruments by simply bisecting the interval six times (since 64 is two to the sixth power).
His measurements were not entirely accurate, though. By his original scale, the actual melting and boiling points would have been noticeably different from 32 °F and 212 °F. Some time after his
death, it was decided to recalibrate the scale with 32 °F and 212 °F as the exact melting and boiling points of plain water. That change was made to easily convert from Celsius to Fahrenheit and
vice versa, with a simple formula. This change also explains why the body temperature once taken as 96 or 100 °F by Fahrenheit is today taken by many as 98.6 °F (it is a direct conversion of 37
°C), although giving the value as 98 °F would be more accurate.
There are at least three other versions of this story, but they appear to be based less on evidence and more on speculation. They are therefore not recounted here.
Conversions and key temperatures on different scales
On the Fahrenheit scale, the freezing point of water is 32 degrees Fahrenheit (°F) and the boiling point is 212 °F (at standard atmospheric pressure). This puts the boiling and freezing points
of water exactly 180Â degrees apart.^[5] Therefore, a degree on the Fahrenheit scale is ^1â„[180] of the interval between the freezing point and the boiling point. On the Celsius scale, the freezing
and boiling points of water are 100 degrees apart. A temperature interval of 1 °F is equal to an interval of ^5â„[9] degrees Celsius. The Fahrenheit and Celsius scales intersect at −40° (i.e.,
−40 °F = −40 °C).
Absolute zero is −273.15 °C or −459.67 °F. The Rankine temperature scale uses degree intervals of the same size as those of the Fahrenheit scale, except that absolute zero is 0 R—the same
way that the Kelvin temperature scale matches the Celsius scale, except that absolute zero is 0Â K.^[5]
The Fahrenheit scale uses the symbol ° to denote a point on the temperature scale (as does Celsius) and the letter F to indicate the use of the Fahrenheit scale (e.g. "Gallium melts at 85.5763
°F"),^[6] as well as to denote a difference between temperatures or an uncertainty in temperature (e.g. "The output of the heat exchanger experiences an increase of 72 °F" and "Our standard
uncertainty is ±5 °F").
For an exact conversion, the following formulas can be applied. Here, f is the value in Fahrenheit and c the value in Celsius:
• f °Fahrenheit to c °Celsius : (f − 32) °F × 5°C/9°F = (f − 32)/1.8 °C = c °C
• c °Celsius to f °Fahrenheit : (c °C × 9°F/5°C) + 32 °F = (c × 1.8) °F + 32 °F = f °F
This is also an exact conversion making use of the identity -40 °F = -40 °C. Again, f is the value in Fahrenheit and c the value in Celsius:
• f °Fahrenheit to c °Celsius : ((f + 40) ÷ 1.8) − 40 = c.
• c °Celsius to f °Fahrenheit : ((c + 40) * 1.8) − 40 = f.
Some key temperatures relating the Fahrenheit scale to other temperature scales are shown in the table below.
│ │ Kelvin │ Celsius │ Fahrenheit │
│ Absolute zero │ │ │ │
│ │ 0 K │ −273.15 °C │ −459.67 °F │
│ (precise, by definition) │ │ │ │
│ Melting point of ice │ 273.15 K │ 0 °C │ 32 °F │
│ Water’s triple point │ │ │ │
│ │ 273.16 K │ 0.01 °C │ 32.018 °F │
│ (precise, by definition) │ │ │ │
│ Water’s boiling point ^A │ 373.1339 K │ 99.9839 °C │ 211.9710 °F │
^A For Vienna Standard Mean Ocean Water (VSMOW) at a pressure of one standard atmosphere (101.325 kPa) when calibrated solely per the two-point definition of thermodynamic temperature.
The Fahrenheit scale was the primary temperature standard for climatic, industrial and medical purposes in most English-speaking countries until the 1960s. In the 1960s and 1970s, governments phased
in the Celsius (formerly centigrade) scale as part of the shift to the metric system of units.
Fahrenheit supporters assert its previous popularity was due to Fahrenheit's user-friendliness. The unit of measure, being only ^5â„[9] the size of the Celsius degree, permits more precise
communication of measurements without resorting to fractional degrees. Also, the ambient air temperature in most inhabited regions of the world tends not to go far beyond the range of 0 °F to 100
°F: therefore, the Fahrenheit scale would reflect the perceived ambient temperatures, following 10-degree bands that emerge in the Fahrenheit system:
• 0s Extremely cold. (0°F ~ -17.8°C)
• 10s Deep frost.
• 20s Light frost.
• 30s Very cold.
• 32 Freezing.
• 40s Cold. Heavy clothing needed.
• 50s Cool. Moderate Clothing required.
• 60s Tepid. Light clothing.
• 70s Warm. Summer clothing.
• 80s Hot. Breathable clothing.
• 90s Very hot. Minimal clothing.
• 98.6 Body Temperature
• 100s Extremely hot. Take precautions against overheating.
Some Celsius supporters argue that their system can be just as natural, for example they might say that 0–10 °C indicates cold, 10–20 °C mild, 20–30 °C warm and 30–40 °C hot.
In the United States, the Fahrenheit system continues to be the accepted standard for nonscientific use. On the other hand, the Celsius scale is widely used in most other parts of the world,
including Europe, Canada, Australia, and New Zealand. Fahrenheit is sometimes used by older generations, especially for measurement of higher temperatures.
See also
ISBN links support NWE through referral fees
External links
All links retrieved March 23, 2024.
Temperature scales
Celsius Fahrenheit Kelvin
Delisle Leyden Newton Rankine Réaumur Rømer
Conversion formulas
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Calculated Metrics Inside Calculated Metrics | Planhat | Help Center
You can reference calculated metrics inside a calculated metric.
This sounds trivial, but unlocks some really powerful use cases:
1. Analyse interpolated data, e.g., when you have non-daily / spotty data (if that sounds theoretical, drop down here to see what it means π € )
2. Time-series analysis of numeric fields (fields, not metrics!), like looking at week-over-week trends, find the min/max or average value over a given period of time for any numeric field in your
customer database
3. Your metric library can become more dynamic, which means updating in one place updates everywhere related (note: with great power comes great responsibility here, be mindful of cascading effects
of updates and circular references)
2. How calculated metrics are processed (π hint: you can now decide the order of metric processing!)
Before we dive into use cases, let's quickly explain how to do it
π Read more here about how to build calculated metrics in general.
To reference another calculated metric inside a calculated metric:
1. Copy the calculated metric ID, which you can find under the calculated metric title and will look like this calculated.625577d94c890d12cbfe6bfd
2. Insert the ID as the property value in a calculated metric operation, like below where the references calculated metric is another time series that you can sum up, take an average over defined
number of days, etc (more on that below)
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "calculated.625577d94c890d12cbfe6bfd"}
Use case #1: Analyse interpolated data
Sometimes when working with metrics you want to perform multiple operations on an entire time-series. This is most often relevant when you have gaps in your data (i.e., it's sent in non-daily) which
leads to you using the "LAST" operator to smooth it over time, and then you want to analyse that smoothed time-series. Examples:
• You send in a time series with a few "gaps" in the data
• You only care about the latest available data point, which might not be daily
Let's imagine the scenario below where you are pushing in 3 different metrics (Log-ins, Temperature, NPS) that all have 1 day of null value. You want to build a calculated metric to show what the
5-day rolling average is.
Day 1 Day 2 Day 3 Day 4 Day 5
Log-ins 7 5 9 5 -
Temp (CΒ°) 22 15 15 - 17
NPS 5 - - - -
Depending on the metric, the null value could be interpreted differently. In the case of Log-ins, it probably means no one logged in that day, and we can assume that null equals 0. In the case of
temperature, looking at the other days' values it seems that null means "no value sent" rather than 0 (maybe the thermometer was broken, there was something wrong with the internet connection, etc).
In the NPS case, a data point is only sent in when updated, but we should still treat Day 2-5 like they have that value.
π Side note: use this setting to choose how Planhat interprets zeros, as 0 or as null.
How does this relate to nested calculated metrics? In the case of Temperature and NPS, you probably want to "smooth" the data by assuming some replacement figure for the null value (since assuming 0
will mess up the 5-day rolling average).
To solve this, you can build a calculated metric which looks at the Temperature or NPS and saves the latest available data point. So for Day 4, that means taking 15CΒ°. For NPS, that means looking
further back and taking the latest available (5). In the Temperature case, the calculated metric would look like this:
// Temperature example: calculated.123456789
{"type": "metricOverTime", "days": 3, "op": "LAST", "prop": "custom.temperature"}
π Calculated Metric extra class: the number of "days" in the "LAST" function can be anything from 1 to infinity, depending on how far back you want to fetch data. If you say days = 1, that means
Planhat will only look back 1 day for a data point - meaning if you have 2 days in a row with null value then you will get a null value. This makes sense if the data is volatile and data farther back
than the day before won't correlate to today. 2-5 is perhaps a reasonable value for Temperature if you live in volatile Sweden - 60 if you live in sunny California!
For NPS, maybe we'll always take the last value available!
This will generate a new time-series as a calculated metric:
Day 1 Day 2 Day 3 Day 4 Day 5
Log-ins 7 5 9 5 0
Temp (CΒ°) 22 15 15 15 17
NPS 5 5 5 5 5
Great - all smoooooth. But now you want to "sum up" these 5 days in order to average them - but this doesn't work. Previously, you couldn't use the "average over 5 days" on this, since that operation
couldn't be performed on calculated metrics. Now, you can!
Again, using the calculated metric example:
// 5-day average of temperature: calculated.987654321
{"type": "metricOverTime", "days": 5, "op": "AVERAGE", "prop": "calculated.123456789"}
Use case #2: Time-series analysis of numeric fields
In calculated metrics, you can reference a numeric field (i.e., point-in-time data) which turns it into a time-series.
This was possible before, but then you couldn't do any "analytics" on it, like:
• Define change over time (e.g., week-over-week change)
• Find the min/max value over a given period of time
• See the average over a period of time
With calculated metrics inside calculated metrics, you can:
The "first" calculated metric which turns field data into a metric could look like:
// Field to metric: calculated.121212123
{"type": "propertyValue", "prop": "company.custom.employee count"}
// Average over last 7 days
{"type": "metricOverTime", "days": 7, "op": "AVERAGE", "prop": "calculated.121212123"}
// Maximum value in last year
{"type": "metricOverTime", "days": 365, "op": "MAX", "prop": "calculated.121212123"}
...and so on!
Note that this does not work retroactively, i.e., Planhat only starts saving down the daily value from the day you set up the metric.
Use case #3: Make your metric library more efficient and dynamic
When you build up a library of metrics to understand your customer operations, you often end up having some depend on others.
Let's assume you have two metrics called "This week's logins" and "Last week's logins" which are based on summing up a custom metric being pushed in from your back-end. Now you want a "Week-over-Week
change in logins" metric which does a week-by-week comparison. Without being able to reference other calculated metrics, this would have looked like:
// Metric 1: THIS WEEK'S LOGINS
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"}
// Metric 2: LAST WEEK'S LOGINS
{"type": "metricOverTime", "days": 14, "op": "SUM", "prop": "custom.logins_per_day"},
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"}]]
// Metric 2: WEEK-OVER-WEEK CHANGE IN LOGINS
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"},
{"type": "metricOverTime", "days": 14, "op": "SUM", "prop": "custom.logins_per_day"},
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"}]]
Now, with the ability to reference other calculated metrics, it both looks cleaner and is more dynamic. Change one calculation, and it updates across. See below:
// Metric 1: THIS WEEK'S LOGINS
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"}
// Metric 2: LAST WEEK'S LOGINS
{"type": "metricOverTime", "days": 14, "op": "SUM", "prop": "custom.logins_per_day"},
{"type": "metricOverTime", "days": 7, "op": "SUM", "prop": "custom.logins_per_day"}]]
// Metric 2: WEEK-OVER-WEEK CHANGE IN LOGINS
{"type": "metricOverTime", "days": 1, "op": "SUM", "prop": "calculated.626a62ecea8fbb3c3ed1086b"},
{"type": "metricOverTime", "days": 1, "op": "SUM", "prop": "calculated.625577d94c890d12cbfe6bfd"}]
π Quick tip: to see which calculated metrics are dependent, spot the arrow icon next to the metric name and hover over it to see which metric(s) they reference or are being used by!
An arrow pointing toward the metric indicates they are using another metric. Whereas an arrow pointing away indicates they are being used in another Metric.
Referencing Calculated Metrics on Related Models
If you're creating a company calculated metric, you can also reference calculated metrics on other, related models such as End Users, Assets, and Projects (just like you can reference custom
metrics), for example:
{"type": "metricOverTime", "days": 1, "op": "SUM", "prop": "enduser.email_events.sent"},
{"type": "metricOverTime", "days": 1, "op": "SUM", "prop": "enduser.calculated.63df8599d800745c0f198456"}
Being able to run calculated metrics on the related object level and then roll these calculations up to the Company level unlocks a number of use cases involving pre-processing at the lower level.
One of the most useful is count conditional (count the unique number of End Users, Assets, and Projects meeting a condition) based on a child-calculated metric, using the SIGN operator.
Example: Unique Active Users
We create a SIGN-based calculated metric on the related object, for example, to indicate which end users are active (have logged in at least once), in the past 14 days:
["SUM", {"type": "metricOverTime", "days": 14, "op": "SUM", "prop": "activities.loggedin"}]
This gives us, for each end user, a 1 or 0 indicating whether they have logged in during the past 14 days. Then, by referencing this calculated metric from the Company level, I can sum up all the 1's
and 0's. Since all the active end users have a value of 1, summing these values each day tells me the total number of active end users (end users who logged in at least once in the past 14 days) on
each company:
{"type": "metricOverTime", "days": 1, "op": "SUM", "prop": "enduser.calculated.id"}
π Quick tip: When combined with the IF function, this is particularly powerful, allowing you to count the number of unique related objects meeting certain criteria, at a Company level. | {"url":"https://support.planhat.com/en/articles/6175098-calculated-metrics-inside-calculated-metrics","timestamp":"2024-11-09T20:50:14Z","content_type":"text/html","content_length":"111801","record_id":"<urn:uuid:8b380fa6-f6de-4716-86ee-27e095a7e1f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00188.warc.gz"} |
Given a matrix nxn
Random commentary about Machine Learning, BigData, Spark, Deep Learning, C++, STL, Boost, Perl, Python, Algorithms, Problem Solving and Web Search
Tuesday, October 16, 2012
Given a matrix nxn
Zero all the columns and rows which contains at least one zero.
Pubblicato da codingplayground a 12:17PM
1 comment:
1. for(int c=0; c < N; c++) {
for(int r=0; r < N; r++) {
if (A[r][c]==0) { markRow(A, r); markCol(A, c); }
for(int c=0; c < N; c++) {
for(int r=0; r < N; r++) {
if (isMarked(A, r, c)) { A[r][c] = 0; }
Now, the question is about what it means to mark a column/row. Some ideas:
1) Assuming a positive matrix, to mark a cell means to replace the content with it's negative value. Then a cell is marked if it's content is negative.
2) Assuming a matrix populated with references to Integer objects, mark a cell means set it to null.
All these assumptions don't cover the case where the matrix contains any kind of primitive integer (positive or negative). In this case it is not possible to identify a "marker". An idea to cover
this case is to use a support matrix of size NxN (let's call it B) that contains zeroes everywhere except in the cells corresponding to the cells in A that need to be be zeroed. The value of such
cells in B will be equals to -1* A[r][c].
Then the solution is A = A + B.
Of course this solution not only require O(n^2) in time but also O(n^2) in space... pretty inefficient!
A second before to press the publish button I had this other idea which I think is good:
List colsWithZeroes = new ArrayList();
for(int c=0; c < N; c++) {
boolean foundZeroInRow = false;
for(int r=0; r < N; r++) {
if (A[r][c]==0) {
foundZeroInRow = true;
zeroPrevInRow(A, r, c);
zeroPrevInCol(A, r, c);
if (colsWithZeroes.contains(c) || foundZeroInRow) {
A[r][c] = 0;
zeroPrevInRow and zeroPrevInCol just set to zero the previous cells on the row or on the column.
This algo is O(n^2).
The main idea is to scan the matrix top-left to bottom-right. Every time a zero is found we say that a zero has been found on the current row. We zero all the *previous* elements on the current
row and on the current column. We also add the current column to the list of columns containing a zero no matter the row. After these operations we move on the next element.
If the current element is not zero, then either:
1) zero the element if we found a zero previously on the row
2) zero the element if the current column is contained in the list of previous columns containing a zero
3) we leave the element as is if none of the previous conditions apply | {"url":"http://codingplayground.blogspot.com/2012/10/given-matrix-nxn.html","timestamp":"2024-11-11T17:14:41Z","content_type":"application/xhtml+xml","content_length":"129418","record_id":"<urn:uuid:73b566c6-64c4-4a8f-9579-944a62802f24>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00121.warc.gz"} |