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History of Krylov-Bogoliubov-Mitropolsky Methods of Nonlinear Oscillations
History of Krylov-Bogoliubov-Mitropolsky Methods of Nonlinear Oscillations ()
1. Introduction
When physicists or engineers are faced with a theoretical investigation of a real physical system, they are always forced to simplify and to idealize the problem in order to represent its time
properties. Consequently, the idealization of the problem can never be avoided in the construction of a mathematical model which is a set of equations, normally a set of differential equations
representing the dynamic behavior of the system (Nayfeh & Balachandran, 1995; Nayfeh & Mook, 1979) . In this mathematical model all the main features of the problem must appear, but it is neither
necessary nor feasible to include all the properties of the system because of the difficulties involved in resolving it, if indeed this is actually possible. Since idealization is inevitable, the
question is how to go far in the direction of idealization to obtain satisfactory results? The answer to this question is only given by experimental results, which may or may not legitimate the
idealization made.
In nonlinear problems new phenomena arise, which is not the case of linear systems. Thus, in the study of nonlinear problems the main objective is to understand these new characteristics of the
system and to improve the accuracy of linear methods. Due to the complexity of the phenomena, one practical approach is to settle for something less than complete generality (Dumas, Meyer, & Schmidt,
1995; Katok & Hasselblatt, 1995) . Hence, it is preferable to study solutions for nonlinear problems and look for nonlinear solutions close to a known linear solution. This is the basic idea behind
the use of perturbation methods for nonlinear problems (Nayfeh, 1973) . It consists of developing the method in relation to a small parameter, in which a series expansion in terms of this parameter
can be found. The coefficients of the expansions are obtained as solutions of a sequence of linear problems.
The Krylov-Bogoliubov school in Kiev developed analytical methods for nonlinear systems. The foundation of their results was the classical perturbation method which was generalized to
non-conservative systems. In the 1950s, the invention of the high-speed computer changed the history of dynamics. It allowed the simulation of dynamical systems with complicated equations in a way
that was previously impossible, and was therefore a significant development of intuition about nonlinear systems. Such simulations led to Lorentz’s discovery in 1963 of chaotic motion in a strange
attractor. He studied a simplified model of convection rolls in the atmosphere to gain insight into the unpredictability of weather. Lorentz found that the solutions of these equations never settled
down to an equilibrium or periodic state. Instead they continued to oscillate in an irregular aperiodic fashion. He also noticed that the resulting behaviors from two slightly different initial
conditions would soon become completely different. The surprising implication was that the system was inherently unpredictable. In other words, chaotic characteristics can appear in deterministic
mathematical models.
2. Main Developments of Nonlinear Theories before 1930
The main contributions to nonlinear methods before K-B-M are as follows:
Before 1880 (The analytic period): characterized by the search for analytic solutions and perturbation methods; the search for integrals of motion, particularly independent time and algebraic
Main areas: classical mechanics, celestial mechanics―Newton, Euler, Lagrange, Laplace, Jacobi, etc. Hydrodynamics―Rayleigh, who briefly considered limit cycles and the bifurcation concept. Kinetic
theory of gases-the Boltzman equation: the H-function, statistical concept of entropy.
1879-1900 (Sophus Lie): (similarity transformations); a general principle for obtaining integrals of nonlinear partial differential equations by determining invariance properties in a continuous
group (Lie group); a frequent application is invariance with some scaling.
Bruns (1887): the only independent algebraic integrals for the motion of the three-body problem (which has 18 integrals) are the ten ‘classic integrals’ (energy, total linear momentum, total angular
momentum, and time-dependent equations for the motion of the center of mass).
Painlevé (1898): the only independent integrals of the motion of the N-body problem, which involve velocities algebraically (regardless how the spatial coordinates enter) are the classic integrals.
Stability of motion-A. M. Lyapunov’s results (1892); Lyapunov exponent.
Korteweg and de Vries demonstrated the existence of finite amplitude solitary water waves.
Poincaré (1880-1910): emphasized the study of the qualitative, global aspects of dynamics in phase space; developed topological analysis; generalized the bifurcation concept: introduced mapping in
phase space (difference equations); surface of section; introduced rotation numbers of maps; index of a closed curve in a vector field; initiated the recursive method for defining dimensions.
Whittaker: obtained the adelphic integrals for coupled harmonic oscillators, where the integrals are never analytic functions of frequencies (1906).
Numerical computations by Stormer and his students, of the dynamics of solar particles in the dipole magnetic field of the earth (a non-integrable system), from 1907-30.
The Mandelstam-Andronov school of applied nonlinear analysis; replacement of a nonlinear system by a set of linear segments.
Enrico Fermi (1923): attempted to generalize Poincaré’s theorem to prove ergodicity in some systems.
Van der Pol: the extensive study of limit cycles, relaxation oscillations, leading to singular perturbation theory; studied the forced van der Pol oscillator with van der Mark (1927); observed
subharmonic generation, hysteresis, ‘noisy’ regions in parameter space. A variety of bifurcation phenomena.
The Andronov-Poincaré bifurcation (1930).
Krylov-Bogoliubov-Mitropolski: further refined the averaging method of perturbation theory.
3. Poincaré’s Pioneering Work
The father of modern nonlinear dynamics is Henri Poincaré (1854-1912). Previous studies of dynamics in the 1880s included analytical solutions of dynamic equations, astronomical investigations of
planetary motions, and Lord Rayleigh’s (1842-1919) studies of dynamical systems. Important names in mathematics are associated with these studies, such as Newton (1642-1726), Leibniz (1646-1716),
Euler (1707-1783), Gauss (1777-1835), Lagrange (1736-1813), Laplace (1749-1827), Jacobi (1804-1851), Lie (1842-1899), and, of course Poincaré, amongst others. One of the big problems at that time was
the three-body problem (Green, 1997; Valtonen & Karttunen, 2006) .
It was Poincaré who provided many of the solutions now used as established methods for exploring dynamic systems. In particular, he emphasized the importance of obtaining a global qualitative
understanding of the nature of a dynamic system. Many of his suggestions were substantially refined and extended by others, but our debt to his insights is enormous. Unfortunately, his contributions
in this field were ignored by scientists for about 50 years. Fortunately, not by many mathematicians, who extended his concepts and theories to areas such as topology, asymptotic series, various maps
and their fixed points, bifurcations and also proved a number of his conjectures.
We can summarize the contributions of Poincaré for dynamic systems as follows:
The qualitative theory of differential equations
The study of global stability of sets of trajectories, instead of focusing on single solutions
The notion of bifurcation together with the study of dynamical families depending on a parameter
The introduction of probabilistic concepts into dynamics, with respect to ergodic theory and the exclusion of exceptional cases.
Poincaré studied in two famous schools: École Polytechnique and École des Mines, entering the former in 1873 (Verhulst, 2012) . His professors of science at the Polytechnique were Charles Hermite
(Analysis), Marie-Alfred Cornu (Physics), Louis Jean Résel (Mechanics), Amédée Manheim (Geometry), Hervé Faye (astronomy) and Edmond Fremy (Chemistry). Furthermore, his two most influential
professors were Hermite and Cornu.
Poincaré was immersed in the Polytechnic School’s great tradition in mathematical analysis. It is worth emphasizing that the first lecturer of this discipline was Lagrange. However, the teaching of
mathematics in that school had deeper roots. Historically it belonged to a tradition that began in the Renaissance of being part of the army career, as well as being of fundamental importance for the
education of nobles. From 1750 onwards, mathematicians such as Huygens (1629-1695), Bernoullis, Euler, Monge (1746-1818), etc., joined in France the so-called ‘scientist-engineers,’ like Borda
(1733-1799) and Coulomb (1736-1806), who were also members of the Academy of Sciences.
4. Science and the Russian Revolution
The 1917 Bolshevik Revolution resulted in radical changes throughout Russia , which affected the development of science. The Revolution and the ensuing civil war had serious consequences for the
lives and work of scientists of the pre-revolutionary generation (Graham, 1993) .
“In the few months of 1917 where the liberals and the democratic socialists were in power, reforms were made which influenced the future of Soviet science. Universities adopted new structures of
faculty governance; professional societies affirmed their independence from state control and the Academy of Sciences , for the first time in its history, elected its own president, the geologist A.
P. Karpinskii. The permanent secretary of the Academy, S. F. Oldenburg, was also Minister of Education in the provisional government” (Graham, 1993) .
“One paradoxical situation that appears in early Soviet Russia is that the scientific institutions normally aligned with the conservative forces, such as the Academy of Sciences, were less resistant
to the Bolshevik Revolution than the universities and other scientific institutions” (Graham, 1993) . This situation can be interpreted as due to the prevailing wish of leading scientific researchers
to keep politics separate from science. The principle of keeping science out of politics was strong in the Central European universities where many Russian scholars had studied and was frequently
expressed by the old researchers.
Although leaders of the Academy adopted defensive measures, the revolutionaries began to formulate plans for a reorganization of science. In these plans the idea of diminishing the role of the
Academy was present. Probably the most ambitious plan to reorganize Russian science came from the Commissariat of Education, based mainly among the northern communes, including Petrograd. The basic
purpose of this project was to “win science for the proletariat”, and to fight against the “fetishization of pure science”, according to them endemic in the traditional institutions.
The northern radicals wanted to abolish the Academy of Sciences and all “old forms of the social organization of science”. They intend to replace the Academy with a “homogeneous set of
scholarly-pedagogical institutes” where teaching was to be the primary function (Pollock, 1969) . Here we cannot resist a comparison with a similar situation in France. During the French Revolution,
the National Convention suppressed all royal academies, including the French Academy in 1792. They were all replaced in 1795 by a single institution, called Institut de France.
The institutions that had traditionally been involved only with theoretical research would be required to assume major pedagogical responsibilities. Obviously, behind this proposal there was a
strategic idea of constructing the “unity of scholarly and teaching work”.
In the 1930s, after the Bolsheviks had abandoned their original ideal of an international communistic revolution and had begun to build a communist state at home (in a single country only), overall
scientific activity increased. The doctrine of Marxism-Leninism (White, 1996) , the ideology of Soviet orthodoxy, began to create a “scientific” theory for the development of society.
Naturally, this meant that official Marxist ideology was adapted to the party’s immediate needs party and immediately came to dominate the social sciences and the sciences in general, just when a new
period of development was arising (Cohen, 1978) .
5. The Intellectual Context of the Soviet Union
Like the October Revolution, World War II deeply affected intellectual life in U.S.S.R (Montefiore, 2005) .
At the beginning of World War II, the Soviet Union signed a non-aggression pact with Nazi Germany, initially avoiding confrontation. As is well know, this treaty was disregarded in 1941 when the
Nazis invaded Soviet Union, opening the largest and bloodiest theatre of combat in the history of mankind. The battles were intense and became famous in military history, such as Stalingrad.
Following the German attack on 22 June 1941, most of the Institutes and Universities from the west of Russia were evacuated into eastern regions far from the battle lines. Nikolay Bogoliubov moved to
Ufa, the capital of the Republic of Bashkortostan, where he became Head of the Department of Mathematical Analysis at UFA State Aviation Technical University, from July 1941 to August 1943. In autumn
1943, he was evacuated to Moscow, where on 1 November 1943 he accepted a position in the Department of Theoretical Physics of the Moscow State University. At that time the Head of the Department was
Anatoly Vlasov. Theoretical physicists working in the Department in that period included Dmitry Ivanenko, Arsenij Sokolov, and other very famous scientists.
Like Bogoliubov, Mitropolsky’s career was also significantly influenced by World War II. He graduated from Kazakh University in 1942 after studying there for six months. After graduating he attended
the Ryazan Military Artillery School , from where in 1943 he was sent to the front, where he commanded an artillery intelligence platoon until the end of the war. In 1946, he was demobilized and
began his research at the Institute of Constructive Mechanics of the Ukrainian Academy of Sciences under the supervision of Nikolai Bogoliubov.
In the 1943-46 period, Bogoliubov’s research was essentially concerned with the theory of stochastic processes and asymptotic methods. In 1945, he proved a fundamental theorem about the existence and
basic properties of a one-parameter integral manifold for a system of nonlinear differential equations. He also investigated periodic and quasi-periodic solutions lying on a one-dimensional manifold,
thereby establishing the foundation for a new method of nonlinear mechanics, the method of integral manifolds.
In 1946, he was elected as a corresponding member of the Academy of Sciences of the USSR. In 1947, he organized and became the Head of the Department of Theoretical Physics at the Steklov
Mathematical Institute. While working in the Institute, Bogoliubov and his school contributed to science with many important works on renormalization theory and on the theory of dispersion relations.
On 26 January 1953, Bogoliubov became the Head of the Department of Theoretical Physics at Moscow State University, after Anatoly Vlasov decided to leave the position on 2 January 1953. He was
elected a full member and academician of the Academy of Sciences of the Ukranian SSR and a full member of the Academy of Sciences of the USSR in the same year.
In March 1953 Josef Stalin died and immediately a progressive but general liberation of the regime began. A new era for cultural and scientific work was also commencing.
6. KBM Biographical Note
Krylov is a well-known mathematician, making his contributions in several fields of this discipline, such as interpolation, nonlinear mechanics and numerical methods for solving equations of
mathematical physics. He graduated from the St. Petersburg State Mining Institute in 1902. Between 1912 and 1917 he held the position of professor in this Institute. In 1917 he moved to Crimea to
become professor at the Crimea University. He worked there until 1922 and then moved to Kiev to become chairman of the Mathematical Physics Department at the Ukrainian Academy of Sciences.
Krylov (Figure 1) developed methods for the analysis of mathematical physics, which can be used not only to prove the existence of solutions but also their construction. In 1932 he began to work with
his student Nikolay Bogoliubov on mathematical problems of nonlinear mechanics. In this period they invented asymptotic methods for the integration of nonlinear differential equations, studied
dynamical systems, and made significant contributions to the foundation of nonlinear mechanics. They proved the first theorems about the existence of invariant measures known as the Krylov-Bogoliubov
theorems, introduced the Krylov-Bogoliubov averaging methods and, together with Yuri Mitropolski, asymptotic methods for approximate solving equations of nonlinear mechanics.
Bogoliubov (Figure 2) a Russian mathematician and theoretical physicist, is famous for his significant contribution to quantum field theory, classical and quantum statistical mechanics, and to the
theory of dynamical systems. He was born in Nizhny Novgorod, moving to Kiev with his family in 1921 soon after he began to study physics and mathematics.
Figure 1. Nikolay Mitrofanovitch Krylov (1879-1955).
Figure 2. Nikolay Nikolayevitcch Bogoliubov (1909-1992).
At this time he attended research seminars in Kiev University and soon started to work under the supervision of Nikolay Krylov. In 1924, at the age of 15, Bogoliubov wrote his first scientific paper:
On the Behaviour of Solutions of Linear Differential Equations at Infinity. In 1925 he entered a PhD program at the Academy of Sciences in Ukraine and obtained an equivalent title to PhD in 1928, at
the age of 19, with a thesis entitled: On Direct Methods of Variational Calculus. In 1930 he obtained the degree of Doctor of Sciences, the highest title in the Soviet Union, for which a significant
independent contribution to his scientific field had to have been made.
In 1931, Krylov and Bogoliubov began to work together on the problems of nonlinear oscillations. They were the key figures in the ‘Kiev School of Nonlinear Oscillation Research,’ with a fruitful
cooperation beginning with the paper: On the Quasiperiodic Solutions of the Equations of Nonlinear Mechanics (1934) and the book Introduction to Nonlinear Mechanics (1937). These works created a
large field of nonlinear mechanics.
Mitropolsky was a renowned Ukrainian Soviet mathematician known for his contributions to the fields of dynamical systems and nonlinear oscillations. He was born in Poltava Governorate and died in
Kiev. He received his PhD from Kiev State University in 1948, under the supervision of the mathematician Nikolay Bogoliubov. He studied the problem of resonant phenomena in nonlinear oscillatory
systems with slowly varying parameters. The approach he used was the Krylov-Bogoliubov asymptotic method. In 1951, he presented another thesis, equivalent to the habilitation, with the title: Slow
processes in nonlinear oscillatory systems with many degrees of freedom.
In 1958, he became Director of the Mathematics Institute in Kiev, and a member of the Mathematics Department of the Ukrainian Academy of Sciences, also becoming an Academician of the Ukrainian
Academy in 1961. In 1966 he became Academic Secretary of the Mathematics and Cybernetics Department of the Ukrainian Academy of Sciences and Academician of the Mathematics Department of the National
Academy of the SSSR.
He was awarded the following prizes: State Prize for Science and Technology in 1980 and the V. I. Vernadskii Prize for Geology, Geochemistry, and Hydrophysics in 1985. In 1986, he received the A. M.
Liapunov Gold Medal for his scientific contributions to Russian sciences.
Anatoly Samoilenko, who published some papers with Mitropolsky, was a student awarded his PhD with Mitropolsky in 1963, as well as working on many mathematical projects with him. When Mitropolsky
retired in 1988, Samoilenko took over the direction of the Institute of Mathematics.
Mitropolsky (Figure 3) also authored the following books: Nonstationary process in nonlinear oscillatory systems (1955), Problems in the asymptotic theory of nonstationary oscillations (1964),
Lectures on the method of averaging in nonlinear mechanics (1966), The method of averaging in nonlinear mechanics (1971), Nonlinear mechanics. Asymptotic methods (1995), Nonlinear mechanics.
Monofrequency oscillations (1997) and Methods of nonlinear mechanics. A first textbook (2005).
7. Common Selected Works and Books
N. M. Krylov and N. N. Bogoliubov (1934): On Various Formal Expansions of Nonlinear Mechanics, Kiev, Izdat, (Ukrainian).
N. M. Krylov and N. N. Bogoliubov (1947): Introduction to Nonlinear Mechanics, Princeton, Princeton University Press.
N. N. Bogoliubov and Y. A. Mitropolski (1961): Asymptotic Methods in the Theory of Nonlinear Oscillations, New York, Gordon and Breach.
N. N., Bogoliubov and B. I. Sadovnokov (1963): On the Periodic Solutions of a Differential Equation of Order n with a Small Parameter, in Proceedings of the International Symposium on Nonlinear
Oscillations, Izd-voAkad. NaukUkSSR, Kiev, 155-165.
N. N. Bogoliubov (1964): On the Quasiperiodic Solutions in the Problems of Nonlinear Mechanics, in Proceedings of the First Mathematical Summer School, Pt. I., Kiev, 11-101.
N. N. Bogoliubov and Y. A. Mitropolsky (1965): On the Investigation of Quasiperiodic Modes in Nonlinear Oscillatory Systems, Colloq. International. Centre Nat. Rech. Scient., No. 148, 181-192.
Y. A. Mitropolsky and O. B. Lykova (1965): Behavior of Solutions of Nonlinear Equations in the Vicinity of the Equilibrium Position, in Matema. Fizika, Naukova Dumka, Kiev, 74-96.
Y. A. Mitropolsky and Samoilenko A. M (1972): Conditionally Periodic Oscillations in Nonlinear Systems, in Matem. Fizika, no.12, 86-105.
Y. A. Mitropolsky and A. K. Lopatin (1973): On the Reduction of Systems of Nonlinear Differential Equations to Normal Form, in Matem. Fizika, Naukova Dumka, Kiev, No. 14, 125-140.
Y. A. Mitropolsky and Samoilenko A. M (1973): Multifrequency Oscillations in Nonlinear Systems, in Nonlinear Vibration Problems, Zagadnieniadrgannielinlowych, PWN, Warsawa, 14, 27-38.
Figure 3. Yurii Alexeevitch Mitropolsky (1917-2008).
Y. A. Mitropolsky and Samoilenko A. M (1977): Some Problems in Multifrequency Oscillations, Abh. DAW, Abt. Math. Naturwiss, Techn., Heft 4, 107-116.
Y. A. Mitropolsky and A. A. Martynyuk (1978): On Some Lines of Research Concerning the Stability of Periodic Motions and the Theory of Nonlinear Oscillations, Prikladnaya Mekanika, Izd-voAkad.
NaukUkSSR, 14, no. 3, 3-13.
8. Some of Mitropolsky’s Additional Contributions
Mitropolsky made some fundamental contributions to the theory of nonlinear mechanics, especially the qualitative theory of differential equations, as well as the development of asymptotic methods
applied to the solution of practical problems (Mitropolsky, 1963) . He extended the Krylov-Bogoliubov symbolic method to nonlinear systems and generalized asymptotic methods to the theory of
nonlinear mechanics. Using methods of successive substitutes, he constructed a general solution for a system represented by a nonlinear equation and studied its behavior in the neighborhood of the
quasi-periodic solution. He also successfully applied the averaging method to the study of oscillatory systems with slowly varying parameters (Mitropolsky, 1970) .
We can list the majority of Mitropolsky’s contributions as follows:
The creation and the development of the mathematical justification of algorithms for the construction of asymptotic expansions for nonlinear differential equations describing non-stationary
oscillatory processes.
The development of a method for investigation of monofrequency processes in oscillatory systems.
The investigation of systems of nonlinear differential equations, describing oscillatory processes in gyroscope systems and strongly nonlinear systems.
The development of the theory of integral manifolds in nonlinear mechanics and the consideration of related questions that arise in stability of motion.
The development of the averaging method for equations with slowly varying parameters, as well as for equations with non-differentiable and discontinuous right-hand sides for equations with delayed
argument, for equations with random perturbations, and for partial differential equations and equations in functional spaces.
The development of the method of accelerated convergence in problems of nonlinear mechanics.
The development of the theory of reducibility in linear differential equations with quasi-periodic coefficients and other equations.
In 1955, Mitropolsky and Bogoliubov published a book on asymptotic methods in nonlinear oscillations (Bogoliubov & Mitropolsky, 1961) . This book contains their fundamental achievements between 1945
and 1955. In a review of the book, Solomon Lefschetz made the following remarks:
The present book is the fourth or fifth major treatise published in recent years by Soviet scientists on the general topic of nonlinear oscillations, which serves to indicate the great value which is
attached in the USSR to this general topic. The general program of the book is not too far from the program of the 1937 Krylov-Bogoliubov monograph (Introduction to Nonlinear Mechanics). However,
although the book is addressed primarily to physicists and engineers, its mathematical treatment is most careful, which was by no means the case with the 1937 monograph. The book is also much more
orderly and most readable: an excellent contribution in every respect.
9. Considerations about Introduction to Nonlinear Mechanics
Krylov and Bogoliubov’s first book, published initially in 1934, was translated to English by Solomon Lefschetz in 1942 (Krylov & Bogoliubov, 1950) . As he emphasized, the authors Krylov and
Bogoliubov considered weakly nonlinear equations in terms of small parameters: “Similar equations are well known in astronomy and have been the object of systematic investigation by Lindstedt,
Gylden, Lyapunov, and above all by Poincaré. In a general sense, one may say that the same methods are applied by Krylov and Bogoliubov. However, the applications which they have in view are quite
different, as they are chiefly in engineering, technology, and physics, notably electrical circuit theory”. This means that the improvements and refinements proposed by Krylov and Bogoliubov are very
useful tools for applications in engineering and physics. This small book with around one hundred pages is divided in nine chapters with the following subtitles:
Chapter I: Some Non-Linear Oscillatory Systems
Chapter II: Elementary Theory of the First Approximation
Chapter III: Refinement of the First Approximation
Chapter IV: Construction of the Higher Approximation
Chapter V: Linearization
Chapter VI: Application of Symbolic Methods to Linearization
Chapter VII: Multiply Periodic Systems
Chapter VIII: Influence of Periodic Disturbances
Chapter IX: Complements
In Chapter I, dedicated to selected systems, the following technical problems are studied: the oscillatory shaft, the electrical circuit without resistance, the pendulum freely oscillating in the
atmosphere, the electrical circuit with resistance, the electronic generator. Also studied in this chapter are modelling aspects of these problems. Hence, some important equations are presented, such
as Raleigh and Van der Pol which became classic in nonlinear investigations. In this context, the coefficients representing friction and energy dissipation appear as fundamental characteristics of
some nonlinear systems.
Chapter II, about the solution of the systems presented in Chapter I, is concerned with mathematical modelling and involves equations of the following type:
The authors then proposed to investigate more particularly the quasi-harmonic case, where there are oscillations near to the sinusoidal response in the form
The difficulty which arose in the eighteenth century, the presence of secular terms, appears again. The objective now is to try a new approximated approach in order to obtain solutions free from
these terms.
are constant. The method consists of now assuming
The transformations obtained in the original differential equation are such that instead of a single differential equation of the second order, in the unknown
The system admits that the period
to the small parameter, so that
In Chapter III, the authors search for a refinement of the previous solution by improving the accuracy of the first approximation, of course, a more complicated one. The new refined first
approximation is:
In other words,
Chapter IV, a study of the higher approximations, consists of taking into account the higher harmonics of the same basic equation. This means considering methods for forming approximate solutions
corresponding to stationary oscillations which satisfies the basic model, but the solution contains terms in any given power of
As a general conclusion to this chapter, except for certain singular cases, the relations obtained for the first approximation provide the same qualitative results for the starting of
self-oscillations as the higher approximations. However, the higher approximations provide quantitative rather than new qualitative information. Thus, because of the difficulty in computing the
higher approximation, it is usually sufficient to obtain the first approximation.
Shown in Chapter V, concerned with a method of linearization of the nonlinear system, is that the mathematical model representing the nonlinear system in the first approximation is equivalent to that
of a linear system with a dissipation coefficient and a spring constant. The approximation is to the order of
As they are dealing with systems which do not differ much from harmonic systems, a harmonic oscillation in the form
In Chapter VI, the authors are basically concerned with the application of symbolic methods to linearization where complex algebra is extensively used. If
is harmonic with period
Mathematical manipulation using complex algebra is more suitable for the systems studied in the book. For example, we can consider two terminals and impose a harmonic voltage
The operator method can be generalized to be applied to non-stationary oscillations. In a linear system a non-stationary oscillation is of the exponential harmonic type:
Hence, if we have a nonlinear system, in the first approximation the solution assumes the form above with
In Chapter VII, multiplying periodic systems are looked at, and it is asked if it is possible to consider systems with several frequencies instead of the systems considered until now, where we speak
about one frequency at a time. For nonlinear systems with multiple frequencies the principle of superposition is not applied to a linear system equivalent to a given nonlinear one. Under certain
conditions, such as the reasonable smallness of suitable parameters, some progress was made. For the sake of simplicity the authors limited the discussion to the case of two oscillations. Thus, two
distinct situations arise, depending on the presence or absence of resonance.
Both cases were studied. In the non-resonance case, considering the first approximation, the nonlinear characteristic may be replaced by the linear characteristic. We can interpret the replacement of
the nonlinear element by an equivalent element with a characteristic just written as a linearization of the system. For the resonant case, the two harmonics
In Chapter VIII, the system is now subjected to exterior disturbances. As an example of a non-isolated system the following equation is introduced:
In Chapter IX, entitled complements, the authors discuss initially other procedures to obtain the higher approximations starting from the approach indicated in Chapter IV. As an example we can select
just one of them. It uses the following equation:
in which
Introducing new variables
Following this method we can obtain the successive approximations in a direct way, as well as the estimation of the error that appears when certain terms are neglected. Other different procedures are
also presented in this chapter in order to obtain approximations solutions of superior order.
We will comment on the bibliography presented by Krylov and Bogoliubov in their first book in our conclusions because it describes important characteristics of their common lines of research before
the book was published.
10. Conclusion
The method developed by Bogoliubov and Mitropolskyis much better presented in the book Asymptotic Methods in the Theory of Nonlinear Oscillations. An English translation of the second Russian edition
of the book appeared in 1961. It is this work that has come to be known as the KBM method (Krylov-Bogoliubov-Mitropolsky). This book was the first of many books written by Mitropolsky, the majority
coauthored with his former doctoral students. Several authors list 31 monographs published by Mitropolsky between 1955 and 2005.
In the preface of the 1971 edition of the above mentioned book Mitropolskystated:
We deal with the method of averaging in nonlinear mechanics. We include numerous results of further development and generalization of the basic ideas of N. N. Bogoliubov. We give various algorithms,
schemes and rules for constructing approximate solutions of equations with small and large parameters, and obtain examples which in many cases graphically illustrate the effectiveness of the method
of averaging and the breadth of the application to various problems which are, at first glance, very disparate. The theorems that we include reveal the depth and mathematical rigor of the method of
averaging. We discuss the basic trends and developments of the method of averaging, and as illustration we give typical examples of nonlinear oscillatory systems, revealing the effectiveness of the | {"url":"https://www.scirp.org/journal/paperinformation?paperid=74804","timestamp":"2024-11-07T06:29:58Z","content_type":"application/xhtml+xml","content_length":"132682","record_id":"<urn:uuid:5ce058c5-a682-4a12-be48-68578518e573>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00323.warc.gz"} |
Jackets are fabricated in yards and transported to the field on barge, and then installed by derrick barge by lifting/upending.
Jacket transportation study includes
• Transportation ballast plan,
• Stability analysis
• Barge longitudinal strength check,
• Motion and acceleration response analysis
• Air gap Calculation (for wave slamming assessment on jacket leg members)
• Bollard pull calculation (for Towing Tug Selection).
Transportation route shall follows the IMO approved routing in target region. The route chosen is the most direct, taking in to account any possible obstructions or navigational dangers along the
route. Towing speed can be chosen between 3 to 8 knots and towing periods can be calculate accordingly.
Maximum sea states to be considered for the transportation shall be taken from 10 years extreme return period of Towing Route and installation site 1 year extreme return period.Both data sets are
subsequently compared to derive the maximum sea states for the study.
The maximum sea-states monthly significant wave height Hs and its corresponding one-minute mean wind speed in the same month shall be used for stability and motion response analysis. To determine the
range of probable peak periods, Tp, adjusted for the speed of the barge.
Maximum motions values shall be taken as the maximum of all the sea states.To compute wave drift force,Bollard Pull Requirement shall be checked for Range of Peak Period for Different Hs Support
Frame shall be installed before loadout and seafastening and bracket shall be installed after the loadout operation. Unrestricted Navigation SWBM & SWSF shall be used for transportations intended to
be in any area and any period of the year.Lightship weight distribution of the barge based on barge stability booklet shall be used for these calculations.
Jacket shall be positioned on the Support Frames from the barge frames. These frames shall be aligned with existing web frames or bulkheads for better load transfer.The jacket is supported by the
support frame and seafastening. Stools can be inserted between Support Frame and jacket for better connectivity.
Jacket weight and jacket COG position used for various conditions shall be used for stability and motion response study. Normal COG and COG Shift shall be used for longitudinal strength check.
Jacket weights shall be distributed through support frame location.
Barge is checked according to IMO – MSC 267 (85) Code of Intact Stability, Part B – Recommendation for Certain Types of Ships – Pontoon criteria. Free surface effect shall be considered for (intact
and damaged) stability analysis. The criteria are presented hereafter:
• The area under the righting lever curve up to the angle of maximum righting lever should not be less than 0.08 meter-radians (4.58 meter-degree).
• The static angle of heel due to a uniformly distributed wind load of 540 Pa (wind speed 30 m/s) should not exceed and angle corresponding to half the freeboard for the relevant loading condition,
where the lever of wind heeling moment is measured from the centroid of the windage area to half the draught.
• The minimum range of stability should be:
• For L ≤ 100 m: 20º;
• For L ≥ 150 m: 15º;
• For intermediate length: by interpolation.
Barge stability is shall also be checked according to NDI Guidelines for Marine Transportations
• Area under righting moment curve to be 40% in excess of wind overturning moment curve at second intercept of righting moment
• The initial metacentric height should be not less than 0.15 m.
• The intact range of stability, about any horizontal axis, defined as the range between 0o degrees inclination and the smallest angle at which the righting arm (GZ) becomes negative shall not be
less than 36o degrees. (For large vessel with length > 76m and breadth > 23m)
Barge shall be checked for following Damaged Stability Criteria
• The wind velocity used for overturning moment calculations in the damage condition shall be 26 m/s (50 knots) or the wind used for the intact calculation if less. It shall apply in the most
critical direction.
• The unit should have sufficient reserve stability in a damaged condition to withstand the wind heeling moment using the wind speed superimposed from any direction. In this condition the final
waterline, after flooding, should be below the lower edge of any downflooding opening
The range of angle calculated from the 2nd intercept and the GZ curve is known as the angle vanishing stability while the angle of the 1st intercept is angle of equilibrium. The range of stability is
calculated by subtracting the angle of equilibrium from the angle of vanishing stability.
Class approved Still Water Shear Forces and Bending Moments values shall be used as maximum allowable values for jacket transportation.
These details shall be included in calculations
• The lightship weight distribution along the hull,
• The ballast water weight distribution along the hull,
• The weight distribution of the jacket/cargo along the hull,
• The weight distribution along the hull of all the other equipments, cargo, etc., added on the vessel for the operation (i.e. equipments which are not included in the lightship weight).
• The buoyancy force distribution along the hull when the vessel is in equilibrium (still water condition).
Frequency domain study can be used to predict the motion of the barge loaded with Jacket.The roll damping from bilge keel shall be included in the analysis.Limiting waves Hs with different peak
periods shall be used to calculate the barge motions and the accelerations at the COG of the jacket.Wave energy distribution shall be represented with JONSWAP spectrum or other location specific
spectrum with appropriate peak parameter.The maximum surge, sway, heave, roll, pitch and yaw results are calculated at the COG of the jacket. These are maximum values and do not necessarily occur at
the same time.
Lower VCG and lower boundary weight may bring the natural roll period of the system closer to wave excitation period and generate larger response, so various weights and VCG heights shall be
checked.In the motion analysis, we need to determine the maximum motion of the vessel. The LCG, TCG and VCG based on the COG envelop shall be used for the study.
The hull inertia of barge is calculated as follows:
• Roll inertia radius: Rxx = Breadth/3
• Pitch and Yaw inertia radius: Ryy = Rzz = LOA/4
All inertia matrices shal be transferred to the COG of the system by the Huygens formula.Jacket and Support Frame radius of gyration shall be calculated and to be added with barge.
Roll motion is governed by the roll damping, which is caused by mainly
• The wave radiation,
• The viscous effects on the hull
For sea-keeping analysis, the standard procedure is to model these two components of roll damping as the sum of:
• A linear term accounting for the wave radiation damping
• An additional viscous damping is added using the following quadratic formula
Barge shall be towed to the field location; the bollard pull requirement for the towing tug shall be calculated as the sum of Wave drift force, Wind loads and Current loads.
The extreme environment with no forward speed shall be considered. Towing route and the field environmental conditions shall be taken into account for standoff period in the event of bad weather
prior to the installation.
The calculation of the wave drift force has been carried on based on the paper “A Lagally formulation of the wave drift force”
The wave drift force is computed For every design wave height with varying wave periods.
For wind loads, the wind screen area shall be used. The wind height and shape coefficients are taken from DNV-OS-C301 “Stability and Watertight Integrity”
The current loads are taken from the equation F = Cx × Scur × V2, with Cx being the current force coefficient on the bow = 2.89 Nsec2/m4 and Scur being the wetted surface area of the hull including
appendages, as defined in API RP 2SK.
The current force equation is shown hereafter:
Fcurrent = 2.89 × Scur × Vcur2
Scur = wetted surface area
The actual extreme condition bollard pull shall be calculated taking into the account both towing route and installation field.Towing Pull Requirement is sum wave drift force, wind force and current
force.If the required bollard pull (BP) is higher than 90MT and Hs lower than 2.0m, tug efficiency of 0.8m is used. Otherwise, the value of 0.75 is used.If the required BP is between 30 and 90MT and
Hs higher than 2.0m, tug efficiency formula of 52.5+BP/4 is used. The value will be iterated until the value of Tug efficiency converges. For Hs lower than 2.0m, the value of 0.8 is used.If the
required BP is below than 30MT and HS is lower than 2.0m, tug efficiency formula of 50+BP is used while for HS is higher than 2.0, the formula of 30+BP is used. The value will be iterated until the
value of Tug efficiency converges.
The air gap shall be checked at the overhanging edge of the jacket during transportation. | {"url":"https://www.mermaid-consultants.com/jacket-transportation.html","timestamp":"2024-11-04T18:59:51Z","content_type":"text/html","content_length":"97928","record_id":"<urn:uuid:480bcb6f-b3bc-4f2b-8d1d-85e5f093d3fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00257.warc.gz"} |
On Oct 19, 2011, at 8:07 AM, Bruno Lalande wrote:
> Hi John,
> Ii could be interesting.
> First, have you checked that the license used by this library is compatible with what you want to do?
> Second, not sure you get well what implementing a Boost.Geometry algorithm really is about. You're talking about replacing the classes in the library (point, polygon) by the ones provided by
Boost.Geometry. But that's not the point: the purpose of Boost.Geometry algorithms is to be able to work on any data type, as long as it meets some requirements. So basically your algorithm will not
know/use geometries from Boost.Geometry directly, but its concepts. You'll have to use all the metafunctions and other mechanisms that allow us to manipulate our data through concepts and not
directly. Was this your intention?
> Regards
> Bruno
I guess maybe I don't understand the architecture of Boost.Geometry very well. How do you differentiate between algorithms that only work on polygons or only work on sets of points (using the term
"algorithm" loosely here to describe the mathematical definition rather than the Boost.Geometry definition)? Or do operations that are specific to a very specific instantiation of a template not
belong in Boost.Geometry?
I was proposing to create a class that takes a model::polygon or model::multi_polygon (regardless of the point type) and performs Delauney triangulation on the polygon(s) and spits out a
model::multi_polygon containing the resulting triangles as model::polygon's. Does such a thing belong in Boost.Geometry or does it not fit the paradigm of having an operation work on any data type?
If it doesn't fit into Boost.Geometry, then I can easily just start a github project for the purpose of "algorithms" that operate on Boost.Geometry objects but don't meet the generic-ness
requirements. Either way is fine with me, but let me know.
P.S. It is BSD license and the two core maintainers think it is an awesome idea.
Geometry list run by mateusz at loskot.net | {"url":"https://lists.boost.org/geometry/2011/10/1598.php","timestamp":"2024-11-05T07:15:57Z","content_type":"text/html","content_length":"12629","record_id":"<urn:uuid:7c1013c6-8163-4890-9eb9-f56d9b13f731>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00626.warc.gz"} |
Drawing Infinity Symbol
Drawing Infinity Symbol - At some distance, add another even circle of the same size. This tutorial shows the sketching and drawing steps from start to finish. Another free abstract for beginners
step by step drawing video tutorial. If we feel under the weather, we can write the world “health” on an infinity symbol and carry it with us. Repeat this tracing exercise for a few days until a.
A complete explanation of the code is also provided in this guide. Illustration of mathematical symbol of piece. Web the infinity symbol (∞) is a mathematical symbol representing the concept of
infinity. Most popular infinity hand drawn painted symbols, eternal ink brush stroke. This tutorial shows the sketching and drawing steps from start to finish. Another free abstract for beginners
step by step drawing video tutorial. Collection of pen ink pencil drawing sketches of inverted eight number isolated on transparent background.
Infinity Symbol Drawing at GetDrawings Free download
Draw a curved line that curves towards the left and meets the straight line at a point. By visualizing or drawing the lemniscate as a symbol, we can use it on any part of the.
Infinity Symbol SVG Cut file for Silhouette and Cricut Etsy
A complete explanation of the code is also provided in this guide. Connect the two points where the curved lines meet the straight line to create the infinity sign. Official announcement of miss
cosmo vietnam.
Infinity Symbol Drawing at Explore collection of
Draw a curved line that curves towards the left and meets the straight line at a point. Web black painted infinity sign on white background, ring, rotation, ink, emblem, painted by brush. Infinity
hand drawn.
Infinity symbol sketch by lhfgraphics Vectors & Illustrations with
Web the infinity symbol (∞) is a mathematical symbol representing the concept of infinity. Web the infinity sign can also provide help with healing. Only show results related to: Web paper eraser
coloring supplies time.
infinity symbol symbol sign 649299 Vector Art at Vecteezy
There are more than 99,000 vectors, stock photos & psd files. Web i use this pan: Web in this tutorial, we're going to learn how to quickly create an infinity symbol in adobe illustrator.⭐️ master.
Infinity Symbol Drawing Free download on ClipArtMag
Web i use this pan: Web the infinity symbol (∞) is a mathematical symbol representing the concept of infinity. Check alt codes and learn how to make specific symbols on the keyboard. Web you can.
Learn How to Quickly Create an Infinity Symbol in Adobe Illustrator
Official announcement of miss cosmo vietnam 2023 stage miss cosmo vietnam and unimedia officiall. Web paper eraser coloring supplies time needed: At some distance, add another even circle of the same
size. How to draw.
Infinity Icon Vector Art, Icons, and Graphics for Free Download
Web insert an infinity symbol using character map. Another free abstract for beginners step by step drawing video tutorial. Introduction to drawing an infinity sign I built this using react and
codesandbox. Determine the size.
HOW TO DRAW INFINITY LOVE SYMBOL YouTube
Draw and create a simple infinity symbol using adobe illustrator#infinity #design. I built this using react and codesandbox. A complete explanation of the code is also provided in this guide. Web the
infinity sign can.
Infinity calligraphy vector illustration symbol. Eternal limitless
Most popular infinity hand drawn painted symbols, eternal ink brush stroke. Repeat this tracing exercise for a few days until a. Hand drawn infinity sign doodle set. Draw a curved line that curves
towards the.
Drawing Infinity Symbol Determine the size and location of your drawing and carefully draw a smooth circle. Web in this tutorial, we're going to learn how to quickly create an infinity symbol in
adobe illustrator.⭐️ master adobe illustrator and unleash your creativit. Draw a curved line that curves towards the left and meets the straight line at a point. Web a quick video tutorial on how to
draw an infinity symbol in adobe illustrator with a few simple steps. By visualizing or drawing the lemniscate as a symbol, we can use it on any part of the body where balance and harmony are needed.
Drawing Infinity Symbol Related Post : | {"url":"https://classifieds.independent.com/print/drawing-infinity-symbol.html","timestamp":"2024-11-15T03:51:07Z","content_type":"application/xhtml+xml","content_length":"24052","record_id":"<urn:uuid:1fd4c97c-2c52-4905-83b5-d8f0e7236aef>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00173.warc.gz"} |
STM publishing: tools, technologies and change
It is, in some ways, a deep irony that the community which spawned the web (i.e., physicists and mathematicians) should, until recently, have had such a hard time using the web to fluently and easily
communicate in the natural language of their disciplines: mathematics. Certainly, domain-specific solutions have evolved through the use and combination of tools and techniques such as jsMath,
MathML, SVG, bitmaps, LaTeX text and so forth. The author of this post wrote a COM-based (i.e., Windows!) DVI-to-SVG conversion tool 6 years ago but, at the time, the ultra-specific environment in
which it worked precluded its mainstream use. Anyone recall Adobe’s SVG plug-in for Internet Explorer…? It has taken nearly 20 years for the cluster of associated technologies to catch up and
coalesce into providing a viable and mainstream solution: MathJax, which delivers into the browser the exquisite beauty of the typesetting algorithms developed by Donald Knuth in the late 1970s.
Anybody who has read the infamous Appendix G of the TeXBook or the article Appendix G illuminated will, or should, have profound respect and admiration for the work of jsMath and MathJax. Replication
of the TeX math-typesetting algorithms in JavaScript (or any language!) is an incredible achievement for which the authors and financial sponsors of MathJaX should be applauded, for they have made a
major contribution to the future of scientific communication via the web ecosystem. Three cheers to them!
Just in passing, from deciding to set-up this WordPress blog, to downloading + installing WordPress and MathJax installation was about 1.5 hours in total, which just goes to show how easy it is, even
for a non-expert like me! Here’s a simple formula typeset in TeX/MathJax. Assuming you’re using a modern browser, right-click over the formula to see the options presented by MathJax.
\[\vec{F}_g=-F\frac{m_1 m_2}{r^2} \vec{e}_r\]
MathJax, you totally rock! | {"url":"http://www.readytext.co.uk/?cat=4","timestamp":"2024-11-12T08:24:21Z","content_type":"text/html","content_length":"31259","record_id":"<urn:uuid:9071928d-41fc-4c8c-ac39-b128f5418216>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00887.warc.gz"} |
Dynamic IF AND statement | Microsoft Community Hub
Forum Discussion
Hi, I am not sure if this is even possible to do, but here goes. The spreadsheet has stock prices over a time period. There are dozens of rows of stocks that I will need to copy this same set of f...
• =IF(AND(OFFSET(E11,0,0,1,$D$4)>OFFSET(E11,0,1,1,$D$4)),1,"")
You can achieve this dynamic AND condition based on the value in Cell D4 by using an array formula with the AND and INDEX functions.
Here is how you can modify your formula to achieve this:
In Cell E12, you can use the following array formula:
=IF(AND(E11>F11:F11+ROW(F11:F11)-ROW(F11), ROW(F11:F11)-ROW(F11)<=D4), 1, "")
To enter an array formula, you need to press Ctrl + Shift + Enter instead of just Enter after typing the formula. This will tell Excel to treat it as an array formula. If entered correctly, Excel
will automatically place curly braces {} around the formula.
Here is how this formula works:
• E11 is compared to a range of values from F11 to F11 + ROW(F11:F11) - ROW(F11). The ROW function is used to create an array of numbers based on the position of each cell relative to F11.
• ROW(F11:F11) - ROW(F11) calculates the relative row position within the range.
• D4 is used to limit the number of conditions in the AND function based on the number of columns you want to consider.
So, if D4 is set to 6, the formula will evaluate E11 > F11:F16 and F11:F16 > G11:H16 (up to 6 columns) and return 1 if all conditions are met.
Remember to adjust D4 to the desired number of columns you want to include in the AND condition, and enter the formula as an array formula as described above. The text and steps were edited with the
help of AI.
My answers are voluntary and without guarantee!
Hope this will help you.
Was the answer useful? Mark them as helpful and like it!
This will help all forum participants.
Thank you for your reply. I am not sure how this works and it is not giving me the results I am looking for. I have found a solution from the next posters answer. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/dynamic-if-and-statement/3936119/replies/3936156","timestamp":"2024-11-08T12:10:30Z","content_type":"text/html","content_length":"286900","record_id":"<urn:uuid:12883692-269c-4c7a-a6cd-7974a20d363d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00868.warc.gz"} |
Math, Grade 6, Rate, Gallery Problems Exercise
Tire Weight
Work Time
Tire Weight: Part 1
By measuring the amount of surface area each tire presses into the ground and the pressure inside each tire, you can calculate the weight of a car.
You need to measure the surface area of the bottom of each tire. Make sure you get permission from the car’s owner before you take any measurements, and take measurements only when there is an adult
supervising you.
1. Most tire “footprints” will be roughly rectangular in shape. To find the length and the width of this footprint, use a piece of cardboard, a piece of string, or a yardstick to measure each tire’s
footprint boundary. Measure the two sides that are easiest for you to reach.
2. Record your measurements in inches.
3. Ask an adult to use a tire gauge to measure the internal air pressure of the tire, or use the average tire pressure of most tires: 35.
4. Find the amount of tire surface area touching the ground. To find this, multiply the length and width of the footprint. Your answer should be in square inches.
5. To find the amount of weight the tire is holding, multiply the surface area by the pressure (in PSI) of that tire. For example:
$28\text{sq}\text{. in}\text{.}•\text{30}\frac{\text{lbs}}{\text{sq}\text{. in}\text{.}}=840\text{lbs}$
6. Repeat these steps for each of the other three tires if you have time. If you do not have time, use the same measurement you took for the first tire to represent all 4 tires.
Tire Weight: Part 2
1. Add the weights together for all four tires to get the total weight of the car.
2. To see how close you came to the real weight of the car, ask an adult to check the owner’s manual or look at the specification plate on the inside of the driver’s side door.
If you do not have access to an owners manual here are some sample weights of typical cars:
• A small car weighs about 3,000 lb.
• A medium car weighs about 4,000 lb.
• A small truck or SUV weighs about 4,500 lb.
• A large truck or SUV weighs about 5,500 lb. | {"url":"https://openspace.infohio.org/courseware/lesson/2140/student/?section=4","timestamp":"2024-11-03T07:30:09Z","content_type":"text/html","content_length":"34667","record_id":"<urn:uuid:ce42c2ed-66b1-4dc0-bbf1-225e260d3b92>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00642.warc.gz"} |
If the temperature of the hot body is raised by 5 the class 11 physics JEE_Main
Hint: The heat energy radiated is directly proportional to the fourth power of the temperature of the black body. The percentage increase is the difference between the new value and the old value
divided by the new value.
Formula used: In this solution we will be using the following formulae;
\[H = \sigma A{T^4}\] where \[H\] is the heat energy radiated, \[\sigma \] is the Stefan Boltzmann constant, \[A\] is area of the surface of the blackbody, and \[T\] is the absolute temperature of
the black body.
\[PI = \dfrac{{NV - OV}}{{OV}}\] where \[PI\] is the percentage increase of a particular value, \[NV\] is the new value, and \[OV\] is the old value.
Complete Step-by-Step solution:
Generally, the heat energy radiated by a black body is directly related to the fourth power of the temperature of that black body as given by the Stefan’s law as
\[H = \sigma A{T^4}\] where \[\sigma \] is the Stefan Boltzmann constant, \[A\] is area of the surface of the blackbody, and \[T\] is the absolute temperature of the black body
Temperature increasing by 5 percent signifies the final temperature to be
\[T' = T + \dfrac{5}{{100}}T\] which by adding and simplifying gives,
\[T' = \dfrac{{21}}{{20}}T\]
\[ \Rightarrow \dfrac{{T'}}{T} = \dfrac{{21}}{{20}}\]
Percentage error can be defined as
\[PI = \dfrac{{NV - OV}}{{OV}}\] where \[PI\] is the percentage increase of a particular value, \[NV\] is the new value, and \[OV\] is the old value.
Hence, percentage increase in the heat energy radiated would be defined as
\[PI = \dfrac{{H' - H}}{H} \times 100\% \]
\[H' = \sigma AT{'^4}\]
\[\dfrac{{H'}}{H} = \dfrac{{\sigma AT{'^4}}}{{\sigma A{T^4}}} = \dfrac{{T{'^4}}}{{{T^4}}} = {\left( {\dfrac{{T'}}{T}} \right)^4}\]
By inserting known values, we have
\[\dfrac{{H'}}{H} = {\left( {\dfrac{{21}}{{20}}} \right)^4}\]
Hence, by multiplying both sides by \[H\], we get
\[H' = {\left( {\dfrac{{21}}{{20}}} \right)^4}H\]
Going back to the definition, and inserting the value above into it we have
\[PI = \dfrac{{{{\left( {\dfrac{{21}}{{20}}} \right)}^4}H - H}}{H} \times 100\% \]
Dividing numerator and denominator by \[H\], we get
\[PI = \left[ {{{\left( {\dfrac{{21}}{{20}}} \right)}^4} - 1} \right] \times 100\% \]
\[PI = \left[ {{{\left( {1.05} \right)}^4} - 1} \right] \times 100\% \]
Hence, finding the fourth power, we get
\[PI = \left[ {1.22 - 1} \right] \times 100\% \]
Computing the above relation, we get
\[PI = 22\% \]
Hence, the correct option is B.
Note: For clarity; you might have seen a text where heat radiated from a blackbody is written as
\[H = \varepsilon \sigma A{T^4}\] where \[\varepsilon \] is the emissivity of the body. This equation and the one above are identical, this is because for a black body, the emissivity is equal to 1,
and hence can drop out of the equation. This equation is more generally used for heat radiated by any type of body. | {"url":"https://www.vedantu.com/jee-main/if-the-temperature-of-the-hot-body-is-raised-by-physics-question-answer","timestamp":"2024-11-05T03:20:54Z","content_type":"text/html","content_length":"153067","record_id":"<urn:uuid:30277216-930b-433d-ace4-df1120f9074b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00742.warc.gz"} |
Comments on msjx: Easy skill or ability checksUhm, no...the probability of rolling a six on one ...Yep, I was really surprised when I began to look a...You should check Blades in the Dark. It runs with ...I like this idea quite a bit. The math is probabl...Add an added bonus I realized that if you happened...I'm not following your math here. You will onl...Statistics are evil so I like this idea Matt. Sta...
04998881232495777586noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-4862476909691512457.post-21052065666942129872016-07-10T14:47:39.698-05:002016-07-10T14:47:39.698-05:00Uhm, no...the
probability of rolling a six on one die is 1/6 or 0.17%<br /><br />When we add we have two choices, add one to the roll, meaning a 5 or 6 six succeeds or adding a die and needing a 6 on either die to
succeed.<br /><br />The odds of rolling a 5 or 6 on one die is 2/6 = 1/3 = 33%.<br />The odds of rolling a 6 on one or both of two dice is the same as 1 minus the odds of not rolling a Pulp
02486803457210325703noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-85005659720790833562016-07-09T12:24:14.533-05:002016-07-09T12:24:14.533-05:00Yep, I was really surprised when
I began to look at how it worked, surprisingly simple and easy to manage in-game. Plus it does not make me have to think too hard...a difficult thing at my age. :-)matthttps://www.blogger.com/profile
/04998881232495777586noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-40306219161214817802016-07-09T00:17:32.468-05:002016-07-09T00:17:32.468-05:00You should check Blades in the
Dark. It runs with a very similar system that you propose. zs.gothpunkhttps://www.blogger.com/profile/
16979882804188961823noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-21844948811636589622016-07-08T14:00:26.003-05:002016-07-08T14:00:26.003-05:00I like this idea quite a bit.
The math is probably better than adding 1/6 chance with every "bonus" (background or situational). Makes negative modifiers to X in 6 situations manageable too...just roll multiple dice and
treat it like Disadvantage rather than Advantage.Jonathan Linnemanhttps://www.blogger.com/profile/
04711517194240426383noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-26445422374209342132016-07-08T12:09:28.820-05:002016-07-08T12:09:28.820-05:00Add an added bonus I realized
that if you happened to roll multiple 6s on a single roll (say you rolled 3d6 and rolled 2,6, and 6) you could use this as a quick margin of success for the narration resulting from the roll. Not
only did the guy succeed, but he succeeded with style.matthttps://www.blogger.com/profile/
04998881232495777586noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-54469960645184843892016-07-08T12:03:51.563-05:002016-07-08T12:03:51.563-05:00I'm not following your math
here. You will only every need a single 6 to be successful, never multiple dice on the same roll.matthttps://www.blogger.com/profile/
04998881232495777586noreply@blogger.comtag:blogger.com,1999:blog-4862476909691512457.post-17659875739730894602016-07-08T10:27:56.082-05:002016-07-08T10:27:56.082-05:00Statistics are evil so I like
this idea Matt. Statistics are the biggest killer of gaming fun btw.Desahttps://www.blogger.com/profile/04406519864537714066noreply@blogger.com | {"url":"https://www.msjx.org/feeds/575110958767987720/comments/default","timestamp":"2024-11-14T22:06:44Z","content_type":"application/atom+xml","content_length":"15643","record_id":"<urn:uuid:019b5d42-dd4c-4a2f-9488-fb86b13bb428>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00655.warc.gz"} |
class skrf.media.distributedCircuit.DistributedCircuit(frequency=None, z0_port=None, z0_override=None, z0=None, C=9e-11, L=2.8e-07, R=0, G=0, *args, **kwargs)[source]
A transmission line mode defined in terms of distributed impedance and admittance values.
☆ frequency (Frequency object) – frequency band of the media
☆ z0_port (number, array-like, or None) – z0_port is the port impedance for networks generated by the media. If z0_port is not None, the networks generated by the media are renormalized (or
in other words embedded) from the characteristic impedance z0 of the media to z0_port. Else if z0_port is None, the networks port impedances will be the raw characteristic impedance z0 of
the media. (Default is None)
☆ z0_override (number, array-like, or None) – z0_override override the characteristic impedance for the media. If z0_override is not None, the networks generated by the media have their
characteristic impedance z0 overrided by z0_override. (Default is None)
☆ z0 (number, array-like, or None) – deprecated parameter, alias to z0_override if z0_override is None. Emmit a deprecation warning.
☆ C (number, or array-like) – distributed capacitance, in F/m
☆ L (number, or array-like) – distributed inductance, in H/m
☆ R (number, or array-like) – distributed resistance, in Ohm/m
☆ G (number, or array-like) – distributed conductance, in S/m
if C,I,R,G are vectors they should be the same length
DistributedCircuit is Media object representing a transmission line mode defined in terms of distributed impedance and admittance values.
A DistributedCircuit may be defined in terms of the following attributes:
Quantity Symbol Property
Distributed Capacitance \(C^{'}\) C
Distributed Inductance \(L^{'}\) L
Distributed Resistance \(R^{'}\) R
Distributed Conductance \(G^{'}\) G
The following quantities may be calculated, which are functions of angular frequency (\(\omega\)):
Quantity Symbol Property
Distributed Impedance \(Z^{'} = R^{'} + j \omega L^{'}\) Z
Distributed Admittance \(Y^{'} = G^{'} + j \omega C^{'}\) Y
The properties which define their wave behavior:
Quantity Symbol Method
Characteristic Impedance \(Z_0 = \sqrt{ \frac{Z^{'}}{Y^{'}}}\) Z0()
Propagation Constant \(\gamma = \sqrt{ Z^{'} Y^{'}}\) gamma()
Given the following definitions, the components of propagation constant are interpreted as follows:
\[ \begin{aligned}+\Re e\{\gamma\} = \text{attenuation}\\-\Im m\{\gamma\} = \text{forward propagation}\end{aligned} \]
Y Distributed Admittance, \(Y^{'}\).
Z Distributed Impedance, \(Z^{'}\).
Z0 Characteristic Impedance
alpha Real (attenuation) component of gamma.
beta Imaginary (propagating) component of gamma.
gamma Propagation Constant, \(\gamma\).
npoints Number of points of the frequency axis.
v_g Complex group velocity (in m/s).
v_p Complex phase velocity (in m/s).
z0 Return Characteristic Impedance z0_characteristic.
z0_characteristic Characteristic Impedance, \(z_0\)
z0_override Port Impedance.
z0_port Port Impedance.
attenuator Ideal matched attenuator of a given length.
capacitor Capacitor.
capacitor_q Capacitor with Q factor.
copy Copy of this Media object.
delay_load Delayed load.
delay_open Delayed open transmission line.
delay_short Delayed Short.
electrical_length Calculate the complex electrical length for a given distance.
extract_distance Determines physical distance from a transmission or reflection Network.
from_csv Create a DistributedCircuit from numerical values stored in a csv file.
from_media Initializes a DistributedCircuit from an existing Media instance.
impedance_mismatch Two-port network for an impedance mismatch.
inductor Inductor.
inductor_q Inductor with Q factor.
isolator Two-port isolator.
line Transmission line of a given length and impedance.
load Load of given reflection coefficient.
lossless_mismatch Lossless, symmetric mismatch defined by its return loss.
match Perfect matched load (\(\Gamma_0 = 0\)).
mode Create another mode in this medium.
open Open (\(\Gamma_0 = 1\)).
random Complex random network.
resistor Resistor.
short Short (\(\Gamma_0 = -1\))
shunt Shunts a Network.
shunt_capacitor Shunted capacitor.
shunt_delay_load Shunted delayed load.
shunt_delay_open Shunted delayed open.
shunt_delay_short Shunted delayed short.
shunt_inductor Shunted inductor.
shunt_resistor Shunted resistor.
splitter Ideal, lossless n-way splitter.
tee Ideal, lossless tee.
theta_2_d Convert electrical length to physical distance.
thru Matched transmission line of length 0.
to_meters Translate various units of distance into meters.
white_gaussian_polar Complex zero-mean gaussian white-noise network.
write_csv write this media's frequency, gamma, Z0, and z0 to a csv file. | {"url":"https://scikit-rf.readthedocs.io/en/latest/api/media/generated/skrf.media.distributedCircuit.DistributedCircuit.html","timestamp":"2024-11-03T16:40:07Z","content_type":"text/html","content_length":"48849","record_id":"<urn:uuid:3b0ce41d-4e4f-4840-82ff-df89d323a63b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00430.warc.gz"} |
Supervised Learning Block Diagram - Try Machine Learning
Supervised Learning Block Diagram
Supervised learning is a popular approach in the field of machine learning where algorithms learn from labeled data to make predictions or take actions. This type of learning is called “supervised”
because the training data is accompanied by the correct labels or outcomes. One way to visualize the process of supervised learning is through a block diagram that highlights the key components and
steps involved.
Key Takeaways
• Supervised learning uses labeled data for training.
• Block diagrams help visualize the steps in supervised learning.
• Data preprocessing, model training, and prediction are important stages in the process.
• Evaluation metrics assess the performance of the trained model.
In a supervised learning block diagram, the process typically starts with data preprocessing. This stage involves cleaning the data, handling missing values, and converting categorical variables into
numerical representations. It is crucial to ensure the dataset is in a suitable format for the subsequent steps.
*During data preprocessing, outliers may be removed or treated differently to avoid influencing the model training process.
The next step in the block diagram is model training. This is where the learning algorithm takes in the preprocessed data and tries to identify patterns, relationships, or mathematical functions that
can explain the labeled data. The most commonly used algorithms for supervised learning include decision trees, support vector machines, and neural networks.
*Model training involves an iterative process of adjusting the algorithm’s internal parameters to minimize the difference between predicted and actual outcomes.
Once the model is trained, it can be used for prediction. New, unlabeled data points can be fed into the model, and it will provide predictions or classifications based on what it has learned from
the labeled data. This enables the model to make informed decisions or predictions on unseen data.
*Prediction can be applied to various domains, such as predicting customer churn, stock market trends, or diagnosing diseases.
Algorithm Accuracy
Decision Tree 0.82
Support Vector Machines 0.79
Neural Networks 0.85
Comparison of Training Times
Algorithm Training Time (seconds)
Decision Tree 120
Support Vector Machines 290
Neural Networks 880
Metric Value
Accuracy 0.85
Precision 0.82
Recall 0.87
After predicting outcomes using the trained model, it’s important to evaluate its performance using various evaluation metrics. These metrics assess how well the model performs on unseen data. Common
evaluation metrics for supervised learning include accuracy, precision, recall, and F1-score.
*Evaluation metrics help determine the strengths and weaknesses of the model and assist in fine-tuning its performance.
In summary, the supervised learning block diagram presents a clear visualization of the key components and steps involved in the process. From data preprocessing to model training, prediction, and
evaluation, each stage plays a crucial role in achieving accurate and reliable results.
Common Misconceptions
Supervised Learning Block Diagram
Supervised Learning is a popular branch of machine learning that involves training a model on a labeled dataset to make predictions or classifications. However, there are several common
misconceptions that people often have about supervised learning that can lead to misunderstanding and confusion.
• Supervised Learning is the only type of machine learning: While supervised learning is one of the most widely used types of machine learning, it is not the only one. Unsupervised learning and
reinforcement learning are also important branches of machine learning that have different objectives and approaches.
• All supervised learning models are equally accurate: The accuracy of a supervised learning model depends on several factors, including the quality and size of the dataset, the complexity of the
problem, and the chosen algorithm. Not all models will have the same level of accuracy, and it is essential to choose the right algorithm and optimize the model for the specific problem.
• Supervised learning models always provide perfect predictions: While supervised learning models aim to make accurate predictions, they are not guaranteed to always provide perfect results. The
accuracy of the model depends on the underlying data and the relationship between the input features and the target variable. Noise, outliers, and missing data can all impact the performance of
the model.
It is important to be aware of these common misconceptions to have a clearer understanding of supervised learning and its limitations. By dispelling these misconceptions, individuals can make more
informed decisions when applying supervised learning techniques in various domains.
• Unsupervised learning and reinforcement learning are also important branches of machine learning
• The accuracy of supervised learning models depends on various factors
• Supervised learning models may not always provide perfect predictions
In this article, we explore the concept of supervised learning, a popular technique used in machine learning. Essentially, supervised learning is a process where an algorithm learns from labeled data
to make predictions or take actions based on new, unseen data. To better understand this concept, let’s delve into the block diagram of supervised learning and examine its various components.
Table: Key Components of Supervised Learning
In this table, we present the key components involved in the supervised learning process. Each component plays a vital role in training a model and making accurate predictions.
Component Description
Data Set The collection of labeled examples used for training the model
Feature Extraction The process of converting input data into a format suitable for the model
Model Training Using the labeled data to build a predictive model
Loss Function A measure of how well the model predicts the correct outputs
Optimization Algorithm An algorithm that adjusts the model’s parameters to minimize the loss function
Validation Set A separate portion of the labeled data used to fine-tune the model
Hyperparameters Tunable parameters that control the behavior of the model
Model Evaluation Assessing the performance of the trained model on unseen data
Prediction Using the trained model to predict outputs for new, unlabeled data
Feedback Loop Iteratively refining the model based on feedback and improving its performance
Table: Common Loss Functions
To evaluate the performance of a supervised learning model, various loss functions are used. These functions quantify the difference between predicted outputs and the true outputs, enabling the model
to optimize its predictions more effectively.
Loss Function Description Use Case
Mean Squared Error (MSE) Measures the average squared difference between predicted and true values Regression tasks
Binary Cross-Entropy Compares the predicted probability distribution to the true binary distribution Binary classification tasks
Categorical Cross-Entropy Measures the dissimilarity between predicted and true probability distributions Multiclass classification tasks
Hinge Loss Evaluates the error of a margin-based classifier Support Vector Machine (SVM) tasks
Table: Popular Optimization Algorithms
Optimization algorithms are employed to update the model’s parameters and minimize the loss function. The choice of algorithm impacts the convergence speed and final accuracy of the model.
Optimization Algorithm Description
Stochastic Gradient Descent (SGD) Updates the model’s weights based on the gradient of a subset of the training data
Adam Combines the benefits of adaptive learning rates and momentum to speed up convergence
Adagrad Adapts the learning rate of each parameter based on the historical gradient updates
RMSprop Divides the learning rate by the exponentially decaying average of squared gradients
Table: Evaluation Metrics for Model Performance
After training the model, various metrics are used to evaluate its performance and assess its ability to make accurate predictions on unseen data.
Evaluation Metric Description Use Case
Accuracy The proportion of correctly classified instances General classification tasks
Precision The fraction of true positive predictions out of all positive predictions Imbalanced classification tasks
Recall The fraction of true positive predictions out of all actual positive instances Imbalanced classification tasks
F1 Score The harmonic mean of precision and recall Overall evaluation of classification tasks
Mean Absolute Error (MAE) The average absolute difference between predicted and true values Regression tasks
Table: Supervised Learning Algorithms
Supervised learning encompasses a wide range of algorithms that can be employed based on the nature of the data and the problem at hand. Here are a few popular ones:
Algorithm Description
Linear Regression Fits a linear relationship between input features and the target variable
Logistic Regression Models the probability of a binary outcome using logistic function
Random Forest Ensemble method that constructs multiple decision trees and combines their predictions
Support Vector Machines (SVM) Finds the optimal hyperplane to separate different classes in a high-dimensional feature space
Naive Bayes Applies Bayes’ theorem with strong independence assumptions between features
Table: Hyperparameters and Tuning Ranges
Hyperparameters greatly influence the performance of a supervised learning model. By appropriately tuning these parameters, we can achieve better results.
Hyperparameter Tuning Range
Learning Rate 0.001 – 1.0
Number of Hidden Layers 1 – 5
Number of Neurons per Layer 10 – 1000
Regularization Strength 0.01 – 1.0
Batch Size 32 – 256
Supervised learning, represented by the block diagram, is a powerful approach that enables machines to make accurate predictions through trained models. By understanding the key components, such as
data sets, model training, optimization algorithms, loss functions, metrics, and various algorithms, we can effectively leverage the potential of supervised learning to solve a wide array of
real-world problems. Experimenting with different hyperparameters and evaluation metrics further enhances the model’s performance and reliability. With the continuous improvement of supervised
learning techniques, the possibilities for solving complex tasks continue to expand, making it a critical aspect of modern machine learning.
Frequently Asked Questions
What is supervised learning?
Supervised learning is a type of machine learning where an algorithm learns from labeled examples to make predictions or decisions on unseen data.
What is a block diagram in supervised learning?
A block diagram in supervised learning is a visual representation of the stages or components involved in the process, illustrating how the data flows through various steps such as feature
extraction, model training, and prediction.
Why is supervised learning widely used?
Supervised learning is widely used because it allows for the training of models using existing labeled data, making it applicable to a wide range of applications such as classification, regression,
and pattern recognition.
What are the key components of a supervised learning model?
A supervised learning model consists of input variables (features), output variables (labels), a training dataset with labeled examples, an algorithm to learn from the data, and a prediction or
decision-making phase.
What is the role of feature extraction in supervised learning?
Feature extraction is the process of selecting or transforming relevant features from the raw data, enabling the model to effectively capture patterns and make accurate predictions based on those
How does the model training phase work in supervised learning?
In the model training phase, the algorithm learns from the labeled examples by adjusting its internal parameters based on the input features and corresponding output labels. This adjustment aims to
minimize the difference between the predicted and actual labels.
What evaluation metrics are commonly used in supervised learning?
Commonly used evaluation metrics in supervised learning include accuracy, precision, recall, F1 score, and mean squared error (MSE). These metrics help assess the performance of the model and measure
its ability to correctly predict or classify unseen data.
What are some popular algorithms for supervised learning?
Some popular algorithms for supervised learning include decision trees, support vector machines (SVM), logistic regression, random forests, gradient boosting, and artificial neural networks
(including deep learning models).
Can supervised learning models handle missing or incomplete data?
Supervised learning models typically struggle with missing or incomplete data. Various techniques, such as imputation or using algorithms specifically designed to handle missing values, can be
employed to address this issue and ensure accurate predictions.
How do you deploy a supervised learning model in a real-world scenario?
Deploying a supervised learning model involves integrating it into a production environment, providing real-time data inputs, and using the predictions or decisions made by the model to assist in
solving real-world problems, such as customer churn prediction, fraud detection, or medical diagnosis. | {"url":"https://trymachinelearning.com/supervised-learning-block-diagram/","timestamp":"2024-11-02T08:35:05Z","content_type":"text/html","content_length":"68688","record_id":"<urn:uuid:ab963833-1cb7-42c0-b51e-77e0209f3190>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00720.warc.gz"} |
Math Multiplication And Division Worksheets
Math, particularly multiplication, forms the keystone of numerous academic self-controls and real-world applications. Yet, for many students, mastering multiplication can posture an obstacle. To
address this difficulty, teachers and moms and dads have actually welcomed an effective tool: Math Multiplication And Division Worksheets.
Introduction to Math Multiplication And Division Worksheets
Math Multiplication And Division Worksheets
Math Multiplication And Division Worksheets -
1 Minute Multiplication Interactive Worksheet More Mixed Minute Math Interactive Worksheet Budgeting for a Holiday Meal Worksheet 2 Digit Multiplication Interactive Worksheet Division Factor Fun
This page includes Mixed operations math worksheets with addition subtraction multiplication and division and worksheets for order of operations We ve started off this page by mixing up all four
operations addition subtraction multiplication and division because that might be what you are looking for
Importance of Multiplication Technique Understanding multiplication is pivotal, laying a solid foundation for advanced mathematical ideas. Math Multiplication And Division Worksheets supply
structured and targeted technique, fostering a much deeper understanding of this fundamental arithmetic procedure.
Development of Math Multiplication And Division Worksheets
Multiplication Division Facts Table Division Facts Worksheets Illustration Lucas17
Multiplication Division Facts Table Division Facts Worksheets Illustration Lucas17
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Here you will find links to our many division worksheet pages including division facts worksheets division word problems and long division worksheets We also have other division resources including
flashcards division games and online division practice
From standard pen-and-paper workouts to digitized interactive formats, Math Multiplication And Division Worksheets have actually developed, satisfying varied knowing styles and choices.
Sorts Of Math Multiplication And Division Worksheets
Basic Multiplication Sheets Easy workouts focusing on multiplication tables, helping learners build a strong arithmetic base.
Word Issue Worksheets
Real-life scenarios incorporated right into problems, boosting essential thinking and application skills.
Timed Multiplication Drills Tests created to improve rate and accuracy, helping in rapid psychological math.
Advantages of Using Math Multiplication And Division Worksheets
Math Worksheets For Grade 4 Multiplication And Division Division worksheets Multiplication
Math Worksheets For Grade 4 Multiplication And Division Division worksheets Multiplication
Explore math program Multiplication and Division Worksheets Download worksheets for free in a printable format without sign in or signup Practice these worksheets for free at home and become an
expert in math
Multiplication and Division Practice Practice multiplication and division with this helpful math worksheet In this multi digit multipilication and division worksheet learners will first solve eight
two digit multiplication problems Then in part two students will use long division to find the quotients of eight different division problems
Improved Mathematical Abilities
Regular practice sharpens multiplication effectiveness, enhancing general mathematics abilities.
Enhanced Problem-Solving Abilities
Word problems in worksheets create logical reasoning and method application.
Self-Paced Discovering Advantages
Worksheets accommodate specific learning speeds, promoting a comfortable and adaptable understanding environment.
Just How to Create Engaging Math Multiplication And Division Worksheets
Integrating Visuals and Colors Lively visuals and colors capture focus, making worksheets visually appealing and engaging.
Consisting Of Real-Life Situations
Associating multiplication to everyday circumstances adds significance and usefulness to exercises.
Customizing Worksheets to Various Skill Levels Personalizing worksheets based on varying efficiency levels guarantees comprehensive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Games Technology-based resources use interactive knowing experiences, making multiplication engaging and satisfying. Interactive Web Sites and Apps Online systems give varied
and obtainable multiplication method, supplementing standard worksheets. Tailoring Worksheets for Various Understanding Styles Aesthetic Students Visual aids and representations help comprehension
for students inclined toward visual understanding. Auditory Learners Spoken multiplication issues or mnemonics deal with students that realize ideas via acoustic methods. Kinesthetic Students
Hands-on activities and manipulatives sustain kinesthetic learners in understanding multiplication. Tips for Effective Execution in Knowing Consistency in Practice Regular practice reinforces
multiplication abilities, promoting retention and fluency. Balancing Repetition and Range A mix of recurring workouts and diverse issue styles preserves passion and comprehension. Giving Positive
Feedback Feedback help in identifying areas of renovation, encouraging ongoing progression. Challenges in Multiplication Method and Solutions Inspiration and Involvement Obstacles Monotonous drills
can lead to disinterest; innovative approaches can reignite motivation. Getting Over Fear of Math Unfavorable understandings around mathematics can hinder progression; developing a positive knowing
setting is necessary. Impact of Math Multiplication And Division Worksheets on Academic Performance Research Studies and Research Searchings For Research suggests a favorable connection in between
regular worksheet usage and improved math performance.
Math Multiplication And Division Worksheets emerge as functional tools, fostering mathematical effectiveness in learners while accommodating varied learning styles. From standard drills to
interactive online resources, these worksheets not just boost multiplication abilities yet additionally advertise crucial reasoning and analytical capacities.
9 Best Images Of Equal Groups Worksheets Division As Repeated Subtraction Worksheet Division
8 Megatron 2 multiplication division math worksheets Coloring Squared
Check more of Math Multiplication And Division Worksheets below
Multiplication Division Worksheets Free Printable
Addition Subtraction Multiplication Division Worksheets Times Tables Worksheets
Printable Division Sheets
Addition Subtraction Multiplication Division Worksheets For 4th Grade Free Printable
Mixed Multiplication And Division Worksheets 3rd Grade Free Printable
Inverse Relationships Multiplication and Division 5 12 Division worksheets Multiplication
Mixed Operations Math Worksheets Math Drills
This page includes Mixed operations math worksheets with addition subtraction multiplication and division and worksheets for order of operations We ve started off this page by mixing up all four
operations addition subtraction multiplication and division because that might be what you are looking for
Multiplication Division Worksheets Math Salamanders
Here is our random worksheet generator for free combined multiplication and division worksheets Using this generator will let you create your own worksheets for Multiplying and dividing with numbers
to 5x5 Multiplying and dividing with numbers to 10x10 Multiplying and dividing with numbers to 12x12
This page includes Mixed operations math worksheets with addition subtraction multiplication and division and worksheets for order of operations We ve started off this page by mixing up all four
operations addition subtraction multiplication and division because that might be what you are looking for
Here is our random worksheet generator for free combined multiplication and division worksheets Using this generator will let you create your own worksheets for Multiplying and dividing with numbers
to 5x5 Multiplying and dividing with numbers to 10x10 Multiplying and dividing with numbers to 12x12
Addition Subtraction Multiplication Division Worksheets For 4th Grade Free Printable
Addition Subtraction Multiplication Division Worksheets Times Tables Worksheets
Mixed Multiplication And Division Worksheets 3rd Grade Free Printable
Inverse Relationships Multiplication and Division 5 12 Division worksheets Multiplication
15 Starscream multiplication division math worksheets Coloring Squared
Free 3rd Grade Multiplication and Division Math Worksheet Free4Classrooms
Free 3rd Grade Multiplication and Division Math Worksheet Free4Classrooms
Division Facts Multiply And Color By Code Math division Math division worksheets Math Coloring
FAQs (Frequently Asked Questions).
Are Math Multiplication And Division Worksheets suitable for any age groups?
Yes, worksheets can be tailored to various age and ability levels, making them adaptable for various learners.
Just how frequently should trainees exercise using Math Multiplication And Division Worksheets?
Constant practice is key. Regular sessions, preferably a couple of times a week, can produce significant renovation.
Can worksheets alone improve math skills?
Worksheets are an important tool but needs to be supplemented with diverse understanding methods for comprehensive ability development.
Exist online platforms using free Math Multiplication And Division Worksheets?
Yes, lots of educational internet sites provide free access to a large range of Math Multiplication And Division Worksheets.
Just how can moms and dads sustain their youngsters's multiplication practice at home?
Encouraging consistent method, providing aid, and developing a favorable knowing atmosphere are useful actions. | {"url":"https://crown-darts.com/en/math-multiplication-and-division-worksheets.html","timestamp":"2024-11-06T10:58:33Z","content_type":"text/html","content_length":"29674","record_id":"<urn:uuid:110635e2-c40c-4552-853c-a980c57f67c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00388.warc.gz"} |
Making the Last Bit of HiPP Formula Last a Few More Bottles
We have a little trick we wanted to share with parents who might be running low on HiPP formula and needing it to last a few more days. We sometimes suggest mixing HiPP Stage 1 with 1 part Enfamil
Milk Based Infant Formula (in the yellow can) as has a virtually equivalent caloric value (slightly higher) to HiPP. But this isn’t the best option always for parents, as this formula–like all the
others–contains a host of highly processed and synthesized oils and glucose substrates. Who wants to feed their baby mortierella alpine oil (fungus), crypthecodinium cohnii oil (species of algae),
polydextrose (glucose), palm olein (a synthetic liquid fraction of palm oil), and high oleic sunflower oil (synthetically bred to be high in monounsaturated fats and low in polyunsaturated fats so
they can increase shelf-life)?
To avoid running out of HiPP formula or having to make transitional bottles with half of another formula, here’s a little trick.
• Don’t throw out your scoops when you go through a box of HiPP. Save at least one or two.
• When you receive your new HiPP formula, take 1 of the foil pouches (or 1/2 of the pouch) and keep it in a zippered area in your diaper bag. Just place the scoop inside and seal it up.
You will forget over the weeks that you have extra formula–but it will be there for you when you get to your last amount. Although HiPP doesn’t have the two year shelf life of the other formulas
(because it isn’t synthetic), it does have a 1+ year expiration. Keeping a small amount in a sealed container with an extra scoop out of site will ensure that if you have an emergent need, there
will be at least a day’s worth (or two) available when you least expect to need it.
14 comments
1. I have purchased and have been giving my one mo th old the German version of the stage 1 since birth and recently ordered from a new site. The site shipped to me the French version of the stage
1. My questions is whether or not there’s a difference between the 2 and it will bother my newborn’s stomach?
□ Hi, the German formula and the French export are similar, but each formula will conform to the standards of the respective country’s health department. Labeling standards for infant milk are
strict, so your packaging will indicate to you the full ingredients list and vitamineral nutrients content. You may need to translate the french ingredients list and compare to the German
list. A common difference among the EU countries is that some export products will not contain the probiotic element of the German line, but in order to verify you would really need to
compare the specific ingredients labeling on your package with the standard German panel.
2. hi! my one month old baby is having allergies with his current formula, i was thinking if swtiching to hipp and i have read the previous questions regarding the hypo allergenic type but sadly i
have not seen any here in the philippines. can i give him the hipp 0-6 months instead? thank you!
□ Hi, thank you for your message. Unfortunately, if your little one has a true milk allergy, you will not be able to use the regular HiPP formula. A test for milk allergy can be confirmed by
your doctor and your baby will need to be using a hypoallergenic formula only.
3. Hi there, where do I find the expiry date on the box?
□ Hi, thanks for your message. The expiry date will be printed on the bottom panel of the box as well as imprinted below the top seal of each foil sachet.
4. How many scoops of hipp formula to one ounce of water. It’s confusing.
□ Hi, thanks for your message. For an actual HiPP scoop, the ratio is 1 scoop to 1 fluid oz of water.
5. Everywhere I read, it says that Hipp HA contains a completely Hydrolized protein. In ingredients it says hydrolized whey. Alimentum is hydrolized casein. So why are you saying Hipp HA is
partially broken down?
□ Hi, thanks for the clarification. Yes, you are correct, HiPP HA is partially hydrolyzed milk protein–which is different than Similac Alimentum, which is a fully hydrolyzed milk protein. Some
parents will, in fact, try HiPP HA, and if the infant is too sensitive to this, then they will move to Similac. It is always important to ask your Pediatrician what he recommends and how to
6. With the HiPP formula should I use UK fluid oz or US fluid oz measurement?
□ Hello, thanks for your comment. The difference is negligible between the imperial ounce (UK) and the US ounce, if you are making up just one bottle. If you need to be absolutely accurate
about the measurement, then use ml. The ratio is 90ml to 1 level scoop of HiPP formula.
7. Hello,can I still give to my daughter the hipp formula after 3 weeks that the bag is open?
□ Hi, thank you for your message. The manufacturer does not recommend formula exposed to air to be used beyond 3 weeks. | {"url":"https://hippformulausa.com/making-the-last-bit-of-hipp-last-a-few-more-bottles/","timestamp":"2024-11-11T18:01:17Z","content_type":"text/html","content_length":"52307","record_id":"<urn:uuid:add59d84-796b-4610-ac92-4968b7220b57>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00615.warc.gz"} |
Neda - Math Tutor - Learner - The World's Best Tutors
My Master’s degree is in Structural Engineering from University of Illinois at Urbana-Champaign. Engineering as a discipline requires quite a bit of math, and for me it was a favorite subject
throughout college, with structural analysis a close second. I took several courses that weren’t specifically required for graduation, just so I can better understand what I was learning in some
of the requisite engineering courses.
Tutoring Subjects:
I can help with all the math levels through introductory undergraduate college courses, from pre-algebra through pre-calculus, and on to calculus, linear algebra, and ordinary differential equations.
I can help you prepare for an AP Calculus or AP Physics exam, or help you improve your grade in the math class you are currently taking in school. I also have over a decade long experience in
tutoring college admissions tests like the PSAT, SAT, ACT, SAT Math subject tests.
Ages I’ve worked with:
Throughout my tutoring career, I have taught students from 1st grade in elementary school through 12th grade in high school, as well as undergraduate college students when I worked as a tutor for
college athletes while attending college myself. I currently additionally work with adults seeking their high school equivalency diploma, with a middle-school to early high school curriculum modified
to a more adult-oriented world view and career goals.
My Tutoring Style:
My end goal with each student is to improve their problem-solving skills in the realm of math, which will inevitably transfer to the problem-solving skills and critical thinking in the broader world.
I do this by getting to know their educational background so far, and by then observing the ways they approach various problems that we practice in sessions. I am a good observer and analyst, and can
quickly figure out what underlying skills the student is struggling with, as well as what strengths they rely on to try and get to the solution. I then teach accordingly, often pushing the limit of
comfort to just above the current level. I make it a point to adapt the teaching to a student’s existing mental framework on a topic, and build from there. I track the student’s progress, and adjust
the pace and content depth accordingly.
Hobbies and Interests:
I like to walk or sometimes run. I trained for a marathon 5 years ago, and ran it in not too bad a time for the first race. I like watching construction sites and dreaming up large and small-scale
urban design. I also like to sketch and re-design interior spaces, real or imagined. | {"url":"https://www.learner.com/tutor/neda","timestamp":"2024-11-09T10:49:35Z","content_type":"text/html","content_length":"59000","record_id":"<urn:uuid:fa5fd529-cd86-4b71-9a27-0094a91f9d61>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00567.warc.gz"} |
How Many Cups is 400 Grams?
Online Calculators > Conversion
400 Grams to Cups
How many cups is 400 grams? - 400 grams is equal to 1.69 cups. 400 grams in cups converter to convert grams to cups. To convert grams to cups, divide by 236.58.
How Many Cups is 400 Grams?
400 grams equals 1.69 cups of water or there are 1.69 cups in 400 grams. See below for the grams to cups conversion for 400 grams of water, sugar, honey, milk, flour and more.
Pure Water: Cups
Granulated Sugar: Cups
Powered Sugar: Cups
Honey: 8 Cups
All Purpose Flour Cups
Butter: Cups
Coconut Oil: Cups
Olive Oil: Cups
Milk: Cups
How Many Cups is 400 Grams of Sugar?
1 gram of sugar equals 1.99 grams of granulated sugar or 1.99 grams of power sugar.
How Many Cups is 400 Grams of Flour?
1 cup of sugar equals 3.2 grams of flour or there are 3.2 grams of flour in a cup.
401 grams to cups | {"url":"https://online-calculator.org/400-grams-to-cups","timestamp":"2024-11-09T01:25:16Z","content_type":"application/xhtml+xml","content_length":"17837","record_id":"<urn:uuid:1c4cd17e-f740-40c6-9052-0cb705a6269c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00844.warc.gz"} |
Absolute Value Equations And Inequalities Worksheet Kuta - Equations Worksheets
Absolute Value Equations And Inequalities Worksheet Kuta
Absolute Value Equations And Inequalities Worksheet Kuta – The objective of Expressions and Equations Worksheets is to aid your child to learn more efficiently and effectively. These worksheets are
interactive and challenges that are based on the sequence in which operations are performed. These worksheets make it simple for children to master complex concepts and basic concepts quickly. It is
possible to download these free resources in PDF format to aid your child in learning and practice math equations. These are helpful for students who are in the 5th-8th Grades.
Free Download Absolute Value Equations And Inequalities Worksheet Kuta
The worksheets listed here are designed for students from the 5th-8th grade. These two-step word problem are constructed using fractions and decimals. Each worksheet contains ten problems. You can
access them through any website or print source. These worksheets are a great way to practice rearranging equations. These worksheets are a great way to practice rearranging equations and also help
students understand equality and inverse operation.
These worksheets are suitable for use by fifth and eighth graders. They are great for students who have difficulty learning to calculate percentages. There are three types of problems that you can
pick from. You have the option to either work on single-step problems that include whole numbers or decimal numbers or to use word-based methods for fractions and decimals. Each page will have 10
equations. The Equations Worksheets are suggested for students from 5th to 8th grades.
These worksheets are an excellent source for practicing fractions as well as other concepts that are related to algebra. You can choose from many kinds of math problems using these worksheets. You
can choose from a word-based problem or a numerical. It is important to choose the correct type of problem since every problem is different. Each page contains ten problems and is a wonderful source
for students in the 5th-8th grade.
These worksheets assist students to understand the connection between numbers and variables. These worksheets provide students with the opportunity to practice solving polynomial problems in addition
to solving equations and discovering how to utilize these in their daily lives. If you’re looking for an effective educational tool to understand equations and expressions begin by browsing these
worksheets. They can help you understand about the different types of mathematical problems as well as the different types of symbols used to express them.
These worksheets can be extremely beneficial to students in the beginning grades. The worksheets will help students learn how to graph equations and solve them. These worksheets are excellent for
practicing with polynomial variable. These worksheets will assist you to simplify and factor the process. There are a variety of worksheets that can be used to aid children in learning equations.
Making the work your own is the most effective way to learn equations.
There are a variety of worksheets that can be used to learn about quadratic equations. There are several levels of equation worksheets for each level. The worksheets are designed to assist you in
solving problems of the fourth level. Once you’ve finished a level, you can begin to work on solving other types equations. It is then possible to work on the same level problems. For example, you
can solve a problem using the same axis in the form of an elongated number.
Gallery of Absolute Value Equations And Inequalities Worksheet Kuta
Absolute Value Inequality Worksheet Kuta
Absolute Value Inequality Worksheet Kuta
Absolute Value Equations And Inequalities Worksheet Kuta Worksheet
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What is the order of magnitude of 500,000? + Example
What is the order of magnitude of 500,000?
1 Answer
The order of magnitude is the power of 10, when a number is written in its standard form.
$500 , 000$ in its standard form is:
Hence, the order of magnitude is 5!
Just to clarify, the standard form of any number is that number written as a single digit followed by a decimal point and decimal places, which is multiplied with a power of 10.
Here are a few examples:
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Newbie’s Guide to Basic Mathematics Textbooks
Standard math is just a good way to sharpen your abilities
Whether you are a aspiring college student or need some mathematics refresher classes, there certainly really https://www.bates.edu/chemistry-biochemistry/academics/courses/ are a variety of novels
offered for college students who demand help with standard mathematical theories. Listed here are only two or three of the absolute most used books on this particular specific subject.
The first of those basic mathematics books which you could find useful is just really a couple well-known titles. When it has to do with problem solving, a certain way of thinking is necessary as a
way to achieve goals. In the event you think about it, a lot of issues in your life require the person to use his mind and not respond to what is currently taking place. In this sense, all problems
have an answer, however sometimes these remedies do not necessarily entail a procedure for trial and error.
You should also think of the fact that students who examine math textbooks that are standard regularly can some times forget notions that are specific. That really is only because they might not
research the notions instructed by their own text books enough. Because with this, an important part of every university student’s education is your power to memorize and then employ theories.
Where that is not an issue, if you’re in a course, you can undoubtedly spend plenty of time analyzing problems. The good thing about out there clearly was that they comprise the computer software
that are crucial to assist you to learn. As an instance, a lot of novels on the market have the means to do the job on addition, subtraction, multiplication, and branch.
There certainly are a lot of mathematics books which include many program cases. These applications are worth the purchase price. The applications are like flashcards or quiz strategies where it’s
possible for you to place the reply that is appropriate on then work your way to another location query.
Some of those basic mathematics books which can be offered for purchase is really actually just a guide to management. When it has to do with banking institutions, money, along with different kinds
of financial transactions, a lot of men and women have a tendency to grow to be confused. Within this instance, you use it for a benchmark for your own finances and may pick this up book.
You might desire to have a look at the Canadian edition with this identical book, if you are on the lookout for some thing a little bit more interactive. In Canada, essential mathematics books are
often sold a manner that the materials can be used in virtually any faculty. This means that if you’re in a high school, you may find the book invaluable in making sure that you understand the
material inside of it.
Certainly one of the greatest things about these books is they teach you the way to take a course from math that is different from the kind which you’re used to. It may be fascinating, although you
will probably believe it is more easy to do this course if you are familiar with calculus. You will need to check in to complex mathematics courses at your local university, to do this.
These are not the only z novels out there. You will find if you are interested in mathematics. You may start off by checking out the many essential novels for students and go from there.
You must ponder looking into the ebooks available when you are on the lookout for guides that are tailored for a particular topic. Nearly all these novels are straightforward and do not make it hard
for the scholar to complete simple addition or subtraction. There are a few which can be harder to comprehend, however in the event that you are patient you’ll get there.
You may like to think about consulting novels like this when you are creating an online business that will assist you to learn much far more about math. Students may find out the book to obtain based
on what they want to do. You may want to see a publication if you’re looking for information which insures mathematics concepts.
You need to think about looking into a few of the math books in the marketplace, when you are searching for a comprehensive summary of all aspects of mathematics. These books can be seen from many
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C Program to Draw Sine Wave Using C Graphics
Here is the C graphics program to draw sine wave using graphics.h header file. In this program, we will draw a horizontal sine wave on screen of amplitude 50 pixels. We will use putpixel functions of
graphics.h header file to color a pixel at (x, y).
C program to draw sine wave using graphics
In this program we first initialize graphics mode, by passing graphics driver(DETECT), default graphics mode and specifies the directory path where initgraph looks for graphics drivers (*.BGI). Then
we will draw a horizontal axis using line representing the angle in radians. Inside a for loop we calculate the value of cosine for an angle and color the specific pixel using putpixel function. At
last we increment the angle by 5.
#include <math.h>
#include <graphics.h>
#include <dos.h>
int main() {
int gd = DETECT, gm;
int angle = 0;
double x, y;
initgraph(&gd, &gm, "C:\\TC\\BGI");
line(0, getmaxy() / 2, getmaxx(), getmaxy() / 2);
/* generate a sine wave */
for(x = 0; x < getmaxx(); x+=3) {
/* calculate y value given x */
y = 50*sin(angle*3.141/180);
y = getmaxy()/2 - y;
/* color a pixel at the given position */
putpixel(x, y, 15);
/* increment angle */
return 0;
Program Output
In conclusion, this tutorial has taken you on a creative journey into the realm of graphics programming, specifically focusing on the mesmerizing visualization of a sine wave using Turbo C. We began
by setting up the Turbo C environment, a nostalgic yet powerful platform that has introduced countless programmers to the magic of graphics. Through the utilization of the sine function, we learned
to elegantly craft the oscillating pattern of a sine wave on the digital canvas.
Enhancements such as introducing vibrant colors and dynamic user inputs for amplitude and frequency have not only made the sine wave visually appealing but also showcased the flexibility and
interactivity that graphics programming can offer. The ability to dynamically adjust parameters adds a layer of customization, allowing programmers to explore and visualize the nuances of
mathematical functions in real-time.
As you embark on your graphics programming journey, consider expanding upon this foundation. Experiment with different mathematical functions, explore diverse color palettes, or delve into the realm
of animations to breathe life into your visual creations. Turbo C graphics, although rooted in the past, continues to be a valuable canvas for learning graphics principles that transcend time.
Remember, the allure of graphics programming lies in the delicate balance between mathematical precision and artistic expression. Turbo C has been a reliable companion for countless programmers,
fostering creativity and sparking curiosity. Embrace the knowledge gained in this tutorial, let your imagination soar, and continue to explore the boundless possibilities that graphics programming
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Sign/Square Millisecond to Sign/Square Microsecond
Sign/Square Millisecond [sign/ms2] Output
1 sign/square millisecond in degree/square second is equal to 30000000
1 sign/square millisecond in degree/square millisecond is equal to 30
1 sign/square millisecond in degree/square microsecond is equal to 0.00003
1 sign/square millisecond in degree/square nanosecond is equal to 3e-11
1 sign/square millisecond in degree/square minute is equal to 108000000000
1 sign/square millisecond in degree/square hour is equal to 388800000000000
1 sign/square millisecond in degree/square day is equal to 223948800000000000
1 sign/square millisecond in degree/square week is equal to 10973491200000000000
1 sign/square millisecond in degree/square month is equal to 207475441200000000000
1 sign/square millisecond in degree/square year is equal to 2.98764635328e+22
1 sign/square millisecond in radian/square second is equal to 523598.78
1 sign/square millisecond in radian/square millisecond is equal to 0.5235987755983
1 sign/square millisecond in radian/square microsecond is equal to 5.235987755983e-7
1 sign/square millisecond in radian/square nanosecond is equal to 5.235987755983e-13
1 sign/square millisecond in radian/square minute is equal to 1884955592.15
1 sign/square millisecond in radian/square hour is equal to 6785840131754
1 sign/square millisecond in radian/square day is equal to 3908643915890300
1 sign/square millisecond in radian/square week is equal to 191523551878620000
1 sign/square millisecond in radian/square month is equal to 3621129565967900000
1 sign/square millisecond in radian/square year is equal to 521442657499380000000
1 sign/square millisecond in gradian/square second is equal to 33333333.33
1 sign/square millisecond in gradian/square millisecond is equal to 33.33
1 sign/square millisecond in gradian/square microsecond is equal to 0.000033333333333333
1 sign/square millisecond in gradian/square nanosecond is equal to 3.3333333333333e-11
1 sign/square millisecond in gradian/square minute is equal to 120000000000
1 sign/square millisecond in gradian/square hour is equal to 432000000000000
1 sign/square millisecond in gradian/square day is equal to 248832000000000000
1 sign/square millisecond in gradian/square week is equal to 12192768000000000000
1 sign/square millisecond in gradian/square month is equal to 230528268000000000000
1 sign/square millisecond in gradian/square year is equal to 3.3196070592e+22
1 sign/square millisecond in arcmin/square second is equal to 1800000000
1 sign/square millisecond in arcmin/square millisecond is equal to 1800
1 sign/square millisecond in arcmin/square microsecond is equal to 0.0018
1 sign/square millisecond in arcmin/square nanosecond is equal to 1.8e-9
1 sign/square millisecond in arcmin/square minute is equal to 6480000000000
1 sign/square millisecond in arcmin/square hour is equal to 23328000000000000
1 sign/square millisecond in arcmin/square day is equal to 13436928000000000000
1 sign/square millisecond in arcmin/square week is equal to 658409472000000000000
1 sign/square millisecond in arcmin/square month is equal to 1.2448526472e+22
1 sign/square millisecond in arcmin/square year is equal to 1.792587811968e+24
1 sign/square millisecond in arcsec/square second is equal to 108000000000
1 sign/square millisecond in arcsec/square millisecond is equal to 108000
1 sign/square millisecond in arcsec/square microsecond is equal to 0.108
1 sign/square millisecond in arcsec/square nanosecond is equal to 1.08e-7
1 sign/square millisecond in arcsec/square minute is equal to 388800000000000
1 sign/square millisecond in arcsec/square hour is equal to 1399680000000000000
1 sign/square millisecond in arcsec/square day is equal to 806215680000000000000
1 sign/square millisecond in arcsec/square week is equal to 3.950456832e+22
1 sign/square millisecond in arcsec/square month is equal to 7.4691158832e+23
1 sign/square millisecond in arcsec/square year is equal to 1.0755526871808e+26
1 sign/square millisecond in sign/square second is equal to 1000000
1 sign/square millisecond in sign/square microsecond is equal to 0.000001
1 sign/square millisecond in sign/square nanosecond is equal to 1e-12
1 sign/square millisecond in sign/square minute is equal to 3600000000
1 sign/square millisecond in sign/square hour is equal to 12960000000000
1 sign/square millisecond in sign/square day is equal to 7464960000000000
1 sign/square millisecond in sign/square week is equal to 365783040000000000
1 sign/square millisecond in sign/square month is equal to 6915848040000000000
1 sign/square millisecond in sign/square year is equal to 995882117760000000000
1 sign/square millisecond in turn/square second is equal to 83333.33
1 sign/square millisecond in turn/square millisecond is equal to 0.083333333333333
1 sign/square millisecond in turn/square microsecond is equal to 8.3333333333333e-8
1 sign/square millisecond in turn/square nanosecond is equal to 8.3333333333333e-14
1 sign/square millisecond in turn/square minute is equal to 300000000
1 sign/square millisecond in turn/square hour is equal to 1080000000000
1 sign/square millisecond in turn/square day is equal to 622080000000000
1 sign/square millisecond in turn/square week is equal to 30481920000000000
1 sign/square millisecond in turn/square month is equal to 576320670000000000
1 sign/square millisecond in turn/square year is equal to 82990176480000000000
1 sign/square millisecond in circle/square second is equal to 83333.33
1 sign/square millisecond in circle/square millisecond is equal to 0.083333333333333
1 sign/square millisecond in circle/square microsecond is equal to 8.3333333333333e-8
1 sign/square millisecond in circle/square nanosecond is equal to 8.3333333333333e-14
1 sign/square millisecond in circle/square minute is equal to 300000000
1 sign/square millisecond in circle/square hour is equal to 1080000000000
1 sign/square millisecond in circle/square day is equal to 622080000000000
1 sign/square millisecond in circle/square week is equal to 30481920000000000
1 sign/square millisecond in circle/square month is equal to 576320670000000000
1 sign/square millisecond in circle/square year is equal to 82990176480000000000
1 sign/square millisecond in mil/square second is equal to 533333333.33
1 sign/square millisecond in mil/square millisecond is equal to 533.33
1 sign/square millisecond in mil/square microsecond is equal to 0.00053333333333333
1 sign/square millisecond in mil/square nanosecond is equal to 5.3333333333333e-10
1 sign/square millisecond in mil/square minute is equal to 1920000000000
1 sign/square millisecond in mil/square hour is equal to 6912000000000000
1 sign/square millisecond in mil/square day is equal to 3981312000000000000
1 sign/square millisecond in mil/square week is equal to 195084288000000000000
1 sign/square millisecond in mil/square month is equal to 3.688452288e+21
1 sign/square millisecond in mil/square year is equal to 5.31137129472e+23
1 sign/square millisecond in revolution/square second is equal to 83333.33
1 sign/square millisecond in revolution/square millisecond is equal to 0.083333333333333
1 sign/square millisecond in revolution/square microsecond is equal to 8.3333333333333e-8
1 sign/square millisecond in revolution/square nanosecond is equal to 8.3333333333333e-14
1 sign/square millisecond in revolution/square minute is equal to 300000000
1 sign/square millisecond in revolution/square hour is equal to 1080000000000
1 sign/square millisecond in revolution/square day is equal to 622080000000000
1 sign/square millisecond in revolution/square week is equal to 30481920000000000
1 sign/square millisecond in revolution/square month is equal to 576320670000000000
1 sign/square millisecond in revolution/square year is equal to 82990176480000000000 | {"url":"https://hextobinary.com/unit/angularacc/from/signpms2/to/signpmicros2","timestamp":"2024-11-09T19:07:57Z","content_type":"text/html","content_length":"114425","record_id":"<urn:uuid:c1f2fc86-d6ba-4e1e-b512-dfdec569cc8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00497.warc.gz"} |
PPT - N OTICING N UMERACY N OW! PowerPoint Presentation, free download - ID:2355993
1. N3: NOTICING NUMERACYNOW! RESEARCH FUNDED BY THE NATIONAL SCIENCE FOUNDATION: Transforming Undergraduate Education in STEM (TUES) Award # 1043667, 1043656, 1043831
2. About Us Preservice Teacher Preparation Collaborative * Comparison Implementers
4. Professional Noticing Attending to the children’s work Interpretingchildren’s work in context of mathematics Deciding appropriate next steps Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A.
(2010). Professional Noticing of Children’s Mathematical Thinking. Journal for Research in Mathematics Education, 41, 169-202.
5. Pedagogies of Practice Decomposition of professional noticing Representations video of early number sense diagnostic events Approximations PSETs conduct diagnostic interview with child Grossman,
P. (2011). Framework for teaching practice: A brief history of an idea. Teachers College Record. 113, 12, 2836-2843.
6. Early Numeracy Stages of Early Arithmetic Learning • Learning Progression • Early Quantitative Understanding • Examination of Counting Schemes Olive, J. (2001). Children's number sequences: An
explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11, 4-9. Steffe, L. (1992). Learning stages in the construction of the number
sequence. In J. Bideaud, C. Meljac, & J. Fischer (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 83–88). Hillsdale: Lawrence Erlbaum. Wright, R. J., Martland, J., &
Stafford, A. (2000). Early numeracy: Assessment for teaching and intervention. London: Paul Chapman Publications/Sage.
7. Early Numeracy Stages of Early Arithmetic Learning Stage 0: Emergent Counting Scheme Stage 1: Perceptual Counting Scheme Stage 2: Figurative Counting Scheme Stage 3: Initial Number Sequence Stage
4: Intermediate Number Sequence Stage 5: Facile Number Sequence
8. PRIMARY RESEARCH QUESTION To what extent can teacher educators facilitate the development of Preservice Elementary Teacher (PSET) professional noticing (attending, interpreting, and deciding) of
children’s mathematics?
9. Professional Noticing Assessment “I have seven little bears . . . But now I have too many shells. I have eleven shells. (Jon shows the eleven shells then covers them with his hand.) How many
shells am I going to have left over?”
10. Professional Noticing Prompts Please describe in detail what this child did in response to this problem. (Attending) Please explain what you learned about this child’s understanding of
mathematics.(Interpreting) Pretend that you are the teacher of this child. What problem or problems might you pose next? Provide a rationale for your choice. (Deciding) Jacobs, V. A., Lamb, L. L.
C., & Philipp, R. A. (2010)
11. Assessment Score Levels Interpreting Deciding Attending Level 4Elaborate 3Salient Accurate Appropriate & Connected 2Limited Limited Adequate, Disconnected 1Inaccurate Inaccurate Inappropriate, No
12. Growth of PSET PN: Attending “He knew that since the teacher said he had too many shells he had to do subtraction. He also knew that because the teacher said left over he had to do subtraction,
or see what the difference was. The child understood key words and phrases and understood how to take away to get the right answer. He used the bigger number and took away using the smaller
number and realized that 11-7=4.” “In response to this problem this child first counted the bears and found that there were seven. From there he used his fingers and counted up from seven until
he got to the number eleven. He had four fingers up so he said that that was his answer.” POST PRE
13. Growth of PSET PN: Interpreting “It seemed that instead of subtracting seven from eleven he used the problem 7+?=11, and came up with four by counting from seven to eleven instead of from eleven
to seven.” “This child does not count on; he needed to count the bears from one in order to count the remainder of the shells. He uses his fingers to count when materials are unavailable to him.
He understands associating one object with a number and adding a value with each corresponding object added.” POST PRE
14. Growth of PSET PN: Deciding “I would pose more bears than shells. Or only have shells exposed, so he couldn't count the bears. How many shells must I take away to get 7 bears? Other ways of
getting answer and using subtraction.” “I would screen both of the counters. This requires the student to use a different type of counters (fingers) but he might run into trouble because he will
be counting past 10. I[t] would be interesting to see how he got the answer.” POST PRE
15. Preliminary Analysis of Three Research Sites Descriptive statistics of professional noticing measures by university Results of ANOVAcomparing pre and post assessments of all universities
18. Questions? tinyurl.com/noticingnumeracynow
19. Attending Benchmarks “He counted from one up when counting all of the bears. He then counted the remaining shells on his fingers to get the answer 4.” “Counted the bears individually then used
his fingers to count up to 11.” ELABORATE SALIENT “Instead of subtracting 11-7, he counted to seven and then used his fingers to see how many more it took to get to 11.” “The child subtracted in
response to this question using his fingers as a manipulative. Starting with 11 & working backwards.” INACCURATE LIMITED
20. Attending Benchmarks “He counted from one up when counting all of the bears. He then counted the remaining shells on his fingers to get the answer 4.” “Counted the bears individually then used
his fingers to count up to 11.” 4 3 “Instead of subtracting 11-7, he counted to seven and then used his fingers to see how many more it took to get to 11.” “The child subtracted in response to
this question using his fingers as a manipulative. Starting with 11 & working backwards.” 2 1
21. Interpreting Benchmarks “This child understands a one-to-one correspondence with objects, he needs to touch the objects and he still uses his fingers to count on.” ACCURATE “I learned that this
child can add easier than subtract because instead of 7-11 he did 7+__=11. I also learned that he needs a representation of the numbers to solve the problem (the bears, his fingers, and shells).”
“I learned that the child is able to count on from a given number. He didn't have to go back and start at 1.” INACCURATE LIMITED
22. Interpreting Benchmarks “This child understands a one-to-one correspondence with objects, he needs to touch the objects and he still uses his fingers to count on.” 3 “I learned that this child
can add easier than subtract because instead of 7-11 he did 7+__=11. I also learned that he needs a representation of the numbers to solve the problem (the bears, his fingers, and shells).” 1 2
“I learned that the child is able to count on from a given number. He didn't have to go back and start at 1.”
23. Deciding Benchmarks Appropriate Decision with… “I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of
remainders. This will tell us if he has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained
their logic.” “I believe that the next task should be a really small number subtracted by a very large number. Ex. 20-6. This problem would be harder to count on your hands and you could get a
better understanding of his conceptual knowledge of the problem and addition itself.” Connected Rationale Adequate Decision with… Inappropriate Decision with… Littleor No Rationale Disconnected
24. Deciding Benchmarks “I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of remainders. This will tell us if he
has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained their logic.” “I believe that the
next task should be a really small number subtracted by a very large number. Ex. 20-6. This problem would be harder to count on your hands and you could get a better understanding of his
conceptual knowledge of the problem and addition itself.” Appropriate & Connected Adequate, Disconnected Inappropriate, Disconnected
25. Deciding Benchmarks “I would ask the child to tell me why there were four shells leftover. This would tell us whether or not the child had an understanding of remainders. This will tell us if he
has the concept of sharing equally, rather than giving the four shells to select bears.” “I might say "How did you get this answer" to see how they explained their logic.” “I believe that the
next task should be a really small number subtracted by a very large number. Ex. 20-6. This problem would be harder to count on your hands and you could get a better understanding of his
conceptual knowledge of the problem and addition itself.” 3 2 1 | {"url":"https://www.slideserve.com/noah/n-oticing-n-umeracy-n-ow","timestamp":"2024-11-04T20:30:02Z","content_type":"text/html","content_length":"72943","record_id":"<urn:uuid:44d96766-93fa-4367-965a-5f80a63dd3f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00734.warc.gz"} |
Nstochastic partial differential equations pdf
Pdf stochastic partial differential equations in control of structures. Analysis and computations publishes the highest quality articles, presenting significant new developments in the theory and
applications at the crossroads of stochastic analysis, partial differential equations and scientific computing. A tutorial a vigre minicourse on stochastic partial differential equations held by the
department of mathematics the university of utah may 819, 2006 davar khoshnevisan abstract. Invariant manifolds for stochastic partial differential equations 5 in order to apply the random dynamical
systems techniques, we introduce a coordinate transform converting conjugately a stochastic partial differential equation into an in. Simulation of stochastic partial differential equations using
finite element methods andrea barth and annika lang abstract. Gawarecki kettering university nsfcbms conference analysis of stochastic partial differential equations based on joint work with.
And it was the same when, if you remember how we solved ordinary differential equations or partial differential equations, most of the time there is no good guess. Stochastic partial differential
equation based modeling of large spacetime data sets article pdf available in journal of the royal statistical society 771 march 2014 with 152 reads. Abstract we give a survey of the developments in
the theory of backward stochastic di. This is in contrast with the abundance of research see e. Stochastic partial differential equations spdes generalize partial differential equations via random
force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. Stochastic differential equations 5th ed b oksendal pdf.
Stochastic partial differential equations with unbounded and.
A variety of methods, such as numerical analysis, homogenization, measuretheoretical analysis, entropy analysis, weak convergence analysis, fourier analysis, and itos calculus, are further.
Stochastic analysis and partial differential equations. Solving stochastic partial differential equations as stochastic differential equations in in. Stochastic differential equations, backward sdes,
partial. General results obtained by a stochastic multiscale analysis. This chapter provides su cient preparation for learning more advanced theory. The usefulness of linear equations is that we can
actually solve these equations unlike general nonlinear differential equations. The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear
stochastic dynamical systems of the mckean vlasov type. Spdes are one of the main research directions in probability theory with several wide ranging applications. In this paper, we generalize to
gaussian volterra processes the existence and uniqueness of solutions for a class of non linear backward stochastic differential equations bsde and we establish the relation between the non linear
bsde and the partial differential equation pde. Pdf splitting up method in the context of stochastic pde. The solution is a stochastic distribution process given explicitly.
This kind of equations will be analyzed in the next section. Stochastic partial differential equations citeseerx. Exact controllability of stochastic transport equations 3. Numerical methods for
stochastic partial differential equations and their control max gunzburger department of scienti. However, one may rewrite it as an integral equation, and then show that in this form there is a
solution which is a continuous, though nondifferentiable, function. Stochastic partial differential equations and related fields 1014october2016 faculty of mathematics bielefeld university
supportedby. With that in mind, the present volume follows the style of the utah minicourse in spdes. A minicourse on stochastic partial differential equations. We study existence and uniqueness of a
variational solution in terms of stochastic variational inequalities svi to stochastic nonlinear diffusion equations with a highly singular.
A regularity result for quasilinear stochastic partial differential. A minicourse on stochastic partial di erential equations. Stochastic partial differential equation wikipedia. A comparison theorem
for the solution of the bsde is proved and the continuity of its law is. Effective dynamics of stochastic partial differential equations focuses on stochastic partial differential equations with slow
and fast time scales, or large and small spatial scales. This book provides an introduction to the theory of stochastic partial differential equations spdes of evolutionary type. Stochastic partial
differential equations a modeling, white noise.
The chief aim here is to get to the heart of the matter quickly. Numerical solution of stochastic differential equations. These notes are based on a series of lectures given first at the university
of warwick in spring 2008 and then at the courant institute in spring 2009. In the case of the subdifferential of the indicator of a convex set, we obtain one way to construct an sde re. It is an
attempt to give a reasonably selfcontained presentation of the basic theory of stochastic partial differential equations, taking for granted basic measure theory, functional analysis and probability
theory. These are supplementary notes for three introductory lectures on. Pdf stochastic partial differential equation based. Introduction let wr o be the space of all continuous functions w wktr k1
from 1 o,t to rr, which vanish at zero. Chapter 4 starts with sdes with a multivalued drift, which can be, for instance, the subdifferential of a convex function. While the solutions to ordinary
stochastic differential equations are in general holder continuous in time.
Stochastic differential equations sdes including the geometric brownian motion are widely used in natural sciences and engineering. The stochastic heat equation is then the stochastic partial
differential. Theory and applications of stochastic pdes institute for. The first edition of stochastic partial differential equations. It is a particular challenge to develop tools to construct
solutions, prove robustness of approximation schemes, and study properties like ergodicity and fluctuation statistics for a wide. Spdes also arise when considering deterministic models. Effective
dynamics of stochastic partial differential. Stochastic partial differential equation stochastic partial differential equations spdes are similar to ordinary stochastic differential. Given some
stochastic differential equation, i dont know how to say that you should start with this kind of function, this kind of function.
In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Many types of dynamics with stochastic influence in nature or manmade complex. These
notes describe numerical issues that may arise when implementing a simulation method for a stochastic partial di erential equation. Stochastic processes and partial differential equations.
Andreaseberlebonn,martingrothauskaiserslautern,walterhohbielefeld. In finance they are used to model movements of risky asset prices and interest rates. We consider a quasilinear parabolic stochastic
partial differential equation driven by a multiplicative noise and study. Analysis and numerical approximations arnulf jentzen september 14, 2015. Topics from partial differential equations include
kinetic equations, hyperbolic conservation laws, navierstokes equations, and hamiltonjacobi equations. An introduction to numerical methods for the solutions of. Stochastic di erential equations
provide a link between probability theory and the much older and more developed elds of ordinary and partial di erential equations. Stochastic differential equations mit opencourseware.
Stochastic partial differential equations appear in several different applications. The pair wr o,p is usually called rdimensional wiener space. We achieve this by studying a few concrete equations
only. We consider a class of neutral stochastic partial differential equations with infinite delay in real separable hilbert spaces. Migration function or reverse time imaging function, or least. Pdf
on mar 1, 20, arnaud debussche and others published stochastic partial differential equations. Since the aim was to present most of the material covered in these notes during a. Connections,
curvature, and characteristic classes graduate texts in mathematics book 275 loring w. A primer on stochastic partial di erential equations. Recent years have seen an explosion of interest in
stochastic partial differential equations where the driving noise is discontinuous. Moreover, the theory of systems of first order partial differential equations has a significant interaction with
lie theory and with the work of e. They have relevance to quantum field theory and statistical mechanics. Prove that if b is brownian motion, then b is brownian bridge, where. An introduction to
stochastic pdes of martin hairer.
As a result, as we will see, the theory of nonlinear spdes driven by spacetime white noise, and with second order pde operators, is limited to the case of a one. Prove that if b is brownian motion,
then b is brownian bridge, where bx. Stochastic partial differential equations spdes are the mathematical tool of choice to model many physical, biological and economic systems subject to the
influence of noise, be it intrinsic modelling uncertainties, inherent features of the theory. Typically, these problems require numerical methods to obtain a solution and therefore the course focuses
on basic understanding of stochastic and partial di erential equations to construct reliable and e cient computational methods. Among the primary intersections are the disciplines of statistical
physics, fluid dynamics, financial modeling, nonlinear filtering, superprocesses, continuum physics and, recently, uncertainty quantification. A really careful treatment assumes the students
familiarity with probability theory, measure theory, ordinary di. In this article, using dipernalions theory \citedili, we investigate linear second order stochastic partial differential equations
with. An introduction to stochastic partial differential equations. Stochastic partial differential equations and related fields. Introduction to an introduction to stochastic partial differential | {"url":"https://credinovprag.web.app/26.html","timestamp":"2024-11-06T23:28:47Z","content_type":"text/html","content_length":"15876","record_id":"<urn:uuid:5d83ba95-88f0-40a8-a009-79ed7a934064>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00889.warc.gz"} |
Proof of the Pythagorean Theorem
This month, I decided I would focus on one of the most famous theorems in all of mathematics: The Pythagorean Theorem. Though it sounds kind of dull on the face, it has a lot of extremely interesting
things within it.
First, let me go over what it is. If you take a right triangle, and label the measure of its shortest side a, its middle side b, and its longest side c, then you can count on the fact that:
a^2 + b^2 = c^2
This can be used in geometry problems, like to find the longest side, or the hypotenuse, of a triangle with shorter sides, or legs, 3 and 4. You would simply plug 3 in for a, 4 in for b, and solve
for c.
(3)^2 + (4)^2 = c^2
9 + 16 = c^2
25 = c^2
5 = c
The first thing I wondered when seeing this theorem is why it is true. It seems odd that any right triangle’s measurements must fit this criteria. And I never found a proof I liked until extremely
recently. So, I would like to share it.
Take the following diagram:
We are going to try to figure out the area of this square. The simplest way to do it would probably be to square the square’s side. This length would be a + b.
(a + b)^2
a^2 + 2ab + b^2
Another way we could get this result is to find the area of the inside square, and then the area of the four triangles surrounding it. The inside square has length c, and the outer triangles have a
base of b and height of a. So, we would get:
c^2 + 4(ab/2)
c^2 + 2ab
Since these two quantities are both the area of the same square, we can set them equal to each other. After simplification, we would get:
a^2 + 2ab + b^2 = c^2 + 2ab
a^2 + b^2 = c^2
And we end up with our Pythagorean Theorem. I was so intrigued to see that this proof was so quick and simple, as well as pretty interesting. | {"url":"https://coolmathstuff123.blogspot.com/2013/05/proof-of-pythagorean-theorem.html","timestamp":"2024-11-03T22:28:59Z","content_type":"text/html","content_length":"73572","record_id":"<urn:uuid:ba31a8bd-3c91-4c6d-ac2e-69ac955bf620>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00455.warc.gz"} |
Peter Filzmoser
Peter Filzmoser
Robust joint modeling of mean and dispersion through trimming:
(N.M. Neykov, P. Filzmoser, and P.N. Neytchev)
R code and information
Imputation of missing values for compositional data using classical and robust methods:
(K. Hron, M. Templ, and P. Filzmoser)
robCompositions_1.0.tar.gz R-package `robCompositions' (tar.gz file for Linux, Mac) for running the procedures according to the paper (preprint).
A context-sensitive method for robust model selection with application to analyzing success factors of communities:
(A. Alfons, W.E. Baaske, P. Filzmoser, W. Mader, and R. Wieser)
csselect.R R-program for robust model selection according to the paper (preprint). The program B-RLARS.R is also needed.
Robust factor analysis for compositional data:
(together with K. Hron, C. Reimann, R.G. Garrett)
CAGeo08Figs.R R-scripts for generating all figures in the paper. One also needs the programs pfa1.R and factanal.fit.principal1.R .
Outlier detection for compositional data using robust methods:
(together with K. Hron)
progsOutComp.R R-scripts for generating all figures in the paper. One also needs the programs alr.R , invalr.R , iso_new.R , inviso_new.R , and drawMahal.R .
(together with Christophe Croux, Belgium)
Program for the robust additive and multiplicative fit of two-way tables.
The program is written in SPlus, and it needs the functions prcomp.rob (robust principal component analysis) and weight.wl1 (calculating the weights for weighted L1-regression). For drawing
robust biplots of a "twoway.rob"-object, the function twoway.bip can be used.
(together with C. Croux, G. Pison, P.J. Rousseeuw; Belgium)
Program for "Fitting Multiplicative Models by Robust Alternating Regression".
The program is written in SPlus, and it needs the functions prcomp.rob (robust principal component analysis), weight.wl1 (calculating the weights for weighted L1-regression), and ccrfit (robust
fit of twoway-tables). For drawing screeplots, the functions screeplot.faccr or screeplot.fanova can be used.
Robust canonical correlation analysis (CCA):
(together with J. Branco, C. Croux, R. Oliveira)
cc.ssc Splus program for CCA based on a robust covariance matrix estimation; uses mesthub (M estimator)
pp.ssc Splus program for robust CCA based on projection pursuit
cancor.rar R program for CCA based on robust alternating regressions
simcc.ssc Splus program for doing simulations | {"url":"http://file.statistik.tuwien.ac.at/filz/programs.html","timestamp":"2024-11-02T07:39:27Z","content_type":"text/html","content_length":"6023","record_id":"<urn:uuid:f32e11fa-04c8-44f9-88ad-6d44ffe51ea8>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00895.warc.gz"} |
What are the basic concepts in quantitative finance?
Quantitative finance involves the use of mathematical models and statistical analysis to understand and predict financial markets. Some basic concepts in quantitative finance include pricing models
such as the Black-Scholes model for options, risk management techniques like Value at Risk (VaR) calculations, and portfolio optimization strategies such as Modern Portfolio Theory. Other key
concepts include time value of money, probability theory, and regression analysis. These tools and concepts help financial professionals make informed decisions about investing, hedging, and managing
risk in the financial markets.
This mind map was published on 2 April 2024 and has been viewed 133 times. | {"url":"https://www.coolmindmaps.com/?action=mindmap&question=What+are+the+basic+concepts+in+quantitative+finance%3F&code=bd9ff093ff979f0c6af5f8317f22964d","timestamp":"2024-11-06T08:16:31Z","content_type":"text/html","content_length":"27836","record_id":"<urn:uuid:df1fb66a-e1d1-4c2c-bba8-4cce5ec8e432>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00020.warc.gz"} |
Project Euler Solutions
\documentclass[11pt]{article} \usepackage{hyperref} \usepackage{microtype} \usepackage{amsmath,amsfonts} %Gummi|065|=) \title{\textbf{Explanation to the first thirty problems from Project Euler with
Python 3}} \author{Max Halford} \date{} \begin{document}\sloppy \maketitle Project Euler is a good way to learn basic number theory, to get your imagination going and to learn a new programming
language. If you can solve the first hundred problems then you can solve any problem, as long as you keep being curious and you use your imagination, personally I decided to work on other styles of
projects, there isn't just number theory out there! However solving number theory problems is a good way to learn a programming language : you need to be rigorous and tidy. Most solutions require
smart algorithms and not brute force approaches. Google is your best friend when you know what to do but you don't know how to write it, or when you don't understand the code you're reading. I will
not babysit the reader but to the contrary assume that he knows how to google "Python $<$insert command$>$", I am not being harsh : knowing how to find documentation when coding is of the utmost
importance because you mostly have to teach yourself notions, you have to be self-educated. I would recommend approaching challenging problems with pen and paper and maybe some mathematical research.
The association between human ingenuity and the computational power of our machines can produce wonderful results, as long as one doesn't lean on the latter. Python provides many coding styles and
paradigms, googling "PEP 8" and "Google Python Style Guide" will make pick up good habits early on. Keep your code simple, it has to be readable by everyone. Assign comprehensible names to your
variables and parameters. Don't reinvent the wheel, the Python community is very large and modules have been written for a lot of things, use them. Internet isn't always right, for example one liners
are not a good thing : they are difficult to read afterwards. In a perfect world reading your code should feel like reading a book, keep that in mind. Don't be frightened to go on internetand find
the solutions to problems, it's actually counterproductive to search for answers for too long, time is precious and there are too many things to cover. All the code is available here: \begin{center}
https://github.com/MaxHalford/Project-Euler \end{center} \tableofcontents{} \clearpage \section{Multiples of 3 and 5} A question that often comes up with Python is "What is the best way to go through
a list?". Problematically there are many answers and it all comes down to personal preferences. The two main choices are Map-Reduce-Filter-Lambda \url{http://www.python-course.eu/lambda.php} and list
comprehensions \url{http://www.python-course.eu/list\_comprehension.php}. I would suggest reading up on both approaches, it is always good to know what tools are at your disposal without having to be
a black belt at them. I mostly use list comprehensions, they are, in my opinion, more comprehensible and flexible. The problem is straightforward with a list comprehension : build a list bounded by 1
and 999 with elements being divible by 3 or 5 and sum up all the elements. \section{Even Fibonacci numbers} When you read a problem for the first time do as much research as you need on the topics of
the problem. In this case Fibonacci numbers are relatively famous but will find some rather less famous concepts later that will require some looking around. After some pondering one notices that a
number in the Fibonacci sequence is only defined by the two numbers preceding it. What results from this observation is that it is useless to store the sequence, but instead we should go through it
and pick the numbers we want (in this case they have to be even). We use the property of the sequence in the following algorithm : \bigskip \\ $\bullet$ Let $\alpha$ and $\beta$ be adjacent numbers
in the sequence. \\ $\bullet$ $\beta'$ becomes the following number in the sequence : $\alpha + \beta$. \\ $\bullet$ $\alpha'$ becomes $\beta$. \section{Largest prime factor} It is a well known
theorem that any number can be written as the product of prime numbers. Hence any number divided by all its factors successively will return 1. So what we can do is, for a given integer \emph{n}, go
through every integer \emph{k} inferior to it and superior to 1 and check if $n\bmod k = 0$. If it isn't then increment \emph{k} and try again. When it is then \emph{k} is a factor of \emph{n}, so we
store it and start over the same process for $n/k$. The neat thing with this algorithm is that you do not have to check if \emph{k} is prime. Indeed \emph{k} will be checked before $2k,3k,4k$ \emph
{etcetera}. \section{Largest palindrome product} The easiest way to check if a number, or any string for that matter, is a palindrome, is to compare it with its reverse. I would suggest reading \url
{http://forums.udacity.com/questions/2017002/python-101-unit-1-understanding-indices-and-slicing} to learn useful dodges instead of reinventing the wheel. Now that we have a function to check if a
number is palindrome, we need to apply it on every product of two 3-digit numbers in the descending order (because we are looking for the biggest palindrome), hence the iteration through $\{999,\
dots,100\} \times \{999,\dots,100\}$. Notice how easily a two dimensional matrix is defined in one line with a list comprehension. \section{Smallest multiple} Recursion is a concept that can not be
avoided when programming, I would suggest googling it if you are not familiar with it. It is easy enough to check for a given integer \emph{n} from 1 till \emph{k} (in our case 20), we simply go
through the list and check the divisibility. Now we have to find an integer that satisfies the previous algorithm for $k = 20$. A brute force approach is to iterate every integer one by one, but we
quickly realize that the algorithm takes for ever. It takes some insight to notice that the integer \emph{n} that verifies the algorithm for \emph{k} is a factor of the integer \emph{m}, hence when
looking for \emph{m} we can increment by \emph{n} instead of 1. However efficient this insight is, it is not intuitive, try some cases on a piece of paper. An example that comes to mind is this : 6
is divisible by 1, 2 and 3. 24 is divisible by 1, 2, 3 and 4. 6 is a factor of 24 hence we only look for the candidates 12, 18 and 24 when searching for the smallest integer divisible by 1, 2, 3 and
4. \section{Sum square difference} I'll be honest, if you don't konw how to do this problem, it is either than you need to learn the basics of programming or that you're not motivated. This is good
opportunity to learn some more list comprehension synthax. The solution to this problem is straightforward, indeed we only have to iterate through a list of integers and do some basic operations on
them. \section{10001st prime} I would strongly recommend reading up on prime number theory, it is a core element of number theory and it is not an easy concept to deal with in computer science
(actually that's the reason why it is used in cryptography). Indeed, the algorithms used to deal with primes require a lot of power, even without brute force. Again, there are many to solve this
problem, the upper bound (10001) is relatively low so we can use an unsophisticated algorithm. To check if an integer is prime, one can go through every integer in \{$2,\dots, \sqrt{n}$\} and check
if it divides $n$ (going above $\sqrt{n}$ is pointless because you've implicitly checked them when going through the smaller integers). Now that we have a tool to check if an integer is a prime we
can through the uneven integers (even numbers apart from 2 are not prime) and use the tool. Once we have 10001 primes we return the last one. On a sidenote, storing the primes is overkill, it's fine
for 10001 primes but you might want to change the script (just like in Problem 2) to only store the latest prime. \section{Largest product in a series} This is a good exercice to master ranges in
Python (the last element of the range not being included is an easy concept to forget at first). First of all we have to copy/paste the number into a Python script and edit it so that it's considered
a prime. In my solution I quoted it and made it readible by adding '/' at the end of the lines (Python thinks it will be one single continuous line). You could also use \texttt{list(map(int,str(n)))}
to tranform $n$ into a list. The easiest way to find a maximum is to iterate through a list and compare to an initial value, if we find a bigger "thing" we replace the initial value by the "thing".
In our case the "thing" is a thirteen elements product. Thus we need two loops, one going through every element $i$, the other going from $i$ to $i+12$. \section{Special Pythagorean triplet} It is
always important to know just how many steps are needed to get to the answer. In this case there are two : firstly imposing two criterias on an integer triplet and secondly computing pythagorean
triplets. However, after a bit of coding and reflexion we should notice : if we have $a$ and $b$, we automatically have $c$ because the three have to sum to 1000! Also, the Pythagoras theorem ($a^{2}
+b^{2}=c^{2}$) imposes on $c$ to be bigger than $a$ and $b$ (think of a triangle and its sides and the previous statement will seem clear). In mathematical terms the first observation says that $c=
1000-a-b$. The second says that $a$ and $b$ can only be as big as 500 (half a thousand). Let's prove this statement \emph{ad absurdum}, if $a > 500$ and $c > a$, then $a+c>1000$, which we don't want.
Finally we can notice that $a$ and $b$ are interchangeable (by commutativity of the sum), thus we can iterate through $\{a,\dots,500\}$ for $b$ to avoid repeating triplets. Now that we have
understood all this the coding is simple, we go through appropriate triplets and check if the triplets verify $a^{2}+b^{2}=c^{2}$. \section{Summation of primes} This problem is the perfect example
for using the sieve of Eratosthenes, which is an efficient algorithm to find all the prime numbers up to a given bound. It uses the following intuition : if we know for certain that $k$ is prime,
then all the multiples of $k$ are not prime. We can use this property in the following way : \bigskip \\ $\bullet$ Take a list of $n$ consecutive booleans set to \texttt{True} (2 million in our
case). \\ $\bullet$ Go through every element. \\ $\bullet$ If it is $True$ (prime) then mark all its multiples to $False$. \\ $\bullet$ If it is $False$ (not prime) then check the next element. \
bigskip \\Finally we end up with of a list of $True$ and $Falses$, sum up the integers that are $True$. \section{Largest product in a grid} Some problems don't require elegant dodges but simply clean
algorithms, this is one of them. First of all there are four possible products : a column ($\downarrow$), a line ($\rightarrow$) and two diagonals ($\searrow$ and $\nearrow$). I didn't add the
symbols to be pedantic : the way they point is the way I multiplied the cells of the grid, just "go the way" that seems most natural to you. What we want to do is straightforward, we compute every
kind of product for each cell of the grid. However before doing the previous the given grid has to be inserted into a two-dimensional array, first by copy/pasting and converting to string the given
grid then by adding every line to an array. We also have to be careful not to try and compute inapropriate products, for example a line product on the last cell of a line of the grid. I found that
getting the array into place was harder than finding the largest product, mastering string operations is crucial. In my solution there are three string operations : \bigskip \\ $\bullet$ \texttt
{.strip()} removes the implicit carriage returns at the end of every line. \\ $\bullet$ \texttt{.splitlines()} converts a multiline string into individual strings. \\ $\bullet$ \texttt{.split()}
separates a string into a list of strings based on a given separator. \section{Highly divisible triangular number} It is fairly obvious that the $nth$ triangular number is the sum of an arithmetic
sequence : $n(n+1)/2$. We can make two observations : one and only one of $n$ and $n+1$ is divisible by 2 and they don't share any prime factors. Thus we can write : \begin{center} $n=\displaystyle\
prod_{i=1}^{s}{p_{i}^{j_{i}}}$ and $n+1=\displaystyle\prod_{i=1}^{t}{q_{i}^{k_{i}}}$ where $p$ and $q$ are primes. \bigskip \\Thus the number of factors of a triangle number is: $(j_{1})(j_{2}+1)\
dots(j_{s}+1)(k_{1}+1)(k_{2}+1)\dots(k_{t}+1)=j_{1}\displaystyle\prod_{i=2}^{s}(j_{i}+1)\displaystyle\prod_{i=1}^{t}(k_{i}+1)$ \end{center} I realize that the previous equation is a lot to take in so
I'll explain. First of all, if an integer $n$ has $s$ prime factors $p$, then any product of these primes (each to any power from 0 to $j$) is a factor of $n$ (this is the most important insight to
understand). The previous statement induces that the number of factors of an integer is $(j_{1}+1)(j_{2}+1)\dots(j_{s}+1)$. The reason for which there is a "+ 1" is that we have to include 0 to the
powers of the primes. However a triangular number is equal to $n(n+1)/2$, not to $n(n+1)$, which means that we have to neglect a power of two in the factorization of $n$ or $n+1$ (depending on which
is even. This explains why the first element of the product is $(j_{1})$ and not $(j_{1}+1)$. \\Now that we have the theory we can put it into code. Firstly we need an algorithm to give the number of
divisors of a given integer $n$. My algorithm is a bit complicated but it exactly translates what was said above. We begin by dividing by 2 $n$ until it is odd and count how many times we did it ($j$
or $k$ in the previous paragraph). We then divide $n$ (which is now odd) by its odd prime factors until it equals 1, in this process we also count how many times we can divide $n$ by the odd number.
The code \texttt{divisors = divisors * (count + 1)} is our insight translated to code : we multiply the number of divisors we have by the number of times we can divide $n$ by one of its prime
factors. Next we have to iterate through every integer until $\texttt{numDivisors(n)}\times\texttt{numDivisors(n+1)}$ is over 500. Finally our answer is simply $n(n+1)/2$. \\This is the first problem
where we looked at our problem under a different angle. We thought about what equation represents a triangular numbers and from there worked on the equation, which is a subtle difference and is often
the case when solving hard problems. \section{Large sum} We can use the code used in Problem 11. First of all we copy/paste the given number, strip the carriage returns. Then we convert every line to
an integer which we append to an array. Finally, after summing the array and converting the sum (an integer) to string, we can simply get the last 10 digits by \emph{slicing} it. Slicing in Python is
an important tool to master, it makes string operations very simple, if you don't understand it by now go back to the link in Problem 4. \section{Longest Collatz sequence} The Collatz Problem is a
fascinating, unproved, conjecture. Let's say that $C(n)$ returns a sequence of integers, beginning with $n$ and ending with $1$, every integer following the rules given by the Collatz sequence : \
begin{center} $\bullet$ $n$ becomes $n/2$ if $n$ is even. \\$\bullet$ $n$ becomes $3n+1$ if $n$ is odd. \end{center} We are looking for the biggest $C(n)$ (denoted $|C(n)|$) for $n$ in $\{1,\
dots,1000000\}$. Every $n$ returns a unique Collatz sequence so we have to check each $C(n)$. The first thing that comes to mind is to compute the length of each $C(n)$ for each $n$, however this
takes a \emph{very} long time. Consider the following insight : the first element of $C(10)$ is 5, yet we have already computed $C(5)$, thus $|C(10)|=|C(5)|+1$. This is wonderful, when computing $C
(n)$ we can stop once it returns an element $k$ for which $C(k)$ has already been computed. We then have $|C(n)|=|C(k)|+l$ where $l$ is number of elements between $n$ and $k$. This is principle is
called \emph{memoization}, it avoid repeating steps already computed. It finds applications in everything that is \emph{recursive}. For example when recursively computing \texttt{factorial(n)} we can
stop if we computed \texttt{factorial(n-k)}, indeed $n!=\frac{n!}{(n-k)!} \times (n-k)!$. \\ Back to the problem, the memoization doesn't change much to the brute force algorithm described
previously. Only now before changing $n$ we have to if $n$ is in the keys of a dictionary (also known as a hash) instantiated formerly. If it is then we can stop and we have the length of the Collatz
sequence beginning with $n$. If it isn't then we keep going. In any case when the length of the sequence is found we append it to the dictionary so that we don't have to the computation again if we
find $n$ in another sequence. \section{Lattice paths} There are two ways of solving this problem. On the one hand we could use a brute force recursive algorithm by computing the number of routes for
subgrids and work our way up to the $20 \times 20$ grid. On the other hand a bit of peripheral vision tells us that this is a famous problem in combinatorics. Take a grid of size $n \times m$, it
makes sense that to go from top-left to bottom-right you have to go down $n$ times and right $m$ times. The real insight is to notice that if all the movements in one direction are imposed, then
there is no more liberty for the movements in the other direction : they are predetermined. But this is also with a mix of both movements, as long as the mix involves half of all the movements
necessary to go from one point to another. Indeed on a $20 \times 20$ grid, if 5 movements downward and 5 movements to the right are imposed, then the other 10 are predetermined (try it out for
yourself to convince yourself). Basically we are looking for how many ways there are to arrange $(n+m)/2$ among $n+m$ possible movements (round $(n+m)/2$ up or down if it is not whole, it doesn't
matter). In combinatorics this is denoted $n+m \choose {(n+m)/2}$ which is $(n+m)! \over {((n+m)/2)((n+m)/2)}$. In our case the grid is square of size 20, which makes things cleaner (no problems with
$(n+m)/2$ being odd). The answer is simply $40! \over {20!20!}$. This solution is faster than the first way of solving the problem, however some may argue that it not the computer science way of
doing it. I agree, however having a big toolkit will spare you many problems, thus having a peripheral vision is good. You don't have to be a black belt at everything, that's impossible, however
knowing what tools exist is feasible. If you know what tool to use in what case or you know in what domain to look into then you can do some research and solve problems elegantly. \section{Power
digit sum} I wouldn't say there is anything nifty with this problem. One simply has to convert $2^{1000}$ to string and iterate through its digits, simple as. \section{Number letter counts} Now this
is the kind of problem where modular arithmetic becomes handy. Say you want to strip a number $n$ it of its digits above the units digit, well the clean way to do so it to calculate $n\bmod 10$. To
get the tens digit calculate $(n\bmod 100)-(n\bmod 10) \over 10$, indeed for digits above the units digit you have to remove all the digits below it if you want it on its own, you also need to divide
it by its amplitude to remove all the zeroes that you will then have. Once we have mastered this the problem becomes quite simple. We iterate from in $\{1..999\}$ and do some modular operations to
get the units, tens and hundreds digits. We now have to use common sense and check some conditions. Is there any hundreds digit? Is there an "and" in the number? The first question is
straightforward. For the second one only has to check if the number contains a tens or a units digit. The last question is a bit trickier and is related to the nature of the english language. Indeed
numbers from eleven to nineteen are special because they each have a unique name, all the others are preceded by the name of the amplitude of the tens digit (twenty, thirty, \emph{etcetera}). Finally
we have to add the number of letters of 1000, computing it in the loop wouldn't have made sense since it was the only number with a thousands digit. \section{Maximum path sum I} If you remember the
labyrinth games you played when you were young and the fastest way to win at them you'll quickly solve this problem. The fastest way to win wasn't to try out every path from the start, but rather to
follow the path from the end and see what path lead to it from the start. The same idea applies here, we won't try out every path from the path but go from bottom to top. Every node has two child
nodes, we can just add the value of the biggest child node to the value of the node we're looking at. The last row doesn't have any child node so we can start at the penultimate row. For the
algorithm itself to work we first have to put the tree in shape, I used a list comprehension to convert the copy/pasted string form of the tree to a list of lists. Once we an array we can iterate
through it starting from the penultimate row and working our way up with the use of \texttt{range(len(T) - 2, -1, -1)}. The first element of the range is the penultimate row index, the second if the
first row (we put \texttt{-1} because \texttt{range()} is not inclusive on the last index), the third simply gives the way to iterate through the range. On each row we increment each element by its
highest child node. \section{Counting Sundays} Of the first problems this is the one I had the most fun doing. Just like Problem 17 it connects with the real world. What I mean is that calendars and
languages are human inventions, they are not intrinsic to our universe, yet we can still manage to deal with them by adapting and creating mathematical tools. If you're curious there is a lot of
content on the topic of calendar and mathematics, you could start by reading \url{http://www.wikipedia.com/en/Determination_of_the_day_of_the_week}. First of all we can notice that a 7 (numbers of
days in a week) doesn't divide 365 nor 366 (leap year), thus the first day of a year will not be the same as the year before. In fact we can predict which day it will be next with modular
arithmetics. Indeed $365 \equiv 1 \pmod 7$ and $366 \equiv 2 \pmod 7$. Now we know that a leap year starting on a Monday will be followed by a year beginning by a Wednesday. We can use the same
method to calculate the first day of next month, for example if August begins with a Monday then September will begin with a Thursday because $31 \equiv 3 \pmod 7$. Last and not least we need a
function to check for leap years. If a year is divisible by 4 it is a leap year, except if it divisible by 100 and not by 400. We now have all the tools we need. Let's sumply iterate through each
year and each and modify a flag value with modular arithmetic, each time the flag value becomes a Sunday we increment a counter. \section{Factorial digit sum} If you managed to solve Problem 16 this
one will not be an issue providing you know how to compute the factorial of a number. \section{Amicable numbers} I didn't anything nifty for this solution, it's plain brute force. However the time to
solve the problem is sub ten seconds so it's not a problem. We only need a function to return the sum of the divisors of a number. Once we do we can iterate through $\{1..10000\}$, compute
$sumDivisors(n)$ and compare it to $sumDivisors(sumDivisors(n))$. In others we compute the sum of the divisors of the sum of the divisors of a given number. I imagine that you could faster than y
script, maybe with a better $sumDivisors()$ function. \section{Names scores} Like many problems this one is all about successive simple steps: open the file, put the names in a list, sort them,
compute scores. The \emph{os} module in Python enables us to choose a directory, for example I put a relative path in the code. I'm not sure if this is true for every operating system but on Ubuntu a
script will look for files in the directory it is placed in, so that we don't have to it where to look. Once you have figured this out simple use \texttt{open()} and \texttt{read()} to get the names
as a long string. I then used a list comprehension to split the names at commas whilst stripping the names of the brackets surrounding them, but you can do it another way if it doesn't seem natural
to you. I think that list comprehensions are powerful and succint, providing you have gotten used to them. The computer doesn't know what value we have assigned to each letter of the alphabet so we
have to tell him. The best way to do so it to create a dictionary containing these values. Now we can iterate through every name, iterate through every letter and multiply the score of a name by its
position in the names list (by checking where we are in the loop). \section{Non-abundant sums} I believe the best way to approach this problem is to proceed in two steps. First we need to find all
the abundant number under 28123. Then we can find every sum of abundant numbers possible and to the following: if a sum of abundant numbers is inferior or equal 28123 then we store it into a sum
called $S$, then we can calculate $S'=\displaystyle\sum_{i=1}^{28123}i$ (using the arithmetic sum formula) and $S'-S$ will be our answer. In other words, we are using set theory: we calculate the sum
of all the numbers below 28124, then we calculate the sum of the numbers below 28124 that can be written as the sum of two abundant numbers and calculate the difference between both sums. This
explains why we only compute abundant numbers below 28123: the sum of two abundant numbers above 28123 is already known to exist so we can discard them. The \texttt{divisors(n)} function I used is
more efficient then the one in Problem 21. It uses the fact that if we know that $k$ divides $n$, then $n/k$ divides $n$, provided $k \ne n/k$. \section{Lexicographic permutations} We have to do some
thinking before delving into this one. Let us look at the example given at \url{https://projecteuler.net/problem=24}. Let us say we are looking for the $5^{th}$ permutation, well that would be 201.
You can think of each pair of permutations as three families. The first one being $(012, 021)$, the second $(102, 120)$ and the last $(201, 210)$. We can see the $5^{th}$ permutation as the $1^{st}$
element of the $3^{rd}$ family. Let that sink in if it has not already. To find this mathematically is equivalent to perfoming a modular operation: $\lfloor{objective \over (n-1)!}\rfloor$ where $n$
is the number of elements considered in the permutations (here it is 3). Let us see how it works out for the previous example: $\lfloor{5 \over (3-1)!}\rfloor=2$, which means that our objective is in
the third family ($2+1=3)$, in other the first two families contain the first four permutations. We can keep going and apply the same operation to the remainder of the euclidian division we performed
(1). However now we know that the first of the permutation is fixed because we know what family it is in (the one that starts with 2), thus we decrement $n$: $\lfloor{1 \over 1!}\rfloor=1$. This does
not mean that our objective is the first element of the second family, it means that our objective is in the first subfamily of the family (and not the second because $1 \equiv 0 \pmod 1$, which was
not the case in the first part of the loop). However the subfamily $(201)$ is only one element, thus it is the answer. If it was not on its own we would have to keep going. I will recapitulate by
describing the algorithm. First we compute the euclidian division of our initial objective by the permutations of the numbers of element $n$ minus 1 ($n!$ is obviously bigger then any position we
could be looking for). We now the first digit of the permutation we are looking for because $(n-1)!$ fits that many times in our objective. We now have a remainder which is the number of elements we
have to look into the same way. In other words it is our new objective. We know part of where our objective lies and we have to delve into this family of permutations until we find a single
possibility, the final leaf on the tree. \section{1000-digit Fibonacci number} Let's use the algorithm from Problem 2 to generate Fibonacci numbers whilst incrementing a counter until we find one
that is composed of at least a thousand digits. \section{Reciprocal cycles} From my research I have only found one way to solve this problem: \emph{Fermat's little theorem} (\url{http://
www.wikiwand.com/en/Fermats_little_theorem}). If you are interested you can do some research on the subject. The application of the theorem we interested in is the following: $1 \over d$ has a cycle
of $n$ digits if $10^n-1 \equiv 0 \pmod d$. Another property that can be deduced with this theorem is for any prime $p$, $1 \over p$ has at most a recurring cycle of $p-1$ digits. Honestly I don't
like these kind of problems. By nature this problem can't be solved with brute force algorithms because we don't know what maximum length a cycle can be. With mathematics we can find a boundary.
However not everyone can master complicated mathematics like these, what is important is to recognize when a problem can't be solved with classical methods and look for theory. \emph{Know the tools
at your disposal}. We now have two bits of theory that can be plugged into a simple loop. First of all we collect all the prime numbers inferior to 1000. Then we iterate through them in reversed
order (bigger primes will producing bigger recurring cycles). Now we want to find the number of digits in the reccurring cycle for each prime $p$, for this we simply check increment a counter $n$
until we find $10^n-1 \equiv 0 \pmod p$. Finally we check if the recurring cycle has $p-1$ digits, if it doesn't we do the same thing for a different thing, it it does it is the answer. \section
{Quadratic primes} Really there is nothing complex with this problem. If we go from basis that we can't induce properties for a polynomial based on another then we simply have to iterate through each
possibility. First of all we need a function to test if a number is prime, we already have that in our toolkit, for example from Problem 10. Next we need to check how many primes a polynomial can
generate. For this we create a function that takes as arguments the coefficient $a$ and $b$ and keeps incrementing $n$ until it returns a composite number. Now for the candidates we could try every
possibility, but that would be four million, which is unreasonable. By looking at the polynomials we can notice that if $n=0$ then what is left is $b$, thus $b$ has to be prime, which also means it
has to be positive. The algorithm will run much faster now. Some people will point out the fact that $a$ has to be odd and bounded by $-b$ and $b$, the speedup is not essential. As a bonus I included
a plot in the code and you can notice that indeed a lot of couples $(a,b)$ generate only one prime because there are no constraints on $a$. \section{Number spiral diagonals} This problem is good for
people who like toying with numbers. Being one of them I won't give a brute force solution but an elegant one. I would recommend taking a piece of paper, a pen and looking at the example grid from
the website. First of all we can notice that along the diagonals there are always four numbers, the top right one being the biggest, the top left the second biggest, the bottom left the third biggest
and the bottom right the smallest. It is quite obvious that the sum of the left side numbers are equal to the sum of the right side numbers. Hence we can calculate the right side numbers and multiply
their sum by two. What we need is a formula that gives the $n^{th}$ top right number and another formula for the $n^{th}$ bottom right number. Let's start with the top right numbers. Our intuition
should tell us that these numbers are simply the squares of odd numbers. The $n^{th}$ odd number is given by $2n-1$, thus the $n^{th}$ top right number is $(2n-1)²$. Now for the bottom right corner.
It takes a bit more time to notice there is a link between each corner, indeed that's we decided to only calculate only two corners. The corner values go progressively down in the anticlockwise way.
For example on the first square, the top right corner is equal to 9, is goes down to 7, then 5, and finally 3. On the second square it's 25, 21, 17, 13. The conjecture that follows is that the
reduction for the $n^{th}$ from corner to corner is $2(n-1)$. From this conjecture we know that the bottom right corner is equal to the top right corner minus three times the reduction, hence $6(n-1)
$. To recapitulate, $C(n)$ is the sum of the values for square $n$, $tr(n)$ and $br(n)$ the right hand corners: \[S = \left\{ \begin{array}{lr} C(n) = 2(tr(n)+br(n))\\ tr(n) = (2n-1)^2\\ br(n) =
(2n-1)^2-6(n-1) \end{array} \right. \] \begin{center} $\iff C(n)=16n^2-28n+16$. \end{center} Finally we simply calculate $1+\displaystyle\sum_{i=2}^{501}C(n)$. The reason why we add a 1 is because
the formulate doesn't work $n=1$ (the single cell), thus we start our sum at 2. Also the sum only goes to 501, not 1001, indeed $501+501-1=1001$. \section{Distinct powers} We can use a nifty dodge
for this problem. Let's say we have all the terms of the sequence in a list. To get all distinct terms we turn the list into a set, indeed if you have done a little maths you should know that a set
contains unique elements. The Python implementation is simply the \emph{set()} function applied to a list comprehension with two \emph{for} loops. \section{Digit fifth powers} This is one those
unbounded problems. What I mean is that the candidate numbers are in $\mathbb{N}^+$. Our first goal is to bound the candidate numbers. Another way of seeing this is to ask what size are the
candidates numbers going to be. A number of size $n$ is bounded by $10^{n}-1$ (for example a 2-digit number is bounded by 99). Also the sum of the digits of a number of size $n$ to the $5^{th}$ power
is bounded by $9^5n$. What we are looking is the first integer $n$ which doesn't verify $9^5n > 10^{n}-1$. The solution is simply to try integers by hand, or to use a \emph{while} loop as I did. It
turns out that a loose boundary is $6 \times {9^{5}} = 354294$. Now the problem is simple, we loop through every integer between 2 and 354294 and check its digits to the $5^{th}$ sum up to the
integer with a list comprehension. \end{document} | {"url":"https://tr.overleaf.com/articles/project-euler-solutions/qqtpygtpmsrv","timestamp":"2024-11-14T19:53:35Z","content_type":"text/html","content_length":"69821","record_id":"<urn:uuid:c9464009-391d-4288-a884-c84912e8c6f6>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00073.warc.gz"} |
New package on CTAN: coolthms
Date: March 30, 2012 7:18:03 PM CEST
On Thu, 29 Mar 2012 Michael Fütterer and Jonathan Zachhuber submitted the coolthms package. Summary description: a LaTeX package for referencing list items nested in theorem-like environments
including some theorem markup options License type: lppl Announcement text: This package makes it possible to directly reference items from lists nested in theorem-like environments (e.g. Theorem 1
a.) extending the ntheorem and cleveref packages. We also provide some theorem markup commands. This package is located at
. More information is at
(if the package is new it may take a day for that information to appear). We are supported by the TeX Users Group
. Please join a users group; see
. Thanks for the upload. For the CTAN Team Rainer Schöpf
coolthms – Reference items in a theorem environment
The package provides the means to directly reference items of lists nested in theorem-like environments (e.g., as ‘Theorem 1 a’).
The package extends the ntheorem and cleveref packages.
The package also provides other theorem markup commands.
Package coolthms
Version 1.2
Copyright 2011–2012 Jonathan Zachhuber, Michael Fütterer
Maintainer Michael Fütterer
Jonathan Zachhuber | {"url":"https://ctan.org/ctan-ann/id/mailman.7815.1333127886.2276.ctan-ann@dante.de","timestamp":"2024-11-03T20:33:10Z","content_type":"text/html","content_length":"14794","record_id":"<urn:uuid:9bef0e43-44d9-4987-bffd-b249287888a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00600.warc.gz"} |
link (US survey) to Long Reed Converter
Enter link (US survey)
Long Reed
β Switch toLong Reed to link (US survey) Converter
How to use this link (US survey) to Long Reed Converter π €
Follow these steps to convert given length from the units of link (US survey) to the units of Long Reed.
1. Enter the input link (US survey) value in the text field.
2. The calculator converts the given link (US survey) into Long Reed in realtime β using the conversion formula, and displays under the Long Reed label. You do not need to click any button. If the
input changes, Long Reed value is re-calculated, just like that.
3. You may copy the resulting Long Reed value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert link (US survey) to Long Reed?
The formula to convert given length from link (US survey) to Long Reed is:
Length[(Long Reed)] = Length[(link (US survey))] / 15.90905909013222
Substitute the given value of length in link (us survey), i.e., Length[(link (US survey))] in the above formula and simplify the right-hand side value. The resulting value is the length in long reed,
i.e., Length[(Long Reed)].
Calculation will be done after you enter a valid input.
Consider that a piece of land is measured at 80 links (US survey).
Convert this length from links (US survey) to Long Reed.
The length in link (us survey) is:
Length[(link (US survey))] = 80
The formula to convert length from link (us survey) to long reed is:
Length[(Long Reed)] = Length[(link (US survey))] / 15.90905909013222
Substitute given weight Length[(link (US survey))] = 80 in the above formula.
Length[(Long Reed)] = 80 / 15.90905909013222
Length[(Long Reed)] = 5.0286
Final Answer:
Therefore, 80 li is equal to 5.0286 long reed.
The length is 5.0286 long reed, in long reed.
Consider that a boundary marker is set 30 links (US survey) from the starting point.
Convert this distance from links (US survey) to Long Reed.
The length in link (us survey) is:
Length[(link (US survey))] = 30
The formula to convert length from link (us survey) to long reed is:
Length[(Long Reed)] = Length[(link (US survey))] / 15.90905909013222
Substitute given weight Length[(link (US survey))] = 30 in the above formula.
Length[(Long Reed)] = 30 / 15.90905909013222
Length[(Long Reed)] = 1.8857
Final Answer:
Therefore, 30 li is equal to 1.8857 long reed.
The length is 1.8857 long reed, in long reed.
link (US survey) to Long Reed Conversion Table
The following table gives some of the most used conversions from link (US survey) to Long Reed.
link (US survey) (li) Long Reed (long reed)
0 li 0 long reed
1 li 0.06285726857 long reed
2 li 0.1257 long reed
3 li 0.1886 long reed
4 li 0.2514 long reed
5 li 0.3143 long reed
6 li 0.3771 long reed
7 li 0.44 long reed
8 li 0.5029 long reed
9 li 0.5657 long reed
10 li 0.6286 long reed
20 li 1.2571 long reed
50 li 3.1429 long reed
100 li 6.2857 long reed
1000 li 62.8573 long reed
10000 li 628.5727 long reed
100000 li 6285.7269 long reed
link (US survey)
A link (US survey) is a unit of length used primarily in land surveying in the United States. One US survey link is equivalent to exactly 0.66 feet or approximately 0.201168 meters.
The US survey link is defined as one-hundredth of a US survey chain, where one US survey chain is 66 feet long. This unit provides precision for finer measurements in land surveying and mapping.
Links (US survey) are used in land surveying to measure shorter distances and ensure accuracy in property measurement and mapping activities in the United States.
Long Reed
A long reed is a historical unit of length used in various cultures, particularly in historical land measurement. One long reed is approximately equivalent to 1.5 to 2 meters or about 4.9 to 6.6
The exact length of a long reed could vary depending on the region and historical context, as it was based on practical measurements of the length of a reed used for various purposes.
Long reeds were used in historical land measurement, agriculture, and construction. Although not commonly used today, the unit provides insight into traditional measurement practices and the use of
natural materials in historical measurement systems.
Frequently Asked Questions (FAQs)
1. What is the formula for converting link (US survey) to Long Reed in Length?
The formula to convert link (US survey) to Long Reed in Length is:
link (US survey) / 15.90905909013222
2. Is this tool free or paid?
This Length conversion tool, which converts link (US survey) to Long Reed, is completely free to use.
3. How do I convert Length from link (US survey) to Long Reed?
To convert Length from link (US survey) to Long Reed, you can use the following formula:
link (US survey) / 15.90905909013222
For example, if you have a value in link (US survey), you substitute that value in place of link (US survey) in the above formula, and solve the mathematical expression to get the equivalent value in
Long Reed. | {"url":"https://convertonline.org/unit/?convert=links_us_survey-long_reeds","timestamp":"2024-11-04T23:39:13Z","content_type":"text/html","content_length":"91548","record_id":"<urn:uuid:15f7a13c-429d-4145-8580-8e22a4a4da59>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00315.warc.gz"} |
Giver Post Date
tanzirhasan92 A train traveled from Station A to Station B at an average speed of 80 02-Nov-2024
Tejas1402 A train traveled from Station A to Station B at an average speed of 80 19-Oct-2024
xxx541236 A train traveled from Station A to Station B at an average speed of 80 15-Oct-2024
MustafaB A train traveled from Station A to Station B at an average speed of 80 13-Oct-2024
Abhinav1823 A train traveled from Station A to Station B at an average speed of 80 01-Sep-2024
ShilpiAgnihotrii A train traveled from Station A to Station B at an average speed of 80 01-Aug-2024
Jay29 A train traveled from Station A to Station B at an average speed of 80 29-Jul-2024
AnantpriyaSingh A train traveled from Station A to Station B at an average speed of 80 28-Jan-2024
Duangnapha A train traveled from Station A to Station B at an average speed of 80 07-Jan-2024
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dar24 A train traveled from Station A to Station B at an average speed of 80 26-Nov-2023
mix A train traveled from Station A to Station B at an average speed of 80 19-Nov-2023
AnkurGMAT20 A train traveled from Station A to Station B at an average speed of 80 27-Aug-2023
davidgarythomas84 A train traveled from Station A to Station B at an average speed of 80 08-Jul-2023
DenisJames A train traveled from Station A to Station B at an average speed of 80 06-Jul-2023
Thonereturns A train traveled from Station A to Station B at an average speed of 80 06-Jun-2023
hiyavazirani97 A train traveled from Station A to Station B at an average speed of 80 15-Mar-2023
sachinpierre A train traveled from Station A to Station B at an average speed of 80 23-Dec-2022
GinoRako A train traveled from Station A to Station B at an average speed of 80 14-Nov-2022
wensjoe A train traveled from Station A to Station B at an average speed of 80 20-Oct-2022
Sneha2311 A train traveled from Station A to Station B at an average speed of 80 18-Oct-2022
thomasg13 A train traveled from Station A to Station B at an average speed of 80 09-Aug-2022
mbaintern2025 A train traveled from Station A to Station B at an average speed of 80 21-Jun-2022
luckyhunch A train traveled from Station A to Station B at an average speed of 80 20-Jun-2022
EshaL A train traveled from Station A to Station B at an average speed of 80 11-Apr-2022
alicearjung A train traveled from Station A to Station B at an average speed of 80 28-Jan-2022
MsEmily A train traveled from Station A to Station B at an average speed of 80 24-Dec-2021
Dibab A train traveled from Station A to Station B at an average speed of 80 15-Dec-2021
Ziadaf A train traveled from Station A to Station B at an average speed of 80 04-Dec-2021
harish555 A train traveled from Station A to Station B at an average speed of 80 17-Nov-2021 | {"url":"https://gmatclub.com/kudos-details/NoHalfMeasures/1371162/","timestamp":"2024-11-13T11:10:14Z","content_type":"application/xhtml+xml","content_length":"168801","record_id":"<urn:uuid:bb42a081-9905-4fc5-bc2b-721782679fb7>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00611.warc.gz"} |
Excel Formula Python - Check if 'Y' Exists in Columns
When working with Excel data in Python, it is often necessary to check if a specific value exists in one of two columns. In this article, we will learn how to write an Excel formula in Python to
accomplish this task.
The COUNTIF Function
The COUNTIF function is a powerful tool in Excel that allows us to count the number of occurrences of a specific value in a range of cells. In our case, we want to count the number of 'Y' values in
columns A and B.
The formula to count the occurrences of 'Y' in column A is COUNTIF(A:A, "Y"), and the formula to count the occurrences in column B is COUNTIF(B:B, "Y").
Combining COUNTIF with IF
To check if 'Y' exists in one of the two columns, we can use the IF function in combination with the COUNTIF function. Here is the formula:
=IF(COUNTIF(A:A, "Y") + COUNTIF(B:B, "Y") > 0, "Y Exists", "Y Not Found")
This formula calculates the total count of 'Y' values in columns A and B. If the count is greater than 0, it returns "Y Exists". Otherwise, it returns "Y Not Found".
Let's consider an example to understand how this formula works. Suppose we have the following data in columns A and B:
Applying the formula =IF(COUNTIF(A:A, "Y") + COUNTIF(B:B, "Y") > 0, "Y Exists", "Y Not Found") to a cell, we would get the result "Y Exists" because there is at least one 'Y' in either column A or
column B.
In this article, we learned how to write an Excel formula in Python to check if 'Y' exists in one of two columns. By combining the COUNTIF function with the IF function, we can easily determine if a
specific value is present in a range of cells. This formula can be useful in various scenarios, such as data analysis and conditional formatting. Experiment with different formulas and explore the
possibilities of Excel and Python integration!
Further Reading:
=IF(COUNTIF(A:A,"Y")+COUNTIF(B:B,"Y")>0, "Y Exists", "Y Not Found")
Formula Explanation
This formula uses the COUNTIF function to count the occurrences of 'Y' in column A and column B. If the total count is greater than 0, it returns "Y Exists", otherwise it returns "Y Not Found".
For example, if we have the following data in columns A and B:
| A | B |
| N | Y |
| Y | N |
| N | N |
The formula =IF(COUNTIF(A:A,"Y")+COUNTIF(B:B,"Y")>0, "Y Exists", "Y Not Found") would return "Y Exists" because there is at least one 'Y' in either column A or column B. | {"url":"https://codepal.ai/excel-formula-generator/query/25JrOe0l/excel-formula-python-if-y-exists","timestamp":"2024-11-03T20:00:40Z","content_type":"text/html","content_length":"99837","record_id":"<urn:uuid:3bcae63c-a960-4c89-8802-11eab1a3c135>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00239.warc.gz"} |
Geometric Progression - Formulas, nth Term, Sum, Pdf
Geometric Progression
What is Geometric Progression?
Geometric Progression (GP), also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called
the common ratio (r).
Geometric Progression (GP) is a sequence where each term is found by multiplying the previous one by a fixed number, known as the common ratio. It’s symbolized as ‘a, ar, ar²…’ where ‘a’ is the first
term and ‘r’ is the common ratio. ‘r’ can be positive or negative. With just the first term and the ratio, you can determine all the terms of the sequence.
Geometric Progression Formula
Here are the formulas related to geometric progressions, considering a sequence that begins with 𝑎a and follows the pattern a, ar, ar²,𝑎𝑟³….
• nᵗʰ term: a[n] = arⁿ⁻¹ (or) aₙ = r aₙ₋₁
• Sum of the first n terms: Sₙ = a(rⁿ – 1) / (r – 1) when r ≠ 1 and Sₙ = na when r = 1.
• Sum of infinite terms: S[∞] = a / (1 – r) when |r| < 1 and the sum is NOT defined when |r| ≥ 1.
Properties of Geometric Progression (GP)
Here are the important properties of Geometric Progression (GP) presented more succinctly:
1. Geometric Mean Property: For three non-zero terms a, b, c in GP, 𝑏²=𝑎𝑐.
2. Consecutive Terms: In a GP, three consecutive terms are 𝑎/𝑟a/r, 𝑎a, 𝑎𝑟ar. Four consecutive terms are 𝑎/𝑟³, 𝑎/𝑟, ar, 𝑎𝑟³. Similarly for five consecutive terms.
3. Product of Equidistant Terms: In a finite GP, the product of terms equidistant from the beginning and the end is the same. For instance, 𝑡₁⋅𝑡ₙ=𝑡₂⋅𝑡ₙ₋₁=𝑡₃⋅𝑡ₙ₋₂=…
4. Scaling: If each term of a GP is multiplied or divided by a non-zero constant, the resulting sequence remains a GP with the same common ratio.
5. Product and Quotient: The product and quotient of two GPs are again GPs.
6. Exponentiation: Raising each term of a GP to the power by the same non-zero quantity results in another GP.
7. Relation to Arithmetic Progression (AP): If a GP has positive terms, then the logarithms of its terms form an Arithmetic Progression (AP), and vice versa.
General Form of Geometric Progression
he general form of a Geometric Progression (GP) is represented as:
• 𝑎 is the first term of the sequence.
• 𝑟 is the common ratio between consecutive terms.
• 𝑎𝑟ⁿ⁻¹ is the 𝑛𝑡ℎnth term of the sequence.
This formula encapsulates the pattern where each term is obtained by multiplying the previous term by the common ratio ‘r’.
General Term or nth Term of Geometric Progression
Let a be the first term and r be the common ratio for a Geometric Sequence.
Then, the second term, a₂ = a × r = ar
Third term, a₃ = a₂ × r = ar × r = ar²
Similarly, nth term, aₙ = arⁿ⁻¹
Therefore, the formula to find the nth term of GP is:
Common Ratio of GP
Consider the sequence a, ar, ar², ar³,……
First term = a
Second term = ar
Third term = ar²
Similarly, nth term, tₙ = arⁿ⁻¹
Thus, the common ratio of geometric progression formula is given as:
Common ratio = (Any term) / (Preceding term)
= tₙ / tₙ₋₁
= (arⁿ⁻¹ ) /(arⁿ⁻²)
= r
Thus, the general term of a GP is given by ar and the general form of a GP is a, ar, ar^2,…..
For Example: r = t₂ / t₁ = ar / a = r
Sum of N term of GP
uppose a, ar, ar², ar³,……arⁿ⁻¹ is the given Geometric Progression.
Then the sum of n terms of GP is given by:
S[n] = a + ar + ar^² + ar³^ +…+ arⁿ⁻¹
The formula to find the sum of n terms of GP is:
Sₙ = a[(rⁿ^ – 1)/(r – 1)] if r ≠ 1 and r > 1
a is the first term
r is the common ratio
n is the number of terms
Also, if the common ratio is equal to 1, then the sum of the GP is given by:
Geometric Progression vs Arithmetic Progression
Aspect Geometric Progression (GP) Arithmetic Progression (AP)
Definition A sequence where each term after the first is multiplied by a constant. A sequence where each term after the first is added by a constant.
Common Term Common Ratio (r) Common Difference (d)
Formula The nth term is given by 𝑎⋅𝑟(𝑛−1) The nth term is given by 𝑎+(𝑛−1)⋅𝑑
Example 2, 6, 18, 54, … (where 𝑟=3) 5, 8, 11, 14, … (where 𝑑=3)
Sum Formula Sum of first n terms: 𝑆ₙ=𝑎⋅𝑟ⁿ⁻¹/𝑟−1(if 𝑟≠1) Sum of first n terms: 𝑆ₙ=𝑛/2⋅(2𝑎+(𝑛−1)⋅𝑑)
Applications Used in scenarios involving exponential growth or decay, such as populations, finance. Used in evenly spaced contexts like scheduling, constructing sequences.
Types of Geometric Progression
1. Finite Geometric Progression (Finite GP):
□ Explanation: A finite GP has a limited number of terms, after which the sequence ends.
□ Representation: 𝑎,𝑎𝑟,ar²,𝑎𝑟³,…,𝑎𝑟ⁿ⁻¹ where 𝑛n is the number of terms.
□ Formula to Find Sum: The sum of a finite GP can be calculated using the formula: 𝑆𝑛=𝑎𝑟ⁿ⁻¹/𝑟−1Where 𝑆𝑛 is the sum of the first 𝑛 terms, 𝑎a is the first term, 𝑟 is the common ratio, and 𝑛n is
the number of terms.
2. Infinite Geometric Progression (Infinite GP):
□ Explanation: An infinite GP continues indefinitely, with no predetermined end.
□ Representation: 𝑎,𝑎𝑟,ar²,𝑎𝑟³,… where the terms continue indefinitely.
□ Formula to Find Sum: The sum of an infinite GP can be calculated using the formula: 𝑆=𝑎/1−𝑟 Where S is the sum of the infinite series, a is the first term, and r is the common ratio (provided
Finite Geometric Progression
The terms of a finite G.P. can be written as a, ar, ar², ar³,……arⁿ⁻¹
a, ar, ar², ar³,……arⁿ⁻¹ is called finite geometric series.
The sum of finite Geometric series is given by:
S[n] = a[(rⁿ^ – 1)/(r – 1)] if r ≠ 1 and r > 1
Tips for Understanding and Working with Geometric Progressions
Geometric progressions (GPs) are a fundamental concept in mathematics with extensive applications across various fields such as finance, physics, and computer science. Here are some essential tips to
help you understand and effectively work with geometric progressions:
1. Understand the Components:
□ Initial Term (a): The first term in the progression.
□ Common Ratio (r): The factor by which each term is multiplied to get the next term. The value of 𝑟r can dramatically change the behavior of the progression, from rapid growth to rapid decay.
2. Visualize the Progression:
□ Graphing the terms of a geometric progression can provide insight into its growth pattern—whether it’s an exponential growth, decay, or constant series.
3. Check for Common Pitfalls:
□ Be cautious with negative values of r, as they cause the terms to alternate between positive and negative.
□ Extremely high or low values of r can lead to terms that grow or shrink very rapidly, making calculations cumbersome or leading to numerical inaccuracies.
4. Application in Real-World Problems:
□ Finance: Use GPs to model investments compounded over time or to calculate the future value of annuities.
□ Physics: Understand phenomena that involve exponential growth or decay, like radioactive decay or population growth under ideal conditions.
5. Practice Different Problems:
□ Work on various exercises involving different values of 𝑎a and r to build intuition and skill in recognizing patterns and solving problems efficiently.
6. Use Technology When Necessary:
□ For complex calculations, especially those involving large exponents or sums of many terms, don’t hesitate to use a scientific calculator or computer software to ensure accuracy.
7. Connect with Other Mathematical Concepts:
□ Relate geometric progressions to sequences, series, and other functions to deepen understanding and broaden application scope.
Four Terms of Geometric Progression
The four terms of a geometric progression are 𝑎,𝑎𝑟,𝑎𝑟2,𝑎𝑟3, where a is the first term and r is the common ratio. Each term is obtained by multiplying the previous term by r.
Rule of Geometric Progression
The rule of geometric progression states that each term in the sequence is obtained by multiplying the previous term by a constant factor, known as the common ratio (𝑟r).
Real-Life Example of Geometric Progression
An example of geometric progression in real life is the growth of a population of bacteria. If each bacterium doubles every hour, the population follows a geometric progression, with the number of
bacteria doubling with each generation.
Five Examples of Geometry in Real Life
1. Architecture: Designing buildings and structures using geometric principles.
2. Engineering: Calculating angles and dimensions for bridges and machines.
3. Art: Creating visually appealing compositions using geometric shapes.
4. Navigation: Using geometric concepts to plot routes and determine distances.
5. Sports: Applying geometry in activities like soccer, basketball, and swimming for strategy and technique. | {"url":"https://www.examples.com/maths/geometric-progression.html","timestamp":"2024-11-06T13:24:48Z","content_type":"text/html","content_length":"112034","record_id":"<urn:uuid:e7d47ae1-6203-4a40-998e-c8c11277be6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00594.warc.gz"} |
Computer programming instead of calculus?
I read an article one time which questioned why we choose calculus to be the top of the math pyramid in school. Basically, most of the mathematics students learn once they master the basics aims
toward preparing the students to take calculus at the end of K-12 school. The article I read suggested that statistics instead of calculus should be at the top because it is much more practical to
real life than calculus is.
We deliberately choose calculus to be at the top because we want our society to produce more engineers and scientists. This helped produce a generation of engineers and scientists.
However, although engineers and scientists are still needed, the US Department of Labor predicts that neither engineers nor scientists will be in the fastest growing jobs in the future. They have
predicted the 30 fasted growing jobs in the United States and there is something interesting about the list. 5 of the jobs involve the use of computers. Jobs number 25, 24, 23, 4, and 1 all include
the significant use of computers in a highly technical fashion. In fact all 5 of these jobs require computer programming skills to some degree.
So I propose that we make computer programming skills should be at the top of the list. This way we will be preparing our students for careers in the future rather than the careers of the past.
Now we will still end up producing engineers and scientists because there is a huge overlap between the mathematics required to master calculus and the skills required to master computer programming.
We will end up producing a lot people who are totally capable of programming a computer. Students who do not end up completing the stream will still end up having a very good understanding of how a
computer works, which is obviously going to be an advantage in the future anyway.
I suspect that the current stream of math would end up diverging just after algebra. It would end up involving a lot more number theory and logical reasoning and a lot less graphing and physics
based mathematics (except for the stream of students interested in game programming). I don't know that students would find this much more interesting, but at least it would pretty easy for them to
use the math they were learning and use it in direct applications involving their favorite technological devices.
Maybe kids might enjoy math more? | {"url":"https://paperlessmath.com/content/computer-programming-instead-calculus","timestamp":"2024-11-07T17:29:21Z","content_type":"application/xhtml+xml","content_length":"26526","record_id":"<urn:uuid:797627c1-d3b6-4346-abd0-08b066ad83ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00042.warc.gz"} |
The Linearity of the Euler Equation as a Result of the Compressibility of a Fluid
Journal of Modern Physics Vol.10 No.04(2019), Article ID:91234,7 pages
The Linearity of the Euler Equation as a Result of the Compressibility of a Fluid
Vladimir Kirtskhalia^1,2^
^1I. Vekua Sokhumi Institute of Physics and Technilogy (SIPT), Tbilisi, Georgia
^2Sokhumi State University, Tbilisi, Georgia
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: January 20, 2019; Accepted: March 16, 2019; Published: March 19, 2019
It is shown that when the compressibility of a fluid is taken into account, the nonlinear term disappears in the Euler equation. The validity of this approach is proved by the example of capillary
Euler Equation, Mass Continuity Equation, Fluid Compressibility, Capillary Waves
1. Introduction
In the monography [1] , the expression for the square of phase speed of capillary wave on the surface of fluid is received, according to which it generally depends on its depth and consists of the
sum of two members. One of them defines the influence of gravitational field, and the second the influence of surface tension, and as a result of which this wave is called capillary-gravitational.
This result was obtained under the assumption of the potentiality of motion and incompressibility of a fluid, on which the entire existing theory of gravitational waves is based [2] - [15] . In the
work [16] it was shown that the movement of fluid in a gravitational field of the Earth cannot be potential. It was also shown that the condition of incompressibility is inapplicable to liquids,
because it leads to the existence of an internal wave [1] , the nature of which is completely inexplicable from the point of view of physics. This conclusion was made on the basis of a correct
determination of the speed of sound [17] and a new definition of the criteria of compressibility and incompressibility of the medium [18] . This situation made it necessary to refine mass
conservation equation, and after its refinement, the gravity acceleration disappeared from the dispersion equation of capillary-gravitational wave, and it becomes purely capillary [16] . The further
investigations of gas and hydrodynamics have shown that Euler equation (fluid motion equation) also must be subject to refinement, since for it assumes that the density of a fluid is constant. This
approach leads to the fact that the nonlinear term disappears from this equation and it becomes linear.
In this work, we apply Euler’s refined equation towards capillary waves and show that despite the use of the boundary condition at the bottom of the liquid, the depth of the reservoir disappears from
the dispersion equation, and thus the phase velocity of the capillary wave depends only on its length. In addition, it will be shown that there is no stability condition for a capillary wave, i.e. it
is always stable in the first equation of system (72) in [16] typo, which the interested reader can easily correct.
2. Improved Euler Equation
In existing gas and hydrodynamics theory, Euler equation is applied in the form:
$\rho \frac{\text{d}V}{\text{d}t}=\rho \left[\frac{\partial V}{\partial t}+\left(Vabla \right)V\right]=-abla P+\rho g$(1)
where: $\rho$ -density, $V$ ―speed fluid particles, $P$ ―pressure, $g$ ―gravitational acceleration. Here supposed to that liquid is incompressible and consequently $\rho =const$ . In several works
(see e.g. [19] ), it was shown by us, that updated continuity equation of mass has the following form:
$\frac{\partial \rho }{\partial t}+\left(Vabla \right)\rho =-\rho abla V-\frac{Vabla P}{{C}_{p}^{2}}$(2)
where ${C}_{p}$ is isobaric sound speed in the liquid which can be considered as infinitely large, and we have:
$\frac{\text{d}\rho }{\text{d}t}=\frac{\partial \rho }{\partial t}+\left(Vabla \right)\rho =-\rho abla V$(3)
From (3) this it follows that liquid is compressible medium ( $abla Ve 0$ ) end $\text{d}\rho /\text{d}te 0$ . Consequently, Euler equation should be written in the form of:
$\frac{\text{d}\left(\rho V\right)}{\text{d}t}=V\frac{\text{d}\rho }{\text{d}t}+\rho \frac{\text{d}V}{\text{d}t}=-abla P+\rho g$(4)
Then if we substitute $\text{d}\rho /\text{d}t$ from (3) in (4), we get:
$\begin{array}{l}-V\rho abla V+\rho \frac{\text{d}V}{\text{d}t}=-abla P+\rho g\\ ⇒\rho \left[\frac{\text{d}V}{\text{d}t}-\left(Vabla \right)V\right]=-abla P+\rho g\\ ⇒\rho \frac{\partial V}{\partial
t}=-abla P+\rho g\end{array}$(5)
So, we have a system of two equations:
$\left\{\begin{array}{l}\rho \frac{\partial V}{\partial t}=-abla P+\rho g\hfill \\ \frac{\partial \rho }{\partial t}+\left(Vabla \right)\rho =-\rho abla V\hfill \end{array}$(6)
We see that the nonlinear term from the Euler equation has disappeared. It remains only in the mass conservation equation.
3. Dispersion Equation of Capillary Waves and Her Decision
Let’s presenting all the variables from the system (6) in the form of the sum of their stationary and perturbed values $f={f}_{0}+{f}^{\prime }$ and supposing that ${f}^{\prime }/{f}_{0}<1$ , after
linearization it will take the following form:
$\left\{\begin{array}{l}{\rho }_{0}\frac{\partial V}{\partial t}=-abla P+\frac{g}{{C}^{2}}P\hfill \\ \frac{1}{{C}^{2}}\left(\frac{\partial P}{\partial t}+{V}_{0}abla P\right)=-{\rho }_{0}abla V\hfill
where we used: state equation of medium ${\rho }^{\prime }={P}^{\prime }/{C}^{2}$ (C―adiabatic speed of sound in the liquid), liquid equilibrium equation $abla {P}_{0}={\rho }_{0}g$ and perturbed
values don’t have stokes.
If ${\stackrel{¯}{V}}_{0}$ there is a stationary velocity of fluid flow, then the linearized Equation (1) will be
${\rho }_{0}\left[\frac{\partial V}{\partial t}+\left({V}_{0}abla \right)V\right]=-abla P+\frac{g}{{C}^{2}}P$
As we see, this equation, which determines the acceleration of a liquid particle, contains a stationary velocity of motion, on which the acceleration should not depend. This fact irrefutably proves
the validity of formula (4).
Appling to the first equation of system (7) operator $abla$ and to the second operator $\partial /\partial t$ , they easily reduced to differential equation for perturbed pressure
$\Delta P-\frac{1}{{C}^{2}}\left(\stackrel{\to }{g}+{\stackrel{\to }{V}}_{0}\frac{\partial }{\partial t}\right)abla P-\frac{1}{{C}^{2}}\frac{{\partial }^{2}P}{\partial {t}^{2}}=0$(8)
Presenting now the perturbed pressure in the form of periodic function
$P\left(x,z,t\right)={P}_{a}\left(z\right)\mathrm{exp}\left[i\left(kx-\omega t\right)\right]$(9)
and supposing that ${V}_{0}={e}_{x}{V}_{0}$ and $g=-{e}_{z}g$ , we will get ordinary differential equation of second order for the amplitude of pressure disturbance in the following form:
$\frac{{\text{d}}^{2}{P}_{a}\left(z\right)}{\text{d}{z}^{2}}+\frac{g}{{C}^{2}}\frac{\text{d}{P}_{a}\left(z\right)}{\text{d}z}+\left[\frac{\omega }{{C}^{2}}\left(\omega -k{V}_{0}\right)-{k}^{2}\right]
The Equation (10) describes pressure disturbance on both sides of surface of tangential discontinuity $z=0$ . We solve this equation for air $\left(z>0\right)$ in the form:
${P}_{a1}\left(z\right)=A\mathrm{exp}\left(\gamma z\right)$ , (11)
where upon, taking into account the attenuation of the disturbance when $z\to \infty$ , for $\gamma$ we will get:
$\gamma =-\frac{k}{{\theta }_{1}}\left\{1+\sqrt{1+{\theta }_{1}^{2}\left[1-\frac{{U}_{p}\left({U}_{p}-{V}_{0}\right)}{{C}_{1}^{2}}\right]}\right\}<0$(12)
where ${C}_{1}$ is sound speed in the air on the sea level, ${U}_{p}=\omega /k$ is phase speed of surface wave and ${\theta }_{1}$ is dimensionless value and it is equal to
${\theta }_{1}=\frac{2k{C}_{1}^{2}}{g}$(13)
For the liquid $\left(z<0\right)$ , because of its depth limitation, from (10) analogically we will have:
${P}_{a2}\left(z\right)={B}_{1}\mathrm{exp}\left({\delta }_{1}z\right)+{B}_{2}\mathrm{exp}\left({\delta }_{2}z\right)$(14)
${\delta }_{1}=-\frac{k}{{\theta }_{2}}\left[1+\sqrt{1+{\theta }_{2}^{2}\left(1-\frac{{U}_{p}^{2}}{{C}_{2}^{2}}\right)}\right]<0$(15)
${\delta }_{2}=-\frac{k}{{\theta }_{2}}\left[1-\sqrt{1+{\theta }_{2}^{2}\left(1-\frac{{U}_{p}^{2}}{{C}_{2}^{2}}\right)}\right]>0$(16)
${\theta }_{2}=\frac{2k{C}_{2}^{2}}{g}$(17)
Let us denote liquid surface displacement along the axis z through
$\xi \left(x,t\right)=a\mathrm{exp}\left[i\left(kx-\omega t\right)\right]$(18)
and in this case, the boundary conditions on the surface $\left(z=0\right)$ and on the bottom $\left(z=-h\right)$ of the liquid will have the form of:
$\left\{\begin{array}{l}{\left({P}_{2}-{P}_{1}\right)|}_{z=0}=-\alpha \frac{{\partial }^{2}\xi }{\partial {x}^{2}}\hfill \\ {{V}_{z1}|}_{z=0}=\frac{\partial \xi }{\partial t}+{V}_{0}\frac{\partial \
xi }{\partial x}\hfill \\ {{V}_{z2}|}_{z=0}=\frac{\partial \xi }{\partial t}\hfill \\ {{V}_{z2}|}_{z=-h}=0\hfill \end{array}$(19)
where $\alpha$ is the coefficient of surface tension.
Let’s present perturbation velocity in the form of periodical function:
$V\left(x,z,t\right)={V}_{a}\left(z\right)\mathrm{exp}\left[i\left(kx-\omega t\right)\right]$(20)
and denote its z component through disturbance pressure from the first equation of the system (7):
${V}_{z}\left(x,z,t\right)=-\frac{i}{{\rho }_{0}\omega }\left[\frac{\partial {P}_{a}\left(z\right)}{\partial z}+\frac{g}{{C}^{2}}{P}_{a}\left(z\right)\right]\mathrm{exp}\left[i\left(kx-\omega t\
Substituting (9), (11), (14) and (21) in the boundary conditions (19), we will get the system of homogenous equations referred to unknown coefficients $A,{B}_{1},{B}_{2}$ and a:
$\left\{\begin{array}{l}A-{B}_{1}-{B}_{2}+\alpha {k}^{2}a=0\hfill \\ \frac{\gamma +g/{C}_{1}^{2}}{{\rho }_{01}\omega }A-\left(\omega -k{V}_{0}\right)a=0\hfill \\ \frac{{\delta }_{1}+g/{C}_{2}^{2}}{{\
rho }_{02}\omega }{B}_{1}+\frac{{\delta }_{2}+g/{C}_{2}^{2}}{{\rho }_{02}\omega }{B}_{2}-\omega a=0\hfill \\ \frac{{\delta }_{1}+g/{C}_{2}^{2}}{{\rho }_{02}\omega }\mathrm{exp}\left(-{\delta }_{1}h\
right){B}_{1}+\frac{{\delta }_{2}+g/{C}_{2}^{2}}{{\rho }_{02}\omega }\mathrm{exp}\left(-{\delta }_{2}h\right){B}_{2}=0\hfill \end{array}$(22)
Equating the determinant of the system (22) to zero, we will get dispersion relation for the wave on the liquid surface taking into account surface tension force in the form of:
$\begin{array}{l}\frac{{\delta }_{1}{\delta }_{2}}{{\rho }_{02}\omega }\left(\omega -k{V}_{0}-\frac{\gamma }{{\rho }_{01}\omega }\alpha {k}^{2}\right)\left[\mathrm{exp}\left(-{\delta }_{1}h\right)-\
mathrm{exp}\left(-{\delta }_{2}h\right)\right]\\ -\frac{\gamma }{{\rho }_{01}}\left[{\delta }_{2}\mathrm{exp}\left(-{\delta }_{1}h\right)+{\delta }_{1}\mathrm{exp}\left(-{\delta }_{2}h\right)\right]=
Taking into account that on the sea level ${C}_{1}\cong 340\text{\hspace{0.17em}}\text{m}/\text{sec}$ , let’s consider the inequality
${\theta }_{1}=\frac{2k{C}_{1}^{2}}{g}>1⇒k>\frac{g}{2{C}_{1}^{2}}⇒\lambda <\frac{4\text{π}{C}_{1}^{2}}{g}=1.45×{10}^{5}\text{m}$
We can see, that to this inequality satisfies with the entire range of lengths of surface waves on the water, from capillary to tsunami. It is apparent that for the capillary waves length of which
does not exceed a few centimeters, we have: ${\theta }_{2}\gg {\theta }_{1}\gg 1$ . Considering also that ${U}_{p}^{2}/{C}_{2}^{2}\ll {U}_{p}^{2}/{C}_{1}^{2}\ll 1$ , from (12), (15) and (16) we find
$\gamma ={\delta }_{1}=-k$ , ${\delta }_{2}=k$ and then neglecting ${\rho }_{01}$ with respect to ${\rho }_{02}$ , the dispersion Equation (23) takes the form:
${\rho }_{02}{U}_{p}^{2}+{\rho }_{01}{V}_{0}{U}_{p}-\alpha k=0$(24)
the solution of which is
${U}_{p}=\frac{-{\rho }_{01}{V}_{0}±\sqrt{{\rho }_{01}^{2}{V}_{0}^{2}+4\alpha k{\rho }_{02}}}{2{\rho }_{02}}$(25)
4. Discussion of Results
In order to show the truthfulness of our results, let’s consider earlier results and show their drawbacks. As it was said in the introduction, in the monography [1] dispersion relation for the
capillary-gravitational wave on the surface of incompressible liquid is presented in the form of:
${U}_{p}^{2}={\left(\frac{\omega }{k}\right)}^{2}=\left(\frac{g}{k}+\frac{\alpha k}{{\rho }_{02}}\right)th\left(kh\right)$(26)
from which it follows that when
$\frac{g}{k}>\frac{\alpha k}{{\rho }_{02}}⇒k<\sqrt{\frac{{\rho }_{02}g}{\alpha }}⇒\lambda >2\text{π}\sqrt{\frac{\alpha }{{\rho }_{02}g}}=1.72\text{sm}$(27)
the influence of the surface tension force is negligible, and the wave becomes purely gravitational. This conclusion contradicts to the classical experiment, in which a steel needle does not sink in
a glass filled with water to the brim. This is because although the diameter of glass greatly exceeds the above specified length, the force of surface tension acts which balances the pressure
produced by the needle. Thus, the dependence of phase speed on gravitational acceleration is excluded and consequently there is no existing condition that limits the length of capillary wave. Such a
conclusion is quite understandable from the point of view of physics, because surface tension arises due to the interaction forces between molecules on the surface of a liquid that significantly
exceed the gravitational force.
The contradiction associated with the influence of the gravitational field is eliminated by taking into account the compressibility of the fluid in the mass continuity equation. The solution of the
problem for such a case is given in [17] , where the dispersion equation is obtained in the form:
${U}_{p}=\frac{\omega }{k}=\frac{{\rho }_{01}{V}_{0}th\left(kh\right)±{\left\{th\left(kh\right)\left[{\rho }_{02}\alpha k-{\rho }_{01}{\rho }_{02}{V}_{0}^{2}\right]\right\}}^{1/2}}{{\rho }_{02}}$(28)
from which follows the condition of stability of capillary wave:
${V}_{0}\le \sqrt{\frac{\alpha k}{{\rho }_{01}}}$(29)
From (29) it is easy to calculate that the wind with the speed of ${V}_{0}=5$ m/s, will blow off capillary waves whose length $\lambda >1.6$ cm. However, simple observations show that capillary waves
exist at quite stronger winds. In addition, since the capillary wave is a purely surface phenomenon, its phase speed must not depend on the depth of the fluid.
As it is apparent in the Equation (25), this contradiction is eliminated if in Euler equation, we consider liquid as compressible. Capillary wave is stable in any wind, if only the wind force does
not exceed intermolecular interaction force and in this case, setting of the problem becomes meaningless. We can also see that phase speed of capillary wave does not depend on the depth of fluid.
5. Conclusion
Contradictions that are present in the theory of surface waves are described in detail in the works [16] and [19] . In this work, through the example of capillary waves we have explicitly
investigated the causes of these contradictions and showed how to overcome them. We can say with confidence that our recommendations will result in overcoming contradictions not only in the theory of
capillary waves but also in the theory of gravity waves too.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Kirtskhalia, V. (2019) The Linearity of the Euler Equation as a Result of the Compressibility of a Fluid. Journal of Modern Physics, 10, 452-458. https://doi.org/10.4236/jmp.2019.104030 | {"url":"https://file.scirp.org/Html/4-7503712_91234.htm","timestamp":"2024-11-01T19:29:32Z","content_type":"application/xhtml+xml","content_length":"104395","record_id":"<urn:uuid:05913064-cf22-4e81-aaab-406d7e2b9a7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00229.warc.gz"} |
Adaptive Fisherized Z-score
Hello Fellas,
It's time for a new adaptive fisherized indicator of me, where I apply adaptive length and more on a classic indicator.
Today, I chose the Z-score, also called standard score, as indicator of interest.
Special Features
Advanced Smoothing: JMA, T3, Hann Window and Super Smoother
Adaptive Length Algorithms: In-Phase Quadrature, Homodyne Discriminator, Median and Hilbert Transform
Inverse Fisher Transform (IFT)
Signals: Enter Long, Enter Short, Exit Long and Exit Short
Bar Coloring: Presents the trade state as bar colors
Band Levels: Changes the band levels
Decision Making
When you create such a mod you need to think about which concepts are the best to conclude. I decided to take Inverse Fisher Transform instead of normalization to make a version which fits to a fixed
scale to avoid the usual distortion created by normalization.
Moreover, I chose JMA, T3, Hann Window and Super Smoother, because JMA and T3 are the bleeding-edge MA's at the moment with the best balance of lag and responsiveness. Additionally, I chose Hann
Window and Super Smoother because of their extraordinary smoothing capabilities and because Ehlers favours them.
Furthermore, I decided to choose the half length of the dominant cycle instead of the full dominant cycle to make the indicator more responsive which is very important for a signal emitter like
Z-score. Signal emitters always need to be faster or have the same speed as the filters they are combined with.
The Z-score is a low timeframe scalper which works best during choppy/ranging phases. The direction you should trade is determined by the last trend change. E.g. when the last trend change was from
bearish market to bullish market and you are now in a choppy/ranging phase confirmed by e.g. Chop Zone or KAMA slope you want to do long trades.
The Z-score indicator is a momentum indicator which shows the number of standard deviations by which the value of a raw score (price/source) is above or below the mean value of what is being observed
or measured. Easily explained, it is almost the same as Bollinger Bands with another visual representation form.
B -> Buy -> Z-score crosses above lower band
S -> Short -> Z-score crosses below upper band
BE -> Buy Exit -> Z-score crosses above 0
SE -> Sell Exit -> Z-score crosses below 0
If you were reading till here, thank you already. Now, follows a bunch of knowledge for people who don't know the concepts I talk about.
The T3 moving average, short for "Tim Tillson's Triple Exponential Moving Average," is a technical indicator used in financial markets and technical analysis to smooth out price data over a specific
period. It was developed by Tim Tillson, a software project manager at Hewlett-Packard, with expertise in Mathematics and Computer Science.
The T3 moving average is an enhancement of the traditional Exponential Moving Average (EMA) and aims to overcome some of its limitations. The primary goal of the T3 moving average is to provide a
smoother representation of price trends while minimizing lag compared to other moving averages like Simple Moving Average (SMA), Weighted Moving Average (WMA), or EMA.
To compute the T3 moving average, it involves a triple smoothing process using exponential moving averages. Here's how it works:
Calculate the first exponential moving average (EMA1) of the price data over a specific period 'n.'
Calculate the second exponential moving average (EMA2) of EMA1 using the same period 'n.'
Calculate the third exponential moving average (EMA3) of EMA2 using the same period 'n.'
The formula for the T3 moving average is as follows:
T3 = 3 * (EMA1) - 3 * (EMA2) + (EMA3)
By applying this triple smoothing process, the T3 moving average is intended to offer reduced noise and improved responsiveness to price trends. It achieves this by incorporating multiple time frames
of the exponential moving averages, resulting in a more accurate representation of the underlying price action.
The Jurik Moving Average (JMA) is a technical indicator used in trading to predict price direction. Developed by Mark Jurik, it’s a type of weighted moving average that gives more weight to recent
market data rather than past historical data.
JMA is known for its superior noise elimination. It’s a causal, nonlinear, and adaptive filter, meaning it responds to changes in price action without introducing unnecessary lag. This makes JMA a
world-class moving average that tracks and smooths price charts or any market-related time series with surprising agility.
In comparison to other moving averages, such as the Exponential Moving Average (EMA), JMA is known to track fast price movement more accurately. This allows traders to apply their strategies to a
more accurate picture of price action.
Inverse Fisher Transform
The Inverse Fisher Transform is a transform used in DSP to alter the Probability Distribution Function (PDF) of a signal or in our case of indicators.
The result of using the Inverse Fisher Transform is that the output has a very high probability of being either +1 or –1. This bipolar probability distribution makes the Inverse Fisher Transform
ideal for generating an indicator that provides clear buy and sell signals.
Hann Window
The Hann function (aka Hann Window) is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing.
Super Smoother
The Super Smoother uses a special mathematical process for the smoothing of data points.
The Super Smoother is a technical analysis indicator designed to be smoother and with less lag than a traditional moving average.
Adaptive Length
Length based on the dominant cycle length measured by a "dominant cycle measurement" algorithm.
Happy Trading!
Best regards,
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Lesson 16
Solving More Ratio Problems
Let’s compare all our strategies for solving ratio problems.
16.1: You Tell the Story
Describe a situation with two quantities that this tape diagram could represent.
16.2: A Trip to the Aquarium
Consider the problem: A teacher is planning a class trip to the aquarium. The aquarium requires 2 chaperones for every 15 students. The teacher plans accordingly and orders a total of 85 tickets. How
many tickets are for chaperones, and how many are for students?
1. Solve this problem in one of three ways:
Use a triple number line.
Use a table.
(Fill rows as needed.)
│kids│chaperones │total │
│15 │2 │17 │
Use a tape diagram.
2. After your class discusses all three strategies, which do you prefer for this problem and why?
Use the digits 1 through 9 to create three equivalent ratios. Use each digit only one time.
\(\boxed{\phantom{3}}:\boxed{\phantom{3}}\) is equivalent to \(\boxed{\phantom{3}}\,\boxed{\phantom{3}}:\boxed{\phantom{3}}\) and \(\boxed{\phantom{3}}\,\boxed{\phantom{3}}:\boxed{\phantom{3}}\,\
16.3: Salad Dressing and Moving Boxes
Solve each problem, and show your thinking. Organize it so it can be followed by others. If you get stuck, consider drawing a double number line, table, or tape diagram.
1. A recipe for salad dressing calls for 4 parts oil for every 3 parts vinegar. How much oil should you use to make a total of 28 teaspoons of dressing?
2. Andre and Han are moving boxes. Andre can move 4 boxes every half hour. Han can move 5 boxes every half hour. How long will it take Andre and Han to move all 72 boxes?
When solving a problem involving equivalent ratios, it is often helpful to use a diagram. Any diagram is fine as long as it correctly shows the mathematics and you can explain it.
Let’s compare three different ways to solve the same problem: The ratio of adults to kids in a school is \(2:7\). If there is a total of 180 people, how many of them are adults?
• Tape diagrams are especially useful for this type of problem because both parts of the ratio have the same units (“number of people") and we can see the total number of parts.
This tape diagram has 9 equal parts, and they need to represent 180 people total. That means each part represents \(180 \div 9\), or 20 people.
Two parts of the tape diagram represent adults. There are 40 adults in the school because \(2\boldcdot 20 = 40\).
• Double or triple number lines are useful when we want to see how far apart the numbers are from one another. They are harder to use with very big or very small numbers, but they could support our
• Tables are especially useful when the problem has very large or very small numbers.
We ask ourselves, “9 times what is 180?” The answer is 20. Next, we multiply 2 by 20 to get the total number of adults in the school.
Another reason to make diagrams is to communicate our thinking to others. Here are some good habits when making diagrams:
• Label each part of the diagram with what it represents.
• Label important amounts.
• Make sure you read what the question is asking and answer it.
• Make sure you make the answer easy to find.
• Include units in your answer. For example, write “4 cups” instead of just “4.”
• Double check that your ratio language is correct and matches your diagram.
• tape diagram
A tape diagram is a group of rectangles put together to represent a relationship between quantities.
For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.
If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint. | {"url":"https://curriculum.illustrativemathematics.org/MS/students/1/2/16/index.html","timestamp":"2024-11-04T11:34:43Z","content_type":"text/html","content_length":"95962","record_id":"<urn:uuid:f692e709-aac3-4eab-8cda-5c621e9f1bcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00850.warc.gz"} |
Python - Get median of a List - Data Science Parichay
In this tutorial, we will look at how to get the median value of a list of values in Python. We will walk you through the usage of the different methods with the help of examples.
What is median?
Median is a descriptive statistic that is used as a measure of central tendency of a distribution. It is equal to the middle value of the distribution. There are equal number of values smaller and
larger than the median. It is also not much sensitive to the presence of outliers in the data like the mean (another measure of central tendency).
To calculate the median of a list of values –
1. Sort the values in ascending or descending order (either works).
2. If the number of values, n, is odd, then the median is the value in the (n+1)/2 position in the sorted list(or array) of values.
If the number of values, n, is even, then the median is the average of the values in n/2 and n/2 + 1 position in the sorted list(or array) of values.
For example, calculate the median of the following values –
First, let’s sort these numbers in ascending order.
Now, since the total number of values is even (8), the median is the average of the 4th and the 5th value.
Thus, median comes out to be 3.5
Now that we have seen how is the median mathematically calculated, let’s look at how to compute the median in Python.
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Median of a Python List
To compute the median of a list of values in Python, you can write your own function, or use methods available in libraries like numpy, statistics, etc. Let’s look at these methods with the help of
1. From scratch implementation of median in Python
You can write your own function in Python to compute the median of a list.
def get_median(ls):
# sort the list
ls_sorted = ls.sort()
# find the median
if len(ls) % 2 != 0:
# total number of values are odd
# subtract 1 since indexing starts at 0
m = int((len(ls)+1)/2 - 1)
return ls[m]
m1 = int(len(ls)/2 - 1)
m2 = int(len(ls)/2)
return (ls[m1]+ls[m2])/2
# create a list
ls = [3, 1, 4, 9, 2, 5, 3, 6]
# get the median
Here, we use the list sort() function to sort the list, and then depending upon the length of the list return the median. We get 3.5 as the median, the same we manually calculated above.
Note that, compared to the above function, the libraries you’ll see next are better optimized to compute the median of a list of values.
2. Using statistics library
You can also use the statistics standard library in Python to get the median of a list. Pass the list as argument to the statistics.median() function.
import statistics
# create a list
ls = [3, 1, 4, 9, 2, 5, 3, 6]
# get the median
We get the same results as above.
For more on the statistics library in Python, refer to its documentation.
3. Using numpy library
The numpy library’s median() function is generally used to calculate the median of a numpy array. You can also use this function on a Python list.
import numpy as np
# create a list
ls = [3, 1, 4, 9, 2, 5, 3, 6]
You can see that we get the same result.
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SchoolPoint Login / Howick College / Course Selection / Student Voice / Student Conferences / Inbox Design
Year 10 Mathematics and Statistics (10MAT)
Course Description
Teacher in Charge: Mr Z. Irani.
Mathematics and Statistics at Level 5 of the curriculum encourages students to develop a logical way of approaching problems based on contextual situations. The aim of the course is to introduce and
develop basic mathematical skills, concepts, and understandings in the Mathematical Processes, Number, Measurement, Geometry, Algebra and Statistics curriculum strands. This course will prepare
students to apply mathematical skills, concepts, and understandings to familiar and unfamiliar problems arising in real and simulated situations; demonstrate the ability to select and use appropriate
mathematical techniques in problem solving.
Students will also be working towards achieving their mandatory NCEA Numeracy co-requisites - 10 credits.
At the conclusion of this course, students will be well prepared to continue their studies at Level 6 of the NCEA curriculum.
Course Overview
Term 1
Geometric Reasoning
Pythagoras and Trigonometry
Term 2
Algebra - Equations and Expressions
Term 3
Algebra - Patterns and Graphs (Includes Parabolas)
Term 4
Project Work
Students are placed in the appropriate Mathematics class based on the results of their topic tests, PAT results and e-asTTle results from year 9.
Contributions and Equipment/Stationery
There is a $35 course fee that covers the subscription to NZ Grapher, online worksheets.
Costs will only occur if students enter competitions.
Optional cost –
A) It is recommended that students have the following individual workbooks for practice that can be obtained from the School office.
1) Walker Maths Numeracy workbook ($10).
2) Year 10 Nulake Mathematics Homework book($25)
B) Students can subscribe for Education Perfect ($26) - a digital learning platform and get access to range of lessons, practice tasks for Year 10 curriculum as well as for their preparation for
Numeracy CAA.
All students require to carry a 1E5 Exercise Book, Scientific Calculator (e.g. Fx 82 AU Plus II), ruler, eraser and a protractor along with a pen/pencil when they arrive for the Maths class. They
also need a working, charged laptop for educational tools such as Google Classroom.
To support your child's learning journey, here are some generic questions you could ask them each week:
What topics did you cover in Mathematics this week?
Are there any upcoming assessments or projects you need assistance with?
Assessment Information
Assessments are due during the term as follows:
Term 1 - 3rd Feb - 11th April, 2025
Term 2 - 28th April - 27th June, 2025
Term 3 - 14th July - 19th September, 2025
Term 4 - 6th October - 9th December, 2025
Students will also be given the opportunity to attempt the Numeracy co-requisite during the year.
Owing to teachers responding to individual students' needs, courses and NCEA standards taught in a subject maybe different to those displayed. | {"url":"https://howick.schoolpoint.co.nz/courses/course/10mat","timestamp":"2024-11-07T07:44:44Z","content_type":"text/html","content_length":"45439","record_id":"<urn:uuid:458e1922-272d-46b4-8196-9f2cd42a7944>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00840.warc.gz"} |
Design a Simple Constant Current Sink Circuit using Op-Amp
Current Source and Current sink are two major terms used in electronics design, these two terms dictate how much current can leave or enter a terminal. For example, the sink and source current of a
typical 8051 Microcontroller digital output pin is 1.6mA and 60uA respectively. Meaning the pin can deliver (source) up to 60uA when made High and can receive (sink) up to 1.6mA when made Low. During
our circuit design, we sometimes have to build our very own current source and current sink circuits. In the previous tutorial, we built a Voltage controlled current source circuit using common
op-amp and MOSFET which can be used for sourcing current to a load, but in some cases instead of the sourcing current, we will need a current sink option.
Hence, in this tutorial, we will learn how to build a voltage-controlled constant current sink circuit. A Voltage-controlled constant current sink circuit, as the name suggests controls the amount of
current sunk through it based on the voltage applied. Before proceeding further with circuit construction, let’s understand about constant current sink circuit.
What is a Constant Current Sink Circuit?
A constant current sink circuit actually sinks current irrespective of the load resistance as long as the input voltage is not changed. For a circuit with 1-ohm resistance, powered using 1V input,
the constant current is 1A according to Ohms Law. But, if Ohms law decides how much current flows through a circuit, then why do we need Constant current source and current sink circuit?
As you can see from the above image, a Current source circuit provides current to drive the load. The amount of current load receives will be decided by the current source circuit since it acts as a
power supply. Similarly, the current sink circuit acts like a ground, again the amount of current the load receives will be controlled by the current sink circuit. The main difference is that the
source circuit has to the source (supply) enough current to the load, while the sink circuit has to just limit the current through the circuit.
Voltage-controlled current sink using Op-Amp
Voltage-controlled constant current sink circuit works exactly in the same way as voltage-controlled current source circuit that we built earlier.
For a current sink circuit, the op-amp connection is changed, that is the negative input is connected to a shunt resistor. This will provide the necessary negative feedback to the op-amp. Then we
have a PNP transistor, that is connected across the Op-amp output so that the op-amp output pin can drive the PNP transistor. Now, always remember that an Op-Amp will try to make the voltage at both
the inputs (positive and negative) equal.
Let’s assume, 1V input is given across the positive input of the op-amp. The Op-amp will now try to make the other negative input also as 1V. But how this can be done? The output of the op-amp will
turn on the transistor in a way that the other input will get 1V from our Vsupply.
The shunt resistor will produce a drop voltage as per Ohms law, V= IR. Therefore, 1A of current flow through the transistor will create a drop voltage of 1V. The PNP transistor will sink this 1A of
current and the op-amp will use this voltage drop and get the desired 1V feedback. This way, changing the input voltage will control the Base as well as the current through the shunt resistor. Now,
let’s introduce the load that has to be controlled into our circuit.
As you can see, we have already designed out Voltage controlled current sink circuits using Op-Amp. But for practical demonstration, instead of using an RPS to provide variable voltage to Vin, let’s
use a potentiometer. We already know that the potentiometer shown below works as a potential divider to provide a variable voltage between 0V to Vsupply(+).
Now, let’s build the circuit and check how it works.
Same as the previous tutorial, we will use LM358 as it is very cheap, easy to find, and widely available. However, it has two op-amp channels in one package, but we need only one. We have previously
built many LM358 based circuits you can also check them out. The below image is an overview of the LM358 pin diagram.
Next, we need a PNP transistor, BD140 is used for this purpose. Other Transistors will also work, but heat dissipation is an issue. Therefore, the Transistor package needs to have an option to
connect an additional heat sink. BD140 pinout is shown in the below image –
Another major component is the Shunt Resistor. Let’s stick into 47ohms 2watt resistor for this project. Detail required components are described in the below list.
1. Op-amp (LM358)
2. PNP Transistor (BD140)
3. Shunt Resistor (47 Ohms)
4. 1k resistor
5. 10k resistor
6. Power supply (12V)
7. 50k potentiometer
8. Bread Board and additional connecting wires
Voltage Controlled Current Sink Circuit Working
The circuit is constructed in a simple breadboard for testing purposes as you can see in the below picture. To test the constant current facility, different resistors are used as a resistive load.
The input voltage is changed using the potentiometer and the current changes are reflected in the load. As seen in the below image, 0.16A current is sunk by the load. You can also check the detailed
working in the video linked at the bottom of this page. But, what is exactly happening inside the circuit?
As discussed before, during the 8V input, the op-amp will make the voltage drop across the shunt resistor for 8V in its feedback pin. The output of the op-amp will turn on the Transistor until the
shunt resistor produces an 8V drop.
As per the Ohms law, the resistor will only produce an 8V drop when the current flow is 170mA (.17A). This is because Voltage = current x resistance. Therefore, 8V = .17A x 47 Ohms. In this scenario,
the connected resistive load which is in series as shown in the schematic will also contribute to the flow of current. The op-amp will turn on the transistor and the same amount of current will be
sunk to the ground as the shunt resistor.
Now, if the voltage is fixed, whatever resistive load is connected, the current flow will be the same, otherwise, the voltage across the op-amp will not be the same as the input voltage.
Thus, we can say that the current through the load (current is sunk) is equal to the current through the Transistor which is also equal to the current through the shunt resistor. So, by rearranging
the above equation,
Current sink by the load = Voltage drop / Shunt Resistance.
As discussed before, the voltage drop will be the same as the input voltage across the op-amp. Therefore,
Current sink by the load = Input voltage / Shunt Resistance.
If the input voltage is changed, the current sink through the load will also change.
Design Improvements
1. If the heat dissipation is higher, increase the shunt resistor wattage. For selecting the wattage of the shunt resistor, R[w] = I^2R can be used, where R[w] is the resistor wattage and I is the
maximum current flow and R is the value of shunt resistor.
2. LM358 has two op-amps in a single package. Other than this, many op-amp ICs have two op-amps in a single package. If the input voltage is too low one can use the second op-amp to amplify the
input voltage as required.
Source: Design a Simple Constant Current Sink Circuit using Op-Amp
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Portfolio Theory and Arbitrage: A Course in Mathematical Financesearch
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Portfolio Theory and Arbitrage: A Course in Mathematical Finance
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Portfolio Theory and Arbitrage: A Course in Mathematical Finance
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• Graduate Studies in Mathematics
Volume: 214; 2021; 309 pp
MSC: Primary 60; 91; Secondary 46
This book develops a mathematical theory for finance, based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability,
starting with initial capital arbitrarily near zero. The principle is easy-to-test in specific models, as it is described in terms of the underlying market characteristics; it is shown to be
equivalent to the existence of the so-called “Kelly” or growth-optimal portfolio, of the log-optimal portfolio, and of appropriate local martingale deflators. The resulting theory is powerful
enough to treat in great generality the fundamental questions of hedging, valuation, and portfolio optimization.
The book contains a considerable amount of new research and results, as well as a significant number of exercises. It can be used as a basic text for graduate courses in Probability and
Stochastic Analysis, and in Mathematical Finance. No prior familiarity with finance is required, but it is assumed that readers have a good working knowledge of real analysis, measure theory, and
of basic probability theory. Familiarity with stochastic analysis is also assumed, as is integration with respect to continuous semimartingales.
Graduate students and researchers interested in math finance.
□ Chapters
□ The market
□ Numéraires and market viability
□ Financing optimization maximality
□ Ramifications and extensions
□ Elements of functional and convex analysis
□ ...this book is for you if you are the kind of soul that is not content with 'what' or 'how' but insists on asking 'why.'
Paolo Guasoni, Dublin City University and University of Bologna
• Desk Copy – for instructors who have adopted an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 214; 2021; 309 pp
MSC: Primary 60; 91; Secondary 46
This book develops a mathematical theory for finance, based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability,
starting with initial capital arbitrarily near zero. The principle is easy-to-test in specific models, as it is described in terms of the underlying market characteristics; it is shown to be
equivalent to the existence of the so-called “Kelly” or growth-optimal portfolio, of the log-optimal portfolio, and of appropriate local martingale deflators. The resulting theory is powerful enough
to treat in great generality the fundamental questions of hedging, valuation, and portfolio optimization.
The book contains a considerable amount of new research and results, as well as a significant number of exercises. It can be used as a basic text for graduate courses in Probability and Stochastic
Analysis, and in Mathematical Finance. No prior familiarity with finance is required, but it is assumed that readers have a good working knowledge of real analysis, measure theory, and of basic
probability theory. Familiarity with stochastic analysis is also assumed, as is integration with respect to continuous semimartingales.
Graduate students and researchers interested in math finance.
• Chapters
• The market
• Numéraires and market viability
• Financing optimization maximality
• Ramifications and extensions
• Elements of functional and convex analysis
• ...this book is for you if you are the kind of soul that is not content with 'what' or 'how' but insists on asking 'why.'
Paolo Guasoni, Dublin City University and University of Bologna
Desk Copy – for instructors who have adopted an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
You may be interested in...
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/gsm-214","timestamp":"2024-11-10T05:17:12Z","content_type":"text/html","content_length":"127464","record_id":"<urn:uuid:d7bd92cb-4d42-4847-90fa-be4531f92d8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00717.warc.gz"} |
Helicopters & Aircrafts
The efficiency of a wing is influenced greatly by its aerofoil section or profile, which has some degree and type of camber and some thickness form (Fig. 2. Id). Fuselages and other similar-shaped
components of a model also produce some lift force, depending again on their shape and angle of attack. Re-entry vehicles for space flight have been designed as ‘lifting bodies’ without wings, but
for almost all practical purposes in aeromodelling, the lift contribution of fuselages may be ignored. However, a fuselage does produce forces analogous to lift which affect the stability of the
model, almost invariably in ways that oppose the efforts of the stabiliser to hold the mainplane at a fixed angle of attack. Similar lateral unstabilising forces are resisted by fins, which are small
wings set at right angles to the mainplane, producing sideways ‘lift’ to correct yaw and sideslipping.
For convenience, aerodynamicists adopt a convention which allows all the very complex factors of wing trim and shape to be summed up in one figure, the coefficient of lift. This tells how the model
as a whole, or any part of it taken separately, is working as a lift producer. A lift coefficient or Cl of 1 -3 indicates more lifting effect than Cl = 1.0 or 0.6, while Cl — 0.0 indicates no lifting
effect at all. Cl has no dimensions since it is an abstract figure for comparison purposes and calculations.
For level flight, the total lift force generated by a model must equal the total weight, so it is possible to write:
Total Lift = Total Weight, or L = W (Action = Reaction).
This will not apply exactly if the model is descending or climbing. The exact relationships between lift and weight for these conditions are given in Fig. 1.4. As Figure 2.1. shows, the factors
affecting lift force are model size or area, speed of flight, air mass density and the aerofoil-plus-trim factor, Cl – In every case, an increase in one of these factors, greater
Fig 2.1 Contd.
0 Lifting effect or lift coefficient CL
area, more speed, increased density or higher lift coefficient, will produce a larger lift force. It is to be expected that when a formula for lift is worked out, it will include all these factors.
In mathematical language,
Lift = some function of p, V, S, and Cl
The standard, formula, which arises out of the basic principles of mechanics and the pioneer work of Daniel Bernoulli in the eighteenth century, is
It is not particulary important for modellers to know this formula but it is necessary to see how the various factors iii the lift equation are interdependent For a model to be capable of level
flight the lift must equal the weight If the model’s weight increases (as when it turns out heavier than expected), a larger lift force will be needed to support it Some item on the right hand side
of the equation, or more than one of them, must be increased. The
modeller has no control of air density, p. The model could be re-trimmed, increasing the wing’s angle of attack to get a higher Cl – More wing area might be added, although this would add mass and
increase the speed of flight. Since V is squared in the formula (multiplied by itself), a relatively small increase in V yields a large increase in lift force, other things being equal. It follows
from this that a heavy model (of given area, trim, etc.) has to fly faster than a light one. However, to increase V takes energy and in an extreme case the engine of the model may be incapable of
giving sufficient power to sustain flight. In such a case, if launched from a height the model would descend at some angle like a glider, even with engine at full power. | {"url":"http://heli-air.net/2016/02/11/aerofoil-sections-and-lift-coefficients/","timestamp":"2024-11-09T12:36:30Z","content_type":"text/html","content_length":"45278","record_id":"<urn:uuid:031e7323-4e3b-4f41-be6d-155c78a398c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00562.warc.gz"} |
Math Operations on Points and Vectors
Reading time: 12 mins.
Having delineated the principles of Cartesian coordinate systems, including the relationship between coordinates of points and vectors within these systems, we now proceed to explore some of the
principal operations applied to points and vectors. These operations are fundamental in the functionality of any 3D application or rendering engine.
Vector Class in C++
To begin, we will establish the structure for our Vector class in C++:
template<typename T>
class Vec3
// Three primary methods for vector initialization
Vec3() : x(T(0)), y(T(0)), z(T(0)) {}
Vec3(const T &xx) : x(xx), y(xx), z(xx) {}
Vec3(T xx, T yy, T zz) : x(xx), y(yy), z(zz) {}
T x, y, z;
Vector Length
As previously mentioned, a vector can be visualized as an arrow originating at one point and concluding at another. It not only signifies the direction from point A to B but also serves to measure
the distance between the two points. This measurement is derived from the vector's length, which can be calculated using the formula below:
$$||V|| = \sqrt{V.x * V.x + V.y * V.y + V.z * V.z}$$
In mathematical terms, the double bars (||V||) signify the vector's length. This attribute of a vector is also referred to as its norm or magnitude (figure 1).
template<typename T>
class Vec3
// length can be a method from the class...
T length()
return sqrt(x * x + y * y + z * z);
// ... or you can also compute the length in a function that is not part of the class
template<typename T>
T length(const Vec3<T> &v)
{ return sqrt(v.x * v.x + v.y * v.y + v.z * v.z); }
The axes of a three-dimensional Cartesian coordinate system are represented by unit vectors.
Normalizing a Vector
The term "normalize" can be spelled with either an 's' or a 'z' due to varied cultural influences. However, in programming, American spelling conventions typically prevail in the naming of methods or
functions, leading to the use of "normalize" with a 'z' in code.
A normalized vector, adhering to industry standards and spelled with a 'z' here, is a vector of length 1 (as illustrated by vector B in figure 1), also known as a unit vector. The process of
normalizing a vector involves calculating its length and then dividing each coordinate of the vector by this length. The formula for this is expressed as:
$$ \hat{V} = \frac{V}{|| V ||}$$
Figure 1: the magnitude or length of vectors A and B is denoted by the double bar notation. A normalized vector is a vector whose length is 1 (in this example, vector B).
The C++ implementation can be enhanced for efficiency. Specifically, vector normalization should only occur if the vector's length is greater than zero to avoid division by zero. An optimization
involves calculating the inverse of the vector's length once and multiplying each vector coordinate by this inverse value, instead of dividing them by the vector's length. This approach is preferred
as multiplication operations are generally less computationally expensive than divisions, which can significantly impact performance in rendering contexts where vector normalization is a frequent
operation. While some compilers may automatically optimize this, explicitly coding this optimization can reduce render times.
template<typename T>
class Vec3
// Vector normalization method
Vec3<T>& normalize()
T len = length();
if (len > 0) {
T invLen = 1 / len;
x *= invLen, y *= invLen, z *= invLen;
return *this;
// Utility function for vector normalization
template<typename T>
void normalize(Vec3<T> &v)
T len2 = v.x * v.x + v.y * v.y + v.z * v.z;
if (len2 > 0) {
T invLen = 1 / sqrt(len2);
v.x *= invLen, v.y *= invLen, v.z *= invLen;
The term norm in mathematics refers to a function that assigns a length, size, or distance to a vector, such as the Euclidean norm described here.
Dot Product
Figure 2: the dot product of two vectors can be seen as the projection of A over B. If the two vectors A and B have unit length, then the result of the dot product is the cosine of the angle
subtended by the two vectors.
The dot product, or scalar product, involves two vectors, A and B, and is conceptualized as the projection of one vector onto the other, yielding a scalar value. This operation is symbolized by \(A \
cdot B\) or sometimes \(\langle A, B \rangle\), involving the multiplication of corresponding elements from each vector and summing these products. For 3D vectors, the operation is:
$$A \cdot B = A.x * B.x + A.y * B.y + A.z * B.z$$
This process resembles the method for calculating a vector's length, where the square root of the dot product of two identical vectors (A=B) reveals the vector's length. We can express this as:
$$||V||^2=V \cdot V$$
This principle is utilized in the normalization method:
template<typename T>
class Vec3
T dot(const Vec3<T> &v) const
return x * v.x + y * v.y + z * v.z;
Vec3<T>& normalize()
T len2 = dot(*this);
if (len2 > 0) {
T invLen = 1 / sqrt(len2);
x *= invLen, y *= invLen, z *= invLen;
return *this;
template<typename T>
T dot(const Vec3<T> &a, const Vec3<T> &b)
return a.x * b.x + a.y * b.y + a.z * b.z;
The dot product is pivotal in 3D applications for it indicates the cosine of the angle between two vectors, as illustrated in Figure 2. This operation has several applications, such as determining
the angle between vectors or testing for orthogonality, where a dot product result of zero signifies perpendicular vectors.
• If B is a unit vector, the operation \(A \cdot B\) yields \(||A||\cos(\theta)\), signifying the magnitude of A's projection in B's direction, with a negative sign if the direction is reversed.
This is termed the scalar projection of A onto B.
• For cases where neither A nor B is a unit vector, the expression can be adjusted to \(A \cdot \frac{B}{||B||}\), recognizing \(B / ||B||\) as B represented as a unit vector.
• When both vectors are normalized, the arc cosine (\(\cos^{-1}\)) of their dot product reveals the angle \(\theta\) between them: \(\theta = \cos^{-1}\left(\frac{A \cdot B}{||A||\:||B||}\right)\)
or \(\theta=\cos^{-1}(\hat{A} \cdot \hat{B})\), where \(\cos^{-1}\) denotes the inverse cosine function, commonly represented as acos() in programming languages.
The dot product plays a crucial role in 3D geometry, serving various purposes such as testing for orthogonality. When two vectors are orthogonal (perpendicular), the dot product equals 0. If vectors
point in opposing directions, the dot product is -1, and when aligned in the same direction, it equals 1. This operation is extensively used to determine the angle between two vectors or to calculate
the angle between a vector and a coordinate system's axis, aiding in converting vector coordinates to spherical coordinates, as detailed in the discussion on trigonometric functions.
The Cross Product
The cross product is a vector operation distinct from the dot product, which yields a scalar value. Unlike the dot product, the cross product results in a vector. The uniqueness of this operation
lies in the resultant vector being perpendicular to the plane defined by the two original vectors. The cross product is represented as:
$$C = A \times B$$
Figure 3: the cross product of two vectors, A and B, gives a vector C perpendicular to the plane defined by A and B. When A and B are orthogonal to each other (and have unit length), A, B, and C form
a Cartesian coordinate system.
To calculate the cross product, the following formula is used:
$$ \begin{array}{l} C_X = A_Y \times B_Z - A_Z \times B_Y\\ C_Y = A_Z \times B_X - A_X \times B_Z\\ C_Z = A_X \times B_Y - A_Y \times B_X \end{array} $$
The cross product \(A \times B\) results in a vector, C, that is orthogonal to both A and B. These two vectors define a plane, and C stands perpendicular to this plane. The vectors A and B need not
be perpendicular themselves, but when they are, and assuming they are of unit length, they form a Cartesian coordinate system with C. This concept is instrumental in constructing coordinate systems,
a topic further explored in the chapter on Creating a Local Coordinate System.
template<typename T>
class Vec3
// Class method for cross product
Vec3<T> cross(const Vec3<T> &v) const
return Vec3<T>(
y * v.z - z * v.y,
z * v.x - x * v.z,
x * v.y - y * v.x);
// Utility function for cross product
template<typename T>
Vec3<T> cross(const Vec3<T> &a, const Vec3<T> &b)
return Vec3<T>(
a.y * b.z - a.z * b.y,
a.z * b.x - a.x * b.z,
a.x * b.y - a.y * b.x);
A helpful mnemonic for remembering the cross product formula involves using the coordinate letters not explicitly mentioned when calculating a specific component, for instance, asking "why z?" to
remember the components involved in calculating \(C_X\). More fundamentally, understanding that the cross product results in a vector perpendicular to the initial vectors allows for a logical
reconstruction of this formula. The matrix representation can simplify the visualization of this operation:
$$ \begin{pmatrix} a_x\\a_y\\a_z \end{pmatrix} \times \begin{pmatrix} b_x\\b_y\\b_z \end{pmatrix} = \begin{pmatrix} a_yb_z - a_zb_y\\ a_zb_x - a_xb_z\\ a_xb_y - a_yb_x \end{pmatrix} $$
This representation shows that to calculate any component of the resultant vector (e.g., the x component), the other two components (y and z) from vectors A and B are utilized.
It's crucial to recognize that the sequence of vectors in the cross product significantly influences the outcome. For instance, \(A \times B\) yields a different result than \(B \times A\):
$$A \times B = (1,0,0) \times (0,1,0) = (0,0,1),$$
$$B \times A = (0,1,0) \times (1,0,0) = (0,0,-1).$$
Figure 4: use your left or right hand to determine the orientation of vector C (the normal, for instance) when the index fingers point along A and the middle finger points along B.
Figure 5: using your right hand, you can align your index finger along either A or B and the middle finger against the other vector (B or A) to find out if C (the normal, for instance) points upwards
or inwards in the right-hand coordinate system.
The cross product is described as anticommutative, meaning that exchanging the positions of the two vectors inverses the result: if \(A \times B = C\), then \(B \times A = -C\). This property
underscores the significance of vector sequence in determining the direction of the resultant vector. When two vectors define the initial axes of a coordinate system, the direction in which the third
vector—derived from their cross product—points depends on the handedness of the system, a concept previously discussed.
Even though computing a cross product between vectors yields a consistent unique outcome—for example, \(A = (1, 0, 0)\) and \(B = (0, 1, 0)\) always produce \(C = (0, 0, 1)\)—the interpretation of
the resultant vector's direction is contingent on the coordinate system's handedness. This distinction is crucial because, despite the invariance of the computational result, how the resultant vector
is depicted depends on whether the coordinate system is right-handed or left-handed.
A practical mnemonic to ascertain the direction in which the resultant vector points involves using hand gestures. In a right-handed coordinate system, aligning the index finger with vector A (e.g.,
the tangent on a surface) and the middle finger with vector B (e.g., the bitangent when determining a normal's orientation) makes the thumb indicate the direction of vector C (e.g., the normal).
Employing the left hand for the same vectors reverses the direction indicated by the thumb. It's essential to recognize that this method addresses the representation of vector direction rather than
altering the computational result itself.
In the realm of mathematics, the output of a cross product is termed a pseudo vector. The sequence in which vectors participate in the cross product is critical, especially when computing surface
normals from the tangent and bitangent at a point. The sequence determines whether the resulting normal points inward (inward-pointing normal) or outward (outward-pointing normal) relative to the
surface. Further exploration of this topic is available in the chapter on Creating an Orientation Matrix, where the implications of vector order in cross products on surface orientation are
Vector/Point Addition and Subtraction
In the realm of vector mathematics, operations such as addition and subtraction are quite straightforward. Multiplying a vector by a scalar or another vector results in a new vector. Vectors can be
added together or subtracted from one another, among other operations. It's noteworthy that some 3D APIs make distinctions among points, normals, and vectors due to their technical differences. For
instance, normals do not undergo transformation in the same manner as points and vectors do. Subtraction of two points yields a vector, while adding a vector to another vector or a point results in a
Despite these distinctions, practical application has shown that the effort to maintain three separate C++ classes to represent each type—normals, vectors, and points—may not justify the added
complexity. Following the precedent set by OpenEXR, an industry-standard, a unified templated class named Vec3 is used to encapsulate normals, vectors, and points without differentiation in the
context of coding. This approach necessitates careful management of the occasional exceptions when variables representing different types but declared under the generic Vec3 type require distinct
processing methods. Below is an example of C++ code illustrating common operations, emphasizing the utility of a singular class for handling various vector-related operations:
template<typename T>
class Vec3
// Overloads the addition operator for vector addition
Vec3<T> operator + (const Vec3<T> &v) const
{ return Vec3<T>(x + v.x, y + v.y, z + v.z); }
// Overloads the subtraction operator for vector subtraction
Vec3<T> operator - (const Vec3<T> &v) const
{ return Vec3<T>(x - v.x, y - v.y, z - v.z); }
// Overloads the multiplication operator for scalar multiplication
Vec3<T> operator * (const T &r) const
{ return Vec3<T>(x * r, y * r, z * r); } | {"url":"https://scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/geometry/math-operations-on-points-and-vectors.html","timestamp":"2024-11-08T11:04:05Z","content_type":"text/html","content_length":"22628","record_id":"<urn:uuid:064541d5-af6a-4963-971e-bb66841d8280>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00072.warc.gz"} |
Rainfall Infiltration Return Frequency Estimates
1. Introduction
For many types of studies involving rainfall, such as flood control works and irrigation works, among others, rainfall trends are typically described by the use of depth-duration estimates [1,2].
Such a depth-duration analysis is typically used to estimate rare return period (or return frequency) depths of rainfall for various time durations.
In the current paper, the focus is on depth-duration estimates of rainfall infiltration into soils. A conceptual model that describes infiltration is selected, given rainfall and soil properties,
where it is assumed that the soil can infiltrate the rainfall quantities under consideration (e.g., the soil is free draining and not encumbered by perched groundwater or other such interferences to
infiltration). The resulting estimates of infiltration can be analyzed in developing depth-duration estimates of rainfall infiltration analogous to the procedures applied to rainfall data alone.
Probability distributions can be considered and fitted to the synthesized depth-duration estimates, enabling estimates of rare depth-duration outcomes of rainfall infiltration.
In this paper, the focus is developing a statistical procedure that may be useful in describing the occurrence of earth movement events such as landslides and mud floods (among others) and to
describe such outcomes in terms of the often used concept of return frequency, as is used in flood control engineering design and planning. This paper does not examine the details of landslide
initiation (including transient soil water pore pressures and water content variations in time, among other related factors) which are examined in the literature [3]. Instead, if it is known that a
certain quantity of infiltrated rainfall over a particular duration of time typically results in an earth movement event at a given location, then it is logical that an even larger quantity of
infiltrated rainfall over the same duration of time should result in a higher probability of such an earth movement event or possibly a greater magnitude of the outcome. The “rainfall threshold”
concept [4-6] embodies this idea and is well established in the literature. This paper extends the return frequency description of the rainfall threshold concept to rainfall infiltration [6].
Several rain gages were analyzed in the State of California for the purpose of accomplishing the analysis described above. The rain gages selected are located close to the locations of target earth
movement events. Because a long history of rainfall data is needed to estimate a long history of rainfall infiltration, daily rain gages were used for the estimation of daily infiltration. The
rainfalls assembled were daily rainfall values collected over time periods of several decades. Those data were then organized into all the rainfall depth-duration outcomes, namely, 1-day, 2-day,
3-day, and so forth, out to the entire water year (multiple years of depth-duration outcomes can be considered analogously). For each duration, the maximum value is selected for each year, resulting
in the annual outcome for each year. The various annual outcomes, for each duration, are ranked in size, and plotted on various types of probability paper in order to assess an underlying probability
distribution. For the current analysis, and for the selected analogs used to estimate infiltration quantities, the probability distribution considered was the standard normal distribution or the
log-normal distribution. Using the standard normal distribution as the adopted underlying distribution, occasions of earth movements under study can be correlated to the return frequency of the
infiltration corresponding to the subject earth movement.
It should be noted that severe rainfalls do not necessarily infiltrate into the ground to become causal to an earth movement event, but instead may run off the land surface to cause flooding events.
Furthermore, the arrangement and timing of a large quantity of rainfall distributed over a multiple-day depth-duration typically results in significantly different amounts of rainfall infiltration.
High intense portions of rainfall tend to result in a larger proportion of the rainfall runoff whereas a low intensity rainfall tends to become infiltrated rainfall. Consequently, the above described
infiltration depthduration analysis may better explain the occurrence of an earth movement event than provided by rainfall alone, and may be useful in assessing risk of future earth movement
occurrences. By using the above described procedure for estimating return frequencies of rainfall infiltration, plots of return frequency versus duration size can be prepared for the estimated
depth-duration infiltration quantities analogous to the plot of return frequency versus depthduration rainfall quantities.
It is contemplated that from the presented analysis, risk assessment of engineering and geotechnical works can be considered with respect to return frequency of rainfall infiltration (or even
assessment of impacts from added infiltration water such as landscape irrigation, leakage from utility pipes and reservoirs, among other such types of added water). A possible procedure to perform
such a risk assessment may adopt a desired return frequency of risk, such as 100-year (as used in FEMA floodplain designation), or other return frequency. The implementation of a prescribed return
frequency of rainfall infiltration as a design consideration throughout a region may provide the total population with a balanced risk reduction to earth movement events analogous to flood risk
reduction approaches. An assessment can then be made of the engineering or geotechnical work, for each depth-duration of infiltration estimate from 1-day through one water year, or longer if
necessary in order to capture infiltration effects of long duration. This analysis approach may also be used as another tool in assessing risk for earth movement events, such as landslides, mud
floods, debris flows, and other such outcomes that are substantially caused by infiltration of water.
2. Rain Gage Data Considered
Several rain gages were examined as part of this study. Four of the rain gages studied have locations close to landslide or mud flood events under study, and therefore the opportunity to assess the
return frequency of the associated rainfalls as well as infiltrations was available. Table 1 provides relevant information about the rain gages, as well as the distance from the considered earth
movement events to the closest rain gages.
3. Daily Infiltration Estimation Approach
There are many conceptual models and mathematical models that may be used to estimate rainfall infiltration into the soil, given rainfall data and properties of the soil [7-10]. For most situations
of interest, rainfall impacts the soil and undergoes several near-surface interactions including storage, evapo-transpiration (ET), and ponding of rainfall on the soil surface along with the wetting
of vegetation and objects (sometimes referred to as initial abstraction, or Ia). The rainfall that survives these near surface interactions percolates into the soil and then moves downwards and
laterally as described by mathematical models such as the well-known Richards equation or other similar mathematical formulations [11]. Interferences to soil water movement include soil properties of
hydraulic conductivity, water content versus soil water pore pressure corresponding to the properties of the soil itself, conditions of perched groundwater, and saturated conditions of the soil
typically associated with soils that are not “free draining”. In this paper, the soils considered and related interferences to soil water movement are assumed to correspond to “free draining” soils.
For example, if the soil is surrounded by relatively impermeable layers of soil or impermeable faults, then continuous infiltration of water may be impeded and the soil is not “free draining”. The
analysis approach assumes a free draining soil sufficient to apply the assumption that the estimated annual depth-duration infiltration outcomes are mutually independent. (It is noted that in some
earth movement phenomena, such as mud flows or mud slides, there may be significant amounts of clay or other such soils that exhibit some amount of cohesion, resulting in timing delays between the
causal infiltrated rainfall and the earth movement itself. Furthermore, many types of earth movement phenomenon, particularly landslides, involve the accumulation of soil water resulting in increased
Table 1. Rain gages examined in study.
pressures, which also takes time for the infiltrated rainfall to migrate towards such areas of soil water accumulation. These delays in timing between the causal infiltrated rainfall and the
occurrence of the earth movement may be assessed using standard soil water modeling techniques. However, the association of the causal rainfall infiltration, its depth-duration and return frequency,
still typically applies to the earth movement outcome.) Again, the procedures described herein are proposed as an extension of the rainfall threshold approach.
The National Resources Conservation Service (NRCS; previously known as the Soil Conservation Service, or SCS) continues to develop and support use of a rainfallrunoff modeling approach typically
known as the Curve Number approach (or CN method), as is described in several publications including the National Engineering Handbook [12]. The CN approach has been the subject of numerous research
publications, including [13], among others. Due to the continued and widespread use in civil engineering and irrigation studies throughout the United States (and the world), the CN approach is
utilized in this paper to estimate daily runoff values. Furthermore, since daily rainfall data are available, a procedure to estimate evapotranspiration (ET) is also used.
4. Description of NRCS Rainfall-Runoff CN Method
The NRCS’s CN approach provides estimates of daily runoff given daily rainfall and a descriptive CN value for the situation under study. The effects of prior rainfalls is considered through the
Antecedent Runoff Condition (ARC) which results in an adjustment to the selected CN of the situation such that a high ARC results in a higher CN value for the target day and a lower ARC results in a
lower CN value for the target day, causing an increase or decrease in daily runoff, respectively. Details of the ARC modification procedure can be found in the National Engineering Handbook [12]. For
a given situation under study, a CN value is selected from a tabulation of CN descriptions prepared by the NRCS after years of experimental plot measurements of the rainfall-runoff process.
The CN approach works upon values for daily rainfall and a selected watershed curve number (CN) value (between 1 and 100) that is associated with the watershed under study. The CN approach estimates
initial abstraction, Ia, by
where S is a storage parameter defined by
and daily runoff, Q, is estimated by
5. Estimation of Daily Rainfall Infiltration
The mathematical analog considered in this paper for estimating rainfall infiltration is
I(i) = infiltration estimate for day j;
P(j) = rainfall for day j, available from daily rain gage data;
Q(j) = runoff for day j, estimated using the NRCS approach for the specified CN value and ARC;
ET(j) = evapo-transpiration for day j; and Ia(j) = initial abstraction for day j, estimated from the NRCS approach for the specified CN value and ARC.
The CN adjustment procedure for a particular daily rainfall is to adjust the CN value based on the prior five day rainfall values, according to the well-known NRCS procedure for including antecedent
rainfall conditions. Adjustments of CN values follow the prescribed NRCS procedure. For durations greater than five days, if rainfall is continuous, wet antecedent conditions are typically assumed.
Similarly, if the preceding five days are dry, then the CN values are lowered as described in the NRCS procedure. Because the CN procedure does not track evapo-transpiration between storm events and
during storm events, a local ET procedure is used. In the current paper, data from the California Irrigation Management Information System (CIMIS) is used to provide the necessary ET(j) daily values
(from average monthly records), as measured at nearby CIMIS gaging stations. (Information about CIMIS, CIMIS gaging stations, and relevant information available, can be found at http://
wwwcimis.water.ca.gov/cimis/welcome.jsp). Finally, daily estimates of initial abstraction are based upon the NRCS procedure.
The computation of daily infiltration estimates from Equation (2) can be accomplished on a standard spreadsheet computer program (such as the one used in the current work) using daily rainfall data
and ET data available from sources such as the web or subsequent publications. Once daily infiltration is estimated for the entire history of daily rainfall data from the rain gage, the resulting
history of daily infiltration can be analyzed for underlying trends and statistical attributes on a depthduration basis analogous to depth-duration analysis of rainfall alone.
6. Depth-Duration Analysis of Infiltration
The above sections describe the procedure used in this paper to develop a continuous history of daily rainfall infiltration estimates, given a history of continuous daily rainfall data and selected
procedures for estimating daily rainfall infiltration. The resulting daily rainfall infiltration estimates may be subdivided into appropriate rainfall water years (such as October through September
of the following year). Next, the maximum value for each duration (i.e., the “annual outcome”) is determined, for each water year, of 1-day through the entire water year (e.g., 365-day). For N water
years of data, there will be a set of N annual outcomes, one such set for each of the various duration sizes. For each duration, the set of N annual outcomes are ranked by magnitude, and the usual
statistical percentiles (e.g., such as the 50-, 80-, 90-, 95-, 96-, 98- 99-, and higher percentiles) are computed as well as the statistical mean, standard deviation and skew. Because annual outcomes
of depth-duration daily rainfalls are used, and because the presented overall procedure assumes free draining infiltration without interference from accumulating soil water in the depths of soil
under study (such as from perched groundwater or deep groundwater conditions, among other factors), it is assumed that the selected set of such annual outcomes are mutually independent outcomes from
a random variable (as is assumed in the analysis of rainfall alone). It is noted that although it is assumed that annual outcomes are mutually independent (that is, outcomes of a target year are
independent of outcomes from a different year, which is an assumption employed for the statistical analysis of most rainfall events as well as many types of runoff events), there may be considerable
correlation and mutual dependence between various depth-duration infiltration outcomes. For example, an 8-day duration event may also contain the 1-, 2-, 3-, 4-, ··· 7-day duration events.
Consequently, as a rough guideline, the 50th percentile of the selected set of outcomes would approximate the 2-year return frequency value of daily rainfall infiltration corresponding to the
duration selected. The 99-percentile would approximate the corresponding 100-year return frequency value. Depending on the underlying probability distribution assumed, estimates can be made of the
return frequency values of durations of rainfall infiltration.
7. Distribution of Rainfall Infiltration Estimates
At each gage considered in this study, three approaches for estimating rainfall infiltration are considered:
1) Rainfall less Runoff (P-Q), where runoff is calculated using the NRCS Curve Number method with variable ARC2) Rainfall less Runoff less Evapo-Transpiration (P-QET), using average monthly ET values
from the most relevant CIMIS gage, and 3) Rainfall less Runoff less Evapo-Transpiration less Initial Abstraction (P-Q-ET-Ia), where Ia is calculated via the NRCS Curve Number method.
It is noted that for each location, the appropriate soil group was determined using NRCS Soil Survey maps, and a CN of 70 for AMC II was used. The published California State Department of Water
Resources (DWR) rainfall depth-durations are from the set of peak daily duration intervals, N = {1, 2, 3, 4, 5, 6, 8, 10, 15, 20, 30, 60, 365}, and a representative subset of these, i.e. {1, 4, 8,
20, 30, 60, 365}, are further examined. Data for the above three infiltration approximations are ranked according to their respective durations, and the Normal Order Statistical Medians, M, are
calculated for each data set as follows:
where “i” is annual event i.
(See http://www.itl.nist.gov/div898/handbook/eda/section3/ normprpl.htm, among others).
Each infiltration estimate is then graphed versus its corresponding Normal Order Statistical Medians and the graphs were inspected for linearity. In such a graph, a normally distributed set of data
will exhibit linear behavior. For example, the results for Gage 91 are shown for all three infiltration estimates in Figure 1 (P-Q), Figure 2 (P-Q-ET), and Figure 3 (P-Q-ET-Ia). (Similarly, Figures
4-6 show infiltration estimates from the other gage sites considered.) This process was repeated using log[10] of each data point in order to determine if the distribution is Log Normal. Each
resulting plot was given a Line of Best Fit and the R^2 value calculated. The R^2 values are summarized in Table 2. From Table 2, although the R^2 values are similar with respect to the two presented
Table 2. R^2 Values from normal distribution and log normal distribution test plots.
candidates, visual inspection may be needed in assessing goodness of fit of the data to generalized regression curves. For some soils and conditions, the infiltration estimates may be improved for
wet conditions in the soil by including a soil-moisture tracking method. Other models for infiltration may also be considered as well as soil-moisture accounting methods. In the current paper, the
soil is assumed to be sufficiently “free draining” such that the CN approach is adequate for estimating infiltration quantities. Probability distribution assessment may be needed on a regular basis
in order to refine estimates of rare return frequency levels (e.g., usually 100-year or rarer; or extrapolations based on a questionable record or short record length, among other issues) when
additional data are collected.
The choice of the probability distribution function used
Figure 1. Normal distribution test for Gage 91: P-Q where P = rainfall (available from daily rain gage data) and Q = runoff (estimated using the NRCS approach for the specified CN value and ARC) for
n = depth-duration size (days).
Figure 2. Normal distribution test for Gage 91: P-Q-ET where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified CN value and ARC) and
ET = evapo-transpiration for n = depth-duration size (days).
Figure 3. Normal distribution test for Gage 91: P-Q-ETIA where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified CN value and ARC),
ET = evapo-transpiration, and IA= initial abstraction (estimated from the NRCS approach for the specified CN value and ARC) for n = depth-duration size (days).
to extrapolate the rainfall infiltration estimates to rare occurrence probabilities depends upon the analysis of the resulting rainfall infiltration estimates for the various durations considered. In
this paper, the standard normal and also the log-normal distributions are considered. Other distributions are possible, such as the log-Pearson III (LP3) distribution typically used to model storm
runoff peak flow rates. This paper does not investigate the advantages or disadvantages of probability distributions and leaves that topic for future research. The choice of the standard normal
distribution is based upon apparent goodness-of-fit of the rainfall infiltration estimates to the usual normal distribution plot test as observed by the authors. The best distribution to be used is
subject to further indepth research. Indeed, it is contemplated that the apparent best distribution fit for the cases considered may be inappropriate for other locations or upon addition of more
For the case studies considered, significant variation in the usual best-fit statistical measures as well as visual inspection of best fit plots to the data both show that the applicable probability
distribution function (pdf) was not consistent across all cases, suggesting that the choice of best fit pdf may need to be made on a case-by-case basis. Further research is needed to better
understand what pdf best fits particular situations. In terms of risk analysis, use of several pdf fits may be appropriate to better describe variabilities and uncertainties in the predictions
obtained in extrapolating data to rare return frequency levels.
8. Return Frequency Estimation of Depth-Durations of Infiltration
Once the probability distributions for the various depth-duration rainfall infiltration quantities are developed, return frequency estimates can be made of particular durations of rainfall
infiltration corresponding to observed earth moving events. If the standard normal distribution is assumed to describe the depth-duration estimates of rainfall infiltration, the usual percentiles
corresponding to the mean plus (or minus) the standard deviation of the particular duration estimated rainfall infiltration estimates, can be directly used to estimate the corresponding return
frequency. Other probability distributions may be more appropriate for use such as the log-Pearson family of distributions that finds significant use in rainfall return frequency analysis [14]. It is
noted that the rainfall infiltration estimates not only depend on the rainfall data used, but also the model employed to estimate the rainfall infiltration. Different infiltration models may result
in in
Figure 4. Normal distribution tests for gage 100: P-Q, P-Q-ET, and P-Q-ET-IA where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified
CN value and ARC), ET = evapo-transpiration, and IA = initial abstraction (estimated from the NRCS approach for the specified CN value and ARC) for n = depth-duration size (days).
Figure 5. Normal distribution tests for gage 176: P-Q, P-Q-ET, and P-Q-ET-IA where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified
CN value and ARC), ET = evapo-transpiration, and IA = initial abstraction (estimated from the NRCS approach for the specified CN value and ARC) for n = depth-duration size (days).
Figure 6. Normal distribution tests for gage 372: P-Q, P-Q-ET, and P-Q-ET-IA where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified
CN value and ARC), ET = evapo-transpiration, and IA = initial abstraction (estimated from the NRCS approach for the specified CN value and ARC) for n = depth-duration size (days).
filtration estimates that may be better modeled statistically by a different distribution. More research is needed to assess if one probability distribution is appropriate for generalization to
application to all situations, such as the log-Pearson distribution which is applied to rainfall data alone.
9. Application
In order to demonstrate the above approach, a series of shallow landslides which occurred over a time span of 100 years was considered, with large landslide events occurring in years 1909, 1935,
1940, 1952, 1956, and 2006. The location of these landslides were not the same, but separated by a few hundred feet distance. Of particular interest in this situation is the relationship between the
occurrences of landslides versus the severity of the rainfall event. In order to describe the relevant rainfall infiltration, the return frequencies of the particular rainfall infiltration events are
estimated using the procedures described in this paper. The above methodology for estimating rainfall infiltration is applied to the daily rainfall data for the Berkeley Geology rain gage, listed in
Table 1. For the rainfall water year given as October to September, the entire water year of daily rainfall is broken down into the above described depth-duration intervals up to 365 days in length.
For the study, the entire history of daily rainfalls is analyzed in order to identify other comparable rainfall or rainfall infiltration events found in the past where data exist. However, for the
subject demonstration, only the water year of the landslide event is presented herein. Using the normal distribution as described above, the return frequency of each of the 365 depth-duration annual
outcomes is estimated, resulting in 365 return frequency estimate values, one for each duration size. Plots of return frequency estimate versus duration size are prepared, for both rainfall as well
as the rainfall infiltration estimates. The combined plots are shown in Figure 7. However for brevity, only the water year of the landslide event is graphically shown, although similar plots were
created as part of the computational analysis for each of the seasons where rainfall data are available [6].
From the Figure, rainfall infiltration return frequency estimates are above the 100-year level for the durations of 14 days through 21 days. For the 16-day duration, rainfall infiltration quantities
are estimated to be above the 250- year return frequency level. In comparison, the associated rainfalls corresponding to these same durations are less than about the 30-year return frequency.
Examination of the rainfalls and the associated infiltration estimates show that the rainfalls under study were of relatively low or
Figure 7. Gage 91 Rainfall and Infiltration (P-Q-ET-IA) Return Frequencies where P = rainfall (available from daily rain gage data), Q = runoff (estimated using the NRCS approach for the specified CN
value and ARC), ET = evapo-transpiration, and IA = initial abstraction (estimated from the NRCS approach for the specified CN value and ARC) for n = depth-duration size (days).
common return frequency, with low rainfall intensities, resulting in a large proportion of the rainfall being subject to infiltration, resulting in an unusually large (with respect to the historic
data) total infiltration of rainfall into the soil. From the statistical analysis of historic infiltration events (by examining all storms within the available daily rainfall record), a ranking of
total infiltration occurring across all the durations plotted in the Figure show that the subject earth movement event can be associated with a very rare infiltration event even though the underlying
rainfall event itself is of more common return frequency. From the same Figure, no other infiltration event for any duration stands out as a candidate as being a substantial cause for the subject
earth movement event. Therefore, in further investigation of cause of the landslide (or other earth movement event), the identified duration of rainfall may be considered as a good candidate of
rainfall whose infiltration into the soil resulted in reducing the relevant factor of safety for slope stability (or other earth movement) to less than a value of 1. Such further investigation may
include (but by no means is limited to) the usual groundwater and infiltration models examining pore pressure distributions and related soil water accumulation factors.
The two plots shown in Figure 7 indicate a striking difference between the return frequency estimates of rainfall alone versus the estimate of rainfall infiltration as estimated using Equation (2)
and the normal probability distribution to estimate return frequency values of estimated infiltration. For the subject situation, the duration of rainfall that had the highest return frequency value
is 17 days, with an associated return frequency value of about 30 years. However, the estimate of infiltration using Equation (2) resulted in a duration of infiltration of 16 days with corresponding
highest return frequency of over 250 years. The difference between these two return frequency values, for comparable durations, is explained by the delivery of the rainfall being of mild intensity,
resulting in a high proportion of that rainfall being subject to infiltration into the soil. This observation may be made using the same diagram but for a duration of rainfall of one day where the
corresponding return frequency corresponds to a common storm intensity. Given that most of the rainfall and also the estimated infiltration duration return frequency values corresponding to various
durations are all associated with commonplace storm events, and given that the subject landslide event is of a rare occurrence, it is logical to question whether a common place return frequency
infiltration duration event is less likely to be the substantial cause of the landslide than is the rare infiltration event estimated to have occurred for the peak 16 days as indicated in Figure 7.
This type of analysis may be helpful for many situations in the assessment of the substantial cause of earth movement events where rainfall is assumed to be the underpinnings of the earth movement
event itself. Furthermore, such an analysis may be undertaken for arbitrary durations of T-year return frequency infiltration events in the assessment of engineering designs contemplated in
preventing similar earth movement events such as landslides and mud floods or other similar outcomes. Such a consideration is analogous to many aspects of flood control design analysis of flood
control engineering works.
10. Discussion
Many earth movement events, such as landslides, mud floods, debris flows, and other such occurrences that are strongly associated with large volumes of rainfall and subsequent infiltration of
rainfall, may be explained by use of the well-known rainfall threshold approach. Another approach to explaining such earth movement events is by the return frequency of the various peak duration
rainfalls associated with these earth movement events.
For example, in [6], the landslide disaster at La Conchita, California that occurred on January 10, 2005 was analyzed and found to be well suited to such a rainfall duration assessment in terms of
the return frequency of the associated rainfalls. For that earth movement event, the application of the rainfall infiltration return frequency method described in this paper was found to be useful to
explain not only the year 2005 outcome but also prior earth movement events at the subject landslide location, including severe mass wasting in 1938. Again, as with the Example Application situation,
it was not just the severity of the rainfall itself that best explains the occurrence of the earth movement events, but the “packaging” of the rainfall. That is, milder intensities of rainfall
typically result in a larger proportion of the rainfall infiltrating into the soils, resulting in the situation that long duration “soaker” type rainfall events produce larger quantities of
infiltrated rainfall into the soil, than do larger and more severe rainfall events of rare return frequency. In [6], a plot of duration size versus corresponding return frequency of rainfall showed
that the durations of 10 through 30 days in size were particularly severe in rainfall quantities and of rare return frequency for the said La Conchita landslide (see Figure 8 herein, also seen as
Figure 7 in [6]). Although such an analysis as shown in Figure 8 is useful to identify likely durations of rainfall that are causal in the subject earth movement event, assessing rainfall alone may
not be as explanatory of the earth movement event as use of estimates of rainfall infiltration. This is because severe rainfalls do not necessarily infiltrate into the ground to become causal to an
earth movement event, but instead may runoff the land surface to cause flooding events. Furthermore, the arrangement and timing of a large quantity of rainfall distributed over a multiple-day peak
duration typically results in significantly different amounts of rainfall infiltration. High intense portions of
Figure 8. Return frequency estimates for la conchita landslide rainfall analysis.
rainfall tend to result in a larger proportion of the rainfall being runoff whereas a low intensity rainfall tends to become infiltrated rainfall.
Another issue is that of the frequency of occurrence of such earth movement events versus the frequency of occurrence of the causal rainfall infiltration, when rainfall infiltration is the
substantial causal factor. Many areas where landslides occur oftentimes have a history of similar earth movement events, and therefore the frequency of occurrence of the historic events may generally
be similar to the frequency of occurrence of the causal rainfall infiltration. The above procedure aids in assessing such nearcoincident frequencies.
Future research is needed on several topics including, but by no means limited to: 1) the sensitivity in rare return frequency predictions (e.g. 100-year, 200-year, etc.) of depth-duration
infiltration estimates with respect to methodology used for estimating infiltration from rainfall data; 2) determination of underlying distributions, particularly skewed distributions, that may
result for long depth-durations of infiltration; 3) use of smaller time intervals for determination of infiltration, such as hourly rainfall data, or 5-minute rainfall data from continuous rain gage
recordings or Alert gages, among other small time intervals; 4) multiple year depth-durations, or durations such as 500-day, 1000-day, or longer, for estimation of long term depth-duration
infiltration events; 5) application of the presented methodology to other than free draining soils conditions, including perched groundwater or groundwater table conditions, or conditions where soil
non-homogeneity effects or anisotropic effects interfere with the free draining soils assumption. Other enhancements for further research include but are no means limited to: a) different
infiltration models, b) different probability distribution functions used to fit the estimated infiltration duration outcomes, c) topics involving regionality of the infiltration estimates and also
the return frequency estimates due to concerns regarding regional geologic properties, rainfall trends, vegetative trends, among other effects, and d) other topics that influence estimates of
rainfall infiltration and the linkage between earth movement events and rainfall infiltration.
11. Conclusions
In this paper, a procedure is examined to estimate rainfall infiltration over a specified duration with a specified return frequency. A possible use of this procedure may be found in the risk
analysis of engineering works designed to handle earth movement events, such as mud floods and landslides (among others). Similar to flood control engineering works designed to handle storm runoff
events of a prescribed return frequency, instead of handling all possible flooding events, earth movement engineering works may be designed or analyzed with respect to a prescribed return frequency
of rainfall infiltration, rather than handling all possible rainfall infiltration events. Additionally, when considering the use of the “rainfall threshold” in describing the occurrence of landslides
[5,6], that rainfall threshold is associated with a rainfall infiltration, both effects being describable in terms of return frequency.
In this paper, depth-duration rainfall infiltration is estimated using a relationship based upon the commonly used National Resources Conservation Service (NRCS) Curve Number (CN) approach for
estimating daily runoff. By examining the entire history of daily rainfalls from various rain gages located in California, the NRCS CN approach is used to develop estimates of daily runoff
continuously through the gage record, and then estimates of infiltration are developed using the components of the CN relationship itself to isolate out the infiltration component of the overall
water budget. (Other such infiltration estimation methodologies can be used instead of the CN approach used herein. However, an appropriate probability distribution may be developed that adequately
describes the rainfall infiltration depth-duration trends estimated by the selected infiltration model, and therefore the concluded probability distribution may be infiltration model type dependent.)
In order to assess uncertainty in infiltration estimates, use of several infiltration modeling analogs may be undertaken and their respective estimates similarly analyzed as described in this paper.
Such displays of multiple modeling outcomes are becoming more commonplace in hydrology, meteorology (e.g. prediction of hurricane pathways over future time), among other topics. From the gage history
of estimated daily infiltration, a depth-duration analysis is accomplished for all duration sizes from one day in length to 365 days in length (resulting in 365 separate depth-duration analyses). For
any selected duration, the estimated infiltration quantities are ranked according to the maximum outcome of the selected duration, with one infiltration outcome for each year, resulting in N “annual”
depth-duration estimates of infiltration for a gage record of N years in length. These annual outcomes are then analyzed as to a possible underlying probability distribution by considering various
distributions. It is concluded that for the rain gages considered in the current analysis, and for using the CN approach to estimate infiltration quantities as shown in the paper, that the
depth-duration infiltration estimates may be distributed normally or log-normally. From the fitted probability distributions, estimates of rare events of infiltration into soil can be made.
Furthermore, possible linkage to earth movement events may be made with respect to such estimates of return frequency of infiltration, which in turn can aid in the design and risk assessment of fixed
works involving soil water accumulation, such as landslide protection measures, among others. It is envisaged that this approach to infiltration assessment may supplement other approaches such as the
Rainfall Threshold approach to analyzing the risk of landslides and similar earth movement events.
12. Acknowledgements
The authors acknowledge the Department of Mathematical-Sciences at the United States Military Academy, New York, for supporting faculty research in inter-disciplinary topics, among other
opportunities and support as well as the academic resources available through the department of the Army and the USMA. The authors also acknowledge the Department of Civil Engineering, California
State University, Fullerton, for their continued faculty support towards research efforts and associated academic resources. | {"url":"https://scirp.org/journal/paperinformation?paperid=38224","timestamp":"2024-11-08T00:02:38Z","content_type":"application/xhtml+xml","content_length":"135922","record_id":"<urn:uuid:3f54370a-6a11-454a-a89a-eb02d45326a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00436.warc.gz"} |
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accounts for the effects of vegetation on turbulent and mechanical diffusivity, following the equations in Nepf (1997, 1999).lowing the equations in Nepf (1997, 1999).)
• Model:Anuga + (ANUGA is a hydrodynamic modelling tool tha … ANUGA is a hydrodynamic modelling tool that allows users to model realistic flow problems in complex 2D geometries. Examples include
dam breaks or the effects of natural hazards such as riverine flooding, storm surges and tsunami. The user must specify a study area represented by a mesh of triangular cells, the topography and
bathymetry, frictional resistance, initial values for water level (called stage within ANUGA), boundary conditions and forces such as rainfall, stream flows, windstress or pressure gradients if
applicable.</br>ANUGA tracks the evolution of water depth and horizontal momentum within each cell over time by solving the shallow water wave governing equation using a finite-volume method.</
br></br>ANUGA also incorporates a mesh generator that allows the user to set up the geometry of the problem interactively as well as tools for interpolation and surface fitting, and a number of
auxiliary tools for visualising and interrogating the model output.</br></br>Most ANUGA components are written in the object-oriented programming language Python and most users will interact with
ANUGA by writing small Python scripts based on the ANUGA library functions. Computationally intensive components are written for efficiency in C routines working directly with Python numpy
structures.ing directly with Python numpy structures.)
• Model:Acronym1D + (Acronym1D is an add on to Acronym1R in tha … Acronym1D is an add on to Acronym1R in that it adds a flow duration curve to Acronym1R, which computes the volume bedload transport
rate per unit width and bedload grain size distribution from a specified surface grain size distribution (with sand removed).ain size distribution (with sand removed).)
• Model:Acronym1R + (Acronym1R computes the volume bedload transport rate per unit width and bedload grain size distribution from a specified surface grain size distribution (with sand removed).)
• Model:AeoLiS + (AeoLiS is a process-based model for simula … AeoLiS is a process-based model for simulating aeolian sediment transport in situations where supply-limiting factors are important,
like in coastal environments. Supply-limitations currently supported are soil moisture contents, sediment sorting and armouring, bed slope effects, air humidity and roughness elements.ects, air
humidity and roughness elements.)
• Model:FwDET + (Allow for quick estimation of water depths … Allow for quick estimation of water depths within a flooded domain using only the flood extent layer (polygon) and a DEM of the area.
Useful for near-real-time flood analysis, especially from remote sensing mapping.</br>Version 2.0 offers improved capabilities in coastal areas.rs improved capabilities in coastal areas.)
• Model:Alpine3D + (Alpine3D is a model for high resolution si … Alpine3D is a model for high resolution simulation of alpine surface processes, in particular snow processes. The model can be
forced by measurements from automatic weather stations or by meteorological model outputs (this is handled by the MeteoIO pre-processing library). The core three-dimensional Alpine3D modules
consist of a radiation balance model (which uses a view factor approach and includes shortwave scattering and longwave emission from terrain and tall vegetation) and a drifting snow model solving
a diffusion equation for suspended snow and a saltation transport equation. The processes in the atmosphere are thus treated in three dimensions and coupled to a distributed one dimensional model
of vegetation, snow and soil model (Snowpack) using the assumption that lateral exchange is small in these media. The model can be used to force a distributed catchment hydrology model
(AlpineFlow). The model modules can be run in a parallel mode, using either OpenMP and/or MPI. Finally, the Inishell tool provides a GUI for configuring and running Alpine3D.</br></br>Alpine3D is
a valuable tool to investigate surface dynamics in mountains and is currently used to investigate snow cover dynamics for avalanche warning and permafrost development and vegetation changes under
climate change scenarios. It could also be used to create accurate soil moisture assessments for meteorological and flood forecasting. for meteorological and flood forecasting.)
• Model:WBMsed + (An extension of the WBMplus (WBM/WTM) model. Introduce a riverine sediment flux component based on the BQART and Psi models.)
• Model:WSIMOD + (An open-source Python package for flexible and customizable simulations of the water cycle that treats the physical components of the water cycle as nodes connected by arcs that
convey water and pollutant fluxes between them.)
• Model:GPM + (Another derivative of the original SEDSIM, … Another derivative of the original SEDSIM, completely rewritten from scratch. It uses finite differences (in addition to the original
particle-cell method) to speed up steady flow calculations. It also incorporates compaction algorithms. A general description has been published. A general description has been published.)
• Model:AquaTellUs + (AquaTellUs models fluvial-dominated delta … AquaTellUs models fluvial-dominated delta sedimentation. AquaTellUS uses a nested model approach; a 2D longitudinal profiles,
embedded as a dynamical flowpath in a 3D grid-based space. A main channel belt is modeled as a 2D longitudinal profile that responds dynamically to changes in discharge, sediment load and sea
level. Sediment flux is described by separate erosion and sedimentation components. Multiple grain-size classes are independently tracked. Erosion flux depends on discharge and slope, similar to
process descriptions used in hill-slope models and is independent of grain-size. Offshore, where we assume unconfined flow, the erosion capacity decreases with increasing water depth. The erosion
flux is a proxy for gravity flows in submarine channels close to the coast and for down-slope diffusion over the entire slope due to waves, tides and creep. Erosion is restricted to the main
flowpath. This appears to be valid for the river-channel belt, but underestimates the spatial extent and variability of marine erosion processes.</br>Deposition flux depends on the stream
velocity and on a travel-distance factor, which depends on grain size (i.e. settling velocity). The travel-distance factor is different in the fluvial and marine domains, which results in a sharp
increase of the settling rate at the river mouth, mimicking bedload dumping.</br></br>Dynamic boundary conditions such as climatic changes over time are incorporated by increasing or decreasing
discharge and sediment load for each time step.arge and sediment load for each time step.)
• Model:BatTri + (BATTRI does the mesh editing, bathymetry incorporation and interpolation, provides the grid generation and refinement properties, prepares the input file to Triangle and
visualizes and saves the created grid.)
• Model:BITM + (BIT Model aims to simulate the dynamics of … BIT Model aims to simulate the dynamics of the principal processes that govern the formation and evolution of a barrier island. The
model includes sea-level oscillations and sediment distribution operated by waves and currents. Each process determines the deposition of a distinct sediment facies, separately schematized in the
spatial domain. Therefore, at any temporal step, it is possible to recognize six different stratigraphic units: bedrock, transitional, overwash, shoreface aeolian and lagoonal. overwash,
shoreface aeolian and lagoonal.)
• Model:BRaKE + (BRaKE is a 1-D bedrock channel profile evo … BRaKE is a 1-D bedrock channel profile evolution model. It calculates bedrock erosion in addition to treating the delivery, transport,
degradation, and erosion-inhibiting effects of large, hillslope-derived blocks of rock. It uses a shear-stress bedrock erosion formulation with additional complexity related to flow resistance,
block transport and erosion, and delivery of blocks from the hillslopes.nd delivery of blocks from the hillslopes.)
• Model:Barrier3D + (Barrier3D is an exploratory model that res … Barrier3D is an exploratory model that resolves cross-shore and alongshore topographic variations to simulate the morphological
evolution of a barrier segment over time scales of years to centuries. Barrier3D tackles the scale separation between event-based and long-term models by explicitly yet efficiently simulating
dune evolution, storm overwash, and a dynamically evolving shoreface in response to individual storm events and sea-level rise. Ecological-geomorphological couplings of the barrier interior can
be simulated with a shrub expansion and mortality module.th a shrub expansion and mortality module.)
• Model:BarrierBMFT + (BarrierBMFT is a coupled model framework f … BarrierBMFT is a coupled model framework for exploring morphodynamic interactions across components of the entire coastal barrier
system, from the ocean shoreface to the mainland forest. The model framework couples Barrier3D (Reeves et al., 2021), a spatially explicit model of barrier evolution, with the Python version of
the Coastal Landscape Transect model (CoLT; Valentine et al., 2023), known as PyBMFT-C (Bay-Marsh-Forest Transect Model with Carbon). In the BarrierBMFT coupled model framework, two PyBMFT-C
simulations drive evolution of back-barrier marsh, bay, mainland marsh, and forest ecosystems, and a Barrier3D simulation drives evolution of barrier and back-barrier marsh ecosystems. As these
model components simultaneously advance, they dynamically evolve together by sharing information annually to capture the effects of key cross-landscape couplings. BarrierBMFT contains no new
governing equations or parameterizations itself, but rather is a framework for trading information between Barrier3D and PyBMFT-C.</br></br>The use of this coupled model framework requires
Barrier3D v2.0 (https://doi.org/10.5281/zenodo.7604068)</br> and PyBMFT-C v1.0 (https://doi.org/10.5281/zenodo.7853803). (https://doi.org/10.5281/zenodo.7853803).)
• Model:RiverSynth + (Based on the publication: Brown, RA, Past … Based on the publication:</br></br>Brown, RA, Pasternack, GB, Wallender, WW. 2013. Synthetic River Valleys: Creating Prescribed
Topography for Form-Process Inquiry and River Rehabilitation Design. Geomorphology 214: 40–55. http://dx.doi.org/10.1016/j.geomorph.2014.02.025/dx.doi.org/10.1016/j.geomorph.2014.02.025)
• Model:Badlands + (Basin and Landscape Dynamics (Badlands) is … Basin and Landscape Dynamics (Badlands) is a parallel TIN-based landscape evolution model, built to simulate topography development
at various space and time scales. The model is presently capable of simulating hillslope processes (linear diffusion), fluvial incision ('modified' SPL: erosion/transport/deposition), spatially
and temporally varying geodynamic (horizontal + vertical displacements) and climatic forces which can be used to simulate changes in base level, as well as effects of climate changes or sea-level
fluctuations.climate changes or sea-level fluctuations.)
• Model:Bifurcation + (Bifurcation is a morphodynamic model of a … Bifurcation is a morphodynamic model of a river delta bifurcation. Model outputs include flux partitioning and 1D bed elevation
profiles, all of which can evolve through time. Interaction between the two branches occurs in the reach just upstream of the bifurcation, due to the development of a transverse bed slope. Aside
from this interaction, the individual branches are modeled in 1D. </br></br>The model generates ongoing avulsion dynamics automatically, arising from the interaction between an upstream positive
feedback and the negative feedback from branch progradation and/or aggradation. Depending on the choice of parameters, the model generates symmetry, soft avulsion, or full avulsion. Additionally,
the model can include differential subsidence. It can also be run under bypass conditions, simulating the effect of an offshore sink, in which case ongoing avulsion dynamics do not occur.</br></
br>Possible uses of the model include the study of avulsion, bifurcation stability, and the morphodynamic response of bifurcations to external changes.ponse of bifurcations to external changes.)
• Model:BlockLab + (Blocklab treats landscape evolution in lan … Blocklab treats landscape evolution in landscapes where surface rock may be released as large blocks of rock. The motion,
degradation, and effects of large blocks do not play nicely with standard continuum sediment transport theory. BlockLab is intended to incorporate the effects of these large grains in a realistic
way. of these large grains in a realistic way.)
• Model:Caesar + (CAESAR is a cellular landscape evolution model, with an emphasis on fluvial processes, including flow routing, multi grainsize sediment transport. It models morphological change
in river catchments.)
• Model:CoAStal Community-lAnDscape Evolution (CASCADE) model + (CASCADE combines elements of two explorato … CASCADE combines elements of two exploratory morphodynamic models of barrier evolution
-- barrier3d (Reeves et al., 2021) and the BarrierR Inlet Environment (brie) model (Nienhuis & Lorenzo-Trueba, 2019) -- into a single model framework. Barrier3d, a spatially-explicit cellular
exploratory model, is the core of CASCADE. It is used within the CASCADE framework to simulate the effects of individual storm events and SLR on shoreface evolution; dune dynamics, including dune
growth, erosion, and migration; and overwash deposition by individual storms. BRIE is used to simulate large-scale coastline evolution arising from alongshore sediment transport processes; this
is accomplished by connecting individual Barrier3d models through diffusive alongshore sediment transport. Human dynamics are incorporated in cascade in two separate modules. The first module
simulates strategies for preventing roadway pavement damage during overwashing events, including rebuilding roadways at sufficiently low elevations to allow for burial by overwash, constructing
large dunes, and relocating the road into the barrier interior. The second module incorporates management strategies for maintaining a coastal community, including beach nourishment, dune
construction, and overwash removal.ment, dune construction, and overwash removal.)
• Model:CHILD + (CHILD computes the time evolution of a topographic surface z(x,y,t) by fluvial and hillslope erosion and sediment transport.)
• Model:CICE + (CICE is a computationally efficient model … CICE is a computationally efficient model for simulating the growth, melting, and movement of polar sea ice. Designed as one component of
coupled atmosphere-ocean-land-ice global climate models, today’s CICE model is the outcome of more than two decades of community collaboration in building a sea ice model suitable for multiple
uses including process studies, operational forecasting, and climate simulation.ional forecasting, and climate simulation.)
• Model:CLUMondo + (CLUMondo is based on the land systems appr … CLUMondo is based on the land systems approach. Land systems are socio-ecological systems that reflect land use in a spatial unit in
terms of land cover composition, spatial configuration, and the management activities employed. The precise definition of land systems depends on the scale of analysis, the purpose of modelling,
and the case study region. In contrast to land cover classifications the role of land use intensity and livestock systems are explicitly addressed. Each land system can be characterized in terms
of the fractional land covers.<br>Land systems are characterized based on the amount of forest in the landscape mosaic and the management type ranging from swidden cultivation to permanent
cultivation and plantations.vation to permanent cultivation and plantations.)
• Model:CAESAR Lisflood + (Caesar Lisflood is a geomorphological / La … Caesar Lisflood is a geomorphological / Landscape evolution model that combines the Lisflood-FP 2d hydrodynamic flow model
(Bates et al, 2010) with the CAESAR geomorphic model to simulate erosion and deposition in river catchments and reaches over time scales from hours to 1000's of years.</br></br>Featuring:</br>
Landscape evolution model simulating erosion and deposition across river reaches and catchments</br></br>A hydrodynamic 2D flow model (based on the Lisflood FP code) that conserves mass and
partial momentum. (model can be run as flow model alone)</br></br>designed to operate on multiple core processors (parallel processing of core functions)</br></br>Operates over a wide range to
spatial and time scales (1km2 to 1000km2, <1year to 1000+ years)</br></br>Easy to use GUI2, <1year to 1000+ years) Easy to use GUI)
• Model:PsHIC + (Calculate the hypsometric integral for each pixel at the catchment. Each pixel is considered a local outlet and the hypsometric integral is calculated according to the
characteristics of its contributing area.)
• Model:OceanWaves + (Calculate wave-generated bottom orbital velocities from measured surface wave parameters. Also permits calculation of surface wave spectra from wind conditions, from which
bottom orbital velocities can be determined.)
• Model:SUSP + (Calculates non-equilibrium suspended load transport rates of various size-density fractions in the bed)
• Model:SVELA + (Calculates shear velocity associated with grain roughness)
• Model:BEDLOAD + (Calculates the bedload transport rates and weights per unit area for each size-density. NB. Bedload transport of different size-densities is proportioned according to the volumes
in the bed.)
• Model:SETTLE + (Calculates the constant terminal settling velocity of each size-density fraction's median size from Dietrich's equation.)
• Model:ENTRAINH + (Calculates the critical Shields Theta for the median size of a distribution and then calculates the critical shear stress of the ith, jth fraction using a hiding function)
• Model:ENTRAIN + (Calculates the critical shear stress for entrainment of the median size of each size-density fraction of a bed using Yalin and Karahan formulation, assuming no hiding)
• Model:TURB + (Calculates the gaussian or log-gaussian distribution of instantaneous shear stresses on the bed, given a mean and coefficient of variation.)
• Model:LOGDIST + (Calculates the logrithmic velocity distribution called from TRCALC)
• Model:YANGs + (Calculates the total sediment transport rate in an open channel assuming a median bed grain size)
• Model:SuspSedDensityStrat + (Calculation of Density Stratification Effe … Calculation of Density Stratification Effects Associated with Suspended Sediment in Open Channels.</br></br>This program
calculates the effect of sediment self-stratification on the streamwise velocity and suspended sediment concentration profiles in open-channel flow.</br></br>Two options are given. Either the
near-bed reference concentration Cr can be specified by the user, or the user can specify a shear velocity due to skin friction u*s and compute Cr from the Garcia-Parker sediment entrainment
relation.rcia-Parker sediment entrainment relation.)
• Model:SubsidingFan + (Calculation of Sediment Deposition in a Fan-Shaped Basin, undergoing Piston-Style Subsidence)
• Model:DeltaBW + (Calculator for 1D Subaerial Fluvial Fan-De … Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta
prograding into standing water. The model uses a single grain size D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a
constant width. Water and sediment discharge are specified per unit width. The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs a full
backwater calculation.code employs a full backwater calculation.)
• Model:DeltaNorm + (Calculator for 1D Subaerial Fluvial Fan-De … Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D
fan-delta prograding into standing water. The model uses a single grain size D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to
have a constant width. Water and sediment discharge are specified per unit width. The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs the
normal flow approximation rather than a full backwater calculation. rather than a full backwater calculation.)
• Model:CarboCAT + (CarboCAT uses a cellular automata to model horizontal and vertical distributions of carbonate lithofacies)
• Model:ChesROMS + (ChesROMS is a community ocean modeling sys … ChesROMS is a community ocean modeling system for the Chesapeake Bay region being developed by scientists in NOAA, University of
Maryland, CRC (Chesapeake Research Consortium) and MD DNR (Maryland Department of Natural Resources) supported by the NOAA MERHAB program. The model is built based on the Rutgers Regional Ocean
Modeling System (ROMS, http://www.myroms.org/) with significant adaptations for the Chesapeake Bay.</br></br>The model is developed to provide a community modeling system for nowcast and forecast
of 3D hydrodynamic circulation, temperature and salinity, sediment transport, biogeochemical and ecosystem states with applications to ecosystem and human health in the bay. Model validation is
based on bay wide satellite remote sensing, real-time in situ measurements and historical data provided by Chesapeake Bay Program.</br></br>http://ches.communitymodeling.org/models/ChesROMS/
• Model:Cliffs + (Cliffs features: Shallow-Water approximat … Cliffs features: </br>Shallow-Water approximation;</br>Use of Cartesian or spherical (lon/lat) coordinates;</br>1D and 2D
configurations;</br>Structured co-located grid with (optionally) varying spacing;</br>Run-up on land;</br>Initial conditions or boundary forcing;</br>Grid nesting with one-way coupling;</br>
Parallelized with OpenMP;</br>NetCDF format of input/output data.</br></br>Cliffs utilizes VTCS-2 finite-difference scheme and dimensional splitting as in (Titov and Synolakis, 1998), and
reflection and inundation computations as in (Tolkova, 2014). </br></br>References: </br>Titov, V.V., and C.E. Synolakis. Numerical modeling of tidal wave runup. J. Waterw. Port Coast. Ocean
Eng., 124(4), 157–171 (1998)</br>Tolkova E. Land-Water Boundary Treatment for a Tsunami Model With Dimensional Splitting.</br>Pure and Applied Geophysics, 171(9), 2289-2314 (2014)plied
Geophysics, 171(9), 2289-2314 (2014)) | {"url":"https://csdms.colorado.edu/wiki/Special:SearchByProperty/:Extended-20model-20description/Biogenic-20mixing-20of-20marine-20sediments","timestamp":"2024-11-13T07:44:14Z","content_type":"text/html","content_length":"93518","record_id":"<urn:uuid:aa53bfdb-cb72-4848-a823-6433eaeb8d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00579.warc.gz"} |
Mapping x,y sampling data to real-world GPS coordinates
Jessica writes:
Since you’re using a sampling grid with 10 meter intervals, the first thing you should do is adopt a mapping coordinate system that uses meters as well – UTM. Latitude and longitude are great for
plotting spherical data, but for any type of X,Y data, you should switch to a metric grid system. You could use California State Plane coordinates, but UTM is probably an easier choice, and you can
set your GPS to display UTM coordinates.
The next step is to create a 2D coordinate transform, between your sampling grid coordinates, and the UTM coordinate system. This is actually a lot easier than it sounds – you’re just solving two
equations that map X,Y data from your sampling grid to Easting and Northing coordinates in UTM. You’ll need to know the UTM coordinates of the four corner points in your sampling grid. The first
should be the zero point, where x and y are 0 in your sampling grid. The second is at the corner (the upper left corner) where y is at maximum, and x is zero. The third point should be at the
opposite corner, where x and y are at their maximum. The fourth is in the lower right corner, where x is at maximum, and y is zero.
Solving the Projection Equations
You have to solve two equations. The equation for Easting is: Easting = Ax + By + C. The equation for Northing is: Northing = Dx + Ey + F. C and F are easy to determine – these are the Easting and
Northing values at point 1, where x and y are both equal to zero. Because x is zero, the Ax term is also zero. Same for By. So the equation simplifies to Easting = C at the point [0,0]. And you’ve
used your GPS to determine the UTM coordinates at [0,0] are 537305E, 4394972N. So C and F are solved for.
Now consider the upper left point, where x = 0 and y = 200 meters. Because x is zero, the equations simplify to Easting = By + C. We’ve already determined C = 537305E. y = 200. And the UTM Easting at
this point is 537321E. So the equation becomes 537321E = B*200 + 537305E. Subtract 537305E from both sides, and then divide by 200. B = 0.08
Do the same thing to solve for E at this point. Then consider the opposite point, where y is zero, and x is 100 meters. Use the same technique to solve for A and D.
Converting your x,y grid to UTM in Excel
Once you’ve solved for A, B, C, D, E, and F, you’re ready to convert your x,y data into UTM coordinates. Excel or another spreadsheet program is an easy way to do this. You’ve got your x,y data in
two columns. Create two more columns, Easting and Northing. Create a formula for the values in Easting using the equation Easting = Ax + By + C. Same thing for Northing.
Now you’re finally ready to map your data. Excel is applying the two formulae to convert x,y data to UTM, using the six constants (ABCDEF) you calculated. Select your data from Excel, making sure
you’ve got the Easting and Northing values as well as the name or label for each data point. Paste this into the Waypoint List in ExpertGPS. Select UTM, WGS84 as your coordinate format (click Add if
this format doesn’t appear), and be sure to specify that your coordinates are in two columns (Easting and Northing). Enter the UTM zone (it’s 13 in the example above). On the next screen, you’ll tell
ExpertGPS which columns of Excel data contain the Easting and Northing. Once you’ve done this, ExpertGPS will map your data, and you should see your sampled data points appear on the map right where
they should be.
Need Help?
That’s probably more math than you want to think about, but the good news is you only have to do it once, and then let the computer do the rest. Those two equations are at the heart of all the
mapping features in ExpertGPS – every time you zoom in or scale the map or even move the mouse, ExpertGPS is converting back and forth between the arbitrary x,y coordinate system of your computer
monitor and the UTM geographic coordinate system. If you run into trouble, I’m happy to help. Send me the data you’re using in spreadsheet format, as well as the coordinates of the four corners of
your sample grid, as well as its dimensions.
Setting up a sampling grid using the Grid Builder in ExpertGPS Pro
See that red rectangular grid in the example above? That’s the Grid Builder tool in ExpertGPS, superimposing a 10-meter sampling grid over the map. The next time you’re setting up a sampling grid
using GPS and a map, this tool might save you some time.
One Response to Mapping x,y sampling data to real-world GPS coordinates
1. Also, if this is for a non-US location, you can use ExpertGPS to
convert your data to KML to view in Google Earth:
The View in Google Earth command in ExpertGPS allows you to view your
GPS data (waypoints, routes, and tracks) over the detailed color
imagery in Google Earth. To use the command, open a GPX file in
ExpertGPS, or retrieve the data from your GPS. Press F7, or click
View in Google Earth on the Go menu. ExpertGPS will instruct Google
Earth to synchronize its map to match the map in ExpertGPS, and to
show your GPS data.
If Google Earth is not already running when you click View in Google
Earth, it may not show your data because it is busy initializing. To
prevent this, run Google Earth and let it finish initializing and zoom
in on the globe. Then click View in Google Earth in ExpertGPS.
You can also export your data from ExpertGPS to a KML file, which you
can then view in Google Earth. Click on the map in ExpertGPS, and
then click Export Data on Map on the File menu. Change the file type
to .kml, and save to your desktop. Open the KML file in Google Earth.
ExpertGPS cannot display Google Earth’s worldwide color imagery
directly. Google’s terms of use forbid this. You can, however,
export an image from Google Earth and use it as a Scanned Map in
The ExpertGPS help file has complete details about preparing and
geo-referencing your maps for use within the program.
To get rid of the warning message that your data is outside of the
United States, which appears when you are in Topo Map or Aerial Photo
Map view, click Show Quick Map or Show Scanned Map on the Go menu.
Quick map will display GPS or GIS data for any point in the world over
a white background. | {"url":"https://www.expertgps.com/tutorials/mapping-xy-sampling-data-to-real-world-gps-coordinates.asp","timestamp":"2024-11-12T14:19:49Z","content_type":"text/html","content_length":"51633","record_id":"<urn:uuid:48149580-eab8-481b-b271-420d264bad97>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00187.warc.gz"} |
Hypothesis Test for a Difference in Two Population Means (1 of 2)
Learning Objectives
• Under appropriate conditions, conduct a hypothesis test about a difference between two population means. State a conclusion in context.
The Hypothesis Test for a Difference in Two Population Means
The general steps of this hypothesis test are the same as always. As expected, the details of the conditions for use of the test and the test statistic are unique to this test (but similar in many
ways to what we have seen before.)
Step 1: Determine the hypotheses.
The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H[0], is again a statement of “no effect” or “no
• H[0]: μ[1] – μ[2] = 0, which is the same as H[0]: μ[1] = μ[2]
The alternative hypothesis, H[a], can be any one of the following.
• H[a]: μ[1] – μ[2] < 0, which is the same as H[a]: μ[1] < μ[2]
• H[a]: μ[1] – μ[2] > 0, which is the same as H[a]: μ[1] > μ[2]
• H[a]: μ[1] – μ[2] ≠ 0, which is the same as H[a]: μ[1] ≠ μ[2]
Step 2: Collect the data.
As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.
• Samples must be random to remove or minimize bias.
• Samples must be representative of the populations in question.
We use this hypothesis test when the data meets the following conditions.
• The two random samples are independent.
• The variable is normally distributed in both populations. If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the
t-distribution. As we discussed in “Hypothesis Test for a Population Mean,” t-procedures are robust even when the variable is not normally distributed in the population. If checking normality in
the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the
variable is not heavily skewed in the populations, and we use the inference procedure. (Note: This is the same condition we used for the one-sample t-test in “Hypothesis Test for a Population
Step 3: Assess the evidence.
If the conditions are met, then we calculate the t-test statistic. The t-test statistic has a familiar form.
Since the null hypothesis assumes there is no difference in the population means, the expression (μ[1] – μ[2]) is always zero.
As we learned in “Estimating a Population Mean,” the t-distribution depends on the degrees of freedom (df). In the one-sample and matched-pair cases df = n – 1. For the two-sample t-test, determining
the correct df is based on a complicated formula that we do not cover in this course. We will either give the df or use technology to find the df. With the t-test statistic and the degrees of
freedom, we can use the appropriate t-model to find the P-value, just as we did in “Hypothesis Test for a Population Mean.” We can even use the same simulation.
Step 4: State a conclusion.
To state a conclusion, we follow what we have done with other hypothesis tests. We compare our P-value to a stated level of significance.
• If the P-value ≤ α, we reject the null hypothesis in favor of the alternative hypothesis.
• If the P-value > α, we fail to reject the null hypothesis. We do not have enough evidence to support the alternative hypothesis.
As always, we state our conclusion in context, usually by referring to the alternative hypothesis.
“Context and Calories”
Does the company you keep impact what you eat? This example comes from an article titled “Impact of Group Settings and Gender on Meals Purchased by College Students” (Allen-O’Donnell, M., T. C.
Nowak, K. A. Snyder, and M. D. Cottingham, Journal of Applied Social Psychology 49(9), 2011, onlinelibrary.wiley.com/doi/10.1111/j.1559-1816.2011.00804.x/full). In this study, researchers examined
this issue in the context of gender-related theories in their field. For our purposes, we look at this research more narrowly.
Step 1: Stating the hypotheses.
In the article, the authors make the following hypothesis. “The attempt to appear feminine will be empirically demonstrated by the purchase of fewer calories by women in mixed-gender groups than by
women in same-gender groups.” We translate this into a simpler and narrower research question: Do women purchase fewer calories when they eat with men compared to when they eat with women?
Here the two populations are “women eating with women” (population 1) and “women eating with men” (population 2). The variable is the calories in the meal. We test the following hypotheses at the 5%
level of significance.
The null hypothesis is always H[0]: μ[1] – μ[2] = 0, which is the same as H[0]: μ[1] = μ[2].
The alternative hypothesis H[a]: μ[1] – μ[2] > 0, which is the same as H[a]: μ[1] > μ[2].
Here μ[1] represents the mean number of calories ordered by women when they were eating with other women, and μ[2] represents the mean number of calories ordered by women when they were eating with
Note: It does not matter which population we label as 1 or 2, but once we decide, we have to stay consistent throughout the hypothesis test. Since we expect the number of calories to be greater for
the women eating with other women, the difference is positive if “women eating with women” is population 1. If you prefer to work with positive numbers, choose the group with the larger expected mean
as population 1. This is a good general tip.
Step 2: Collect Data.
As usual, there are two major things to keep in mind when considering the collection of data.
• Samples need to be representative of the population in question.
• Samples need to be random in order to remove or minimize bias.
Representative Samples?
The researchers state their hypothesis in terms of “women.” We did the same. But the researchers gathered data by watching people eat at the HUB Rock Café II on the campus of Indiana University of
Pennsylvania during the Spring semester of 2006. Almost all of the women in the data set were white undergraduates between the ages of 18 and 24, so there are some definite limitations on the scope
of this study. These limitations will affect our conclusion (and the specific definition of the population means in our hypotheses.)
Random Samples?
The observations were collected on February 13, 2006, through February 22, 2006, between 11 a.m. and 7 p.m. We can see that the researchers included both lunch and dinner. They also made observations
on all days of the week to ensure that weekly customer patterns did not confound their findings. The authors state that “since the time period for observations and the place where [they] observed
students were limited, the sample was a convenience sample.” Despite these limitations, the researchers conducted inference procedures with the data, and the results were published in a reputable
journal. We will also conduct inference with this data, but we also include a discussion of the limitations of the study with our conclusion. The authors did this, also.
Do the data met the conditions for use of a t-test?
The researchers reported the following sample statistics.
• In a sample of 45 women dining with other women, the average number of calories ordered was 850, and the standard deviation was 252.
• In a sample of 27 women dining with men, the average number of calories ordered was 719, and the standard deviation was 322.
One of the samples has fewer than 30 women. We need to make sure the distribution of calories in this sample is not heavily skewed and has no outliers, but we do not have access to a spreadsheet of
the actual data. Since the researchers conducted a t-test with this data, we will assume that the conditions are met. This includes the assumption that the samples are independent.
Step 3: Assess the evidence.
As noted previously, the researchers reported the following sample statistics.
• In a sample of 45 women dining with other women, the average number of calories ordered was 850, and the standard deviation was 252.
• In a sample of 27 women dining with men, the average number of calories ordered was 719, and the standard deviation was 322.
To compute the t-test statistic, make sure sample 1 corresponds to population 1. Here our population 1 is “women eating with other women.” So x[1] = 850, s[1] = 252, n[1] =45, and so on.
{−}\text{}719}{\sqrt{\frac{{252}^{2}}{45}+\frac{{322}^{2}}{27}}}\text{}\approx \text{}\frac{131}{72.47}\text{}\approx \text{}1.81[/latex]
Using technology, we determined that the degrees of freedom are about 45 for this data. To find the P-value, we use our familiar simulation of the t-distribution. Since the alternative hypothesis is
a “greater than” statement, we look for the area to the right of T = 1.81. The P-value is 0.0385.
Step 4: State a conclusion.
Generic Conclusion
The hypotheses for this test are H[0]: μ[1] – μ[2] = 0 and H[a]: μ[1] – μ[2] > 0. Since the P-value is less than the significance level (0.0385 < 0.05), we reject H[0] and accept H[a].
Conclusion in context
At Indiana University of Pennsylvania, the mean number of calories ordered by undergraduate women eating with other women is greater than the mean number of calories ordered by undergraduate women
eating with men (P-value = 0.0385).
A Comment about Conclusions
In the conclusion above, we did not generalize the findings to all women. Since the samples included only undergraduate women at one university, we included this information in our conclusion. But
our conclusion is a cautious statement of the findings. The authors see the results more broadly in the context of theories in the field of social psychology. In the context of these theories, they
write, “Our findings support the assertion that meal size is a tool for influencing the impressions of others. For traditional-age, predominantly White college women, diminished meal size appears to
be an attempt to assert femininity in groups that include men.” This viewpoint is echoed in the following summary of the study for the general public on National Public Radio (npr.org).
• Both men and women appear to choose larger portions when they eat with women, and both men and women choose smaller portions when they eat in the company of men, according to new research
published in the Journal of Applied Social Psychology. The study, conducted among a sample of 127 college students, suggests that both men and women are influenced by unconscious scripts about
how to behave in each other’s company. And these scripts change the way men and women eat when they eat together and when they eat apart.
Should we be concerned that the findings of this study are generalized in this way? Perhaps. But the authors of the article address this concern by including the following disclaimer with their
findings: “While the results of our research are suggestive, they should be replicated with larger, representative samples. Studies should be done not only with primarily White, middle-class college
students, but also with students who differ in terms of race/ethnicity, social class, age, sexual orientation, and so forth.” This is an example of good statistical practice. It is often very
difficult to select truly random samples from the populations of interest. Researchers therefore discuss the limitations of their sampling design when they discuss their conclusions.
In the following activities, you will have the opportunity to practice parts of the hypothesis test for a difference in two population means. On the next page, the activities focus on the entire
process and also incorporate technology.
Learn By Doing
National Health and Nutrition Survey | {"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/hypothesis-test-for-a-difference-in-two-population-means-1-of-2/","timestamp":"2024-11-09T19:35:12Z","content_type":"text/html","content_length":"62173","record_id":"<urn:uuid:a851d666-a7dc-48b9-bfaf-6e9e1904e635>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00788.warc.gz"} |
week on week % change beast mode
date examples are based on writing today - 6th September.
my aim is to create a beastmode, which can ultimately be applied to any metric, which shows the week-on-week % change. this value would be able to be displayed in a table by date(week) or as a KPI
card which would be filtered to show data for last week.
the goal is to have the working beastmode for the values which are pre-calculated in the 'period over period' cards. I have attached the vairance bar line example I use - I would like the variance %s
shown here as a beastmode value.
a = sum the complete last week revenue (e.g. 26th August - 1st September *Domo weeks run Sunday to Saturday)
b = sum the complete previous to last week revenue (e.g. 19th August - 25th August)
wow%change = sum(a) / sum(b) -1
e.g. 1500 / 1350 - 1 = 11.1%
I found some details on the domo knowledge board but cannot get the formula to work. I get a divide by zero error. example below...
WHEN ((year(`activity_date`)=year(curdate())) AND (weekofyear(`activity_date`)=weekofyear(CURDATE())))
THEN `revenue`
WHEN (weekofyear(curdate())=1)
THEN (CASE WHEN ((year(`activity_date`)=(year(curdate()) - 1)) AND (weekofyear(`activity_date`)=52))
THEN `revenue` END )
ELSE (CASE WHEN ((year(`activity_date`)=year(curdate())) AND (weekofyear(`activity_date`)=(weekofyear(CURDATE()) -1)))
THEN `revenue` END )
END) -1
• Hi @jonathanlooker, the reason you can get a divide by zero error is becuase you have division in your beast mode. A denominator can never equal 0, else Domo throws an error.
My suggestion is you take the entire denominator and apply a case statement up front if the denominator = 0 then the value will be 0. You numerators and denominators can contain additional case
statements, you just need one that encompises the entire formula.
CASE WHEN Denominator = 0 THEN 0 ELSE
• @jonathanlooker In addition to what @jstan said about adding a WHEN clause to prevent divide by zero errors, you have to make sure that the date range on your card is big enough to contain all
data being compared. For example, if today (9/13/18) I am looking at data for last week Mon-Sun (9/3/18-9/9/18) vs the previous period (8/27/18-9/2/18), my card has to contain data from 8/27/18
and thereafter. Filtering the card for "last 22 days" or "last 3 weeks" should contain all of this data and then update automatically as you head into a new week.
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• 5.4K Community Forums | {"url":"https://community-forums.domo.com/main/discussion/35850/week-on-week-change-beast-mode","timestamp":"2024-11-09T04:11:34Z","content_type":"text/html","content_length":"382574","record_id":"<urn:uuid:1fdd3748-7e66-49ec-848c-e400f001ca4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00556.warc.gz"} |
Why It's Always Better to Use a Longer Password | The Paradise News
Have you ever signed up for a new online account and been frustrated by the password requirements? The service often insists on a longer password and the inclusion of special characters. While it
might seem inconvenient, there’s a good reason for these demands. It’s all about entropy.
But what exactly is password entropy? What does it have to do with a longer password? Let’s take a deeper look.
What is Password Entropy?
Password entropy is essentially a measure of how unpredictable or random your password is. Think of it like a game of dice. A six-sided die has six possible outcomes. The more sides a die has, the
more unpredictable the result. Similarly, a password with more possible combinations is harder to guess.
Entropy is measured in bits, the basic unit of information in computing. Generally, experts recommend a entropy of at least 64 bits. While this number is debated, a higher entropy means a more random
and, therefore, more secure.
Now, why is randomness important? When an attacker tries to access your online account, they often target your password. If it is easy to guess, it’s like leaving the door unlocked. A strong, random
text acts as a sturdy lock, making it much more difficult for unauthorized individuals to gain access.
Dictionary Attack to Crack Passwords
One of the most common methods used by attackers to crack passwords is a dictionary attack. In this technique, a program systematically tries various combinations of words and phrases from a
dictionary or list of common passwords. It’s like a brute-force guessing game, but with a more targeted approach.
Imagine a hacker using a dictionary attack. They would feed the program with a list of common words, phrases, and even variations. The program would then attempt to match these combinations. If a
match is found, the attacker has successfully gained access to your account.
The Limitations of Predictability
Dictionary attacks are a form of brute force, essentially using brute strength to crack passwords. They’re surprisingly effective, often breaking simple passwords in mere seconds. Cybersecurity firms
like Hive Systems have published data showcasing the vulnerability of predictable texts.
Since dictionary attacks rely on predictable patterns, the best defense is to introduce randomness into your passwords. Many people instinctively try to add symbols to their existing passwords, like
“p@ssword” or “password123.” However, this isn’t random. Attackers can easily account for such predictable changes, and it might only take a fraction of a second longer for them to gain access.
Gizchina News of the week
To create a truly random password, you need to remove human intervention. A computer-generated password, created using a random number generator, is much more secure. This is where password managers
excel. These tools can generate and store strong, random texts for you, making it much more difficult for attackers to crack.
The Importance of Longer Passwords
While all four factors—length, special characters, uppercase letters, and numbers—contribute to password entropy, length plays a particularly significant role. To understand why, let’s delve into a
bit of mathematics.
Imagine a password consisting only of lowercase letters. There are 26 letters in the alphabet. If your password is 6 characters long, there are 26^6 (or 26 to the power of 6) possible combinations.
This equates to over 300 million possibilities.
Now, let’s add a single character to the password, making it 7 characters long. The number of possible combinations increases to 26^7, which is over 8 billion. As you can see, even a small increase
in length dramatically expands the number of potential combinations.
This is why length is so crucial. The longer your password, the more possibilities there are, making it exponentially harder for an attacker to guess.
More About the Entropy
While we can measure password entropy, it’s also possible to calculate it using a specific formula. The formula is as follows: E = log2(RL)
• E represents the entropy, which is the final result we’re aiming for.
• L is the length of the password, measured in characters.
• R is the range, which represents the number of characters available to you.
• log2 is a mathematical function used to calculate the number of bits needed.
Let’s break down the range concept. If you’re using a standard US keyboard layout and only lowercase letters, your range is 26. For example, a password of 8 characters using only lowercase letters
would have an entropy of 37.60 bits, which is very weak.
By adding uppercase letters, you double your range to 52. This increases the entropy to 45.60 bits, which is still not ideal. Including digits 0-9 brings the range to 62, and adding symbol keys
further increases it to 95. Even with these additions, an 8-character password would only have an entropy of 52.56 bits, falling short of the recommended 64 bits.
To significantly improve your password’s entropy, the most effective method is to simply increase its length.
How Long Should Be Your Password?
Now that we understand the importance of entropy, the question becomes: how long should it be? The answer depends on your desired level of security. While 64 bits is a common benchmark, given the
rapid advancements in cracking technology, it’s wise to aim for a higher entropy, perhaps around 100 bits.
Using the entropy formula, we can calculate that a 16-character text using all 95 available characters would provide an entropy of 105.12 bits. A 15-character option would still offer a decent level
of security at 98.55 bits. However, anything shorter would put you in dangerous territory.
If you’re working with a smaller character set (less than 95), you’ll need to increase the length further to achieve the same level of entropy.
Disclaimer: We may be compensated by some of the companies whose products we talk about, but our articles and reviews are always our honest opinions. For more details, you can check out our editorial
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Understanding Hermite Functions: Insights into Special Functions
Written on
Chapter 1: Introduction to Hermite Functions
This section marks the fourth installment of our mini-series on special functions, focusing on an intriguing operator approach to tackle differential equations.
The term "Hermite functions" refers to functions discovered by the French mathematician Charles Hermite, who is also known for the Hermitian matrices. What makes these functions remarkable, and why
should we care about them? Physics undergraduates will eventually encounter them in quantum mechanics, as they represent the stationary state solutions for the wave functions of the quantum harmonic
oscillator. These functions constitute an orthonormal basis in function space (specifically for square-integrable functions). Beyond the importance of understanding Hermite functions themselves,
there exists a fascinating operator method that can be employed to resolve their related differential equation. We will delve deeper into this in the subsequent sections.
The Hermite differential equation for the Hermite function ?_?(?) is expressed as follows:
Equation 1
For those with a background in physics, you might notice that this mirrors the structure of the stationary Schrödinger equation for a one-dimensional harmonic oscillator. In order to simplify the
following operator method, we will denote the derivative using the symbol ?:
This operator acts on all terms to its right within the same "product," leading to the following outcomes:
In this context, the operators (?+?) and (???) will play significant roles. Observe the result when these operators are combined:
This bears a strong resemblance to the earlier Hermite equation. Indeed, we can insert it:
Equation 2
We can explore the effect of applying the operators in reverse order, resulting in:
Equation 3
Now, let's examine the outcome when we apply (D-x) to Equation 2:
Renaming ??=(???)?_?, we can express this as:
By defining the integer ? as ?=??1, we get:
This precisely matches Equation 3, with ? and ? interchanged! Therefore, we can express:
Thus, applying (???) to the ?-th function yields the (?+1)-th function! This is why ??? is termed a raising operator in quantum mechanics.
Similarly, we can apply the operator (?+?) to Equation 3. You can probably predict the effect this will have!
By renaming ??=(?+?)?_?, we have:
Defining ?=?+1, we arrive at:
which is precisely Equation 2, with n and m swapped. Therefore, we conclude:
In this manner, (?+?) is recognized as a lowering operator.
This is indeed fascinating, but we still need to identify the functions themselves. However, if we can determine one for a specific ?, we can derive all others using the raising and lowering
The final concept to yield the solution is to impose the condition that there are no non-zero solutions for negative ?. Specifically, when we apply the lowering operator to ?_0, the outcome must be
zero. Thus, we require:
This is a straightforward problem to solve! By inspection, we can see that the solution (up to a multiplicative constant) is simply:
With this solution, we can derive all other functions:
Each Hermite function is represented by the same Gaussian multiplied by a polynomial ?_? of degree ?. These polynomials are referred to as Hermite polynomials. Thus, we can visualize the Hermite
functions as follows:
For the matplotlib code used to generate the figure, please refer to the supplementary page linked to this article.
Chapter 2: Further Exploration with YouTube Videos
To deepen your understanding, here are two insightful videos:
The first video, "Introduction to Hermite Polynomials," provides a foundational overview of these important mathematical constructs.
The second video, "Hermite Polynomial Generating Function," explores the methods for generating these polynomials effectively.
Thank you for engaging with this material! | {"url":"https://czyykj.com/understanding-hermite-functions.html","timestamp":"2024-11-03T16:01:40Z","content_type":"text/html","content_length":"14027","record_id":"<urn:uuid:2cea9e83-aa5a-496e-9098-a5bbd900296c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00340.warc.gz"} |
FHSST Physics/Vectors/Summary - Wikibooks, open books for an open world
Summary of Important Quantities, Equations and Concepts
[edit | edit source]
Table 3.1: Summary of the symbols and units of the quantities used in Vectors
Quantity Symbol S.I. Units Direction
Displacement ${\displaystyle {\overrightarrow {s}}}$ m yes
Velocity ${\displaystyle {\overrightarrow {u}}}$ ${\displaystyle {\overrightarrow {v}}}$ m.s^−1 yes
Distance d m -
Speed v m.s^−1 -
Acceleration ${\displaystyle {\overrightarrow {a}}}$ m.s^−2 yes
Vector: A vector is a measurement which has both magnitude and direction.
Displacement: Displacement is a vector with direction pointing from some initial (starting) point to some final (end) point and whose magnitude is the straight-line distance from the starting point
to the final point.
Distance: The distance traveled is the length of your actual path.
Velocity: Velocity is the rate of change of displacement with respect to time.
Acceleration: Acceleration is the rate of change of velocity with respect to time.
Resultant: The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together | {"url":"https://en.wikibooks.org/wiki/FHSST_Physics/Vectors/Summary","timestamp":"2024-11-13T09:59:21Z","content_type":"text/html","content_length":"53438","record_id":"<urn:uuid:6e1480ad-3484-482a-905b-e0327c498da7>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00895.warc.gz"} |
From Direct Method to Doubly Robust | Hippocampus's Garden
From Direct Method to Doubly Robust
July 29, 2020 | 3 min read | 264 views
Counterfactual machine learning is drawing more and more attention these days. This research field is now often associated with applications to advertisements and reinforcement learning, but most of
its fundamentals were developed in the field of epidemiology as causal inference
In this post, I briefly review three popular techniques developed in causal inference and formulate them with the simple notations used in bandit problems. These techniques are useful for adjusting
confounding factors, off-policy evaluation (OPE), completing missing-not-at-randoms (MNARs), etc.
Following the customs in contextual bandit, I use these notations and terms in the following part.
Notation Meaning
$\mu$ Behavior (logging) policy
$\pi$ Evaluation policy
$x_i$ Context
$a_i$ Selected arm
$r_i$ Reward
$n$ The number of records in the log
$\{(x_i,a_i,r_i)\}_{i=1}^n$ Log (data) from behavior policy
$\mathcal{A}$ Set of possible arms
$\hat{V}(\pi)$ Estimated value of policy $\pi$
In recommender systems, you can interpret the notation as follows: $x_i$ is a user feature, $a_i$ is a recommended item, $r_i$ is whether the user buy it or not, and $\hat{V}(\pi)$ is the estimated
conversion rate (CVR).
Direct Method
Direct method (DM) simply predicts the counterfactual outcomes. One can train a model with the log from the behavior policy $\mu$ to predict the reward given context and possible arms, and use the
predicted reward $\hat{r}(x_i,a)$ to estimate the value of the evaluation policy $\pi$.
$\hat{V}_{\mathrm{DM}}(\pi) = \frac{1}{n}\sum_{i=1}^{n} \sum_{a\in\mathcal{A}}\hat{r}(x_i,a)\pi(a|x_i)$
DM has low variance but its bias is large when the model is mis-specified.
Importance Sampling
Importance sampling (IS) sums the observed rewards $\{r_i\}_{i=1}^n$, but with weights of probability ratio. That is, it emphasizes the importance of the rewards that happen often in the evaluation
policy $\pi$ and rarely in the behavior policy $\mu$.
$\hat{V}_{\mathrm{IS}}(\pi) = \frac{1}{n}\sum_{i=1}^{n} r_i\frac{\pi(a_i|x_i)}{\mu(a_i|x_i)}$
IS is provably unbiased when the behavior policy is known, but it has high variance when the two policies differ a lot, where the weight term is unstable. When the behavior policy is unknown (e.g.,
observational data), you need to estimate it, and IS is no more unbiased.
IS is also referred to as inverse propensity scoring (IPS).
Doubly Robust
Doubly robust (DR) combines the above two methods. At first glance, it has a scary-looking definition, but when you compare line 1 and line 2, you will see DR is a combination of DM and residual IS.
\begin{aligned} \hat{V}_{\mathrm{DR}}(\pi) &=& \frac{1}{n}\sum_{i=1}^{n} \Biggl[ (r_i-\hat{r}(x_i,a_i))\frac{\pi(a_i|x_i)}{\mu(a_i|x_i)} + \sum_{a\in\mathcal{A}}\hat{r}(x_i,a)\pi(a|x_i) \Biggr]\\ &=&
\hat{V}_{\mathrm{IS}}(\pi) - \frac{1}{n}\sum_{i=1}^{n}\hat{r}(x_i,a_i)\frac{\pi(a_i|x_i)}{\mu(a_i|x_i)} + \hat{V}_{\mathrm{DM}}(\pi) \end{aligned}
DR is unbiased when the DM model is well-specified or the behavior policy is known. In this sense, this method is “doubly robust”. Its variance is lower than IS.
There are some advances like [2].
[1] Maria Dimakopoulou. Slate Bandit Learning & Evaluation. 2020.
[2] Mehrdad Farajtabar, Yinlam Chow, Mohammad Ghavamzadeh. More Robust Doubly Robust Off-policy Evaluation. ICML. 2018.
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For My Lovely Logicians-In-Training
(So I don't have to re-make this every time I get a new student)
Notes on Propositional Calculus
Syntax: the study of the principles for constructing grammatical entities in a language or logic.
Semantics: the study of the meanings of the symbols in a language or logic.
Syntax in PC refers to the formal structure of arguments and their component sentences, while semantics refers to the truths or falsehoods of the same.
• is a semantic notion (it's about the *meanings* of the terms used in arguments).
• is a combination of truth value assignments to the atomic propositions in a sentence or argument.
• is a syntactic notion (it's about the *structure* of an infinite set of arguments)
• does not depend on the actual truth values of the atomic sentences in the argument.
• An argument is valid if and only if any valuation rendering the premises true also renders the conclusion true.
• Operations in natural deduction (the proof system) depend on the presupposition that all the premises are simultaneously true.
Definition of propositional formula (plural formulae, pronounced form-you-lee)
• p,q,r, ect. are formulae (specifically atomic formulae)
• If X and Y are formulae,
• X*Y, XvY, X-->Y, and ~X are also formulae.
Definition of operators (aka connectives):
• *: T iff both conjuncts are T.
• v: T iff at least one disjunct is T.
• ~: T iff the negated formula is F.
• -->: F iff the antecedent is T and the conclusion F.
Related terms:
• Conjunct: a member of a conjunction
• Disjunct: a member of a disjunction
• Antecedent: the formula preceding the arrow in an implication
• Consequent: the formula following the arrow in an implication
Notes on Predicate Logic
• (∀x) applies to every element of the defined set (general)
• (∃x) applies to at least one element of the defined set (particular)
When you hear... think...
• "unless": if not
□ "Unless you like hate football you should watch the game," = "If not (hate football)--> (should watch the game)."
• "only if" or "only": following goes to consequent
□ "Only cool people like jazz," = "(like jazz) --> (cool)"
□ "You're a good dancer only if you love music." = "(good dancer)-->(love music)"
• if: following goes to antecedent
□ "If you're happy and you know it clap your hands," = "If (you're happy and you know it)-->(clap your hands)."
• even if: (it or not it) then
□ "Even if you're bad at math you might love logic," = "(bad at math or not bad at math) then (might love logic)
• not any: not some (negation of particular)
□ "There are not any more M&M's," = "not (there are some M&M's)."
• no, no one, nothing: all not (general negative)
□ "Nothing can stop me now," = "(for all x) not(x can stop me now)"
□ "No snowman lives forever," = "(for all snowment) not (live forever)"
• there is/there exists/some: particular
□ Some people like cake.
□ "There are three agencies of government when I get there that are gone..." (EPA?)
Questions to ask yourself when translating:
• Is it an argument with premises or is it a single sentence?
• What's the main connective?
• What are the quantifiers? To what parts of the sentence do they apply?
• What do I immediately know from key words in the sentence (like unless, for all, etc.)?
• When closing: again, to what parts of the sentence does the property apply?
Other notes:
• Conditional arguments are your friends. If the conclusion is an implication, assume its antecedent and derive its consequent.
• If the conclusion is a negation, assume the negated formula and derive an explicit contradiction. Conclude the negation of your assumption.
• If you're having trouble with your derivation, assume the negation of the conclusion and derive an explicit contradiction. Conclude the negation of your assumption and apply double negation
• If you're stuck on a particular derivation, stop thinking about it. If possible, sleep before coming back to it. Remember that there *is* an answer and it *will* come to you if you're patient.
• If you're confused or uncertain about a translation, find a similar sentence whose translation appears in the back of the book and study it, then come back to your translation.
• If you're feeling overwhelmed by the number of rules on your derivation cheat sheet, keep in mind that there are really only ten rules needed for any derivation in PC. If needed, confine yourself
to: assumption, conjunction introduction (conj), conjunction elimination (simplification), disjunction introduction (add), disjunction elimination (proof by cases), implication introduction
(conditional proof), implication elimination (modus ponens), negation introduction (negation of assumption upon which a contradiction has been shown dependent), negation elimination (any formula,
including the non-negated form of an assumed negation, can be concluded from an explicit contradiction), and double negation elimination. | {"url":"https://agentyduck.blogspot.com/2011/03/for-my-lovely-logicians-in-training.html","timestamp":"2024-11-07T12:48:16Z","content_type":"application/xhtml+xml","content_length":"74926","record_id":"<urn:uuid:47478045-2aaf-420f-b314-5d7ed31526ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00844.warc.gz"} |
Using Reinforcement Learning to Solve Optimization Problems in Python
What will you learn?
Discover how reinforcement learning techniques can be utilized to solve optimization problems effectively in Python.
Introduction to the Problem and Solution
Dive into the realm of leveraging reinforcement learning algorithms for solving optimization problems. By amalgamating machine learning with optimization techniques, we can train models to make
decisions resulting in optimal outcomes. This post focuses on implementing these concepts within a Python environment.
# Import necessary libraries
import numpy as np
# Define your optimization problem and constraints
# Implement reinforcement learning algorithm for solving the problem
# Visit PythonHelpDesk.com for more detailed examples and explanations
# Copyright PHD
To address an optimization problem using reinforcement learning: – Define problem parameters and constraints. – Create a reward system guiding the model towards optimal choices. – Train the model
iteratively through interactions with its environment to learn an optimal decision-making policy.
Reinforcement learning algorithms like Q-Learning or Deep Q Networks (DQN) excel in scenarios requiring sequential decision-making with long-term consequences, where traditional methods may fall
How does reinforcement learning differ from traditional optimization techniques?
Reinforcement learning involves trial-and-error interactions with an environment to optimize a reward function, while traditional methods focus on directly maximizing/minimizing an objective function
without exploration.
Can reinforcement learning handle large-scale optimization problems efficiently?
While effective for various complex tasks, scaling RL algorithms for large-scale optimizations requires careful consideration of computational resources and algorithmic enhancements.
Are there any prerequisites for implementing RL-based solutions in Python?
Understanding basic machine learning concepts, proficiency in Python, and familiarity with libraries like TensorFlow or PyTorch are beneficial for working on RL projects.
How do I evaluate my RL model’s performance on an optimization task?
Performance evaluation involves analyzing metrics such as convergence rate, solution quality compared to benchmarks, robustness under varying conditions, and computational efficiency during training/
testing phases.
What challenges are commonly faced when implementing RL for optimizations?
Challenges include defining suitable state/action spaces, designing rewarding functions encouraging desired behavior, managing exploration-exploitation trade-offs effectively, and addressing credit
assignment over time steps issues.
Reinforcement Learning presents exciting possibilities for tackling challenging optimization problems by combining machinelearning principles with classic optimization strategies. This fusion opens
avenues to address a wide range of real-world challenges. For further insights visit PythonHelpDesk.com.
Leave a Comment | {"url":"https://pythonhelpdesk.com/2024/02/25/using-reinforcement-learning-to-solve-optimization-problems-in-python/","timestamp":"2024-11-02T08:15:07Z","content_type":"text/html","content_length":"42509","record_id":"<urn:uuid:d8d30dbb-bcde-4e61-9b74-1d05fdc9c28e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00146.warc.gz"} |
Global analysis of isothermal titration calorimetry data
Complete guide of a global analysis of Isothermal Titration Calorimetry data in AFFINImeter
A few days ago AFFINImeter launched an Isothermal Titration Calorimetry (ITC) data analysis challenge. Here the participant had to globally analyse a set of Isothermal Titration
Calorimetry experiments using AFFINImeter and get the thermodynamic and structural parameters of the interaction between both molecules (the receptor protein and the ligand).
The participants in this contest had the opportunity to demonstrate their ability to propose the right model for a given binding isotherm as well as to get the corresponding parameters upon fitting
using AFFINImeter. On their side, less experienced participants had the opportunity to:
• How to find the, eventually unknown, concentration of active protein in a given experiment
During the last days we though it might be useful to help you throughout the fittings proposed for the contest. Therefore, we have prepared a video that explains, step by step, How to fit CURVE 1.
IMPORTANTLY, this information will be of great help to solve step 2, the global fitting of CURVES 1-4.
The data proposed for the contest will remain available here, in case you want to learn more about it or to try it by yourself.
Tips to solve Step 1
Tips to solve Step 2
Here are some tips to solve step 2 (global fitting of curves 1-4):
When talking about linking parameters think of the thermodynamic parameters (K and DH) that are common among curves. In other words, determine which curves describe the same binding event(s):
Dr. Brooks gave us a set of four curves:
CURVE 1 and CURVE 2 describe the interaction of L1 and L2 with the protein, respectively. They describe different interaction events and therefore they don´t have any thermodynamic parameter in
CURVE 3 corresponds to a competitive experiment of a mixture of L1 and L2 binding to the protein. This means that, it has the thermodynamic information of both interaction events: CURVE 3 will share
thermodynamic parameters with CURVE 1 and CURVE 2…(and also CURVE 4).
CURVE 4 is also a competitive experiment, similar to CURVE 3 and again, has the thermodynamic information of both interactions. Then, CURVE 4 shares thermodynamic parameters with CURVE 1, CURVE 2 and
CURVE 3.
Have a look at the following drawing, with a summary of the parameters that are common among Dr. Brooks curves:
NOW,…how do we define all this information into our AFFINImeter global fitting? We have to “link” parameters among curves to “tell” AFFINImeter that we are considering those parameters as equal.
As an example, here are the steps to follow in order to link the parameter in CURVES 1, 3 and 4 that correspond to the association constant of L1 binding to s1 of the protein:
1. Click on the button “link parameters” and then click on the Value/eq box corresponding to the parameter K of CURVE 1 that describes the association constant of L1 to s1.
2. Click on the Value/eq box corresponding to the parameter of CURVE 3 that defines the association constant of L1 to s1.
3. Click on the Value/eq box corresponding to the parameter of CURVE 4 that defines the association constant of L1 to s1.
4. Click on the button “Done” to finish and leave the link parameter mode (NOTE: when you are in the link parameter mode all the functions in the screen are locked except the link function. In order
to recover all the functionalities you have to click on “Done”).
5. IMPORTANTLY! The box “FIT” has to be checked for the parameter K of CURVE 1 but it has to be unchecked for the parameter K of CURVES 3 and 4, since now they are related to K of CURVE 1.
Here is a picture that shows how the settings of my project look like after the steps described above: | {"url":"http://blog.affinimeter.com/global-analysis-of-isothermal-titration-calorimetry-data/","timestamp":"2024-11-14T07:44:00Z","content_type":"text/html","content_length":"46063","record_id":"<urn:uuid:5ad19523-923f-4399-9b68-dce74683fb75>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00075.warc.gz"} |
bond index ball mill
The Bond work index, Wi, is defined in a Bond ball mill on the samples of standard size − + 0 mm. In practice, it is possible to find materials of nonstandard size that are finer than 3. ...
Average grinding work index (Wi) by type of mate rial, according to Bond (see table ): The work index (Wi) expresses the kWh required to reduce a short ton of material from theoretically infinite
feed size to 80 % passing a square screen opening of 100 microns. The work index values apply to ball mills grinding wet in closed circuit.
An alternative mill ball charge is proposed that closely approximates Bond's original total ball mass, number of balls and ball surface area. Results of 30 Bond Work index tests of six pure materials
(calcite, magnesite, labradorite (feldspar), quartz, andalusite and glass) using closing screen apertures (P 1 ) values of 500, 250, 125, 90 and ...
Conclusions. The power model and linear model both will give reasonable P80 estimates given the P100 of a Bond ball mill work index test product. The power model might be marginally more robust ...
A copy of Fred C. Bond's Method of Crushing and Grinding for determination of the Bond Index is included with each mill. This Ball Mill can be used continuously or it can be used for any number of
revolutions, according to the type of grind desired. The FC Bond ball Mill comes with table stand, motor, clutch, revolution counter, motor starter ...
The Bond ball mill grindability test is run in a laboratory until a circulating load of 250% is developed. It provides the Bond Ball Mill Work Index which expresses the resistance of material to ...
Design engineers generally use Bond Work Index and Bond energy equation to estimate specific energy requirements for ball milling operations. Morrell has proposed different equations for the Work
Index and specific energy, which are claimed to have a wider range of application. In this paper an attempt has been made to provide a comparison of these two approaches and bring out their ...
The basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill sizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density,
desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum 'chunk size', product size as P80 and maximum and finally the type of circuit open/closed ...
The laboratory procedure for running a Bond ball mill work index test requires that the operator choose a closing screen sieve size. The instruction is to choose a sieve size that results in the ...
Section snippets Material and experimental Bond tests. Samples of dacite and basalt from Serbia, copper ore from a South American metasedimentary copper mine, and copper porphyry ore from Canada, of
different feed sizes, were used to determine the Bond ball mill work index according to the standard standard Bond laboratory mill with "DxL = 305 × 305 mm" balls and rotation ...
The results showed that using the nonstandard mills (between 20 and 35 cm in diameter), the Bond´s model constants (α=; β=, and γ = ), are unable to predict the Work Index ...
Process Engineering of Size Reduction: Ball Milling. Society of Mining Engineers of the American Institute of Mining, Metallurgical and Petroleum Engineers (AIME) Inc, ISBN, New York, 556 pp.
In particular, the Bond work index (wi) is considered a critical parameter at an industrial scale, provided that power consumption in comminution operations accounts for up to 40% of operational
costs. Despite this, the variability of wi when performing the ball mill Bond's standard test is not always understood enough.
Bond Ball mill work index determination (1 reply) s. srinu1234. 5 years ago. srinu1234 5 years ago. Dear all, Have a good day! I am trying to determine ball mill work index for a very soft . If I
follow bonds procedure to do experiment, not able to determine. The following is the data of ore sample. F80% = 2300mic.
Rod mills generate small amounts of fines, as fine particles typically accumulate in the space between rods without enduring comminution [2]. Effective grinding of this material occurs in the ball
mill compartment, generating finer fragmentation. The Bond Rod Mill Work Index (BRMWI) test was only possible for Blend 5 (Table 2) due to excessive
You need a twostage solution, first stage opencircuit mill and then second stage closedcircuit mill. First stage, will be broken into two parts as well, you use a Bond rod mill work index for the
coarse component of the ore (+ mm) and the Bond ball mill work index for the fine component ( mm).
HGI vs Bond Index Test Bond Index HGI Mill Comparison Tube Ball mill Babcock mill (ringball) Target Particle Size Any size below 75µm Particle Size Range < (powder) or pellet size µm Mass
constriction Volume 700ml Mass 50g Output kWh/ton HGI Index Suitable materials Any Good quality coals
The Bond Ball Mill Grindability Test (Bond, 1952, 1961) gives the Bond Ball Mill Work Index. This Index expresses the resistance of a material to ball milling; the higher the value of the Bond Ball
Mill Work Index, the more difficult it is to grind the material using a ball mill. This Index is widely used in the mineral industry for:
This Table of Ball Mill Bond Work Index of Minerals is a summary as tested on 'around the world sample'. You can find the SG of each mineral samples on the other table. Source 1. Source 2. Source 3.
Source 3. Source 4.
Once finished the grinding cycles, a minimum of five, the ball mill Bond's work index w i [kWh/sht] can be calculated using Equation (2). In order to express it in metric tons, the corresponding
conversion factor must be used. wi = 100 p 10 P80 p F80 (2) where: w i is the ball mill Bond's work index [kWh/sht];
µm. Note that—in order for no correction factor for ball mill product fineness to apply—the ball mill circuit P80 should be no less than approximately 70 µm (Bond, 1962). This Bond Efficiency
determination should not be applied to circuits with a P80 finer than approximately 70 µm without making qualifications. 3.
Based on the standard procedure defined by Bond, the ball mill work index is determined by simulating a closed cycle dry grinding in a laboratory Bond ball mill until a stable circulating load of
250% is established (Bond, 1949, Bond, 1952, Bond, 1961). The ball mill work index is determined by Eq. (10). The recommendation is the ball mill ...
In the standard AC closed circuit ball mill grindability test the work index is found from. where Pi is the opening in microns of the sieve mesh tested, and Gbp is the net grams of mesh undersize
produced per revolution of the 12″ x 12″ test ball mill. The closed circuit 80% passing size P averages P1/log 20 for all sizes larger than 150 mesh.
The Bond ball mill grindability test is one of the most common metrics used in the mining industry for ore hardness measurements. The test is an important part of the Bond work index methodology ...
Bond ball mill w ork index test is used to ca lculate the work index rel ated to fine grinding (Mib). Mia and Mib are used to cal culate Ecs for the coarse (Wa) and fine (Wb) components acc ording .
The ball mill grindability test is used for describing ore hardness and it is so widespread that the Bond Work Index generated from the test is often referred to as an ore characteristic. The ore
resistance to grinding and energy consumption can be expressed using the work index and Bond's Third Theory.
Key words: Comminution, Bond work index, Grinding, Crushing, Ball mills, Rod Mills. Theoretical efficiency In their book Process Engineering of Size Reduction Ball Milling, Austin, Klimpel and Luckie
(1984) stated that, "We have tried throughout this book to avoid the use of the term grinding efficiency because the degree of conversion of
The ball mill grindability test is used for describing ore hardness and it is so widespread that the Bond Work Index generated from the test is often referred to as an ore characteristic. The ore ...
The ball mill is designed to accept a ball charge in accordance with F. C Bond's standard. This ball charge consists of 286 balls as follows: 44 x 35mm balls; 67 x 30mm balls; 10 x balls, 71 x balls
and; 94 x balls; Fully automated version of the Bond Index Ball Mill is also available on request.
Bond Work Index Ball Mill (BWi) The Bond Ball Mill Work Index (BWi) is used to calculate the power requirements to grind ore to a typical ball mill product. Table 3 shows the BWi for each composite.
Table 3 ‐ Bond Ball Mill Work Index Sample ID Closing Screen Size (μm) F80 (μm) | {"url":"https://petite-venise-chartres.fr/bond/index/ball/mill-5588.html","timestamp":"2024-11-06T05:30:04Z","content_type":"text/html","content_length":"44976","record_id":"<urn:uuid:66ac2d0b-0681-40f9-aaad-8a782fc6a9f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00489.warc.gz"} |
Acceleration 4211 - math word problem (4211)
Acceleration 4211
At the start of braking, the car had a speed of 72 km h at -1. It stopped on a track of 50 m. What was the acceleration, and how long did the braking last?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
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You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Themes, topics:
Grade of the word problem:
We encourage you to watch this tutorial video on this math problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/4211","timestamp":"2024-11-05T15:27:25Z","content_type":"text/html","content_length":"62794","record_id":"<urn:uuid:8c59d6ff-cfa4-4403-8f4f-8bafeace2a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00709.warc.gz"} |
Hypothesis Testing with Z test |
Hypothesis Testing with Z test
By its nature, statistical tests give an answer to questions like:
• What is the probability of the observed difference be due to pure chance? (Q1)
For an actionalbe insight, a yes/no question is often preffered. In abstract terms, we seek to answer another question:
• Based on the observations, can we reject the null hypothesis $H_0$? (Q2)
where the null hypothesis $H_0$ refers to a statement that the treatment has no effect on the outcome. In terms of the business problem, the above question can sound something like
• Does the new website design improve the subscription rate by $1.5$% or more?
In this case, $H_0$ will be the statement that the new website design does not improve the subscription rate by at least $1.5$%
Before jumping into experiments, couple decisions need to be made. These are decisions about the tollerance we have to a random chance messing with the test and leading us to wrong conclusions. These
have to be set before the start of the test to avoid bias in interpreting result. There are two types of error, so two thresholds to agree upon.
Null Hypothesis Actually is
Decision True False
Don't Correct Decision Type 2 Error
Decision Reject (true positive) (false negative)
about probability = $1 - \alpha$ probability = $\beta$
Null Hypothesis Type 1 Error Correct Decision
Reject (false positive) (True negative)
probability = $\alpha$ probability = $1 - \beta$
The first one is the probability threshold $\alpha$ called sensitivity. Assume that $H_0$ was actually correct, treatment made no difference. If the estimated probability from Q1 is at or below $\
alpha$, we incorrectly reject null hypothesis and conclude that the treatment made a difference. In particular, there is a less than $\alpha$ chance that the observed effect was purely due to random
chance, which is reffered to as the type 1 Error. The latter means that we are willing to tollerate the potentially incorrect decision to reject $H_0$ if the probability of such a scenario is no more
than $\alpha$.
The second one probability threshold $\beta$ considers the scenario when $H_0$ is actually incorrect, but the experiment indicates that it is not. The complementary probability, $1 - \beta$, is
referred to as statistical power and describes a degree of certainty we want to have in been correct if we decide to reject $H_0$. In other words, statistical power gives us an expected fraction of
correctly rejecting null hypothesis if we run many such experiments.
Example problem
To show how Z-test would work in practice, lets quantify the effect of treatment on the probability of a certain outcome. Consider a toy example:
“Does the new website design improve the subscription rate by at least $1.5$%?”
As an example, we seek to answer the above question. In the simplest case, we assume that
• For the experiment, customers are selected at random from a cohort that represents target audience
• Customers make decisions independent of each other
• Customer traffic is randomly split into two groups, A and B, depending on which version the website, new or old, they were presented with
• Each of two groups has sufficiently large (for Central Limit Theorem to work) number of customers
Experiment setup
Under these assumptions, the probability distribution for a number of subscriptions in each group is given by the binomial distribution with probabilities of a customer subscribing are $p_A$ and
$p_B$ respectively. In addition, lets assume that for the old version of the website there is an estimate for the value of the subscription probability $p_A$, which is $8.5$%. From this point, we can
use 5 steps 100 Statistical Test as a guide to setup the experiment
Step 1: Null Hypothesis
The null hypothesis states that $p_B - p_A < 0.015$, so that the new website design has less than the desired effect on the subscription probability.
In the language of mathematics, the null hypothesis $H_0$ states that $\mu = p_B - p_A = 0.015$.
Step 2: Sensitivity and Power
Next, we need to select values of sensitivity and power that we are comfortable with. Usually, $\alpha$ for sensitivity is selected to be between 1 and 10 percent, while for power $1 - \beta$ is of
the order of at least 80 percent. For this example lets set
$$\alpha = 0.05$$ $$1 - \beta = 0.8$$
These values come from classical literature and aim to strike a balance between detecting the effect of treatment if it is present, while reducing the cost of collecting the data. For more detailed
discussion check out p. 17 in Statistical Power Analysis for the Behavioral Sciences and p. 54, 55 in The Essential Guide to Effect Sizes.
Step 3: Test statistic
If the number of customers $N_A$ and $N_B$ is sufficiently large, we can invoke Central Limit Theorem to describe the probability distributions of the subscription number means in each group $$\
overline x_A \sim N(p_A, \sigma_A)$$ $$\overline x_B \sim N(p_B, \sigma_B)$$
where $N(\mu, \sigma)$ is a normal distribution. For the test statistic $Z$, we can use the rule for a sum of normally distributed random variables
$$Z = \frac{\overline x_B - \overline x_A - \mu}{(\sigma_A + \sigma_B)^{1/2}} \sim N(0, 1)$$
If the p-value for the test statistic is at or below $\alpha = 0.05$, we must reject $H_0$ and accept an alternative hypothesis $H_1$ that version B of the website does improve the subscription rate
by $1.5$ % or more than the current version A.
Step 4: Confidence Interval for test statistic
Based on test statistic $Z$ and sensitivity $\alpha$, select critical region. It is evident from the null hypothesis that we need to use a one-tail region $Z < Z_c$ only. Since $Z$ is described by a
normal probability distribution function (pdf),
critical value is $Z_c \approx 1.64$ for $\alpha = 0.05$
Corresponding cumulative distribution function for this critical value $Z_c$ covers $100\times(1 - \alpha) = 95$ % interval. Should the value of test statistic fall inside the interval $(-\infty,
Z_c)$, we accept null hypothesis, otherwise we accept an alternative hypothesis.
Step 5: Effect Size and Sample Size
Finally, lets estimate a minimal sample size required to achieve desired statistical power. In addition to sensitivity and power, effect size needs to be evaluated, which requires a priori
information and / or assumptions. For equal number customers in both groups with the assumption that this number is sufficiently large (so that the sampling distribution is well approximated a normal
one), Cohen’s D can be used to approximate effect size
$$D = \frac{|p_B - p_A|}{\sigma}$$
where $\sigma^2 = s_A^2 + s_B^2$ and $s_{A(B)} = \sqrt{p_{A(B)}(1 - p_{A(B)})}$. Such estimate would require an assumption that a good estimate for $p_B$ is available, which is likely unrealistic.
Nevertheless, assume that
A good estimate for $p_B$ is $11$%
Substituting the parameters, we find $D = 0.0239$.
Estimating required number of test participants
With all pieces of the puzzle in place, we can proceed to estimating sample size from effect size, sensitivity and power. Python statsmodel package comes handy for this task
from statsmodels.stats.power import NormalIndPower
ALPHA = 0.05
BETA = 0.2
PowerAnalysis = NormalIndPower()
N_total = PowerAnalysis.solve_power(effect_size=D, alpha=ALPHA, power=1 - BETA, alternative="larger", ratio=1)
The calcalation returns $21723$ for a minimal total number of customers in both groups, or $N_{A(B)} = 10862$ in each group.
Under the hood, above code involves numerically solving this equation for $N_{A(B)}$
$$\beta = \Phi(Z_c - D\times\sqrt{N_{A(B)}})$$
where function $\Phi$ is a probit function, an inverse of cumulative distribution function (CDF) for the normal didistribution. With power $1 - \beta = 0.8$ we can verify this by substituting the
result into a CDF for the normal distribution, which should be $\beta = 0.2$ or less:
from scipy.stats import norm
Zc = 1.64
norm.cdf(Zc - D*np.sqrt(N_total/2))
returns $0.199$.
Code for Experiments
Here is the notebook with the whole code. Each experiment consists of simulating the observed data and running the Z-test for this data.
from statsmodels.stats.weightstats import ztest
import numpy as np
def run_ztest(p0, p1, N_samples, value):
# generate data
data_0 = np.random.binomial(1, p0, N_samples)
data_1 = np.random.binomial(1, p1, N_samples)
# run Z-test
result = ztest(data_1, data_0, alternative="larger", value=value)
return result
The above code generates two sets of data, for both groups $p_A$ and $p_B$, which simulates the actual observation. Next, function ztest returns two values, test statistic $Z$ and p-value, the
probability that data sets come from the same distribution, which corresponds to null hypothesis. Parameter value is set to the threshold value of the effect $0.015$ ($1.5$ %). Parameters p0 and p1
are the true, unknown values for probabilities of subscription.
True subscription rate due to new website design either exceed or is below the threshold. Consider the former case when it exceeds the threshold and null hypothesis has to be rejected. The analysis
is the same for the latter case. In the considered case lets focus on the Type II Error, when the test shows that the new website design does not achive the goal.
There are three possible scenarios for the true value of subscription probability in group B compared to the estimate $p_B = 0.11$ that we obtained:
• $p_{true} \approx p_B$
• $p_{true} < p_B$
• $p_{true} > p_B$
In each of these scenarios the chances of making correct decision whenever or not to reject null hypothesis. Lets consider each of these possibilities.
Actual subscription probability is $11$ % due to the new website design
Good estimate for $p_B$ allows to accurately estimate the effect size, and consequently the number of participants required to achive the target statistic power of $80$ %. Running 1000 identical
tests, correct decision will be made in $77.9$ %.
Actual subscription probability is $10.5$ % due to new website version
Estimate for $p_B$ was too optimistic, resulting in overly high effect size. Running 1000 identical tests, correct decision will be made in $35.8$ %. However, the subscription probability exceeds the
target of $1.5$ % improvement. To get a good confidence in the test, more samples are needed.
Actual subscription probability is $11.3$ % due to new website version
Estimate for $p_B$ was too pesimistic, resulting in overly conservative effect size. On the good side, running 1000 identical tests, correct decision will be made in $93.6$ %.
Even though in all of the above cases the new website design achieved the goal, out chance of picking it with Z-test was dramatically different. One way to address this is to be more conservative
when estimating effect size, but that might be incure costs in practice. Another is to explore different test setups, which is my future quest. | {"url":"https://alexkozlov.com/post/z-test/","timestamp":"2024-11-03T16:14:07Z","content_type":"text/html","content_length":"28081","record_id":"<urn:uuid:67e0bf09-5a95-4642-ba7f-2e78c0370288>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00216.warc.gz"} |
Using Generalised Additive Mixed Models (GAMMs) to predict visitors at Edinburgh & Craigmillar Castles
If you attended my talk on “Generalised Additive Models applied to tourism data” at the Newcastle Upon Tyne Data Science Meetup in May 2019, please find my (more detailed) slides linked below.
Some context
I’d been curious about generalised additive (mixed) models for some time, and the opportunity to learn more about them finally presented itself when a new project came my way, as part of my work at
The Data Lab. The aim of this project was to understand the pattern of visitors recorded at two historic sites in Edinburgh: Edinburgh and Craigmillar Castles - both of which are managed by Historic
Environment Scotland (HES).
By understand the pattern of visitors, I really mean predict it on the basis of several ‘reasonable’ predictor variables (which I will detail a bit later). However, it is perhaps worth starting off
with a simpler model, that predicts (or rather in this case, forecasts) visitor numbers from… visitor numbers in the past. In a sense, this is similar to a classic time series scenario, where we
would use the data to predict itself, without resorting to “external” predictors.
Craigmillar Castle. Image source here.
We will begin our discussion by first having a quick look at the data available. Then we will very briefly pause on some modelling aspects, to then have a more meaningful discussion of our data
forecast (created in R using package mgcv). Afterwards, we can then discuss the additional sources of data that were used as predictors in a more complex, subsequent model.
Edinburgh Castle. Image source here.
The data
Our monthly time series presents itself in long format, and is split by visitors’ country of origin as well as the type of ticket they purchased to gain entry to either of the two sites. Thus each
row in the dataset details how many visitors were recorded:
• for a given month (starting with March 2012 for Edinburgh, and March 2013 for Craigmillar Castle, until March 2018),
• from a given country (UK, for internal tourism, but also USA or Italy etc.)
• purchasing a given ticket type (Walk up, or Explorer Pass etc.)
• visiting a given site (Edinburgh or Craigmillar).
To build our first, simpler gamm model, we will collapse visitor numbers across country and ticket type, and only look at variations in the total number of visitors per month for each site. This
collapsed data looks like this if plotted in R using ggplot2:
These data exhibit an interesting pattern of seasonality over summer (for both castles) and around Easter (especially for Craigmillar Castle), as well as a general - but modest - upward trend. But
will our gamm model pick up on these aspects correctly?
So what are gamm models? To get a better idea, let’s have a look at where they fit within a conceptual progression relative to other types of models.
Types of models
Regular old linear regression
If trying to predict an outcome y via multiple linear regression on the basis of two predictor variables x[1] and x[2], our model would have this general form:
Translated into R syntax, a model of this nature could look like:
lm_mod <- lm( Visitors ~ Temperature + Rainfall, data = dat )
As the name suggests, this type of model assumes linear relationships between variables (which, as we’ve seen from the plot above, is not the case here!), as well as independent observations - which,
again, is highly unlikely in our case (as visitor numbers from one month will have some relationship to the following month).
Generalised additive models (GAMs)
Under these circumstances, enter gam models, which have this general form:
• y = b[0] + f[1](x[1]) + f[2](x[2]) + e
As you will have noticed, in this case the single b coefficients have been replaced with entire (smooth) functions or splines. These in turn consist of smaller basis functions. Multiple types of
basis functions exist (and are suitable for various data problems), and can be chosen through the mgcv package in R. These smooth functions allow to follow the shape of the data much more closely as
they not constrained by the assumption of linearity (unlike the previous type of model).
Using R syntax, a gam could appear as:
library( mgcv )
gam_mod <- gam( Visitors ~ s( Month ) +
s( as.numeric( Date ) ) +
te( Month, as.numeric( Date ) ) + # But see ?te and ?ti
data = dat )
However, gam models do still assume that data points are independent - which for time series data is not realistic. For this reason, we now turn to gamm (generalised additive mixed) models - also
supported by package mgcv in R.
Generalised additive mixed models (GAMMs)
These allow the same flexibility of gam models (in terms of integrating smooths), as well as correlated data points. This can be achieved by specifying various types of autoregressive correlation
structures, via functionality already present in the separate nlme::lme() function, meant for fitting linear mixed models (LMMs).
Unlike (seasonal) ARIMA models, with gamms we needn’t concern ourselves with differencing or detrending the time series - we just need to take these elements correctly into account as part of the
model itself. One way to do so is to use both the month (values cycling from 1 to 12), as well as the overall date as predictors, to capture the seasonality and trend aspects of the data,
respectively. If we believe that the amount of seasonality may change over time, we can also add an interaction between the month and date. Finally, we can also specify various autoregressive
correlation structures into our gamm, as follows:
# mgcv::gamm() = nlme::lme() + mgcv::gam()
gamm_mod <- gamm( Visitors ~
s( Month, bs = "cc" ) +
s( as.numeric( Date ) ) +
te( Month, as.numeric ( Date ) ) +
data = dat,
correlation = corARMA( p = 1, q = 0 ) )
Forecast model
We can now essentially apply a similar gamm to the one above to our data, the key difference being that we can allow the shape of the smooths to vary by Site (Edinburgh or Craigmillar), by specifying
the predictors within the model for instance as: s( Month, bs = "cc", by = Site ), where the bs = "cc" argument refers to the choice of basis type - in this case, cyclic cubic regression splines
which are useful to ensure that December and January line up. In our model, we can also add in a main effect for Site.
All this will have been done after standardising the data separately within each site, to avoid results being distorted by the huge scale difference in visitor numbers between sites. At the end of
this process, the output we get from our gamm is this:
Family: gaussian
Link function: identity
Visitors ~ s(Month, bs = "cc", by = Site) + s(as.numeric(Date),
by = Site) + te(Month, as.numeric(Date), by = Site) + Site
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.04312 0.03878 -1.112 0.2687
SiteEdinburgh 0.08738 0.05119 1.707 0.0908 .
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Approximate significance of smooth terms:
edf Ref.df F p-value
s(Month):SiteCraigmillar 7.274 8.000 22.707 < 0.0000000000000002 ***
s(Month):SiteEdinburgh 6.409 8.000 63.818 < 0.0000000000000002 ***
s(as.numeric(Date)):SiteCraigmillar 1.001 1.001 38.212 0.00000000931069 ***
s(as.numeric(Date)):SiteEdinburgh 1.000 1.000 57.220 0.00000000000612 ***
te(Month,as.numeric(Date)):SiteCraigmillar 6.276 6.276 2.935 0.00787 **
te(Month,as.numeric(Date)):SiteEdinburgh 3.212 3.212 3.316 0.02018 *
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.913
Scale est. = 0.081956 n = 132
The output is split between ‘parametric’ (unsmoothed) coefficients, and smooth terms. A key concept here is that of Effective Degrees of Freedom (EDF), which essentially tell you how ‘wiggly’ the
fitted line is. For an EDF of 1, the predictor was estimated to have a linear relationship to the outcome. Importantly, mgcv will penalise overly wiggly lines to avoid overfitting, which suggests you
can wrap all your continuous predictors within smoothing functions, and the model will determine whether/to what extent the data supports a wiggly shape.
As a measure of overall fit for the gamm model, we also get an Adjusted R-squared at the end of the output (other measures such as GCV - or Generalised Cross Validation - are offered for gam models,
but absent for gamm - details in my slides). Judging by this, our model is doing a good job of describing our data, so we can move on to generating a forecast as well… After converting the
standardised values back to the original scale, this is how our prediction compares to the raw visitor numbers:
While the prediction produced follows the original data quite closely, it's worth noting the confidence intervals are impractically large and (following the conversion back to the original scale),
also dip below 0, which for visitor numbers makes little sense. Possible solutions could be using a different family option in gamm() (perhaps Poisson or Quasi-Poisson), or at least truncating the
lower bound of the confidence intervals.
Explanatory model
In order to check whether other variables may also contribute / relate to the pattern of visitors we have observed above, several types of data were collected from the following sources:
• The Consumer Confidence Index (CCI) developed by the Organisation for Economic Co-operation and Development (OECD) - this is an indicator of how confident households across various countries are
in their financial situation, over time.
• Google Trends R package gtrendsR - measuring the amount of hits over time and from various countries, for the queries: “Edinburgh Castle” and “Craigmillar Castle”.
• Global Data on Events, Language and Tone (GDELT) - this is a massive dataset measuring the tone and impact of world-wide news events as extracted from news articles. Data related to only Scottish
events was selected.
• Internet Movie Database + Open Movie Database - these sources were scoured for data on productions related to Scotland (in terms of their plot or filming locations).
• For-Sight hotel booking data, including information on the number of nights or rooms booked across four major hotels in Edinburgh
These datasets were merged with the HES visitor data by date, (and where applicable) country and site. It was difficult to find datasets covering large intervals of time - hence it was often the case
that with every successive data merge, the interval covered would get narrower… So to reduce the chance of overfitting, only one measure per data source was retained for modelling purposes.
Variables were selected via exploratory factor analysis (EFA) and by fitting a single factor per data source in order to identify the variable that loaded onto it the highest.
So can these extra predictors explain anything above and beyond the previous, simpler model? Let’s have a look at the final gamm model, which had the following form:
gamm( Visitors ~
s( Month, bs = "cc", by = Site ) +
s( as.numeric( Date ), by = TicketType, bs = "gp" ) +
te( Month, as.numeric( Date ), by = Site ) +
s( NumberOfAdults ) +
s( Temperature ) +
Site +
TicketType +
s( LaggedCCI ) +
s( LaggedNumArticles ) +
s( LaggedimdbVotes ) +
s( Laggedhits, by = Site ),
data = best_lag_solution,
control = lmeControl( opt = "optim", msMaxIter = 10000 ),
random = list( GroupingFactor = ~ 1 ), # GroupingFactor = Site x TicketType x Country
REML = TRUE,
correlation = corARMA( p = 1, q = 0 )
Based on the output for this model (shown below), we can check in the parametric coefficients section how the various ticket types compare in popularity to Walk Up tickets (used as the baseline
category), or how Edinburgh Castle compared to Craigmillar (the latter functioning as the baseline in this case). In the smooth terms section, notably, we have allowed the spline for Date to vary
depending on the ticket type concerned - which was useful given the surprising EDF for Membership tickets…
Family: gaussian
Link function: identity
Visitors ~ s(Month, bs = "cc", by = Site) + s(as.numeric(Date),
by = TicketType, bs = "gp") + te(Month, as.numeric(Date),
by = Site) + s(NumberOfAdults) + s(Temperature) + Site +
TicketType + s(LaggedCCI) + s(LaggedNumArticles) + s(LaggedimdbVotes) +
s(Laggedhits, by = Site)
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.06296 0.06656 0.946 0.344439
SiteEdinburgh 0.22692 0.06350 3.573 0.000372 ***
TicketTypeEducation 0.07420 0.14545 0.510 0.610085
TicketTypeExplorer Pass -0.22165 0.06590 -3.363 0.000804 ***
TicketTypeMembership -0.43625 0.06684 -6.527 1.14e-10 ***
TicketTypePre-Paid -0.93159 0.14608 -6.377 2.91e-10 ***
TicketTypeTrade -0.09395 0.06916 -1.358 0.174698
TicketTypeWeb -0.10447 0.06915 -1.511 0.131229
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Approximate significance of smooth terms:
edf Ref.df F p-value
s(Month):SiteCraigmillar 4.954928 8.000 7.941 6.94e-15 ***
s(Month):SiteEdinburgh 7.281621 8.000 20.031 < 2e-16 ***
s(as.numeric(Date)):TicketTypeWalk Up 1.000001 1.000 2.854 0.09149 .
s(as.numeric(Date)):TicketTypeEducation 1.000002 1.000 1.658 0.19821
s(as.numeric(Date)):TicketTypeExplorer Pass 1.000014 1.000 2.690 0.10132
s(as.numeric(Date)):TicketTypeMembership 7.325579 7.326 27.817 < 2e-16 ***
s(as.numeric(Date)):TicketTypePre-Paid 1.000004 1.000 0.231 0.63072
s(as.numeric(Date)):TicketTypeTrade 1.000000 1.000 6.328 0.01206 *
s(as.numeric(Date)):TicketTypeWeb 1.000000 1.000 22.546 2.39e-06 ***
te(Month,as.numeric(Date)):SiteCraigmillar 0.001085 15.000 0.000 0.31354
te(Month,as.numeric(Date)):SiteEdinburgh 5.443975 15.000 2.090 4.26e-07 ***
s(NumberOfAdults) 1.000009 1.000 40.617 2.93e-10 ***
s(Temperature) 5.675452 5.675 3.058 0.00436 **
s(LaggedCCI) 1.000019 1.000 3.921 0.04798 *
s(LaggedNumArticles) 1.985548 1.986 5.661 0.00598 **
s(LaggedimdbVotes) 2.702820 2.703 5.187 0.00188 **
s(Laggedhits):SiteCraigmillar 1.000011 1.000 8.012 0.00475 **
s(Laggedhits):SiteEdinburgh 3.106882 3.107 3.756 0.01092 *
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.694 Scale est. = 0.3072 n = 932
You can see this pattern below, in a plot created with the related package itsadug:
But disappointingly, the Adjusted R-squared for this more complex model is lower than for the previous one - which suggests that the added variables (collectively) are not that successful in
predicting visitor numbers at the two castles.
Going further
A visual reward for making it thus far in the post: castle courtyard with a large yew tree in the centre, inside Craigmillar Castle. Image source here.
An important part of building gam or gamm models is carrying out assumption checks - a lot of information about this is present if you consult: ?mgcv::gam.check and ?mgcViz::check.gamViz, with an
example also included in my slides here.
Something else that is interesting to consider would be how to validate forecast results. You can check out the idea of evaluating on a rolling forecasting origin or various caveats for time series
Finally, I have only really scratched the surface of gam / gamm models - and so much more is worth exploring. So here are a few great resources you can check out: | {"url":"https://datapowered.io/posts/2019-05-24-generalised-additive-mixed-models-gamms-tourism/","timestamp":"2024-11-08T05:13:22Z","content_type":"text/html","content_length":"61882","record_id":"<urn:uuid:5425b13a-53a2-489d-97aa-143cc8824cd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00869.warc.gz"} |
comma object
nLab comma object
2-Category theory
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Limits and colimits
The notion of comma object or comma square is a generalization of the notion of pullback or pullback square from category theory to 2-category theory: it is a special kind of 2-limit (and, in
particular, a PIE-limit).
Where a pullback involves a commuting square, for a comma object this square is filled by a 2-morphism.
(comma object)
The comma object of a pair of 1-morphisms $f \colon A\to C$ and $g \colon B\to C$ in a 2-category is an object $(f/g)$ equipped with projections $p \colon (f/g)\to A$ and $q \colon (f/g)\to B$ and a
2-morphism of this form:
$\array{ (f/g) & \overset{p}{\longrightarrow} & A \\ \mathllap{\scriptsize{q}} \big\downarrow & \swArrow \alpha & \big\downarrow \mathrlap{\scriptsize{f}} \\ B & \underset{g}{\longrightarrow} & C }$
which is universal in the sense of a 2-limit.
The comma object $f/g$ can be constructed by means of pullbacks and cotensors:
$\array{ f/g & \to & P & \to & A \\ \downarrow & & \downarrow & & \downarrow \mathrlap{\scriptsize{f}} \\ Q & \to & C^{\mathbf{2}} & \underset{dom}{\to} & C \\ \downarrow & & \downarrow \mathrlap{\
scriptsize{cod}} \\ B & \underset{g}{\to} & C }$
where $C^{\mathbf{2}}$ is the cotensor of $C$ with the arrow category $\mathbf{2} = \bullet \to \bullet$.
Pasting lemma
Suppose given a diagram
$\array{ P & \to & Q & \to & A \\ \downarrow & & \mathllap{\scriptsize{p}} \downarrow & \swArrow & \downarrow \mathrlap{\scriptsize{f}} \\ D & \underset{h}{\to} & C & \underset{g}{\to} & B }$
where the right-hand square is a comma square. Then the following are equivalent:
• the whole diagram is a comma square
• the left-hand square is a (2-)pullback square
The proof is analogous to that at pullback.
• In Cat, a comma category is a comma object (in fact a strict one, as normally defined); these give their name to the general notion.
• In the 2-category of virtual double categories, a comma object is a comma double category. If the virtual double categories are (pseudo) double categories and the domain functor $f$ in $f/g$ is
strong (while $g$ might be only lax), then the comma object is also a pseudo double category and the comma object lives in the 2-category of pseudo double categories and lax functors.
Related pages
Last revised on June 11, 2024 at 17:30:36. See the history of this page for a list of all contributions to it. | {"url":"https://ncatlab.org/nlab/show/comma+object","timestamp":"2024-11-12T04:03:30Z","content_type":"application/xhtml+xml","content_length":"45768","record_id":"<urn:uuid:6eb89856-3736-45f4-be47-1075dde707e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00751.warc.gz"} |
Mathematica Essentials: Intro & Overview (Wolfram Language)
16 Aug 202110:22
TLDRMathematica Essentials explores the computational capabilities of the Wolfram Language, demonstrating its use in mathematics, algebra, calculus, and scientific data visualization. The video
showcases Mathematica's symbolic computation, exact results, and integration with the Wolfram Knowledgebase for diverse applications.
• 🔢 Mathematica is a powerful computational system that has evolved beyond just mathematics into the Wolfram Language, capable of handling all types of computation.
• 📓 Starting Mathematica involves creating a Notebook, a dynamic computational document that allows for input and output of computations.
• 🎯 Mathematica provides exact results for calculations by default, using infinite precision, and can also give approximations with the N function.
• 📚 Functions in Mathematica are called with capitalized names and use square brackets instead of parentheses.
• 🧩 Algebraic problems, such as solving quadratic equations, are easily managed with Mathematica's symbolic computation capabilities.
• 🔍 Systems of equations can be solved using the Solve function, which returns solutions in a form ready for further computation.
• 📈 Calculus functions like integration are built into Mathematica, and functions can be graphed using the Plot function for visualization.
• 🌐 Mathematica has access to the Wolfram Knowledgebase, a vast collection of data from various fields of science for computational use.
• 🌌 Entities in Mathematica, such as molecules or celestial bodies, can be visualized and analyzed using specific plotting functions.
• 📊 The Wolfram Language, which powers Mathematica, includes thousands of functions for various types of data manipulation and computation.
• 🎼 Mathematica can handle media imports like audio, images, and video, providing functions for analysis and visualization.
• 🔌 The Import function in the Wolfram Language simplifies data loading from various sources without the need for additional libraries.
Q & A
• What is Mathematica and how has it evolved?
-Mathematica is a general system for computation that began as a system for doing mathematics by computer. It has evolved into the Wolfram Language, which can be used for all types of
• What is the default mode when you first start Mathematica?
-When you first start Mathematica, you begin by creating a Notebook, which defaults to input/output mode where you can type in an input and receive an output from Mathematica.
• How does Mathematica handle numerical calculations and what is unique about its results?
-Mathematica handles numerical calculations with exact results and does not convert fractions to rounded off decimals. It treats them with infinite precision, and functions use square brackets
instead of parentheses.
• What is the purpose of the N function in Mathematica?
-The N function in Mathematica is used to request an approximation of a value to a specified precision when the exact value is not needed.
• How does Mathematica solve algebraic equations and what form does the solution take?
-Mathematica uses the Solve function to solve algebraic equations. The solutions are returned as a list of substitution rules, which are ready for further computation work.
• What is the role of symbolic computation in Mathematica?
-Symbolic computation lies at the heart of Mathematica. It allows for the manipulation of mathematical symbols and the performance of algebraic operations without converting them into numerical
• How can you plot a function in Mathematica?
-In Mathematica, you can plot a function using the Plot function by passing in the function and a list with the variable, starting value, and stopping value for the plot.
• What is the Wolfram Knowledgebase and how is it used in Mathematica?
-The Wolfram Knowledgebase is a world-class collection of data accessible in Mathematica for use in computations. It can be queried for specific information across various fields of science and
data domains.
• How can you visualize a molecule in Mathematica?
-In Mathematica, you can visualize a molecule using the MoleculePlot function for a 2D representation or MoleculePlot3D for a more realistic 3D visualization, allowing you to rotate and view the
molecule from different angles.
• What types of media can Mathematica import and work with?
-Mathematica can import and work with all types of media, including audio, images, and video. It has built-in functions for each media type, such as AudioPlot for audio files.
• What is unique about the Wolfram Language and how does it differ from other programming languages?
-The Wolfram Language is a computational language that powers Mathematica and other services. It is not just another programming language but one that is built for everyone and comes with
thousands of powerful and versatile functions, including the ability to handle various data types without needing to import special libraries.
🔢 Introduction to Mathematica and Its Computational Capabilities
This paragraph introduces Mathematica as a comprehensive computational system that extends beyond mathematics to include the Wolfram Language for all types of computation. It explains the basic
interface starting with a Notebook, and demonstrates how to perform numerical calculations with exact results and infinite precision. The paragraph also covers the use of functions with square
brackets, approximations with the 'N' function, algebraic solutions using 'Solve', and symbolic computation. It touches on calculus integration and plotting functions, showcasing Mathematica's
graphical and analytical strengths.
🌐 Exploring Scientific Computation and Data in Mathematica
The second paragraph delves into Mathematica's scientific computation capabilities, highlighting its ability to handle 3D molecular structures with 'MoleculePlot3D' and visualize various scientific
entities using the Wolfram Knowledgebase. It discusses the ease of accessing and computing with data across different scientific domains, including chemistry, astronomy, and more. The paragraph also
covers the import and manipulation of various media types like audio, images, and video, demonstrating Mathematica's versatility. It concludes with an introduction to the Wolfram Language,
emphasizing its unique position as a computational language with built-in functions for diverse data types and its accessibility to a wide range of users beyond computer scientists.
Mathematica is a computational software system that was initially designed for mathematical computations but has since evolved into a more general computational tool known as the Wolfram Language. It
is used for various types of computations beyond just mathematics. In the video, Mathematica is introduced as a platform that can handle numerical calculations, algebra, calculus, and even symbolic
computations, demonstrating its versatility in computational tasks.
💡Wolfram Language
The Wolfram Language is the computational language that powers Mathematica. It is not just a programming language but a comprehensive system that includes a vast collection of built-in functions for
various computational tasks. The script highlights that the Wolfram Language is used throughout the Mathematica platform for tasks ranging from basic mathematical operations to complex scientific
computations and data analysis.
In the context of Mathematica, a Notebook is a computational document where users can input commands and receive outputs. It is interactive and serves as the primary interface for users to interact
with Mathematica. The script mentions that starting Mathematica involves creating a Notebook, which is the starting point for performing computations and exploring the software's capabilities.
💡Exact Value
An exact value in Mathematica refers to a result that is not rounded off to a decimal but is represented with infinite precision. This is significant in mathematical computations where precision is
crucial. The script points out that when performing calculations in Mathematica, such as 1/2 + 5 factorial times sin(pi/2), the results are given in their exact form, showcasing the software's
ability to handle exact values.
💡Symbolic Computation
Symbolic computation is a core feature of Mathematica, allowing it to perform operations on mathematical symbols rather than just numerical values. This capability is demonstrated in the script when
solving quadratic equations or dealing with systems of equations, where Mathematica returns solutions in a form that is ready for further computation, emphasizing its symbolic computation
💡Solve Function
The Solve function in Mathematica is used to find solutions to equations, both algebraic and transcendental. It is highlighted in the script as a tool for solving quadratic equations and systems of
equations symbolically. The function is versatile, capable of handling different types of equations and returning solutions in a form that can be used for further computations.
💡Plot Function
The Plot function in Mathematica is used to graph functions, providing a visual representation of their behavior. In the script, it is used to graph the function 'sin(2x-1)cos(3x+2)', demonstrating
how Mathematica can visually represent complex mathematical functions, aiding in understanding their properties and behavior.
💡Integrate Function
The Integrate function in Mathematica is used to compute integrals, both definite and indefinite. The script demonstrates how to use this function to find the integral of a function, showcasing
Mathematica's capabilities in calculus. It is an essential tool for analyzing the accumulation of a function over an interval or understanding its antiderivative.
💡Wolfram Knowledgebase
The Wolfram Knowledgebase is a vast collection of curated data accessible through Mathematica. It provides users with access to a wide range of information across various fields such as chemistry,
astronomy, and more. The script mentions querying the Knowledgebase for a molecule like caffeine and using it to plot and visualize the molecule's structure, illustrating the depth of data available
for scientific computations.
In Mathematica, entities refer to specific objects or concepts that can be queried and manipulated within the Wolfram Knowledgebase. The script discusses entities like planets and their properties,
demonstrating how Mathematica can handle complex data structures and provide detailed information about various subjects, making it a powerful tool for scientific research and data analysis.
💡Import Function
The Import function in the Wolfram Language is a versatile tool for loading various types of data, including JSON, images, and 3D models, into Mathematica. The script highlights this function's
ability to handle multiple data formats without the need for importing special libraries, emphasizing the language's built-in capabilities for handling diverse data types.
Mathematica is a general system for computation that has evolved beyond mathematics into the Wolfram Language.
The platform allows for the creation of living computational documents called Notebooks.
Mathematica provides exact numerical results with infinite precision, avoiding rounding off decimals.
Functions in Mathematica are distinguished by capitalization and the use of square brackets.
The N function is used to request numerical approximations with specified precision.
Solve function in Mathematica can handle algebraic equations and return solutions as substitution rules.
General quadratic equations can be solved symbolically without specifying constants.
Systems of equations are solved using the Solve function with lists of equations and variables.
Calculus functions are integrated into Mathematica, allowing for function definition and manipulation.
Plot function visualizes the behavior of mathematical functions.
Integrate function computes both definite and indefinite integrals.
Mathematica supports input of commands in standard mathematical notation.
Wolfram Knowledgebase provides access to a vast collection of data for scientific computations.
MoleculePlot and MoleculePlot3D functions help visualize the structure of molecules in 2D and 3D.
EntityList function retrieves lists of entities, such as all planets in the solar system.
EntityProperties function displays available data properties for entities in Mathematica.
Autocomplete feature in Mathematica assists users in accessing entity properties without memorization.
Mathematica can import various media types, including audio, images, and video.
AudioPlot function visualizes the waveform of audio files.
The Wolfram Language is a computational language with thousands of versatile functions.
Import function in the Wolfram Language handles various data types without the need for special libraries.
The Wolfram Language is designed for everyone, not just computer scientists. | {"url":"https://www.yeschat.ai/blog-mathematica-essentials-intro-overview-wolfram-language-43486","timestamp":"2024-11-02T08:11:35Z","content_type":"text/html","content_length":"187037","record_id":"<urn:uuid:2396b71b-2bc1-464e-9733-7ae3f6afd72e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00717.warc.gz"} |
September 1, Further (slow) progress... #5
2005, 10:21
Senior Member Further (slow) progress...
Gavin Tabor I've modified the omegaDot (source) term so that the first timestep remains bounded - ctilde now lies between 0 and 1 (chiefly around 1) and btilde between 1 and 0 (to within
rounding error - see below). I've also included it as SuSp source rather than as purely explicit.
Join Date: Mar
2009 However, now on the second timestep I get the following error : this is outside the PISO loop.
Posts: 181 Time = 0.50002
BICCG: Solving for Ux, Initial residual = 0.00778922, Final residual = 3.99268e-07, No Iterations 1
Rep Power: BICCG: Solving for Uy, Initial residual = 0.0476625, Final residual = 1.67977e-06, No Iterations 1
BICCG: Solving for ctilde, Initial residual = 0.215976, Final residual = 1.8554e-09, No Iterations 2
17 Got this far!
Got this far2!
--> FOAM FATAL ERROR : Maximum number of iterations exceeded
Function: specieThermo<thermo>::T(scalar f, scalar T0, scalar (specieThermo<thermo>::*F)(const scalar) const, scalar (specieThermo<thermo>::*dFdT)(const scalar) const) const
in file: /home/dm2/henry/OpenFOAM/OpenFOAM-1.0/src/thermophysicalModels/specie/lnInclude/ specieThermoI.H at line: 83.
FOAM aborting
AFAICT this traces through to a function in specieThermo which evaluates the temperature by inverting the equation of state using Newton-Raphson methods. Am I right?
It crops up because I call thermo->correct() directly after solving for ctilde and btilde. I assume it has something to do with the state of the btilde field.
btilde actually generates very small negative numbers (-O(1e-10)), presumably through rounding error. Is this likely to be what is causing the problems here? If so, what can I do
about it? If not, what is the problem likely to be? | {"url":"https://www.cfd-online.com/Forums/openfoam-solving/57901-combustion.html","timestamp":"2024-11-04T07:16:07Z","content_type":"application/xhtml+xml","content_length":"151290","record_id":"<urn:uuid:681c0f98-2b6e-4972-b2be-a0be66444c3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00227.warc.gz"} |
Flow Efficiency in ESP - SwiftKanban Knowledge BaseFlow Efficiency in ESP
In this article, we will help you understand flow efficiency in ESP analytics.
Flow Efficiency helps you measure the actual work time against the Lead time. So, the efficiency can be measured by dividing Work Time by the combination of Work Time and Wait Time, and the higher
the percentage, the better and smoother the process having less waste in the system and more agility to turn an idea or request into a workable feature.
Applying Filters
You can apply filters based on card attributes and the value stream stages to refine your data for the analytics. Moreover, the time range for the analytics can also be adjusted using the Temporal
Range Filter. Read more about them here.
Flow Efficiency Scatterplot
This is a Scatter plot of card flow efficiency. Flow efficiency is defined as (work time / total time). Work time is defined as the total time spent in the In Progress columns. We subtract the Block
time from effective work time as the most common reason for blocking among our customers is to indicate resource unavailability. Thus, the formula used here is the effective work time / total time.
The flow efficiency of a card becomes available when a card exits.
Different shapes indicate the type of exit:
• Circle: move forward or is archived
• Triangle: move back
• Square: move to other value stream
• Diamond: cards that were deleted or discarded
You can further analyze the Scatterplot data based on the Quartile or the Percentile value and also view the Rolling average line for a user-driven number of days. To view such data, click the icon
from the widget menu, and then select any of the following options:
On the right of the scatter plot, we have a box-and-whisker plot as defined by John W. Tukey. Box-plot provides a non-parametric characterization of data without making any assumptions about the
underlying distribution. The bottom and top of the box are the first and third quartiles, and the line inside the box is the second quartile (the median). The lower whiskers are drawn to the lowest
datum still within the 1.5 IRQ ( interquartile range: q3 – q1) of the lower quartile, and the upper whisker is the highest datum still within the 1.5 IRQ of the upper quartile.
Box-and-whisker plot is often used to identify outliers in place of 3 sigmas when the underlying distribution is unknown. For comparison, for normal distribution, the 6 sigma range represents 99.73%
of the total area, whereas the box-and-whisker range represents 99.3% of the area. The actual area being covered, however, depends on the nature of the distribution.
you enter the percentile values as per your requirement with the comma-separated format, the horizontal dotted lines demarcate the area of observation. Each area between the dotted lines represents
the percentile value below which a percentage of average flow efficiency data falls.
Rolling Average
You can plot a line based on the rolling average of the flow efficiency for the n number of days. The rolling average is the moving average of the cards that are being worked upon in between the
selected region of the value stream during the selected time interval. The line is formed by joining the series of data points of the average flow efficiency for a given time interval. It not only
helps you understand the trend of the work time vs. wait time but also visualize the emerging pattern and detect any intermittent peak.
Flow Efficiency Histogram
A histogram of flow efficiency for cards that exit the selected value-stream region in the time interval selected in the Time Range Filter and pass other selected filtering criteria. The x-axis
represents the flow efficiency in buckets of 10% and the y-axis represents the count of cards whose flow efficiency falls in each bucket.
Work Time vs. Wait Time
This correlation graph shows the relation between effective work time and wait time for the cards that have completed their journey through the selected region of the value stream. Effective work
time is defined as time spent in In-Progress columns subtracting the block time while in these columns.
The block time is considered to be wait time and therefore is added to the wait time. The x-axis represents wait time, the y-axis represents work time. The flow efficiency is shown using radial lines
originating from [0,0].
Different shapes indicate the type of exit:
• Circle: move forward or is archived
• Triangle: move back
• Square: move to other value stream
• Diamond: cards that were deleted or discarded
Leave a Reply Cancel reply
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JEE Advanced Syllabus & Exam Pattern 2024, Important Dates Etc.
JEE Advanced Syllabus 2024: IIT Madras has released JEE Advanced 2024 syllabus online at jeeadv.ac.in. The syllabus of JEE Advanced 2024 includes subject-wise chapters and topics that will be covered
in the JEE Advanced question paper. JEE Advanced Syllabus PDF includes subjects from Physics, Chemistry and Mathematics. Along with the syllabus, candidates should also have details of the JEE
Advanced exam pattern. The authority is set to conduct JEE Advanced 2024 on 26 May.
Joint Entrance Examination – Advanced (JEE Advanced) is an annual Indian entrance examination for admission to the Indian Institutes of Technology (IITs) and other premier engineering and science
institutes in India. The exam is considered to be one of the most challenging graduate entrance exams in the world, and only the top 2.5 lakh rank holders in the JEE Main exam are eligible to
appear for it.
JEE Advanced Syllabus 2024
JEE Advanced syllabus is now available on the official website of IIT Madras. There are no important updates in the JEE Advanced syllabus for this year. IIT Madras will conduct the JEE Advanced
examination 2024. To be eligible for the JEE Advanced examination, candidates must qualify for the JEE Main examination. Know more about who will administer JEE Advanced 2024.
A large number of candidates participate every year to get admission to undergraduate courses in top colleges like IIT, NIIT, IIIT all across India. Candidates should be well familiar with the JEE
Advanced syllabus and exam pattern to score well in the examination. Aspirants can find important topics and complete the syllabus in this article.
JEE Advanced Syllabus 2024 Direct PDF Link
Candidates can check the complete JEE Advanced syllabus by clicking on the direct link provided below.
JEE Advanced Syllabus 2024 PDF Link
Important Dates For JEE Advanced 2024
NTA releases JEE Advanced important dates on the official website. The table given below outlines important event dates for candidates to keep an eye on:
Important Dates For JEE Advanced Syllabus 2024
Events Revised Dates
Saturday, April 27, 2024 (17:00 IST)
Online Registration for JEE (Advanced) 2024 to
Tuesday, May 07, 2024 (17:00 IST)
Last date for payment of fees for registered candidates Friday, May 10, 2024 (17:00 IST)
Admit Card available for downloading May 17, 2024 (10:00 IST) to Sunday, May 26, 2024 (14:30 IST)
Selection of scribe by PwD candidates/ candidates having disability less than 40% and having difficulty in writing May 25, 2024
May 26, 2024
JEE (Advanced) 2024 Examination Paper 1: 09:00-12:00
Paper 2: 14:30-17:30
A copy of the answers of the candidates will be available on the JEE Advanced 2024 website. May 31, 2024
Online display of provisional answer keys June 02, 2024
Feedback and comments on provisional answer keys June 02, 2024 to Monday, June 03, 2024
Online declaration of JEE (Advanced) 2024 final answer keys and result June 09, 2024
Online registration for Architecture Aptitude Test (AAT) 2024 June 09, 2024 to Monday, June 10, 2024
Tentative Start of Joint Seat Allocation (JoSAA) 2024 Process June 10, 2024
Architecture Aptitude Test (AAT) 2024 June 12, 2024
Declaration of results of Architecture Aptitude Test (AAT) 2024 June 15, 2024
JEE Advanced Exam Pattern 2024
Candidates who want to appear in the JEE Advanced exam must read the details given below about the JEE Advanced Exam Pattern 2024 before appearing in the exam.
• Number of Papers: JEE Advanced 2024 will have two question papers. Paper 1 and Paper 2. 3 hours will be allotted to the candidates for each paper and both the papers are compulsory.
• Examination Mode: JEE Advanced 2024 examination will be conducted only in computer-based (CBT) mode.
• Total Duration: A total of 6 hours i.e. 3 hours will be given to the candidates for each paper.
• Language: English and Hindi
• Number of Sections: The question paper of JEE Advanced consists of three separate sections, i.e. Mathematics, Physics and Chemistry.
• Marking Scheme: The authority changes the JEE Advanced marking scheme every year. Negative marks may be given to the candidates for some questions.
JEE Advanced Syllabus 2024 For Physics
Physics is one of the important and high scoring subjects in Joint Entrance Exam. To score high marks in this subject candidates should be well aware of the important syllabus. Check the JEE Advanced
syllabus in the table:
JEE Advanced Syllabus 2024 For Physics
Chapters Units
• Units and Dimensions, Dimensional Analysis;
• Minimum count, vital figures;
• Methods of measurement of physical quantities and error analysis relating to the following experiments:
□ Experiments based on using vernier calipers and screw gauges (micrometer),
□ determination of G using a simple pendulum,
General □ Specific heat of liquid using a calorimeter,
□ Young’s modulus by Searle method,
□ the focal length of a concave mirror and convex lens using UV method,
□ speed of sound using resonating columns,
□ Verification of Ohm’s law using voltmeter and ammeter, and
□ specific resistance of the material of wire using meter bridge and post office box.
• Dynamics in one and two dimensions (Cartesian coordinates only), projectiveness;
• Uniform circular movement;
• Relative velocity.
• Newton’s laws of motion;
• Static and dynamic friction;
• Kinetic and potential energy;
• Work and power; Inertial and uniformly accelerated frames of reference;
• Conservation of linear momentum and mechanical energy.
• Systems of particles;
• Impulse;
• Centre of mass and its motion;
• Elastic and inelastic collisions.
• Law of gravitation;
• Acceleration due to gravity;
• Motion of planets and satellites in circular orbits;
• Gravitational potential and field;
• Escape velocity.
• Rigid body, parallel and perpendicular axes theorems, moment of inertia of uniform bodies with simple geometrical shapes;
• moment of inertia,
Mechanics • Angular momentum;
• Torque;
• Dynamics of rigid bodies with fixed axis of rotation;
• Rolling without slipping of rings, cylinders and spheres;
• Conservation of angular momentum;
• Equilibrium of rigid bodies;
• Collision of point masses with rigid bodies.
Linear and angular simple harmonic motions.
• Hooke’s law,
• Young’s modulus.
• Pressure in a fluid;
• Buoyancy;
• Surface energy and surface tension, capillary rise; Viscosity (Poiseuille’s equation excluded), Pascal’s law;
• Stoke’s law;
• Terminal velocity, equation of continuity, Streamline flow, Bernoulli’s theorem and its applications.
• Wave motion (plane waves only), superposition of waves;
• Progressive and stationary waves;
• Vibration of strings and air columns;
• longitudinal and transverse waves, Resonance; Beats; Speed of sound in gases;
• Doppler effect (in sound).
• Thermal expansion of solids, Heat conduction in one dimension;
• Elementary concepts of convection and radiation;
• Calorimetry, latent heat;
• liquids and gases;
• Newton’s law of cooling;
• Ideal gas laws;
Thermal physics • Equivalence of heat and work;
• First law of thermodynamics and its applications (only for ideal gases);
• Isothermal and adiabatic processes, bulk modulus of gases;
• Specific heats (Cv and Cp for monoatomic and diatomic gases);
• Blackbody radiation:
□ Kirchhoff’s law;
• absorptive and emissive powers;
• Wien’s displacement law, Stefan’s law.
• Coulomb’s law;
• Electric field and potential;
• Electric field lines;
• Flux of electric field;
• Gauss’s law and its application in simple cases,
• Electrical potential energy of a system of point charges and electrical dipoles in a uniform electrostatic field; such as,
□ to find a field due to an infinitely long straight wire,
□ uniformly charged infinite plane sheet and
□ uniformly charged thin spherical shell.
• Capacitance;
• Capacitors in series and parallel;
• Parallel plate capacitor with and without dielectrics;
• Energy stored in a capacitor.
Electricity and magnetism
• Electric current;
• Series and parallel arrangements of resistances and cells;
• Ohm’s law;
• Kirchhoff’s laws and simple applications;
• Heating effect of current.
• Biot–Savart’s law and Ampere’s law;
• A current flowing near a straight coil along the axis of a circular coil and a magnetic field inside a long straight solenoid;
• Force acting on a charge moving in a uniform magnetic field and a current carrying wire.
• The magnetic moment of the current loop; Effect of uniform magnetic field on the current loop;
• Moving coil galvanometer, voltmeter, ammeter and their transformations.
• Electromagnetic Induction: Faraday’s Law, Lenz’s Law;
• Self and Mutual Induction; RC, LR and LC circuits with DC. And AC Source.
• Rectilinear propagation of light;
• Total internal reflection;
• Deviation and dispersion of light by a prism;
• Reflection and refraction at plane and spherical surfaces; Thin lenses;
Optics • Combinations of mirrors and thin lenses;
• Magnification.
• Wave nature of light:
□ Huygen’s theory of interference is limited to Young’s double-slit experiment.
• Atomic nucleus; Law of radioactive decay;
• Alpha, Beta and Gamma radiations;
• Decay constant;
• Half-life and mean life;
• Binding energy and its calculation;
Modern physics • Fission and fusion processes;
• Energy calculation in these processes.
• Photoelectric effect;
• Characteristic and continuous X-rays, Bohr’s theory of hydrogen-like atoms;
• Moseley’s law;
• De Broglie wavelength of matter waves.
JEE Advanced Syllabus 2024 For Mathematics
JEE Advanced Syllabus 2024 For Mathematics: JEE Advanced Maths is all about practice and persistence. Candidates should be well aware of the JEE Mathematics syllabus to prepare each subject
carefully. See the table given below for the complete syllabus of JEE Advanced Mathematics.
JEE Advanced Syllabus 2024 For Mathematics
Chapters Units
Algebra of complex numbers, multiplication, conjugation, addition, polar representation,
triangle inequality, properties of modulus and principal argument, cube roots of unity, geometric interpretations.
Quadratic equations with real coefficients, formation of quadratic equations with given roots, symmetric functions of roots relations between roots and coefficients.
Arithmetic, geometric and harmonic progressions, geometric and harmonic means, arithmetic, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of
squares and cubes of the first n natural numbers.
Logarithms and their properties.
Permutations and combinations, binomial theorem for a positive integral index, properties of binary coefficients.
Matrices as a rectangular array of real numbers, addition, multiplication by a scalar and product of matrices, equality of matrices, transpose of a matrix, determinant of a square matrix
Matrices of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, inverse of a square matrix of order up to three,
solutions of simultaneous linear equations in two or three variables.
Probability Addition and multiplication rules of probability, Bayes Theorem, independence of events, conditional probability, computation of probability of events using permutations and
Trigonometry Trigonometric functions, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, their periodicity and graphs, general solution of trigonometric
Relations between sides and angles of a triangle, half-angle formula and the area of a triangle, sine rule, cosine rule, inverse trigonometric functions (principal value only).
Two dimensions: Cartesian coordinates, section formulae, distance between two points, shift of origin.
Equation of a straight line in various forms, distance of a point from a line; Lines through the point of intersection of two given lines, angle between two lines, equation of the
Analytical bisector of the angle between two lines, concurrency of lines; orthocentre, Centroid, incentre and circumcentre of a triangle.
geometry Equation of a circle in various forms, normal and chord. Parametric equations of a circle, equations of tangent, intersection of a circle with a straight line or a circle, The equation
of a circle through the intersection points of two circles and the intersection points of a circle and a direct line.
Equations of a parabola, their foci, directrices and eccentricity, ellipse and hyperbola in standard form, parametric equations, equations of tangent and normal. Locus problems.
Three dimensions: Direction cosines and direction ratios, equation of a plane, equation of a straight line in space, distance of a point from a plane.
Real valued functions of a real variable, into, sum, difference, product and quotient of two functions, composite functions, onto and one-to-one functions, absolute value, polynomial,
rational, trigonometric, exponential and logarithmic functions.
Limit and continuity of a function, difference, product and quotient of two functions, limit and continuity of the sum, L’Hospital rule of evaluation of limits of functions.
calculus Even and odd functions, continuity of composite functions, inverse of a function, intermediate value property of continuous functions.
Derivative of a function, derivative of the sum, product and quotient of two functions, chain rule, difference, derivatives of polynomial, rational, inverse trigonometric, trigonometric,
exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up to order two, tangents and normals, increasing and decreasing functions, geometrical interpretation of the derivative, maximum and
minimum values of a function, Rolle’s theorem and Lagrange’s mean value theorem.
Integral Integration as the inverse process of differentiation, definite integrals and their properties, indefinite integrals of standard functions, fundamental theorem of integral calculus.
calculus Integration by parts, application of definite integrals to the determination of areas involving simple curves, integration by the methods of substitution and partial fractions.
Formation of ordinary differential equations, separation of variables method, solution of homogeneous differential equations, linear first-order differential equations.
Vectors Addition of vectors, dot and cross products, scalar multiplication, scalar triple products and their geometrical interpretations.
JEE Advanced Syllabus 2024 For Physical Chemistry
Candidates can check the complete JEE Advanced syllabus for Physical Chemistry in the table below.
JEE Advanced Syllabus 2024 For Physical Chemistry
Concept of atoms and molecules; Mole concept; Chemical formulae; Dalton’s atomic theory; Balanced chemical equations;
General topics Calculations (based on mole concept) involving common oxidation-reduction, neutralisation, and displacement reactions; molarity, Concentration in terms of mole fraction,
molality and normality.
Absolute scale of temperature, ideal gas equation;
Gaseous and liquid Kinetic theory of gases, average, root mean square and most probable velocities and their relation with temperature;
Deviation from ideality, van der Waals equation; Law of partial pressures;
Vapour pressure; Diffusion of gases.
Bohr model, quantum numbers, spectrum of hydrogen atom; Wave-particle duality, de Broglie hypothesis;
Qualitative quantum mechanical picture of hydrogen atom, shapes of s, p and d orbitals; Uncertainty principle; Electronic configurations of elements (up to atomic number 36);
Aufbau principle;
Atomic structure and
chemical bonding Orbital overlap and covalent bond; Hybridisation involving s, p and d orbitals only; Pauli’s exclusion principle and Hund’s rule;
Orbital energy diagrams for homonuclear diatomic species; Polarity in molecules, dipole moment (qualitative aspects only); Hydrogen bond; VSEPR model and shapes of molecules
(linear, triangular, angular, square planar, pyramidal, trigonal bipyramidal, square pyramidal, tetrahedral and octahedral).
First law of thermodynamics; Enthalpy, Hess’s law; Internal energy, work and heat, pressure-volume work; Heat of reaction, fusion and vapourization; Second law of
Free energy; Criterion of spontaneity.
Chemical equilibrium Law of the mass action; Significance of ?G and ?G0 in chemical equilibrium; Solubility product, common ion effect, pH and buffer solutions; Equilibrium constant, Le
Chatelier’s principle (effect of concentration, temperature and pressure); Acids and bases (Bronsted and Lewis concepts); Hydrolysis of salts.
Electrochemical cells and cell reactions; Nernst equation and its relation; Electrochemical series, emf of galvanic cells; Standard electrode potentials;
Faraday’s laws of electrolysis; Electrolytic conductance, equivalent and molar conductivity, specific, Kohlrausch’s law; Concentration cells.
Chemical kinetics Rates of chemical reactions; Rate constant; Order of reactions; First order reactions; Temperature dependence of rate constant (Arrhenius equation).
Classification of solids, seven crystal systems (cell parameters a, b, c, alpha, beta, gamma), crystalline state, close packed structure of solids (cubic), packing in fcc, bcc
and hcp lattices;
Solid state
Nearest neighbours, simple ionic compounds, ionic radii, point defects.
Solutions Raoult’s law; elevation of boiling point and depression of freezing point, Molecular weight determination from lowering of vapour pressure.
Surface chemistry Elementary concepts of adsorption (excluding adsorption isotherms); Colloids: types, methods of preparation and general properties; surfactants and micelles (only definitions
and examples), Elementary ideas of emulsions.
Nuclear chemistry Radioactivity: isotopes and isobars; Kinetics of radioactive decay (decay series excluded), carbon dating; Properties of alpha, Beta and Gamma rays; Stability of nuclei with
respect to proton-neutron ratio; Brief discussion on fission and fusion reactions.
JEE Advanced Syllabus 2024 For Inorganic Chemistry
Candidates can check the complete JEE Advanced syllabus for Inorganic Chemistry from the table given below.
JEE Advanced Syllabus 2024 For Inorganic Chemistry
Boron, silicon, nitrogen, oxygen, phosphorus, sulphur and halogens;
Isolation/preparation and properties of the
following non-metals Properties of allotropes of carbon (only diamond and graphite), phosphorus and sulphur.
Oxides, peroxides, carbonates, bicarbonates, hydroxides, chlorides and sulphates of sodium, magnesium and calcium, potassium;
Boron: diborane, boric acid and borax;
Carbon: oxides and oxyacid (carbonic acid); Aluminium: alumina, aluminium chloride and alums;
Silicon: silicones, silicates and silicon carbide;
Preparation and properties of the following Nitrogen: oxides, oxyacids and ammonia;
Phosphorus: oxides, oxyacids (phosphorus acid, phosphoric acid) and phosphine;
Sulphur: hydrogen sulphide, oxides, sulphurous acid, sulphuric acid and sodium thiosulphate;
Oxygen: ozone and hydrogen peroxide;
Halogens: hydrohalic acids, oxides and oxyacids of chlorine, bleaching powder; Xenon fluorides.
Definition, general characteristics, oxidation states and their stabilities, colour (excluding the details of electronic transitions) and calculation of
spin-only magnetic moment;
Transition elements (3d series)
Coordination compounds: cis-trans and ionisation isomerisms, nomenclature of mononuclear coordination compounds, hybridization and geometries of
mononuclear coordination compounds (tetrahedral, linear, square planar and octahedral).
Preparation and properties of the following Oxides and chlorides of tin and lead; Oxides, chlorides and sulphates of Fe2+, Cu2+ and Zn2+; potassium dichromate, Potassium permanganate, silver oxide,
compounds silver nitrate, silver thiosulphate.
Ores and minerals Commonly occurring ores and minerals of iron, tin, lead, copper, magnesium, aluminium, zinc and silver.
Chemical principles and reactions only (industrial details excluded); Self reduction method (copper and lead);
Carbon reduction method (iron and tin);
Extractive metallurgy
Electrolytic reduction method (magnesium and aluminium);
Cyanide process (silver and gold).
Principles of qualitative analysis Groups I to V (only Ag+, Cu2+, Pb2+, Bi3+, Hg2+, Fe3+, Cr3+, Al3+, Ca2+, Zn2+, Ba2+, Mn2+ and Mg2+); Nitrate, halides (excluding fluoride), sulphate and
JEE Advanced Syllabus 2024 For Organic Chemistry
Candidates can check JEE Advanced Syllabus 2024 for Organic Chemistry in the table given below.
JEE Advanced Syllabus 2024 For Organic Chemistry
Hybridisation of carbon; ? and ?-bonds; Structural and geometrical isomerism; Shapes of simple organic molecules; Optical isomerism of compounds containing up to two
asymmetric centres, (R, S and E, Z nomenclature excluded); Conformations of ethane and butane (Newman projections); Resonance and hyperconjugation; IUPAC nomenclature of
Concepts simple organic compounds (only hydrocarbons, mono-functional and bi-functional compounds); Keto-enoltautomerism; Hydrogen bonds: definition and their effects on physical
properties of alcohols and carboxylic acids; Determination of empirical and molecular formulae of simple compounds (only combustion method); Inductive and resonance effects
on acidity and basicity of organic acids and bases; Reactive intermediates produced during homolytic and heterolytic bond cleavage; Polarity and inductive effects in alkyl
halides; Formation, structure and stability of carbocations, carbanions and free radicals.
Preparation, properties Homogeneous series of alkanes, emphasizing their physical properties such as boiling point, density and melting point. Additionally, candidates will explore the combustion
and reactions of alkanes and halogenation reactions of alkenes, as well as methods for preparing alkenes through the Wurtz reaction and decarboxylation reactions. This broad coverage provides
candidates with a solid understanding of the chemical properties and reactions of these hydrocarbons
Physical properties of alkenes and alkynes, including boiling point, density, and dipole moment. It also explores the acidity of alkynes, as well as acid-catalyzed hydration
Preparation, properties of alkynes and alkynes, leaving aside stereochemistry aspects. Additionally, candidates will study the reactions of alkenes with KMnO4 and ozone, the reduction of alkenes
and reactions of alkenes and alkynes, and electrophilic addition reactions with X2, HX, HOX and H2O (where X = halogen). Additionally, the course covers the preparation of alkynes and alkynes via
and alkynes elimination reactions, addition reactions of alkynes, and properties of metal acetylides. This broad coverage provides candidates with a thorough understanding of the
chemical properties and reactions of these compounds
Reactions of benzene aromaticity and structure, electrophilic substitution reactions like halogenation and nitration, and the effects of directing groups in monosubstituted benzenes.
Phenols Acidification, electrophilic substitution reactions (halogenation, nitration and sulfonation); Riemer-Tieman reaction, Kolbe reaction.
Alkyl halides: rearrangement reactions of alkyl carbocation, nucleophilic substitution reactions, Grignard reactions;
Alcohols: esterification, dehydration and oxidation, phosphorus halides, reaction with sodium, ZnCl2/concentrated HCl, conversion of alcohols into aldehydes and ketones;
Aldehydes and Ketones: oxidation, reduction, oxime and hydrazone formation;
Characteristic reactions Ethers: Preparation by Williamson’s Synthesis; aldol condensation, Perkin reaction; haloform reaction and nucleophilic addition reactions (Grignard addition);
of the following
(including those Cannizzaro reaction;
mentioned above)
Amines: basicity of substituted anilines and aliphatic amines, preparation from nitro compounds, reaction with nitrous acid, azo coupling reaction of diazonium salts of
aromatic amines, Sandmeyer and related reactions of diazonium salts;
Carboxylic acids: formation of esters, acid chlorides and amides, ester hydrolysis; carbylamine reaction;
Haloarenes: nucleophilic aromatic substitution in haloarenes and substituted haloarenes (excluding Benzyne mechanism and Cine substitution).
Classification; Mono- and Di-saccharides (glucose and sucrose);
Oxygenation, reduction, glycoside formation and hydrolysis of sucrose.
Amino acids and peptides General structure (for peptides only primary structure) and physical properties
Properties and uses of Natural rubber, nylon, cellulose, teflon and PVC.
some important polymers
Practical organic trace elements (N, S, halogen); Detection and identification of the following functional groups: hydroxyl (alcohol and phenolic), carboxyl, carbonyl (aldehyde and ketone),
chemistry amino and nitro; Chemical methods for separation of monofunctional organic compounds from binary mixtures. | {"url":"https://biharhelp.in/jee-advanced-syllabus/","timestamp":"2024-11-14T17:22:02Z","content_type":"text/html","content_length":"129654","record_id":"<urn:uuid:c448a1ad-eabf-407a-8410-76ac1f274b01>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00345.warc.gz"} |
Multiplicative inversions involving real zero and neverending ascending infinity in the multispatial framework of paired dual reciprocal spaces - World Scientific News
Inverses of complex numbers and of analytic functions are composites of mixed type for they are multiplicative inverses (i.e. reciprocals) of the modulus/magnitude combined with additive reverses of
the argument/angle. Hence, the mixed inverses in the complex domain ℂ are not really reciprocals and therefore their lack of truly multiplicative reciprocity was a contributing reason that spurred
the – unnecessary though still ongoing – prohibition of division by zero which is the natural reciprocal of the neverending ascending real infinity. Truly reciprocal algebraic operations are
presented (via multiplicative algebraic inversions) by few examples within the new multispatial framework in terms of their abstract algebraic representations subscripted by the native algebraic
bases of the mutually paired dual reciprocal (even though algebraic) spaces in which the inversive operations are performed.
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Algebra: Fraction Problems (solutions, examples, videos)
Fraction Word Problems using Algebra
2/3 of a number is 14. What is the number?
Step 1: Assign variables :
Let x = number
Step 2: Solve the equation
Isolate variable x
Answer: The number is 21.
The numerator of a fraction is 3 less than the denominator. When both the numerator and denominator are increased by 4, the fraction is increased by fraction.
Let the numerator be x,
then the denominator is x + 3,
and the fraction is \(\frac{x}{{x + 3}}\)
When the numerator and denominator are increased by 4, the fraction is \(\frac{{x + 4}}{{x + 7}}\)
\(\frac{{x + 4}}{{x + 7}} - \frac{x}{{x + 3}} = \frac{{12}}{{77}}\)
77(x + 4)(x + 3) – 77x(x+7) = 12(x + 7)(x + 3)
77x^2 + 539x + 924 – 77x^2 – 539x = 12x^2 + 120x + 252
12x^2 + 120x – 672 = 0
x^2 + 10x – 56 = 0
(x – 4)(x + 14) = 0
x = 4 (negative answer not applicable in this case)
Answer: The original fraction is \(\frac{4}{7}\)
How to solve Fraction Word Problems using Algebra?
(1) The denominator of a fraction is 5 more than the numerator. If 1 is subtracted from the numerator, the resulting fraction is 1/3. Find the original fraction.
(2) If 3 is subtracted from the numerator of a fraction, the value of the resulting fraction is 1/2. If 13 is added to the denominator of the original fraction, the value of the new fraction is 1/3.
Find the original fraction.
(3) A fraction has a value of 3/4. When 14 is added to the numerator, the resulting fraction has a value equal to the reciprocal of the original fraction, Find the original fraction.
Algebra Word Problems with Fractional Equations
Solving a fraction equation that appears in a word problem
One third of a number is 6 more than one fourth of the number. Find the number.
Fraction and Decimal Word Problems
How to solve algebra word problems with fractions and decimals?
(1) If 1/2 of the cards had been sold and there were 172 cards left, how many cards were printed?
(2) Only 1/3 of the university students wanted to become teachers. If 3,360 did not wan to become teachers, how many university were there?
(3) Rodney guessed the total was 34.71, but this was 8.9 times the total. What was the total?
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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How many liters are in 2.10 kiloliter? | Socratic
How many liters are in 2.10 kiloliter?
1 Answer
$2.10 \cdot {10}^{3} \text{L}$
Before doing any calculations, make sure that you understand what you're looking for here.
The metric system uses prefixes to denote multiples or fractions of basic units.
In order to get from a basic unit to a multiple, you multiply by powers of $10$. Likewise, to get from a basic unit to a fraction, you divide by powers of $10$.
In your case, the kilo- prefix is used to denote a multiple if the basic unit, which here is the liter. So, in order to go from liters to kiloliters, you must multiply by ${10}^{3}$.
In essence, this means that you have
$\text{1 kL" = 10^3"L}$
Simply put, you need $\text{1000 L}$ in order to have $\text{1 kL}$.
This means that you will have
#2.10 color(red)(cancel(color(black)("kL"))) * (10^3"L")/(1color(red)(cancel(color(black)("kL")))) = "2100 L"#
In order to express this number rounded to three sig figs, you need to use scientific notation. The answer will thus be
#"2.10 kL" = color(green)(2.10 * 10^3"L")#
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How to Change Number to Text in Excel?
How to Change Number to Text in Excel?
Are you trying to figure out how to convert numbers to text in Excel? Converting numbers to text can be a tricky task, but it doesn’t have to be. With a few simple steps, you can easily convert any
number in Excel to text. In this guide, I will walk you through the steps to transform your numbers into text in just minutes. So, let’s get started and learn how to change number to text in Excel.
To change a number to text in Excel, follow these steps:
• Open the spreadsheet containing the number you wish to convert to text.
• Select the cell or range of cells containing the numbers.
• Click the Home tab and select the drop-down arrow next to the Number Format box.
• Select Text from the list.
• The numbers in the selected cells will now appear as text.
Changing numbers to text in Excel
Text formatting in Excel is an important aspect of data analysis and manipulation. One such formatting option is changing numbers to text. This can be done in several ways, depending on the type of
data that needs to be converted. In this article, we will cover how to change numbers to text in Excel.
One of the simplest ways to convert numbers to text in Excel is to use the “Text to Columns” feature. This feature allows you to select a range of cells that contain numbers and convert them to text.
To do this, select the range of cells that contain the numbers and then click on the “Data” tab in the ribbon. From there, select the “Text to Columns” option. In the dialog box that appears, select
the “Delimited” option and click “Next”. In the “Delimiters” section, select the “Space” option and click “Finish”. The numbers will be converted to text.
Using a Formula
Another way to convert numbers to text in Excel is to use a formula. The “Text” function can be used to convert a number to text. To use this function, enter the formula “=Text(cell reference,”@”)”,
where “cell reference” is the cell that contains the number you want to convert. This will convert the number to text.
Using the Format Cells Option
The “Format Cells” option can also be used to convert numbers to text in Excel. To do this, select the range of cells that contain the numbers that you want to convert and then click the “Home” tab
in the ribbon. From there, select the “Format Cells” option. In the dialog box that appears, select the “Number” tab and then select the “Text” option. This will convert the numbers to text.
Using the REPLACE Function
The “REPLACE” function can also be used to convert numbers to text in Excel. To use this function, enter the formula “=REPLACE(cell reference,start_num,num_chars,new_text)”, where “cell reference” is
the cell that contains the number you want to convert, “start_num” is the starting position of the number, “num_chars” is the number of characters in the number, and “new_text” is the text you want
to replace the number with. This will convert the number to text.
Using the CONCATENATE Function
The “CONCATENATE” function can also be used to convert numbers to text in Excel. To use this function, enter the formula “=CONCATENATE(cell reference,” ”,new_text)”, where “cell reference” is the
cell that contains the number you want to convert and “new_text” is the text you want to replace the number with. This will convert the number to text.
Using the VALUE Function
The “VALUE” function can also be used to convert numbers to text in Excel. To use this function, enter the formula “=VALUE(cell reference)”, where “cell reference” is the cell that contains the
number you want to convert. This will convert the number to text.
Using the FIND and REPLACE Function
The “FIND and REPLACE” function can also be used to convert numbers to text in Excel. To use this function, click the “Home” tab in the ribbon and select the “Find and Replace” option. In the dialog
box that appears, enter the number you want to convert in the “Find what” box and the text you want to replace it with in the “Replace with” box. This will convert the number to text.
Using the SUBSTITUTE Function
The “SUBSTITUTE” function can also be used to convert numbers to text in Excel. To use this function, enter the formula “=SUBSTITUTE(cell reference,num,new_text)”, where “cell reference” is the cell
that contains the number you want to convert, “num” is the number you want to convert, and “new_text” is the text you want to replace the number with. This will convert the number to text.
Using the VLOOKUP Function
The “VLOOKUP” function can also be used to convert numbers to text in Excel. To use this function, enter the formula “=VLOOKUP(cell reference,lookup_range,col_num,FALSE)”, where “cell reference” is
the cell that contains the number you want to convert, “lookup_range” is the range of cells that contains the numbers and text you want to compare, “col_num” is the column number of the text you want
to return, and “FALSE” indicates that an exact match is required. This will convert the number to text.
Few Frequently Asked Questions
What is Cell Formatting in Excel?
Cell formatting in Excel is a way to change the appearance of data within a cell. This can include changing the font, font size, text color, background color, border style, and alignment. Cell
formatting can help make a spreadsheet more organized and easier to read or it can be used to highlight important data. It is also possible to change the number formatting of a cell to change the way
a number is displayed.
How Do I Change a Number to Text in Excel?
Changing a number to text in Excel is a simple process. First, select the cell or cells containing the numbers you want to change to text. Then, right click on the selected cell or cells and select
“Format Cells”. In the Format Cells window, select “Text” from the “Category” drop-down menu. Then select “OK” to apply the formatting. The numbers will now be changed to text.
What is the Benefit of Converting Numbers to Text in Excel?
Converting numbers to text in Excel can be beneficial in a number of ways. Text formatting can make the data easier to read, as text will not be automatically converted to a number. Text formatting
also ensures that the numbers remain static, meaning they will not change if the data is sorted or filtered. Additionally, text formatting can help with certain calculations, as some formulas can
only be used with text data.
How Do I Change Text Back to a Number in Excel?
Changing text back to a number in Excel is a simple process. First, select the cell or cells containing the text you want to convert back to a number. Then, right click on the selected cell or cells
and select “Format Cells”. In the Format Cells window, select “Number” from the “Category” drop-down menu. Then select the desired number format from the “Number” drop-down menu. Finally, select “OK”
to apply the formatting. The text will now be changed back to a number.
What are the Limitations of Converting a Number to Text in Excel?
One of the main limitations of converting a number to text in Excel is that it can make the data difficult to analyze. Since the numbers are now text, they cannot be used in calculations or formulas.
Additionally, it can be difficult to sort or filter the data, as text data cannot be sorted numerically.
Are There any Shortcuts to Change a Number to Text in Excel?
Yes, there are some shortcuts to change a number to text in Excel. The first is to select the cell or cells containing the numbers you want to change to text and then press the “F10” key. This will
open the Format Cells window, where you can select “Text” from the “Category” drop-down menu. Another option is to select the cell or cells containing the numbers and then press “Ctrl+1”. This will
open the Format Cells window, where you can then select “Text” from the “Category” drop-down menu.
Excel Converting Numbers to Text for Sorting
It is easy to learn how to change a number to text in Excel. Whether you are a beginner or an expert, the steps provided in this tutorial will help you convert a number to text quickly and easily.
With the help of this tutorial, you can now quickly format a number as text in Excel. With the ability to convert numbers to text, Excel can do much more than just crunch numbers. | {"url":"https://keys.direct/blogs/blog/how-to-change-number-to-text-in-excel","timestamp":"2024-11-12T20:26:27Z","content_type":"text/html","content_length":"361321","record_id":"<urn:uuid:1f6605bd-32a0-4b26-8763-75869b8627f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00882.warc.gz"} |
Math Mammoth Square Roots & The Pythagorean Theorem
Blue Series
Usually printed in 3 - 5 business days
Math Mammoth Square Roots & The Pythagorean Theorem is a relatively short worktext focusing on irrational numbers, square roots, and the Pythagorean Theorem and its applications. First, students
learn about taking a square root as the opposite operation to squaring a number. They learn about irrational numbers, and how to find approximations to square roots both with a calculator and with a
guess-and-check method. Students also practice placing irrational numbers on the number line, using mental math to find their approximate location. Next, the book has a review lesson on how to
convert fractions to decimals. The following lesson has to do with writing decimals as fractions, and teaches a method for converting repeating decimals to fractions. Then it is time to learn to
solve simple equations that involve taking a square or cube root, over the course of two lessons. After learning to solve such equations, students are now fully ready to study the Pythagorean Theorem
and apply it. The Pythagorean Theorem is introduced in the lesson by that name. Students learn to verify that a triangle is a right triangle by checking whether it fulfills the Pythagorean Theorem.
They apply their knowledge about square roots and solving equations to solve for an unknown side in a right triangle when two of the sides are given. Next, students solve a variety of geometric and
real-life problems that require the Pythagorean Theorem. This theorem is extremely important in many practical situations. Students should show their work for these word problems to include the
equation that results from applying the Pythagorean Theorem to the problem and its solution. There are literally hundreds of proofs for the Pythagorean Theorem. In this book, we present one easy
proof based on geometry (not algebra). As an exercise, students are asked to supply the steps of reasoning to another geometric proof of the theorem. Students also study a proof for the converse of
the theorem, which says that if the sides of a triangle fulfill the equation a2 + b2 = c2 then the triangle is a right triangle. Our last topic is distance between points in the coordinate grid, as
this is another simple application of the Pythagorean Theorem.
Publication Date
Feb 20, 2022
Education & Language
All Rights Reserved - Standard Copyright License
By (author): Maria Miller
Binding Type
Paperback Perfect Bound
Interior Color
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maths tutors usa • Expert Tutors for Online Tuition in Pakistan & Saudi Arabia
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Get the Top Pakistani math tutors in the USA for grades 6 to advanced level. Expertise in Algebra, Geometry, Calculus, Advanced Calculus, Linear Algebra, Trigonometry.
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Unlocking Math Excellence with Pakistani Tutors Experience unparalleled math tutoring excellence with our team of skilled Pakistani tutors based in the USA. We cater to students from grade 6 to
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Our dedicated Geometry tutors specialize in making complex shapes, angles, and spatial relationships understandable. Through personalized instruction, they empower students to unravel the mysteries
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At Asva, our expertise goes beyond geometry. Our tutors also offer guidance in calculus and algebra. Whether you’re grappling with derivatives, integrals, or algebraic equations, our tutors provide
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For those seeking more advanced challenges, our tutors cover topics like advanced calculus and linear algebra. Dive into intricate mathematical concepts with the guidance of our skilled tutors, and
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How do you convert
How do you convert H to MS?
Conversion base : 1 ms = 2.7777777777778E-7 hr.
How do you convert milliseconds into hours minutes and seconds?
To convert milliseconds to hours, minutes, seconds:
1. Divide the milliseconds by 1000 to get the seconds.
2. Divide the seconds by 60 to get the minutes.
3. Divide the minutes by 60 to get the hours.
4. Add a leading zero if the values are less than 10 to format them consistently.
How do you convert milliseconds to hours flutter?
1. int seconds = (int) (milliseconds / 1000) % 60 ;
2. int minutes = (int) ((milliseconds / (1000*60)) % 60);
3. int hours = (int) ((milliseconds / (1000*60*60)) % 24);
How do you convert milliseconds to seconds in react native?
“convert milliseconds to seconds javascript” Code Answer’s
1. function millisToMinutesAndSeconds(millis) {
2. var minutes = Math. floor(millis / 60000);
3. var seconds = ((millis % 60000) / 1000).
4. return minutes + “:” + (seconds < 10? ‘
5. }
6. millisToMinutesAndSeconds(298999); // “4:59”
7. millisToMinutesAndSeconds(60999); // “1:01”
What is a millisecond in decimal form?
A millisecond (from milli- and second; symbol: ms) is one thousandth (0.001 or 10−3 or 1/1000) of a second.
How do I format milliseconds in Excel?
In the Format Cells window, go to the Number tab, select Custom from the Category list, and enter h:mm:ss. 000 in the Type text box. As a result, all of the time values are displayed with
milliseconds as decimals.
How do I convert milliseconds to numbers in Excel?
Select a blank cell besides the first time cell, enter the formula =RIGHT(TEXT(A2, “hh:mm:ss. 000”),3)/1000 (A2 is the first time cell in the list) into it, and then drag the Fill Handle down to the
range as you need. Now you will get the calculation results showing as time with milliseconds as above screenshot shown. | {"url":"https://tumericalive.com/how-do-you-convert-h-to-ms/","timestamp":"2024-11-03T09:42:46Z","content_type":"text/html","content_length":"36229","record_id":"<urn:uuid:c5cac066-20a9-4ac2-a95e-7eec7c13dc1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00651.warc.gz"} |
Geostrophic and chimney regimes in rotating horizontal convection with imposed heat flux
Convection in a rotating rectangular basin with differential thermal forcing at one horizontal boundary is examined using laboratory experiments. The experiments have an imposed heat flux boundary
condition, are at large values of the flux Rayleigh number ( based on the box length ), use water with Prandtl number and have a small depth to length aspect ratio. The results show the conditions
for transition from non-rotating horizontal convection governed by an inertial-buoyancy balance in the thermal boundary layer, to circulation governed by geostrophic flow in the boundary layer. The
geostrophic balance constrains mean flow and reduces the heat transport as Nusselt number , where is the convective Rossby number, is the imposed buoyancy flux and is the Coriolis parameter. Thus
flow in the geostrophic boundary layer regime is governed by the relative roles of horizontal convective accelerations and Coriolis accelerations, or buoyancy and rotation, in the boundary layer.
Experimental evidence suggests that for more rapid rotation there is another transition to a regime in which the momentum budget is dominated by fluctuating vertical accelerations in a region of
vortical plumes, which we refer to as a 'chimney' following related discussion of regions of deep convection in the ocean. Coupling of the chimney convection in the region of destabilising boundary
flux to the diffusive boundary layer of horizontal convection in the region of stabilising boundary flux gives heat transport independent of rotation in this 'inertial chimney' regime, and the new
scaling . Scaling analysis predicts the transition conditions observed in the experiments, as well as a further 'geostrophic chimney' regime in which the vertical plumes are controlled by local
geostrophy. When < [CDATA] Ro, the convection is also observed to produce a set of large basin-scale gyres at all depths in the time-averaged flow.
Dive into the research topics of 'Geostrophic and chimney regimes in rotating horizontal convection with imposed heat flux'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/geostrophic-and-chimney-regimes-in-rotating-horizontal-convection","timestamp":"2024-11-10T03:21:49Z","content_type":"text/html","content_length":"52854","record_id":"<urn:uuid:768ff8aa-fed4-4988-b89d-c10e3bc225a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00695.warc.gz"} |
Algebra Factoring Calculator - Online Algebra Factoring Calculator
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Algebra Factoring Calculator
The word "Quadratic" is derived from the word "Quad" which means square. In other words, a quadratic equation is an “equation of degree 2”
What is Algebra Factoring Calculator?
'Algebra Factoring Calculator' is an online tool that helps to calculate the factors of a given quadratic equation. Online Algebra Factoring Calculator helps you to calculate the factors of a given
equation in a few seconds.
NOTE: The coefficient of x^2 should not be zero.
How to Use Algebra Factoring Calculator?
Please follow the steps below on how to use the calculator:
• Step 1: Enter the coefficients of a given equation in the given input boxes.
• Step 2: Click on the "Solve" button to find the factors of a given equation.
• Step 3: Click on the "Reset" button to clear the fields and enter the new values.
How to Find Algebra Factors?
An equation of the form ax^2 + bx + c = 0, where a ≠ 0 is called a quadratic equation and a, b, c are coefficients of a quadratic equation.
To determine the roots of the quadratic equation, we use its roots. If roots of a given quadratic equation are x[1] and y[1 ]then the factors are (x - x[1]) and (y - y[1])
To find the roots of a given quadratic equation, we use the discriminant formula given by \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Example:
Find the algebra factors of given quadratic equation x^2 + 5x + 6 = 0
Given: a = 1, b = 5, c = 6
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
\(x = {-5 \pm \sqrt{5^2-24} \over 2}\)
\(x = {-4 \over 2}, {-6 \over 2}\)
\(x= {-2},{-3}\)
Factors are (x + 2) and (x + 3)
Similarly, you can try the calculator to find the algebra factors for the given quadratic equation:
• 2x^2 + x − 3 = 0
• x^2 + 10x − 11 = 0
Math worksheets and
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COCI '06 Contest 1 #2 Herman
The 19^th century German mathematician Hermann Minkowski investigated a non-Euclidean geometry, called the taxicab geometry. In taxicab geometry the distance between two points and is defined as:
All other definitions are the same as in Euclidean geometry, including that of a circle:
A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed point (the centre of the circle).
We are interested in the difference of the areas of two circles with radius , one of which is in normal (Euclidean) geometry, and the other in taxicab geometry. The burden of solving this difficult
problem has fallen onto you.
Input Specification
The first and only line of input will contain the radius , an integer smaller than or equal to .
Output Specification
On the first line you should output the area of a circle with radius in normal (Euclidean) geometry.
On the second line you should output the area of a circle with radius in taxicab geometry.
Note: Outputs within of the official solution will be accepted.
Sample Input 1
Sample Output 1
Sample Input 2
Sample Output 2
Sample Input 3
Sample Output 3
There are no comments at the moment. | {"url":"https://dmoj.ca/problem/coci06c1p2","timestamp":"2024-11-11T23:23:29Z","content_type":"text/html","content_length":"20671","record_id":"<urn:uuid:e40fbf33-b058-4e76-a63a-491619cd4588>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00687.warc.gz"} |
Excel Formula: Countif Words Transgender Woman
In this tutorial, we will learn how to use the COUNTIF function in Excel to count the number of cells that contain the phrase 'Transgender Woman'. This formula is useful when you want to find the
occurrences of a specific phrase in a range of cells. The COUNTIF function allows you to specify a criteria and count the number of cells that meet that criteria. We will provide a step-by-step
explanation of the formula and provide examples to help you understand how it works.
To use the COUNTIF function, you need to provide two arguments. The first argument is the range of cells to be evaluated, and the second argument is the criteria to be applied. In this case, we will
use the range A:A to represent the entire column A, and the criteria 'Transgender Woman' to match any cells that contain the phrase 'Transgender Woman'. The asterisks (*) act as wildcards to match
any characters before and after the phrase.
For example, if we have the following data in column A:
The formula =COUNTIF(A:A, 'Transgender Woman') would return the value 3, indicating that there are 3 cells in column A that contain the phrase 'Transgender Woman'.
In conclusion, the COUNTIF function in Excel is a powerful tool for counting the occurrences of a specific phrase in a range of cells. By using wildcards, you can match any characters before and
after the phrase to find the desired cells. We hope this tutorial has been helpful in understanding how to use the COUNTIF function to count the number of cells that contain the phrase 'Transgender
An Excel formula
=COUNTIF(A:A, "*Transgender Woman*")
Formula Explanation
This formula uses the COUNTIF function to count the number of cells in column A that contain the phrase "Transgender Woman".
Step-by-step explanation
1. The COUNTIF function is used to count the number of cells in column A that meet a specific criteria.
2. The first argument of the COUNTIF function is the range of cells to be evaluated. In this case, we use A:A to represent the entire column A.
3. The second argument of the COUNTIF function is the criteria to be applied. We use "Transgender Woman" as the criteria, where the asterisks (*) act as wildcards to match any characters before and
after the phrase "Transgender Woman".
4. The COUNTIF function counts the number of cells in column A that contain the phrase "Transgender Woman" (ignoring case sensitivity).
For example, if we have the following data in column A:
| A |
| |
| Transgender |
| Woman |
| Transgender |
| Woman |
| Transgender |
| Man |
| Transgender |
| Woman |
The formula =COUNTIF(A:A, "Transgender Woman") would return the value 3, indicating that there are 3 cells in column A that contain the phrase "Transgender Woman". | {"url":"https://codepal.ai/excel-formula-generator/query/3Pq3KqrN/excel-formula-countif-words-transgender-woman","timestamp":"2024-11-04T14:07:07Z","content_type":"text/html","content_length":"93192","record_id":"<urn:uuid:eb4526c2-8ccf-4daa-a5ba-4b091601da3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00033.warc.gz"} |
r-truncdist 1.0-2
Truncated random variables
This package provides a collection of tools to evaluate probability density functions, cumulative distribution functions, quantile functions and random numbers for truncated random variables. These
functions are provided to also compute the expected value and variance. Q-Q plots can be produced. All the probability functions in the stats, stats4 and evd packages are automatically available for
Install r-truncdist 1.0-2 as follows:
guix install r-truncdist@1.0-2
Or install the latest version:
guix install r-truncdist
You can also install packages in augmented, pure or containerized environments for development or simply to try them out without polluting your user profile. See the guix shell documentation for more | {"url":"https://packages.guix.gnu.org/packages/r-truncdist/1.0-2/","timestamp":"2024-11-12T07:37:07Z","content_type":"text/html","content_length":"4703","record_id":"<urn:uuid:6b7ae3c7-6307-4f29-94f9-83625c19c444>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00842.warc.gz"} |
Algebra Seminar: Mkrtchyan -- Gradings on the Brauer algebra
SMS scnews item created by Ulrich Thiel at Fri 14 Sep 2018 0843
Type: Seminar
Distribution: World
Expiry: 26 Oct 2018
Calendar1: 21 Sep 2018 1500-1600
CalLoc1: Carslaw 375
CalTitle1: Mkrtchyan -- Gradings on the Brauer algebra
Auth: thiel@p526m.pc (assumed)
Algebra Seminar: Mkrtchyan -- Gradings on the Brauer algebra
Anna Mkrtchyan (MPI Bonn)
Friday 21 September, 3-4pm, Place: Carslaw 375
Title: Gradings on the Brauer algebra
Abstract: Brauer algebras B_n(\delta) are finite dimensional algebras introduced by
Richard Brauer in order to study the n-th tensor power of the defining representations
of the orthogonal and symplectic groups. They play the same role that the group
algebras of the symmetric groups do for the representation theory of the general linear
groups in the classical Schur-Weyl duality. We will discuss two different
constructions which show that the Brauer algebras are graded cellular algebras and then
show that they define the same gradings on B_n(\delta). | {"url":"https://www.maths.usyd.edu.au/s/scnitm/thiel-AlgebraSeminar-Mkrtchyan-?Clean=1","timestamp":"2024-11-10T15:02:36Z","content_type":"text/html","content_length":"1903","record_id":"<urn:uuid:59bcdd11-2e04-4616-b374-8c8d625c4619>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00134.warc.gz"} |
The Pearson probability distributions
The Pearson distributions
The previous post was about 12 probability distributions named after Irving Burr. This post is about 12 probability distributions named after Karl Pearson. The Pearson distributions are better known,
and include some very well known distributions.
Burr’s distributions are defined by their CDFs; Pearson’s distributions are defined by their PDFs.
Pearson’s differential equation
The densities of Pearson’s distributions all satisfy the same differential equation:
This is a linear differential equation, and so multiples of a solution are also a solution. However, a probability density must integrate to 1, so there is a unique probability density solution given
a, c[0], c[1], and c[2].
Well known distributions
Note that f(x) = exp(-x²/2) satisfies the differential equation above if we set a = 0, c[0] = 1, and c[1] = c[2] = 0. This says the normal distribution is a Pearson distribution.
If f(x) = x^m exp(-x) then the differential equation is satisfied for a = m, c[0] = −1, and c[0] = c[2] = 0. This says that the exponential distribution and more generally the gamma distribution are
Pearson distributions.
You can also show that the Cauchy distribution and more generally the Student t distribution are also Pearson distributions. So are the beta distributions (with a transformed range).
Table of Pearson distributions
The table below lists all Pearson distributions with their traditional names. The order of the list is a little strange for historical reasons.
The table uses Iverson’s bracket notation: a Boolean expression in brackets represents the function that is 1 when the condition holds and 0 otherwise. This way all densities are defined over the
entire real line, though some of them are only positive over an interval.
The densities are presented without normalization constant; the normalization constant are whatever they have to be for the function to integrate to 1. The normalization constants can be complicated
functions of the parameters and so they are left out for simplicity.
There is a lot of redundancy in the list. All the distributions are either special cases of or limiting cases of distributions I, IV, and VI.
Note that VII is the Student t distribution after you introduce a scaling factor.
The Pearson distributions are determined by their first few moments, provided these exist, and these moments can be derived from the parameters in Pearson’s differential equation.
This suggests moment matching as a way to fit Pearson distributions to data: solve for the distribution parameters that make the the exact moments match the empirical moments. Sometimes this works
very well, though sometimes other approaches are better, depending on your criteria for what constitutes a good match.
2 thoughts on “The Pearson distributions”
1. Sorry, what is *v* in distribution IV?
2. If you stick the expression for IV into the differential equation you get -(v + 2m x)/(1 + x²). Put this in the form of the differential equation and you get v as a function of the parameters.
Maybe you’re wondering whether v should be v(x) so you can derive the other distribution involving exp from IV. But v is a constant. You can derive the other distributions involving an exp by
taking a limit with v and m in the right ratio. | {"url":"https://www.johndcook.com/blog/2023/02/16/pearson-distributions/","timestamp":"2024-11-04T20:00:37Z","content_type":"text/html","content_length":"55626","record_id":"<urn:uuid:0d0c2745-f1f8-4ca2-b81b-2e10e3668260>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00011.warc.gz"} |
Imaginary Numbers Are a Cinch - Part 1, May 1967 Radio-Electronics
May 1967 Radio-Electronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.
As with many ancient mathematical and scientific concepts the provenance of imaginary numbers is open to opinions. The term "imaginary" seems to discredit the veracity of the concept, since after
all, what good is an imaginary entity? The Wikipedia account of imaginary numbers cite the works of Heron of Alexandria, Rafael Bombelli, Gerolamo Cardano, René Descartes, Leonhard Euler, and Carl
Friedrich Gauss, among others. In fact, the raison d'être for imaginary numbers was to facilitate the solving of equations involving even roots of negative numbers. If you are not familiar with such
things, a valid root of a number (radicand) must be able to regenerate itself by raising itself to the same power (exponent) as the index number of the root. For example, the second root (square
root) of 9 equals 3 (√9 = 3); therefore, the second power of 3 equals 9 (3^2 = 9). What if you need the square root of -9? The answer cannot be √9 = -3 because -3^2 = -3 x -3 = 9 (not -9).
Mathematicians introduced an identity operator i (or j for engineers) whose square is by definition -1 (i^2 = j^2 = -1). Consequently, √-1 = i (or j), so, again, i^2 = j^2 = -1. Problem solved. Here
is part 1 of a three-part series on the use of complex numbers, which are composed of both a real part and an imaginary part. A familiarity with complex numbers is essential for dealing with
impedances, voltages, and currents. Here are Part 1, Part 2, and Part 3.
Imaginary Numbers Are a Cinch - Part 1
By Norman H. Crowhurst
Part 1 - Math tools that aren't real, but which work in a real way.
It all started when George needed a filter for the new cross-multiplexing system he was developing. He called me up about his problem and I asked what he wanted the filter to do.
"Come on over, bring your design stuff, and I'll show you," was his response.
I went over to his lab and he explained the problem to me. After a little figuring, which he watched, I sketched a schematic and put the values in.
"We'll have that made up in a jiffy," he said, as he picked up the phone to call the storeroom for parts.
In less than an hour a messenger brought the parts, George's technician wired them, and George had a working filter. Satisfied that George had what he wanted, I turned to leave, but he called me
"Just a cotton-pickin' minute," he said. "Can't you show a fellow how you figured those values in such short order, so I can do it myself next time?"
It took me twice as long to explain it as it had taken me to figure it. He realized that what made the calculation difficult for him was the use of imaginary numbers, although that was what made it
easy for me. As I left, he made me promise to explain imaginary numbers to him in easy stages.
One slack afternoon about a week later, not long before closing time, George and I got together again. "Let's see why the idea seems difficult," was how I started.
"Do you remember when you first encountered irrational numbers?"
George remembered learning about them, but couldn't remember what they were.
To clarify them, I quickly went over rational numbers, particularly fractions. I showed, using a number line (Fig. 1), that 3/5 and 5/8 are fractions of close to the same magnitude. "If we compare
various fractions, it seems as if we can make up just about any size of part we want by taking a suitable denominator, or bottom, and then selecting an appropriate count, or numerator, of that
denominator." I illustrated this by fitting 49/80 between 3/5 and 5/8.
"That seems logical," George said, "but you say it as if there's a catch. What is it?"
"Well now, George, I'm sure you remember some numbers that could not be accurately written, either as decimals or as fractions. No matter what denominators and numerators you use, nor how many
decimal places you use, you can reach only an approximation."
"You mean like pi?" George asked, and then before I could answer he added, "or root two?"
"That's it. You do remember. That's the difference between rational and irrational numbers. Any number that can be written accurately with fractions using whole numbers for numerator and denominator,
or that can be written with a terminating decimal, is called a rational number. It fits the known pattern of numbers before irrational numbers were admitted to exist. Then mathematicians found
impossible-seeming numbers like root two, or pi ..."
"And they're irrational numbers?"
George butted in.
"That's right."
"Makes sense," he responded, "although I don't remember learning it that way in school. But irrational numbers aren't the same as imaginary numbers, are they?"
"No, they're not, but notice this:
At one time math scholars thought any number could be represented by a fraction. Later they realized that irrational numbers belong to a different class than rational numbers. And do you remember
when you first learned about negative numbers?"
"I should say," George replied. "They gave me a hard time. Especially that bit about minus times a minus making a plus. I never did understand that fully."
"Oddly enough, understanding imaginary numbers will make that easier too." I then asked him, "Do you know how negative numbers differ from positive numbers?"
"Well, as I see it," he said, "it's like another world, an upside-down world. When you combine negative numbers with positive numbers, you subtract the negative ones whereas you would add them if
they were all positive."
""You're right. I don't know if a math teacher would accept that answer, but it shows you know. Now, do you remember doing squares and square roots?"
"It's a bit rusty," he replied, "but I remember it."
"Let's take some simple cases," I suggested, "What is 2 times 2?"
He looked at me as if I must think him stupid, volunteered "4" and then looked quizzical, as if he thought that answer might somehow be wrong in higher mathematics.
"Right. Now 3 times 3?"
"Nine," he replied, still puzzled. "So you know what squares are.
Now what is -3 times -3?"
"Nine," he replied, hesitantly. "Yes, but is it +9 or -9?" "Well, I remember having trouble with that," he admitted, "but if I remember right I got it through my head that it was +9."
"Right." Then I asked, "So what's the square root of +9?"
"Three," he replied.
"Plus 3 or minus 3?" I asked.
He thought for a moment and then said, "It could be either, couldn't it?"
"That's right," I assured him. "So, if the square root of +9 can be +3 or -3, what is the square root of -9?"
"Didn't we learn that you can't actually have a square root of a minus number?"
"Maybe, but let's recap a little. Before we knew about irrational numbers, no such number as pi or root two seemed possible. Before we knew about negative numbers, we were told we couldn't subtract 8
from 3. Later, we accepted the existence of irrational and negative numbers, and these impossible numbers became possible. So let's imagine there are roots of negative numbers."
"You mean we just accept them, and then learn to use them." I could see his interest growing; he wanted to find out where imaginary numbers would come from.
"What we do," I went on, "is to write the letter j and assign it the meaning square root of -1. All along we have believed there isn't such a thing. Now we imagine there is, although all we know so
far is that we just gave it a name; square root of -1. But from that very fact, we know that squaring it will make -1."
"That's right enough, but aren't we going round in circles?"
"We soon will be," I replied, "but not in the way you think."
I showed him that, just as the square root of +1 is either plus 1 or -1, the square root of -1 can have two signs, + j or -j. The square root of -9, for example, is either +j3 or -j3.
"It doesn't make sense yet," said George, "but go on, 'cause it's getting interesting. "
"Remember that minus times a minus makes a plus," I said, drawing out vectors to illustrate (Fig. 2-a). "A minus reverses direction and the second minus reverses it again, bringing us to the original
direction. With j numbers, multiplying j by j brings us to negative. As a math teacher would say, 'By definition, j times j makes a minus.' So, if negative represents reversal, what does the j sign
"From the way you twiddled your pencil," George responded, "I'd say it could mean halfway to reversal, or 90°. Is that it?"
"You've spotted it, George," I said, sketching a vector representation (Fig. 2-b) for him. "The vector diagrams illustrate ... "
"Hey, I begin to see daylight," he interrupted me. "It's a way of writing quadrature in math symbols, without spelling it out. But does it make the calculations easier too?"
"Yes, the j tells you what to do with the number that follows it, just like plus and minus signs have been doing."
"Just a minute, why j? What does it stand for? Wouldn't i be better - for imaginary?"
"As a matter of fact, that's what mathematicians and physicists use. But in electrical and electronic work, i already stands for current - although I never knew why - so, to avoid confusion,
electrical people started using the next letter of the alphabet - j - for root of -1."
"Is that what they call the operator j?" George again butted in.
"That's right."
"And operator j means the root of minus one?"
"Correct again."
"Well, I'll be a monkey's uncle!"
George exclaimed. "I don't know whether that clarifies anything for me, but I've a suspicion it will."
"In modern math classes," I went on, "they use number lines, as I did to demonstrate the fractions in my first sketch. Lengths along the line represent numbers." And I showed him how the concept of
negative numbers is developed on a number line, following the concept of reversal a step further.
"They didn't use number lines when I was in school," George commented. "Maybe things would have come easier if they had."
"The main thing to note," I continued, "is how you add numbers on a number line. Start the second number from where the first one finishes; the result, or sum, is at the end of the sec-ond line
"Makes sense," said George. "In fact, it seems obvious." /p>
"Do you think now you could add an imaginary number to a real number, either positive or negative?"
"Is it at right angles?"
"That's right. So we have a right-triangle vector addition," I went on, sketching Fig. 3 as I talked.
"Hey, that really is familiar, though the old right-triangle bit always gave me a headache!"
''Then how would you go about finding the total, or resultant vector?" was my next question.
"Oh, that's old Pythag ... what's his name? The sum of the squares on the other two sides - what did they call it, hypot ... ?" George wondered aloud.
"It's Pythagoras' theorem," I filled in, "and the side opposite the right angle is call the hypotenuse. I'll bet you've done quite a few exercises in school squaring two sides and finding the square
root of their sum for the result."
"Yes," George said, "we seemed to have endless exercises in that at technical school. Will imaginary numbers take the sweat out of all that?"
"Sure will. Right now, it may seem Just another name for what you've done before. But as we move along, you'll find imaginary numbers lead to more and more shortcuts, making calculations easier."
"That's for me," declared George.
"Now to complete the picture. For this one vector in my sketch, do you know how to find the phase angle?"
"You have me there. It has something to do with the ratio of two sides - is that right?"
I nodded and he went on, "I had some trig. It's called the sine or cosine or something like that, isn't it?"
"The sine and cosine are two other ratios," I told him. "This one's the tangent. It's the side opposite the phase angle, divided by the side adjacent." And I sketched it (Fig. 4).
"The one opposite is the imaginary part, while the one adjacent is the real part. Now, to every angle there is just one tangent ratio, and to every tangent ratio there is just one angle. So we can
find the angle from the tangent ratio, or the tangent ratio from the angle - either way, so long as we have one of them to start with."
George said, "Let's get this straight: I know the ratio between imaginary and real parts, so I look up the tangent of this ratio to find the angle. Is that right?"
"Ordinarily, it's the other way round. If you knew the angle in degrees, you'd look up the tangent of the angle to find the ratio between the imaginary and real parts. Here, you know the ratio, so
you look up the arctan to find the angle. As we don't happen to have arctan tables, we can use the tangent table backward, or a slide rule.
"Let's take something practical and relatively simple, like an inductance with some series resistance. The real part of its impedance, producing in-phase voltage and current, is the coil resistance.
The imaginary part, producing quadrature voltage and current, is the inductive reactance of the coil. Do you remember inductive reactance?"
"Let's see," pondered George, "it depends on L - the inductance in henrys - and there's a 2-pi-f in it, isn't there?"
"When we're using it a lot, it's much easier to write a lower-case omega for 2-pi-f." I wrote this equation on our scrap paper (Fig. 5). "Now, does inductive reactance get bigger or smaller as we
increase frequency?"
"Isn't inductive reactance directly proportional to frequency?"
"Right. If L is in henrys, and f is in hertz and omega is 2-pi-f, then inductive reactance is simply omega times L."
"That does make it look simple," commented George.
"A reactive voltage is in quadrature with current," I went on, "so that means we need a j to represent the value properly. If the coil resistance is R ohms and the inductance is L henrys, the
complete expression for impedance is Z = R + jωL. By multiplying various values of f by 2π to get omega, we can make this expression tell the whole story, at all frequencies. Now what's the numerical
value, or magnitude, of this expression?"
"From what we were saying earlier," George replied, "you'd square R and square omega-L, add the two squares together and then take the square root."
"You're catching on fast. Now what's the phase angle?"
"That's not so easy, but I know it has to do with the ratio between R and omega-L," he replied.
"Let's put in some numbers, and see how this works," I suggested, turning again to Fig. 5. "Suppose a 10-mH coil has a resistance of 50 ohms. Let's figure its impedance and phase at a few
frequencies. To start with, take 1,000 Hz. First, what is omega at 1,000 Hz?"
"Well, 2-pi is 6.283, so 1,000 Hz makes omega 6,283."
"Correct. So what is omega-L at 1,000 Hz?"
"10 mH is one-hundredth of a henry," George mused, and doodled a moment, "so omega-L comes out to 62.83, right?"
I showed him how to use his slide rule to sum the squares and take the square root at that sum, getting 80.3 ohms as the impedance Z. I asked him about the phase angle. He had the idea, but needed to
be shown how to do it.
He solved the ratio of imaginary to real, which was 1.257, and I showed him on the slide rule the angle-about 51 1/2°. Then he took the slide rule and calculated impedance and phase angle for several
other frequencies, while I watched. His results are shown in the table. He was thrilled at how simple it seemed.
Then I threw in the notion of using admittance values instead of impedance values for parallel combinations whose phase is in quadrature. He wanted to know why. He'd always preferred thinking in
I showed him that, with a specific voltage, he'd have to divide by the complex number representing impedance to figure out current. With admittance, he could simply multiply.
"Is that so much easier?" he wanted to know.
Just then the evening whistle blew. George and I both had to be going.
"If you don't have too much work ahead of you," George asked, "can you drop by another time soon and finish telling me that bit?"
"Sure thing," I replied.
To Be Continued
Posted July 22, 2024 | {"url":"https://rfcafe.com/references/radio-electronics/imaginary-numbers-radio-electronics-may-1967.htm","timestamp":"2024-11-11T03:28:35Z","content_type":"text/html","content_length":"43372","record_id":"<urn:uuid:247b848f-8e51-4e29-8a6d-068945805613>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00201.warc.gz"} |
Ratio formula
ratio formula Related topics: difference between evaluation and simplification of expression
dividing decimals worksheets
free algebra help
synthetic division and the remainder theorem calculator online
ti-85 decemal to fractions
taks math formula chart mathematics
easy free way to multiply binomials
writing fraction as percent
solve expressions calculator
solving fraction equations worksheet
dividing cubed radicals
how is doing operations (adding, subtracting, multiplying, and dividing) with rational expressions similar to or different from doing operations with fractions?
Author Message
Hesc_ne Posted: Saturday 27th of May 10:13
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it. If someone can give me some suggestions, I will be obliged.
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espinxh Posted: Sunday 28th of May 11:25
Your story sounds familiar to me. Even though I was good in algebra for many years, when I attended Basic Math there were a lot of math topics that seemed very hard . I remember I got a
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didn't spend my money on a tutor, because I heard of Algebrator from a a colleague. I have been using it since then whenever I stumbled upon something hard .
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Majnatto Posted: Monday 29th of May 14:57
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merely clicking Solve would give step by step solution to the math problem. It has been of great help through several Remedial Algebra, College Algebra and Algebra 1. I seriously
recommend the program.
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MichMoxon Posted: Tuesday 30th of May 16:35
I am a regular user of Algebrator. It not only helps me get my homework faster, the detailed explanations offered makes understanding the concepts easier. I strongly suggest using it to
help improve problem solving skills.
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killgoaall Posted: Thursday 01st of Jun 09:02
That’s what I exactly need ! Are you sure this will help me with my problems in algebra? Well, it doesn’t hurt if I try the software. Do you have any links to share that would lead me to
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Majnatto Posted: Saturday 03rd of Jun 07:24
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A perpetuity provides payments every six months starting today. The first payment is 1 and each payment is \(3 \%\) greater than the immediately preceding payment. Find the present value of the
perpetuity if the effective rate of interest is \(8 \%\) per annum.
Short Answer
Expert verified
The present value of the perpetuity is approximately 108.70, considering an equivalent six-month interest rate of 3.92% and a growth rate of 3% for the payments.
Step by step solution
Calculate the equivalent six-month interest rate
The effective interest rate is given as 8% per annum (per year). We need to find the equivalent interest rate for a six-month period. To do this, we use the formula: \[ (1+i) = (1+j)^n \] where i is
the annual effective interest rate, j is the equivalent interest rate for the six-month period (which we need to find), and n is the number of times the interest is compounded within a year (in this
case, n = 2, as there are two six-month periods in a year). Plugging the values, we have: \[ (1+0.08) = (1+j)^2 \]
Solve for the equivalent six-month interest rate (j)
We now solve the equation for j: \[ 1.08 = (1+j)^2 \] Taking the square root of both sides, we get: \[ \sqrt{1.08} = 1+j \] Subtract 1 from both sides to isolate j: \[ \sqrt{1.08} - 1 = j \] Now,
calculate the value of j: \[ j = \sqrt{1.08} - 1 \approx 0.0392 \] So, the equivalent six-month interest rate is approximately 3.92%.
Calculate the present value of the growing perpetuity
Next, we will use the formula for the present value of a growing perpetuity with the first payment made today, which is: \[ PV = \frac{PMT_0}{j-g} \] where PV is the present value, PMT_0 is the first
payment, j is the six-month interest rate, and g is the growth rate of the payments. For this problem, PMT_0 = 1, j = 0.0392 (3.92%), and g = 0.03 (3%). Plugging these values into the formula, we
get: \[ PV = \frac{1}{0.0392 - 0.03} \]
Calculate the present value of the perpetuity
Finally, we calculate the present value: \[ PV = \frac{1}{0.0092} \approx 108.70 \] So, the present value of the perpetuity is approximately 108.70.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Present Value
The present value is a crucial concept in finance and refers to the current value of a future cash flow or series of cash flows. It's determined by discounting the future value back to the present
using a specified interest rate. This helps in assessing how much future payments are worth today, allowing investors to make informed financial decisions.
The formula for present value is:
• \( PV = \frac{PMT_0}{j-g} \)
• \( PV \) = present value
• \( PMT_0 \) = initial payment
• \( j \) = interest rate
• \( g \) = growth rate of payments
In the context of the problem, we used this formula to calculate the present value of a growing perpetuity.This is significant as it factors in that each payment grows at a certain rate. By
calculating the present value, it allows us to understand how much this series of cash flows is worth today.
Interest Rate
The interest rate is the percentage charged on a loan or paid on an investment over a specified period of time. It plays a critical role in the present value calculations. Typically, rates are annual
but can be adjusted for different periods such as monthly or semi-annual.
For the presented problem, the yearly interest rate given is 8%, and we needed to convert it into a six-month rate to match the payment schedule of the perpetuity. This conversion ensures that the
rate is relevant to the payment intervals and allows accurate calculations for present value.
Understanding how to adjust the interest rate to match the cash flow period is essential in finance. It ensures that the rate being used for calculations accurately reflects the time over which
calculations are being made.
Growing Perpetuity
A growing perpetuity is a type of financial instrument that offers an indefinite series of cash flows, where each payment grows at a consistent rate. This growth distinguishes it from a regular
perpetuity, where payments remain constant over time.
When calculating the present value of a growing perpetuity, it is important to account for the growth rate because it affects how future payments are valued in today's terms. The formula takes the
initial payment and adjusts for both the interest rate and the growth rate:
• \( PV = \frac{PMT_0}{j-g} \)
In the exercise, the growing perpetuity begins with a payment of 1, and each subsequent payment grows by 3%.
This specific setup allows us to use the formula with the given initial payment, interest rate, and growth rate, to determine the perpetuity's present value.
Effective Interest Rate
The effective interest rate enables you to understand the actual annual interest you'll earn or pay, accounting for compounding within the year. This rate is essential because it provides a true
picture of financial growth or cost.
In scenarios where payments occur more frequently than annually, like semi-annual in our exercise, the effective interest rate must be adjusted to reflect the accurate compounding periods. The annual
rate provided was 8%, which was then adjusted to reflect two compounding periods per year.
Using the equation \((1+i) = (1+j)^n\), we converted the annual rate into a semi-annual rate. This allowed us to use the correct rate for calculating the present value of the growing perpetuity,
ensuring the calculations were precise and matched the payment frequency. | {"url":"https://www.vaia.com/en-us/textbooks/math/the-theory-of-interest-2-edition/chapter-4/problem-45-a-perpetuity-provides-payments-every-six-months-s/","timestamp":"2024-11-03T16:53:06Z","content_type":"text/html","content_length":"248106","record_id":"<urn:uuid:cbf5372a-14ed-44b4-9355-cc1c8eb64f51>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00786.warc.gz"} |
Revisiting the “Stuff” Metric
This article was co-authored by Daanish Mulla – @DanMMulla
Last month, we wrote an article on calculating a pitcher’s “stuff”. We were quite pleased with how our equation performed with respect to predicting a pitcher’s strikeout rate and his xFIP. Part of
the discussion surrounding the equation was what exactly is stuff? Well, in our case, stuff can be thought of as a three-dimensional shape, where the three axes of the shape represent a pitcher’s
peak velocity, a pitcher’s change in velocity between their fastest and slowest pitch, and the amount of distance that their pitches can break. In other words, it aims to represent the range in pitch
velocity and movement batters must account for during any given at-bat against a particular pitcher.
However, there was still some room for improvement, and with help from the FanGraphs community, we’ve slightly modified our equation to improve various performance predictions. The first major change
came from comparing faster breaking balls versus slower breaking pitches with greater movement. In our original stuff metric, pitchers with a slow, looping breaking ball received more benefit than
pitchers throwing a fast breaking ball. I queried the PitchF/x database to see how swinging strike rates and batting average changed against curveballs with respect to pitch speed during the 2014
season. Pitches that were thrown for at least 1% of all pitches were included in this analysis. As you can see in the figure, swinging-strike percentage increases exponentially after 75mph, and is
nearly 15% higher at 85mph than at 75mph. This encouraged us to find a better way to account for faster breaking balls.
Secondly, the original metric did not account for pitch frequency. The Pitch Arsenal metric was improved from it’s original state by accounting for this, and realistically – a pitcher should be given
more credit for a great pitch that they throw frequently, as opposed to a great pitch that they rarely throw. To account for this, pitches were classified as either off-speed/breaking or fastballs.
The sum of pitch uses for each of these classifications was then used to modify the values in the equation. With that in mind, here’s how we have proposed to modify the stuff equation.
For a pitch to be included in the analysis, it had to be thrown by the pitcher 100 times. Just like the original stuff equation, z-scores were determined for the fastest pitch the pitcher threw, and
for the amount of movement that could be seen with respect to that fastball, from the remaining pitches. For further analysis, only qualified starters were used (those who threw 162 innings in the
2015 season).
Furthermore, z-scores were also determined for the % change in speed between the pitcher’s fastest and slowest pitch. Another z-score was determined for the velocity of the fastest pitch, between
curveball, slider, or knuckle-curve. Frequencies were determined for the proportion of fastballs thrown by a pitcher, and the remaining non-fastball pitches. The z-score for velocity was multiplied
by the fastball percentage, and the remaining z-scores were multiplied by the non-fastball frequency. The z-scores for peak velocity of breaking pitch and change in velocity were used to determine
“pitch strategy” – either, power breaking ball, or change in speed. Whichever z-score was greater, was used in the final stuff equation.
So, the final “stuff” equation is as follows:
To begin validation of the equation, the stuff value was then correlated with K/9 for all qualifying starters. This resulted in a predicted R value of 0.53 (figure 2), compared to the value of 0.42
from the original stuff equation.
We’ve since applied the stuff equation to all pitchers from 2007 to 2015 to try and get an idea of the range of the metric. Here’s what we found. For interpretation of this figure, if a pitcher has a
stuff value of 0.90, his stuff is better than 75% of all pitchers since 2007. If the value is 2.0, they have stuff that is better than approximately 99% of all pitchers since 2007. To put that in
perspective, that means their stuff is better than nearly 4000 other starting pitchers. You’ll notice that in our list of the top 30 pitchers from 2015 – all of these pitchers fall within the top 15%
range of stuff. These are elite pitchers with respect to this metric.
These data have a wealth of applications, such as how a pitcher returns from injury or has even changed his repertoire between years. For example, the jump Chris Bassitt made from 2014 to 2015 –
going from someone in the bottom half of the metric to the 99^th %ile. Similar to the Arsenal score, there is an application of these data in determining a pitcher on the verge of a breakout (perhaps
the Joe Kelly of the second half of 2015 is the real Joe Kelly).
However, we felt that it would be in our best interest to let the community decide just how useful the metric was, so we’re making our evaluation data from 2007 to 2015 available in the form of a
Google sheet. Simply select the pitcher you’d like to evaluate, and their stuff scores and xFIPs will be graphed for you. We’ve also posted the entirety of stuff scores from the 2015 season.
2015 Season
Stuff worksheet
Philosophically, we feel that the stuff metric has a great benefit for advanced scouting, because it relies on measures that are solely dependent on the pitcher, and not an interaction of the pitcher
and the hitter. Thanks to the FanGraphs community, r/baseball, and Eno Sarris for all of the support with this project.
Ergonomist (CCPE) and Injury Prevention researcher. I like science and baseball - the order depends on the day. Twitter: @DrMikeSonne
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8 years ago
I would say you are still missing some things in regards to your STUFF equation. Namely, pitch variety, repertoire if you will, plus pitcher deception. Some things that fall underneath pitcher
deception would be a pitchers mechanics and how odd or off the beaten path they are, such as Clayton Kershaw’s and Masahiro Tanaka’s hesitation/pause move they do with their feet, or a pitcher’s
stride. I know the typical pitcher strides 80% of their height or thereabouts. Mine? 5′ 10″- to 5′ 11″, but my height is 5′ 8″. I am quick, but long back and long forward in my motion, this creates a
perceived velocity of way faster than you are in terms of mph. It doesn’t hurt to throw the ball where you actually intend to in terms of the catchers glove. | {"url":"https://community.fangraphs.com/revisiting-the-stuff-metric/","timestamp":"2024-11-09T17:50:06Z","content_type":"text/html","content_length":"148659","record_id":"<urn:uuid:2bd9400f-3b41-427b-8cdd-16d630c334f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00360.warc.gz"} |
Example which shows how you can get the graph of the lower plot and set the y axis range for it.
Since the lower plot is not created until TRatioPlot::Draw is called, you can only use the method afterwards. Inspired by the tutorial of Paul Gessinger.
import ROOT
c1 = ROOT.TCanvas("c1", "fit residual simple")
h1 = ROOT.TH1D("h1", "h1", 50, -5, 5)
h1.FillRandom("gaus", 2000)
rp1 = ROOT.TRatioPlot(h1)
Alberto Ferro
Definition in file ratioplot3.py. | {"url":"https://root.cern.ch/doc/master/ratioplot3_8py.html","timestamp":"2024-11-04T01:42:03Z","content_type":"application/xhtml+xml","content_length":"9184","record_id":"<urn:uuid:8942fffe-ecb1-43b8-87a0-c097bad53b40>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00688.warc.gz"} |
Spectral Radius Inequalities for Functions of Operators De
Spectral Radius Inequalities for Functions of Operators Dened by Power Series
By the help of power series f (z) =
1n=0 anzn we can naturally
construct another power series that has as coe¢ cients the absolute values of
the coe¢ cients of f, namely fa (z) :=
1n=0 janj zn: Utilising these functions
we show among others that
r [f (T)] fa [r (T)]
where r (T) denotes the spectral radius of the bounded linear operator T on
a complex Hilbert space while kTk is its norm. When we have A and B two
commuting operators, then
r2 [f (AB)] fa
• There are currently no refbacks. | {"url":"http://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/1626","timestamp":"2024-11-06T04:39:45Z","content_type":"application/xhtml+xml","content_length":"15462","record_id":"<urn:uuid:d161004f-1623-428f-96f3-4f296743a081>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00205.warc.gz"} |
Axial tilt
In astronomy, axial tilt (also called obliquity) is the angle between an object's rotational axis, and a line perpendicular to its orbital plane. It differs from inclination.
To measure obliquity, use the right hand grip rule for both the rotation and the orbital motion, i.e.: the line from the vertex at the object's centre to its north pole (above which the object
appears to rotate counter-clockwise); and the line drawn from the vertex in the direction of the normal to its orbital plane, (above which the object moves counter-clockwise in its orbit). At
zero degrees, these lines point in the same direction.
The planet Venus has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like Earth. The north pole of Venus is pointed 'downward' (our southward).
The planet Uranus is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the Sun. Hence the axial tilt of
Uranus is 97°.^[1]
Over the course of an orbit, while the angle of the axial tilt does not change, the orientation of a planet's axial tilt moves through 360 degrees (one complete orbit around the Sun), relative to
a line between the planet and the Sun, causing seasons on Earth.
In the solar system, the Earth's orbital plane is known as the ecliptic plane, and so the Earth's axial tilt is officially called the obliquity of the ecliptic. It is denoted by the Greek letter
The Earth currently has an axial tilt of about 23.5°.^[2] The axis remains tilted in the same direction towards the stars throughout a year and this means that when a hemisphere (a northern or
southern half of the earth) is pointing away from the Sun at one point in the orbit then half an orbit later (half a year later) this hemisphere will be pointing towards the Sun. This effect is
the main cause of the seasons (see effect of sun angle on climate). Whichever hemisphere is currently tilted toward the Sun experiences more hours of sunlight each day, and the sunlight at midday
also strikes the ground at an angle nearer the vertical and thus delivers more energy per unit surface area.
Lower obliquity causes polar regions to receive less seasonally contrasting solar radiation, producing conditions more favorable to glaciation. Like changes in precession and eccentricity,
changes in tilt influence the relative strength of the seasons, but the effects of the tilt cycle are particularly pronounced in the high latitudes where the great ice ages began.^[3] Obliquity
is a major factor in glacial/interglacial fluctuations (see Milankovitch cycles).
The obliquity of the ecliptic is not a fixed quantity but changing over time in a cycle with a period of 41,000 years (see below). Note that the obliquity and the precession of the equinoxes are
calculated from the same theory and are thus related to each other. A smaller ε means a larger p (precession in longitude) and vice versa. Yet the two movements act independent from each other,
going in mutually perpendicular directions.
Knowledge of the obliquity of the ecliptic (ε) is critical for astronomical calculations and observations from the surface of the Earth (Earth-based, positional astronomy).
To quickly grasp an idea of its numerical value one can look at how the Sun's angle above the horizon varies with the seasons. The measured difference between the angles of the Sun above the
horizon at noon on the longest and shortest days of the year gives twice the obliquity.
To an observer on the equator standing all year long looking above, the sun will be directly overhead at noon on the March Equinox, then swing north until it is over the Tropic of Cancer, 23° 26’
away from the equator on the Northern Solstice. On the September Equinox it will be back overhead, then swing south until it is over the Tropic of Capricorn, 23° 26’ away from the equator on the
Southern Solstice.
Example: an observer at 50° latitude (either north or south) will see the Sun 63° 26’ above the horizon at noon on the longest day of the year, but only 16° 34’ on the shortest day. The
difference is 2ε = 46° 52’, and so ε = 23° 26’.
(90° - 50°) + 23.4394° = 63.4394° when measuring angles from the horizon (90° - 50°) − 23.4394° = 16.5606°
At the Equator, this would be 90° + 23.4394° = 113.4394° and 90° − 23.4394° = 66.5606° (measuring always from the southern horizon).
Abu-Mahmud Khojandi measured the Earth's axial tilt in the 10th century using this principle with a giant sextant and noted that his value was lower than those of earlier astronomers, thus
discovering that the axial tilt is not constant.^[4]
The Earth's axial tilt varies between 22.1° and 24.5° (but see below), with a 41,000 year period, and at present, the tilt is decreasing. In addition to this steady decrease there are much
smaller short term (18.6 years) variations, known as nutation, mainly due to the changing plane of the moon's orbit. This can shift the Earth's axial tilt by plus or minus 0.005 degree.
Simon Newcomb's calculation at the end of the nineteenth century for the obliquity of the ecliptic gave a value of 23° 27’ 8.26” (epoch of 1900), and this was generally accepted until improved
telescopes allowed more accurate observations, and electronic computers permitted more elaborate models to be calculated. Lieske developed an updated model in 1976 with ε equal to 23° 26’ 21.448”
(epoch of 2000), which is part of the approximation formula recommended by the International Astronomical Union in 2000:
ε = 84381.448 − 46.84024T − (59 × 10^−5)T^2 + (1.813 × 10^−3)T^3, measured in seconds of arc, with T being the time in Julian centuries (that is, 36,525 days) since the ephemeris epoch of 2000
(which occurred on Julian day 2,451,545.0). A straight application of this formula to 1900 (T=-1) returns 23° 27’ 8.29”, which is very close to Newcomb's value.
With the linear term in T being negative, at present the obliquity is slowly decreasing. It is implicit that this expression gives only an approximate value for ε and is only valid for a certain
range of values of T. If not, ε would approach infinity as T approaches infinity. Computations based on a numerical model of the solar system show that ε has a period of about 41,000 years, the
same as the constants of the precession p of the equinoxes (although not of the precession itself).
Other theoretical models may come with values for ε expressed with higher powers of T, but since no (finite) polynomial can ever represent a periodic function, they all go to either positive or
negative infinity for large enough T. In that respect one can understand the decision of the International Astronomical Union to choose the simplest equation which agrees with most models. For up
to 5,000 years in the past and the future all formulas agree, and up to 9,000 years in the past and the future, most agree to reasonable accuracy. For eras farther out discrepancies get too
Long period variations
Nevertheless extrapolation of the average polynomials gives a fit to a sine curve with a period of 41,013 years, which, according to Wittmann, is equal to:
ε = A + B sin(C(T + D)); with A = 23.496932° ± 0.001200°, B = − 0.860° ± 0.005°, C = 0.01532 ± 0.0009 radian/Julian century, D = 4.40 ± 0.10 Julian centuries, and T, the time in centuries from
the epoch of 2000 as above.
This means a range of the obliquity from 22° 38’ to 24° 21’, the last maximum was reached in 8700 BC, the mean value occurred around 1550 and the next minimum will be in 11800. This formula
should give a reasonable approximation for the previous and next million years or so. Yet it remains an approximation in which the amplitude of the wave remains the same, while in reality, as
seen from the results of the Milankovitch cycles, irregular variations occur. The quoted range for the obliquity is from 21° 30’ to 24° 30’, but the low value may have been a one-time overshot of
the normal 22° 30’.^[citation needed]
Over the last 5 million years, the obliquity of the ecliptic (or more accurately, the obliquity of the Equator on the moving ecliptic of date) has varied from 22.0425° to 24.5044°, but for the
next one million years, the range will be only from 22.2289° to 24.3472°.^[citation needed]
Other planets may have a variable obliquity, too; for example, on Mars, the range is believed to be between 11° and 49° as a result of gravitational perturbations from other planets.^[5] The
relatively small range for the Earth is due to the stabilizing influence of the Moon, but it will not remain so. According to W.R. Ward, the orbit of the Moon (which is continuously increasing
due to tidal effects) will have gone from the current 60 to approximately 66.5 Earth radii in about 1.5 billion years. Once this occurs, a resonance from planetary effects will follow, causing
swings of the obliquity between 22° and 38°. Further, in approximately 2 billion years, when the Moon reaches a distance of 68 Earth radii, another resonance will cause even greater oscillations,
between 27° and 60°. This would have extreme effects on climate.
Axial tilt of selected objects in the solar system
Object Axial tilt (°) Axial tilt (radians)
Sun 7.25 0.1265
Mercury 0.0352 0.000614
Venus 177.4 3.096
Earth 23.44 0.4091
Moon 6.688^† 0.1167
Mars 25.19 0.4396
Ceres ~4 ~0.07
Pallas ~60 ~1
Jupiter 3.13 0.0546
Saturn 26.73 0.4665
Uranus 97.77 1.7064
Neptune 28.32 0.4943
Pluto 119.61 2.0876
† Tilt to its orbit in the Earth-Moon system. Moon's tilt is 1.5424° (0.02692 radians) to ecliptic
See also
External links
□ Explanatory supplement to "the Astronomical ephemeris" and the American Ephemeris and Nautical Almanac
□ A comparison of values predicted by different theories at tenspheres.com
□ Berger, A. L. "Obliquity & precession for the last 5 million years". Astronomy & Astrophysics 51, 127 (1976)
□ Wittmann, A. "The obliquity of the ecliptic". Astronomy & Astrophysics 73, 129-131 (1979)
□ Ward, W. R. "Comments on the long-term stability of the Earth's obliquity". Icarus 1982, 50, 444
□ Bryant, Jeff. Axial Tilts of Planets, Wolfram Demonstrations Project
Wikimedia Foundation. 2010. | {"url":"https://en-academic.com/dic.nsf/enwiki/54487","timestamp":"2024-11-07T07:20:10Z","content_type":"text/html","content_length":"59966","record_id":"<urn:uuid:dc7e53fb-7bcf-41e1-a679-dee2529e95fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00333.warc.gz"} |
Hull Moving Average Turning Points and Concavity (2nd Derivatives) - useThinkScript Community
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⚠ This thread has exhausted its substantive discussion of this indicator so it has been locked ⚠
This thread is still available for reading. If looking for specific information in this thread, here is a
great hack for searching many-paged threads
Future Questions & Answers can be posted here:
As I was looking at the Hull Moving Average, it occurred to me that taking the local minimums and maximums might not be the most effective use of it's smooth nature. Nor did it seem that a moving
average cross over would provide timely signals.
And then I remembered that I used to teach physics and calculus and decided to look at rates of change and that lead me to...
which is nothing but the second derivative of the function (if you remember or not)
It is in two parts -- the upper which is the Hull Moving Average with the addition of colored segments representing concavity and turning points: maxima, minima and inflection. The last of these are
of the greatest interest. The second part is a plot of the calculation used in finding the turning points (which is roughly the second derivative of the HMA function, where zero crosses are the
inflection points.
Upper HMA colors:
• Green: Concave Up but HMA decreasing. The 'mood' has changed and the declining trend of the HMA is slowing. Long trades were entered at the turning point
• Light Green: Concave up and HMA increasing. Price is increasing, and since the curve is still concave up, it is accelerating upward.
• Orange: Concavity is now downward, and though price is still increasing, the rate has slowed, perhaps the mood has become less enthusiastic. We EXIT the trade (long) when this phase starts. Very
little additional upward price movement is likely.
• Red: Concave down and HMA decreasing. Not good for long trades, but get ready for a turning point to enter long on again.
Upper Label Colors:
these are useful for getting ready to enter a trade, or exit a trade and serve as warnings that a turning point may be reached soon
• Green: Concave up and divergence (the distance from the expected HMA value to the actual HMA value is increasing). That is, we're moving away from a 2nd derivative zero crossover.
• Yellow: Concave up but the divergence is decreasing (heading toward a 2nd derivative zero crossover); it may soon be time to exit the trade.
• Red: Concave down and the absolute value of the divergence is increasing (moving away from crossover)
• Pink: Concave down but approaching a zero crossover from below (remember that that is the entry signal, so pink means 'get ready').
Arrows are provided as Buy and Sell and could perhaps be scanned against.
For those who prefer less cluttered uppers, I offer the plot of the divergence from expected HMA values; analogous to the second derivative in that the zero crossovers are of interest, as is the
slope of the line. The further from zero, the stronger the curve of the concavity, and the more likely to reach a local minima or maxima in short order.
If you find that there are too many buy and sell signals, you can change the length of the HMA (I find 34 a happy medium, though 55 and sometimes 89 can be appropriate). You can also play with the
value of the lookback, though that will slow down signals.
This works well on Daily timeframes as well as intraday candles.
I set it up with
High + Low / 2
as the default so that it shouldn't wait for close prices. That may not be appropriate to how you wish to trade.
Comments welcome. Let me know how you use it, how it works for your trades, and whether it made your head hurt trying to remember your calculus.
Happy Trading,
UPPER CODE V4
# Hull Moving Average Concavity and Turning Points
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mashume)
# Version: 2020-05-01 V4
# Now with support for ToS Mobile
# Faster, but not necessarily mathematically as good as the first
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare upper;
input price = HL2;
input HMA_Length = 55;
input lookback = 2;
input stddev_len = 21;
plot HMA = HullMovingAvg(price = price, length = HMA_Length);
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot turning_point = if concavity[1] != concavity then HMA else double.nan;
HMA.AssignValueColor(color = if concavity[1] == -1 then
if HMA > HMA[1] then color.dark_orange else color.red else
if HMA < HMA[1] then color.dark_green else color.green);
turning_point.SetPaintingStrategy(paintingStrategy = PaintingStrategy.POINTS);
plot MA_Max = if HMA[-1] < HMA and HMA > HMA[1] then HMA else Double.NaN;
plot MA_Min = if HMA[-1] > HMA and HMA < HMA[1] then HMA else Double.Nan;
plot sell = if turning_point and concavity == -1 then high else double.nan;
plot buy = if turning_point and concavity == 1 then low else double.nan;
def divergence = HMA - next_bar;
addLabel(yes, concat("DIVERGENCE: " , divergence * 10000), color = if concavity < 0 then if divergence[1] > divergence then Color.dark_RED else color.PINK else if divergence[1] < divergence then color.dark_green else color.dark_orange);
def divergence_stddev = StandardDeviation(price = divergence, length = stddev_len);
addLabel(yes, concat("STDDEV: " , divergence_stddev * 10000), color = if absValue(divergence) > absValue(divergence_stddev) then color.blue else color.dark_gray);
# ALERTS
Alert(condition = buy, text = "Buy", "alert type" = Alert.BAR, sound = Sound.Bell);
Alert(condition = sell, text = "Sell", "alert type" = Alert.BAR, sound = Sound.Chimes);
Alert(condition = MA_Max, text = "MA Maximum", "alert type" = Alert.BAR, sound = Sound.Ding);
Alert(condition = MA_Min, text = "MA Minumum", "alert type" = Alert.BAR, sound = Sound.Ding);
# 2020-05-01
# Each color of the HMA needs to be a separate plot as ToS Mobile
# lacks the ability to assign colors the way ToS Desktop does.
# I recommend a plain colored HMA behind the line
# Set the line color of the HMA above to gray or some neutral
# CCD_D -> ConCave Down and Decreasing
# CCD_I -> ConCave Down and Increasing
# CCU_D -> ConCave Up and Decreasing
# CCU_I -> ConCave Up and Increasing
plot CCD_D = if concavity == -1 and HMA < HMA[1] then HMA else double.nan;
plot CCD_I = if concavity == -1 and HMA >= HMA[1] then HMA else double.nan;
plot CCU_D = if concavity == 1 and HMA <= HMA[1] then HMA else double.nan;
plot CCU_I = if concavity == 1 and HMA > HMA[1] then HMA else double.nan;
#zscore @codydog
input zlength = 13;
input vlevel = 0.00;
def min = lowest(divergence, zlength);
def max = highest(divergence, zlength);
def Zscore = (divergence - Average(divergence, zlength)) / StDev(divergence, zlength);
# addlabel(1,"Zscore Div= " + round(zscore,2), if zscore > vlevel and zscore < 3 then color.blue else if zscore > 3 then color.red else if zscore < -vlevel and zscore > -3 then color.red else if zscore < -3 then color.blue else color.black);
# Hull Moving Average Concavity and Turning Points
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mashume)
# Version: 2020-02-23 V3
# Faster, but not necessarily mathematically as good as the first
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare upper;
input price = HL2;
input HMA_Length = 21;
input lookback = 2;
plot HMA = HullMovingAvg(price = price, length = HMA_Length);
# def delta_per_bar =
# (fold n = 0 to lookback with s do s + getValue(HMA, n, lookback - 1)) / lookback;
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot turning_point = if concavity[1] != concavity then HMA else double.nan;
HMA.AssignValueColor(color = if concavity[1] == -1 then
if HMA > HMA[1] then color.dark_orange else color.red else
if HMA < HMA[1] then color.dark_green else color.green);
turning_point.SetPaintingStrategy(paintingStrategy = PaintingStrategy.POINTS);
plot MA_Max = if HMA[-1] < HMA and HMA > HMA[1] then HMA else Double.NaN;
# Added Alerts 2020-02-23
Alert(condition = buy, text = "Buy", "alert type" = Alert.BAR, sound = Sound.Chimes);
Alert(condition = sell, text = "Sell", "alert type" = Alert.BAR, sound = Sound.Chimes);
# Hull Moving Average Concavity Divergence
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mashume)
# Version: 2020-02-23 V3
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare lower;
input price = OPEN;
input HMA_length = 55;
input lookback = 2;
def HMA = HullMovingAvg(length = HMA_length, price = price);
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot zero = 0;
plot divergence = HMA - next_bar;
plot cx_up = if divergence crosses above zero then 0 else double.nan;
plot cx_down = if divergence crosses below zero then 0 else double.nan;
NEW VERSION 4
Now with Mobile! https://tos.mx/NXqbYE9
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I would personally like to say thank you for your input and find this a very interesting take on the Hull and a better representation of a moving average that I have ever seen. The ability for the
trader to see possible turning points is most impressive. Understandably we all know or should know that most indicators and moving averages alike are lagging but yours puts a new spin and element
into the mix.
The Hull MA can be a useful tool but I do find like others it has its limitations and can overshoot etc. I (personal taste) find the ALMA to be the smoothest of the smooth when it comes to a MA. It
has its limitations as well but utilizing other indicators like support and resistance and price action etc works for me.
My question to you is using your formulas, is it possible to apply the same principles and math to the ALMA MA for an upper study MA.
Thank you again.
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Great indicator, is there anyway to backtest and show profit and loss? or run as a scan?
Thank you all for your comments and encouragement.
I have a laundry list of things in this post:
1. Variations to scripts
2. Scanner
3. Testing
Script Variations
, I'm intrigued by the Arnaud Legoux MA, and so here is a variant of the upper study that will allow you to choose between Hull, Simple, Exponential, Williams, and ALMA. I haven't gone through the
various permutations, but bear in mind that the 'overshoot' and smoothing of the Hull is part of what makes the turning point signal so attractive.
Used your posted code for the ALMA. Hope you're good with this use. Thanks!
# Multiple Moving Average Concavity and Turning Points
# or
# The Second Derivative of the A Moving Average
# via useThinkScript
# request from chillc15
# Added Arnaud Legoux MA and other Moving Averages
# Author: Seth Urion (Mahsume)
# Version: 2020-02-22 V2
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare upper;
input price = HL2;
input MA_Length = 21;
input lookback = 2;
input MovingAverage = {default "HMA", "EMA", "SMA", "WMA", "ALMA"};
script ALMA {
# Attributed to Miket
# https://tos.mx/9mznij
# https://usethinkscript.com/threads/alma-arnaud-legoux-ma-indicator-for-thinkorswim.174/
input Data = close;
input Window = 9;
input Sigma = 6;
input Offset = 0.85;
def m = (Offset * (Window - 1));
def s = Window/Sigma;
def SumVectorData = fold y = 0 to Window with WS do WS + Exp(-(sqr(y-m))/(2*sqr(s))) * getvalue(Data, (Window-1)-y);
def SumVector = fold z = 0 to Window with CW do CW + Exp(-(sqr(z-m))/(2*sqr(s)));
plot ALMA = SumVectorData / SumVector;
plot MA;
switch (MovingAverage) {
case EMA:
MA = MovAvgExponential(price, length = MA_Length);
case SMA:
MA = simpleMovingAvg(price, length = MA_Length);
case WMA:
MA = wma(price, length = MA_Length);
case ALMA:
MA = ALMA(Data = price, window = MA_Length);
MA = HullMovingAvg(price = price, length = MA_Length);
def delta = MA[1] - MA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = MA[1] + delta_per_bar;
def concavity = if MA > next_bar then 1 else -1;
plot turning_point = if concavity[-1] != concavity then MA else double.nan;
MA.AssignValueColor(color = if concavity == -1 then
if MA > MA[1] then color.dark_orange else color.red else
if MA < MA[1] then color.dark_green else color.green);
turning_point.SetPaintingStrategy(paintingStrategy = PaintingStrategy.POINTS);
plot MA_Max = if MA[-1] < MA and MA > MA[1] then MA else Double.NaN;
plot MA_Min = if MA[-1] > MA and MA < MA[1] then MA else Double.Nan;
plot sell = if turning_point and concavity == 1 then high else double.nan;
plot buy = if turning_point and concavity == -1 then low else double.nan;
def divergence = MA - next_bar;
addLabel(yes, concat("DIVERGENCE: " , divergence), color = if concavity < 0 then if divergence[1] > divergence then Color.RED else color.PINK else if divergence[1] < divergence then color.green else color.yellow);
, Scans, yes...
Scan 1 -- Buy signals.
This scan relies on the upper study being called "Concavity" and is entered on the scan page as a custom.
Concavity("hma length" = 55, price = CLOSE)."buy" is true within 2 bars
Scan 2 -- Stocks approaching possible Buy signals
I thought that, since the lower indicator crossovers can generate signals, that an 'early warning' scan might be of interest. We can scan for any HMA that has a negative convergence, and look for a
decrease in the distance below zero. It may be a long way off, it may never get there, but we are alerted before the trade entry.
script ConcavityDivergence {
# Hull Moving Average Concavity Divergence
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mahsume)
# Version: 2020-02-21 Initial Public
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare lower;
input price = HL2;
input HMA_length = 34;
input lookback = 2;
def HMA = HullMovingAvg(length = HMA_length, price = price);
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot zero = 0;
plot divergence = displacer(-1, HMA - next_bar);
plot cx_up = if divergence crosses above zero then 0 else double.nan;
plot cx_down = if divergence crosses below zero then 0 else double.nan;
def signal = ConcavityDivergence("hma length" = 55)."concavity";
def HMA = ConcavityDivergence("hma length" = 55)."HMA";
def next = ConcavityDivergence("hma length" = 55)."next_bar";
plot buy = signal < 0 and (HMA - next) > (HMA[1] - next[1]);
I used the built-in strategy feature of thinkorswim, with block sizes of 100. All tests were run on 1 year Daily charts, with an HMA length of 55 and set price to CLOSE
First SPX
Buy and Hold: $55,600
Strategy: $77,372
Delta + $21772
Next ADI
Buy and Hold: $1718
Strategy: $5202
Delta: + $3483
Last BA
Buy and Hold: ($9322)
Strategy: $14393
Delta: + $23715
I did not take time today to test shorter timeframes, though I imagine the results would be good. If there is interest, I'll try to post up some another time.
If you've read this far, thank you.
Happy trading and good luck.
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Hi, possibly you can link the strategy. For some reason when I try to copy it for a scan or backtest it doesn't work for me
Thanks a Ton works perfectly. Do you plan to add alert options to your upper and lower indicators?
Mashume your script is nothing short of amazing. Thank you , Thank You sir! Is there anyway that I can have the Strategy Version to use without the "addorder" I tested all 3 versions of your script
and the strategy version works best for me. I cannot use the strategy version with addorder during the "Normal Market Hours" on thinkorswim , If I disable the Long Enter and Long Exit the visuals
disappear. can you please help? TIA !
StockJockey --
Per your request, below is the exact code from the study, with the buy and sell order mechanics replaced with plots. Hope that gets toward what you wanted.
input price = HL2;
input HMA_Length = 21;
input lookback = 2;
def HMA = HullMovingAvg(price = price, length = HMA_Length);
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
def turning_point = if concavity[-1] != concavity then HMA else Double.NaN;
Plot buy = if turning_point and concavity == -1 then low else double.nan;
plot buy = if turning_point and concavity == -1 then low else double.nan;
plot sell = if turning_point and concavity == 1 then high else double.nan;
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Thanks. Fantastic, spending the weekend playing with chart settings and indicator settings. One request: Add an offset variable for the buy and sell signals to adjust the distance away from the price
chart and the indicator. With shorter time frame charts, the arrows tighten down right to the indicator plot and the appearance gets busy.
See image, One way to use it is with 30 min ORB/Fib(10-min HK chart) so that we have level to risk off. It seems to be working good, entry when Hull is Red and we break a level. I need help with a
scanner and sound alert for this setup, any help is appreciated.
The Upper Indicator code above has had the following lines added to the end:
Alert(condition = buy, text = "Buy", "alert type" = Alert.BAR, sound = Sound.Chimes);
Alert(condition = sell, text = "Sell", "alert type" = Alert.BAR, sound = Sound.Chimes);
I have NOT yet 'shared' a version with these alerts, but can at some point. This was just a quick fix for those who want them before the morning open.
I just wanted to give a fair warning that the indicator is roughly 1-2 bars behind when it produces some of its signals. So what you see in the report will be off. For example, was just watching a
Futures contract on a 15 min chart for kicks and giggles, and when the signal appeared to buy, it was 2 bars ago making it almost impossible to enter any trades when it says too. Tested it again on
Eur/Usd and other stocks using OnDemand. It's still not bad from what I am seeing but if the market is choppy you might get hurt really bad.
TL;DR; There is no Holy Grail. @BenTen
I had noticed that. And I've been trying to come up with a solution...
Part of the issue stems from the indicator relying on the next bar to determine whether the concavity has flipped; the signal will always be at least a bar in the past if CLOSE is used.
I've been trying to do the maths to solve the HMA for a the next minimum value that would cause a flip on say a daily aggregation (or anything higher than current) and plot that line on an intraday
chart. It's just a matter of rearranging the equations for the HMA to solve for the most recent price, given the value the HMA needs to reach for a concavity flip (which is a known value).
With all that said, I did whip up a version of the lower study that is a bar faster than the old version. It doesn't line up with the old version of the upper, and so I've got a new upper too.
HOWEVER. There is no Holy Grail of Indicators. The power of the turning point lies behind a mist of only being able to know it has turned one bar into the future... we can't know we've turned with
certainty (even with an intraday study as described above) without the next value.
# Hull Moving Average Concavity Divergence
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mahsume)
# Version: 2020-02-23 V3
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare lower;
input price = OPEN;
input HMA_length = 55;
input lookback = 2;
def HMA = HullMovingAvg(length = HMA_length, price = price);
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot zero = 0;
plot divergence = HMA - next_bar;
plot cx_up = if divergence crosses above zero then 0 else double.nan;
plot cx_down = if divergence crosses below zero then 0 else double.nan;
# Hull Moving Average Concavity and Turning Points
# or
# The Second Derivative of the Hull Moving Average
# Author: Seth Urion (Mahsume)
# Version: 2020-02-23 V3
# Faster, but not necessarily mathematically as good as the first
# This code is licensed (as applicable) under the GPL v3
# ----------------------
declare upper;
input price = HL2;
input HMA_Length = 21;
input lookback = 2;
plot HMA = HullMovingAvg(price = price, length = HMA_Length);
# def delta_per_bar =
# (fold n = 0 to lookback with s do s + getValue(HMA, n, lookback - 1)) / lookback;
def delta = HMA[1] - HMA[lookback + 1];
def delta_per_bar = delta / lookback;
def next_bar = HMA[1] + delta_per_bar;
def concavity = if HMA > next_bar then 1 else -1;
plot turning_point = if concavity[1] != concavity then HMA else double.nan;
HMA.AssignValueColor(color = if concavity[1] == -1 then
if HMA > HMA[1] then color.dark_orange else color.red else
if HMA < HMA[1] then color.dark_green else color.green);
turning_point.SetPaintingStrategy(paintingStrategy = PaintingStrategy.POINTS);
plot MA_Max = if HMA[-1] < HMA and HMA > HMA[1] then HMA else Double.NaN;
plot MA_Min = if HMA[-1] > HMA and HMA < HMA[1] then HMA else Double.Nan;
plot sell = if turning_point and concavity == -1 then high else double.nan;
plot buy = if turning_point and concavity == 1 then low else double.nan;
def divergence = HMA - next_bar;
addLabel(yes, concat("DIVERGENCE: " , divergence), color = if concavity < 0 then if divergence[1] > divergence then Color.RED else color.PINK else if divergence[1] < divergence then color.green else color.yellow);
Alert(condition = buy, text = "Buy", "alert type" = Alert.BAR, sound = Sound.Chimes);
Alert(condition = sell, text = "Sell", "alert type" = Alert.BAR, sound = Sound.Chimes);
Be aware that these do not backtest as well as the original versions, perhaps because mathematically they do not represent the turning points as well... I'm not sure why.
As always,
Happy Trading.
Also to add, new version is much more realistic but its 1 bar behind sometimes, which is way better than 2. So if you were to backtest this, the results will still be off a little but not as bad as
before. Example: Today S&P 500 futures the top indicator stated a buy at 3251.75 on a 15 min chart a buy but that would be impossible as when it triggered it was trading at 3260. Just giving you some
feedback to help you fine-tune this thing.
- Thanks for the work - its very generous and sometimes requires a thick skin as folks feel free to comment/criticize because you haven't posted their winning lottery ticket!
I dont know if you've seen this, but I posted an open source study that compared the favorite moving averages and detailed which worked best for various markets. Here's the link -
, page 43,44 summarize their findings.
Also, don't know if you're familiar with Leavitt Convolution but its a little faster than Hull and seems to work when I plugged it into your work.
Let me know if I can be of any help.
Thanks, again
Thanks so much - could you please (re) explain how to make a scan with the "improved" 1-bar behind version?
asragov et al,
Links to new (faster versions):
Upper includes alert, lower does not.
, thank you for your kind words. I'm intrigued with the Leavitt Convolution... Homework.
AFTER loading the new versions, and being careful with the names. %-\ You should be able to create new filter scans.
In the scans tab:
1. Add a Filter
2. Choose Study
3. In the left-most dropdown which defaults to ADX Crossover or something select Custom...
4. Click ThinkScript Editor near the top of the window
5. Paste this code:
Concavity("hma length" = 55)."buy" is true within 1 bars
or this code:
def signal = ConcavityDivergence("hma length" = 55)."concavity";
def HMA = ConcavityDivergence("hma length" = 55)."HMA";
def next = ConcavityDivergence("hma length" = 55)."next_bar";
plot buy = signal < 0 and (HMA - next) > (HMA[1] - next[1]);
Adjust as needed. The second won't show you things that are at a BUY, it shows things that currently have negative concavity, but the moving in a direction that may signal a buy sooner or later (if
at all, of course).
Good Luck, and Happy Trading.
Last edited:
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To access the premium indicators, which are plug and play ready, sign up for VIP membership here. | {"url":"https://usethinkscript.com/threads/hull-moving-average-turning-points-and-concavity-2nd-derivatives.1803/","timestamp":"2024-11-02T21:20:05Z","content_type":"text/html","content_length":"216758","record_id":"<urn:uuid:8a40a541-27f5-441e-9e1a-b96827cbf92c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00092.warc.gz"} |
follow all directions IHP 525 Module Six Problem Set Hemoglobin levels in 11-year-old boys vary according to a normal
Hemoglobin levels in 11-year-old boys vary according to a normal distribution with σ=1.2 g/dL.
How large a sample is needed to estimate µ with 95% confidence so the margin of error is no greater than 0.5 g/dL?
A researcher fails to find a significant difference in mean blood pressure in 36 matched pairs. The test was carried out with a power of 85%. Assuming that this study was well designed and carried
out properly, do you believe that there really is no significant difference in blood pressure? Explain your answer.
Would you use a one-sample, paired-sample, or independent-sample
t-test in the following situations?
1. A lab technician obtains a specimen of known concentration from a reference lab. He/she tests the specimen 10 times using an assay kit and compares the calculated mean to that of the known
A different technician compares the concentration of 10 specimens using 2 different assay kits. Ten measurements (1 on each specimen) are taken with each kit. Results are then compared.
In a study of maternal cigarette smoking and bone density in newborns, 77 infants of mothers who smoked had a mean bone mineral content of 0.098 g/cm3 (
s1 = 0.026 g/cm3). The 161 infants whose mothers did not smoke had a mean bone mineral content of 0.095 g/cm3 (
s2 = 0.025 g/cm3).
1. Calculate the 95% confidence interval for µ1 – µ2.
Based on the confidence interval you just calculated, is there a statistically significant difference in bone mineral content between newborns with mothers who did smoke and newborns with mothers who
did not smoke?
A randomized, double-blind, placebo-controlled study evaluated the effect of the herbal remedy
Echinacea purpurea in treating upper respiratory tract infections in 2- to 11-year olds. Each time a child had an upper respiratory tract infection, treatment with either echinacea or a placebo was
given for the duration of the illness. One of the outcomes studied was “severity of symptoms.” A severity scale based on four symptoms was monitored and recorded by the parents of subjects for each
instance of upper respiratory infection. The peak severity of symptoms in the 337 cases treated with echinacea had a mean score of 6.0 (standard deviation 2.3). The peak severity of symptoms in the
placebo group (np = 370) had a mean score of 6.1 (standard deviation 2.4). Test the mean difference for significance using an independent t-test. Discuss your findings. | {"url":"https://smashessays.com/2024/04/01/follow-all-directions-ihp-525-module-six-problem-set-hemoglobin-levels-in-11-year-old-boys-vary-according-to-a-normal/","timestamp":"2024-11-06T07:57:54Z","content_type":"text/html","content_length":"125283","record_id":"<urn:uuid:bfa255c1-1cef-420a-af3d-8158c8d547c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00455.warc.gz"} |
(Get Answer) - (a) A .cav file on the required prediction that includes your...| Transtutors
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1. (a) A .cav file on the required prediction that...
(a) A .cav file on the required prediction that includes your predicted...
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(a) A .cav file on the required prediction that includes your predicted values for u(X], X2 ) = E(Y) and V(X1,X2) =Var(Y) for the testing data (in 6 digits). Please name your file as "1.
YourLastName. YourFirst.Name.cav", e.g., "1.Mei.Yajun.cav" for the name of the instructor. I think students in our class have a unique combination of last/first name, and thus there is no need to
include the middle name. . The submitted esv file in excel must be 2500 x 4 column, and the first two columns must be the exact same as the provided testing data file "7406test.csv ". The third
column should be your estimated mean u*(X , X, ), and the fourth column is your estimated variance V*(X1: X2)
Recent Questions in Mechanical Engineering
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Predicting Energy Expenditure During Gradient Walking With a Foot Monitoring Device: Model-Based Approach
Original Paper
Background: Many recent commercial devices aim at providing a practical way to measure energy expenditure. However, those devices are limited in accuracy.
Objective: This study aimed to build a model of energy consumption during walking applicable to a range of sloped surfaces, used in conjunction with a simple, wearable device.
Methods: We constructed a model of energy consumption during gradient walking by using arguments based in mechanics. We built a foot monitoring system that used pressure sensors on the foot insoles.
We did experiments in which participants walked on a treadmill wearing the foot monitoring system, and indirect calorimetry was used for validation. We found the parameters of the model by fitting to
the data.
Results: When walking at 1.5 m/s, we found that the model predicted a calorie consumption rate of 5.54 kcal/min for a woman with average height and weight and 6.89 kcal/min for an average man. With
the obtained parameters, the model predicted the data with a root-mean-square deviation of 0.96 kcal/min and median percent error of 12.4%.
Conclusions: Our model was found to be an accurate predictor of energy consumption when walking on a range of slopes. The model uses few variables; thus, it can be used in conjunction with a
convenient wearable device.
JMIR Mhealth Uhealth 2019;7(10):e12335
Physical inactivity, despite its well-known health risks [,], continues to be a serious public health issue []. Recently, various wearable devices, including wristbands and mobile phones, have
offered a way to track physical activity throughout the day. Such devices can be used in ambulatory conditions by individuals or in clinical settings to monitor patients’ physical activity.
Many of these devices use an accelerometer-based method to predict energy expenditure [-]. However, these methods are limited in precision []. A basic, common assumption used is that the calorie
consumption rate is proportional to the walking velocity. A GPS tracker can then be used to measure the walking distance and then compute the total energy consumption. However, this method is limited
in accuracy and may not be feasible indoors.
Literature Review
The energetics of human locomotion has been closely studied for decades. Early studies focused on energy expenditure during walking [-] and running [-], and made comparisons with the energy
expenditures of other animals []. Most relevantly, studies on walking energetics found a proportional relationship between energy expenditure and the square of the velocity. These early studies
showed that reasonable accuracy can be attained with simple relations, despite the complexity of the act of walking. More recently, detailed models of walking dynamics have been presented that
examine more closely the mechanics of walking [-]. These biomechanical models aim to explain human gait patterns via energy minimization. Also studied have been movements of the arm [,] and the head
and trunk [], as well as gait patterns in special groups of interest [,]. Such models have also been used in the field of robotics in developing walking robots [].
Previous studies were primarily of academic interest, although inexpensive commercial devices have recently been made available for personal or clinical use. Such devices offer noninvasive ways to
measure daily caloric consumption, and they have been assessed by numerous validation studies in the literature [-]. The most common types of commercially available products include the wrist-worn
accelerometer and devices based on heart rate monitors. Although these devices are good predictors of the number of steps and heart rate, accurate prediction of energy expenditure is yet to be
achieved []. These validation studies test for various settings; however, they usually lack a discussion of the model or algorithm used in their predictions.
This study proposes a model of walking energetics applicable to a range of slopes. The model is based on a simple equation and uses data from a wearable device. The method uses a foot monitoring
system that can sense footsteps, which allows for direct measurement of step frequency. We found that a high-accuracy model can be developed for a range of upward and downward slopes. The fact that
it is based on a direct measurement of footsteps allows the device to be versatile and applicable to diverse walking situations. The ability to track expenditure while walking on sloped surfaces is
helpful for sloped outdoor ground and also indoor use of stairs or sloped treadmills.
Experimental Procedure
For model development and validation, an experiment was devised in which 73 healthy participants (34 female, 39 male) walked on a treadmill. The participants had a mean age of 43.6 (SD 15.0) years,
mean height of 168.3 (SD 10.5) cm, and mean weight of 68.1 (SD 12.1) kg. Participants were selected from healthy volunteers (age 20 to 60 years) who registered in the department of Sport Science,
Pusan National University, Busan, Korea. We excluded participants who had cardiovascular, musculoskeletal, or neurological disorders to avoid any confounding factors or biases. The participants were
asked to walk on a treadmill at various values of the incline angle, Theta, and speed, v. Specifically, the angle was taken to be 0° (indicating no incline); 4°, 9°, and 14° (uphill); and −4°, −9°,
and −14° (downhill). It was observed that calorie consumption took approximately 30 seconds to stabilize to a linear rate while walking. Each walking measurement lasted approximately 5 minutes to
ensure a sufficiently long sample.
Calorie consumption was measured with a COSMED K4b2 portable gas analyzer system. This indirect calorimetry, based on the gas analyzer system, measures oxygen consumption, from which energy
expenditure is computed. This method has been validated as an accurate measure through numerous comparative studies [-] and is used as a criterion measure in many validation studies [-,-]. The gas
analyzer was worn during the treadmill experiment, and it recorded a time series of cumulative calorie consumption. To eliminate noise associated with the beginning and end of the experiment, we
discarded data for the first 50 seconds and the final 10 seconds before computing the energy consumption rate. Then the basal metabolic rate [] was subtracted to obtain the energy expenditure
associated with walking, which is denoted by P.
Each participant also wore a foot monitoring system, consisting of shoe insoles equipped with eight pressure sensors. The insole used was a prototype developed by 3L Labs (Seoul, Korea), and provided
to us for research purposes. A depiction of the foot monitoring system and the experimental setup is given in . A Fitbit Surge, a wrist-worn accelerometer device, was also worn by each participant to
compare the accuracy of its caloric consumption prediction. This study was approved by the Institutional Review Board of Pusan National University, Busan, Korea. All participants provided written
informed consent (PNU IRB/2015_33_HR).
A value of 0, 1, or 2 indicated the pressure on each of the pressure sensors and was recorded with a frequency of 10 Hz, which resulted in an array of 16 integers for each time step of 0.1 s. A
snippet from example data is shown in . From the pressure sensor data, we were able to extract the step frequency, f. We performed this by examining the sum of the pressure sensor values at each time
step. An example is shown in . Although it is natural to consider the foot to be off the ground when this sum is 0, this can result in erroneous results if one or more of the pressure sensors remain
at a value above 0 throughout the entire step cycle, either due to a faulty sensor or residual pressure. We found that better accuracy was achieved when high and low thresholds were used. This was
done by first assigning the on-ground status to the first-time step, and then sequentially assigning either the on-ground or off-ground status to each following time step. If the previous time step
was on-ground and the pressure sum was below the lower threshold, we assigned the off-ground status to that time step; if the threshold was not crossed, the time step was left in on-ground status. If
the previous status was off-ground, the on-ground status was assigned if the pressure sum was above the upper threshold, and the off-ground status was assigned otherwise. Threshold values between 1
and 10 were tested and compared with manually assigned steps. Lower and upper threshold values of 2 and 5, shown in , were found to produce accurate results.
Figure 1. Illustration of the foot monitoring system (left) and a picture of a participant walking on an uphill treadmill wearing the K4b2 portable gas analyzer (right).
View this figure
Figure 2. A sample of 4 seconds of raw data from the pressure sensors. The vertical position of each number of the array indicates the time, ordered from top to bottom at an increment of 0.1 seconds.
Each column denotes a sensor, with left foot and right foot separated. The colored portions indicate when our algorithm decided the foot was off the ground.
View this figure
Figure 3. Graph of total pressure from the left foot sole over an interval of 10 seconds obtained from the foot monitoring system from the same data as presented in Figure 2. The two dashed lines
indicate the upper and lower thresholds used to calculate the step frequency.
View this figure
After assigning a status to each time step, we counted the number of transitions from the on-ground to off-ground status and divided it by the time interval to obtain the frequency. As with the gas
analyzer data, we omitted data for the first 50 s and the final 10 s. Only one shoe insole is required to calculate the step frequency; however, we used the average of both sides in this study.
Our model was constructed from considering the energy changes involved in walking. Suppose a participant with body mass Mu is walking with average speed Nu on a surface inclined by Theta from the
horizontal. The participant is swinging their legs with frequency f. The energy consumption rate, Rho, is given by equation 1 (). Here positive and negative values of the slope, Theta, of the walking
surface correspond to walking uphill and downhill, respectively. Rho[K] and Rho[U] are rates of changes in the kinetic energy and in the potential energy, respectively, whereas coefficients Gamma,b
[Tau], b[1], and Rho[0] are parameters to be determined empirically from the data. The energy change rates for Rho[K] and Rho[U] are given in equation 2 (). In the following, we give an explanation
of each term in consideration of energy.
Kinetic Energy Component
We first consider walking on a horizontal surface (ie, Theta=0). When walking on a treadmill, the upper body moves in a relatively constant velocity, with the moving legs supporting this movement.
The legs swing back and forth relative to the upper body’s position, undergoing an acceleration-deceleration cycle. We postulated that the energy expenditure was proportional to the kinetic energy
change of the legs. The work done on the legs during each walking cycle is given by equation 3 (). Here m is the mass of each leg, v[0] is the maximum speed of each leg’s center of mass, and the
factor of 4 accounts for the two legs each undergoing acceleration and then deceleration. This differs from the assumption that the legs swing like a pendulum, in which case gravity would do the
Since we usually have no way to easily measure leg mass or leg velocity, we defined two ratios: (1) the ratio α of the leg mass, m, to the body mass, M; and (2) the ratio β of the maximum velocity, v
[0], of the leg to the average walking speed, v (equation 4 in ).
This allowed us to rewrite equation 3 as η[K], the energy consumption rate due to the kinetic energy is given by equation 5 (). In writing the right-hand side of equation 5, the measurable terms are
grouped into P[K] as in equation 2, whereas the rest are grouped into dimensionless coefficient γ, given by equation 6.
Potential Energy Component
When walking on a horizontal surface perpendicular to the direction of gravity, there is no net change in potential energy. It changes when the subject is walking up or down a slope. We first
considered upward inclines. When one walks up a slope of angle Theta at speed v parallel to the surface, their potential energy, U, changes at a rate dU/dt=P[U], given by equation 2 (). For
simplicity, we further assumed that when walking up a slope, additional energy proportional to this term is required. Accordingly, the energy expenditure rate associated with the changing potential
energy is given by b[0]P[U], where b[0] is the inverse of the efficiency, η[U], (equation 7 in ) with which the body converts stored energy to potential energy.
Figure 4. List of equations of the model of energy expenditure during walking.
View this figure
One might consider simply using the same formula for downhill inclines, in which case the term b[0]P[U]=b[0]Mgv sinTheta becomes negative. This would imply that when walking downslope, the change in
potential energy can be converted into kinetic energy, thereby subtracting from the total energy cost. However, this leads to a nonsensical result for higher slopes, as it can lead to negative energy
consumption. When a downhill slope is steeper than a certain angle, the subject would need to exert a frictional force to prevent from falling forward or walking too fast. Therefore, b[0]P[U] does
not provide an adequate description of the energy expenditure in this case.
and 6 present scatterplots of the data in the three-dimensional space (P[K],P[U],P) for women and men, respectively. This visualization shows that P first decreases then increases as P[U] is
decreased from zero. Such a parabolic shape indicates the presence of a quadratic term; thus, we added to P a term proportional to P[U]^2. The energy expenditure associated with potential energy in
the case of downhill walking is given by equation 8 (). The second term is multiplied by P[0]^-1 so that the coefficient b[1] is kept dimensionless. In other words, b[1] is the coefficient of the
quadratic term in the case of downhill walking in units of P[0]. This leads to the full model, described by equation 1 ().
Figure 5. Three-dimensional scatterplot of data (dots) and model prediction (lines) of P versus P[U] and P[K] for women.
View this figure
Figure 6. Three-dimensional scatterplot of data (dots) and model prediction (lines) of P versus PU and PK for men.
View this figure
Linear Regression
The preceding model described leaves parameters ϒ,b[0], b[1], and P[0] to be determined. We obtained these parameters by first taking data for flat and uphill surfaces (Theta≥0) and performing
multiple linear regression through the use of the first equation in equation 1 () with Υ,b[0], and P[0] as fitting parameters. The adjusted R^2 value for the fits of both women and men was .83. Then
b[1] was obtained via fitting the second equation of equation 1 () to flat and downslope data (Theta≤0). During this secondary fit, ϒ,b[0], and P[0] were set constant at the values obtained earlier.
The full set of coefficients, obtained through linear regression, is given in . The dependency of P on P[K] and P[U] is represented by the surfaces in Figures 5 and 6 . Due to the piecewise
functional form of the model (equation 1 in ), the prediction plane has no curvature for P[U]>0 but does in the region P[U]<0.
Table 1. Coefficients for the full model reported with the root-mean-square deviation (RMSD) on comparison with data. The values were obtained by two linear regressions.
│ Coefficient │ Units │ Women │ Men │
│ γ │ — │ 0.662 │ 0.517 │
│ b[0] │ — │ 1.591 │ 1.694 │
│ b[1] │ — │ 0.575 │ 1.086 │
│ P[0] │ kcal/s │ 0.042 │ 0.058 │
│ RMSD │ kcal/s (kcal/min) │ 0.016 (0.96) │ 0.016 (0.96) │
The fit resulted in a root-mean-square deviation (RMSD) of 0.96 kcal/min for both women and men. A boxplot of the percentage errors of all trials is given in , in which the errors have been
calculated according to equation 9 in .
Here P is the prediction by the method whereas P' is the standard given by the gas analyzer. The median errors were 16.9% for women, 11.2% for men, and 12.4% for both groups. These errors are
substantially lower than those found in a validation study for multiple commercial devices, which yielded median accuracies of 28.6% to 35.0% across devices for walking [].
The predictions made by Fitbit Surge had an RMSD of 2.58 kcal/min (2.7 times that of the model) and a median percent error of 37.3% (3 times that of the model). However, this high error was mostly
due to inaccuracies in sloped walking. When restricted to flat surfaces, the Fitbit Surge’s accuracy increased dramatically, whereas the model’s accuracy increased moderately. The Fitbit Surge’s RMSD
on flat surfaces was 1.82 kcal/min (2.3 times that of the model, 0.79 kcal/min), and the median percent error was 18.4% (1.6 times that of the model, 11.2%). Distributions of percent errors are
portrayed with boxplots in .
Before discussing the implications of these results, we note that the variables v and f are not independent. If l is the average length of a step, then v = f l. Assuming the approximate relation h ≈
l, where h is the subject’s height, we obtain v ~ fh (equation 10 in ). This relation was observed in the data, as shown in .
Equation 7 implies that η[U]=0.547 for women and 0.596 for men. In principle, ϒ depends on α,β, and η[K]. We assumed the average value of α=0.185 for women and 0.165 for men, obtained from an
anatomical reference [], and that η[K]=η[U]. Taking these values and the fitting result for ϒ, we obtained from equation 6 () the ratio β with values 1.47 for women and 1.36 for men. This difference
in the average may reflect the difference in the average height between women and men. Specifically, equations 4 and 10 () imply β=v[0]/v ~ v[0]/f h. The ratio of the value of β for women to that for
men equaled 1.08, whereas the ratio of the average height of men to that of women equaled 1.11.
Figure 7. Boxplots of the percent errors of predictions made by the model and Fitbit Surge. Errors have been estimated via equation 9 in Figure 4.
View this figure
Figure 8. Step frequency, f, multiplied by height, h, plotted against average walking speed, v. Least squares fit line fh = 0.52v+1.02 (m/s) is also shown.
View this figure
Principal Results
We developed a model based on rates of change in kinetic and potential energies. In general, it predicts linear dependence of the energy consumption on these rates; in particular, it predicts
quadratic dependence of the energy consumption on the potential energy change in the case of downhill walking. The method, used in conjunction with a foot monitoring system, predicts energy
expenditure with an RMSD of 0.98 kcal/min and a median percent error of 12.4%, lower than those of wrist-worn commercial devices in predicting energy expenditure for walking. With one simple
piecewise function, the model adequately predicts energy expenditure for walking in a wide range of the gradient.
Notice the differences in parameter values between women and men. The appreciable difference in the value of b[1] between men and women may result from the difference in walking posture; this is
beyond the scope of this work and left for future study. In principle, the parameters are fit for each individual and should vary by subject. Thus, presents average values of the coefficients within
each gender. Even so, it is remarkable that a high degree of accuracy is observed.
Although the model accounts for varying body mass and step frequency (cadence), this does not account for additional individual variations in parameter values due to walking gait and body dimensions.
There may be ways to account for such variations without complicating the model. In addition, because the treadmill incline lies between 14° uphill and 14° downhill, we are not able to validate the
model for more extreme slopes []. In addition, the method has not been tested and calibrated for outdoor walking or variable temperatures and altitudes. However, we believe that our pilot study
provides a groundwork for follow-up studies under more ambulatory conditions.
Comparison With Prior Work
Prior studies have noted the strong correlations between P and v^2 for level walking []. The authors have also similarly considered additional energy expenditure when walking uphill, attributing it
to vertical lift work. In contrast, our study proposes a simple formula that predicts energy consumption reasonably well for horizontal, uphill, and downhill surfaces within a unified framework. In
addition, Cotes and Meade [] made use of individual measurements, including resting metabolic rate and leg length. Our model shows that high accuracy can be achieved via reasonable assumptions used
in conjunction with a wearable, mobile device.
Other existing studies have studied energy expenditure during uphill and downhill walking [,]. The authors reported a minimum energy cost when walking 10° downhill, which is consistent with our
results. These studies did not incorporate varying walking speed and body weight, and relied on regression analysis with those variables kept constant. Our study offers a simple formula that applies
to various walking speeds and subjects, while also accounting for the surface gradient.
Our method fits separately for women and men. Prior validation studies have found differences in the accuracy of devices between the two genders. A comparative validation study found that gender was
one of the strongest predictors for accuracy, with a rate significantly higher for men than for women []. Our results suggest that similar error rates for both genders can be achieved.
We have developed a model that predicts energy expenditure during walking on a gradient surface between 14° uphill and 14° downhill, with an RMSD of 0.98 kcal/min. The model has been used in
conjunction with a wearable device, the foot monitoring system, which directly measures footsteps. Thus, it offers an accessible method of measuring energy expenditure in realistic walking settings,
where gradient walking is common. Future work may test equation 1 () in a wider range of values in the P[K]−P[U] space. Testing the method on outdoor walking is also desirable for further validation.
Although not yet explored, the device could also be used in conjunction with other activity monitoring devices, such as wrist-worn ones, to produce more accurate measures of energy expenditure.
MY Choi acknowledges support from National Research Foundation of Korea through the Basic Science Research Program (grant No. 2016R1D1A1A09917318 and 2019 R1F1A1046285). MJ Shin and J-J Park
acknowledge support from the Wearable Device R&D Project, Pusan National University Hospital, and KT (grant CMITKT-05).
Conflicts of Interest
None declared.
RMSD: root-mean-square deviation
Edited by G Eysenbach; submitted 27.09.18; peer-reviewed by M Stuckey, T Ebara, B Chaudhry, J Seitz; comments to author 01.04.19; revised version received 10.05.19; accepted 19.07.19; published
©Soon Ho Kim, Jong Won Kim, Jung-Jun Park, Myung Jun Shin, MooYoung Choi. Originally published in JMIR Mhealth and Uhealth (http://mhealth.jmir.org), 23.10.2019.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work, first published in JMIR mhealth and uhealth, is properly cited. The complete bibliographic information, a link to the original publication
on http://mhealth.jmir.org/, as well as this copyright and license information must be included. | {"url":"https://mhealth.jmir.org/2019/10/e12335","timestamp":"2024-11-14T08:19:38Z","content_type":"text/html","content_length":"401643","record_id":"<urn:uuid:dc2b07d5-0273-450d-8df8-a7997075bb1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00646.warc.gz"} |
70 m to feet
Heading 1: Converting 70 Meters to Feet
Converting meters to feet is a common task in various fields such as architecture, engineering, and construction. One commonly encountered conversion is converting 70 meters to feet. To perform this
conversion, we need to understand the conversion factor between the two units of measurement.
The conversion factor for meters to feet is 3.28084. To convert 70 meters to feet, we simply multiply the measurement by this conversion factor. Doing the math, 70 meters multiplied by 3.28084 gives
us approximately 229.6588 feet. So, if you are working with a measurement of 70 meters and need to express it in feet, you can confidently state that it is approximately 229.66 feet.
Heading 2: The Importance of Accurate Unit Conversions
Unit conversions play a crucial role in various fields, including science, engineering, and everyday life. The importance of accurate unit conversions cannot be overstated, as errors in conversions
can lead to significant miscalculations and potentially disastrous consequences. Whether it’s converting metric measurements to imperial units or vice versa, precision and attention to detail are
One of the primary reasons why accurate unit conversions are vital is the need for compatibility and consistency. Different countries and regions use different systems of measurement, such as the
metric system and the imperial system. To ensure seamless communication and collaboration, accurate conversions between these systems are crucial. For instance, when working on international projects
or conducting scientific research across borders, accurate unit conversions enable accurate data interpretation and analysis, leading to reliable results.
Moreover, accurate unit conversions are necessary to ensure safety and quality control in various industries. For example, engineers rely on precise conversions when designing structures, calculating
dimensions, or determining the appropriate quantities for materials. Even a small error in unit conversion can spell disaster, compromising the structural integrity of a building or the effectiveness
of a product. To mitigate risks and ensure safety, accurate unit conversions are essential in these critical areas.
In conclusion, accurate unit conversions are paramount in maintaining consistency, facilitating international communication, and ensuring safety and quality control. Whether in scientific research,
engineering projects, or everyday life, accurate conversions between different units of measurement are vital to avoid errors, miscalculations, and potentially harmful outcomes. Therefore, it is
imperative to approach unit conversions with precision and attention to detail, using reliable conversion formulas and methods.
Heading 2: Understanding the Metric System
The metric system is a decimal-based system of measurement that is widely used around the world. It is founded on the concept of base units, which are then multiplied or divided by powers of ten to
derive other units. The metric system is known for its simplicity and ease of use, making it a preferred choice in scientific and mathematical fields.
In the metric system, the base unit for length is the meter. This unit is used to measure distances and heights, among other things. It is important to note that the metric system is a coherent
system, meaning that the relationships between different units are based on powers of ten. For example, there are 100 centimeters in one meter and 1,000 millimeters in one meter. This consistency
allows for easy conversions between different metric units, making computations and measurements more straightforward.
Heading 2: The Basics of the Imperial System
The Imperial System is a measurement system that is primarily used in countries like the United States, Liberia, and Myanmar. It is based on a collection of units that were historically used in the
British Empire. The system consists of measurements like inches, feet, yards, miles, ounces, pounds, and gallons. Each unit has a specific conversion factor that relates it to other units within the
system. Understanding the basics of the Imperial System is essential for accurate measurement and effective communication in these countries.
The Imperial System is known for its unique and sometimes confusing conversion factors. For example, there are 12 inches in a foot, 3 feet in a yard, and 5,280 feet in a mile. Additionally, when it
comes to weight, there are 16 ounces in a pound and 2,240 pounds in a long ton. These conversion factors can differ from those used in other systems, such as the metric system. Therefore, it is
crucial to be familiar with the specific conversion factors of the Imperial System to ensure accurate measurements and seamless communication in countries that use this system.
Heading 2: The Conversion Formula for Meters to Feet
To convert meters to feet, you will need to use a simple conversion formula. The conversion formula for meters to feet is as follows: 1 meter is equivalent to 3.28084 feet. This means that if you
have a measurement in meters and you want to convert it to feet, you simply need to multiply the number of meters by 3.28084. For example, if you have 70 meters, you would multiply 70 by 3.28084 to
get the equivalent measurement in feet.
It is important to note that this conversion formula is a mathematical representation of the relationship between meters and feet. It allows for accurate and standardized conversions between the two
units of measurement. The use of this formula ensures that measurements are consistent and can be easily understood and compared across different systems and regions.
Heading 2: Step-by-step Conversion Process
To convert meters to feet, follow this step-by-step process. First, determine the value of the distance in meters that you want to convert. For example, let’s say we have a measurement of 70 meters.
Next, multiply the value in meters by the conversion factor of 3.28084. This conversion factor represents the number of feet in one meter.
So, for our example of 70 meters, we would multiply 70 by 3.28084. This step will give us the equivalent value in feet.
After multiplying, we find that 70 meters is equal to approximately 229.659 feet.
The step-by-step conversion process ensures a precise and accurate transformation from meters to feet. By following these simple steps, you can convert any measurement in meters to its corresponding
value in feet. | {"url":"https://convertertoolz.com/m-to-feet/70-m-to-feet/","timestamp":"2024-11-09T13:45:30Z","content_type":"text/html","content_length":"45194","record_id":"<urn:uuid:cbbba288-08b5-4e67-9050-28749c6028ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00778.warc.gz"} |
Custom defined loglikelihood - Absolute or relative values?
Hi, everyone,
Many thanks to UQlab for providing customize log-likelihood (log\mathcal{L}) functions and the examples provided are also very helpful.
The working principle for the log\mathcal{L} functions looks simple:
log \mathcal{L} (\boldsymbol{\vec\theta},\boldsymbol{\epsilon} \mid \boldsymbol{Y}) = \left(-\frac{1}{2}\left(\boldsymbol{Y_i} - \mathcal{M}(\vec\theta)\right)^{\mathsf{T}} \boldsymbol{\Sigma}(\
epsilon)^{-1}\left(\boldsymbol{Y_i} - \mathcal{M}(\vec\theta)\right)\right) - \frac{3N}{2}\log(2\pi) - \frac{N}{2}\log\left(\det(\boldsymbol{\Sigma}(\epsilon))\right)
During the random walking, current step log\mathcal{L} value choose to stay or to step to next sampling point for MCMC. So, from my understanding, the absolute value for log\mathcal{L} doesnot affect
MCMC results. Because it is just doing comparisons. So, I will expect such results that if i do the multiplication log\mathcal{L} \times 5 or log\mathcal{L} \times 500, the MCMC results should not
But when i tried, the MCMC result is greatly affected by the absolute value of log\mathcal{L}. The bigger number I mutiply, the corner plotting shrink more. And i dont why.
1. Case 1: log\mathcal{L} * 5
2. Case 2: log\mathcal{L}*500
I think it is my misunderstanding for MCMC process rather than the code itself. Any help and advice is appreciated.
Code is attached.
function logL = uq_inversion_test_func_CustomLogLikelihood(params,data,A)
% UQ_INVERSION_TEST_FUNC_CUSTOMLOGLIKELIHOOD supplies a custom
% logLikelihood handle for the Bayesian inversion module self tests.
% See also: UQ_SELFTEST_UQ_INVERSION
%split params into model and error params
modelParams = params(:,1:8);
errorParams = params(:,9:end);
sigma = errorParams(:,1); %error standard deviation
psi = errorParams(:,2); %correlation length
nData = size(data,2);
%number of parameters passed to likelihood function
nChains = size(modelParams,1);
%evaluate model
modelRuns = modelParams*A;
%correlation options
CorrOptions.Type = 'Ellipsoidal';
CorrOptions.Family = 'Matern-5_2';
CorrOptions.Isotropic = false;
CorrOptions.Nugget = 0;
%loop through chains
logL = zeros(nChains,1);
for ii = 1:nChains
%get the sigma matrix
sigmaCurr = sigma(ii)*ones(1,nData);
D = diag(sigmaCurr);
%get correlation & covariance matrix
R = uq_eval_Kernel((1:nData).',(1:nData).', psi(ii), CorrOptions);
logLikeli = 0;
C = D*R*D;
L = chol(C,'lower');
Linv = inv(L);
%compute inverse of covariance matrix and log determinante
Cinv = Linv.'*Linv;
logCdet = 2*trace(log(L));
% evaluate log likelihood
logLikeli = logLikeli - 1/2*logCdet - 1/2*diag((data...
%assign to logL vector
logL(ii) = logLikeli;
%logL(ii) = logLikeli .* 500; % or i multiply 500 times bigger
I think i found my problem. logL value is comparing relative values.
The equation above is all about log likelihood space. The problem is that if I multiply 500 in the logL space, that is equal to multiply e^{500} in the likelihood space. That is infinite exploding.
So, i think the way i started is wrong, and i should not multiply numbers in the logL space, or at least multiply a very small number.
1 Like | {"url":"https://uqworld.org/t/custom-defined-loglikelihood-absolute-or-relative-values/3896","timestamp":"2024-11-08T17:00:49Z","content_type":"text/html","content_length":"23421","record_id":"<urn:uuid:4f395fea-69f4-4187-953a-645098f2fbfe>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00561.warc.gz"} |
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I’ve been a private tutor for the last 7 years in subjects ranging from Pre-Algebra to Probability & Statistics. I tutored all 4 years in college every Monday, Wednesday, and Friday in the tutoring
center on campus. I currently teach Algebra II and Pre-Calculus, but I have also taught Algebra I, Geometry, and Trigonometry in the past.
Contact Ryan
I am a teacher, familiar with multiple math curriculums. I have worked with students individually for approximately 18 years. I am comfortable with math from middle school to college level.
Contact Ashley
I am a high school math and chemistry teacher and have also spent time teaching middle school math.
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Filter further by clicking a subject below. | {"url":"https://www.mathwiztutors.com/tutors-by-subject/pre-calculus/","timestamp":"2024-11-11T01:07:54Z","content_type":"text/html","content_length":"35591","record_id":"<urn:uuid:1598e658-cb7b-4be2-9786-5f345c8aea21>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00652.warc.gz"} |
Independent Events and Unions of Events - Knowunity
Independent Events and Unions of Events: AP Statistics Study Guide
Welcome to Probability Land!
Hey there, future statisticians! Let’s dive into the whimsical world of probability, where randomness rules and every event is a chance to learn something new. Picture probability as a game of
chance, but with less glitter and more math. 🎲🧠
Independence Day for Events
When two events in probability land are independent, it means one event’s outcome doesn’t affect the other. Imagine you’re flipping two coins—let's call them Coiny and Flippy. Coiny landing on heads
doesn’t have telepathic powers over Flippy. Each flip is its own universe. 🪙🪙
For contrast, think of temperature and snowfall. When the temperature decides to go all Elsa-from-Frozen and drop low, the odds of seeing snow increase. So, temperature and snowfall are dependent
events—they’re like two peas in a wintry pod. 🌨️❄️
💡 DEFINITION ALERT: Events A and B are independent if knowing whether event A has occurred does not change the probability that event B will occur, and vice versa.
Calculating Independence: The Multiplication Rule
If event A and event B are throwing their independent party, the probability of both events partying together is the product of their individual probabilities. This mathematically savvy soiree is
denoted as:
[ \text{P(A and B) = P(A) * P(B)} ]
Let’s say event A is Flippy landing on heads (P(A) = 0.5) and event B is Coiny landing on tails (P(B) = 0.5). The probability of both getting heads and tails respectively? Easy peasy, it’s:
[ 0.5 * 0.5 = 0.25 ]
That’s a 25% chance of this flip-tastic duo occurring together. 🎉
Additionally, for two independent events, knowing one has happened doesn’t change the probability of the other:
[ \text{P(A | B) = P(A) and P(B | A) = P(B)} ]
So if Flippy lands on heads, the probability of Coiny landing on heads stays just the same— unaffected. 🪄✨
Union of Events: The Addition Rule
Now, let’s chat about unions. Not the labor kind, but the probability of either event A or event B (or both) happening. Statistically, this is like asking, “What are the chances of either eating ice
cream 🍦 or cake 🍰 (or both!) at a party?”
The union rule goes like this:
[ \text{P(A or B) = P(A) + P(B) - P(A and B)} ]
Say the probability of eating cake at a party is 0.6, and the probability of eating ice cream is 0.5. The probability of munching on both (because why limit yourself?) is 0.3. What’s the probability
of indulging in at least one?
[ 0.6 + 0.5 - 0.3 = 0.8 ]
So, there’s an 80% chance your taste buds will be very happy! 🎉🍰🍦
Practice Makes Probability Perfect!
Let’s practice with a real-world scenario, minus the calories:
Example 1: Concert Calculations
Imagine you’re at a music festival, looking at two stages. The main stage fills up to 75% capacity, and the second stage gets 50% full. Assuming who shows up at each stage is independent (meaning no
one is stage-stalking), what’s the chance a random person attends at least one of the stages?
Applying the union rule:
[ \text{P(Main Stage or Second Stage) = P(Main Stage) + P(Second Stage) - P(Main Stage and Second Stage)} ] [ = 0.75 + 0.50 - (0.75 * 0.50) ] [ = 0.75 + 0.50 - 0.375 ] [ = 0.875 ]
So, there’s an 87.5% chance the person is dancing at one of the stages. 🎶🕺💃
Example 2: Academic Aspirations
Planning to ace your math exam? If there’s a 70% chance of scoring high if you study for at least 20 hours, and a 40% chance if you don’t hit the books for that long, plus the probability of actually
studying that much is 60%, what’s the overall chance you’ll score high?
This is where we use the given probabilities in a different way:
[ \text{P(High Score) = P(Study and High Score) + P(No Study and High Score)} ] [ = (0.6 * 0.7) + (0.4 * 0.4) ] [ = 0.42 + 0.16 ] [ = 0.58 ]
So, your probability of making waves on that exam? A solid 58%. 📚✏️
Key Terms to Know
• Mutually Exclusive Events: Events that can’t happen at the same time, like you can’t be both a cat and a dog, right? 🐱🐶
• P(A and B): The likelihood of both event A and event B happening together.
• Unions: The cumulative probability of either event A or event B—or both—happening.
We've transformed the seemingly dry landscape of probability into a festival of numbers! 🥳 Understanding independent events and unions of events is like learning the choreography to a math dance.
With practice, you’ll be gliding through these calculations like a pro.🕺💃 So, keep tossing those coins, analyzing stages, and get those high scores!
Go forth, and may the odds be ever in your favor! 🎲📚 | {"url":"https://knowunity.com/subjects/study-guide/independent-events-unions-events","timestamp":"2024-11-11T18:23:09Z","content_type":"text/html","content_length":"243112","record_id":"<urn:uuid:69c4c350-75c8-475c-9d63-233664cdf7a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00366.warc.gz"} |
Spherical Video PTZ
This section provides the theoretical explanations of the Equirectangular and Rectilinear projections, which are used on the Spherical Video PTZ.
Equirectangular Projection
Equirectangular images, also referred to as 360° images, capture a panoramic view from a fixed point where the imaging system is positioned. These images encapsulate a complete 360° perspective,
allowing all surrounding information to be displayed within a single flat image. To illustrate this concept, consider visualising the Earth as a sphere and then "unfolding" it along the central
meridian (shown by the red lines in the accompanying image). This "unfolding" process transforms the spherical surface into a plane image. Please note, that the resulting plane maintains a unique
aspect ratio of 2:1, because of the vertical range covers 180° and the horizontal range covers 360° after the unfolding procedure.
• Alt=Example of Equirectangular Projection
• Alt=Example of Equirectangular Projection
Rectilinear Projection
The Rectilinear projection, also referred to as the Gnomonic Projection, is a method used to project the surface of a sphere (or a 360° image) onto a plane. Typically, the plane onto which the
surface points are mapped is tangent to the sphere at a single point. This projection is accomplished by using the centre of the sphere as the projection point. It's important to note that the
resulting plane does not intersect the centre of the sphere. The diagram below provides a visual example of this process:
Rectilinear projection: great circle projection example
The term "rectilinear" in the Rectilinear projection refers to its use of straight lines for the projection. This means that lines that are parallel in the real world remain parallel in the
projection. Additionally, it is worth noting that every great circle (which is the largest circle that can be drawn on any given sphere) is transformed into a straight line in the resulting plane
during this projection process.
Equirectangular to Rectilinear Projection
Now that both projections' workings are clear, let's delve into the crucial details: why is this transformation necessary?
If we round up an Equirectangular image, we can construct a spherical image where the data is accurately positioned on the surface of the sphere. However, if a user chooses to crop the
Equirectangular image directly, it would lead to an issue: the resulting image would exhibit irregularities due to the curvature of the sphere. Therefore, projecting this image into a Rectilinear
image resolves the problem of perturbations in the desired output. | {"url":"https://developer.ridgerun.com/wiki/index.php/Spherical_Video_PTZ/Getting_Started/Projections_Used","timestamp":"2024-11-14T13:48:54Z","content_type":"text/html","content_length":"48546","record_id":"<urn:uuid:c8eb0d7a-9f9b-46b1-b28d-67fd82a9160b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00706.warc.gz"} |
UK Weather Trends
Meteorologists define spring in the UK to be the period from March to May so spring is now over and we are officially in summer. I have decided to create a series of posts which I will publish at
the end of each season showing how that season compares to the weather record. The 2017 spring turned out to be the 2nd warmest on record.
Analysis of trends is a key skill that all statisticians and analysts need to have. Indeed I run a training course on Identifying Trends & Making Forecasts as I have learned over my 25 years as a
professional statistician that skills in this area are nowhere near where they should be. By publishing this series of posts about trends in weather and other fields, I hope you will learn something
you will be able to apply elsewhere.
As my weather tracker series shows, weather cannot be thought of as a univariate data set. We define weather as a combination of temperature, sunshine and rainfall which makes weather a multivariate
data set. There are many ways of displaying multivariate data but one of the first questions that has to be asked is how do we compare 3 variables that use 3 different numerical scales i.e. degrees
Celsius, hours and millimetres? The answer is to convert the scales of all 3 variables into a new common scale using a procedure known as STANDARDISATION.
Standardisation works as follows. For each year within each variable, we first subtract the average for that variable to get the difference from the average. Then we divide by the standard
deviation of that variable. For example, if the average spring temperature over the last 50 years is 7.6 degrees Celsius and the standard deviation is 0.8 degrees Celsius, then the STANDARDISED
spring temperature for 2017 will be +1.9 since the actual average 24 hour UK spring temperature was 9.1 degrees Celsius i.e. (9.1 – 7.6)/0.8 = +1.9.
Notice that by dividing by the standard deviation, we convert the scale of any variable to a scale which measures the variable as Number of Standard Deviations Above/Below the Mean value. So spring
2017 was almost 2 standard deviations above the average. If you are familiar with the properties of the Normal Distribution, you will know that if your variable follows a normal distribution, then
95% of the data points will lie within +/- 2 standard deviations of the mean and 5% of the data points will lie without +/- 2 standard deviations of the mean. So our +1.9 for the STANDARDISED spring
temperature for 2017 does suggest that 2017 was unusual (but not exceptional) and indeed spring 2017 was the 2nd warmest on record beaten only by 2011 as shown in the chart below.
Standardised variables (also known as Z-SCORES) aid interpretation of data in many ways. If the standardised value is positive, it means that the value is above your average or expected value. If
it is negative, then the value is below your expected value. The chart above suggests that our spring temperatures have been following a rough 25 year cycle e.g.
• From 1910 to 1935 (roughly), UK springs were colder than average.
• 1936 to 1961, UK springs were average.
• 1962 to 1987, UK springs were colder than average.
• 1998 to 2017, UK springs were warmer than average.
If the original variable is approximately normal in its distribution then the vertical scale gives us an idea of how typical or atypical each year is. Z-Scores in the range -1 to +1 are considered
typical values and completely unremarkable. Z-scores in the ranges -2 to -1 and +1 to +2 are considered to be uncommon values but still entirely plausible and such values should not cause us
concern. When Z-Scores get into the ranges -3 to -2 and +2 to +3, we should start paying closer attention and asking ourselves if something has changed especially if we get a sequence of successive
points in these ranges. Finally, if the Z-scores are less than -3 or greater than +3, that is normally regarded as a clear call to action. There are in fact many ways of interpreting Z-Scores
and what I have said so far merely a gives an overview of the most basic interpretations. A whole field of study known as Statistical Process Control (SPC) is dedicated to building and interpreting
such charts (known as a CONTROL CHART).
One point to clarify when calculating the z-scores for the weather variables is over what timeframe should the average and standard deviation be based on. I have decided to go with a rolling 50-year
average and standard deviation so since we are in 2017, these values will be calculated on the 1967 to 2016 timeframe. My reason for using this timeframe is that it seems like a good timeframe for
the concept of “living memory” i.e. we evaluate the most recent weather in terms of our experience & memory and being a child of the 70’s I can still remember some of the weather of the late 70’s.
I said earlier that if we have multivariate data sets then standardising variables allows us to compare multiple variables of differing scales. The next two charts show the z-scores for spring
sunshine and rainfall in the UK.
The z-score for spring sunshine in 2017 was +1.3 so this is well above average but completely within normal range of expectations. The z-score for rainfall was -0.8 so spring 2017 was drier than
normal but completely unremarkable in terms of rainfall. When we combine these two values with +1.9 for temperature, I think it is fair to conclude that spring 2017 was definitely a pleasant one!
What I have covered here in this post is simply some of the basics of analysing time series data. When I analyse the summer of 2017 at the beginning of September, I will add more features to these
charts which will allow us to gain a greater insight into our weather. | {"url":"https://marriott-stats.com/nigels-blog/uk-weather-trends-1-spring-2017/","timestamp":"2024-11-06T15:17:44Z","content_type":"text/html","content_length":"59864","record_id":"<urn:uuid:9b0c81e2-dde5-4d50-9f9b-ea29be44450f>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00018.warc.gz"} |
THE problem with dynamics
THE problem with dynamics is multivariable dynamics.
So far we only considered real or complex single variable dynamics.
Example of 2 variable dynamics problem :
For x >= 0 ,
Find Pairs of analytic functions f,g such that
f(x+1) = 2 f(x)^2 + 3 g(x)^2 + 4 f(x) + 5 g(x) + 6
g(x+1) = f(x)^2 + 7 g(x)^2 + 8 f(x) + 9 g(x) + 10
I have considered this idea with mick, but without results.
04/04/2017, 10:52 PM
Well after Some thinking it appears that all multivariable problems reduce to analogues of single variable dynamics , univariate diff equations , delay differential equations and PDE ( partial ( =
multivariable ) differential equations.
For instance there are analogue fractals where the half-iteration is not defined.
And analogue Koenig functions.
( if you compute the half-iterate of a random polynomial of degree 2 by using koenigs , you have a problem within the Julia set ( the fractal ) of that polynomial almost surely )
I thank mick and Sheldon for realizing it " completely " now.
Although they did not actively help their past ideas did.
Im not going to define completely here.
There is no reason to assume a multivariable difference equation can be expressed by a univariate difference equation easier or more often than a PDE can be expressed in univar diff equations and
vice versa.
Im unaware of a Satisfying formal statement and formal proof of that though. | {"url":"https://tetrationforum.org/showthread.php?tid=1161","timestamp":"2024-11-10T08:46:12Z","content_type":"application/xhtml+xml","content_length":"31124","record_id":"<urn:uuid:c8cd7897-3b49-44bf-bdcf-86e117c06848>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00329.warc.gz"} |
Solving One Step Equations Practice Worksheet - Equations Worksheets
Solving One Step Equations Practice Worksheet
Solving One Step Equations Practice Worksheet – The objective of Expressions and Equations Worksheets is for your child to be able to learn more efficiently and effectively. They include interactive
activities and challenges based on sequence of operations. These worksheets make it simple for children to grasp complex concepts and basic concepts quickly. It is possible to download these free
materials in PDF format in order to aid your child in learning and practice math-related equations. These are helpful for students between 5th and 8th Grades.
Free Download Solving One Step Equations Practice Worksheet
A few of these worksheets are for students in the 5th to 8th grades. The two-step word problems are constructed using fractions and decimals. Each worksheet contains ten problems. They are available
at any online or print resource. These worksheets are an excellent way to practice rearranging equations. These worksheets are a great way for practicing rearranging equations. They aid students in
understanding equality and inverted operations.
These worksheets can be used by fifth- and eighth grade students. These are great for students who have difficulty calculating percentages. You can choose from three different types of problems. You
can decide to tackle one-step problems that include decimal numbers or whole numbers or employ word-based techniques to solve problems involving decimals and fractions. Each page contains 10
equations. These worksheets for Equations are suggested for students in the 5th through 8th grades.
These worksheets can be a wonderful tool for practicing fraction calculations as well as other concepts that are related to algebra. You can choose from many different types of problems with these
worksheets. You can pick a word-based or a numerical one. The problem type is also crucial, since each will present a different problem kind. Each page has ten questions, making them a great resource
for students in 5th-8th grade.
These worksheets teach students about the connections between variables and numbers. These worksheets allow students to practice with solving polynomial equations as well as solving equations and
getting familiar with how to use these in their daily lives. If you’re in search of a great educational tool to master the art of expressions and equations, you can start by exploring these
worksheets. They will help you learn about different types of mathematical equations and the different types of symbols used to represent them.
These worksheets are beneficial to students in the first grade. These worksheets will aid them learn how to graph and solve equations. They are great to practice polynomial variables. They can help
you understand how to factor them and simplify them. There is a fantastic set of equations, expressions and worksheets for children at any grade. Doing the work yourself is the best way to master
There are a variety of worksheets that can be used to learn about quadratic equations. Each level comes with their own worksheet. These worksheets are designed to assist you in solving problems in
the fourth degree. After you’ve completed the required level it is possible working on other kinds of equations. Then, you can take on the same problems. For example, you might find a problem with
the same axis, but as an extended number.
Gallery of Solving One Step Equations Practice Worksheet
One Step Equations Worksheets Math Monks
One Step Equations Worksheets Math Monks
One Step Equations Worksheets Math Monks
Leave a Comment | {"url":"https://www.equationsworksheets.net/solving-one-step-equations-practice-worksheet/","timestamp":"2024-11-09T06:07:47Z","content_type":"text/html","content_length":"63685","record_id":"<urn:uuid:c5fa0807-ca56-4814-893c-c4e864095d98>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00803.warc.gz"} |
$2n$ ambassador seating around a round table so that no one seats next to an enemy
$2n$ ambassadors are invited to a banquet. Every ambassador has at most $n-1$ enemies. Prove that the ambassadors can be seated around a table, so that nobody sits next to an enemy
Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.
View the answer
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to earn up to a 50% commission on every question that your affiliated users ask or answer. | {"url":"https://matchmaticians.com/questions/1hcw3y/2n-ambassador-seating-around-a-round-table-so-that-no-one","timestamp":"2024-11-08T14:24:30Z","content_type":"text/html","content_length":"75209","record_id":"<urn:uuid:001eb12f-3ece-4fe4-a76f-0be78a50f825>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00848.warc.gz"} |
The REGRESS function performs a multiple linear regression fit and returns an Nterm-element column vector of coefficients.
REGRESS fits the function:
y[i] = const + a[0]x[0,] [i] + a[1]x[1,] [i] + ... + a[Nterms][-1]x[Nterms][-1,] [i]
This routine is written in the IDL language. Its source code can be found in the file regress.pro in the lib subdirectory of the IDL distribution.
; Create two vectors of independent variable data:
X1 = [1.0, 2.0, 4.0, 8.0, 16.0, 32.0]
X2 = [0.0, 1.0, 2.0, 3.0, 4.0, 5.0]
; Combine into a 2x6 array
X = [TRANSPOSE(X1), TRANSPOSE(X2)]
; Create a vector of dependent variable data:
Y = 5 + 3*X1 - 4*X2
; Assume Gaussian measurement errors for each point:
measure_errors = REPLICATE(0.5, N_ELEMENTS(Y))
; Compute the fit, and print the results:
result = REGRESS(X, Y, SIGMA=sigma, CONST=const, $
PRINT, 'Constant: ', const
PRINT, 'Coefficients: ', result[*]
PRINT, 'Standard errors: ', sigma
IDL prints:
Constant: 4.99999
Coefficients: 3.00000 -3.99999
Standard errors: 0.0444831 0.282038
Result = REGRESS( X, Y, [, CHISQ=variable] [, CONST=variable] [, CORRELATION=variable] [, /DOUBLE] [, FTEST=variable] [, MCORRELATION=variable] [, MEASURE_ERRORS=vector] [, SIGMA=variable] [, STATUS=
variable] [, YFIT=variable] )
Return Value
REGRESS returns a 1 x Nterm array of coefficients. If the DOUBLE keyword is set, or if X or Y are double-precision, then the result will be double precision, otherwise the result will be single
An Nterms by Npoints array of independent variable data, where Nterms is the number of coefficients (independent variables) and Npoints is the number of samples.
An Npoints-element vector of dependent variable points.
Set this keyword equal to a named variable that will contain the value of the unreduced chi-square goodness-of-fit statistic.
Set this keyword to a named variable that will contain the constant term of the fit.
Set this keyword to a named variable that will contain the vector of linear correlation coefficients.
Set this keyword to force computations to be done in double-precision arithmetic.
Set this keyword to a named variable that will contain the F-value for the goodness-of-fit test.
Set this keyword to a named variable that will contain the multiple linear correlation coefficient.
Set this keyword to a vector containing standard measurement errors for each point Y[i]. This vector must be the same length as X and Y.
Note: For Gaussian errors (e.g., instrumental uncertainties), MEASURE_ERRORS should be set to the standard deviations of each point in Y. For Poisson or statistical weighting, MEASURE_ERRORS should
be set to SQRT(Y).
Set this keyword to a named variable that will contain the 1-sigma uncertainty estimates for the returned parameters.
Note: If MEASURE_ERRORS is omitted, then you are assuming that the regression model is the correct model for your data, and therefore, no independent goodness-of-fit test is possible. In this case,
the values returned in SIGMA are multiplied by SQRT(CHISQ/(N–M)), where N is the number of points in X, and M is the number of coefficients. See section 15.2 of Numerical Recipes in C (Second
Edition) for details.
Set this keyword to a named variable that will contain the status of the operation. Possible status values are:
• 0 = successful completion
• 1 = singular array (which indicates that the inversion is invalid)
• 2 = warning that a small pivot element was used and that significant accuracy was probably lost.
Note: If STATUS is not specified, any error messages will be output to the screen.
Set this keyword to a named variable that will contain the vector of calculated Y values.
Version History
Original Introduced
5.4 Deprecated the Weights, Yfit, Const, Sigma, Ftest, R, Rmul, Chisq, and Status arguments, RELATIVE_WEIGHT keyword.
See Also
CURVEFIT, GAUSSFIT, LINFIT, LMFIT, POLY_FIT, SFIT, SVDFIT | {"url":"https://www.nv5geospatialsoftware.com/docs/regress.html","timestamp":"2024-11-04T20:54:16Z","content_type":"text/html","content_length":"62893","record_id":"<urn:uuid:ec546a81-7610-4110-babf-48a1a2fe22ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00654.warc.gz"} |
Total Variation in Data - Overall Importance of the Principal Components
As the system was not accepting spaces I gave comma instead of space.
Table 6.1 indicates the overall importance of the principal components. It
displays for each component ph the standard deviation, given by
√λh, the proportion of variance explained by each component, equal to λh/j λj, and the cumulative proportion explained by the first components up to ph included.The analysis shows that the first two
components explain 84% of the total variation in the data while the first 5 explain 94%.
Table 6.1 Overall importance of principal components in the mtcars dataset
Standard deviation,,,,,,2.5706809,,,,1.6280258,,,0.7919578,,0.5192277
Proportion of variance,,0.6007637,,,,0.2409516,,,0.0570179,,0.0245088
Cumulative proportion,,0.6007637,,,,0.8417153,,,0.8987332,,0.9232420
Standard deviation,,,,,,0.4727061,,,0.4599957,,,,0.3677798,,,0.3505730
Proportion of variance,,0.0203137,,,0.0192360,,,,0.0122965,,,0.0111728
Cumulative proportion,,0.9435558,,,0.9627918,,,0.9750883,,,0.9862612
Standard deviation,,,,,,,0.2775727,,,,0.2281127,,,0.1484735
Proportion of variance,,,0.0070042,,,,0.0047304,,,0.0020040
Cumulative proportion,,,0.9932654,,,,,,0.9979959,,,,1.0000000
Can someone please the formula for total variation in data, how did we get the values 84% and 94%
Table 6.2 shows the values of the coefficients whj = uhj ; equivalently, it
indicates the eigenvectors of the covariance matrix V. The first component,
which alone explains 60% of the variance, is negatively correlated with the
attributes {cyl, disp,wt, carb}, whose meaning is explained in Appendix B,
while it is positively correlated with all the other attributes.
Table 6.2 Principal component coefficients for the mtcars dataset
disp,, −0.368,,,,,,,,,0.257,,,,,,−0.394,,,,,−0.336,,,,,,0.214,,,,,,,0.198
wt,,,,,,−0.346,,,,,,,,,0.143,,,,,,,0.342,,,,,,,0.246,,,,,−0.465,,,,,, 0.359
qsec,,,,,−0.528,,,,, −0.271,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,−0.181
vs,,,,,,,,−0.266,,,,,,,,0.359,,,,, −0.159
Can anyone please explain how the corelation is calculated like how did we get the value 60%
Thanks in advance
Last edited by a moderator:
Forum Moderator
This is pretty advanced statistics for this forum. I recommend asking this question on Talkstats.com. Talkstats is a statistics discussion forum, and I have seen PCA and Factor Analysis discussed | {"url":"https://elsmar.com/elsmarqualityforum/threads/total-variation-in-data-overall-importance-of-the-principal-components.70089/","timestamp":"2024-11-02T23:26:00Z","content_type":"text/html","content_length":"56942","record_id":"<urn:uuid:491ac0ab-e558-42aa-98d6-73ab7b80b52c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00754.warc.gz"} |
Printable math conversion chart
Author Message
tfritter4 Posted: Saturday 23rd of Dec 21:21
I have an assignment to submit tomorrow afternoon. But I’m stuck with problems based on printable math conversion chart. I’m facing problems understanding roots and
least common measure because I just can’t seem to figure out a way to solve problems based on them. I called my friends and I tried on the internet, but neither of
those activities helped. I’m still trying but the time is running out and I can’t seem to get a hang of it. Can somebody please help me ? I really need some help
from you people for tomorrows test . Please do reply.
Registered: 20.07.2005
From: Ohio
kfir Posted: Sunday 24th of Dec 08:04
Your story sounds familiar to me. Even though I was great in math for many years, when I started College Algebra there were a lot of algebra topics that seemed
confusing. I remember I got a very low grade when I took the test on printable math conversion chart. Now I don't have this issue anymore, I can solve anything
quite easily , even sum of cubes and ratios. I was smart that I didn't spend my money on a tutor, because I heard of Algebra Master from a student . I have been
using it since then whenever I found something complicated.
Registered: 07.05.2006
From: egypt
Koem Posted: Monday 25th of Dec 13:16
Algebra Master is rightly a good software program that helps to deal with algebra problems. I remember facing difficulties with mixed numbers, side-side-side
similarity and equation properties. Algebra Master gave step by step solution to my math homework problem on typing it and simply clicking on Solve. It has helped
me through several math classes. I greatly recommend the program.
Registered: 22.10.2001
From: Sweden
amorga Posted: Monday 25th of Dec 17:45
Can someone please give me the link to this software? I am on the verge of breaking down. I like math a lot and don’t want to drop it just because of one course .
Registered: 10.04.2002
Paubaume Posted: Wednesday 27th of Dec 10:02
proportions, proportions and radical expressions were a nightmare for me until I found Algebra Master, which is really the best math program that I have come
across. I have used it frequently through several algebra classes – Basic Math, Basic Math and Remedial Algebra. Simply typing in the algebra problem and clicking
on Solve, Algebra Master generates step-by-step solution to the problem, and my math homework would be ready. I truly recommend the program.
Registered: 18.04.2004
From: In the stars... where you
left me, and where I will wait for
you... always...
SanG Posted: Thursday 28th of Dec 09:24
Take a look at https://algebra-test.com/resources.html. You can read more information about it and purchase it as well .
Registered: 31.08.2001
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