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A look at intrinsic broadband noise spectral density
You can think of noise as an unwanted signal. This signal creates an error by combining with the desired signal in your circuit. Exterior sources can couple into your circuits, such as your 50 or 60
Hz DC mains signal or your cell phone. The starting point in your circuit’s noise evaluation is to reach a theoretical determination of the analog noise levels. In this blog, we will work strictly on
analog intrinsic noise.
An amplifier generates voltage and current noise in the 1/f and broadband frequency range. Additionally, the standard resistor generates thermal noise that is flat across the same frequency spectrum.
A closed-loop amplifier circuit in a gain of 2 volts per volts (V/V) grows an input signal from ± 1 V to ± 2V at the output (Figure 1).
Figure 1. An amplifier with a 2 kΩ resistor in the feedback loop and a 1 k resistor from the inverting terminal to the ground creates a positive signal gain of 2 V/V to the output. (Image: Texas
In the top right graph, the input signal is a ± 1V sinusoidal signal. Given an amplifier circuit gain of +2 V/V, the output signal in the middle diagram is a ± 2V sinusoidal signal. In an ideal
world, the amplifier and surrounding resistor’s noise is zero. But this is not the case. The bottom Figure 1 graph shows the actual circuit output with a significant amount of intrinsic noise.
Circuit noise sources
Let’s see if we can track down the source of the noise in Figure 1. The next step is to show the amplifier circuit with all amplifier and resistor noise sources. Semiconductor devices and resistors
generate predictable intrinsic noise. The Figure 1 amplifier circuit noise comes from the input voltage noise (V[A-N]) and current noise (V[A-I]). The resistor noise (V[R1_N], V[R2_N]) contributes to
the V[OUT] total output noise (Figure 2).
Figure 2. The position of the voltage amplifier noise is at the non-inverting input, and the current amplifier noise straddles the inverting and non-inverting inputs. (Image: Bonnie Baker)
It is important to note that the amplifier voltage noise model is at the non-inverting input of the amplifier. The amplifier current noise flows through R[1] and R[2], and the amplifier gains the
current-resistance voltage to the output of the amplifier.
The amplifier’s voltage and current noise have 1/f noise at low frequencies, except for zero-drift amplifiers. Past the 1/f corner frequency, the amplifier produces a broadband noise (Figure 3).
Figure 3. The intrinsic spectral density noise of Texas Instruments’ OPA828 operational amplifier illustrates low-frequency 1/f and higher frequency broadband noise. (Image: Texas Instruments)
The resistor’s thermal electron agitation sets the conductor’s noise level. The synonyms of the resistor and the broadband noise category are white noise, Johnson noise, thermal noise, and resistor
noise. In 1928 J. B Johnson found the conductor’s existing nonperiodic voltage. He also found a temperature related to the magnitude of the conductor’s voltage. The ideal calculated magnitude of the
resistance thermal noise voltage is dependent on the Boltzmann’s constant, Kelvin temperature, and resistance value (Equation 1).
For example, for a Celsius temperature equaling 10°C and a 1 MW resistor, the thermal noise voltage is:
One would hope that you could add the individual noise sources together. That equation makes sense if the signals are correlated. However, noise sources are not correlated, so the appropriate formula
for adding noise sources is the square-root-of the-sum-of the-squares (RSS). The equations below summarize the noise calculation of this circuit.
Figure 4 shows an oscilloscope view of broadband or white noise waveform in the time domain when we examine the broadband noise. In general, broadband noise is in the middle-to-high frequency range
or frequencies greater than 1 kHz.
Figure 4. On the left is an oscilloscope view of broadband noise. The diagram on the right captures each peak value from the left and adds the noise occurrences together. (Image: Bonnie Baker)
In Figure 4, the right diagram illustrates a normal statistical distribution and represents the cumulation of the data peaks on the left. The labels for the lines are m, s, and C. m equals the mean
of all collected signals on the right diagram. The data between the green lines capture ±s (2s) or 68.3% of the samples. The data between the blue lines, C, capture the samples that land between the
peak-to-peak (P-P) specification.
The design engineer defines P-P specification, which depends on a particular design requirement. Equation 7 calculates the Figure 4 C limits.
For example, if P(-1<x<+1) = 0.3, the probability that x is between -1 and 1 is 30%.
If the designer prefers an rms response, 2s is appropriate for that application. This choice identifies the data occurrence between the s lines is 68.3% of the time (Figure 5).
Figure 5. The Probability Distribution function determines the percentage of data occurrences inside the limits. (Image: Bonnie Baker)
Figure 5 has seven different crest factor limits that show the number of standard deviations per the measurement probability. The industry P-P standard is either 6.0 s or 6.6 s per the product
datasheet specifications. Keep in mind that the tails of the Gaussian curve are infinite. There is always a possibility that noise is outside the designer’s “a” and “b” range.
Start your circuit’s noise evaluation by identifying the value of the semiconductor and resistive noise sources. An amplifier generates voltage and current noise across the 1/f and broadband regions.
The resistor generates thermal noise that is flat across the frequency spectrum. The noise contribution of these semiconductor and resistive elements combines with the RSS equation.
Noise Reduction Techniques in Electronic Systems, Henry W. Ott, John Wiley & Sons, ISBN 0-471-85068-3
“Analysis and Measurement of Intrinsic Noise In Op Amp Circuits Part II: Introduction To Op Amp Noise”, Texas Instruments, Art Kay,
“Squash 1/f noise with zero-drift amplifiers”, Baker, Bonnie, EEWorld
“Including the often overlooked 1/f Current Noise”, Baker, Bonnie, EEWorld, July 28, 2021
“Negotiate through 1/f noise challenges towards sample sizes into the 100s of years” Baker, Bonnie, EEWorld, July 12, 2021 | {"url":"https://www.analogictips.com/a-look-at-intrinsic-broadband-noise-spectral-density-faq/","timestamp":"2024-11-13T18:49:11Z","content_type":"text/html","content_length":"108154","record_id":"<urn:uuid:672a1d47-9109-4930-827c-0dafa1bdfd51>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00135.warc.gz"} |
Spring 2025 Course Booklet Course Info on SIS Course Advising for New Students Archives
Course Descriptions
The courses below include descriptions of all undergraduate and graduate courses offered by the Department of Mathematics, though some courses may be taught more often than others. Descriptions for
special topics seminars are updated each semester.
Visit the undergraduate and graduate pages for course requirements for specific programs. For up-to-date information on course offerings, schedules, room locations and registration, please visit the
Student Information System (SIS).
Undergraduate Courses
MATH 0014 Introduction To Finite Mathematics. Topics selected from financial mathematics, matrix algebra, linear inequalities and linear programming, counting arguments, and statistics and
probability. Recommendations: High school geometry and algebra. (Math 30 is not a prerequisite.) Engineering students are not permitted to take MATH 14 for credit.
MATH 15 Mathematics In Antiquity. (Cross-listed as CLS 15) History of mathematics in Babylonian, Egyptian, Greek, and other ancient civilizations. Number systems and computational techniques;
achievements in elementary algebra, geometry, and number theory; famous results, proofs and constructions. Emphasis on solving problems in the style and spirit of each culture. Engineering students
are not permitted to take MATH 15 for credit.
MATH 16 Symmetry. A mathematical treatment of the symmetries of wallpaper patterns. The main goal is to prove that the symmetries of these patterns fall into seventeen distinct types. In addition,
students will learn to identify the symmetries of given patterns (with special emphasis on the periodic drawings of M.C. Escher) and to draw such patterns. Three lectures, one section.
Recommendations: High school geometry. Engineering students are not permitted to take MATH 16 for credit.
MATH 19 The Mathematics Of Social Choice. Introduction to mathematical methods for dealing with questions arising from social decision making. Topics vary but usually include ranking, determining the
strength of, and choosing participants in multicandidate and two-candidate elections, and apportioning votes and rewards to candidates. Recommendations: High school algebra. Engineering students are
not permitted to take MATH 19 for credit.
MATH 21 Introductory Statistics. Descriptive data analysis, sampling and experimentation, basic probability rules, binomial and normal distributions, estimation, regression analysis, one and two
sample hypothesis tests for means and proportions. The course may also include contingency table analysis, and nonparametric estimation. Applications from a wide range of disciplines.
Recommendations: High school algebra and geometry.
MATH 30 Introduction to Calculus. Functions and their graphs, limits, derivatives, techniques of differentiation. Applications of derivatives, curve sketching, extremal problems. Integration:
indefinite and definite integrals, some techniques of integration, Fundamental Theorem of Calculus. Logarithmic and exponential functions with applications. Recommendations: High school geometry and
algebra. MATH 30 is a one-semester calculus course and is not adequate preparation for MATH 34. Students will receive an additional two credits (with grade) for passing MATH 32 after receiving
credit for MATH 30. MATH 32 must be taken at Tufts and for a grade.
MATH 32 Calculus I. Differential and integral calculus: limits and continuity, the derivative and techniques of differentiation, extremal problems, related rates, the definite integral, Fundamental
Theorem of Calculus, derivatives and integrals of trigonometric functions, logarithmic and exponential functions. Recommendations: High school geometry, algebra, and trigonometry. Students will
receive an additional two credits (with grade) for passing MATH 32 after receiving credit for MATH 30. MATH 32 must be taken at Tufts and for a grade.
MATH 34 Calculus II. Applications of the integral, techniques of integration, separable differential equations, improper integrals. Sequences, series, convergence tests, Taylor series. Polar
coordinates, complex numbers. Students may count only one of MATH 34 and MATH 36 for credit. Recommendations: MATH 32.
MATH 39 Honors Calculus I-ii. (Formerly MATH 17). The first course of the two-semester sequence of honors calculus. Intended for students who have had at least the AB syllabus of advanced
placement mathematics in secondary school. Stresses the theoretical aspects of the subject, including proofs of basic results. Topics include: convergence of sequences and series; continuous
functions, Intermediate Value and Extreme Value Theorems; definition of the derivative, formal differentiation, finding extrema, curve-sketching, Mean Value Theorems; basic theory of the Riemann
integral, Fundamental Theorem of Calculus and formal integration, improper integrals; Taylor series, power series and analytic functions. Recommendations: AB syllabus of advanced placement
mathematics. Students who receive credit for MATH 39 (formerly MATH 17) cannot receive credit for MATH 30, 32, or 34 (formerly MATH 5, 11, or 12). Upon successful completion of MATH 39, all
students receive two credits
MATH 42 Calculus III. Vectors in two and three dimensions, applications of the derivative of vector-valued functions of a single variable. Functions of several variables, continuity, partial
derivatives, the gradient, directional derivatives. Multiple integrals and their applications. Line integrals, Green's theorem, divergence theorem, Stokes’ theorem. Prerequisite: MATH 34 or 39.
MATH 44 Honors Calculus III. Analysis of real- and vector-valued functions of one or more variables using tools from linear and multilinear algebra; stress is on theoretical aspects of the
subject, including proofs of basic results. Topics include: geometry and algebra of vectors in 3-space, parametrized curves and arc length, linear transformations and matrices; Jacobian and
gradient of a real-valued function, Implicit Function Theorem, extrema, Taylor's Theorem and Lagrange multipliers; multiple integrals, differential forms and vector fields, line integrals,
parametrized surfaces and surface integrals, exact and closed forms, vector calculus.
Recommendations: MATH 39 or permission of instructor. Students who receive credit for MATH 44 cannot receive credit for MATH 42.
MATH 51 Differential Equations. An introduction to linear differential equations with constant coefficients, linear algebra, and Laplace transforms. Recommendations: MATH 42 or 44.
MATH 65 Bridge to Higher Mathematics. Introduction to rigorous reasoning, applicable across all areas of mathematics, to prepare for proof-based courses at the 100 level. Writing proofs with precise
reasoning and clear exposition. Topics may include induction, functions and relations, combinatorics, modular arithmetic, graph theory, and convergence of sequences and series of real
numbers. Recommendations: MATH 34 or permission.
MATH 70 Linear Algebra. Introduction to the theory of vector spaces and linear transformations over the real or complex numbers, including linear independence, dimension, matrix multiplication,
similarity and change of basis, inner products, eigenvalues and eigenvectors, and some applications. Recommendations: MATH 34 or 39 or permission of instructor. Students may count only one of MATH
70 and MATH 72 for credit.
MATH 87 Mathematical Modeling And Computation. A survey of major techniques in the use of mathematics to model physical, biological, economic, and other systems; topics may include derivative-based
optimization and sensitivity analysis, linear programming, graph algorithms, probabilistic modeling, Monte-Carlo methods, difference equations, and statistical data fitting. This course includes an
introduction to computing using a high-level programming language, and studies the transformation of mathematical objects into computational algorithms. Prerequisites: (1) MATH 34 , 36, or 39, and
(2) Math 70 or 72, or permission of instructor. Recommendations: MATH 34, MATH 36 or MATH 39, or consent.
MATH 102 Math-Education: From Numbers to Functions. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts
and the pedagogical insights behind the following topics: real numbers, fractions and their multiple representations, introduction to functions: the intuitive and formal definition of function,
composition of functions, representations through tables, graphs, dynagraphs, algebraic and verbal expressions, the vertical line criterion, composition of functions, examples of functions coming
from arithmetic operations as well as functions commonly used in mathematics and science, functional approach to division with remainder, decimals and decimal representation of rational numbers,
divisibility for integers and decomposition into product of powers of primes. Teaching projects with school age students are an integral part of this course. Offered on line with a face-to -face
component. Permission of instructor.
MATH 103 Math-Education: Transformations and Equations. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical
concepts and the pedagogical insights behind the following topics: transformations of the plane with an emphasis on the comparison with arithmetic operations and the action of transformations on the
graphs of functions. Geometric and algebraic interpretations of equations. The use of transformations in the solutions of linear and quadratic equations. Divisibility for integers and polynomials,
the euclidean algorithm for the greatest common divisor, divisibility and factorization of polynomials and it solution in the solution of polynomial equations. Teaching projects with school age
students are an integral part of the course. This course is offered on line with a face-to -face component. Permission of instructor.
MATH 104 Math-Education: Change and Invariance. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and
the pedagogical insights behind the following topics: Helping students with word problems. Functions of several variables. Linear systems of equations and their solutions. Limits of sequences and of
functions, limits at infinity. Slope and rate of change for non-linear functions. The derivative function and applications. Teaching projects with school age students are an integral part of the
course. This course is offered on line with a face-to -face component. Permission of instructor.
MATH 110 Special Topics in Mathematics Education. Intended for education students. Meets with a mid-level mathematics course emphasizing proofs (such as Math 63, 70, and 72). Additional content
connects the mathematics to the students' teaching. Students have extra pedagogical responsibilities to be determined with the mathematics instructor and the instructor in the Education Department.
The grade in the mathematics course will count for 75% of the course grade, and to pass, the student must receive at least a B+ in the mathematics course. Does not count for any degree in the
Mathematics Department nor for A&S Distribution Credit in Mathematical Sciences. Permission of Instructor.
MATH 112 Topics In The History Of Mathematics. The evolution of mathematical concepts and techniques from antiquity to modern times. Recommendations: MATH 34 or 39 or permission of instructor.
MATH 123 Mathematical Aspects of Data Analysis. Dimension reduction and data compression via principal component analysis, and the singular value decomposition; k-means clustering; clustering via
diffusion on weighted graphs; support vector machines; tensor data analysis; kernel trick. Homework includes programming. Prerequisite: MATH 42, and MATH 70 or MATH 72. Some prior programming
experience desirable, but not required.
MATH 125 Numerical Analysis. (Cross-listed as CS 125.) Analysis of algorithms involving computation with real numbers. Interpolation, methods for solving linear and nonlinear systems of equations,
numerical integration, numerical methods for solving ordinary differential equations. Recommendations: MATH 51 and programming ability in a language such as C, C++, Fortran, or Matlab.
MATH 126 Numerical Linear Algebra. (Cross-listed as CS 126) The two basic computational problems of linear algebra: solution of linear systems and computation of eigenvalues and eigenvectors.
Recommendations: MATH 70 or 72 and CS 11.
MATH 133 Complex Variables. Introduction to the theory of analytic functions of a single complex variable, analytic functions, Cauchy's integral theorem and formula, residues, series expansions of
analytic functions, conformal representation, entire and meromorphic functions, multivalued functions. Recommendations: MATH 42 or 44, or permission of instructor.
MATH 135 Real Analysis I. An introduction to analysis. Metric spaces (with Euclidean spaces as the primary example), compactness, connectedness, continuity and uniform continuity, uniform
convergence, the space of continuous functions on a compact set, contraction mapping lemma with applications. Recommendations: either MATH 0042 or 0044 and MATH 0065; or permission of instructor
MATH 136 Real Analysis II. Applications of ideas from MATH 135 to further, in-depth study of functions on Euclidean spaces. Derivatives as linear maps, differentiable mappings, inverse and implicit
function theorems. Further topics such as theory of the Riemann and Lebesgue integral, Hilbert spaces, and Fourier series. Recommendations: either MATH 0070 or 0072 and MATH 0135; or permission of
MATH 145 Abstract Algebra I. An introduction to the basic concepts of abstract algebra, including groups and rings. Recommendations: MATH 0065 and either MATH 0070 or 0072; or permission of
MATH 146 Abstract Algebra II. Further topics in groups and rings. Field extensions and Galois theory. Recommendations: Either MATH 0070 or 0072 and either MATH 0145 or 0245; or permission of
MATH 151 Mathematical Neuroscience. (Cross listed w/ BIO 151) Mathematical and computational study of systems of differential equations modeling nerve cells (equilibria, limit cycles, bifurcations),
neuronal networks (intrinsic rhythmic synchronization, entrainment by external inputs), and learning (synaptic plasticity), and of the potential function of rhythmic synchrony for signaling among
neuronal networks and for plasticity. Prerequisite: Math 51 or instructor’s consent.
MATH 153 Ordinary Differential Equations. Upper-level course in ordinary differential equations from an applied point of view. Equilibria, limit cycles, and their stability. Saddle-node, pitchfork,
transcritical, Hopf, and homoclinic bifurcations. Chaotic dynamics, strange attractors, and fractal dimension. Strong emphasis on examples from the natural sciences. Prerequisite: Math 42 or Math 44,
and at least one of the following three: Math 51, Math 70, Math 72.
MATH 155 Partial Differential Equations I. Introduction to partial differential equations, with emphasis on linear first- and second-order wave equations, diffusion equations, and the Laplace and
Poisson equations. Prerequisites: MATH 70 or MATH 72, and MATH 51 or MATH 153. MATH 155 and ME 150 cannot both be taken for credit.
MATH 156 Partial Differential Equations II. Introduction to the theory of nonlinear partial differential equations, including the method of characteristics, weak solutions, shocks and jump
conditions, nonlinear wave equations, nonlinear diffusion and reaction-diffusion equations, applications to fluid dynamics. Prerequisite: MATH 155 or permission of instructor.
MATH 164 The Mathematics of Poverty and Inequality. Mathematical description of wealth distribution (some distribution theory, Lorenz curves), and the quantification of inequality (Hoover index, Gini
coefficient, Theil indices, Sen's properties of inequality metrics). Agent-based models of wealth distribution, random walks, Wiener processes, Boltzmann and Fokker-Planck equations, and their
application to models of wealth distribution. Wealth condensation and weak solutions. Upward mobility and first-passage times. Methods of mathematical modeling and comparison with empirical
observations are emphasized throughout. Prerequisites: MATH 42: Calculus III or equivalent; and MATH 51: Differential Equations or equivalent; or instructor permission. Recommended but not strictly
necessary: MATH 135: Real Analysis or equivalent; and MATH 165: Probability or equivalent.
MATH 165 Probability. Probability, conditional probability, random variables and distributions, expectation, special distributions, joint distributions, laws of large numbers, and the central limit
Recommendations: MATH 42 or 44, or permission of instructor.
MATH 166 Statistics. Statistical estimation, sampling distributions of estimators, hypothesis testing, regression, analysis of variance, and nonparametric methods. Recommendations: MATH 165 or
permission of instructor.
MATH 171 Point-set Topology. Introduction to the basic definitions and constructions of topology, with a goal of providing ideas and tools that are essential for further study of many branches of
modern mathematics. The definition of a topological space, examples of topological spaces, continuous functions, compactness, connectedness, and separability. Other topics may include homeomorphisms,
subspaces, the quotient topology, and countability axioms. Prerequisite: MATH 0065 or permission of instructor.
MATH 175 Algebraic Topology. Applications of algebra to the study of topological objects, with emphasis on surfaces. Surfaces as manifolds, homotopy of curves, fundamental group, simple
connectedness, covering spaces, genus, Euler Characteristic, orientability, and the classification of compact surfaces. Recommendations: MATH 135 and 145.
MATH 181 Computational Geometry. (Cross-listed as CS 163.) Design and analysis of algorithms for geometric problems. Topics include proof of lower bounds, convex hulls, searching and point location,
plane sweep and arrangements of lines, Voronoi diagrams, intersection problems, decomposition and partitioning, farthest-pairs and closest-pairs, rectilinear computational geometry. Recommendations:
CS 160 or permission of instructor.
MATH 185 Differential Geometry. Study of basic notations of differential geometry in the context of curves and surfaces. Curvature and torsion, implicit function theorem, coordinate systems, first
and second fundamental forms, geodesics, Gauss-Bonnet theorem. Recommendations: MATH 70 or 72, and 135, or permission of instructor.
MATH 190 Advanced Special Topics. Content and prerequisites vary from semester to semester. Topics covered in recent years have included mathematical neuroscience, Lie algebras, and nonlinear
dynamics and chaos.
MATH 191 Computation Theory. (Cross-listed as CS 170). Models of computation: Turing machines, pushdown automata, and finite automata. Grammars and formal languages, including context-free languages
and regular sets. Important problems, including the halting problem and language equivalence theorems.
Recommendations: CS 15 and MATH 61.
MATH 192 Seminars In Mathematics. Attendance at department seminars and colloquia. May include research, teaching-based, and/or student-run seminars with significant math content and/or an outside
speaker. Attendance at 10 seminars required for passing grade with up to two outside Tufts allowed if approved by instructor. Prerequisites: Graduate Standing or consent.
MATH 195 Senior Honors Thesis A. Thesis course for thesis honors candidates; see Thesis Honors Program for details. Open to seniors. This is a yearlong course. Each semester counts as 4 credits
towards a student’s credit load. Students will earn 8 credits at the end of the second semester.
MATH 196 Senior Honors Thesis B. Thesis course for thesis honors candidates; see Thesis Honors Program for details. Open to seniors. This is a yearlong course. Each semester counts as 4 credits
towards a student’s credit load. Students will earn 8 credits at the end of the second semester.
Graduate Courses
MATH 220 Special Topics in Numerical Analysis. A special topics course in the field of Numerical Analysis or Numerical Linear Algebra. Topics change from year to year, and the course may be taken
more than once for credit.
MATH 225 Numerical Analysis. (Cross-list w/ CS 226) Analysis of algorithms involving computation with real numbers. Interpolation, approximation, orthogonal polynomials, methods for solving linear
and nonlinear systems of equations, integration including Gaussian quadrature, ordinary differential equations including A-stability, introduction to methods for hyperbolic partial differential
equations: upwinding, Lax-Friedrichs, Lax-Wendroff. Strong theoretical emphasis. Prerequisites: Math 51 or 153, Math 70 or 72, and graduate standing; or permission of instructor.
MATH 226 Numerical Linear Algebra. (Cross-list w/ CS 228) Design and analysis of algorithms for solving linear systems of equation, least squares problems, and eigenvalue problems, with a strong
emphasis on matrix theory. Unitary matrices (including projections, rotations, and reflections). Matrix factorizations (including LU, Cholesky, QR, and the singular value decomposition).
Conditioning; stability; perturbation analysis; operation counts. Krylov subspace methods (including theoretical analysis, preconditioning, and the connection to Gaussian quadrature).
Applications. Prerequisites: Math 70 or 72 and graduate standing; or permission of instructor.
MATH 229 Graph Algorithms. Mathematical theory and implementation of graph algorithms and their applications. Topics include basic spectral graph theory, shortest path, spanning trees, coloring,
maximal independent set, matching, aggregations, sparsifiers, randomized algorithms, and multilevel methods. Prerequisites: Math 125 or 225, Math 126 or 226, and graduate standing; or permission of
MATH 230 Special Topics in Analysis. A special topics course in the field of Analysis. Topics change from year to year, and the course may be taken more than once for credit.
MATH 233 Complex Analysis. Analytic functions, power series. Integration in the complex plane, Cauchy's integral theorem and formulas. Entire functions. Singularities. Conformal mapping, Riemann
mapping theorem. Prerequisites: Math 135 and graduate standing; or permission of instructor.
MATH 235 Analysis. Measure and integration: sigma-algebras, measurable sets and functions, Lebesgue measure and integration, Monotone/Dominated Convergence, Lp-spaces, Fubini-Tonelli theorem, bounded
variation, absolute continuity, Radon Nikodym theorem, Carathéodory extension and abstract measure. Real functions and functionals: Banach spaces, Hilbert spaces, and topological vector spaces,
linear functionals and representation theorems. Prerequisites: Math 135, Math 136, and graduate standing; or permission of instructor.
MATH 237 Functional Analysis. Topological vector spaces, seminorms and local convexity, Banach Steinhaus theorem, open mapping theorem, Hahn-Banach theorem, duality. Test functions and distributions,
localization and supports of distributions. Fourier transforms, inversion, tempered distributions, Paley-Wiener theorem, Sobolev's lemma. Banach algebras, Gelfand transforms, spectral theory of
bounded linear operators on Hilbert spaces. Prerequisites: Math 235 and graduate standing, or permission of instructor.
MATH 245 Algebra I. General properties of groups, rings, modules over a principal ideal ring, and field extensions. Prerequisites: Math 145 and graduate standing; or permission of instructor.
MATH 246 Algebra II. Foundational results in commutative algebra, algebraic geometry, and homological algebra. Prerequisites: Math 245; or permission of instructor.
MATH 250 Special Topics in Differential Equations. A special topics course in the field of Differential Equations (either Ordinary or Partial). Topics change from year to year, and the course may
be taken more than once for credit.
MATH 255 Partial Differential Equations I. The theory of the Laplace, heat, and wave equations: Fundamental solutions, mean-value formulas, properties of solutions, Green's functions, energy methods,
Duhamel's principle. Quasilinear first-order PDEs, including the Hamilton-Jacobi equation, the method of characteristics, hyperbolic conservation laws and systems thereof, shocks and entropy
conditions. Other selected topics. Prerequisites: Math 51 or 153, Math 135, and graduate standing; or permission of instructor.
MATH 256 Partial Differential Equations II. Boundary-value problems of Sturm-Liouville type, separation of variables, special functions. Similarity solutions, transform methods, power-series
solutions, the Cauchy-Kovalevskaya Theorem. Topics in functional analysis, including L^p spaces and derivatives and existence of weak solutions to second-order elliptic equations and linear evolution
equations. Interior and boundary regularity. Topics in the calculus of variations. Other selected topics. Prerequisites: Math 255; or permission of instructor.
MATH 257 Numerical Partial Differential Equations. Mathematical theory and implementation of computational methods for the solution of partial differential equations (PDEs). Topics include
finite-difference methods for hyperbolic PDEs, finite element methods for elliptic PDEs, and iterative linear solvers for large systems of linear equations. Analysis of consistency, stability, and
accuracy using variational formulations and functional analysis. Prerequisites: Math 135 and 151/251 or consent
MATH 260 Special Topics in Probability and Statistics. A special topics course in the field of Probability and Statistics. Topics change from year to year, and the course may be taken more than once
for credit.
MATH 270 Special Topics in Topology. A special topics course in the field of Topology. Topics change from year to year, and the course may be taken more than once for credit.
MATH 275 Algebraic Topology I. An introduction to the algebraic invariants assigned to topological spaces. Topics include topological manifolds, classification of surfaces, homotopy type, fundamental
group, covering spaces, CW and/or simplicial complexes, introduction to homology. Prerequisites: graduate standing; or permission of instructor.
MATH 276 Algebraic Topology II. Homology, homological algebra, cohomology and Poincare duality. Group cohomology if time permits. Prerequisites: Math 275; or permission of instructor.
MATH 280 Special Topics in Differential Geometry. A special topics course in the field of Differential Geometry and/or Manifolds. Topics change from year to year, and the course may be taken more
than once for credit.
MATH 281 Advanced Computational Geometry. (Cross-listed as CS 263.) Design and analysis of sequential, parallel, probabilistic, and approximation algorithms for geometry problems. Geometric data
structures, complexity, searching, computation, and applications. Selected advanced topics.
Recommendations: CS 163 or permission of instructor.
MATH 285 Manifolds. Key examples of manifolds such as spheres, tori, projective spaces. Topics in manifolds, including quotients, submanifolds, regular level sets, Lie groups, and smooth maps between
manifolds. Topics in tangent spaces, including differential and rank of a smooth map, regular level set theorem (implicit function theorem), vector fields, integral curves, and the Lie algebra of a
Lie group. Topics in differential forms and integration, including: wedge product, pullback of forms, exterior derivatives, orientation, integral of an n-form, and Stokes' theorem. Prerequisites:
Math 135, Math 136, Math 145, and graduate standing; or permission of instructor.
MATH 286 Differential Geometry. Topics on Riemannian Manifolds, including Riemannian metric, curves
and surfaces in three dimensions, affine connections, and Theorema Egregium. Connections and curvature using differential forms, geodesics, the exponential map, distance and volume, Gauss–Bonnet
Theorem, and the De Rham Cohmology. Additional topics as time permits. Prerequisites: Math 285; or permission of instructor.
MATH 287 Lie Groups. Real and complex Lie groups, relations between Lie groups and Lie algebras, exponential map, the adjoint representation, homogeneous manifolds, semisimplicity, maximal tori, root
space decompositions, compact forms, Cartan decompositions, and the classification of simple Lie algebras. Prerequisites: Math 285; or permission of instructor.
MATH 290 Graduate Special Topics. A special topics course in any generic field of Mathematics. Topics change
from year to year, and the course may be taken more than once for credit.
MATH 291 Graduate Development Seminar. Graduate-student-led training for teaching and public speaking about mathematics, and other professional skills. May involve group discussions with faculty
about pedagogy and communication of research level mathematics to a broader audience. Intended for first-year PhD students in Mathematics. Meets once a week for 75 minutes. Students are also
expected to attend either a research seminar or a colloquium each week, when offered. Math 291 is offered in the Fall and Math 292 in the Spring. Recommendations: PhD standing or consent of
MATH 292 Graduate Development Seminar. Graduate-student-led training for teaching and public speaking about mathematics, and other professional skills. May involve group discussions with faculty
about pedagogy and communication of research level mathematics to a broader audience. Intended for first-year PhD students in Mathematics. Meets once a week for 75 minutes. Students are also
expected to attend either a research seminar or a colloquium each week, when offered. Math 291 is offered in the Fall and Math 292 in the Spring. Recommendations: PhD standing or consent of
MATH 293 One-on-One Course. Guided individual study of an approved topic.
MATH 294 Internship in Mathematics. Study of approved topics in Mathematics in concert with an internship in a related outside the University. Prerequisites: Graduate Standing and Permission of
MATH 295 Master’s Thesis I. Guided research on a topic that has been approved as suitable for a master's thesis.
MATH 296 Master’s Thesis II. Guided research on a topic that has been approved as suitable for a master's thesis.
MATH 297 PhD Thesis I. Guided research on a topic suitable for a doctoral dissertation.
MATH 298 PhD Thesis II. Guided research on a topic suitable for a doctoral dissertation.
MATH 401 Master's Continuation, Part-time. No description at this time.
MATH 402 Master's Continuation, Full-time. No description at this time.
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NFA And DFA - My Online Vidhya
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NFA and DFA
NFA and DFA are two types of automata used in theoretical computer science to model and recognize formal languages.
NFA stands for “nondeterministic finite automaton”. An NFA is a mathematical model that consists of a finite set of states, a set of input symbols, a transition function that maps a state and an
input symbol to a set of possible next states, a designated start state, and a set of accepting states. The main difference between an NFA and a DFA is that an NFA is allowed to have multiple
possible next states for a given input symbol and current state. This nondeterminism can make NFAs more powerful than DFAs in some cases, but it also makes them more difficult to analyze and
DFA stands for “deterministic finite automaton”. A DFA is a mathematical model that is similar to an NFA, but with one key difference: for each state and input symbol, there is exactly one possible
next state. This determinism makes DFAs easier to understand and implement than NFAs, and allows them to be analyzed more easily using formal methods. However, this determinism also makes DFAs less
expressive than NFAs, which means that there are some languages that can be recognized by an NFA but not by any DFA.
Examples of languages that can be recognized by an NFA but not by any DFA:
1. L = {0^n 1^n | n ≥ 0}. This language consists of all strings of the form 0^n1^n, where n is a nonnegative integer. This language can be recognized by an NFA, but it cannot be recognized by any
2. L = {w | w contains an equal number of 0s and 1s}. This language consists of all strings that contain the same number of 0s and 1s. This language can be recognized by an NFA, but it cannot be
recognized by any DFA.
3. L = {ww^R | w is a string over some alphabet, w^R is the reverse of w}. This language consists of all strings that are palindromes. This language can be recognized by an NFA, but it cannot be
recognized by any DFA.
Examples of languages that can be recognized by both an NFA and a DFA:
1. L = {0, 1}. This language consists of the strings “0” and “1”. This language can be recognized by both an NFA and a DFA, because it is a very simple language that does not require any special
features like nondeterminism.
2. L = {w | w is a string of 0s and 1s that contains the substring “101”}. This language consists of all strings that contain the substring “101”. This language can be recognized by both an NFA and
a DFA, because it can be represented by a relatively simple automaton that does not require any nondeterminism.
3. L = {w | w is a string of balanced parentheses}. This language consists of all strings of balanced parentheses, like “()”, “()()”, “(())”, and so on. This language can be recognized by both an
NFA and a DFA, because it can be represented by an automaton that uses a stack to keep track of the balance of parentheses.
First, let’s consider the language L = {0^n 1^n | n ≥ 0}. This language consists of all strings of the form 0^n1^n, where n is a nonnegative integer. For example, the strings “01”, “0011”, and
“000111” are in L, but the strings “001”, “011”, and “00011011” are not. This language can be recognized by an NFA, but it cannot be recognized by any DFA.
The reason for this is that the language requires the automaton to keep track of the number of 0s it has seen, in order to ensure that there are the same number of 1s. This requires the use of
nondeterminism, because the automaton needs to be able to remember multiple possible values for the number of 0s it has seen so far.
Next, let’s consider the language L = {w | w contains an equal number of 0s and 1s}. This language consists of all strings that contain the same number of 0s and 1s. For example, the strings “01”,
“0011”, “000111”, and “101010” are in L, but the strings “001”, “011”, and “00011011” are not.
This language can be recognized by an NFA, but it cannot be recognized by any DFA. The reason for this is similar to the previous example: the automaton needs to keep track of the number of 0s and 1s
it has seen, and there are multiple possible values for these counts at any given point in the input. This requires the use of nondeterminism.
Finally, let’s consider the language L = {ww^R | w is a string over some alphabet, w^R is the reverse of w}. This language consists of all strings that are palindromes. For example, the strings
“racecar”, “aba”, and “aa” are in L, but the strings “hello”, “abc”, and “abbaa” are not. This language can be recognized by an NFA, but it cannot be recognized by any DFA.
The reason for this is that palindromes can be of arbitrary length, and an automaton that recognizes palindromes needs to be able to “remember” the beginning of the string in order to compare it with
the end. This requires the use of nondeterminism, because the automaton needs to be able to guess where the middle of the string is and then check whether the beginning and end match up.
Construct a NFA and DFA state diagram that recognizes the following language : L = {w | w is a palindrome }
To construct the NFA and DFA state diagrams that recognize the language L = {w | w is a palindrome}, we can follow these steps:
1. Start by creating a state for each letter in the input alphabet ∑, which we will assume to be {0, 1} for simplicity.
2. Add an additional state to represent the start and end of the input string, which we will call q0.
3. Connect each letter state to itself with a transition on the corresponding letter in the input alphabet.
4. Connect the start/end state q0 to each letter state with a transition on the empty string ε.
5. Add additional transitions to handle the possibility of the input string being of odd length, by connecting each letter state to a “mirror” state on the opposite side of the start/end state q0.
6. Mark the mirror states as accepting states, since they represent palindromes.
With these steps in mind, here are the NFA and DFA state diagrams:
NFA state diagram:
q0 --------> a0
| 0/ε |
| |
| 1/ε |
v v
a1 ----------> a2
In this diagram, the start/end state q0 connects to both letter states a0 and a1 on the empty string ε. Each letter state has transitions on both 0 and 1, either looping back to itself or
transitioning to its mirror state on the opposite side of q0. The mirror states are represented by a2, which is marked as an accepting state since it represents a palindrome.
DFA state diagram:
------> a0 <------
| |
v v
a1 <------ q0 ----> a2*
| |
v v
------> a3 <------
In this diagram, the start/end state q0 connects to two letter states, a1 and a2*. The states a0, a1, a2*, and a3 represent the four possible “halves” of a palindrome, depending on whether the input
string length is odd or even. Each letter state has transitions on both 0 and 1, either looping back to itself or transitioning to the next state in the sequence. The state a2* is marked as an
accepting state, since it represents a palindrome.
One thought on “NFA and DFA”
1. Greetings! Very useful advice in this particular article! Its the little changes that will make the biggest changes. Thanks for sharing! | {"url":"https://myonlinevidhya.com/nfa-and-dfa/","timestamp":"2024-11-12T06:29:07Z","content_type":"text/html","content_length":"163729","record_id":"<urn:uuid:e4f8e701-3ea6-4910-8957-0405bac86e32>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00668.warc.gz"} |
To launch the demonstration, click on the large button in the top-right corner of this page labelled "Demo". After a little while, two windows should appear; one will be labelled "Controls" and the
other "Triangles".
In the "Triangles" window, you will see a triangle and it's corresponding midpoint triangle. You can drag the blue vertex of the triangle around; the areas of the two triangles and their ratio will
automatically be calculated and displayed in the "Controls" window.
In addition, it is possible to move the other two points of the triangle by typing in their coordinates (in the boxes labelled "A" and "B") in the "Controls" window.
You may notice that the areas are sometimes negative. This is because we are calculating the oriented area. If you are not familiar with this concept, you may want to read the introduction to
oriented area. | {"url":"http://www.math.brown.edu/tbanchof/midpoint/demos/trianglearea.html","timestamp":"2024-11-11T10:33:53Z","content_type":"text/html","content_length":"7814","record_id":"<urn:uuid:aa6c2260-1b3b-487f-900b-c7fb8635a259>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00859.warc.gz"} |
The expected returns earned from investment - Global Essay Writers
The expected returns earned from investment in the stock of two companies, Company A and Company B, are shown in the following table. Use the table to complete parts (a) through (c) below.
Demand for Product Probability of Demand Expected Return: Stock A Expected Return: Stock B
Strong 0.3 40% 20%
Normal 0.45 20% 5%
Weak 0.25 0% (5%)
(a) Compute the expected rates of return for each stock.
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(b) Compute the standard deviations for each stock.
(c) Compute the coefficient of variation for each stock. Based on the coefficient of variation, which stock has the higher risk for investment?
2. The expected returns earned from investment in the stock of two companies, Company A and Company B, are shown in the following table. Assume a two-stock portfolio with $25,000 in Company A and
$75,000 in Company B. Compute the expected return on the portfolio.
Demand for Product Probability of Demand Expected Return: Stock A Expected Return: Stock B
Strong 0.3 40% 20%
Normal 0.45 20% 5%
Weak 0.25 0% (5%)
3. Suppose you have a portfolio consisting of three stocks. You invest a total of $200,000 in the stocks. The investments and beta for the stocks are shown in the following table. Use the table to
complete parts (a) through (c) below.
Stock Investment Beta
1 $60,000 1.25
2 $40,000 (0.5)
3 $100,000 1.5
(a) Assume the risk-free rate is 5.5% and the expected return for the market is 10%. Estimate the appropriate required rate of return for each stock.
(b) Compute the portfolio beta.
(c) Find the portfolio’s required rate of return, assuming the same risk-free rate and expected return for the market as in part (a).
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A New Approach to Optimization
Optimizations in a traditional compiler are applied sequentially, with each optimization destructively modifying the program to produce a transformed program that is then passed to the next
optimization. We present a new approach for structuring the optimization phase of a compiler. In our approach, optimizations take the form of equality analyses that add equality information to a
common intermediate representation. The optimizer works by repeatedly applying these analyses to infer equivalences between program fragments, thus saturating the intermediate representation with
equalities. Once saturated, the intermediate representation encodes multiple optimized versions of the input program. At this point, a profitability heuristic picks the final optimized program from
the various programs represented in the saturated representation. Our proposed way of structuring optimizers has a variety of benefits over previous approaches: our approach obviates the need to
worry about optimization ordering, enables the use of a global optimization heuristic that selects among fully optimized programs, and can be used to perform translation validation, even on compilers
other than our own. We present our approach, formalize it, and describe our choice of intermediate representation. We also present experimental results showing that our approach is practical in terms
of time and space overhead, is effective at discovering intricate optimization opportunities, and is effective at performing translation validation for a realistic optimizer.
Emergent Optimizations
Emergent optimizations are advanced optimizations derived from simple rules. They are a frequent occurence in our approach, so for no additional effort Peggy finds many advanced classical
optimizations we never had to explicitly code. Because they are not explicityly coded, emergent optimizations are often non-classical. An example of this is what we have called "Partial Inlining",
where the equality information provided by the inliner is exploited without electing to inline the method. This is particularly common for functions where the stateful component is complex but the
returned value is a simple function of the parameters, or sometimes vice-versa. The optimized program will still call the function for its stateful effect, but will have inlined and optimized the
return value.
Program Expression Graphs
Program expression graphs (PEGs) are an intermediate representation we designed specifically for equality reasoning. PEGs are a complete representation of a function, making it unnecessary to keep
the original control flow graph. PEGs are referentially transparent, allowing common subexpressions to be identified and allowing equalities to propagate via congruence closure. PEGs have no
intermediate variables, which turns out to be a subtly crucial property since it enables optimizations to apply across would-be block boundaries with the same ease as within blocks, giving rise to
emergent branch and loop optimizations. Simply the process of converting to and from PEGs produces optimizations such as constant propagation, copy propagation, common subexpression elimination,
partial redundancy elimination, unused assignment elimination, unreachable code elimination, branch fusion, loop fusion, loop invariant branch hoisting and sinking, loop invariant code hoisting and
sinnking, and loop unswitching. An E-PEG is a PEG with equality annotation edges between PEG nodes representing the same value. E-PEGs are the representation used by our equality saturation engine in
our optimization and translation validation processes.
Reducing Blow-Up
Although our equality saturation has exponential blow-up, the effect is reduced dramatically by the suprising amount of redundancy in this process. For example, there are an infinite number of ways
to express a+b+c+d+0 using monoid axioms, 1800 of which have "0" occur once or fewer times. E-PEGs need only 86 nodes divided into 16 equivalence classes, and these also express all equivalent
subexpressions. This difference grows dramatically as you increase the number of variables. Even when there is exponential or infinite growth, our implementation processes new expressions in a
breadth-first manner. This prevents our search from getting trapped in a rat hole, fairly distributing its exploration across the entire PEG.
Reversion Choices
Jason Riedy reminded me of a research avenue. When converting a PEG back to a CFG, we make a lot of choices. To summarize them: we compute a value at most once and only when we absolutely need to,
and we fuse as much as we can. We made the first choice because we wanted to revert with as little information as possible about the language, including which operations are safe. This forces us to
only compute a value when we have to, and only computing a value at most once leads to partial redundancy elimination. The second choice leads to branch and loop fusion regardless of the sizes of the
pertinent blocks. From a PEG it tends to be very obvious which branchs and loops can be split or fused. It would be interesting to explore heuristics which, given a PEG to revert to a CFG, suggests
which branches and loops to split or fuse, which values to redundantly recompute, and which values to compute even when it may never be used. These issues seem to be orthogonal to the equality
inference stage, although they would have some implications for our cost model used to select the optimal PEG.
Loop Unrolling
Loop unrolling is an optimization which highlights one of the shortcomings of PEGs, although not necessarily of equality saturation in general. It has to do with the bigger problem of loop
reindexing. In PEGs, a loop value is the sequence of non-loop values made as the loop iterates. Loop unrolling, then, constructs a new sequence consisting of the values at even loop iteration counts.
Loop peeling is the original sequence minus the first value. In our current implemetation, we have an operation for constructing the peeled sequence, which we use in both equality saturation and PEG
reversion. For equality saturation, we include axioms such as peel(θ(A,B)) = B. It would be interesting to apply the same approach with loop unrolling using operators like even and odd, with axioms
such as even(θ(A,B)) = θ(A,odd(B)) and odd(θ(A,B)) = even(B). This process would wrap up nicely in the same way loop-induction-variable strength reduction wraps up. The possible problem I foresee is
not the explosion but how this may interact with the global profitability heuristic. Yet another avenue to explore eventually. Thanks again to Jason Riedy for bringing this up.
Questions, Comments, and Suggestions
If you have any questions, comments, or suggestions, please e-mail us. We would be glad to know of any opinions people have or any clarifications we should make. | {"url":"https://www.cs.cornell.edu/~ross/publications/eqsat/","timestamp":"2024-11-03T16:51:50Z","content_type":"text/html","content_length":"10398","record_id":"<urn:uuid:0e4d1384-39c8-4077-996c-b680d535d760>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00840.warc.gz"} |
Which situation describes the mileage of a car as 40 miles per gallon of gasoline?  A. A car can travel 2 miles on 80 gallons of gasoline.  B. A car can travel 42 miles on 2 gallons of gasoline.  C.
Which situation describes the mileage of a car as 40 miles per gallon of gasoline? A. A car can travel 2 miles on 80 gallons of gasoline. B. A car can travel 42 miles on 2 gallons of gasoline.
C. A car can travel 80 miles on 2 gallons of gasoline. D. A car can travel 80 miles on 40 gallons of gasoline.
Find an answer to your question 👍 “Which situation describes the mileage of a car as 40 miles per gallon of gasoline? A. A car can travel 2 miles on 80 gallons of ...” in 📗 Mathematics if the
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Vec2 | 8th Wall
Interface representing a 2D vector. A 2D vector is represented by (x, y) coordinates, and can represent a point in a plane, a directional vector, or other types of data with three ordered dimensions.
Vec2 objects are created with the ecs.math.vec2 Vec2Factory, or through operations on other Vec2 objects.
Code Example
const {vec2} = ecs.math
const a = vec2.xy(1, 2)
const b = vec2.scale(3) // b is 3, 3
const c = a.plus(b).times(vec2.xy(5, 4)).normalize() // c is (1, 1)
The Vec2Source interface represents any object that has x and y properties and hence can be used as a data source to create a Vec2. In addition, Vec2Source can be used as an argument to Vec2
algorithms, meaning that any object with {x: number, y: number} properties can be used.
Vec2Source has the following enumerable properties:
readonly x: number Access the x component of the vector.
readonly y: number Access the y component of the vector.
Vec2 objects are created with the ecs.math.vec2 Vec2Factory, with the following methods:
vec2.from({x, y}: {x: number, y: number}) => Vec2
Create a Vec2 from a Vec2, or other object with x, y properties.
vec2.one: () => Vec2
Create a Vec2 with all elements set to one. This is equivalent to vec2.from({x: 1, y: 1}).
vec2.scale: (s: number) => Vec2
Create a Vec2 with all elements set to the scale value s. This is equivalent to vec2.from({x: s, y: s}).
vec2.xy: (x: number, y: number) => Vec2
Create a Vec2 from x, y values. This is equivalent to vec2.from({x, y}).
vec2.zero: () => Vec2
Create a Vec2 with all components set to zero. This is equivalent to vec2.from({x: 0, y: 0}).
Vec2 has the following enumerable properties:
readonly x: number Access the x component of the vector.
readonly y: number Access the y component of the vector.
Immutable API
The following methods perform computations based on the current value of a Vec2, but do not modify its contents. Methods that return Vec2 types return new objects. Immutable APIs are typically safer,
more readable, and less error-prone than mutable APIs, but may be inefficient in situations where thousands of objects are allocated each frame. In cases where object garbage collection is a
performance concern, prefer the Mutable API (below). | {"url":"https://www.8thwall.com/docs/ja/studio/api/ecs/math/vec2/","timestamp":"2024-11-01T21:58:43Z","content_type":"text/html","content_length":"63247","record_id":"<urn:uuid:40392971-a5bb-4aba-bb5a-0c48972ae47f>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00053.warc.gz"} |
Leasing Interest - Online leasing
This article is not intended to replace the textbooks on Financial Mathematics. Our task is a more practical one. Here we will try to bring some clarity into a topic rarely debated, quite confused
and suspected by many as a “conspiracy” – the topic of the leasing interest.
Where does the mystery of leasing interest come from?
The first reason is the lack of legal requirements, today and in the past, for the disclosure of leasing interest or any other form of profitability by lessors. The logic “if they do not ask, I will
not tell,” is popular among lessors around the world. Accordingly, many of the lease agreements do not even mention interest and interest payments at all. Rather they regulate only leasing
installments. Some confusion arises from the fact that monthly lease payments include both interest and a part of the principal repayment.
Another reason for the “mystery” around the lease rates comes from the most popular way of leasing distribution – at the point of sale. Imagine how a standard car dealerSee Supplier... More, who can
talk for hours about the different cars he sells, comments leasing interest rate, advises on the calculation method and expectations interest rate fluctuations over the next five years. Without
wanting to offend the professional car dealers, it is often difficult for them to comment and sometimes even understand a question on leasing interest derived from a lease contract drafted for the
asset they sell. This is the starting point of the suspicion that “they are hiding something”.
The third reason for the “mystery” are the lessors themselves. In a fair marketing effort to present their service as the most competitive, which is available on the market, they sometimes openly
claim that the lease service has “no interest” or introduces easy-to-use concepts such as “leasing appreciation” / popular in Bulgaria through the first year of the new century / or “leasing factor”
/ still very popular in the US/. Both concepts, though easy to calculate, are not exactly “leasing interest rates”, although, as numbers, they are presented as similar percentage or decimal values.
The fact is that not all lessees are financially literate and they find it difficult to understand even the most conscious lessorOne of the parties to the... More who tries to explain the floating
rate, the interest rate risk, the interest rate index to which the leasing interest is linked and the possibilities for its fluctuation. In parallel, it should be noted that the formula for
calculating a monthly annuity installment is not the easiest and is difficult to use without a specialized financial calculator or a computer. Accordingly, “in order not to confuse them” leasing
companiesA company whose main activity is... More rarely show the leasing interest to their clients. This again raises doubts about a “conspiracy”.
Last but not least, the reason for the enigma around the leasing interest rates is the competition among the lessors themselves. Leasing interest, as the cost of the leasing service, is an important
parameter for comparing one lease offer with another. But when margins are tight, lessors begin to transfer some of their returns to various other non-interest receivables – lesseeThe lessee is the
user of... More fees, tax concessions, trade commissions, profits from various additional to the leasing services. With the inclusion of all these different sources of income, the boundaries of the
lease rates are greatly diluted. This is how “interest-free” leasing is achieved as well as leasing advertisements for rates significantly lower than the currently prevailing market interest rates /
bank loans and other similar services/.
In the absence of information on leasing interest, foul play is possible by lessors and charge of unacceptably high rates. The legislator tries to protect lessees from unethical practices by lessors.
In Bulgaria (as well as in most European countries) there is regulated protection for individuals (the logic is that it is the most difficult for them to find their way in the labyrinth of lease
rates). The term “Annual Percentage of Costs” is introduced, which somewhat brings together various sources of income for lessors, such as leasing interest and leasing fees. APC is far from perfect
because it includes elements such as “insurance premium” and cannot include others such as “trade discounts”. APR rather adds to the “mystery” with diversity among lease rates.
Who needs leasing interest?
One of the favorite phrases of Milton Friedman is that “there is no such thing as a free lunch“.
The same is true for leasing too. The use of the leasing assetLease asset is the subject of... More by the lesseeThe lessee is the user of... More needs to be compensated.
The payments under the leasing contract always have two component parts:
1. Payment of the amortization of the leasing assetLease asset is the subject of... More
2. Payment of the financial interest
For example, if a lesseeThe lessee is the user of... More receives a vehicle worth 12,000, he pays an initial installment of 2,000, and for the remainder, he contracts for a full payout lease (no
residual valueThe residual value is present in... More) for 10 months, then the depreciation component in the monthly installments will be 1,000 / 10,000 divided by 10 /. Thus, the asset will be
financially depreciated at the end of the lease – the tenth month in our example. Obviously, the lessorOne of the parties to the... More would have recovered the funds invested at the start of the
lease, but that would be a weak motivation for him. He also needs the second component – the payment of the financial costs, ie. payment of interest for the use of the financial resource, which was
blocked in the leasing assetLease asset is the subject of... More. This is exactly the leasing interest.
The leasing interest is the main source of profit for the leasing companiesA company whose main activity is... More and is an absolute analog to the credit interest with bank credits.
How to calculate the monthly leasing installment using the leasing interest
The short answer is – the monthly leasing installment can be computed using the following formula:
MI is the monthly installment
LA is leasing amount
RV is residual valueThe residual value is present in... More
r is the monthly interest rate of the leasing interest
n in the number of payments
An alternative way for computation has been automated by Microsoft using the built-in formulas in Excel nor net present value /NPV/ and internal rate of return /IRR/.
Now you know from experience why do the lessors often use simpler alternatives for calculation of the monthly installments, such as “leasing appreciation” or “leasing factor”, which were mentioned
Monthly interest explained
Let us start with the easiest formula for simple interest:
I=P * r * t
I is the nominal amount of interest
Р is the principal /the amount which was leased, the so-called “leasing amount” /
r is the annual interest rate
t is the number of year of the leasing
This is the formula for the “simple flat rate”.
If interest was paid at the end of the period /hence “simple”/ and the leasing had a tenor of one year with a 5% leasing interest rate and a principal of 10,000, also due in bulk at the end of the
period /hence „flat“/, then the leasing interest would be:
This formula and the respective calculations would be quite true for a leasing contract, had the leasing period been one year and the lessorOne of the parties to the... More wanted full pay-back of
the leased value /principal of 10 000/ and interest for the leased resource /500/ payable on the last day of the year.
It is quite obvious, that in real life there are no such transactions like the one described above. At the same time, the example strongly helps us understand the principles of calculation, which we
shall further use.
Leasing appreciation
Due to the relatively complex method for calculation of the monthly installments of a lease contract, the lessors in Bulgaria adopted the term “leasing appreciation”. The leasing appreciation is the
percent with which the leasing transactions increases the price of the leasing assetLease asset is the subject of... More /compared with its purchase price/. If we continue the example from above,
the leasing nominal leasing appreciation is 500 or expressed as a percentage: 500/10000 = 5%. In this example, the leasing appreciation is the same as the leasing interest /we should not forget that
the leasing interest was calculated as a simple flat rate/.
Should the leasing contract be for a longer than one year period, then the above result is divided by the respective number of years. The formula for calculation of leasing appreciation is:
I is the amount of interest for the whole leasing period
Р is the principal /the leasing amount/
t is the number of years of the leasing term
The reasons for using the ratio “leasing appreciation” are two. The first is the relative ease of calculation and explanation /to potential lessees/. The second is more a marketing one – this is a
relatively low percentage ratio /compared with all others/ which can be advertised, simultaneously hiding the real leasing interest, which is considerably higher.
Why is the leasing interest higher?
In our simplified example, we accepted that the payment of the interest and repayment of the principal to occur with a bullet payment at the end of the leasing contract. In reality, all leasing
contracts stipulate repayment in monthly installments, e.g. each month interest and principal is being paid. Therefore with each following payment, the principal decreases and respectively the
interest decreases too /in real terms/.
For example, if the principal is 10,000 and the interest is 5%, and the leasing term is divided into ten equal installments, with the first payment of 1500, the lesseeThe lessee is the user of...
More will be paying 1000 principal and 500 interest /see the formula for simple interest above/. Therefore for the second payment, the remaining principal will be reduced to 9,000, and the interest
for the second payment will decrease to 450. Thus with the second payment of 1500, only 450 will be used for interest payment and the remaining 1050 will be used for principal repayment. In this way,
for the third installment, the principal will have dropped to 8,950 and so on. With each installment, the remaining principal decreases progressively, with the decrease of the interest component.
Now, let us bring our example a little closer to reality. Here is how the two ratios will look for 12-month leasing: 10% annual interest, 10,000 principal, 10% initial installment and equal leasing
installments with 0 residual valueThe residual value is present in... More:
In order to calculate the leasing amount, we deduct the initial installment. Thus the leasing amount /the principal/ will be 10,000 – (10,000*10%) = 9,000
The monthly installment will be 494.88 payable 12 times
The total interest paid for the whole period will be 494.88
The annual leasing appreciation will be 5.50%, with the leasing interest rate being 10%.
The example emphasizes the second reason /further to the ease of calculation/ for marketing of the leasing appreciation – its value is almost half of the value of the leasing interest rate.
Although it does not carry much information and hinders the comparison with other financial products /e.g. bank credits/, the leasing appreciation is still widely used on the Bulgarian market.
Leasing Factor
The Leasing /or Money/ Factor is a little more complex analog of the leasing appreciation, which is widely used throughout the USA, especially in the car leasing industry. The reasons for its
popularity are the same as the ones described above, bringing the popularity of the ration “leasing appreciation”.
Aiming to find an easy way for calculation of the monthly installment or the monthly interest, some unknown, but smart lessorOne of the parties to the... More /or car dealerSee Supplier... More/ from
the computer-less-past discovered, that if one calculates the monthly interest on the average outstanding leasing amount /the average principal due under the leasing contract/ he will get an amount
which is extremely close to the average leasing interest calculated using the complicated method as described above. Here are the calculations for our example:
The interest rate for one month is the annual interest rate divided by 12 or:
Then the monthly leasing interest is calculated as simple interest on the average outstanding leasing amount /principal/:
or in our example the average monthly interest will be:
and the annual interest will amount to 37.5*12 = 450. This amount is very close to the one as calculated using the complex formula above.
For even greater facility of the car dealers, the leasing companiesA company whose main activity is... More simply forward to them the so-called „leasing factor“ or „money factor“, which is
Leasing Calculators
With the increased use of computers in our lives, the leasing installments are calculated from specialized leasing programs. Often for the use by future lessees, the leasing companiesA company whose
main activity is... More provide online calculators, where the leasing interest rate is seldom mentioned, but the readers know already that it is always present and due with each leasing installment.
In the online leasing calculators instead of quoting the leasing interest rate, it is incorporated in the different leasing programs or is asset-specific – one interest rate for cars, another for
equipment, etc.
Here are a few examples of good online leasing calculators:
Leasing calculator of UBB Interlease: https://interlease.bg/calculator.php
Leasing calculator of SoGeLease: http://www.sogelease.bg/bg/instrumenti/kalkulator.html
Leasing calculator of Raiffeisen Leasing: https://www.rlbg.bg/bg/
Leasing calculator of Addventure Leasing: addventure.leasing/calculator
Leasing Interest for Operational Leasing
It is quite clear that all the above methods for calculation of leasing interest are applicable for financial leasingA leasing contract, under which, the... More. Here the lessorOne of the parties to
the... More purchases the leasing assetLease asset is the subject of... More for a specific purchase price, then grants it to the lesseeThe lessee is the user of... More for a specific term, during
which the lesseeThe lessee is the user of... More returns the invested amount /leasing amount or principal/ as well as the interest for the time during which the investment was blocked in the leasing
assetLease asset is the subject of... More. How then should one calculate the interest for an operational lease contract?
If under financial leasingA leasing contract, under which, the... More the disclosure of the leasing interest is rare, under operational leasing this never occurs. All lessors talk about “rent”,
“rent installment”, “additional service packages” and never mention the word “interest”.
Although it seems complicated it really is quite simple. It helps if we review the grading of the different types of leasing:
Under a closed end financial leasingA leasing contract, under which, the... More contract the lessorOne of the parties to the... More invests in an asset and expects the return in full of his
investment by the lesseeThe lessee is the user of... More, together with the respective interest due until the full repayment of the investment.
Under the open ended financial leasingA leasing contract, under which, the... More, the lessorOne of the parties to the... More invest in an asset and expects the repayment only partially of the
invested amount plus interest by the lesseeThe lessee is the user of... More. The full return of the investment will happen upon the payment of the residual valueThe residual value is present in...
More – be it by the lesseeThe lessee is the user of... More (if she should use the purchase option), be it from a third party (should the lesseeThe lessee is the user of... More simply return the
leasing assetLease asset is the subject of... More).
Under the net operational leasing (without any additional services) the calculations are the same, as with open ended financial leasingA leasing contract, under which, the... More. The difference
being that under operational leasing the asset will certainly be remarketed to a third party (the lesseeThe lessee is the user of... More does not have the right to acquire the leasing assetLease
asset is the subject of... More.
It is rather often when the operational leasing includes some additional services. With cars, for instance, these may include insurance, road fees, taxes, maintenance, consumables, tires, return
vehicle and even fuel. How should one calculate the interest on these?
Again this is not very complicated. If in all examples above the initial investment was a single upfront payment in the beginning of the leasing period, under operational leasing with additional
services, the investment will be made more or less although the life of the leasing contract. Further to the purchase of the leasing assetLease asset is the subject of... More in the beginning of the
leasing transaction, the lessorOne of the parties to the... More will make additional payments through the leasing period – for insurance, for repairs, for consumables, for tires, etc. The latter
smaller investments, just like the main investment in the leasing assetLease asset is the subject of... More, also need to be compensated to the lessorOne of the parties to the... More in the leasing
installments, including the respective interest upon them for the respective holding period.
Since the lessorOne of the parties to the... More needs to determine a specific rent installment upfront, for the operational leasing contract, he will have to make educated assumptions for all
service expenses based on the millage of the car, the price of the insurance, etc. Now you know why operational leasing contracts have limits on the allowed annual millage.
The calculation of the leasing interest for operational lease contracts with additional services /without the use of specialized software/ can be made easily in Excel, using the function for Internal
Rate of Return /which in this case is the leasing interest/.
Other Sources of Revenue for Lessors
The leasing market is strongly competitive. Further to many competing lessors, the interest rates are curbed also by competing products, such as the bank credits. This intense competition usually
leads to the decrease of the interest margins for the leasing companiesA company whose main activity is... More. Aiming at being profitable, lessors seek other alternative sources of revenue, which
are circumstantial to the leasing transactions.
The most obvious are the captiveCaptive Leasing Company is a company... More leasing companiesA company whose main activity is... More. Their main purpose is not so much a financial profit, as
support to the mother-dealer and his sales. Here one can often come across a lesing company, which is selling its service at interest rate levels equal to, and sometimes bellow, the price of funding.
CaptiveCaptive Leasing Company is a company... More leasing companiesA company whose main activity is... More often offer “interest free leasing” or package additional free services in the leasing
transaction. As anyone can guess, the price of these services is subsidized by the mother-dealer, who decreases part of his profit for the respective promotion. The leasing interest exist in the
interest-free leasing too. The only difference is the party which will pay it.
Another, very easy to spot alternative source of revenue for the lessors are the various fees, which they charge for various “services” – „fee for document handling“, „leasing administration fee“,
„fee for the registration of the leasing assetLease asset is the subject of... More“, „fee for …“. All these fees increase the positive cash flow to the leasing companiesA company whose main activity
is... More and compliment the revenues from the interest rate margin.
Another serious source of revenue is the management of the market risk of the residual valueThe residual value is present in... More in leasing transactions which have a residual valueThe residual
value is present in... More. For example /from an actual leasing deal/ if a new car has been leased for 3 years with allowed annual 15,000 km. mileage, full insurance cover and residual valueThe
residual value is present in... More of 30%, it should be clear to everyone that the lessorOne of the parties to the... More is ready to make an additional revenue of at least 10% from the purchase
price of the car upon remarketing of the residual valueThe residual value is present in... More. If the lesseeThe lessee is the user of... More accepts this – the transaction and the additional
revenue will happen.
One very popular in the world, but not often found in Bulgaria, additional source of revenue for the leasing companiesA company whose main activity is... More are tax allowances. Not that the legal
framework is unfavorable in Bulgaria, the reason is that the local leasing market is still immature for their utilization.
Yet another additional source of additional income are the various penalty fees and interest. Some of these are so high, that they belittle the revenue generated by the leasing interest. For example,
the penalty for exceeding the allowed mileage under an operational leasing contract can reach BGN 1.00 for each kilometer above the contracted limit – a penalty which exceeds the price for taxi
services. The penalty annual interest rates often reach a three digit number /even with today’s low interest rate levels/.
Other additional sources of revenue for lessors may include trade discounts from the vendors of the leasing assets, insurance commissions from the insurance companies, commissions from various
sub-contractors for services linked with the leasing transaction. Leasing companiesA company whose main activity is... More should not be deemed as greedy. These additional revenue streams help the
leasing companiesA company whose main activity is... More to:
1. Exist on the market /even with negligible interest margins/;
2. To offer better terms than bank credits;
3. To underwrite bigger client risk than banks;
4. To offer various promotions to the lessees;
5. To invest in better customer service for their clients.
If need be, all these additional revenue streams to the leasing transactions may also be calculated as “leasing interest” or “leasing return”. All which is necessary is to add the respective positive
cash payments on their respective dates in the life of the leasing contract /similar to the negative cash payments for the services under operational leasing, as described above/.
We do hope that we have shed some light on the leasing interest topic which should now be a bit less “mysterious”.
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Graffiti and Math
Cartesian Coordinates
Graffiti artists often work on a piece in sketchbooks before they actually begin painting it. The sketchbooks sometimes use a grid to help plan out the design. Sometimes they go beyond planning, and
create visual effects that look as if they were stretching or folding the grid. In the picture on the right, you can see someone who actually shows the folded grid, although this is rare. More
commonly, you might see graffiti writers use the actual brickwork as a grid itself, as we can see in this picture below.
Whether it's a grid in a sketchbook, or a grid of bricks on a wall, these grids are much like the Cartesian coordinate system in mathematics. Let's see how we can map the Graffiti artist's grid onto
the Cartesian coordinate system.
Coordinates and Lines
In the Graffiti Grapher software we will use Cartesian coordinates to locate the start and finish of each line. Each coordinate is a pair of numbers. The X coordinate tells you how far left or right.
The Y coordinate tells you how far up or down. Cartesian coordinates use both negative and positive numbers, so don't forget to use the "-" sign.
The most important lines in Graffiti Grapher are the borders. Every two borders gives you a "shape." A collection of shapes is called a "group." Here is a group of two shapes. Shape 1 has green
borders, shape 2 has red borders.
Polar Coordinates
Now that you've seen how to indicate location based on a pair of values (Cartesian Coordinates), it's time to look at a different way to express location on a grid - Polar Coordinates. Polar
Coordinates use an angle and a distance from a center point (known as the origin) to determine location.
With an origin consisting of x and y coordinates, a distance r, and an angle a, you describe a polar coordinate.
Arcs and Polar Coordinates
Using polar coordinates, we can draw the curved shape of an arc. Using the go to methods we discussed before, you can set the orgin of the arc anywhere. Then we can define the diameter of the arc,
and the angle of the arc. For example, if sweep is 15 degrees, then the arc will go from 0 to 15, and if the sweep is 360 it will make a full circle.
Arcs and Spirals
Some of the curves in graffiti are arcs of spirals. The radius for the arc of a circle never changes, but the radius of a spiral arc changes as the arc moves from one endpoint to another. Depending
on how quickly the radius changes we call the arc different things.
Some arcs change radius by a factor of about 3 (2.7), we call these log spirals. These spirals grow at a rate controlled by a value C. They have a starting radius of “size” and start facing a certain
direction and then go around in an arc until it reaches a given angle. As you draw you may want to change the width of your pen. You can use negative pen growth to shrink the pen, and positive to
make it grow.
Color Picker
When using the pen, you can use the "set pen color" block to change the pen color.
To change the color, simply click the color square, and pick a color from anywhere on the screen (even out side of the color spectrum box!) | {"url":"https://csdt.org/culture/graffiti/math.html","timestamp":"2024-11-09T01:33:29Z","content_type":"text/html","content_length":"10362","record_id":"<urn:uuid:40895801-d9a0-45b6-8b8f-bd10d2f60045>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00128.warc.gz"} |
Simpsons Variables Worksheet Answers
Simpsons Variables Worksheet Answers - He creates three groups of 50 workers each and assigns each. Explain whether the data supports the advertisements claims about its product. He creates two
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Experimental Variables Worksheet Bart Simpson Controls And Variables
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Related Post: | {"url":"https://beoala.website/en/simpsons-variables-worksheet-answers.html","timestamp":"2024-11-09T23:18:00Z","content_type":"text/html","content_length":"33264","record_id":"<urn:uuid:5f9fe0dc-a2ed-4ee8-aa03-478a26a98102>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00430.warc.gz"} |
Analytic Theory of Probability | work by Laplace | Britannica
Analytic Theory of Probability
work by Laplace
Learn about this topic in these articles:
discussed in biography
• In Pierre-Simon, marquis de Laplace
…Théorie analytique des probabilités (Analytic Theory of Probability), first published in 1812, in which he described many of the tools he invented for mathematically predicting the probabilities
that particular events will occur in nature. He applied his theory not only to the ordinary problems of chance but also to…
Read More
normal distribution
• In normal distribution
Pierre-Simon Laplace, in his Théorie analytique des probabilités (1812; “Analytic Theory of Probability”), into the first central limit theorem, which proved that probabilities for almost all
independent and identically distributed random variables converge rapidly (with sample size) to the area under an exponential function—that is, to a normal distribution.…
Read More | {"url":"https://www.britannica.com/topic/Analytic-Theory-of-Probability","timestamp":"2024-11-05T01:31:18Z","content_type":"text/html","content_length":"48589","record_id":"<urn:uuid:93474f78-5391-4eaf-ab05-9df4caba8571>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00627.warc.gz"} |
A (not so) safe betting strategy for winning at roulette
A couple of years ago I was on a trip to Budapest with a couple of friends. While roaming the streets we were passing by a casino and my friend insisted that there was a perfect strategy that would
only lead to winning at roulette tables. Curious as I was I had him explain his theory. The system basically works as follows:
First, you place a coin on red. If red wins, take your winning and start over. Otherwise, you double your bet after every loss, so that the first win would recover all previous losses plus win a
profit equal to the original stake. If there were no constraints this could actually work. I usually get suspicious when I hear “guaranteed wins” in the context of gambling. My first doubt was, that
the chance of getting either red or black were not equally fifty percent since there was also the zero. Another thing is that at a roulette tables there is usually a limit.
I thought the best way to convince my friend that his system was not so perfect as he thought, was to simulate the whole process and show him the outcome.
Our gambler starts with a budget of 1000 coins and bets 1 coin initially. He also has a winning target, that if he reaches it, he withdraws from the table and goes home happily. The table limit is
1200 coins per bet. For simplicity I also assume that if zero comes up it counts as a lost bet.
The graph shows clearly that the higher the gambler sets it target the lower is the probability of reaching it. If you want to play with the system, alter some parameters or extend it to a different
betting strategy here is my code: | {"url":"https://blog.tafkas.net/2009/06/09/a-not-so-safe-betting-strategy-for-winning-at-roulette/","timestamp":"2024-11-13T05:51:49Z","content_type":"text/html","content_length":"24154","record_id":"<urn:uuid:4ff98faa-0e52-4e12-89e8-92455c568bc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00772.warc.gz"} |
Categories, Logic and Physics
Categories, Logic and the Foundations of Physics 1
Imperial College London, Wednesday January 9, 2008
The full schedule and announcement is given here.
12.00-13.00 Chris Isham, Imperial College London
Topos theory in the formulation of theories of physics
Video, Slides
(no abstract provided)
13.00-13.30 Chris Heunen, University of Nijmegen
A topos for algebraic quantum theory
Video, Slides
Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra A induces a topos in which the amalgamation of all its
commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum X in the
topos, which plays the role of a quantum phase space of the system. States on A become probability integrals on X, and self-adjoint elements of A define functions from X to the pertinent internal
real numbers (the interval domain), allowing for a state-proposition pairing. Thus the quantum theory defined by A is turned into a classical theory by restriction to its associated topos.
13.30-14.15 Lunch
14.15-15.15 Samson Abramsky, University of Oxford
Categorical quantum mechanics: The "monoidal" approach
Video, Slides
(no abstract provided)
15.15-15.45 Ross Duncan, University of Oxford
Classical structures, MUBs, and pretty pictures
Video, Slides
(no abstract provided)
15.45-16.30 General discussion session
16.30-17.00 Coffee break
17.00-17.30 Jamie Vicary, Imperial College London
A categorical framework for the quantum harmonic oscillator
Video, Slides
I will describe a categorical approach to the construction of symmetric Fock space, the state space of the quantum harmonic oscillator. Many of the conventional mathematical tools used to study this
system — such as raising and lowering operators, and coherent states — emerge naturally from the category theory, and satisfy the usual equations. However, the formalism is more general than the
conventional approach, and I will describe how to construct an infinite variety of 'exotic' Fock spaces. I will finish with the question: "Where has the 'quantumness' come from?"
17.30-18.00 Louis Crane, Kansas State University
Relational topology and quantum gravity
We explore arguments for replacing the absolute point set by a sheaf over the site of observation as a foundation for quantum gravity. Time permitting, we consider apparent geometry as a formulation
for relational geometry.
18.15-19.00 Pub session | {"url":"http://categorieslogicphysics.wikidot.com/meeting1","timestamp":"2024-11-14T17:27:51Z","content_type":"application/xhtml+xml","content_length":"26221","record_id":"<urn:uuid:28094f72-6cbc-4521-b388-b56d2d0b34ee>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00305.warc.gz"} |
Utilities - OpenZeppelin Docs
This document is better viewed at https://docs.openzeppelin.com/contracts/api/utils
Miscellaneous contracts and libraries containing utility functions you can use to improve security, work with new data types, or safely use low-level primitives.
Because Solidity does not support generic types, EnumerableMap and EnumerableSet are specialized to a limited number of key-value types.
import "@openzeppelin/contracts/utils/math/Math.sol";
Standard math utilities missing in the Solidity language.
tryAdd(uint256 a, uint256 b) → bool success, uint256 result internal
Returns the addition of two unsigned integers, with an success flag (no overflow).
trySub(uint256 a, uint256 b) → bool success, uint256 result internal
Returns the subtraction of two unsigned integers, with an success flag (no overflow).
tryMul(uint256 a, uint256 b) → bool success, uint256 result internal
Returns the multiplication of two unsigned integers, with an success flag (no overflow).
tryDiv(uint256 a, uint256 b) → bool success, uint256 result internal
Returns the division of two unsigned integers, with a success flag (no division by zero).
tryMod(uint256 a, uint256 b) → bool success, uint256 result internal
Returns the remainder of dividing two unsigned integers, with a success flag (no division by zero).
ternary(bool condition, uint256 a, uint256 b) → uint256 internal
Branchless ternary evaluation for a ? b : c. Gas costs are constant.
This function may reduce bytecode size and consume less gas when used standalone. However, the compiler may optimize Solidity ternary operations (i.e. a ? b : c) to only compute one branch when
needed, making this function more expensive.
average(uint256 a, uint256 b) → uint256 internal
Returns the average of two numbers. The result is rounded towards zero.
ceilDiv(uint256 a, uint256 b) → uint256 internal
Returns the ceiling of the division of two numbers.
This differs from standard division with / in that it rounds towards infinity instead of rounding towards zero.
mulDiv(uint256 x, uint256 y, uint256 denominator) → uint256 result internal
Calculates floor(x * y / denominator) with full precision. Throws if result overflows a uint256 or denominator == 0.
mulDiv(uint256 x, uint256 y, uint256 denominator, enum Math.Rounding rounding) → uint256 internal
Calculates x * y / denominator with full precision, following the selected rounding direction.
invMod(uint256 a, uint256 n) → uint256 internal
Calculate the modular multiplicative inverse of a number in Z/nZ.
If n is a prime, then Z/nZ is a field. In that case all elements are inversible, except 0. If n is not a prime, then Z/nZ is not a field, and some elements might not be inversible.
If the input value is not inversible, 0 is returned.
If you know for sure that n is (big) a prime, it may be cheaper to use Fermat’s little theorem and get the inverse using Math.modExp(a, n - 2, n). See invModPrime.
invModPrime(uint256 a, uint256 p) → uint256 internal
Variant of invMod. More efficient, but only works if p is known to be a prime greater than 2.
From Fermat’s little theorem, we know that if p is prime, then a(p-1) ≡ 1 mod p. As a consequence, we have a * a(p-2) ≡ 1 mod p, which means that a**(p-2) is the modular multiplicative inverse of a
in Fp.
this function does NOT check that p is a prime greater than 2.
modExp(uint256 b, uint256 e, uint256 m) → uint256 internal
Returns the modular exponentiation of the specified base, exponent and modulus (b ** e % m)
Requirements: - modulus can’t be zero - underlying staticcall to precompile must succeed
The result is only valid if the underlying call succeeds. When using this function, make sure the chain you’re using it on supports the precompiled contract for modular exponentiation at address
0x05 as specified in EIP-198. Otherwise, the underlying function will succeed given the lack of a revert, but the result may be incorrectly interpreted as 0.
tryModExp(uint256 b, uint256 e, uint256 m) → bool success, uint256 result internal
Returns the modular exponentiation of the specified base, exponent and modulus (b ** e % m). It includes a success flag indicating if the operation succeeded. Operation will be marked as failed if
trying to operate modulo 0 or if the underlying precompile reverted.
The result is only valid if the success flag is true. When using this function, make sure the chain you’re using it on supports the precompiled contract for modular exponentiation at address 0x05 as
specified in EIP-198. Otherwise, the underlying function will succeed given the lack of a revert, but the result may be incorrectly interpreted as 0.
modExp(bytes b, bytes e, bytes m) → bytes internal
Variant of modExp that supports inputs of arbitrary length.
tryModExp(bytes b, bytes e, bytes m) → bool success, bytes result internal
Variant of tryModExp that supports inputs of arbitrary length.
sqrt(uint256 a) → uint256 internal
Returns the square root of a number. If the number is not a perfect square, the value is rounded towards zero.
This method is based on Newton’s method for computing square roots; the algorithm is restricted to only using integer operations.
sqrt(uint256 a, enum Math.Rounding rounding) → uint256 internal
Calculates sqrt(a), following the selected rounding direction.
log2(uint256 value) → uint256 internal
Return the log in base 2 of a positive value rounded towards zero. Returns 0 if given 0.
log2(uint256 value, enum Math.Rounding rounding) → uint256 internal
Return the log in base 2, following the selected rounding direction, of a positive value. Returns 0 if given 0.
log10(uint256 value) → uint256 internal
Return the log in base 10 of a positive value rounded towards zero. Returns 0 if given 0.
log10(uint256 value, enum Math.Rounding rounding) → uint256 internal
Return the log in base 10, following the selected rounding direction, of a positive value. Returns 0 if given 0.
log256(uint256 value) → uint256 internal
Return the log in base 256 of a positive value rounded towards zero. Returns 0 if given 0.
Adding one to the result gives the number of pairs of hex symbols needed to represent value as a hex string.
log256(uint256 value, enum Math.Rounding rounding) → uint256 internal
Return the log in base 256, following the selected rounding direction, of a positive value. Returns 0 if given 0.
import "@openzeppelin/contracts/utils/math/SignedMath.sol";
Standard signed math utilities missing in the Solidity language.
ternary(bool condition, int256 a, int256 b) → int256 internal
Branchless ternary evaluation for a ? b : c. Gas costs are constant.
This function may reduce bytecode size and consume less gas when used standalone. However, the compiler may optimize Solidity ternary operations (i.e. a ? b : c) to only compute one branch when
needed, making this function more expensive.
average(int256 a, int256 b) → int256 internal
Returns the average of two signed numbers without overflow. The result is rounded towards zero.
import "@openzeppelin/contracts/utils/math/SafeCast.sol";
Wrappers over Solidity’s uintXX/intXX/bool casting operators with added overflow checks.
Downcasting from uint256/int256 in Solidity does not revert on overflow. This can easily result in undesired exploitation or bugs, since developers usually assume that overflows raise errors.
SafeCast restores this intuition by reverting the transaction when such an operation overflows.
Using this library instead of the unchecked operations eliminates an entire class of bugs, so it’s recommended to use it always.
toUint248(uint256 value) → uint248 internal
Returns the downcasted uint248 from uint256, reverting on overflow (when the input is greater than largest uint248).
Counterpart to Solidity’s uint248 operator.
• input must fit into 248 bits
toUint240(uint256 value) → uint240 internal
Returns the downcasted uint240 from uint256, reverting on overflow (when the input is greater than largest uint240).
Counterpart to Solidity’s uint240 operator.
• input must fit into 240 bits
toUint232(uint256 value) → uint232 internal
Returns the downcasted uint232 from uint256, reverting on overflow (when the input is greater than largest uint232).
Counterpart to Solidity’s uint232 operator.
• input must fit into 232 bits
toUint224(uint256 value) → uint224 internal
Returns the downcasted uint224 from uint256, reverting on overflow (when the input is greater than largest uint224).
Counterpart to Solidity’s uint224 operator.
• input must fit into 224 bits
toUint216(uint256 value) → uint216 internal
Returns the downcasted uint216 from uint256, reverting on overflow (when the input is greater than largest uint216).
Counterpart to Solidity’s uint216 operator.
• input must fit into 216 bits
toUint208(uint256 value) → uint208 internal
Returns the downcasted uint208 from uint256, reverting on overflow (when the input is greater than largest uint208).
Counterpart to Solidity’s uint208 operator.
• input must fit into 208 bits
toUint200(uint256 value) → uint200 internal
Returns the downcasted uint200 from uint256, reverting on overflow (when the input is greater than largest uint200).
Counterpart to Solidity’s uint200 operator.
• input must fit into 200 bits
toUint192(uint256 value) → uint192 internal
Returns the downcasted uint192 from uint256, reverting on overflow (when the input is greater than largest uint192).
Counterpart to Solidity’s uint192 operator.
• input must fit into 192 bits
toUint184(uint256 value) → uint184 internal
Returns the downcasted uint184 from uint256, reverting on overflow (when the input is greater than largest uint184).
Counterpart to Solidity’s uint184 operator.
• input must fit into 184 bits
toUint176(uint256 value) → uint176 internal
Returns the downcasted uint176 from uint256, reverting on overflow (when the input is greater than largest uint176).
Counterpart to Solidity’s uint176 operator.
• input must fit into 176 bits
toUint168(uint256 value) → uint168 internal
Returns the downcasted uint168 from uint256, reverting on overflow (when the input is greater than largest uint168).
Counterpart to Solidity’s uint168 operator.
• input must fit into 168 bits
toUint160(uint256 value) → uint160 internal
Returns the downcasted uint160 from uint256, reverting on overflow (when the input is greater than largest uint160).
Counterpart to Solidity’s uint160 operator.
• input must fit into 160 bits
toUint152(uint256 value) → uint152 internal
Returns the downcasted uint152 from uint256, reverting on overflow (when the input is greater than largest uint152).
Counterpart to Solidity’s uint152 operator.
• input must fit into 152 bits
toUint144(uint256 value) → uint144 internal
Returns the downcasted uint144 from uint256, reverting on overflow (when the input is greater than largest uint144).
Counterpart to Solidity’s uint144 operator.
• input must fit into 144 bits
toUint136(uint256 value) → uint136 internal
Returns the downcasted uint136 from uint256, reverting on overflow (when the input is greater than largest uint136).
Counterpart to Solidity’s uint136 operator.
• input must fit into 136 bits
toUint128(uint256 value) → uint128 internal
Returns the downcasted uint128 from uint256, reverting on overflow (when the input is greater than largest uint128).
Counterpart to Solidity’s uint128 operator.
• input must fit into 128 bits
toUint120(uint256 value) → uint120 internal
Returns the downcasted uint120 from uint256, reverting on overflow (when the input is greater than largest uint120).
Counterpart to Solidity’s uint120 operator.
• input must fit into 120 bits
toUint112(uint256 value) → uint112 internal
Returns the downcasted uint112 from uint256, reverting on overflow (when the input is greater than largest uint112).
Counterpart to Solidity’s uint112 operator.
• input must fit into 112 bits
toUint104(uint256 value) → uint104 internal
Returns the downcasted uint104 from uint256, reverting on overflow (when the input is greater than largest uint104).
Counterpart to Solidity’s uint104 operator.
• input must fit into 104 bits
toUint96(uint256 value) → uint96 internal
Returns the downcasted uint96 from uint256, reverting on overflow (when the input is greater than largest uint96).
Counterpart to Solidity’s uint96 operator.
• input must fit into 96 bits
toUint88(uint256 value) → uint88 internal
Returns the downcasted uint88 from uint256, reverting on overflow (when the input is greater than largest uint88).
Counterpart to Solidity’s uint88 operator.
• input must fit into 88 bits
toUint80(uint256 value) → uint80 internal
Returns the downcasted uint80 from uint256, reverting on overflow (when the input is greater than largest uint80).
Counterpart to Solidity’s uint80 operator.
• input must fit into 80 bits
toUint72(uint256 value) → uint72 internal
Returns the downcasted uint72 from uint256, reverting on overflow (when the input is greater than largest uint72).
Counterpart to Solidity’s uint72 operator.
• input must fit into 72 bits
toUint64(uint256 value) → uint64 internal
Returns the downcasted uint64 from uint256, reverting on overflow (when the input is greater than largest uint64).
Counterpart to Solidity’s uint64 operator.
• input must fit into 64 bits
toUint56(uint256 value) → uint56 internal
Returns the downcasted uint56 from uint256, reverting on overflow (when the input is greater than largest uint56).
Counterpart to Solidity’s uint56 operator.
• input must fit into 56 bits
toUint48(uint256 value) → uint48 internal
Returns the downcasted uint48 from uint256, reverting on overflow (when the input is greater than largest uint48).
Counterpart to Solidity’s uint48 operator.
• input must fit into 48 bits
toUint40(uint256 value) → uint40 internal
Returns the downcasted uint40 from uint256, reverting on overflow (when the input is greater than largest uint40).
Counterpart to Solidity’s uint40 operator.
• input must fit into 40 bits
toUint32(uint256 value) → uint32 internal
Returns the downcasted uint32 from uint256, reverting on overflow (when the input is greater than largest uint32).
Counterpart to Solidity’s uint32 operator.
• input must fit into 32 bits
toUint24(uint256 value) → uint24 internal
Returns the downcasted uint24 from uint256, reverting on overflow (when the input is greater than largest uint24).
Counterpart to Solidity’s uint24 operator.
• input must fit into 24 bits
toUint16(uint256 value) → uint16 internal
Returns the downcasted uint16 from uint256, reverting on overflow (when the input is greater than largest uint16).
Counterpart to Solidity’s uint16 operator.
• input must fit into 16 bits
toUint8(uint256 value) → uint8 internal
Returns the downcasted uint8 from uint256, reverting on overflow (when the input is greater than largest uint8).
Counterpart to Solidity’s uint8 operator.
• input must fit into 8 bits
toUint256(int256 value) → uint256 internal
Converts a signed int256 into an unsigned uint256.
• input must be greater than or equal to 0.
toInt248(int256 value) → int248 downcasted internal
Returns the downcasted int248 from int256, reverting on overflow (when the input is less than smallest int248 or greater than largest int248).
Counterpart to Solidity’s int248 operator.
• input must fit into 248 bits
toInt240(int256 value) → int240 downcasted internal
Returns the downcasted int240 from int256, reverting on overflow (when the input is less than smallest int240 or greater than largest int240).
Counterpart to Solidity’s int240 operator.
• input must fit into 240 bits
toInt232(int256 value) → int232 downcasted internal
Returns the downcasted int232 from int256, reverting on overflow (when the input is less than smallest int232 or greater than largest int232).
Counterpart to Solidity’s int232 operator.
• input must fit into 232 bits
toInt224(int256 value) → int224 downcasted internal
Returns the downcasted int224 from int256, reverting on overflow (when the input is less than smallest int224 or greater than largest int224).
Counterpart to Solidity’s int224 operator.
• input must fit into 224 bits
toInt216(int256 value) → int216 downcasted internal
Returns the downcasted int216 from int256, reverting on overflow (when the input is less than smallest int216 or greater than largest int216).
Counterpart to Solidity’s int216 operator.
• input must fit into 216 bits
toInt208(int256 value) → int208 downcasted internal
Returns the downcasted int208 from int256, reverting on overflow (when the input is less than smallest int208 or greater than largest int208).
Counterpart to Solidity’s int208 operator.
• input must fit into 208 bits
toInt200(int256 value) → int200 downcasted internal
Returns the downcasted int200 from int256, reverting on overflow (when the input is less than smallest int200 or greater than largest int200).
Counterpart to Solidity’s int200 operator.
• input must fit into 200 bits
toInt192(int256 value) → int192 downcasted internal
Returns the downcasted int192 from int256, reverting on overflow (when the input is less than smallest int192 or greater than largest int192).
Counterpart to Solidity’s int192 operator.
• input must fit into 192 bits
toInt184(int256 value) → int184 downcasted internal
Returns the downcasted int184 from int256, reverting on overflow (when the input is less than smallest int184 or greater than largest int184).
Counterpart to Solidity’s int184 operator.
• input must fit into 184 bits
toInt176(int256 value) → int176 downcasted internal
Returns the downcasted int176 from int256, reverting on overflow (when the input is less than smallest int176 or greater than largest int176).
Counterpart to Solidity’s int176 operator.
• input must fit into 176 bits
toInt168(int256 value) → int168 downcasted internal
Returns the downcasted int168 from int256, reverting on overflow (when the input is less than smallest int168 or greater than largest int168).
Counterpart to Solidity’s int168 operator.
• input must fit into 168 bits
toInt160(int256 value) → int160 downcasted internal
Returns the downcasted int160 from int256, reverting on overflow (when the input is less than smallest int160 or greater than largest int160).
Counterpart to Solidity’s int160 operator.
• input must fit into 160 bits
toInt152(int256 value) → int152 downcasted internal
Returns the downcasted int152 from int256, reverting on overflow (when the input is less than smallest int152 or greater than largest int152).
Counterpart to Solidity’s int152 operator.
• input must fit into 152 bits
toInt144(int256 value) → int144 downcasted internal
Returns the downcasted int144 from int256, reverting on overflow (when the input is less than smallest int144 or greater than largest int144).
Counterpart to Solidity’s int144 operator.
• input must fit into 144 bits
toInt136(int256 value) → int136 downcasted internal
Returns the downcasted int136 from int256, reverting on overflow (when the input is less than smallest int136 or greater than largest int136).
Counterpart to Solidity’s int136 operator.
• input must fit into 136 bits
toInt128(int256 value) → int128 downcasted internal
Returns the downcasted int128 from int256, reverting on overflow (when the input is less than smallest int128 or greater than largest int128).
Counterpart to Solidity’s int128 operator.
• input must fit into 128 bits
toInt120(int256 value) → int120 downcasted internal
Returns the downcasted int120 from int256, reverting on overflow (when the input is less than smallest int120 or greater than largest int120).
Counterpart to Solidity’s int120 operator.
• input must fit into 120 bits
toInt112(int256 value) → int112 downcasted internal
Returns the downcasted int112 from int256, reverting on overflow (when the input is less than smallest int112 or greater than largest int112).
Counterpart to Solidity’s int112 operator.
• input must fit into 112 bits
toInt104(int256 value) → int104 downcasted internal
Returns the downcasted int104 from int256, reverting on overflow (when the input is less than smallest int104 or greater than largest int104).
Counterpart to Solidity’s int104 operator.
• input must fit into 104 bits
toInt96(int256 value) → int96 downcasted internal
Returns the downcasted int96 from int256, reverting on overflow (when the input is less than smallest int96 or greater than largest int96).
Counterpart to Solidity’s int96 operator.
• input must fit into 96 bits
toInt88(int256 value) → int88 downcasted internal
Returns the downcasted int88 from int256, reverting on overflow (when the input is less than smallest int88 or greater than largest int88).
Counterpart to Solidity’s int88 operator.
• input must fit into 88 bits
toInt80(int256 value) → int80 downcasted internal
Returns the downcasted int80 from int256, reverting on overflow (when the input is less than smallest int80 or greater than largest int80).
Counterpart to Solidity’s int80 operator.
• input must fit into 80 bits
toInt72(int256 value) → int72 downcasted internal
Returns the downcasted int72 from int256, reverting on overflow (when the input is less than smallest int72 or greater than largest int72).
Counterpart to Solidity’s int72 operator.
• input must fit into 72 bits
toInt64(int256 value) → int64 downcasted internal
Returns the downcasted int64 from int256, reverting on overflow (when the input is less than smallest int64 or greater than largest int64).
Counterpart to Solidity’s int64 operator.
• input must fit into 64 bits
toInt56(int256 value) → int56 downcasted internal
Returns the downcasted int56 from int256, reverting on overflow (when the input is less than smallest int56 or greater than largest int56).
Counterpart to Solidity’s int56 operator.
• input must fit into 56 bits
toInt48(int256 value) → int48 downcasted internal
Returns the downcasted int48 from int256, reverting on overflow (when the input is less than smallest int48 or greater than largest int48).
Counterpart to Solidity’s int48 operator.
• input must fit into 48 bits
toInt40(int256 value) → int40 downcasted internal
Returns the downcasted int40 from int256, reverting on overflow (when the input is less than smallest int40 or greater than largest int40).
Counterpart to Solidity’s int40 operator.
• input must fit into 40 bits
toInt32(int256 value) → int32 downcasted internal
Returns the downcasted int32 from int256, reverting on overflow (when the input is less than smallest int32 or greater than largest int32).
Counterpart to Solidity’s int32 operator.
• input must fit into 32 bits
toInt24(int256 value) → int24 downcasted internal
Returns the downcasted int24 from int256, reverting on overflow (when the input is less than smallest int24 or greater than largest int24).
Counterpart to Solidity’s int24 operator.
• input must fit into 24 bits
toInt16(int256 value) → int16 downcasted internal
Returns the downcasted int16 from int256, reverting on overflow (when the input is less than smallest int16 or greater than largest int16).
Counterpart to Solidity’s int16 operator.
• input must fit into 16 bits
toInt8(int256 value) → int8 downcasted internal
Returns the downcasted int8 from int256, reverting on overflow (when the input is less than smallest int8 or greater than largest int8).
Counterpart to Solidity’s int8 operator.
• input must fit into 8 bits
toInt256(uint256 value) → int256 internal
Converts an unsigned uint256 into a signed int256.
• input must be less than or equal to maxInt256.
toUint(bool b) → uint256 u internal
Cast a boolean (false or true) to a uint256 (0 or 1) with no jump.
SafeCastOverflowedUintDowncast(uint8 bits, uint256 value) error
Value doesn’t fit in an uint of bits size.
SafeCastOverflowedIntDowncast(uint8 bits, int256 value) error
Value doesn’t fit in an int of bits size.
import "@openzeppelin/contracts/utils/cryptography/ECDSA.sol";
Elliptic Curve Digital Signature Algorithm (ECDSA) operations.
These functions can be used to verify that a message was signed by the holder of the private keys of a given address.
tryRecover(bytes32 hash, bytes signature) → address recovered, enum ECDSA.RecoverError err, bytes32 errArg internal
Returns the address that signed a hashed message (hash) with signature or an error. This will not return address(0) without also returning an error description. Errors are documented using an enum
(error type) and a bytes32 providing additional information about the error.
If no error is returned, then the address can be used for verification purposes.
The ecrecover EVM precompile allows for malleable (non-unique) signatures: this function rejects them by requiring the s value to be in the lower half order, and the v value to be either 27 or 28.
hash must be the result of a hash operation for the verification to be secure: it is possible to craft signatures that recover to arbitrary addresses for non-hashed data. A safe way to ensure this
is by receiving a hash of the original message (which may otherwise be too long), and then calling MessageHashUtils.toEthSignedMessageHash on it.
Documentation for signature generation: - with Web3.js - with ethers
recover(bytes32 hash, bytes signature) → address internal
Returns the address that signed a hashed message (hash) with signature. This address can then be used for verification purposes.
The ecrecover EVM precompile allows for malleable (non-unique) signatures: this function rejects them by requiring the s value to be in the lower half order, and the v value to be either 27 or 28.
hash must be the result of a hash operation for the verification to be secure: it is possible to craft signatures that recover to arbitrary addresses for non-hashed data. A safe way to ensure this
is by receiving a hash of the original message (which may otherwise be too long), and then calling MessageHashUtils.toEthSignedMessageHash on it.
tryRecover(bytes32 hash, bytes32 r, bytes32 vs) → address recovered, enum ECDSA.RecoverError err, bytes32 errArg internal
Overload of ECDSA.tryRecover that receives the r and vs short-signature fields separately.
recover(bytes32 hash, bytes32 r, bytes32 vs) → address internal
Overload of ECDSA.recover that receives the r and `vs short-signature fields separately.
tryRecover(bytes32 hash, uint8 v, bytes32 r, bytes32 s) → address recovered, enum ECDSA.RecoverError err, bytes32 errArg internal
Overload of ECDSA.tryRecover that receives the v, r and s signature fields separately.
recover(bytes32 hash, uint8 v, bytes32 r, bytes32 s) → address internal
Overload of ECDSA.recover that receives the v, r and s signature fields separately.
import "@openzeppelin/contracts/utils/cryptography/P256.sol";
Implementation of secp256r1 verification and recovery functions.
The secp256r1 curve (also known as P256) is a NIST standard curve with wide support in modern devices and cryptographic standards. Some notable examples include Apple’s Secure Enclave and Android’s
Keystore as well as authentication protocols like FIDO2.
verify(bytes32 h, bytes32 r, bytes32 s, bytes32 qx, bytes32 qy) → bool internal
Verifies a secp256r1 signature using the RIP-7212 precompile and falls back to the Solidity implementation if the precompile is not available. This version should work on all chains, but requires the
deployment of more bytecode.
verifyNative(bytes32 h, bytes32 r, bytes32 s, bytes32 qx, bytes32 qy) → bool internal
Same as verify, but it will revert if the required precompile is not available.
Make sure any logic (code or precompile) deployed at that address is the expected one, otherwise the returned value may be misinterpreted as a positive boolean.
verifySolidity(bytes32 h, bytes32 r, bytes32 s, bytes32 qx, bytes32 qy) → bool internal
Same as verify, but only the Solidity implementation is used.
recovery(bytes32 h, uint8 v, bytes32 r, bytes32 s) → bytes32 x, bytes32 y internal
isValidPublicKey(bytes32 x, bytes32 y) → bool result internal
Checks if (x, y) are valid coordinates of a point on the curve. In particular this function checks that x < P and y < P.
import "@openzeppelin/contracts/utils/cryptography/RSA.sol";
RSA PKCS#1 v1.5 signature verification implementation according to RFC8017.
This library supports PKCS#1 v1.5 padding to avoid malleability via chosen plaintext attacks in practical implementations. The padding follows the EMSA-PKCS1-v1_5-ENCODE encoding definition as per
section 9.2 of the RFC. This padding makes RSA semantically secure for signing messages.
pkcs1Sha256(bytes data, bytes s, bytes e, bytes n) → bool internal
Same as pkcs1Sha256 but using SHA256 to calculate the digest of data.
pkcs1Sha256(bytes32 digest, bytes s, bytes e, bytes n) → bool internal
Verifies a PKCSv1.5 signature given a digest according to the verification method described in section 8.2.2 of RFC8017 with support for explicit or implicit NULL parameters in the DigestInfo (no
other optional parameters are supported).
For security reason, this function requires the signature and modulus to have a length of at least 2048 bits. If you use a smaller key, consider replacing it with a larger, more secure, one.
This verification algorithm doesn’t prevent replayability. If called multiple times with the same digest, public key and (valid signature), it will return true every time. Consider including an
onchain nonce or unique identifier in the message to prevent replay attacks.
This verification algorithm supports any exponent. NIST recommends using 65537 (or higher). That is the default value many libraries use, such as OpenSSL. Developers may choose to reject public keys
using a low exponent out of security concerns.
import "@openzeppelin/contracts/utils/cryptography/EIP712.sol";
EIP-712 is a standard for hashing and signing of typed structured data.
The encoding scheme specified in the EIP requires a domain separator and a hash of the typed structured data, whose encoding is very generic and therefore its implementation in Solidity is not
feasible, thus this contract does not implement the encoding itself. Protocols need to implement the type-specific encoding they need in order to produce the hash of their typed data using a
combination of abi.encode and keccak256.
This contract implements the EIP-712 domain separator (_domainSeparatorV4) that is used as part of the encoding scheme, and the final step of the encoding to obtain the message digest that is then
signed via ECDSA (_hashTypedDataV4).
The implementation of the domain separator was designed to be as efficient as possible while still properly updating the chain id to protect against replay attacks on an eventual fork of the chain.
This contract implements the version of the encoding known as "v4", as implemented by the JSON RPC method eth_signTypedDataV4 in MetaMask.
In the upgradeable version of this contract, the cached values will correspond to the address, and the domain separator of the implementation contract. This will cause the _domainSeparatorV4
function to always rebuild the separator from the immutable values, which is cheaper than accessing a cached version in cold storage.
constructor(string name, string version) internal
Initializes the domain separator and parameter caches.
The meaning of name and version is specified in EIP-712:
• name: the user readable name of the signing domain, i.e. the name of the DApp or the protocol.
• version: the current major version of the signing domain.
These parameters cannot be changed except through a smart contract upgrade.
_hashTypedDataV4(bytes32 structHash) → bytes32 internal
Given an already hashed struct, this function returns the hash of the fully encoded EIP712 message for this domain.
This hash can be used together with ECDSA.recover to obtain the signer of a message. For example:
bytes32 digest = _hashTypedDataV4(keccak256(abi.encode(
keccak256("Mail(address to,string contents)"),
address signer = ECDSA.recover(digest, signature);
eip712Domain() → bytes1 fields, string name, string version, uint256 chainId, address verifyingContract, bytes32 salt, uint256[] extensions public
_EIP712Name() → string internal
The name parameter for the EIP712 domain.
By default this function reads _name which is an immutable value. It only reads from storage if necessary (in case the value is too large to fit in a ShortString).
import "@openzeppelin/contracts/utils/cryptography/MessageHashUtils.sol";
Signature message hash utilities for producing digests to be consumed by ECDSA recovery or signing.
The library provides methods for generating a hash of a message that conforms to the ERC-191 and EIP 712 specifications.
toEthSignedMessageHash(bytes32 messageHash) → bytes32 digest internal
Returns the keccak256 digest of an ERC-191 signed data with version 0x45 (personal_sign messages).
The digest is calculated by prefixing a bytes32 messageHash with "\x19Ethereum Signed Message:\n32" and hashing the result. It corresponds with the hash signed when using the eth_sign JSON-RPC
The messageHash parameter is intended to be the result of hashing a raw message with keccak256, although any bytes32 value can be safely used because the final digest will be re-hashed.
toEthSignedMessageHash(bytes message) → bytes32 internal
Returns the keccak256 digest of an ERC-191 signed data with version 0x45 (personal_sign messages).
The digest is calculated by prefixing an arbitrary message with "\x19Ethereum Signed Message:\n" + len(message) and hashing the result. It corresponds with the hash signed when using the eth_sign
JSON-RPC method.
toDataWithIntendedValidatorHash(address validator, bytes data) → bytes32 internal
Returns the keccak256 digest of an ERC-191 signed data with version 0x00 (data with intended validator).
The digest is calculated by prefixing an arbitrary data with "\x19\x00" and the intended validator address. Then hashing the result.
toTypedDataHash(bytes32 domainSeparator, bytes32 structHash) → bytes32 digest internal
Returns the keccak256 digest of an EIP-712 typed data (ERC-191 version 0x01).
The digest is calculated from a domainSeparator and a structHash, by prefixing them with \x19\x01 and hashing the result. It corresponds to the hash signed by the eth_signTypedData JSON-RPC method as
part of EIP-712.
import "@openzeppelin/contracts/utils/cryptography/SignatureChecker.sol";
Signature verification helper that can be used instead of ECDSA.recover to seamlessly support both ECDSA signatures from externally owned accounts (EOAs) as well as ERC-1271 signatures from smart
contract wallets like Argent and Safe Wallet (previously Gnosis Safe).
isValidSignatureNow(address signer, bytes32 hash, bytes signature) → bool internal
Checks if a signature is valid for a given signer and data hash. If the signer is a smart contract, the signature is validated against that smart contract using ERC-1271, otherwise it’s validated
using ECDSA.recover.
Unlike ECDSA signatures, contract signatures are revocable, and the outcome of this function can thus change through time. It could return true at block N and false at block N+1 (or the opposite).
isValidERC1271SignatureNow(address signer, bytes32 hash, bytes signature) → bool internal
Checks if a signature is valid for a given signer and data hash. The signature is validated against the signer smart contract using ERC-1271.
Unlike ECDSA signatures, contract signatures are revocable, and the outcome of this function can thus change through time. It could return true at block N and false at block N+1 (or the opposite).
import "@openzeppelin/contracts/utils/cryptography/Hashes.sol";
Library of standard hash functions.
commutativeKeccak256(bytes32 a, bytes32 b) → bytes32 internal
Commutative Keccak256 hash of a sorted pair of bytes32. Frequently used when working with merkle proofs.
Equivalent to the standardNodeHash in our JavaScript library.
import "@openzeppelin/contracts/utils/cryptography/MerkleProof.sol";
These functions deal with verification of Merkle Tree proofs.
The tree and the proofs can be generated using our JavaScript library. You will find a quickstart guide in the readme.
You should avoid using leaf values that are 64 bytes long prior to hashing, or use a hash function other than keccak256 for hashing leaves. This is because the concatenation of a sorted pair of
internal nodes in the Merkle tree could be reinterpreted as a leaf value. OpenZeppelin’s JavaScript library generates Merkle trees that are safe against this attack out of the box.
Consider memory side-effects when using custom hashing functions that access memory in an unsafe way.
This library supports proof verification for merkle trees built using custom commutative hashing functions (i.e. H(a, b) == H(b, a)). Proving leaf inclusion in trees built using non-commutative
hashing functions requires additional logic that is not supported by this library.
verify(bytes32[] proof, bytes32 root, bytes32 leaf) → bool internal
Returns true if a leaf can be proved to be a part of a Merkle tree defined by root. For this, a proof must be provided, containing sibling hashes on the branch from the leaf to the root of the tree.
Each pair of leaves and each pair of pre-images are assumed to be sorted.
This version handles proofs in memory with the default hashing function.
processProof(bytes32[] proof, bytes32 leaf) → bytes32 internal
Returns the rebuilt hash obtained by traversing a Merkle tree up from leaf using proof. A proof is valid if and only if the rebuilt hash matches the root of the tree. When processing the proof, the
pairs of leaves & pre-images are assumed to be sorted.
This version handles proofs in memory with the default hashing function.
verify(bytes32[] proof, bytes32 root, bytes32 leaf, function (bytes32,bytes32) view returns (bytes32) hasher) → bool internal
Returns true if a leaf can be proved to be a part of a Merkle tree defined by root. For this, a proof must be provided, containing sibling hashes on the branch from the leaf to the root of the tree.
Each pair of leaves and each pair of pre-images are assumed to be sorted.
This version handles proofs in memory with a custom hashing function.
processProof(bytes32[] proof, bytes32 leaf, function (bytes32,bytes32) view returns (bytes32) hasher) → bytes32 internal
Returns the rebuilt hash obtained by traversing a Merkle tree up from leaf using proof. A proof is valid if and only if the rebuilt hash matches the root of the tree. When processing the proof, the
pairs of leaves & pre-images are assumed to be sorted.
This version handles proofs in memory with a custom hashing function.
verifyCalldata(bytes32[] proof, bytes32 root, bytes32 leaf) → bool internal
Returns true if a leaf can be proved to be a part of a Merkle tree defined by root. For this, a proof must be provided, containing sibling hashes on the branch from the leaf to the root of the tree.
Each pair of leaves and each pair of pre-images are assumed to be sorted.
This version handles proofs in calldata with the default hashing function.
processProofCalldata(bytes32[] proof, bytes32 leaf) → bytes32 internal
Returns the rebuilt hash obtained by traversing a Merkle tree up from leaf using proof. A proof is valid if and only if the rebuilt hash matches the root of the tree. When processing the proof, the
pairs of leaves & pre-images are assumed to be sorted.
This version handles proofs in calldata with the default hashing function.
verifyCalldata(bytes32[] proof, bytes32 root, bytes32 leaf, function (bytes32,bytes32) view returns (bytes32) hasher) → bool internal
Returns true if a leaf can be proved to be a part of a Merkle tree defined by root. For this, a proof must be provided, containing sibling hashes on the branch from the leaf to the root of the tree.
Each pair of leaves and each pair of pre-images are assumed to be sorted.
This version handles proofs in calldata with a custom hashing function.
processProofCalldata(bytes32[] proof, bytes32 leaf, function (bytes32,bytes32) view returns (bytes32) hasher) → bytes32 internal
Returns the rebuilt hash obtained by traversing a Merkle tree up from leaf using proof. A proof is valid if and only if the rebuilt hash matches the root of the tree. When processing the proof, the
pairs of leaves & pre-images are assumed to be sorted.
This version handles proofs in calldata with a custom hashing function.
multiProofVerify(bytes32[] proof, bool[] proofFlags, bytes32 root, bytes32[] leaves) → bool internal
Returns true if the leaves can be simultaneously proven to be a part of a Merkle tree defined by root, according to proof and proofFlags as described in processMultiProof.
This version handles multiproofs in memory with the default hashing function.
Consider the case where root == proof[0] && leaves.length == 0 as it will return true. The leaves must be validated independently. See processMultiProof.
processMultiProof(bytes32[] proof, bool[] proofFlags, bytes32[] leaves) → bytes32 merkleRoot internal
Returns the root of a tree reconstructed from leaves and sibling nodes in proof. The reconstruction proceeds by incrementally reconstructing all inner nodes by combining a leaf/inner node with either
another leaf/inner node or a proof sibling node, depending on whether each proofFlags item is true or false respectively.
This version handles multiproofs in memory with the default hashing function.
Not all Merkle trees admit multiproofs. To use multiproofs, it is sufficient to ensure that: 1) the tree is complete (but not necessarily perfect), 2) the leaves to be proven are in the opposite
order they are in the tree (i.e., as seen from right to left starting at the deepest layer and continuing at the next layer).
The empty set (i.e. the case where proof.length == 1 && leaves.length == 0) is considered a no-op, and therefore a valid multiproof (i.e. it returns proof[0]). Consider disallowing this case if
you’re not validating the leaves elsewhere.
multiProofVerify(bytes32[] proof, bool[] proofFlags, bytes32 root, bytes32[] leaves, function (bytes32,bytes32) view returns (bytes32) hasher) → bool internal
Returns true if the leaves can be simultaneously proven to be a part of a Merkle tree defined by root, according to proof and proofFlags as described in processMultiProof.
This version handles multiproofs in memory with a custom hashing function.
Consider the case where root == proof[0] && leaves.length == 0 as it will return true. The leaves must be validated independently. See processMultiProof.
processMultiProof(bytes32[] proof, bool[] proofFlags, bytes32[] leaves, function (bytes32,bytes32) view returns (bytes32) hasher) → bytes32 merkleRoot internal
Returns the root of a tree reconstructed from leaves and sibling nodes in proof. The reconstruction proceeds by incrementally reconstructing all inner nodes by combining a leaf/inner node with either
another leaf/inner node or a proof sibling node, depending on whether each proofFlags item is true or false respectively.
This version handles multiproofs in memory with a custom hashing function.
Not all Merkle trees admit multiproofs. To use multiproofs, it is sufficient to ensure that: 1) the tree is complete (but not necessarily perfect), 2) the leaves to be proven are in the opposite
order they are in the tree (i.e., as seen from right to left starting at the deepest layer and continuing at the next layer).
The empty set (i.e. the case where proof.length == 1 && leaves.length == 0) is considered a no-op, and therefore a valid multiproof (i.e. it returns proof[0]). Consider disallowing this case if
you’re not validating the leaves elsewhere.
multiProofVerifyCalldata(bytes32[] proof, bool[] proofFlags, bytes32 root, bytes32[] leaves) → bool internal
Returns true if the leaves can be simultaneously proven to be a part of a Merkle tree defined by root, according to proof and proofFlags as described in processMultiProof.
This version handles multiproofs in calldata with the default hashing function.
Consider the case where root == proof[0] && leaves.length == 0 as it will return true. The leaves must be validated independently. See processMultiProofCalldata.
processMultiProofCalldata(bytes32[] proof, bool[] proofFlags, bytes32[] leaves) → bytes32 merkleRoot internal
Returns the root of a tree reconstructed from leaves and sibling nodes in proof. The reconstruction proceeds by incrementally reconstructing all inner nodes by combining a leaf/inner node with either
another leaf/inner node or a proof sibling node, depending on whether each proofFlags item is true or false respectively.
This version handles multiproofs in calldata with the default hashing function.
Not all Merkle trees admit multiproofs. To use multiproofs, it is sufficient to ensure that: 1) the tree is complete (but not necessarily perfect), 2) the leaves to be proven are in the opposite
order they are in the tree (i.e., as seen from right to left starting at the deepest layer and continuing at the next layer).
The empty set (i.e. the case where proof.length == 1 && leaves.length == 0) is considered a no-op, and therefore a valid multiproof (i.e. it returns proof[0]). Consider disallowing this case if
you’re not validating the leaves elsewhere.
multiProofVerifyCalldata(bytes32[] proof, bool[] proofFlags, bytes32 root, bytes32[] leaves, function (bytes32,bytes32) view returns (bytes32) hasher) → bool internal
Returns true if the leaves can be simultaneously proven to be a part of a Merkle tree defined by root, according to proof and proofFlags as described in processMultiProof.
This version handles multiproofs in calldata with a custom hashing function.
Consider the case where root == proof[0] && leaves.length == 0 as it will return true. The leaves must be validated independently. See processMultiProofCalldata.
processMultiProofCalldata(bytes32[] proof, bool[] proofFlags, bytes32[] leaves, function (bytes32,bytes32) view returns (bytes32) hasher) → bytes32 merkleRoot internal
Returns the root of a tree reconstructed from leaves and sibling nodes in proof. The reconstruction proceeds by incrementally reconstructing all inner nodes by combining a leaf/inner node with either
another leaf/inner node or a proof sibling node, depending on whether each proofFlags item is true or false respectively.
This version handles multiproofs in calldata with a custom hashing function.
Not all Merkle trees admit multiproofs. To use multiproofs, it is sufficient to ensure that: 1) the tree is complete (but not necessarily perfect), 2) the leaves to be proven are in the opposite
order they are in the tree (i.e., as seen from right to left starting at the deepest layer and continuing at the next layer).
The empty set (i.e. the case where proof.length == 1 && leaves.length == 0) is considered a no-op, and therefore a valid multiproof (i.e. it returns proof[0]). Consider disallowing this case if
you’re not validating the leaves elsewhere.
import "@openzeppelin/contracts/utils/ReentrancyGuard.sol";
Contract module that helps prevent reentrant calls to a function.
Inheriting from ReentrancyGuard will make the nonReentrant modifier available, which can be applied to functions to make sure there are no nested (reentrant) calls to them.
Note that because there is a single nonReentrant guard, functions marked as nonReentrant may not call one another. This can be worked around by making those functions private, and then adding
external nonReentrant entry points to them.
If EIP-1153 (transient storage) is available on the chain you’re deploying at, consider using ReentrancyGuardTransient instead.
If you would like to learn more about reentrancy and alternative ways to protect against it, check out our blog post Reentrancy After Istanbul.
nonReentrant() modifier
Prevents a contract from calling itself, directly or indirectly. Calling a nonReentrant function from another nonReentrant function is not supported. It is possible to prevent this from happening by
making the nonReentrant function external, and making it call a private function that does the actual work.
_reentrancyGuardEntered() → bool internal
Returns true if the reentrancy guard is currently set to "entered", which indicates there is a nonReentrant function in the call stack.
import "@openzeppelin/contracts/utils/ReentrancyGuardTransient.sol";
This variant only works on networks where EIP-1153 is available.
nonReentrant() modifier
Prevents a contract from calling itself, directly or indirectly. Calling a nonReentrant function from another nonReentrant function is not supported. It is possible to prevent this from happening by
making the nonReentrant function external, and making it call a private function that does the actual work.
_reentrancyGuardEntered() → bool internal
Returns true if the reentrancy guard is currently set to "entered", which indicates there is a nonReentrant function in the call stack.
import "@openzeppelin/contracts/utils/Pausable.sol";
Contract module which allows children to implement an emergency stop mechanism that can be triggered by an authorized account.
This module is used through inheritance. It will make available the modifiers whenNotPaused and whenPaused, which can be applied to the functions of your contract. Note that they will not be pausable
by simply including this module, only once the modifiers are put in place.
whenNotPaused() modifier
Modifier to make a function callable only when the contract is not paused.
• The contract must not be paused.
whenPaused() modifier
Modifier to make a function callable only when the contract is paused.
• The contract must be paused.
import "@openzeppelin/contracts/utils/Nonces.sol";
Provides tracking nonces for addresses. Nonces will only increment.
_useNonce(address owner) → uint256 internal
Returns the current value and increments nonce.
_useCheckedNonce(address owner, uint256 nonce) internal
Same as _useNonce but checking that nonce is the next valid for owner.
This set of interfaces and contracts deal with type introspection of contracts, that is, examining which functions can be called on them. This is usually referred to as a contract’s interface.
Ethereum contracts have no native concept of an interface, so applications must usually simply trust they are not making an incorrect call. For trusted setups this is a non-issue, but often unknown
and untrusted third-party addresses need to be interacted with. There may even not be any direct calls to them! (e.g. ERC-20 tokens may be sent to a contract that lacks a way to transfer them out of
it, locking them forever). In these cases, a contract declaring its interface can be very helpful in preventing errors.
import "@openzeppelin/contracts/utils/introspection/IERC165.sol";
Interface of the ERC-165 standard, as defined in the ERC.
Implementers can declare support of contract interfaces, which can then be queried by others (ERC165Checker).
For an implementation, see ERC165.
supportsInterface(bytes4 interfaceId) → bool external
Returns true if this contract implements the interface defined by interfaceId. See the corresponding ERC section to learn more about how these ids are created.
This function call must use less than 30 000 gas.
import "@openzeppelin/contracts/utils/introspection/ERC165.sol";
Implementation of the IERC165 interface.
Contracts that want to implement ERC-165 should inherit from this contract and override supportsInterface to check for the additional interface id that will be supported. For example:
function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
return interfaceId == type(MyInterface).interfaceId || super.supportsInterface(interfaceId);
supportsInterface(bytes4 interfaceId) → bool public
import "@openzeppelin/contracts/utils/introspection/ERC165Checker.sol";
Library used to query support of an interface declared via IERC165.
Note that these functions return the actual result of the query: they do not revert if an interface is not supported. It is up to the caller to decide what to do in these cases.
supportsERC165(address account) → bool internal
Returns true if account supports the IERC165 interface.
supportsInterface(address account, bytes4 interfaceId) → bool internal
Returns true if account supports the interface defined by interfaceId. Support for IERC165 itself is queried automatically.
getSupportedInterfaces(address account, bytes4[] interfaceIds) → bool[] internal
Returns a boolean array where each value corresponds to the interfaces passed in and whether they’re supported or not. This allows you to batch check interfaces for a contract where your expectation
is that some interfaces may not be supported.
supportsAllInterfaces(address account, bytes4[] interfaceIds) → bool internal
Returns true if account supports all the interfaces defined in interfaceIds. Support for IERC165 itself is queried automatically.
Batch-querying can lead to gas savings by skipping repeated checks for IERC165 support.
supportsERC165InterfaceUnchecked(address account, bytes4 interfaceId) → bool internal
Assumes that account contains a contract that supports ERC-165, otherwise the behavior of this method is undefined. This precondition can be checked with supportsERC165.
Some precompiled contracts will falsely indicate support for a given interface, so caution should be exercised when using this function.
Interface identification is specified in ERC-165.
Data Structures
import "@openzeppelin/contracts/utils/structs/BitMaps.sol";
Library for managing uint256 to bool mapping in a compact and efficient way, provided the keys are sequential. Largely inspired by Uniswap’s merkle-distributor.
BitMaps pack 256 booleans across each bit of a single 256-bit slot of uint256 type. Hence booleans corresponding to 256 sequential indices would only consume a single slot, unlike the regular bool
which would consume an entire slot for a single value.
This results in gas savings in two ways:
• Setting a zero value to non-zero only once every 256 times
• Accessing the same warm slot for every 256 sequential indices
get(struct BitMaps.BitMap bitmap, uint256 index) → bool internal
Returns whether the bit at index is set.
setTo(struct BitMaps.BitMap bitmap, uint256 index, bool value) internal
Sets the bit at index to the boolean value.
import "@openzeppelin/contracts/utils/structs/EnumerableMap.sol";
Library for managing an enumerable variant of Solidity’s mapping type.
Maps have the following properties:
• Entries are added, removed, and checked for existence in constant time (O(1)).
• Entries are enumerated in O(n). No guarantees are made on the ordering.
contract Example {
// Add the library methods
using EnumerableMap for EnumerableMap.UintToAddressMap;
// Declare a set state variable
EnumerableMap.UintToAddressMap private myMap;
The following map types are supported:
• uint256 → address (UintToAddressMap) since v3.0.0
• address → uint256 (AddressToUintMap) since v4.6.0
• bytes32 → bytes32 (Bytes32ToBytes32Map) since v4.6.0
• uint256 → uint256 (UintToUintMap) since v4.7.0
• bytes32 → uint256 (Bytes32ToUintMap) since v4.7.0
• uint256 → bytes32 (UintToBytes32Map) since v5.1.0
• address → address (AddressToAddressMap) since v5.1.0
• address → bytes32 (AddressToBytes32Map) since v5.1.0
• bytes32 → address (Bytes32ToAddressMap) since v5.1.0
Trying to delete such a structure from storage will likely result in data corruption, rendering the structure unusable. See ethereum/solidity#11843 for more info.
In order to clean an EnumerableMap, you can either remove all elements one by one or create a fresh instance using an array of EnumerableMap.
set(struct EnumerableMap.Bytes32ToBytes32Map map, bytes32 key, bytes32 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.Bytes32ToBytes32Map map, bytes32 key) → bool internal
Removes a key-value pair from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.Bytes32ToBytes32Map map, bytes32 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.Bytes32ToBytes32Map map) → uint256 internal
Returns the number of key-value pairs in the map. O(1).
at(struct EnumerableMap.Bytes32ToBytes32Map map, uint256 index) → bytes32 key, bytes32 value internal
Returns the key-value pair stored at position index in the map. O(1).
Note that there are no guarantees on the ordering of entries inside the array, and it may change when more entries are added or removed.
tryGet(struct EnumerableMap.Bytes32ToBytes32Map map, bytes32 key) → bool exists, bytes32 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.Bytes32ToBytes32Map map, bytes32 key) → bytes32 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.Bytes32ToBytes32Map map) → bytes32[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.UintToUintMap map, uint256 key, uint256 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.UintToUintMap map, uint256 key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.UintToUintMap map, uint256 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.UintToUintMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.UintToUintMap map, uint256 index) → uint256 key, uint256 value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.UintToUintMap map, uint256 key) → bool exists, uint256 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.UintToUintMap map, uint256 key) → uint256 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.UintToUintMap map) → uint256[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.UintToAddressMap map, uint256 key, address value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.UintToAddressMap map, uint256 key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.UintToAddressMap map, uint256 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.UintToAddressMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.UintToAddressMap map, uint256 index) → uint256 key, address value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.UintToAddressMap map, uint256 key) → bool exists, address value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.UintToAddressMap map, uint256 key) → address internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.UintToAddressMap map) → uint256[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.UintToBytes32Map map, uint256 key, bytes32 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.UintToBytes32Map map, uint256 key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.UintToBytes32Map map, uint256 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.UintToBytes32Map map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.UintToBytes32Map map, uint256 index) → uint256 key, bytes32 value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.UintToBytes32Map map, uint256 key) → bool exists, bytes32 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.UintToBytes32Map map, uint256 key) → bytes32 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.UintToBytes32Map map) → uint256[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.AddressToUintMap map, address key, uint256 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.AddressToUintMap map, address key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.AddressToUintMap map, address key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.AddressToUintMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.AddressToUintMap map, uint256 index) → address key, uint256 value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.AddressToUintMap map, address key) → bool exists, uint256 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.AddressToUintMap map, address key) → uint256 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.AddressToUintMap map) → address[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.AddressToAddressMap map, address key, address value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.AddressToAddressMap map, address key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.AddressToAddressMap map, address key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.AddressToAddressMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.AddressToAddressMap map, uint256 index) → address key, address value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.AddressToAddressMap map, address key) → bool exists, address value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.AddressToAddressMap map, address key) → address internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.AddressToAddressMap map) → address[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.AddressToBytes32Map map, address key, bytes32 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.AddressToBytes32Map map, address key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.AddressToBytes32Map map, address key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.AddressToBytes32Map map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.AddressToBytes32Map map, uint256 index) → address key, bytes32 value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.AddressToBytes32Map map, address key) → bool exists, bytes32 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.AddressToBytes32Map map, address key) → bytes32 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.AddressToBytes32Map map) → address[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.Bytes32ToUintMap map, bytes32 key, uint256 value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.Bytes32ToUintMap map, bytes32 key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.Bytes32ToUintMap map, bytes32 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.Bytes32ToUintMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.Bytes32ToUintMap map, uint256 index) → bytes32 key, uint256 value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.Bytes32ToUintMap map, bytes32 key) → bool exists, uint256 value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.Bytes32ToUintMap map, bytes32 key) → uint256 internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.Bytes32ToUintMap map) → bytes32[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
set(struct EnumerableMap.Bytes32ToAddressMap map, bytes32 key, address value) → bool internal
Adds a key-value pair to a map, or updates the value for an existing key. O(1).
Returns true if the key was added to the map, that is if it was not already present.
remove(struct EnumerableMap.Bytes32ToAddressMap map, bytes32 key) → bool internal
Removes a value from a map. O(1).
Returns true if the key was removed from the map, that is if it was present.
contains(struct EnumerableMap.Bytes32ToAddressMap map, bytes32 key) → bool internal
Returns true if the key is in the map. O(1).
length(struct EnumerableMap.Bytes32ToAddressMap map) → uint256 internal
Returns the number of elements in the map. O(1).
at(struct EnumerableMap.Bytes32ToAddressMap map, uint256 index) → bytes32 key, address value internal
Returns the element stored at position index in the map. O(1). Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
tryGet(struct EnumerableMap.Bytes32ToAddressMap map, bytes32 key) → bool exists, address value internal
Tries to returns the value associated with key. O(1). Does not revert if key is not in the map.
get(struct EnumerableMap.Bytes32ToAddressMap map, bytes32 key) → address internal
Returns the value associated with key. O(1).
keys(struct EnumerableMap.Bytes32ToAddressMap map) → bytes32[] internal
Return the an array containing all the keys
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the map grows to a point where copying to memory
consumes too much gas to fit in a block.
import "@openzeppelin/contracts/utils/structs/EnumerableSet.sol";
Library for managing sets of primitive types.
Sets have the following properties:
• Elements are added, removed, and checked for existence in constant time (O(1)).
• Elements are enumerated in O(n). No guarantees are made on the ordering.
contract Example {
// Add the library methods
using EnumerableSet for EnumerableSet.AddressSet;
// Declare a set state variable
EnumerableSet.AddressSet private mySet;
As of v3.3.0, sets of type bytes32 (Bytes32Set), address (AddressSet) and uint256 (UintSet) are supported.
Trying to delete such a structure from storage will likely result in data corruption, rendering the structure unusable. See ethereum/solidity#11843 for more info.
In order to clean an EnumerableSet, you can either remove all elements one by one or create a fresh instance using an array of EnumerableSet.
add(struct EnumerableSet.Bytes32Set set, bytes32 value) → bool internal
Add a value to a set. O(1).
Returns true if the value was added to the set, that is if it was not already present.
remove(struct EnumerableSet.Bytes32Set set, bytes32 value) → bool internal
Removes a value from a set. O(1).
Returns true if the value was removed from the set, that is if it was present.
contains(struct EnumerableSet.Bytes32Set set, bytes32 value) → bool internal
Returns true if the value is in the set. O(1).
length(struct EnumerableSet.Bytes32Set set) → uint256 internal
Returns the number of values in the set. O(1).
at(struct EnumerableSet.Bytes32Set set, uint256 index) → bytes32 internal
Returns the value stored at position index in the set. O(1).
Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
values(struct EnumerableSet.Bytes32Set set) → bytes32[] internal
Return the entire set in an array
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the set grows to a point where copying to memory
consumes too much gas to fit in a block.
add(struct EnumerableSet.AddressSet set, address value) → bool internal
Add a value to a set. O(1).
Returns true if the value was added to the set, that is if it was not already present.
remove(struct EnumerableSet.AddressSet set, address value) → bool internal
Removes a value from a set. O(1).
Returns true if the value was removed from the set, that is if it was present.
contains(struct EnumerableSet.AddressSet set, address value) → bool internal
Returns true if the value is in the set. O(1).
length(struct EnumerableSet.AddressSet set) → uint256 internal
Returns the number of values in the set. O(1).
at(struct EnumerableSet.AddressSet set, uint256 index) → address internal
Returns the value stored at position index in the set. O(1).
Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
values(struct EnumerableSet.AddressSet set) → address[] internal
Return the entire set in an array
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the set grows to a point where copying to memory
consumes too much gas to fit in a block.
add(struct EnumerableSet.UintSet set, uint256 value) → bool internal
Add a value to a set. O(1).
Returns true if the value was added to the set, that is if it was not already present.
remove(struct EnumerableSet.UintSet set, uint256 value) → bool internal
Removes a value from a set. O(1).
Returns true if the value was removed from the set, that is if it was present.
contains(struct EnumerableSet.UintSet set, uint256 value) → bool internal
Returns true if the value is in the set. O(1).
length(struct EnumerableSet.UintSet set) → uint256 internal
Returns the number of values in the set. O(1).
at(struct EnumerableSet.UintSet set, uint256 index) → uint256 internal
Returns the value stored at position index in the set. O(1).
Note that there are no guarantees on the ordering of values inside the array, and it may change when more values are added or removed.
values(struct EnumerableSet.UintSet set) → uint256[] internal
Return the entire set in an array
This operation will copy the entire storage to memory, which can be quite expensive. This is designed to mostly be used by view accessors that are queried without any gas fees. Developers should
keep in mind that this function has an unbounded cost, and using it as part of a state-changing function may render the function uncallable if the set grows to a point where copying to memory
consumes too much gas to fit in a block.
import "@openzeppelin/contracts/utils/structs/DoubleEndedQueue.sol";
A sequence of items with the ability to efficiently push and pop items (i.e. insert and remove) on both ends of the sequence (called front and back). Among other access patterns, it can be used to
implement efficient LIFO and FIFO queues. Storage use is optimized, and all operations are O(1) constant time. This includes clear, given that the existing queue contents are left in storage.
The struct is called Bytes32Deque. Other types can be cast to and from bytes32. This data structure can only be used in storage, and not in memory.
DoubleEndedQueue.Bytes32Deque queue;
pushBack(struct DoubleEndedQueue.Bytes32Deque deque, bytes32 value) internal
Inserts an item at the end of the queue.
popBack(struct DoubleEndedQueue.Bytes32Deque deque) → bytes32 value internal
Removes the item at the end of the queue and returns it.
pushFront(struct DoubleEndedQueue.Bytes32Deque deque, bytes32 value) internal
Inserts an item at the beginning of the queue.
popFront(struct DoubleEndedQueue.Bytes32Deque deque) → bytes32 value internal
Removes the item at the beginning of the queue and returns it.
front(struct DoubleEndedQueue.Bytes32Deque deque) → bytes32 value internal
Returns the item at the beginning of the queue.
back(struct DoubleEndedQueue.Bytes32Deque deque) → bytes32 value internal
Returns the item at the end of the queue.
at(struct DoubleEndedQueue.Bytes32Deque deque, uint256 index) → bytes32 value internal
Return the item at a position in the queue given by index, with the first item at 0 and last item at length(deque) - 1.
clear(struct DoubleEndedQueue.Bytes32Deque deque) internal
Resets the queue back to being empty.
The current items are left behind in storage. This does not affect the functioning of the queue, but misses out on potential gas refunds.
length(struct DoubleEndedQueue.Bytes32Deque deque) → uint256 internal
Returns the number of items in the queue.
import "@openzeppelin/contracts/utils/structs/CircularBuffer.sol";
A fixed-size buffer for keeping bytes32 items in storage.
This data structure allows for pushing elements to it, and when its length exceeds the specified fixed size, new items take the place of the oldest element in the buffer, keeping at most N elements
in the structure.
Elements can’t be removed but the data structure can be cleared. See clear.
Complexity: - insertion (push): O(1) - lookup (last): O(1) - inclusion (includes): O(N) (worst case) - reset (clear): O(1)
• The struct is called Bytes32CircularBuffer. Other types can be cast to and from bytes32. This data structure can only be used in storage, and not in memory.
contract Example {
// Add the library methods
using CircularBuffer for CircularBuffer.Bytes32CircularBuffer;
// Declare a buffer storage variable
CircularBuffer.Bytes32CircularBuffer private myBuffer;
setup(struct CircularBuffer.Bytes32CircularBuffer self, uint256 size) internal
Initialize a new CircularBuffer of given size.
If the CircularBuffer was already setup and used, calling that function again will reset it to a blank state.
The size of the buffer will affect the execution of includes function, as it has a complexity of O(N). Consider a large buffer size may render the function unusable.
clear(struct CircularBuffer.Bytes32CircularBuffer self) internal
Clear all data in the buffer without resetting memory, keeping the existing size.
push(struct CircularBuffer.Bytes32CircularBuffer self, bytes32 value) internal
Push a new value to the buffer. If the buffer is already full, the new value replaces the oldest value in the buffer.
count(struct CircularBuffer.Bytes32CircularBuffer self) → uint256 internal
Number of values currently in the buffer. This value is 0 for an empty buffer, and cannot exceed the size of the buffer.
length(struct CircularBuffer.Bytes32CircularBuffer self) → uint256 internal
Length of the buffer. This is the maximum number of elements kept in the buffer.
last(struct CircularBuffer.Bytes32CircularBuffer self, uint256 i) → bytes32 internal
Getter for the i-th value in the buffer, from the end.
Reverts with Panic.ARRAY_OUT_OF_BOUNDS if trying to access an element that was not pushed, or that was dropped to make room for newer elements.
includes(struct CircularBuffer.Bytes32CircularBuffer self, bytes32 value) → bool internal
Check if a given value is in the buffer.
import "@openzeppelin/contracts/utils/structs/Checkpoints.sol";
This library defines the Trace* struct, for checkpointing values as they change at different points in time, and later looking up past values by block number. See Votes as an example.
To create a history of checkpoints define a variable type Checkpoints.Trace* in your contract, and store a new checkpoint for the current transaction block using the push function.
push(struct Checkpoints.Trace224 self, uint32 key, uint224 value) → uint224 oldValue, uint224 newValue internal
Pushes a (key, value) pair into a Trace224 so that it is stored as the checkpoint.
Returns previous value and new value.
Never accept key as a user input, since an arbitrary type(uint32).max key set will disable the library.
lowerLookup(struct Checkpoints.Trace224 self, uint32 key) → uint224 internal
Returns the value in the first (oldest) checkpoint with key greater or equal than the search key, or zero if there is none.
upperLookup(struct Checkpoints.Trace224 self, uint32 key) → uint224 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
upperLookupRecent(struct Checkpoints.Trace224 self, uint32 key) → uint224 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
This is a variant of upperLookup that is optimised to find "recent" checkpoint (checkpoints with high keys).
latest(struct Checkpoints.Trace224 self) → uint224 internal
Returns the value in the most recent checkpoint, or zero if there are no checkpoints.
latestCheckpoint(struct Checkpoints.Trace224 self) → bool exists, uint32 _key, uint224 _value internal
Returns whether there is a checkpoint in the structure (i.e. it is not empty), and if so the key and value in the most recent checkpoint.
at(struct Checkpoints.Trace224 self, uint32 pos) → struct Checkpoints.Checkpoint224 internal
Returns checkpoint at given position.
push(struct Checkpoints.Trace208 self, uint48 key, uint208 value) → uint208 oldValue, uint208 newValue internal
Pushes a (key, value) pair into a Trace208 so that it is stored as the checkpoint.
Returns previous value and new value.
Never accept key as a user input, since an arbitrary type(uint48).max key set will disable the library.
lowerLookup(struct Checkpoints.Trace208 self, uint48 key) → uint208 internal
Returns the value in the first (oldest) checkpoint with key greater or equal than the search key, or zero if there is none.
upperLookup(struct Checkpoints.Trace208 self, uint48 key) → uint208 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
upperLookupRecent(struct Checkpoints.Trace208 self, uint48 key) → uint208 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
This is a variant of upperLookup that is optimised to find "recent" checkpoint (checkpoints with high keys).
latest(struct Checkpoints.Trace208 self) → uint208 internal
Returns the value in the most recent checkpoint, or zero if there are no checkpoints.
latestCheckpoint(struct Checkpoints.Trace208 self) → bool exists, uint48 _key, uint208 _value internal
Returns whether there is a checkpoint in the structure (i.e. it is not empty), and if so the key and value in the most recent checkpoint.
at(struct Checkpoints.Trace208 self, uint32 pos) → struct Checkpoints.Checkpoint208 internal
Returns checkpoint at given position.
push(struct Checkpoints.Trace160 self, uint96 key, uint160 value) → uint160 oldValue, uint160 newValue internal
Pushes a (key, value) pair into a Trace160 so that it is stored as the checkpoint.
Returns previous value and new value.
Never accept key as a user input, since an arbitrary type(uint96).max key set will disable the library.
lowerLookup(struct Checkpoints.Trace160 self, uint96 key) → uint160 internal
Returns the value in the first (oldest) checkpoint with key greater or equal than the search key, or zero if there is none.
upperLookup(struct Checkpoints.Trace160 self, uint96 key) → uint160 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
upperLookupRecent(struct Checkpoints.Trace160 self, uint96 key) → uint160 internal
Returns the value in the last (most recent) checkpoint with key lower or equal than the search key, or zero if there is none.
This is a variant of upperLookup that is optimised to find "recent" checkpoint (checkpoints with high keys).
latest(struct Checkpoints.Trace160 self) → uint160 internal
Returns the value in the most recent checkpoint, or zero if there are no checkpoints.
latestCheckpoint(struct Checkpoints.Trace160 self) → bool exists, uint96 _key, uint160 _value internal
Returns whether there is a checkpoint in the structure (i.e. it is not empty), and if so the key and value in the most recent checkpoint.
at(struct Checkpoints.Trace160 self, uint32 pos) → struct Checkpoints.Checkpoint160 internal
Returns checkpoint at given position.
import "@openzeppelin/contracts/utils/structs/Heap.sol";
Heaps are represented as a tree of values where the first element (index 0) is the root, and where the node at index i is the child of the node at index (i-1)/2 and the parent of nodes at index 2*i+1
and 2*i+2. Each node stores an element of the heap.
The structure is ordered so that each node is bigger than its parent. An immediate consequence is that the highest priority value is the one at the root. This value can be looked up in constant time
(O(1)) at heap.tree[0]
The structure is designed to perform the following operations with the corresponding complexities:
• peek (get the highest priority value): O(1)
• insert (insert a value): O(log(n))
• pop (remove the highest priority value): O(log(n))
• replace (replace the highest priority value with a new value): O(log(n))
• length (get the number of elements): O(1)
• clear (remove all elements): O(1)
This library allows for the use of custom comparator functions. Given that manipulating memory can lead to unexpected behavior. Consider verifying that the comparator does not manipulate the Heap’s
state directly and that it follows the Solidity memory safety rules.
pop(struct Heap.Uint256Heap self) → uint256 internal
Remove (and return) the root element for the heap using the default comparator.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
pop(struct Heap.Uint256Heap self, function (uint256,uint256) view returns (bool) comp) → uint256 internal
Remove (and return) the root element for the heap using the provided comparator.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
insert(struct Heap.Uint256Heap self, uint256 value) internal
Insert a new element in the heap using the default comparator.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
insert(struct Heap.Uint256Heap self, uint256 value, function (uint256,uint256) view returns (bool) comp) internal
Insert a new element in the heap using the provided comparator.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
replace(struct Heap.Uint256Heap self, uint256 newValue) → uint256 internal
Return the root element for the heap, and replace it with a new value, using the default comparator. This is equivalent to using pop and insert, but requires only one rebalancing operation.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
replace(struct Heap.Uint256Heap self, uint256 newValue, function (uint256,uint256) view returns (bool) comp) → uint256 internal
Return the root element for the heap, and replace it with a new value, using the provided comparator. This is equivalent to using pop and insert, but requires only one rebalancing operation.
All inserting and removal from a heap should always be done using the same comparator. Mixing comparator during the lifecycle of a heap will result in undefined behavior.
import "@openzeppelin/contracts/utils/structs/MerkleTree.sol";
Each tree is a complete binary tree with the ability to sequentially insert leaves, changing them from a zero to a non-zero value and updating its root. This structure allows inserting commitments
(or other entries) that are not stored, but can be proven to be part of the tree at a later time if the root is kept. See MerkleProof.
A tree is defined by the following parameters:
• Depth: The number of levels in the tree, it also defines the maximum number of leaves as 2**depth.
• Zero value: The value that represents an empty leaf. Used to avoid regular zero values to be part of the tree.
• Hashing function: A cryptographic hash function used to produce internal nodes. Defaults to Hashes.commutativeKeccak256.
Building trees using non-commutative hashing functions (i.e. H(a, b) != H(b, a)) is supported. However, proving the inclusion of a leaf in such trees is not possible with the MerkleProof library
since it only supports commutative hashing functions.
setup(struct MerkleTree.Bytes32PushTree self, uint8 treeDepth, bytes32 zero) → bytes32 initialRoot internal
Calling this function on MerkleTree that was already setup and used will reset it to a blank state.
Once a tree is setup, any push to it must use the same hashing function. This means that values should be pushed to it using the default push function.
The zero value should be carefully chosen since it will be stored in the tree representing empty leaves. It should be a value that is not expected to be part of the tree.
setup(struct MerkleTree.Bytes32PushTree self, uint8 treeDepth, bytes32 zero, function (bytes32,bytes32) view returns (bytes32) fnHash) → bytes32 initialRoot internal
Same as setup, but allows to specify a custom hashing function.
Once a tree is setup, any push to it must use the same hashing function. This means that values should be pushed to it using the custom push function, which should be the same one as used during the
Providing a custom hashing function is a security-sensitive operation since it may compromise the soundness of the tree.
Consider verifying that the hashing function does not manipulate the memory state directly and that it follows the Solidity memory safety rules. Otherwise, it may lead to unexpected behavior.
push(struct MerkleTree.Bytes32PushTree self, bytes32 leaf) → uint256 index, bytes32 newRoot internal
Insert a new leaf in the tree, and compute the new root. Returns the position of the inserted leaf in the tree, and the resulting root.
Hashing the leaf before calling this function is recommended as a protection against second pre-image attacks.
This variant uses Hashes.commutativeKeccak256 to hash internal nodes. It should only be used on merkle trees that were setup using the same (default) hashing function (i.e. by calling the default
setup function).
push(struct MerkleTree.Bytes32PushTree self, bytes32 leaf, function (bytes32,bytes32) view returns (bytes32) fnHash) → uint256 index, bytes32 newRoot internal
Insert a new leaf in the tree, and compute the new root. Returns the position of the inserted leaf in the tree, and the resulting root.
Hashing the leaf before calling this function is recommended as a protection against second pre-image attacks.
This variant uses a custom hashing function to hash internal nodes. It should only be called with the same function as the one used during the initial setup of the merkle tree.
import "@openzeppelin/contracts/utils/Create2.sol";
Helper to make usage of the CREATE2 EVM opcode easier and safer. CREATE2 can be used to compute in advance the address where a smart contract will be deployed, which allows for interesting new
mechanisms known as 'counterfactual interactions'.
See the EIP for more information.
deploy(uint256 amount, bytes32 salt, bytes bytecode) → address addr internal
Deploys a contract using CREATE2. The address where the contract will be deployed can be known in advance via computeAddress.
The bytecode for a contract can be obtained from Solidity with type(contractName).creationCode.
• bytecode must not be empty.
• salt must have not been used for bytecode already.
• the factory must have a balance of at least amount.
• if amount is non-zero, bytecode must have a payable constructor.
computeAddress(bytes32 salt, bytes32 bytecodeHash) → address internal
Returns the address where a contract will be stored if deployed via deploy. Any change in the bytecodeHash or salt will result in a new destination address.
computeAddress(bytes32 salt, bytes32 bytecodeHash, address deployer) → address addr internal
Returns the address where a contract will be stored if deployed via deploy from a contract located at deployer. If deployer is this contract’s address, returns the same value as computeAddress.
import "@openzeppelin/contracts/utils/Address.sol";
Collection of functions related to the address type
sendValue(address payable recipient, uint256 amount) internal
Replacement for Solidity’s transfer: sends amount wei to recipient, forwarding all available gas and reverting on errors.
EIP1884 increases the gas cost of certain opcodes, possibly making contracts go over the 2300 gas limit imposed by transfer, making them unable to receive funds via transfer. sendValue removes this
because control is transferred to recipient, care must be taken to not create reentrancy vulnerabilities. Consider using ReentrancyGuard or the checks-effects-interactions pattern.
functionCall(address target, bytes data) → bytes internal
Performs a Solidity function call using a low level call. A plain call is an unsafe replacement for a function call: use this function instead.
If target reverts with a revert reason or custom error, it is bubbled up by this function (like regular Solidity function calls). However, if the call reverted with no returned reason, this function
reverts with a {Errors.FailedCall} error.
Returns the raw returned data. To convert to the expected return value, use abi.decode.
• target must be a contract.
• calling target with data must not revert.
functionCallWithValue(address target, bytes data, uint256 value) → bytes internal
Same as functionCall, but also transferring value wei to target.
• the calling contract must have an ETH balance of at least value.
• the called Solidity function must be payable.
functionStaticCall(address target, bytes data) → bytes internal
functionDelegateCall(address target, bytes data) → bytes internal
verifyCallResultFromTarget(address target, bool success, bytes returndata) → bytes internal
Tool to verify that a low level call to smart-contract was successful, and reverts if the target was not a contract or bubbling up the revert reason (falling back to {Errors.FailedCall}) in case of
an unsuccessful call.
verifyCallResult(bool success, bytes returndata) → bytes internal
Tool to verify that a low level call was successful, and reverts if it wasn’t, either by bubbling the revert reason or with a default {Errors.FailedCall} error.
import "@openzeppelin/contracts/utils/Arrays.sol";
Collection of functions related to array types.
sort(uint256[] array, function (uint256,uint256) pure returns (bool) comp) → uint256[] internal
Sort an array of uint256 (in memory) following the provided comparator function.
This function does the sorting "in place", meaning that it overrides the input. The object is returned for convenience, but that returned value can be discarded safely if the caller has a memory
pointer to the array.
this function’s cost is O(n · log(n)) in average and O(n²) in the worst case, with n the length of the array. Using it in view functions that are executed through eth_call is safe, but one should be
very careful when executing this as part of a transaction. If the array being sorted is too large, the sort operation may consume more gas than is available in a block, leading to potential DoS.
Consider memory side-effects when using custom comparator functions that access memory in an unsafe way.
sort(uint256[] array) → uint256[] internal
Variant of sort that sorts an array of uint256 in increasing order.
sort(address[] array, function (address,address) pure returns (bool) comp) → address[] internal
Sort an array of address (in memory) following the provided comparator function.
This function does the sorting "in place", meaning that it overrides the input. The object is returned for convenience, but that returned value can be discarded safely if the caller has a memory
pointer to the array.
this function’s cost is O(n · log(n)) in average and O(n²) in the worst case, with n the length of the array. Using it in view functions that are executed through eth_call is safe, but one should be
very careful when executing this as part of a transaction. If the array being sorted is too large, the sort operation may consume more gas than is available in a block, leading to potential DoS.
Consider memory side-effects when using custom comparator functions that access memory in an unsafe way.
sort(address[] array) → address[] internal
Variant of sort that sorts an array of address in increasing order.
sort(bytes32[] array, function (bytes32,bytes32) pure returns (bool) comp) → bytes32[] internal
Sort an array of bytes32 (in memory) following the provided comparator function.
This function does the sorting "in place", meaning that it overrides the input. The object is returned for convenience, but that returned value can be discarded safely if the caller has a memory
pointer to the array.
this function’s cost is O(n · log(n)) in average and O(n²) in the worst case, with n the length of the array. Using it in view functions that are executed through eth_call is safe, but one should be
very careful when executing this as part of a transaction. If the array being sorted is too large, the sort operation may consume more gas than is available in a block, leading to potential DoS.
Consider memory side-effects when using custom comparator functions that access memory in an unsafe way.
sort(bytes32[] array) → bytes32[] internal
Variant of sort that sorts an array of bytes32 in increasing order.
findUpperBound(uint256[] array, uint256 element) → uint256 internal
Searches a sorted array and returns the first index that contains a value greater or equal to element. If no such index exists (i.e. all values in the array are strictly less than element), the array
length is returned. Time complexity O(log n).
The array is expected to be sorted in ascending order, and to contain no repeated elements.
Deprecated. This implementation behaves as lowerBound but lacks support for repeated elements in the array. The lowerBound function should be used instead.
lowerBound(uint256[] array, uint256 element) → uint256 internal
Searches an array sorted in ascending order and returns the first index that contains a value greater or equal than element. If no such index exists (i.e. all values in the array are strictly less
than element), the array length is returned. Time complexity O(log n).
upperBound(uint256[] array, uint256 element) → uint256 internal
Searches an array sorted in ascending order and returns the first index that contains a value strictly greater than element. If no such index exists (i.e. all values in the array are strictly less
than element), the array length is returned. Time complexity O(log n).
lowerBoundMemory(uint256[] array, uint256 element) → uint256 internal
upperBoundMemory(uint256[] array, uint256 element) → uint256 internal
unsafeAccess(address[] arr, uint256 pos) → struct StorageSlot.AddressSlot internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeAccess(bytes32[] arr, uint256 pos) → struct StorageSlot.Bytes32Slot internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeAccess(uint256[] arr, uint256 pos) → struct StorageSlot.Uint256Slot internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeMemoryAccess(address[] arr, uint256 pos) → address res internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeMemoryAccess(bytes32[] arr, uint256 pos) → bytes32 res internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeMemoryAccess(uint256[] arr, uint256 pos) → uint256 res internal
Access an array in an "unsafe" way. Skips solidity "index-out-of-range" check.
Only use if you are certain pos is lower than the array length.
unsafeSetLength(address[] array, uint256 len) internal
Helper to set the length of an dynamic array. Directly writing to .length is forbidden.
this does not clear elements if length is reduced, of initialize elements if length is increased.
unsafeSetLength(bytes32[] array, uint256 len) internal
Helper to set the length of an dynamic array. Directly writing to .length is forbidden.
this does not clear elements if length is reduced, of initialize elements if length is increased.
import "@openzeppelin/contracts/utils/Base64.sol";
Provides a set of functions to operate with Base64 strings.
encodeURL(bytes data) → string internal
Converts a bytes to its Bytes64Url string representation. Output is not padded with = as specified in rfc4648.
string _TABLE internal constant
import "@openzeppelin/contracts/utils/Strings.sol";
toString(uint256 value) → string internal
Converts a uint256 to its ASCII string decimal representation.
toStringSigned(int256 value) → string internal
Converts a int256 to its ASCII string decimal representation.
toHexString(uint256 value) → string internal
Converts a uint256 to its ASCII string hexadecimal representation.
toHexString(uint256 value, uint256 length) → string internal
Converts a uint256 to its ASCII string hexadecimal representation with fixed length.
toHexString(address addr) → string internal
Converts an address with fixed length of 20 bytes to its not checksummed ASCII string hexadecimal representation.
toChecksumHexString(address addr) → string internal
Converts an address with fixed length of 20 bytes to its checksummed ASCII string hexadecimal representation, according to EIP-55.
import "@openzeppelin/contracts/utils/ShortStrings.sol";
This library provides functions to convert short memory strings into a ShortString type that can be used as an immutable variable.
Strings of arbitrary length can be optimized using this library if they are short enough (up to 31 bytes) by packing them with their length (1 byte) in a single EVM word (32 bytes). Additionally, a
fallback mechanism can be used for every other case.
contract Named {
using ShortStrings for *;
ShortString private immutable _name;
string private _nameFallback;
constructor(string memory contractName) {
_name = contractName.toShortStringWithFallback(_nameFallback);
function name() external view returns (string memory) {
return _name.toStringWithFallback(_nameFallback);
toShortString(string str) → ShortString internal
Encode a string of at most 31 chars into a ShortString.
This will trigger a StringTooLong error is the input string is too long.
toShortStringWithFallback(string value, string store) → ShortString internal
Encode a string into a ShortString, or write it to storage if it is too long.
toStringWithFallback(ShortString value, string store) → string internal
Decode a string that was encoded to ShortString or written to storage using {setWithFallback}.
byteLengthWithFallback(ShortString value, string store) → uint256 internal
Return the length of a string that was encoded to ShortString or written to storage using {setWithFallback}.
This will return the "byte length" of the string. This may not reflect the actual length in terms of actual characters as the UTF-8 encoding of a single character can span over multiple bytes.
import "@openzeppelin/contracts/utils/SlotDerivation.sol";
Library for computing storage (and transient storage) locations from namespaces and deriving slots corresponding to standard patterns. The derivation method for array and mapping matches the storage
layout used by the solidity language / compiler.
contract Example {
// Add the library methods
using StorageSlot for bytes32;
using SlotDerivation for bytes32;
// Declare a namespace
string private constant _NAMESPACE = "<namespace>" // eg. OpenZeppelin.Slot
function setValueInNamespace(uint256 key, address newValue) internal {
_NAMESPACE.erc7201Slot().deriveMapping(key).getAddressSlot().value = newValue;
function getValueInNamespace(uint256 key) internal view returns (address) {
return _NAMESPACE.erc7201Slot().deriveMapping(key).getAddressSlot().value;
Consider using this library along with StorageSlot.
This library provides a way to manipulate storage locations in a non-standard way. Tooling for checking upgrade safety will ignore the slots accessed through this library.
erc7201Slot(string namespace) → bytes32 slot internal
Derive an ERC-7201 slot from a string (namespace).
offset(bytes32 slot, uint256 pos) → bytes32 result internal
Add an offset to a slot to get the n-th element of a structure or an array.
deriveArray(bytes32 slot) → bytes32 result internal
Derive the location of the first element in an array from the slot where the length is stored.
deriveMapping(bytes32 slot, address key) → bytes32 result internal
Derive the location of a mapping element from the key.
deriveMapping(bytes32 slot, bool key) → bytes32 result internal
Derive the location of a mapping element from the key.
deriveMapping(bytes32 slot, bytes32 key) → bytes32 result internal
Derive the location of a mapping element from the key.
deriveMapping(bytes32 slot, uint256 key) → bytes32 result internal
Derive the location of a mapping element from the key.
deriveMapping(bytes32 slot, int256 key) → bytes32 result internal
Derive the location of a mapping element from the key.
deriveMapping(bytes32 slot, string key) → bytes32 result internal
Derive the location of a mapping element from the key.
import "@openzeppelin/contracts/utils/StorageSlot.sol";
Library for reading and writing primitive types to specific storage slots.
Storage slots are often used to avoid storage conflict when dealing with upgradeable contracts. This library helps with reading and writing to such slots without the need for inline assembly.
The functions in this library return Slot structs that contain a value member that can be used to read or write.
Example usage to set ERC-1967 implementation slot:
contract ERC1967 {
// Define the slot. Alternatively, use the SlotDerivation library to derive the slot.
bytes32 internal constant _IMPLEMENTATION_SLOT = 0x360894a13ba1a3210667c828492db98dca3e2076cc3735a920a3ca505d382bbc;
function _getImplementation() internal view returns (address) {
return StorageSlot.getAddressSlot(_IMPLEMENTATION_SLOT).value;
function _setImplementation(address newImplementation) internal {
require(newImplementation.code.length > 0);
StorageSlot.getAddressSlot(_IMPLEMENTATION_SLOT).value = newImplementation;
Consider using this library along with SlotDerivation.
getAddressSlot(bytes32 slot) → struct StorageSlot.AddressSlot r internal
Returns an AddressSlot with member value located at slot.
getBooleanSlot(bytes32 slot) → struct StorageSlot.BooleanSlot r internal
Returns a BooleanSlot with member value located at slot.
getBytes32Slot(bytes32 slot) → struct StorageSlot.Bytes32Slot r internal
Returns a Bytes32Slot with member value located at slot.
getUint256Slot(bytes32 slot) → struct StorageSlot.Uint256Slot r internal
Returns a Uint256Slot with member value located at slot.
getInt256Slot(bytes32 slot) → struct StorageSlot.Int256Slot r internal
Returns a Int256Slot with member value located at slot.
getStringSlot(bytes32 slot) → struct StorageSlot.StringSlot r internal
Returns a StringSlot with member value located at slot.
getStringSlot(string store) → struct StorageSlot.StringSlot r internal
Returns an StringSlot representation of the string storage pointer store.
getBytesSlot(bytes32 slot) → struct StorageSlot.BytesSlot r internal
Returns a BytesSlot with member value located at slot.
import "@openzeppelin/contracts/utils/TransientSlot.sol";
Library for reading and writing value-types to specific transient storage slots.
Transient slots are often used to store temporary values that are removed after the current transaction. This library helps with reading and writing to such slots without the need for inline
• Example reading and writing values using transient storage:
contract Lock {
using TransientSlot for *;
// Define the slot. Alternatively, use the SlotDerivation library to derive the slot.
bytes32 internal constant _LOCK_SLOT = 0xf4678858b2b588224636b8522b729e7722d32fc491da849ed75b3fdf3c84f542;
modifier locked() {
Consider using this library along with SlotDerivation.
asAddress(bytes32 slot) → TransientSlot.AddressSlot internal
Cast an arbitrary slot to a AddressSlot.
asBoolean(bytes32 slot) → TransientSlot.BooleanSlot internal
Cast an arbitrary slot to a BooleanSlot.
asBytes32(bytes32 slot) → TransientSlot.Bytes32Slot internal
Cast an arbitrary slot to a Bytes32Slot.
asUint256(bytes32 slot) → TransientSlot.Uint256Slot internal
Cast an arbitrary slot to a Uint256Slot.
tload(TransientSlot.AddressSlot slot) → address value internal
Load the value held at location slot in transient storage.
tstore(TransientSlot.AddressSlot slot, address value) internal
Store value at location slot in transient storage.
tload(TransientSlot.BooleanSlot slot) → bool value internal
Load the value held at location slot in transient storage.
tstore(TransientSlot.BooleanSlot slot, bool value) internal
Store value at location slot in transient storage.
tload(TransientSlot.Bytes32Slot slot) → bytes32 value internal
Load the value held at location slot in transient storage.
tstore(TransientSlot.Bytes32Slot slot, bytes32 value) internal
Store value at location slot in transient storage.
tload(TransientSlot.Uint256Slot slot) → uint256 value internal
Load the value held at location slot in transient storage.
tstore(TransientSlot.Uint256Slot slot, uint256 value) internal
Store value at location slot in transient storage.
tload(TransientSlot.Int256Slot slot) → int256 value internal
Load the value held at location slot in transient storage.
import "@openzeppelin/contracts/utils/Multicall.sol";
Provides a function to batch together multiple calls in a single external call.
Consider any assumption about calldata validation performed by the sender may be violated if it’s not especially careful about sending transactions invoking multicall. For example, a relay address
that filters function selectors won’t filter calls nested within a multicall operation.
Since 5.0.1 and 4.9.4, this contract identifies non-canonical contexts (i.e. msg.sender is not {_msgSender}). If a non-canonical context is identified, the following self delegatecall appends the
last bytes of msg.data to the subcall. This makes it safe to use with ERC2771Context. Contexts that don’t affect the resolution of {_msgSender} are not propagated to subcalls.
import "@openzeppelin/contracts/utils/Context.sol";
Provides information about the current execution context, including the sender of the transaction and its data. While these are generally available via msg.sender and msg.data, they should not be
accessed in such a direct manner, since when dealing with meta-transactions the account sending and paying for execution may not be the actual sender (as far as an application is concerned).
This contract is only required for intermediate, library-like contracts.
import "@openzeppelin/contracts/utils/Packing.sol";
Helper library packing and unpacking multiple values into bytesXX.
library MyPacker {
type MyType is bytes32;
function _pack(address account, bytes4 selector, uint64 period) external pure returns (MyType) {
bytes12 subpack = Packing.pack_4_8(selector, bytes8(period));
bytes32 pack = Packing.pack_20_12(bytes20(account), subpack);
return MyType.wrap(pack);
function _unpack(MyType self) external pure returns (address, bytes4, uint64) {
bytes32 pack = MyType.unwrap(self);
return (
address(Packing.extract_32_20(pack, 0)),
Packing.extract_32_4(pack, 20),
uint64(Packing.extract_32_8(pack, 24))
import "@openzeppelin/contracts/utils/Panic.sol";
Helper library for emitting standardized panic codes.
contract Example {
using Panic for uint256;
// Use any of the declared internal constants
function foo() { Panic.GENERIC.panic(); }
// Alternatively
function foo() { Panic.panic(Panic.GENERIC); }
panic(uint256 code) internal
Reverts with a panic code. Recommended to use with the internal constants with predefined codes.
import "@openzeppelin/contracts/utils/Comparators.sol";
Provides a set of functions to compare values. | {"url":"https://docs.openzeppelin.com/contracts/5.x/api/utils","timestamp":"2024-11-05T09:08:37Z","content_type":"text/html","content_length":"497900","record_id":"<urn:uuid:16fd8405-4120-45b6-acdb-87f4ed1afe56>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00354.warc.gz"} |
GCF Calculator
Greatest Common Factor aka Greatest Common Divisor calculator is an online maths tool that finds the GCF of two or several numbers by all three GCF methods.
Through GCF (GCD) calculator, you can select the method of your choice. The GCF calculation through the selected method will be shown with a detailed explanation.
You can simply click on the other method to see its process, that is, you will not have to enter the values all over from the start.
What is GCF?
Short form for Greatest Common Factor, GCF is the one factor that has the highest value among the common factors.
Other names of GCF include HCF and GCD.
How to find GCF?
It is a little tricky to find GCF, that is why the GCF finder above is recommended. But for your information and concept, we will see how to find GCF in detail.
The famous methods used for calculating GCF are:
• Prime factorization
• Division method
• List of factors
You can see the example of prime factorization in the HCF and LCM calculator. In this post, we will learn the remaining two methods through examples.
1. Division method:
Find the GCF of 14 and 20 through the division method.
Step 1: Use the higher value as a dividend and the lower value as the divisor.
Larger value = 20
Lower value = 14
Step 2: Divide by the nearest multiple.
The nearest value to 20 in 14’s time's table is 14 itself.
Step 3: Continue until no more can be solved. The last divisor is GCF (HCF).
2. List of Factors:
What is the greatest common factor of 12, 16, 29? Use the list of factors method.
Step 1: Write all the factors of 12, 16, and 29.
Factors of 12 = 1, 2, 3, 4, 6, 12.
Factors of 16 = 1, 2, 4, 8, 16.
Factors of 29 = 1, 29.
Step 2: Look for the common factors.
There is only one common factor among these three numbers i.e 1.
Step 3: Identify the greatest common factor.
Since there is no other factor, 1 will be the GCF of 12, 16, and 29. | {"url":"https://www.lcm-calculator.com/gcf-calculator","timestamp":"2024-11-10T14:00:50Z","content_type":"text/html","content_length":"25661","record_id":"<urn:uuid:c5a8af9f-af67-4b4a-bf62-31f366fec6ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00781.warc.gz"} |
Algebra and Number Theory – Feature Column
An automorphic form is, in the simplest sense, like a trigonometric function. Trigonometric functions are inescapable in both mathematics and physics, so it makes sense that we would see
generalizations of them in physics applications... Strung Out on Automorphic Forms Holley Friedlander Dickinson College Automorphic forms are highly symmetric functionsRead More →
Welcome to the Fold
Tagged: lattices (groups), origami, rings
A natural question in the context of origami mathematics is: What if we make the paper infinitely large? Welcome to the Fold Sara Chari Saint Mary's College of Maryland Adriana Salerno Bates College
and the National Science Foundation Origami—from the Japanese words for “fold” (oru) and “paper” (kami)—is the artRead More →
Lattices, Plane and Not-So-Plane
By: Courtney Gibbons
Tagged: cryptography, lattices (posets), posets, semigroups
Suppose this blog’s editor Ursula and I want to create a shared secret to use as a password to access this blog post... Lattices, Plane and Not-So-Plane Courtney Gibbons Hamilton College In January
2023, Bill Casselman wrote a great column introducing readers to lattices that imagined them as vectors inRead More →
Elliptic curves come to date night
In: 2024, Algebra and Number Theory, Geometry and Topology, Math and Social Sciences, Ursula Whitcher
Tagged: Bernd Sturmfels, economics, elliptic curves, game theory, İrem Portakal, Wolfgang Spohn
Willa is an economist and Cara is a mathematician, so together they have decided to turn the problem of which game to play into a separate meta-game. Because they both love numbers, Willa and Cara
start by creating matrices... Elliptic curves come to date night Ursula Whitcher Mathematical Reviews (AMS)Read More →
What I Think About When I Think About Voting
In: 2023, Algebra and Number Theory, Discrete Math and Combinatorics, Math and Social Sciences, Sarah Wolff
Tagged: persi diaconis, representation theory, voting
Inevitably, I think back to my favorite result in mathematics: when Diaconis used the representation theory of the symmetric group to show us that psychologists just don’t get along… What I Think
About When I Think About Voting Sarah Wolff Denison University It’s November. Here in Ohio, that means cozyRead More →
Putting a period on mathematical physics
Tagged: integrals, periods, physics
One of the fundamental forces in the universe is the weak force. The weak force is involved in holding atoms together or breaking them apart... Putting a period on mathematical physics Ursula
Whitcher Mathematical Reviews (AMS) You've heard of periods at the ends of sentences and periods of sine waves.Read More →
A Different Sense of Distance
Tagged: p-adic numbers, primes
There are other ways to compare rational numbers, especially if one happens to enjoy number theory... A Different Sense of Distance Maria Fox Oklahoma State University Dedicated to my Dad, Dr. Barry
R. Fox. The idea of distance is central to so much of the mathematics we do and teachRead More →
Perspectives on Polynomials (it’s a witch!)
By: Courtney Gibbons
Tagged: partial fractions, polynomials, witch of agnesi
Polynomials, it turns out, are useful for more than just input-output assignments! Perspectives on Polynomials (it’s a witch!) Courtney Gibbons Hamilton College It was a dark and stormy night… Okay,
it was probably more like 3:30 in the afternoon on a crisp fall day back when I was teaching CalcRead More →
What will they do when quantum computers start working?
Tagged: cryptography, lattices (groups), quantum cryptography
Mathematically, the most intriguing of the new proposals use lattices for message encryption… What will they do when quantum computers start working? Bill Casselman University of British Columbia
Commercial transactions on the internet are invariably passed through a process that hides them from unauthorized parties, using RSA public key encryptionRead More →
Hyperoperations, Distributivity, and the Unreasonable Effectiveness of Multiplication
Tagged: hyperoperations, Lie groups
In 1915, the paper “Note on an Operation of the Third Grade” by Albert A. Bennett appeared in the Annals of Mathematics. A terse two-page note, it was largely neglected until the early 2000s…
Hyperoperations, Distributivity, and the Unreasonable Effectiveness of Multiplication Anil Venkatesh Adelphi University Iterated Operations Everyone knowsRead More → | {"url":"https://mathvoices.ams.org/featurecolumn/category/algebra-and-number-theory/","timestamp":"2024-11-10T17:55:15Z","content_type":"text/html","content_length":"108921","record_id":"<urn:uuid:ec30b563-2fb0-45a1-860d-dbe01695edb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00468.warc.gz"} |
Found 10533 definitions mentioning List. Of these, 16 match your pattern(s).
Loogle searches Lean and Mathlib definitions and theorems.
You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code
extension, the lean.nvim integration or the Zulip bot.
Loogle finds definitions and lemmas in various ways:
1. By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
2. By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
3. By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.
The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
4. By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.
As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the
form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.
The #lucky button will directly send you to the documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.
This is Loogle revision 791b5a0 serving mathlib revision 16c9d8c | {"url":"https://loogle.lean-lang.org/?q=(?a%20-%3E%20?b)%20-%3E%20List%20?a%20-%3E%20List%20?b","timestamp":"2024-11-10T10:55:32Z","content_type":"text/html","content_length":"16012","record_id":"<urn:uuid:f86c45fc-400d-4158-b14f-7207e6672dc9>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00404.warc.gz"} |
Free Printable Subtraction Worksheets
Free Printable Subtraction Worksheets - Subtraction Worksheets by Grades
Are you looking for free printable subtraction worksheets to help your students/kids practice and master this critical math skill?
If so, here are some fantastic subtraction worksheets for Grade 1 to Grade 6 that you can download and print for free. These worksheets are designed to suit different levels of difficulty and
learning styles so that you can find the perfect ones for children.
Advantages to learning math with subtraction worksheets
• There are many advantages to learning math with subtraction worksheets.
□ Subtraction worksheets are a fun and easy way to practice math skills and improve mental calculation abilities.
□ Subtraction worksheets can help to learn different methods and strategies for solving subtraction problems, such as borrowing, regrouping, or using number lines.
□ Subtraction worksheets can also help to develop logical thinking and problem-solving skills, as they drive to apply the rules of subtraction to different situations and scenarios in real-life
□ Subtraction worksheets can boost confidence and motivation in math, as you can see your progress and achievements through the feedback and rewards you get from completing them.
Subtraction worksheets are a great way to prepare for exams and tests, as they can help to review and reinforce what students/kids have learned in class or from other sources.
• These are only some of the worksheets available for this section.
For more practice, browse contents using * ALL WORKSHEETS BY CATEGORY *.
Free printable subtraction worksheets by Grades
Subtraction Worksheets for Grade 1
Subtraction worksheets for Grade 1 are ideal for beginners who are just learning how to subtract single-digit numbers. They include subtraction facts from 0 to 10, easy subtraction word problems, and
subtraction with pictures. The pictures help kids visualize the concept of taking away and make math more fun and engaging.
• Subtraction Worksheets for Grade 2
Subtraction worksheets for Grade 2 are suitable for kids who are ready to move on to subtracting two-digit numbers. They cover topics such as regrouping (or borrowing), subtracting multiples of
10, subtracting across zeros, and subtracting money. They also have challenging word problems requiring fundamental logical thinking and problem-solving skills.
• Subtraction Worksheets for Grade 3
Subtraction worksheets for Grade 3 are designed for kids who are learning how to subtract three-digit numbers. They include exercises on regrouping, subtracting hundreds, large numbers, and
decimals. They also have some real-world scenarios that involve subtraction, such as measuring lengths, weights, and temperatures.
Subtraction Worksheets for Grade 4
Subtraction worksheets for Grade 4 are advanced supports for subtracting four-digit numbers and beyond. They cover topics such as regrouping with multiple zeros, subtracting fractions, subtracting
mixed numbers, and subtracting negative numbers. They also have some complex word problems that involve multiple steps and operations.
• Subtraction Worksheets for Grade 5
Subtraction worksheets for Grade 5 are here to help kids to master subtraction at a higher level. They include exercises on subtracting large numbers with up to six digits, subtracting decimals
with up to three decimal places, subtracting fractions with unlike denominators, and subtracting percentages. They also have some algebraic expressions that involve subtraction, such as
simplifying and evaluating.
• Subtraction Worksheets for Grade 6
Subtraction worksheets for Grade 6 are the perfect tool that prepares kids for more advanced math topics.
They include exercises on subtracting integers, rational numbers, irrational numbers, and algebraic terms.
They also have exciting word problems involving ratios, proportions, rates, and unit conversions.
• Second Grade
Third Grade
Fourth Grade
Easy to difficult subtraction worksheets for all Grades
Here are four types of activities expressly designed to fulfill easy to difficult subtraction worksheets for all Grades with our best subtraction worksheets with pictures, subtraction worksheets with
a number line, long subtraction worksheets, and real-life subtraction worksheets with word problems.
• Subtraction Worksheets with Pictures
Subtraction worksheets with pictures are great for beginners who are just learning to subtract by counting objects. They help students visualize the concept of taking away and finding the
The worksheets have colorful pictures of animals, fruits, toys, and other familiar items that students can count and subtract.
For example, one worksheet might have 10 apples and ask students to subtract 4 apples by crossing them out and writing the answer.
These worksheets are suitable for kindergarten and Grade 1 students.
Subtraction Worksheets with Number Lines
Subtraction worksheets with number lines are helpful for students who are ready to move on from counting objects to using a number line. A number line is a horizontal line with numbers marked at
equal intervals. It helps students see the relationship between numbers and use strategies like counting on or counting back to subtract.
The worksheets have subtraction problems with numbers up to 20 or 100, depending on the Grade level. Students have to use the number line provided to find the answer.
For example, one worksheet might have 15 - 7 = ? and a number line from 0 to 20. Students must start from 15 and jump back 7 spaces to land on 8, which is the answer.
These worksheets are suitable for Grade 1 and Grade 2 students.
Long Subtraction Worksheets
Long subtraction worksheets are challenging for students who are learning to subtract larger numbers in columns.
They require students to regroup or borrow when the subtrahend is larger than the minuend in a given place value.
For example, one worksheet might have 456 - 178 = ?
Explanation of how to subtract 456 - 178 =
Students have to start from the ones place and subtract 8 from 6, but they cannot do that because 8 is larger than 6.
The next step is for them to borrow 1 ten from the tens place and add it to the ones place, making it 16 - 8 = 8. Then they move on to the tens place and subtract 7 from 5, but they cannot do that
either because 7 is larger than 5.
So they have to borrow 1 hundred from the hundreds place and add it to the tens place, making it 15 - 7 = 8.
Finally, they move on to the hundreds place and subtract 1 from 3, which is easy: 3 - 1 = 2. The final answer is 288.
These worksheets are suitable for Grade 3 and Grade 4 students.
• Real-Life Subtraction Worksheets with Word Problems
Real-Life subtraction worksheets with word problems are engaging for students who want to apply their subtraction skills to real-life situations.
They have word problems that involve subtraction of money, time, distance, weight, or other quantities. They also test students' reading comprehension and problem-solving skills.
For example, one worksheet might have this word problem: Lisa bought a dress for $75 and a pair of shoes for $35. How much money did she spend in total? How much money did she have left if she
started with $150?
Students must read the question carefully and determine what they need to subtract.
They have to subtract $75 and $35 from $150 to find the total amount spent and the remaining amount.
So, the answer is $110 spent and $40 left.
These worksheets are suitable for Grade 4, Grade 5, and Grade 6 students.
Secret to understanding subtraction facts - develop best subtraction skills
An ideal secret to understanding subtraction facts depends on kid’s firm engagement in our subtraction math strategies worksheets.
Here, they will develop best subtraction skills in a very simple way. For sure kids, you’ll feel confident in areas that require quick calculations of numbers in the head, such as the balance you’ll
collect after having bought something from a store, shop etc.
In addition, a solid foundation in these worksheets is equally useful as you get to more complex math operations, such as subtracting decimals, fractions etc. | {"url":"https://mathskills4kids.com/subtraction-practice","timestamp":"2024-11-08T02:35:21Z","content_type":"text/html","content_length":"77348","record_id":"<urn:uuid:030a4f46-46f8-4fc2-90d0-2d8aee4c7adb>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00725.warc.gz"} |
revolving mills rod mills and ball mills
A ball mill also known as pebble mill or tumbling mill is a milling machine that consists of a hallow cylinder containing balls; mounted on a metallic frame such that it can be rotated along its
longitudinal axis. The balls which could be of different diameter occupy 30 – 50 % of the mill volume and its size depends on the feed and mill size.
The rod mill usually uses 50-100mm diameter steel rod as grinding medium, while the ball mill uses steel ball as grinding medium. The length of steel rod is 25-50mm shorter than the cylinder, and it
is usually made of high carbon steel with carbon content of 0.8% – 1%; the loading capacity of rod is about 35% – 45% of the effective volume ...
We have parts suited for all types of mills: autogenous and semi-autogenous, ball, stirred, and rod mills, as well as high-pressure grinding rolls, and vertical mills. We also supply parts for select
non- Outotec mills.
industries of revolving grinding mill. CSJ rough mill is applied by the industries of pharmaceutics chemical metallurgy foodstuff and architecture etc It specially grinds the hard and solid granule
materials including the breaking of plastics materials and cooper wires as well as the pretreatment coordinated machine for fine grinding process.
Welcome to the premier industrial source for Rod Mills. The companies featured in the following listing offer a comprehensive range of Rod Mills, as well as a variety of related products and
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As a ball mills supplier with 22 years of experience in the grinding industry, we can provide customers with types of ball mill, vertical mill, rod mill and AG/SAG mill for grinding in a variety of
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Tega Mill Linings provide optimal grinding solutions in major mineral processing plants all over the world. Tega rubber lining system is the preferred lining system for secondary ball mills; regrind
mills, rod mills and scrubbers. Tega reinforced lifters have an integrated aluminium track to accommodate the fixing clamp.
Rod mill is an efficient tool for crushing and grinding ore and other materials and widely-used in many industries such mineral preparation, metallurgy, chemical industry, building material, thermal
power industry, and etc., and rod mills always used as the preliminary crushing equipment before the grinding ball mills.
Grinding mills ball mill rod mill design parts 911 metallurgistower percentage of critical speed is used for attrition grinding when a fine product is desiredhe graph below will be helpful in
determining percentage of critical speed when internal mill diameter and rpm are known grinding mill is a revolving cylinder.
Ball Mills or Rod Mills in a complete range of sizes up to 10′ diameter x 20′ long, offer features of operation and convertibility to meet your exact needs. They may be used for pulverizing and
either wet or dry grinding systems. Mills are available in both light-duty and heavy-duty construction to meet your specific requirements.
Mining and processing plants use rotating drum mills (pic. 1). Depending on the form of the grinding media there are ball mills, rod mills, pebble mills and autogenous grinding mills. Grinding media
– iron and steel balls with diameter of 15-120 mm, steel or cast iron cylpebs dimensions (diameter and length) of 16 to 25 and 30 and 40 mm ...
Construction of Ball Mill: The ball mill consists of a hollow metal cylinder mounted on a shaft and rotating about its horizontal axis. The cylinder can be made of metal, porcelain, or rubber. Inside
the cylinder balls or pebbles are placed. The balls occupy between 30 and 50% of the volume of the cylinder. The diameter of the balls depends on ...
The mill product can either be finished size ready for processing, or an intermediate size ready for final grinding in a rod mill, ball mill or pebble mill. AG/SAG mills can accomplish the same size
reduction work as two or three stages of crushing and screening, a rod mill, and some or all of the work of a ball mill.
Mills Grinding Mills Revolving metzgerei-graf.de. Revolving mills rod mills and ball mills.Rod mill power vs ball millszm rod mill is a type of grinding mill whose grinding media is steel bar while
ball mill is steel ball 1 rod mill are usually applied to grinding the wsn ore and rare metal in the reselection or magneticoredressing plant in order to avoid the damage caused by the over …
[Overflow Rod Mills Figure 1 Peripheral Discharge Rod Mills Figures 2 and 3 Compartment Mills Rod and Ball Figure 6 Ball Figure 6a Pebble Mill Figure 6b Overflow Ball Mills Figure 8 Diaphragm (Grate
Discharge) Ball Mills Figure 9] I. GENERAL MILL DESIGN A. Liners The interior surface of grinding mills exposed to grinding media and/or the ...
How do you decide between using a ball mill or a rod mill? Many investigators have attributed the selective grinding of rods to line contact. Other things should beCommon types of grinding mills
include Ball Mills and Rod Mills. This includes all rotating mills with heavy grinding media loads. This article focuses on ball and rod mills ...
revolving grinding mills goldenretrieverpl. Grinding Mills Revolving Grinding mills ball mill rod mill design parts 911 metallurgistower percentage of critical speed is used for attrition grinding
when a fine product is desiredhe graph below will be helpful in determining percentage of critical speed when internal mill diameter and rpm are known grinding mill is a revolving cylinderget
Grinding Mills: Ball Mill & Rod Mill Design & Jul 10,, 1 Grinding Mill RPM. We sell all types of Grinding Mills, Rod Mills, Pebble Mills, SAG Mills, Ball Mills, if you are, A Grinding Ball mill
consists of a hollow cylindrical shell rotating about its axis, 105 Ft x 12 Ft Allis Chalmers Ball Mill, No Liners, 800hp Motor, gears in, 4,000 Volts, 115 Amp Arm, Amp Field …
rod mill vs ball mill. BMZsk The ball and rod mill The ballmill GM and rodmill TM are destined for the wet milling of ores and mineral raw materialDescription The ball and rodmills are drums rotating
on hollow journals around their horizontal axis. …
Ball/Rod mill Literature . The Ball/Rod mills are meant for producing fine particle size reduction through attrition and compressive forces at the grain size level. They are the most effective
laboratory mills for batch-wise, rapid grinding of medium-hard …
A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from its
position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula. The mill speed is typically defined as the percent of the …
Ball mills are fine grinders, have horizontal ball mill and vertical ball mill, their cylinders are partially filled with steel balls, manganese balls, or ceramic balls. The material is ground to the
required fineness by rotating the cylinder causing friction and impact.
A rod mill is an ore grinding mechanism that uses a number of loose steel rods within a rotating drum to provide its attrition or grinding action. An ore charge is added to the drum, and as it
rotates, friction between the tumbling rods breaks the ore down into finer particles. Although similar in operation, a rod mill is often more effective ...
A Ball mill consists of a hollow cylindrical shell rotating about its axis. The axis of the shell may be either horizontal or at a small angle to the horizontal. Ball Mills, Rod Mills, Pebble Mills,
Mine Hoists, Crushers, Pumps, Synchronous Motors, DC Motors, Diesel Generators, Natural Gas Generators and more, call us today and let us help. Get ...
Ball mill： Ball mills are the most widely used one. Rod mill： The rod mill has the highest efficiency when the feed size is <30mm and the discharge size is about 3mm with uniform particle size and
light overcrushing phenomenon. SAG mill: When the weight of the SAG mill is more than 75%, the output is high and the energy consumption is low ...
Rotary ball mill is composed of feeding part, discharging part, turning part, transmission part (reducer, small transmission gear, motor, electric control) and other main parts. The hollow shaft of
the rotary ball mill is made of cast steel, and the lining can be removed and replaced. The large rotary gear is processed by casting hobbing, and the cylinder body is equipped with a …
revolving mills rod mills and ball mills. rod mill power vs ball mill – SZM Rod mill is a type of grinding mill whose grinding media is steel bar while ball mill is steel ball 1 Rod mill are usually
applied to grinding the wsn ore and rare metal …
A Ball mill consists of a hollow cylindrical shell rotating about its axis. The axis of the shell may be either horizontal or at a small angle to the horizontal. ... Ball Mills, Rod Mills, Pebble
Mills, Mine Hoists, Crushers, Pumps, Synchronous Motors, DC Motors, Diesel Generators, Natural Gas Generators and more, call us today and let us help ...
8.3.2.2 Ball mills. The ball mill is a tumbling mill that uses steel balls as the grinding media. The length of the cylindrical shell is usually 1–1.5 times the shell diameter ( Figure 8.11). The
feed can be dry, with less than 3% moisture to minimize ball …
Grinding Mills: Ball Mill & Rod Mill Design & Parts. Common types of grinding mills include Ball Mills and Rod Mills. This includes all rotating mills with heavy grinding media loads. This article
focuses on ball and rod mills excluding SAG and AG mills. Although their concepts are very similar, they are not discussed here.
Revolving Mills Rod Mills And Ball Mills. Revolving mills rod mills and ball millsmon types of grinding mills include ball mills and rod mills this includes all rotating mills with heavy grinding
media loads this article focuses on ball and rod mills excluding sag and ag mills although their concepts are very similar they are not discussed here photographs of a glass …
A ball mill is a type of grinder used to grind or blend materials for use in mineral dressing processes, paints, pyrotechnics, ceramics, and selective laser sintering.It works on the principle of
impact and attrition: size reduction is done by impact as the balls drop from near the top of the shell. A ball mill consists of a hollow cylindrical shell rotating about its axis. T
revolving mills rod mills and ball mills Scale-up methodology for tumbling ball mill based on… This paper provides a method to scale-up horizontal tumbling ball mills, i.e. to determine the
dimensions of the rotating drum and the drum rotational speed.
revolving mills rod mills and ball mills Hitlers Hollywood. revolving mills rod mills and ball mills Ball and Rod Mills JeanPaul Duroudier in Size Reduction of Divided Solids 2016 A variant of the
rotating or tumbling ball mill is the planetary ball mill This variant imparts a higher degree of energy in an attempt to create finer or more homogeneous powder size distributions More … | {"url":"https://www.zamieszkajna1maja.pl/20172/revolving/mills/rod/mills/and/ball/mills.html","timestamp":"2024-11-07T15:42:45Z","content_type":"text/html","content_length":"38074","record_id":"<urn:uuid:43dcb099-175a-4891-af8d-9c524f234f29>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00406.warc.gz"} |
Gwersi PCAME a Dewch i Feddwl Mathemateg (9 i 11 oed)
Dewch i Feddwl Mathemateg (9 i 11 oed) Gwers 6 Siapiau Symudwyr
Who knows what a crystal is? What is special about how it grows? A new type of crystal was discovered by a space expedition. It always starts in a square shape but can change to become a different
quadrilateral shape. We need to work out the secret of exactly how the crystal changes its shape so we can predict what shapes it could turn into.
Episode 1
Introduce the children to the square shaped aliens using a large pin board and an elastic band, or on an interactive whiteboard tool, by showing a 2 x 2 square on the top left of a 5 x 5 grid,
labelled ABCD, and asking them to describe what they can see. Encourage them to focus on the position of the shape on the grid as well as the properties:
Move point B one space to the right and tell the children that some crystals with this new shape have been seen. How could you describe how the shape has changed? What might the secret of the shape
change be? Pairs discuss the questions, then feedback ideas, which are listed. Use questions eg. there? where? to play upon ambiguities that arise from initial descriptions, as the children need to
struggle to explain their actions efficiently to see the need for using a common language.
Establish that the shape can move one vertex horizontally or vertically in any direction, and this move can be more than one space, but that diagonal moves are not permitted.
Emphasise that the crystals always have four sides.
Ask the pupils to repeat the rules to you so that you can write them up on the board. For example:
• The shape always starts as a 2 by 2 square in the top left hand corner.
• It is always a quadrilateral.
• One corner/vertex can move horizontally or vertically in any direction.
• A corner/vertex can move more than one space.
• Diagonal moves are not allowed.’
Reinforce by asking children what instruction they would give to change a given shape back to a square, and/or model an incorrect move and asking if it is permitted.
Hand out pegboards, elastic bands and Resource Sheet A for recording.
Group Discussion
Working in pairs, use a pin board or dotty paper to investigate how many different shapes the aliens could be if using the above rules. For each new shape you find, you must have a written
description of how your shape changed. How will you record the move you have made for each alien?
Selected pairs to share their findings. Establish there are only three possibilities
ie. these trapeziums in different orientations
(one vertex, one move and one vertex two moves)
How did you record the moves you did? Collate ideas on the board for comparison and agree a coding system. For example:
• Use the A/B/C/D labels as above or label corners TL (top left) etc
• Use V/H (for vertical/horizontal) or ↑/↓
• Use a number for distance.
So a move might be B1V or TL2↓. Consider the systematic sequencing of the coding, so, is it best to always use the same order eg start point-distance-direction or is a random order acceptable?
Episode 2
A second and more advanced crystal has also been located. These crystals have been classified as level two crystals. Explore what this might mean by showing an oblong level two rectangle transposed
over the original square and asking children to describe the change.
They will recognise that two moves have been made (vertex B two moves right and vertex D two moves right) How could this alien change back to the square?
Ask the pupils to write the rule for this second crystal. For example:
• The shape always starts as a 2 by 2 square in the top left hand corner.
• It is always a quadrilateral.
• Two corners/vertices can move horizontally or vertically in any direction.
• The corner/vertex can move more than one space.
• Diagonal moves are not allowed.’
How many different shapes could a level two alien be? Pairs predict, then feedback with reasons.
Group Discussion
Work in pairs to investigate how many different shapes a level two alien can make. Record the moves made for each shape using the agreed coding system. (Some children will need to continue to use the
pin boards, in which case a second elastic band of a different colour is useful, whilst most use several copies of Resource Sheet A.)
What different level two alien shapes did you find? Pairs feedback the different shapes and describe how they shifted, using the coding system. Identify shapes (children may be using 2D shape names,
though not necessarily), which are the same, but in a different position (ie. translations). Children should recognise that in this system, the 2 x 2 square provides the starting point ie. the zero
position. NB the focus for discussion here is not on finding all the possibilities, but checking the efficiency of the coding system they have developed.
Have we found all possibilities? How can we be sure? Focus in on any use of a system to record the different possible moves. Some pupils may have listed these systematically, rather than relying on
drawing alone.
Episode 3
A collection of crystals has been found, but these include new types which can make three, and even four, moves! The task for the scientists is to identify the crystals by working out the shifts they
have made.
Can we work out whether shapes have been made by level one/two/three/ or four crystals? Reinforce link to one/two/three/four moves from the original square shape.
Group Discussion
Children work in pairs to identify the crystals on Resource Sheet B, according to their level description, by working out how they have shifted.
Feedback findings by pairs of children giving instructions for transforming some of shapes on Resources Sheet B from the original square.
Could you predict accurately which level the shape would be? How?
Can you see any patterns?
Prompt: What did you notice about the number of moves for shape 4 compared to shape 8?
Will a bigger rectangle need more than 3 moves?
Can you describe a link between the number of moves and the size of the oblong? | {"url":"https://community.letsthink.org.uk/pcamelessonswelsh/chapter/lesson-plan-28/","timestamp":"2024-11-03T10:06:29Z","content_type":"text/html","content_length":"113504","record_id":"<urn:uuid:3acfcdda-7bca-4baa-8baf-167a51bcbf71>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00705.warc.gz"} |
How to sum variables?
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
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Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Summing variables
One method for summing a number of variables is to use the addition operator
1 + 4 + 7 = ? m (12 m)
An alternative method is to use the sum function
Sum( 1, 4, 7 ) = ? m (12 m)
You can of course also sum variables
L1 = 1 m
L2 = 4 m
L3 = 3 m
Sum( L1, L2, L3 ) = ? m (8 m)
In the above example there are known to be three values for L[n] But how can you write the expression so that it will work for any number of L values, i.e. 1-n?
You could use the GetVar function to ensure that if a particular L value didn't exist a value of zero is used so as not to affect the end result, the example below will correctly sum the values of L
[n] for 1, 2 or 3 values, however it is still limited to a maximum of the specified variables.
Sum( GetVar("L1", 0m), GetVar("L2", 0m), GetVar("L3", 0m) ) = ? m (8m)
To write an expression that will sum any number of variables in a series you have to write a sequence of expressions that will create a string of the expression and then evaluate that string
NumItems = 3
Eval("Sum( L[1] )", StrReplace( StrListRange(NumItems), ",", ",L" )) = ? m (8m)
NumItems is a variable that defines how many items there are to sum from 1-n.
How it works
• Create a comma separated list of the indices of the variables to sum
StrListRange(NumItems)= "1,2,3"
• Replace all the commas in the string with ,L
StrReplace( "1,2,3", ",", ",L" ) = “1,L2,L3”
• Insert the string in to the string “Sum( L_{[1]} )” and evaluate that string to get the final result
Eval("Sum( L[1] )", “1,L2,L3" )) = 8m
To amend the sum expression for different variables change the text in red as appropriate for the variable name you need.
L1, L2, L3, L4, …
Eval("Sum( L[1] )", StrReplace( StrListRange(NumItems), ",", ",L" )) = ?
L_{1}, L_{2}, L_{3}, L_{4}, …
Eval("Sum( L_{[1]} )", StrReplace( StrListRange(NumItems), ",", "},L_{" )) = ?
A_{1}, A_{2}, A_{3}, A_{4}, …
Eval("Sum( A_{[1]} )", StrReplace( StrListRange(NumItems), ",", "},A_{" )) = ?
One final improvement to the use of the sum expression is to assign the expression to an expression variable. An expression variable stores the actual expression rather than the result of the
expression, this allows you to easily reuse the expression variable as if it were a function. To specify that the value of a variable should be stored as an expression prefix the variable name with
the $ symbol.
$L_{sum} = Eval("Sum( L_{[1]} )", StrReplace( StrListRange(NumItems), ",", "},L_{" ))
L_{1} = 1 m
L_{2} = 4 m
L_{3} = 3 m
NumItems = 1
L_{sum} = ? m (1 m)
NumItems = 3
L_{sum} = ? m (8 m)
L_{2} = 8 m
L_{sum} = ? m (12 m) | {"url":"https://support.tekla.com/article/how-to-sum-variables","timestamp":"2024-11-04T11:32:18Z","content_type":"text/html","content_length":"39712","record_id":"<urn:uuid:e570de39-e797-42b1-97fc-2750de8a92ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00445.warc.gz"} |
Tohoku University - Page 2
Researchers at Tohoku University, the University of Messina and the University of California, Santa Barbara (UCSB) have developed a scaled-up version of a probabilistic computer (p-computer) with
stochastic spintronic devices that is suitable for hard computational problems like combinatorial optimization and machine learning.
The constructed heterogeneous p-computer consisting of stochastic magnetic tunnel junction (sMTJ) based probabilistic bit (p-bit) and field-programmable gate array (FPGA). ©Kerem Camsari, Giovanni
Finocchio, and Shunsuke Fukami et al.
A p-computer harnesses naturally stochastic building blocks called probabilistic bits (p-bits). Unlike bits in traditional computers, p-bits oscillate between states. A p-computer can operate at
room-temperature and acts as a domain-specific computer for a wide variety of applications in machine learning and artificial intelligence. Just like quantum computers try to solve inherently quantum
problems in quantum chemistry, p-computers attempt to tackle probabilistic algorithms, widely used for complicated computational problems in combinatorial optimization and sampling.
Read the full story Posted:
Dec 08,2022 | {"url":"http://www.spintronics-info.com/tags/tohoku-university?page=1","timestamp":"2024-11-12T10:03:21Z","content_type":"text/html","content_length":"43327","record_id":"<urn:uuid:65d85cb0-09f5-4369-ab63-6896604dfe29>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00114.warc.gz"} |
Feedback between node and network dynamics can produce real-world network properties
Real-world networks are characterized by common features, including among others a scale-free degree distribution, a high clustering coefficient and a short typical distance between nodes. These
properties are usually explained by the dynamics of edge and node addition and deletion. In a different context, the dynamics of node content within a network has been often explained via the
interaction between nodes in static networks, ignoring the dynamic aspect of edge addition and deletion. We here propose to combine the dynamics of the node content and of edge addition and deletion,
using a threshold automata framework. Within this framework, we show that the typical properties of real-world networks can be reproduced with a Hebbian approach, in which nodes with similar internal
dynamics have a high probability of being connected. The proper network properties emerge only if an imbalance exists between excitatory and inhibitory connections, as is indeed observed in real
networks. We further check the plausibility of the suggested mechanism by observing an evolving social network and measuring the probability of edge addition as a function of the similarity between
the contents of the corresponding nodes. We indeed find that similarity between nodes increases the emergence probability of a new link between them. The current work bridges between multiple
important domains in network analysis, including network formation processes, Kaufmann Boolean networks and Hebbian learning. It suggests that the properties of nodes and the network convolve and can
be seen as complementary parts of the same process.
• Hebbian learning
• Neural networks
• Scale-free
• Social networks
• Stochastic processes
Dive into the research topics of 'Feedback between node and network dynamics can produce real-world network properties'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/feedback-between-node-and-network-dynamics-can-produce-real-world","timestamp":"2024-11-03T23:35:20Z","content_type":"text/html","content_length":"51531","record_id":"<urn:uuid:5fe370a2-5651-432c-ae91-9f4c7af4a153>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00384.warc.gz"} |
Unit Conversion
Metric to Imperial Conversion
This site provides unit conversion, metric conversion, Unit Conversion, unit conversion calculator, Unit Conversion Calculator, unit conversions, unit converter, Units, units conversion, units
conversions, units converter.
Weights and Measures
Weights & Measures, weights & measures, weights and measures, units of measure.
Metric to US Conversion
US Customary, US to metric, US to Metric, US to S.I., US to SI, US to Imperial, US to British, US to English, U.S., Unit, US, metric to US, Metric to US, metric to British, Metric to British, metric
to English, Metric to English, Metric to Imperial, metric to Imperial, Metric to imperial, metric to imperial, metric to S.I., metric to SI, metric units conversion.
Imperial to Metric Conversion
British, British to metric, British to Metric, British to metric, Conversion, conversion, English, English to metric, English to Metric, Imperial, imperial, Imperial to Metric, Imperial to metric,
imperial to Metric, imperial to metric, Imperial Units, imperial units, Metric, metric, Old English measures, S.I., S.I. to Metric, S.I. to metric, S.I. Units, SI, SI to metric, and SI Units
Units of Measure
A brief but by no means complete list of categories of units usage, units of measure supported include, Acceleration, Angle, Angular Acceleration, Angular Velocity, Area, Capacitance, Charge,
Current, Density, Dynamic Viscosity, Energy, Flow, Mass & Volume, Force, Fraction, Mass & Vol., Illumination, Inductance, Kinematic Viscosity, Length, Length (Ancient), Length (Foreign), Linear
Density, Luminance, Luminous Intensity, Magnetic Flux, Magnetic Induction, Mass, Mileage/Fuel Usage, Moment of Inertia, Number, Potential, Power, Pressure, Radioactivity, Solid Angle, Specific Heat,
Speed, Surface Tension, Temperature, Thermal Conductivity, Time, Torque, Volume, Volume (Ancient), Volume (Foreign), Weight, Weight (Ancient), and Weight (Foreign).
Sample Units
acre, anker, are, bag, barrel, bin, binary, bushel, butt, cable, centare, chain, coombe, count, cubit, cup, deal, drachm, dram, ell, fathom, firkin, foot, foot pound second, fps, gallon, gill, hand,
hectare, hide, hogshead, homestead, iu, jeroboam, jigger, kilderkin, kilogram, kilogramme, knot, league, line, link, liter, litre, meter, metre, nail, nails, nautical mile, pace, palm, peck, perch,
pint, pipe, point, pole, pottle, pound, puncheon, quart, quarter, quartersection, rod, rood, second, section, strike, tierce, township, tun.
Applications & Areas of Use
agriculture, algebra, anglo saxon measures, apparel, archeology, arithmetic, astronomical distance, astronomy, atomic number, atomic weight, automotive, boiling point, bra sizes, breadth,
calculation, calculator, calculators, canadian measures, chemicals, chemistry, clothing, clothing sizes, communications, computation, compute, computers, concentration, constants, construction,
cooking measures, data transfer, decimals, division, domestic science, dress sizes, duodecimals, education, electrical, electrical color codes, electron configuration, electron shells, electronics,
elements, engineering, engineering units, equivalents, factors, farming, figures, figuring, finance, finance interest, fineness, food measures, fineness, foreign length, foreign volume, foreign
weight, fuel consumption, fuel rate, fuel usage, gas consumption, gas usage, gasoline consumption, gasoline usage, geometry, hardware, heat index, hex, hexadecimal, hurricane force, information,
international units, internet, kitchen measures, liquor, meteorology, number, octal, old english, old english measures, periodic table, periodic table of elements, physical constants, physics, pitch,
planetary distances, planetary mass, planets, publishing, purity, research, resources, roman numerals, roof pitch, rope density, rope weight, science, shoe sizes, size, sizes, sizing, spirits,
sporting measures, surface area, telecommunications, thread density, thread weight, tire sizes, tools, type sizes, typography, volume flow rates, volume fraction, weather, wire density, wire weight,
yarn counts, yarn density, yarn weight.
Unit,Units,Metric,Imperial,US Customary,SI,S.I.,SI Units,Conversion,fps,Foot, Pound,Second,metre,meter,kilogram,kilogramme,Converter,Unit Conversion,Units Conversion, Unit Conversions,Metric
Units,Imperial Units,Unit Conversion Calculator,Tables,Units,Measures, Acceleration,Angle,Angular Acceleration,Angular Velocity,Area,Capacitance,Charge,Current,Density, Dynamic
Viscosity,Energy,Fineness or Purity,Flow Rates (Mass & Volume),Force,Illumination, Inductance,Kinematic Viscosity,Length,Linear Density,Luminance,Luminous Intensity,Magnetic Flux, Mass,Mileage,Moment
of Inertia,Number,Potential,Power,Pressure,Radioactivity,Solid Angles, Specific Heat,Speed,Surface Tension,Temperature,Thermal Conductivity,Time,Torque,Volume, Yarn Counts,Old English
Measures,rod,pole,perch,cable,chain,nautical mile,cubit,ell,fathom,hand, knot,league,line,link,nail,pace,palm,point,quarter,bag,barrel,firkin,kilderkin,hogshead,puncheon,
tierce,liter,litre,meter,metre,gallon,gill,jigger,jerboam,peck,cup,dram,drachm,pipe,butt,tun, anker,gill,pottle,peck,bushel,coombe,strike,puncheon,quart,pint,deal,tun,acre,are,centare,
hectare,hide,homestead,quartersection,rood,section,township,Old English measures
Your browser does not support or has Javascript disabled. This site relies heavily on Javascript and you will not be able to make use of the facilities without it. | {"url":"http://footrule.com/","timestamp":"2024-11-14T21:49:24Z","content_type":"text/html","content_length":"17485","record_id":"<urn:uuid:2878cd1b-a050-40ed-b649-5b30886a54af>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00261.warc.gz"} |
Linear Constraints
What Are Linear Constraints?
Several optimization solvers accept linear constraints, which are restrictions on the solution x to satisfy linear equalities or inequalities. Solvers that accept linear constraints include fmincon,
intlinprog, linprog, lsqlin, quadprog, multiobjective solvers, and some Global Optimization Toolbox solvers.
Linear Inequality Constraints
Linear inequality constraints have the form A·x ≤ b. When A is m-by-n, there are m constraints on a variable x with n components. You supply the m-by-n matrix A and the m-component vector b.
Pass linear inequality constraints in the A and b arguments.
For example, suppose that you have the following linear inequalities as constraints:
x[1] + x[3] ≤ 4,
2x[2] – x[3] ≥ –2,
x[1] – x[2] + x[3] – x[4] ≥ 9.
Here, m = 3 and n = 4.
Write these constraints using the following matrix A and vector b:
$\begin{array}{l}A=\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& -2& 1& 0\\ -1& 1& -1& 1\end{array}\right],\\ b=\left[\begin{array}{c}4\\ 2\\ -9\end{array}\right].\end{array}$
Notice that the “greater than” inequalities are first multiplied by –1 to put them in “less than” inequality form. In MATLAB^® syntax:
A = [1 0 1 0;
0 -2 1 0;
-1 1 -1 1];
b = [4;2;-9];
You do not need to give gradients for linear constraints; solvers calculate them automatically. Linear constraints do not affect Hessians.
Even if you pass an initial point x0 as a matrix, solvers pass the current point x as a column vector to linear constraints. See Matrix Arguments.
For a more complex example of linear constraints, see Set Up a Linear Program, Solver-Based.
Intermediate iterations can violate linear constraints. See Iterations Can Violate Constraints.
Linear Equality Constraints
Linear equalities have the form Aeq·x = beq, which represents m equations with n-component vector x. You supply the m-by-n matrix Aeq and the m-component vector beq.
Pass linear equality constraints in the Aeq and beq arguments in the same way as described for the A and b arguments in Linear Inequality Constraints.
Related Topics | {"url":"https://www.mathworks.com/help/optim/ug/linear-constraints.html","timestamp":"2024-11-12T14:04:50Z","content_type":"text/html","content_length":"70860","record_id":"<urn:uuid:d4cc2d2c-874c-4bd9-9f4f-92eebcecd6f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00075.warc.gz"} |
Name Syntax Description
This console command will give you the resource with the specified ID. You can also specify an amount of the resource you wish to add - e.g. 100.
GiveResource giveresource [resource You must have at least one of the resource you wish to add in order for this to work. Use the additem command to add a resource if you don't already
id] [amount] have one.
additem [item id] This console command will give you the item with the specified ID. You can optionally specify an amount of the item that you wish to add. See IDs at
AddItem [amount] commands.gg/xcom2/additem.
GiveHackReward givehackreward [hack This console command will give you the hack reward with the specified ID. See IDs at commands.gg/xcom2/givehackreward.
reward id] Mission
GiveActionPoints giveactionpoints [amount] This console command will give the unit that you currently have selected the specified amount of action points.
This console command will give you a scientist of the specified level. Note that you can only have one scientist at once, so this will replace your
GiveScientist givescientist [level] existing scientist.
This console command will give you an engineer of the specified level. Note that you can only have one engineer at once, so this will replace any
GiveEngineer giveengineer [level] existing engineer that you have.
GiveTech givetech [tech id] This console command will research the technology with the specified ID. See all technology IDs at commands.gg/xcom2/givetech.
This console command will give you the facility with the specified ID at the specified position. See commands.gg/xcom2/givefacility for facility IDs
givefacility [facility and position IDs (3-14). If there is an existing facility, debris, or ongoing construction at the position you specify, This console command will
GiveFacility id] [avenger/map index] not do anything. The game may not update instantly after using This console command - switching screens (e.g. going into the Geoscape and back) will
apply changes.
setsoldierstat [stat id] This console command will set the stat of the specified solider to the specified value. See commands.gg/xcom2/setsoldierstat for stat IDs. If you
SetSoldierStat [value] [soldier name] [0 are using the WOTC DLC, you will also need to specify the 0/1 argument at the end of the command.
/ 1] Avenger
makesoldieraclass This console command sets the class of the soldier with the specified name. Note that this will demote your soldier to squaddie rank, and that you
MakeSoldierAClass ["soldier name"] [class should make a save before using This console command, as some classes can break your game. See class IDs at commands.gg/xcom2/makesoldieraclass.
id] Avenger
RemoveFortressDoom removefortressdoom This console command removes the specified amount of doom from the Avatar Project.
[amount] Avenger
forcecompleteobjective This command will complete the specified objective for your current mission. See commands.gg/xcom2/forcecompleteobjective for objective IDs.
ForceCompleteObjective [objective id] Complete all of your mission objectives with this command before using the endbattle command to win your current mission.
endbattle endbattle [0 / 1] This command will end your current mission. If all objectives are completed, you will win the mission, otherwise you will fail. Complete all mission
objectives with the ForceCompleteObjective command before using this to win your mission.
bondsoldiers ["soldier This console command bonds (true), or unbonds (false), the two specified soldiers.
BondSoldiers name"] ["soldier name"] Avenger War of the Chosen
[true / false]
SkipAI skipai This console command makes the AI skip a turn (i.e. this will end the AIs turn and make it your turn).
This console command enables and disables (toggles) god mode for all of your squad (not AI). When in god mode, your squad will not have to reload,
PowerUp powerup and will not take any damage (unlimited health).
TakeNoDamage takenodamage This console command enables and disables (toggles) invincibility for all of your soliders.
This console command enables and disables (toggles) unlimited action points for both you and the AI (well, APs will not be used up). Note that if
ToggleUnlimitedActions toggleunlimitedactions you use this on the AIs turn, the AI will not run out of action points.
This console command enables and disables (toggles) 100% chance for both your squad and the AI to hit critical hits - i.e. with this enabled, every
ForceCritHits forcecrithits hit will be a critical hit.
GiveContinentBonus givecontinentbonus This console command will give you the specified continent bonus. See continent bonus IDs at commands.gg/xcom2/givecontinentbonus.
[continent bonus id] Avenger
This console command enables and disables (toggles) unlimited ammunition for both your squad and the AI. Turn on at the start of your turn, and off
ToggleUnlimitedAmmo toggleunlimitedammo at the end, to avoid giving the AI unlimited ammo.
ToggleFOW togglefow This console command enables and disables (toggles) Fog of War (FoW). Fog of War is the fog that covers the map in out-of-reach places.
This console command enables and disables (toggles) the concealment of your squad. Aliens that are already aware of any solider(s) location, will
ToggleSquadConcealment togglesquadconcealment remain aware regardless of whether this is on or off.
TTC ttc This console command will teleport the unit that you currently have selected to the location in the game that your mouse cursor is over.
TATC tatc This console command will teleport all units to the location in the map that your mouse cursor is over.
LevelUpBarracks levelupbarracks [amount] This console command will level up all soldiers in your barracks by the specified amount of levels.
HealAllSoldiers healallsoldiers This command heals all soldiers currently in your barracks to full HP.
GiveFactionSoldiers givefactionsoldiers This console command will give you a soldier from each Faction in the War of the Chosen. The three Factions are Reaper, Skirmisher, and Templar.
Avenger War of the Chosen
GiveAbilityCharges giveabilitycharges NOTE: This command has been reported as "buggy" - make sure you save your game before using it. This command should add 100 charges to all of your
abilities, excluding class abilities.
RestartLevel restartlevel This command will restart your current mission.
This console command will restart your current mission with the same seed that it was generated with. This means all of the random aspects of the
RestartLevelWithSameSeed restartlevelwithsameseed game (e.g. spawn locations) will be the exact same as they were when you first started.
pause pause This command will toggle the pause state for the game (i.e. if paused, it will unpause, if not paused, it will pause).
sloMo slomo [multiplier] This command will fast forward or slow down the game - i.e. it will change the speed the game runs at. A multiplier of 2 would make everything in
the game be twice as fast, a multiplier of 0.5 would make the game run in slowmotion, half as fast as it usually would.
screenshot screenshot This command will take a screenshot of your game and save it to Documents\my games\XCOM2\XComGame\Screenshots\PCConsole.
listtextures listtextures This command lists all texture files that are currently loaded by the game.
listsounds listsounds This command lists all sound files that are currently loaded by the game.
ToggleRain togglerain This command will enable or disable (toggle) rain.
ChangeList changelist This command will print to the console log a list of the most recent changes in the game. | {"url":"https://commands.gg/xcom2","timestamp":"2024-11-13T15:23:34Z","content_type":"text/html","content_length":"66049","record_id":"<urn:uuid:09e51970-3653-4465-a579-e4899102fa1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00036.warc.gz"} |
Errorea "h"-z ala "h"-rik gabe?
Error with or without h?
Text created by automatic translator Elia and has not been subsequently revised by translators.
Elia Elhuyar
Matematikan lizentziatua eta doktoregaia
In some languages it has sound, in others it does not. In Basque, although it is there when it is cold, it does feel, but it is not heard. If the wind has it in Basque and, although its worm is
heard, there is no "h". The mistake is not in Basque, but it can be carried with mathematics. And the "H" also has its silent function in mathematics.
Error with or without h?
01/04/2010 | Alberdi Celaya, Elisabete | Matematikan PhD graduate
Although in some languages it is not pronounced, "h" performs its function in any language, since a word is not the same without "h" or "h". It is not the same to catch something (turning it into a
thread) as to act on something; that something is “concave” (strings) or that someone is “contrary”. Similarly, the error is also different without h or h. Sometimes the word with h and without h can
be connected even if they are of different meaning. It is about listening to hear in English and if you lose the “h” becomes ear, it seems that you lose the “h” to not disturb the ear. The error that
is calculated in h and without h are also connected, since both are measures of something that has not been done correctly. Also important is the place that occupies the letter "H" in the word: the
function can be concave or convex; the child, on the other hand, is Chinese or proja. The commutative property of multiplication makes mathematics indifferent to the position of “h” in error, as it
depends on whether or not “h” exists.
Different error estimation
Graphs of estimates and actual errors resulting from the resolution of two differential equations of analytical result known with a 5(4) Runge-Kutta method. The superior differential equation is
resolved with a tolerance of 0.01 and the lower of 0.1. Method 1 should take small steps not to overcome the established tolerance. In addition, despite the lower tolerance used in the 1st, there is
a greater accumulation of errors than in the 2nd. Graf. : Elisabete Alberdi.
If a differential equation cannot be solved analytically, it can be solved by numerical methods. The analytical result is accurate, while the result obtained by the numerical method is approximate
and is built step by step. The error in releasing a differential equation by numerical method is the difference between the exact and approximate result. A concept that theoretically has no more
difficulties than a subtraction, when it must be calculated in practice becomes a complex concept. But what is the problem? We do not know the exact results of many differential equations. Therefore,
by ignoring the exact result, it is impossible to calculate the error we are making when using the numerical method. Although we do not know the error, we know that the approximate result obtained
using high order numerical methods is more accurate than that obtained with lower order numerical methods. However, whenever a step is taken by the numerical method, a measure is usually used that
tells us whether the result obtained in that step is valid or not. What is this measure if we have said that the error cannot be calculated without knowing the concrete result?
The measure used is the estimation of the error. A very useful error estimation is calculated as the difference of results obtained using numerical methods of successive order, based on the
elimination of the result obtained by using a numerical method of order ( n+1), which is known as local extrapolation. When we use it, at each step we demand that the difference between the results
obtained by two consecutive methods be less than a previously established tolerance. If the condition is met, the result given by the order method ( n+1) is accepted, otherwise it will be necessary
to repeat the step operations and test it with another step less than that used previously. If we wanted to use local extrapolation of any of the measurements to be carried out in a laboratory, two
measurements of the amount to be measured using two different precision devices would be made. Since the precisions of these devices must be consecutive, when a higher precision device is available,
the second one will be the one with greater precision among the devices with less precision than the first. In this way, we would ensure that we are using devices with successive precision. The
measure should be repeated until the difference between the two measures is less than a given value and, if the condition is met, the result of the most accurate device will be accepted. As can be
seen in the laboratory example, ignorance of the actual result is paid by doubling the number of calculations.
Local extrapolation is therefore a safe but expensive resource at the same time. Since its potential is based on comparing the results obtained with two different methods, it will always mean an
increase in the number of calculations. Taking advantage of the certainty of local extrapolation and accepting the increase in the number of calculations, many of the creators or parents of numerical
methods have focused on the design of methods that could make this number of calculations not double, that is, they have aimed to give birth to two methods of successive order with the smallest
possible differences. In this sense, by using a different set of constants in a single operation, the methods that perform a method of order n or ( n+1) are very useful, since they manage to optimize
the difference between two methods of successive order. Example of this are the Runge-Kutta methods introduced. In them, it is sufficient that, in addition to the operations performed to obtain the
result by the order method ( n+1), a single operation is carried out to obtain the result provided by the order method n. They therefore offer an economical option to obtain results from different
orders. There is only another cheaper option than getting both at the price of one. But that is impossible, because different methods must be differentiated into something.
Estimation with h without h
Steps sizes given by the algorithm corresponding to a Runge-Kutta 5(4) method using error estimates with or without h in three differential equations. When using error estimation with “H”, larger
step sizes can be given. Graf. : Elisabete Alberdi.
In numerical methods, the letter "h" is used to indicate the step size. Knowing the exact result of the initial point, a step of size "h" is given in which the approximate result of the differential
equation is calculated by the operations that the numerical method requires us. Using the approximate result obtained at the new point the process is repeated. To measure the error in each step,
estimates of two types are usually used: one, called estimation per step unit, pure difference between the results obtained by consecutive methods (without h); and another, called estimation per
step, which is obtained by multiplying the difference between the results by the size of the step (“h”). Of course, the estimate with h and without h are not the same. Suppose we have an estimate of
the error per step unit. Since the sizes of the steps usually used are smaller than the unit, the estimate obtained by multiplying this estimate by “h” is less than the one we have previously.
Consequently, it can happen that an estimate not inferior to the tolerance established without multiplying by “h”, after multiplying by “h”, below the tolerance, that is, that the smallness of the
step has helped to pass the barrier. In addition, the estimation multiplied by “h” gives a special advantage to small steps, since by multiplying the estimate by a lower “h”, the new estimate will be
lower. In other words, the estimation per step unit (without h) places the barrier at the same height. The estimate obtained by multiplying by “H” lowers the barrier to all steps. The smallest steps
are the small ones.
Sometimes the drop in the barrier will not be affected, as there will be estimates that would pass the initial height of the barrier without help. But there will be leaps that cannot exceed the
initial height of the barrier and that will be affected by the descent of the barrier. Consequently, the step that should be repeated using another estimate will be considered valid. Repeating a step
involves testing with smaller steps, and if the steps are small, more is needed. Therefore, when we use the estimates in which we are lowering the barrier, fewer steps will be taken and therefore
greater than those that do not fall. The coin, however, has another aspect, since it may happen that the descent of the barrier does not have the desired result. This can negatively influence the
final error and drastically reduce the quality of the approximate result we get. In general, the estimate that is calculated without h will lead to the need to take smaller steps, but the
accumulation of real error will also be lower.
Estimation error in h or without h
Differential equations can be rigid or not rigid. The practical definition of the rigid differential equation is that of an equation that must take many steps to an algorithm. Although the word
"much" does not indicate specific numbers, 100 steps are not many and 3,000 are many. When the problem is rigid, it is preferred to use an estimation without “h”, since with the estimation with “h” a
much better result will be obtained than that obtained with the estimation without “h”. Although the problem is not rigid, the result obtained with an estimation without h will be, in most cases,
better than that obtained with h, but the result --difference between both - will not be so spectacular, that is, the additional work that involves the use of an estimation without h in non-rigid
differential equations will not generate much brightness in the solution.
Differences between approximate results and analytical results obtained using the Runge-Kutta 5(4) method and estimates with or without h. In case 1 the real error by points decreases, in the 2 has
incidents and in the 3 increases. Graf. : Elisabete Alberdi.
In languages it is common for a word to lose or gain the "h" according to the time. Examples of this are the construction of walls sometimes with or without h, or the healing of patients in hospitals
with or without h. However, the function or meaning of the words that have stuck or removed the “h” by the time has remained the same. The subject of errors with h and without h in mathematics does
not depend on the time: there has always been coexistence between both, and both are necessary because the function of one and the other has never been the same. As we consult in the dictionary
whether a word has an “h” or not, to decide whether or not to use an “h” in the estimation of the error we will have to “consult” the problem to be released, since the key to the success we will
obtain in the result will be that the estimation of the error has an “h” or not.
Differences between approximate and analytical results obtained using the Runge-Kutta 5(4) method and estimates with or without h. In the rigid problem it is observed that the estimation without h is
much better than that with h. Graf. : Elisabete Alberdi.
Butcher, J. C. Numerical methods for ordinary differential equations, John Wiley Sons Ltd. Chichester (2008).
Dormand, J. R.; Prince, P. J. A family of embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics, 6 (1), (1980), 19-26.
Hairer, E.; Norsett, S. P.; Wanner, G.: Solving Ordinary differential equations I, Nonstiff problems, Springer, 1993.
Higham, Desmond J.: Global error versus tolerance for explicit Runge-Kutta methods, IMA Journal of Numerical Analysis (1991) 11, 457-480.
Lambert, J. D. Numerical Methods for ordinary differential systems, John Wiley Sons Ltd. Chichester, 1991.
Shampine, L. F.; Gladwell, I.; Thompson, S.: Solving ODEs with Matlab. Cambridge University Press (2003).
Shampine, L. F.; Reichelt, M. W: The MATLAB ODE suite, SIAM J. Sci. Comput. 18 (1) (1997) 1-22.
Skeel, Robert D.: Thirteen ways to estimate global error, Numer. Math. 48, (1986) 1-20.
The MathWorks Inc.: http://www.mathworks.com.
Alberdi Celaya, Elisabete | {"url":"https://aldizkaria.elhuyar.eus/gai-librean/errorea-h-z-ala-h-rik-gabe/en","timestamp":"2024-11-04T19:02:00Z","content_type":"text/html","content_length":"46932","record_id":"<urn:uuid:c6b2db30-0e8c-447e-b713-b52cb529eeca>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00329.warc.gz"} |
Replace Non-Coprime Numbers in Array | CodingDrills
Replace Non-Coprime Numbers in Array
Given an array of numbers nums, replace pairs of adjacent non-coprime numbers in the array with their least common multiple (LCM) until no more such pairs exist. Return the modified array.
Ada AI
I want to discuss a solution
What's wrong with my code?
How to use 'for loop' in javascript?
javascript (node 13.12.0) | {"url":"https://www.codingdrills.com/practice/replace-non-coprime-numbers","timestamp":"2024-11-10T11:42:09Z","content_type":"text/html","content_length":"13681","record_id":"<urn:uuid:00af6f14-4e30-4437-9d45-a2af9e725237>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00238.warc.gz"} |
Richa is sitting in a merry-go-round. the time period of circ-Turito
Are you sure you want to logout?
Richa is sitting in a merry-go-round. The time period of circular motion if it forms a circle of 56 m radius. The speed of rotation is 40 m/s.
A. 2.8 pi s
B. 280 pi s
C. 0.14 pi s
D. 28 pi s
Using formula for time period of uniform circular motion.
The correct answer is: 2.8 pi s
Speed, v = 40 m/s
Time period, t = ?
Radius, r = 56 m
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/Physics-richa-is-sitting-in-a-merry-go-round-the-time-period-of-circular-motion-if-it-forms-a-circle-of-56-m-radi-q6feb2177","timestamp":"2024-11-12T06:21:31Z","content_type":"application/xhtml+xml","content_length":"438577","record_id":"<urn:uuid:b1be2fd1-09c0-40ce-8900-5773a676f460>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00077.warc.gz"} |
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Could a 5-D black hole break Einstein's theory?
Poor Einstein. Where gravitational waves created by two black holes just gave us a massive proof for the physicist’s theory of general relativity, that theory could be broken – by another black hole.
Using supercomputers, a group of UK scientists have just created a black hole that defies Einstein’s theory. Admittedly, this hole is really a ring, and had to be simulated in five dimensions.
Theoretical physicists have long argued over whether “naked singularities” could exist.
Your average singularity – gravity so intense space and time break down – is predicted by Einstein’s theory of relativity to occur at the core of every black hole. But the theory’s equations only
hold up if that singularity is tucked in by an event horizon – the “point of no return” where no light or matter can escape from the black hole’s gravitational tug. This rule is known as “cosmic
If a singularity could exist outside of an event horizon, it would be a visible representation of an object collapsed to an infinite density, as light would no longer be sucked away from around it.
At the same time it would break the known laws of physics. Now that’s something worth seeing.
“If naked singularities exist, general relativity breaks down,” said study author Saran Tunyasuvunakool. “And if general relativity breaks down, it would throw everything upside down, because it
would no longer have any predictive power – it could no longer be considered as a standalone theory to explain the universe.”
The scientists simulated Einstein’s complete general relativity theory in five dimensions using supercomputers, to see if it would be possible to create a naked singularity.
They modelled a black hole bizarrely shaped like a thin ring. If the ring was thin enough, the simulation showed, instead of collapsing into a black hole sphere, the ring wobbled and “bulges”
connected by strings formed along it.
As the strings became thinner with time, they eventually became so thin they pinched off – how a thin stream of water from a tap breaks into little droplets. The bulges then broke off as a series of
miniature black holes, forming a naked singularity at the pinch points, the authors say.
Not convinced a 5-D black ring is possible in the first place?
This is extreme theoretical physics so, in one sense, anything goes, but probing many dimensions is legitimate enough – general relativity never specified how many dimensions there are in the
Universe, even though we can only directly perceive three. Add the fourth dimension, time, and you end up with spacetime. But according to some branches of physics, such as string theory, the
Universe could be made up of as many as 11 dimensions.
“We’re pushing the limits of what you can do on a computer when it comes to Einstein’s theory,” said Tunyasuvunakool. “But if cosmic censorship doesn’t hold in higher dimensions, then maybe we need
to look at what’s so special about a four-dimensional universe that means it does hold.”
The work was published in Physical Review Letters. | {"url":"https://cosmosmagazine.com/science/physics/could-a-5-d-black-hole-break-down-einsteins-theory/","timestamp":"2024-11-09T04:54:25Z","content_type":"text/html","content_length":"87566","record_id":"<urn:uuid:e307e7c6-3fe5-42ca-8f84-c44df67cab77>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00349.warc.gz"} |
Mathematics - iGCSE Mathematics A Edexcel | Exam Revision Content | Adapt app
Orders and Operations
Play audio lesson
Orders and Operations
• "Orders and Operations" refers to the sequence in which calculations in a mathematical expression should be carried out.
• BIDMAS/BODMAS is an acronym used to remember the order of operations: Brackets, Indices/Powers or Order, Division and Multiplication (from left to right), Addition and Subtraction (from left to
• Always perform operations in brackets first, no matter their positioning in the equation.
• After brackets, treat powers or indices next. Review exponents (squared, cubed etc.) and roots (square root, cube root etc.) calculations.
• Division and multiplication are on the same 'tier'. They should be solved from left to right in the equation or expression.
• The last operations you should handle are addition and subtraction (again, left-to-right).
• If single numbers appear next to brackets, this implies multiplication. For example, 2(3+4) equals 2*7=14.
• A common pitfall is to incorrectly conduct operations from right-to-left or forgetting to handle operations in brackets first. Ensure you can accurately apply BIDMAS/BODMAS to avoid these errors.
• Practice using the order of operations with fractions, decimals and negative numbers to enhance your understanding.
• Algebra also uses the rules of order of operations. Remember that letters (variables) can take the place of unknown numbers.
• Applying the rules of order consistently ensures that you always carry out calculations correctly, which is key to success in your Numbers revision. | {"url":"https://getadapt.co.uk/revision-content/mathematics/igcse-mathematics-a-edexcel","timestamp":"2024-11-03T07:35:52Z","content_type":"text/html","content_length":"144122","record_id":"<urn:uuid:6d3bd4f8-39a8-42b0-80ea-b92f3cde4c10>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00583.warc.gz"} |
Multiplication Practice Worksheet
Mathematics, particularly multiplication, develops the keystone of countless scholastic techniques and real-world applications. Yet, for several students, mastering multiplication can pose a
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Multiplication Timed Tests The Curriculum Corner 123 Free Printable Multiplication Speed
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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
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Our multiplication worksheets are free to download easy to use and very flexible These multiplication worksheets are a great resource for children in Kindergarten 1st Grade 2nd Grade 3rd Grade 4th
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Exactly how frequently should pupils exercise utilizing Multiplication Practice Worksheet?
Consistent technique is crucial. Regular sessions, preferably a few times a week, can yield significant improvement.
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Worksheets are an important tool yet needs to be supplemented with varied discovering methods for extensive ability growth.
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What is the differential equation for half-life?
τ=1kln2. (Figure 4.1. 2 ). The half-life is independent of t0 and Q0, since it is determined by the properties of material, not by the amount of the material present at any particular time.
What are the 5 types of nuclear decay?
The 5 types of radioactive decay are:
• α decay.
• β decay.
• γ decay.
• Positron emission.
• Electron capture.
What is the decay constant formula?
Suppose N is the size of a population of radioactive atoms at a given time t, and dN is the amount by which the population decreases in time dt; then the rate of change is given by the equation dN/dt
= −λN, where λ is the decay constant.
How do you calculate decay rate?
In mathematics, exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula y=a(1-b)x wherein y is the
final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed.
How do you calculate decay from half-life?
The time required for half of the original population of radioactive atoms to decay is called the half-life. The relationship between the half-life, T1/2, and the decay constant is given by T1/2 =
What are the three types of nuclear decay?
17.3: Types of Radioactivity: Alpha, Beta, and Gamma Decay
• Alpha Decay.
• Beta Decay.
• Gamma Radiation.
What is the formula for decay constant?
What is the nuclear decay constant?
decay constant, proportionality between the size of a population of radioactive atoms and the rate at which the population decreases because of radioactive decay.
What is decay constant in nuclear physics?
Definition. The decay constant (symbol: λ and units: s−1 or a−1) of a radioactive nuclide is its probability of decay per unit time. The number of parent nuclides P therefore decreases with time t as
dP/P dt = −λ. The energies involved in the binding of protons and neutrons by the nuclear forces are ca.
How do you calculate growth and decay rate?
The constant k is called the continuous growth (or decay) rate. In the form P(t) = P0bt, the growth rate is r = b − 1. The constant b is sometimes called the growth factor.
How is nuclear radioactive decay defined?
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A
material containing unstable nuclei is considered radioactive.
Is decay rate the same as half-life?
The rate of decay, or activity, of a sample of a radioactive substance is the rate of decrease in the number of radioactive nuclei per unit time. The half-life of a reaction is the time required for
the reactant concentration to decrease to one-half its initial value.
What are the different types of nuclear decay?
Three of the most common types of decay are alpha decay (α-decay), beta decay (β-decay), and gamma decay (γ-decay), all of which involve emitting one or more particles.
How do you calculate nuclear decay?
– Remove all known sources of radioactivity from the room. – Set the counter to zero. – Switch on and start a stop clock. – After 20 minutes switch off. Record the count. – Divide the count by 20 to
calculate the count rate per minute.
Which equation represents a spontaneous nuclear decay?
nuclear fusion which balanced equation represents a spontaneous radioactive decay carbon 14 emits a beta particle and it changed into nitrogen-14 which nuclear emission is negatively charged a beta
particle which nuclear emission has no charge and no mass gamma ray
What are the four types of nuclear decay in order?
Nuclear decay. an unstable nucleus undergoes a change and a reduction in energy in order to become more stable. What are the four types of nuclear decay? alpha decay, beta decay, positron emission,
electron capture. Alpha decay. nucleus emits an alpha particle. Alpha particle.
How to write a nuclear equation for alpha decay?
These are the equations that define how radioactive substances decay over time and are of the form N =N1 *e^-lambda*t, where t is time, N1 is the intital condition at t =0, and lambda is the decay
constant, inherent to every species. | {"url":"https://www.goodgraeff.com/what-is-the-differential-equation-for-half-life/","timestamp":"2024-11-12T03:17:00Z","content_type":"text/html","content_length":"54663","record_id":"<urn:uuid:94b2ec0a-571f-4afc-8e1e-7d53c8c30f62>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00171.warc.gz"} |
For queuing systems with moving servers, the control policy which means delays of a beginning service is introduced. In the capacity of efficiency index of systems is taken a customer’s average
waiting time before service. Although it seems that it is a paradoxical idea to introduce delays of beginning service, it is shown that for some systems it gives a gain in a customer’s average
waiting time before service. The class of queuing systems for which it is advisable to introduce delays is described. The form of an optimal function minimizing the efficiency index is found.
It is shown that if the intervals between neighbor services have exponential distribution, then the gain in a customer’s average waiting time before service equals 10% and independent of parameter of
exponential distribution. For uniform distribution such gain equals 3.5% and also independent of parameter of uniform distribution. The criterion to define for which systems the gain is greater are
given. Some open problems and numerical examples demonstrating theoretical results are given.
Keywords: queues with moving servers; a customer’s average waiting time; delay of beginning service; optimal function
1. Asmussen S. “Applied Probab. and Queues”. Springer-Verlag (2020).
2. Baccelli F and Bremaud P. “Elements of Queueing Theory, 2003”. Springer-Verlag, Berlin (2003).
3. Belyaev YuK., et al. “Markov approximation of stochastic model of motion on a two-road lane”. М., МАДИ (2002): 32 (in Russian).
4. Blank M. “Ergodic properties of a simple deterministic traffic flow model”. J. Stat. Phys 111 (2003): 903-930.
5. Borovkov AA. “Asymptotic methods in queuing theory”. John Willey Son, NY (1984).
6. Bozejko W and Bocewicz G. “Modelling and Performance Analysis of Cyclic Systems”. Springer, Studies in Systems, Decision and Control 241 (2020).
7. Cox D. “Renewal theory”. Chapman and Hall (1967).
8. Franken P., et al. “Queues and Point Processes”. Wiley, Chichester (1982).
9. Gazis DC. “Traffic Theory”. Berlin: Springer (2002).
10. Gnedenko BV and Kovalenko IN. “Introduction to Queuing Theory”. Birkhauser (1989).
11. Haight F. “Mathematical Theory of Traffic Flow”. Academic Press (1968).
12. Hajiyev AH. Mammadov TSh. “Mathematical models of moving particles and their Applic”. Lambert, Academic Publish, Germany (2013): 134.
13. Hajiyev AH and Mammadov TSh. “Cyclic queues with delays”. RAS, Doklady, Mathem 1 (2009).
14. Hajiyev AH and Mammadov TSh. “Mathematical models of moving particles and its application”. Theory of Probab. Appl 56.4 (2011): 1-14.
15. Kerner BS. “Introduction to Modern Traffic Flow Theory and Control”. Berlin: Springer (2009).
16. Khinchin AYa. “Mathematical theory of queues”. М (1963) (in Russian).
17. Kleinrock L. “Queuing systems”. John Wiley Sons 1.2 (2008).
18. Long Z., et al. “Dynamic Scheduling of Multiclass Many-Server Queues with Abandonment: The Generalized cµ/h Rule”. Oper. Res 68 (2020): 1218-1230.
19. Lee H-S and Srinivasan MM. “The shuttle dispatch problem with compound Poisson arrivals: controls at two terminals”. Queuing Systems 6 (1990): 207-222.
20. Moeschlin O and Poppinga C. “Controlling traffic lights at a bottleneck with renewal arrival processes”. Proc. Inst. Math. And Mech. Azerb. National Acad. Sci 14 (2001): 187-194.
21. Nagatani T. “The physics of traffic jams”. Rep. Prog. Phys 65 (2002): 1311-1356.
22. Nagel K, Wagner R and Woesler R. “Still flowing: Approaches to traffic flow and traffic jam modelling”. Oper. Research 51.5 (2003): 681-710.
23. Newell GF. “A simplified theory of kinematic waves in highway traffic, part I: General theory”. Transp. Research Part B. Methodological (1993).
24. Newell GF. “Applications of queuing theory”. London: Chapman and Hall (1982).
25. Renyi A. “On two mathematical models of the traffic on a divided highway”. J. Appl. Probab 1 (1964): 311-320.
26. Ross SM. “Average delay in queues with non-stationary Poisson arrivals”. J. Appl. Probab 15 (1978): 602-609.
27. Saati TL. “Elements of queueing theory, with applications”. McGrow Hill Book Comp (1961).
28. Shiryaev AN. “Probability”. Springer-Verlag, New York- Berlin-Heidelberg (2013): 580.
29. Zhou W, Huang W and Zhang R. “A two-stage queueing network on form postponement supply chain with correlated demands”. Appl. Math. Model 38 (2014): 2734-2743. | {"url":"https://primerascientific.com/journals/psen/PSEN-05-161","timestamp":"2024-11-06T01:35:24Z","content_type":"text/html","content_length":"26646","record_id":"<urn:uuid:d218dcef-7b89-4135-a024-d523bf800ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00160.warc.gz"} |
Alternatives to model diagnostics for statistical inference?
Consider the problem of making statistical inferences, as opposed to predictions. The product of statistical inference is a probabilistic statement about a population quantity, for example a 100(1-)%
confidence interval for a population median. In this context, the principal reason for diagnostics is to comfort ourselves about the quality of such inferences. For example, we would like to ensure
that the stated confidence level of an interval is roughly the population coverage. We can't compute the population coverage directly because we generally don't have access to an entire population.
Hence, we often attempt to verify the modeling assumptions that lead us to theoretically correct coverage.
In the prediction framework, we use model diagnostics to verify that the model fits well, which has a direct bearing on the quality of predictions. For example, a line generally does not approximate
a quadratic curve. However, it is possible to make accurate inferences about a linear approximation to a quadratic curve. Hence, model fit is not required to make quality inferences. Rather, the
requirement is that the associated probability statements are correct.
Assessing model diagnostics is an indirect mechanism to comfort ourselves about the quality of inferences. As an alternative, we might attempt a more direct check, for example, by constructing an
empirical estimate of coverage. We may then go further and adjust, or calibrate, the confidence interval to have the correct empirical coverage. These ideas are fundamental parts of the 'double
bootstrap', and 'iterated Monte Carlo' methods. For the sake of argument, I will state that this type of empirical check and calibration is sufficient to fully replace model diagnostics for
statistical inference. It is also my hypothesis that model diagnostics have been historically favored to iterative Monte Carlo methods (the double bootstrap appeared in the late 1980's) because the
latter is more computationally intensive. Current computational tools mitigate, but do not eliminate this concern.
I will present examples with R code in a later post.
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Max Int in Python: Understanding Maximum Integer Limits
When working with integers in Python, you should know the maximum value your code can handle. This will depend on whether you are using Python 2 or Python 3.
Python 2 has a Max Int constant (sys.maxint) that defines the maximum integer value. Python 3 has removed the maximum limit of integers and is only constrained by the system resources that the code
runs on.
This article explains the concept in the older and newer versions of Python. You’ll learn how to access and use the limits in Python 2 and 3 with sample code. You’ll also learn how to avoid errors
and memory overload with large numbers.
Let’s get started!
Quick Explanation of Integers in Python
Mathematical integers are whole numbers that can be positive, negative, or zero. They have unlimited precision, which means they can grow as large as the system’s memory can handle.
These three numbers are integers:
In contrast, floats represent real numbers and are written with a decimal point. A float can also be expressed in scientific notation. Here are examples of floats:
Python 2 Versus Python 3
One of the major changes from Python 2 to Python 3 was in handling integers. Most developers will work with Python 3 now, but you may encounter older code that works with large integers. It’s useful
to understand the differences between the two versions.
Integers in Python 2
Python 2 has two numeric types that can represent integers: int and long. The int type is limited by the maximum and minimum values it can store. The maximum is available with the constant sys.maxint
The long type can store larger numbers than the maximum integer size. If an operation on plain int values produces a value over sys.maxint, the interpreter automatically converts the data type to
Integers in Python 3
Python 3 does not have this limitation of size. The maxint constant was removed from the sys module in Python 3 when the int and long data types were merged.
The plain int type in Python 3 is unbounded, which means that it can store any integer value without a need for a separate long integer type.
This makes it more straightforward for programmers to deal with integers without worrying about the maximum possible value or switching between int and long.
Python’s Max Int: What It is and Why It Matters
Python’s max int refers to the maximum integer value that a Python interpreter can handle.
Some languages like C or Java have a fixed maximum size for integers based on 32-bit or 64-bit storage. Python is different in that it dynamically adjusts the number of bits based on the value to be
Python’s integers can keep growing in size as long as your machine has memory to support it. This is referred to as “arbitrary precision.”
This doesn’t mean Python can handle infinite numbers! There is always a practical limit because the system’s memory is finite.
However, this limit is generally so large that for most practical applications, it might as well be infinite.
How to Use Sys.MaxInt in Python 2
In Python 2, you can look at the maximum integer value defined by the sys.maxint constant like this:
import sys
print("The maximum integer value is: ", sys.maxint)
The constant is often used to define the upper limit for loops. This sample code ensures that the loop index doesn’t go beyond the maximum integer size.
import sys
for i in range(sys.maxint):
# do some stuff
You can also check user input to ensure that a number does not exceed the max value.
How to Use Sys.MaxSize in Python 3
You can use sys.maxsize in Python 3 as a replacement for sys.maxint in Python 2.
It’s important to understand that this doesn’t represent the maximum integer value that Python 3 can handle. The maxsize property represents the max value of an integer that can be used as an index
for Python’s built-in data structures, such as lists and strings.
This value depends on the available memory, so it may change between different systems or configurations.
The exact value of sys.maxsize is usually 2**31 – 1 on a 32-bit platform and 2**63 – 1 on a 64-bit platform. These are the maximum values that can be used for fixed-size integers on those platforms.
Here is an example of a function that uses sys.maxsize to avoid creating a list so large that it will fail due to lack of memory:
import sys
def create_list(input_number):
if input_number > sys.maxsize:
print("the requested size is too large.")
large_list = [0] * input_number
Remember to import the sys module before using sys.maxsize. It’s not a built-in keyword but part of the sys module.
How to Find the Maximum Integer in a Data Structure
In Python 2 and 3, you can use the max() function to find the highest value in an iterable data structure such as a list, tuple, or set.
Here is an example of finding the largest integer in a list:
numbers = [1, 9999, 35, 820, -5]
max_value = max(numbers)
This sample code will print the number 9999.
The counterpart is the min() function which returns the minimum value.
Finding the largest values within a range is important when running calculations like linear regression. If very large values exceed the integer limits, you can run into inaccuracies or errors in
3 Tips for Avoiding Maximum Integer Issues
Python’s flexibility does bring several disadvantages.
Operations involving large integers can be slower due to the overhead of managing arbitrary precision.
Large integers can also significantly increase the memory consumption of your program, potentially leading to memory errors.
Here are three tips for avoiding problems:
Tip 1: Choose Appropriate Data Types
There are many scenarios when the exact size of your integer values is not crucial. Consider using a smaller, fixed-size data type when this is the case.
This avoids needlessly consuming memory and slowing your application.
Tip 2: Use Efficient Programming Practices
Be aware of operations that handle large integer numbers and design algorithms with this in mind.
This could involve breaking down calculations into smaller parts or using approximations where the exact precision of a large number is not necessary.
Tip 3: Track Memory Usage
Keep track of the memory usage of your Python program and optimize your code to reduce memory footprint.
This could include deleting large variables when they are no longer needed, or using tools or libraries designed for handling large datasets efficiently.
Final Thoughts
Understanding the maximum integer value that your Python code can handle is essential for writing robust and efficient programs. This article explored the concept in both Python 2 and Python 3.
You learned how to access and utilize these maximum integer values in both Python versions. Whether you’re working with Python 2 or 3, remember our tips on optimizing your code to avoid memory
Armed with this knowledge, you’re well-equipped to harness the full power of Python’s integer handling capabilities!
A comprehensive guide to utilizing the Matplotlib library for data visualization and analysis in Python.
Mastering Data Analytics with Matplotlib in Python | {"url":"https://blog.enterprisedna.co/python-max-int/","timestamp":"2024-11-10T01:47:48Z","content_type":"text/html","content_length":"505940","record_id":"<urn:uuid:b56790e5-321e-4e2a-8f0a-1071c983c5ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00059.warc.gz"} |
Matthew Fahrbach
Matthew is a Staff Research Scientist at Google in the
Algorithms and Optimization
group. He received his PhD in computer science from the Georgia Institute of Technology, where he was advised by Dana Randall. Prior to that, he studied computer science and mathematics at the
University of Kentucky. He is the recipient of a FOCS 2020 Best Paper Award, NSF Graduate Research Fellowship, and Barry Goldwater Scholarship. His research interests broadly include algorithms,
discrete mathematics, machine learning, and optimization.
Authored Publications
PriorBoost: An Adaptive Algorithm for Learning from Aggregate Responses
Adel Javanmard
Proceedings of the 41st International Conference on Machine Learning (2024), pp. 21410-21429
Preview abstract This work studies algorithms for learning from aggregate responses. We focus on the construction of aggregation sets (called \emph{bags} in the literature) for event-level loss
functions. We prove for linear regression and generalized linear models (GLMs) that the optimal bagging problem reduces to one-dimensional size-constrained $k$-means clustering. Further, we
theoretically quantify the advantage of using curated bags over random bags. We propose the \texttt{PriorBoost} algorithm, which iteratively forms increasingly homogenous bags with respect to
(unseen) individual responses to improve model quality. We also explore label differential privacy for aggregate learning, and provide extensive experiments that demonstrate that \PriorBoost
regularly achieves optimal quality, in contrast to non-adaptive algorithms for aggregate learning. View details
Practical Performance Guarantees for Pipelined DNN Inference
Kuikui Liu
Proceedings of the 41st International Conference on Machine Learning (2024), pp. 1655-1671
Preview abstract This work optimizes pipeline parallelism of machine learning model inference by partitioning computation graphs into $k$ stages and minimizing the running time of the bottleneck
stage. We design practical algorithms for this NP-complete problem and prove they are nearly optimal in practice by comparing against lower bounds obtained from solving novel mixed-integer
programming (MIP) formulations. We apply these algorithms and lower-bound techniques to production models to achieve substantial improvements in the approximation guarantees, compared to simple
combinatorial lower bounds. For example, our new MIP formulations improve the lower bounds enough to drop the geometric mean approximation ratio from $2.175$ to $1.082$ across production data with $k
=16$ pipeline stages. This work shows that while bottleneck partitioning is theoretically hard, in practice we have a handle on the algorithmic side of the problem and much of the remaining challenge
is in developing more accurate cost models to give to the partitioning algorithms. View details
Learning Rate Schedules in the Presence of Distribution Shift
Adel Javanmard
Proceedings of the 40th International Conference on Machine Learning (2023), pp. 9523-9546
Preview abstract We design learning rate schedules that minimize regret for SGD-based online learning in the presence of a changing data distribution. We fully characterize the optimal learning rate
schedule for online linear regression via a novel analysis with stochastic differential equations. For general convex loss functions, we propose new learning rate schedules that are robust to
distribution shift, and we give upper and lower bounds for the regret that only differ by constants. For non-convex loss functions, we define a notion of regret based on the gradient norm of the
estimated models and propose a learning schedule that minimizes an upper bound on the total expected regret. Intuitively, one expects changing loss landscapes to require more exploration, and we
confirm that optimal learning rate schedules typically increase in the presence of distribution shift. Finally, we provide experiments for high-dimensional regression models and neural networks to
illustrate these learning rate schedules and their cumulative regret. View details
Approximately Optimal Core Shapes for Tensor Decompositions
Mehrdad Ghadiri
Proceedings of the 40th International Conference on Machine Learning (2023), pp. 11237-11254
Preview abstract This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give
an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we
prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree
tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm
that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster. View details
Sequential Attention for Feature Selection
Taisuke Yasuda
Lin Chen
Proceedings of the 11th International Conference on Learning Representations (2023)
Preview abstract Feature selection is the problem of selecting a subset of features for a machine learning model that maximizes model quality subject to a budget constraint. For neural networks,
prior methods, including those based on L1 regularization, attention, and other techniques, typically select the entire feature subset in one evaluation round, ignoring the residual value of features
during selection, i.e., the marginal contribution of a feature given that other features have already been selected. We propose a feature selection algorithm called Sequential Attention that achieves
state-of-the-art empirical results for neural networks. This algorithm is based on an efficient one-pass implementation of greedy forward selection and uses attention weights at each step as a proxy
for feature importance. We give theoretical insights into our algorithm for linear regression by showing that an adaptation to this setting is equivalent to the classical Orthogonal Matching Pursuit
(OMP) algorithm, and thus inherits all of its provable guarantees. Our theoretical and empirical analyses offer new explanations towards the effectiveness of attention and its connections to
overparameterization, which may be of independent interest. View details
Unified Embedding: Battle-Tested Feature Representations for Web-Scale ML Systems
Ben Coleman
Ruoxi Wang
Lichan Hong
Advances in Neural Information Processing Systems (2023), pp. 56234-56255
Preview abstract Learning high-quality feature embeddings efficiently and effectively is critical for the performance of web-scale machine learning systems. A typical model ingests hundreds of
features with vocabularies on the order of millions to billions of tokens. The standard approach is to represent each feature value as a d-dimensional embedding, which introduces hundreds of billions
of parameters for extremely high-cardinality features. This bottleneck has led to substantial progress in alternative embedding algorithms. Many of these methods, however, make the assumption that
each feature uses an independent embedding table. This work introduces a simple yet highly effective framework, Feature Multiplexing, where one single representation space is used for many different
categorical features. Our theoretical and empirical analysis reveals that multiplexed embeddings can be decomposed into components from each constituent feature, allowing models to distinguish
between features. We show that multiplexed representations give Pareto-optimal space-accuracy tradeoffs for three public benchmark datasets. Further, we propose a highly practical approach called
Unified Embedding with three major benefits: simplified feature configuration, strong adaptation to dynamic data distributions, and compatibility with modern hardware. Unified embedding gives
significant improvements in offline and online metrics compared to highly competitive baselines across five web-scale search, ads, and recommender systems, where it serves billions of users across
the world in industry-leading products. View details
Subquadratic Kronecker Regression with Applications to Tensor Decomposition
Mehrdad Ghadiri
Proceedings of the 36th Annual Conference on Neural Information Processing Systems (2022), pp. 28776-28789
Preview abstract Kronecker regression is a highly-structured least squares problem $\min_{\mathbf{x}} \lVert \mathbf{K}\mathbf{x} - \mathbf{b} \rVert_{2}^2$, where the design matrix $\mathbf{K} = \
mathbf{A}^{(1)} \otimes \cdots \otimes \mathbf{A}^{(N)}$ is a Kronecker product of factor matrices. This regression problem arises in each step of the widely-used alternating least squares (ALS)
algorithm for computing the Tucker decomposition of a tensor. We present the first \emph{subquadratic-time} algorithm for solving Kronecker regression to a $(1+\varepsilon)$-approximation that avoids
the exponential term $O(\varepsilon^{-N})$ in the running time. Our techniques combine leverage score sampling and iterative methods. By extending our approach to block-design matrices where one
block is a Kronecker product, we also achieve subquadratic-time algorithms for (1) Kronecker ridge regression and (2) updating the factor matrices of a Tucker decomposition in ALS, which is not a
pure Kronecker regression problem, thereby improving the running time of all steps of Tucker ALS. We demonstrate the speed and accuracy of this Kronecker regression algorithm on synthetic data and
real-world image tensors. View details
Edge-Weighted Online Bipartite Matching
Runzhou Tao
Zhiyi Huang
Journal of the ACM, 69 (2022), 45:1-45:35
Preview abstract Online bipartite matching is one of the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for
the unweighted problem that achieves an optimal competitive ratio of 1 - 1/e. Aggarwal et al. (SODA 2011) later generalized their algorithm and analysis to the vertex-weighted case. Little is known,
however, about the most general edge-weighted problem aside from the trivial 1/2-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long standing 1/2
barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013) that restricts beating a 1/2 competitive ratio for the more
general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in
the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and
selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative
measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems. View details
A Fast Minimum Degree Algorithm and Matching Lower Bound
Robert Cummings
Animesh Fatehpuria
Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (2021), pp. 724-734
Preview abstract The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half
a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. We present a simple but novel combinatorial algorithm for computing an exact
minimum degree elimination ordering in $O(nm)$ time. Our approach uses a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^
+}, \Delta m^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges encountered by the algorithm and $\Delta$ is the maximum degree of the input graph. Furthermore, we show
there cannot exist a minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. Our fine-grained reduction uses a
new sparse, low-degree graph construction called \emph{$U$-fillers}, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. View details
Faster Graph Embeddings via Coarsening
Gramoz Goranci
Richard Peng
Sushant Sachdeva
Chi Wang
Proceedings of the 37th International Conference on Machine Learning (2020), pp. 2953-2963
Preview abstract Graph embeddings are a ubiquitous tool for machine learning tasks on graph-structured data (e.g., node classification and link prediction). Computing embeddings for large-scale
graphs, however, is often prohibitively inefficient, even if we are only interested in a small subset of relevant vertices. To address this, we present an efficient graph coarsening algorithm based
on Schur complements that only computes the embeddings of the relevant vertices. We prove these embeddings are well approximated by the coarsened graph obtained via Gaussian elimination on the
irrelevant vertices. As computing Schur complements can be expensive, we also give a nearly linear time algorithm to generate a coarsened graph on the relevant vertices that provably matches the
Schur complement in expectation. In our experiments, we investigate various graph prediction tasks and demonstrate that computing embeddings of the coarsened graphs, rather than the entire graph,
leads to significant time and space savings without sacrificing accuracy. View details | {"url":"https://research.google/people/106910/","timestamp":"2024-11-08T23:40:33Z","content_type":"text/html","content_length":"154096","record_id":"<urn:uuid:d302ca9b-b580-4f55-86f7-bd147bb8d1fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00349.warc.gz"} |
Slope of a Graph, Table, Equation, and 2 Points Stained Glass Activities
There are 4 worksheets on slope given a graph, table, two points, and standard form. Each worksheet contains 10 problems. The 4 worksheets are: (1) Slope Given a Graph, (2) Slope Given a Table, (3)
Slope Given 2 Points, and (4) Slope Given Standard Form.
These worksheets help with self-assessment due to creating a stained glass look. Students will use a ruler or straight edge to connect the letter of the question to its answer. After connecting 10
lines, students should see a letter in each section. Students will color each section according to the color code.
CLICK HERE, to purchase! | {"url":"http://www.commoncorematerial.com/2024/04/slope-of-graph-table-equation-and-2.html","timestamp":"2024-11-10T21:36:12Z","content_type":"text/html","content_length":"105797","record_id":"<urn:uuid:450a974d-cf5b-47a3-b52b-5271f8d6dfb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00103.warc.gz"} |
Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.
Algebra 1 Example: Lizzy’s mother uses the function C(p)=450+7.75p, where C(p) represents the total cost of a rental space and p is the number of people attending, to help budget Lizzy’s 16th
birthday party. Lizzy’s mom wants to spend no more than $850 for the party. Graph the function in terms of the context.
Clarification 1
: Key features are limited to domain, range, intercepts and rate of change.
Clarification 2: Instruction includes the use of standard form, slope-intercept form and point-slope form.
Clarification 3: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.
Clarification 4: Within the Algebra 1 course, notations for domain, range and constraints are limited to inequality and set-builder.
Clarification 5: Within the Mathematics for Data and Financial Literacy course, problem types focus on money and business.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
• Coordinate Plane
• Domain
• Function Notation
• Range
• Rate of Change
• Slope
• $x$-intercept
• $y$-intercept
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students determined and interpreted the slope and
-intercept of a two-variable linear equation in slope-intercept form from a real-world context. In Algebra I, students solve real-world problems that are modeled with linear functions when given
equations in all forms, as well as tables and written descriptions, and they determine and interpret the domain, range and other key features. Students will additionally, interpret key features and
identify any constraints. In later courses, students will graph and solve problems involving linear programming, systems of equations in three variables and piecewise functions.
• This benchmark is a culmination of MA.912.AR.2. Instruction here should feature a variety of real-world contexts.
• Instruction includes representing domain, range and constraints using words, inequality notation and set-builder notation.
□ Words
☆ If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.”
□ Inequality Notation
☆ If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2.
□ Set-Builder Notation
☆ If the domain is all values of $x$ less than or equal to zero, it can be represented as {$x$|$x$ ≤ 0} and is read as “all values of $x$ such that $x$ is less than or equal to zero.”
• Instruction includes the use of $x$-$y$ notation and function notation.
• This benchmark presents the first opportunity for students to represent constraints in the domain and range of functions. Students should develop an understanding that linear graphs, without
context, have no constraints on their domain and range. When specific contexts are modeled by linear functions, parts of the domain and range may not make sense and need to be removed, creating
the need for constraints.
• Instruction includes the understanding that a real-world context can be represented by a linear two-variable equation even though it only has meaning for discrete values.
□ For example, if a gym membership cost $10.00 plus $6.00 for each class, this can be represented as $y$ = 10 + 6$c$. When represented on the coordinate plane, the relationship is graphed using
the points (0,10), (1,16), (2,22), and so on.
• For mastery of this benchmark, students should be given flexibility to represent real-world contexts with discrete values as a line or as a set of points.
• Instruction directs students to graph or interpret a representation of a context that necessitates a constraint. Discuss the meaning of multiple points on the line and announce their meanings in
the associated context (MTR.4.1). Allow students to discover that some points do not make sense in context and therefore should not be included in a formal solution (MTR.6.1). Ask students to
determine which parts of the line create sensible solutions and guide them to make constraints to represent these sections.
• Instruction includes the use of technology to develop the understanding of constraints.
• Instruction includes the connection to scatter plots and lines of fit (MA.912.DP.2.4) and the connection to systems of equations or inequalities (MA.912.DP.9.6).
Common Misconceptions or Errors
• Students may express initial confusion with the meaning of $f$($x$) for functions written in function notation.
• Students may assign their constraints to the incorrect variable.
• Students may miss the need for compound inequalities in their constraints. Students may not include zero as part of the domain or range.
□ For example, if a constraint for the domain is between 0 and 10, a student may forget to include 0 in some contexts, since they may assume that one cannot have zero people, for instance.
Strategies to Support Tiered Instruction
• Teacher provides equations in both function notation and $x$-$y$ notation written in slope-intercept form and models graphing both forms using a graphing tool or graphing software (MTR.2.1).
□ For example, $f$($x$) = $\frac{\text{2}}{\text{3}}$$x$ + 6 and $y$ = $\frac{\text{2}}{\text{3}}$$x$ + 6, to show that both $f$($x$) and $y$ represent the same outputs of the function.
• Instruction provides opportunities for identifying the domain and range on the $x$- and $y$- axis respectively using a highlighter.
□ For example, Tim bought 2 cubic feet of fertilizer and uses a little everyday on his lawn for 6 months, and the amount of fertilizer decreases at a constant rate as shown on the graph. The
domain of the function in this context is 0 ≤ $x$ ≤ 6.
• Teacher provides context to visualize and determine if it would make sense for the function to extend to a given area.
□ For example, if Garrison bought a house in 2014 and the price increases at a constant rate, he can model this by graphing a linear function where $x$ represents the time since 2014. The
domain could include negative values if he wanted to show the estimated price of the house before 2014.
□ For example, if the temperature in Alaska is at 14 degrees Fahrenheit at 6:00 am and drops at a constant rate, this can be modeled by graphing a linear function where $t$ represents the time
since 6:00 am. The range could include negative numbers to show the temperature below 0 degrees Fahrenheit.
Instructional Tasks
Instructional Task 1 (MTR.7.1)
• The population of St. Johns County, Florida, from the year 2000 through 2010 is shown in the graph below. If the trend continues, what will be the population of St. Johns County in 2025?
Instructional Task 2 (MTR.7.1)
• Devon is attending a local festival downtown. He plans to park his car in a parking garage that operates from 7:00 a.m. to 10:00 p.m. and charges $5 for the first hour and $2 for each additional
hour of parking.
□ Part A. Create a linear graph that represents the relationship between the price and number of hours parked.
□ Part B. What is an appropriate domain and range for the given situation?
Instructional Items
Instructional Item 1
• Suppose you fill your truck’s tank with fuel and begin driving down the highway for a road trip. Assume that, as you drive, the number of minutes since you filled the tank and the number of
gallons remaining in the tank are related by a linear function. After 40 minutes, you have 28.4 gallons left. An hour after filling up, you have 26.25 gallons left.
□ Part A. Graph this relationship.
□ Part B. Determine how many hours it will take for you to run out of fuel.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
This benchmark is part of these courses.
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
Given a mathematical and/or real-world problem that is modeled with linear functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this
Related Resources
Vetted resources educators can use to teach the concepts and skills in this benchmark.
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Experts
Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
STEM Lessons - Model Eliciting Activity
Movie Theater MEA:
In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that
decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
Testing water for drinking purposes:
The importance of knowing what drinking water contains. How to know what properties are present in different bottled water. Knowing the elements present in water that is advantageous to growth and
development of many things in the body. To know what to be alert for in water and to understand the importance of water in general.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
Turning Tires Model Eliciting Activity:
The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of
the MEA is applying surface area concepts and algebra through modeling.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
Computer Repair:
Students are given a linear function in context and asked to interpret its parameters in the context of a problem.
Constraints on Equations:
Students are asked to determine the constraint on a profit equation and to interpret solutions as being viable or not in the context of the problem.
Lunch Account:
Students are given a linear function in context and asked to interpret its parameters in the context of a problem.
Original Student Tutorials Mathematics - Grades 9-12
Linear Functions: Jobs:
Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Original Student Tutorial
Linear Functions: Jobs:
Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.
Type: Original Student Tutorial
Perspectives Video: Expert
Problem Solving with Project Constraints:
<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help
him solve problems!</p>
Type: Perspectives Video: Expert
Problem-Solving Tasks
Coffee and Crime:
This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.
Type: Problem-Solving Task
Cash Box:
The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only
in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of
the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or
Type: Problem-Solving Task
This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted
domain for interpreting more precisely what the equation is modeling.
Type: Problem-Solving Task
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
Perspectives Video: Expert
Problem Solving with Project Constraints:
<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help
him solve problems!</p>
Type: Perspectives Video: Expert
Problem-Solving Tasks
Coffee and Crime:
This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.
Type: Problem-Solving Task
Cash Box:
The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only
in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of
the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or
Type: Problem-Solving Task
This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted
domain for interpreting more precisely what the equation is modeling.
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Worden Discussion Forum
Registered Could you please translate the following information to TC2000, and plot the data as an indicator (6/100HV).
Joined: 1/ a. Compute for the 6-day historical volatility:
13/2005 1. Take the natural logarithm of Today's close divided by Yesterday's close;
Posts: 9 2. Take the standard deviation of the result of(1) for 6 trading days;
3. Multiply the result of (2) by 100, then multiply all by the square root of 256.
b. Compute for the 100-day historical volatility.
c. Divide the 6-dayHV by 100-dayHV
If plotting an indicator is not possible, may be an PCF scan for the 6/100HV ratio can be done?
Thank you.
This looks possible (but very involved). Give us some time and check back soon.
- Craig
Worden Here to Help!
Joined: 10
Registered Yep, please post them so I can give it a try on my system. Please indicate if it can be done as a scan or an indicator.
Joined: 1/ Thank you very much guys.
Posts: 9
This is what I came up with:
Worden 6/100 HV Ratio:
Trainer SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2-(LOG(C/C6)^2)/6)/6)/SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5
/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/
Joined: 10 C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG
/7/2004 (C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2
Posts: +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/
65,138 C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG
(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2
+LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/
C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG
Please keep in mind my warnings that it can really slow things down.
Personal Criteria Formulas
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Registered Hi Bruce,
Joined: 1/ Can I ask you just to compute for just the 50-day historical volatility? Here's the formula again:
Posts: 9 HV(50-day)= standard deviation(ln(today's close/yesterday's close),50)*100*square root(256)
where ln = natural logarithm
1) Divide today's close by yesterday's close
2) Take the natural logarithm of #1
3) Take the standard deviation of #2 for 50 trading days
4)Multiply #3 by 100
5)Multiply #4 by the squae root of 256
May be this will plot easier on the system.
I'll write a formula and post it when I'm done. An optimization to reduce the size and computation time was pointed out by bustermu in this thread:
Worden PCF formulation for calculating Standard Deviation
I'll rewrite the first formula for you as well.
Joined: 10 -Bruce
/7/2004 Personal Criteria Formulas
Posts: TC2000 Support Articles
Try this:
Worden 50-Period HV:
Trainer 1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG
(C12/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2
Joined: 10 +LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/
/7/2004 C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG
Posts: (C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2-(LOG(C/C50)^2)/50)/50)
I also shortened up the 6HV/100HV Ratio formula and made the changes to the original post.
Personal Criteria Formulas
TC2000 Support Articles
Registered Hi Bruce,
Joined: 1/ Thanks for all of these.
Posts: 9
Registered Hi! Bruce
User Greetigs from California. I was looking for the 100 HV or volatility standar deviation as an indicator when I came across your posting dated Feb 11/05.
Joined: 10 As I need the 100 HV (nor the 50)
/7/2004 1)Shall I replace the 50 for 100 in the formula posted? and
Posts: 16 2)Enter the formula as a PCF.
3)But how can I use it as an indicator?
Any help would be very much appreciate it.
By the way, it is a pleasure and a learning experience to read all the postings that you trainers have in this site. I wish I could find an index for all of them so I would not miss
reading any of them.
Try this:
Worden 1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG
Trainer (C12/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2
+LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/
Joined: 10 C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG
/7/2004 (C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2
Posts: +LOG(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/
65,138 C71)^2 +LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG
(C82/C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2
+LOG(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
To use it as an indicator, select Chart Template | Add Indicator | Indicator and paste the formula into the Indicator Formula window.
Personal Criteria Formulas
TC2000 Support Articles
Registered Hi Bruce,
Joined: 1/ Firstly may I say that the support that the worden trainers are providing are first class.
Posts: 10 Following on from the posting that was made on the 24th March on the Historical Volatility, I would be greatful if you could kindly assist me with the following.
I am looking for the formula to :
1) plot as indicators where the 6 day HV is above or below the 50% 100 day HV and the 10 day HV is above or below the 50% 100 day HV.
2) As a PCF to scan the market to highlight those stocks where the 6 day HV is above or below the 50% 100 day HV and where the 10 day HV is above or below the 50% 100 day HV.
Thank you in advance for you help.
Best regards
Sanjay (from the UK)
Bruce will be taking a look at this for you, Sanjay. Check back within the next 24 hours or so for a response.
- Doug
Worden Teaching Online!
Joined: 10
Registered Thanks very much Doug.
Joined: 1/ Rgds
Posts: 10 Sanjay
10/100 HV Ratio:
SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2-(LOG(C/C10)^2)/10)/10)/SQR((LOG(C/C1)^2
Worden +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG(C13/
Trainer C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2 +LOG
(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/C37)^2
Joined: 10 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG(C48/
/7/2004 C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2 +LOG
Posts: (C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/C72)^2
65,138 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG(C83/
C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2 +LOG
(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
The 6/100 HV Ratio was already posted above.
To write a Boolean condition where the shorter period’s HV is greater than 50% of the longer period’s HV, just add a “>.5” onto the end. To write a Boolean condition where the shorter
period’s HV is less than 50% of the longer period’s HV, just add a “<.5” onto the end.
You might also want to use the techniques in the following video to use the formulas unchanged:
Constructing more versatile and reusable Personal Criteria Formulas
Personal Criteria Formulas
TC2000 Support Articles
Registered Hi Bruce,
Joined: 1/ Thank you for the quick repsonse. However, with the formula that you have provideb for the 10/100 and the 6/100 earlier, these only plot single indicator lines. What I am specifically
29/2005 looking for is where two lines are plotted showing where the 6 HV or 10 HV is above or below the 50% 100HV. On that note am I correct in assuming that say for the 10/50%100 HV I would
Posts: 10 have to plot two indicators as follows:
SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2-(LOG(C/C10)^2)/10)/10)
50% 100 HV :
SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/
C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG
(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2
+LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/
C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG
(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2
+LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/
C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG
(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)*0.5
Best regards
You could write and plot three formulas (6HV, 10HV and 50% 100HV), but comparing heights and crossovers using this method would not make much sense because each of the three indicators
would plot on its own scale (as a matter of fact, the chart looks exactly the same when plotting 100HV or 50% 100HV).
Trainer There are a variety of methods that can be used to create valid comparisons (you can plot the ratio or the difference for example). However, without a way to fix the scale for Custom
Indicators, every way I've thought of so far results in a single line being plotted.
Joined: 10
/7/2004 Since it is possible to fix the scale to Price, it is possible to create multiple lines with valid crossovers by adding a Custom Indicator value to Price. But, the Custom Indicator values
Posts: would have to be fairly small compared to Price for the indicators to be visible on the screen and the plotted indicator line would probably be dominated by Price itself.
65,138 -Bruce
Personal Criteria Formulas
TC2000 Support Articles
Registered Bruce,
Joined: 1/ There are various ways to fix any scale you choose for Custom Indicators. Any that I know may have to be adjusted for a particular symbol, zoom, right-edge, etc.
Posts: Probably the easiest way is to check "Plot using price scale" and replace the PCF by:
where a and b are numbers. The a and b can be chosen so that the Indicator occupies any portion of the screen you please. Use the same a and b for any collection of Custom Indicators and
they will all be plotted to the same scale.
Jim Murphy
Registered Hi Bruce / Bustermu,
Joined: 1/ Thanks for the updates, but please forgive my inexperience, and
Posts: 10 Bruce,
Could you kindly further ( preferably with some illustrations) as you have mentioned on
1) "(...... difference for example)."
2) "Since it is possible to fix the scale to Price, it is possible to create multiple lines with valid crossovers by adding a Custom Indicator value to Price. But, the Custom Indicator
values would have to be fairly small compared to Price for the indicators to be visible on the screen and the plotted indicator line would probably be dominated by Price itself."
Could you kindly further ( preferably with some illustrations) as you have mentioned on
Probably the easiest way is to check "Plot using price scale" and replace the PCF by:
where a and b are numbers. The a and b can be chosen so that the Indicator occupies any portion of the screen you please. Use the same a and b for any collection of Custom Indicators and
they will all be plotted to the same scale.
Apologies for going on about this, but in a nutshell what I am looking for is where the indicator window show the 6 HV or 10 HV crossing the 50% 100 HV, and an PCF to highlight those
candidates above or below.
Once again thank you in advance for your support an patience with my inexperience.
1) If you have plotted the 6/100 HV Ratio and 10/100 HV Ratio as Custom Indicators, you already know what they look like. The 6/100 HV Difference and 10/100 HV Difference use almost the
Worden same formulas. You just replace the "/" between the short and long HV with a "-" (although in this case, you'll have to add the "1600*" portion of HV back in since it was factored out of
Trainer the Ratio version).
Joined: 10 6-50%100HV Difference:
/7/2004 1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2-(LOG(C/C6)^2)/6)/6)-800*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^
Posts: 2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2
65,138 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/
C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG
(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2
+LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/
C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG
(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2
+LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/
C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
10-50%100HV Difference:
1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2-(LOG(C/C10)^2)/10)/10)-800*SQR((LOG(C/
C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG
(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2
+LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/
C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG
(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2
+LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/
C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG
(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2
+LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
2) I really don’t know of a better way to describe it, but a demonstration usually helps.
I’m assuming you already have two Custom Indicators from when you plotted and posted formulas for 10HV and 50% 100 HV in your Saturday, April 02, 2005 9:34:32 AM post (if not, create
them). Adjusting the Zoom or Scrolling back in time will change the crossovers (go ahead and try it).
Now, make two changes to each formula. Add a “C+” to the beginning of each formula and select Plot Using Price Scale. The indicators no longer look anything like your old Custom
Indicators, but the crossovers no longer change when Zooming and Scrolling (again, try it out). If the "HV" is large compared to price, it will not be visible because it will plot outside
the chart boundary.
Personal Criteria Formulas
TC2000 Support Articles
Registered QUOTE
Joined: 1/ (sanjayp95)
2,645 Could you kindly further ( preferably with some illustrations) as you have mentioned on
Probably the easiest way is to check "Plot using price scale" and replace the PCF by:
I will illustrate the procedure with V and XAVGV5 as Custom Indicators. Please do the following setup:
Symbol: SP-500
Right-Edge: 04/01/05
Time Frame: Daily
Zoom: 6
Bottom Window:
Custom Indicator 1:
PCF: 0.00408059*V+1131.61
Check "Plot using price scale"
Custom Indicator 2:
PCF: 0.00408059*XAVGV5+1131.61
Check "Plot using price scale"
The two Custom Indicators are plotted to the same scale which was chosen to be full-scale for V. Please observe the effects of changing Right-Edge, Zoom, Time Frame, and, finally, Symbol.
Jim Murphy
I don’t know of any working method in TeleChart to fix the scale for Custom Indicators without selecting "Plot using price scale".
Trainer Your formula conversion:
Joined: 10 a*PCF+b
Posts: using constants is interesting, but the limitation of having to adjust a and b for each symbol for most of the Custom Indicators I would want to use it on is significant and more severe
65,138 than would be acceptable for me to use regularly.
I chose C for b because; at least with a PCF with values significantly smaller than C, it makes the chart visible at the rather significant expense of having the indicators pretty much
look like a plot of C (but the crossovers are valid). If the PCF values are not small enough, I can usually choose a constant value for a that will work for most symbols.
I haven’t had a chance to think about it very much, but using MAXH and MINL to construct a and b might minimize (although not eliminate, as happens when using constants) the effect of
price on the indicator while still allowing most charts to visible when switching symbols (assuming the indicator range for different symbols is relatively stable). I’m sure the following
would still need to be adjusted for different zooms (zoom factor=z):
Zero Centered Indicators:
Zero (or b) Bottomed Indicators:
If the expected range of the PCFs is known, it is relatively simple to determine a and b. If the range is less predictable, the MAX and MIN values for the formulas over z could be used to
dynamically generate a, but in many (maybe even most) cases, determining the MAX and MIN for the formulas in question, while possible, probably isn't practical.
Personal Criteria Formulas
TC2000 Support Articles
Registered QUOTE
Joined: 1/ (Bruce_L)
Posts: bustermu,
2,645 I don’t know of any working method in TeleChart to fix the scale for Custom Indicators without selecting "Plot using price scale".
Your formula conversion:
using constants is interesting, but the limitation of having to adjust a and b for each symbol for most of the Custom Indicators I would want to use it on is significant and more severe
than would be acceptable for me to use regularly.
There are various methods of fixing the scale of Custom Indicators under the conditions:
1) "Center Zero Line" checked.
2) "Plot using price scale" checked.
3) Neither 1) nor 2).
For 2), one can use a*PCF+b as previously discussed. For 1) and 3), one can insert artifical spikes. None would be considered "working" methods if one intends to scan a WatchList with
By way of confession, I chose V as my illustration so that I could test:
to find a and b, respectively. One could always find the maximum and minimun of any PCF from its plot and plug them into the above to obtain a suitable a and b.
Jim Murphy
Registered Hi Bruce & Jim Murphy,
Joined: 1/ Thanks the fog has almost cleared from my head.
Posts: 10 Bruce,
In the second part of your feedback you mentioned add "C+" and select " plot using price scale". When doing this both lines are literally idential plots. Am I doing something wrong. FYA
here are the formulas that I am using with your feedback incorporated.
Indicator 1 ( with plot using price scale checked):
C +SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2-(LOG(C/C10)^2)/10)/10)
Indicator 2 ( with plot using price scale checked):
C +SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12
/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2
+LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/
C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG
(C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2
+LOG(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/
C71)^2 +LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG
(C82/C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2
+LOG(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)*0.5
PS thanks for the difference formulas, getting closer to my end objecive for the strategy in mind !
You already mentioned the spike methods you have explored couldn't be considered working methods, so this is probably not news, but I have tested several spike methods to fix the scale
Worden and have not found a decent general method to insert artificial spikes for 1) or 3) without already knowing a past value (or specific condition) for the PCF during the displayed period
Trainer (which usually ends up being symbol specific).
Joined: 10 2) If one knows MAXH, MINL, MAX(PCF) and MIN(PCF) for the displayed period, determining a and b is fairly simple. Unfortunately, this is usually symbol specific as well.
/7/2004 -Bruce
Posts: Personal Criteria Formulas
65,138 TC2000 Support Articles
You have the formulas correct. I was looking at some stocks that were about a dollar when I pasted the formulas into TeleChart to run some tests and made the suggestion. In this
Worden particular example, for stocks more than a few dollars the indicators are usually going to be too dominated by price to visually determine crossovers.
I'm not sure I realized how dominated by price this particular example would be, but minimizing this effect is the topic my Sunday, April 03, 2005 1:45:18 PM post explores.
Joined: 10 -Bruce
/7/2004 Personal Criteria Formulas
Posts: TC2000 Support Articles
Registered Bruce,
Joined: 1/ I suggest one use specific conditions on price and volume but not on the PCF's to position the spike. If you do not know the PCF range, you will have to clip (or limit) in order to stop
1/2005 it from moving outside the bounds set by the spikes.
2,645 One can set the scale independent of symbol with limiters if it is known that the PCF's will always reach the limits.
All of this could, in principle, be done independent of the symbol if the date designation, e.g., C'mm/dd/yy', worked properly in Custom Indicators.
Jim Murphy
Registered QUOTE
Joined: 1/ (Bruce_L)
Posts: 10
I'm not sure I realized how dominated by price this particular example would be, but minimizing this effect is the topic my Sunday, April 03, 2005 1:45:18 PM post explores.
Hi Bruce,
Thanks for the clarification so far, but not sure how one proceed from here based on what you mentioned above. Any more thoughts on how to overcome the dominance by price for stock more
than a few dollars ?
Best regards
Registered Jim ( Bustermu),
Joined: 1/ Thanks for the further clarification.
Posts: 10 Best regards
My suggestion on how to proceed would be to not try doing it with two lines (or plot the two lines, but don't compare them to each other). The single line formulas produce accurate
Worden information about their relative values and crossovers. When the 6/100 HV Ratio or 10/100 HV Ratio crosses .5, or the 6-50%100HV Difference or 10-50%100HV Difference crosses 0, you know
Trainer the 6HV or 10HV has crossed the 50% 100HV.
Joined: 10 Personal Criteria Formulas
/7/2004 TC2000 Support Articles
Registered Bruce,
Joined: 1/ That works for me! Actually I had the same thoughts in anticipation of your feedback that the line could not be plotted to show a clear distinction betwen the two ratios.
Posts: 10 If a solution presents itself in the near future I would greatly appreciate a heads up.
Finally I would like to say a " big up " to you Bruce. Your responses have been beyond first class!, and I look forward to tapping in to your knowledge base for future guidances.
Best regards
Registered Hello every one.
Joined: 4/ I would be appreciate if some one could tell me or direct me
16/2005 to web where i can find charting about historical volatility ratio.
Posts: 1
Best regards
nisim pan.
I would do a Google search for STOCKS HISTORICAL VOLATILITY RATIO.
- Craig
Worden Here to Help!
Joined: 10
Registered When I add indicators that are already saved formulas from the clipboard why does the title not show up ? I am only seeing the word "formula" and a small amount of the actual formula. It
User is very confusing to understand which indicator I have actually chosen.
Joined: 3/
Posts: 4
That's currently the way the indicators show up. It has been added to the suggestion list that the program allow names to be assigned to custom indicators. You can feel free to add your
suggestion over in the Comments/Suggestions forum.
Trainer In the future, it would be better for you to start a new topic rather than add a totally unrelated comment or question to the bottom of an existing thread. You might want to take a look
at this video to get the most out of using the forums:
Joined: 10
/1/2004 Learn how to use the forums: post a new topic, reply, Search existing topics
Posts: - Doug
4,308 Teaching Online!
Registered Can tc200 be set up to highlight a price bar when it meets certain requirements such as being an inside bar?
Joined: 1/
Posts: 1
In a manner, yes.... check out this video:
Worden Visually Backtesting Specific Symbols
Trainer - Craig
Here to Help!
Joined: 10
Registered So exacly how do I plot 3 seperate lines...
Joined: 10 1.) 6 day Hisorical Volatility
/27/2005 2.) 10 day Historical volatility
Posts: 71 3.) 100 day Historical bolatility
and identify that the 6 and 10 are down 50% below the 100.
plot each of these as CUSTOM INDICATORS with the ZERO LINE option checked. When both are under the center zero line you know the 6 and 10 are below the 100*.5
Worden 6-50%100HV Difference:
Trainer 1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2-(LOG(C/C6)^2)/6)/6)-800*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^
2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2
Joined: 10 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2 +LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/
/1/2004 C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG
Posts: (C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2
18,819 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2 +LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/
C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG
(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2
+LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2 +LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/
C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
10-50%100HV Difference:
1600*SQR((LOG(C/C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2-(LOG(C/C10)^2)/10)/10)-800*SQR((LOG(C/
C1)^2 +LOG(C1/C2)^2 +LOG(C2/C3)^2 +LOG(C3/C4)^2 +LOG(C4/C5)^2 +LOG(C5/C6)^2 +LOG(C6/C7)^2 +LOG(C7/C8)^2 +LOG(C8/C9)^2 +LOG(C9/C10)^2 +LOG(C10/C11)^2 +LOG(C11/C12)^2 +LOG(C12/C13)^2 +LOG
(C13/C14)^2 +LOG(C14/C15)^2 +LOG(C15/C16)^2 +LOG(C16/C17)^2 +LOG(C17/C18)^2 +LOG(C18/C19)^2 +LOG(C19/C20)^2 +LOG(C20/C21)^2 +LOG(C21/C22)^2 +LOG(C22/C23)^2 +LOG(C23/C24)^2 +LOG(C24/C25)^2
+LOG(C25/C26)^2 +LOG(C26/C27)^2 +LOG(C27/C28)^2 +LOG(C28/C29)^2 +LOG(C29/C30)^2 +LOG(C30/C31)^2 +LOG(C31/C32)^2 +LOG(C32/C33)^2 +LOG(C33/C34)^2 +LOG(C34/C35)^2 +LOG(C35/C36)^2 +LOG(C36/
C37)^2 +LOG(C37/C38)^2 +LOG(C38/C39)^2 +LOG(C39/C40)^2 +LOG(C40/C41)^2 +LOG(C41/C42)^2 +LOG(C42/C43)^2 +LOG(C43/C44)^2 +LOG(C44/C45)^2 +LOG(C45/C46)^2 +LOG(C46/C47)^2 +LOG(C47/C48)^2 +LOG
(C48/C49)^2 +LOG(C49/C50)^2 +LOG(C50/C51)^2 +LOG(C51/C52)^2 +LOG(C52/C53)^2 +LOG(C53/C54)^2 +LOG(C54/C55)^2 +LOG(C55/C56)^2 +LOG(C56/C57)^2 +LOG(C57/C58)^2 +LOG(C58/C59)^2 +LOG(C59/C60)^2
+LOG(C60/C61)^2 +LOG(C61/C62)^2 +LOG(C62/C63)^2 +LOG(C63/C64)^2 +LOG(C64/C65)^2 +LOG(C65/C66)^2 +LOG(C66/C67)^2 +LOG(C67/C68)^2 +LOG(C68/C69)^2 +LOG(C69/C70)^2 +LOG(C70/C71)^2 +LOG(C71/
C72)^2 +LOG(C72/C73)^2 +LOG(C73/C74)^2 +LOG(C74/C75)^2 +LOG(C75/C76)^2 +LOG(C76/C77)^2 +LOG(C77/C78)^2 +LOG(C78/C79)^2 +LOG(C79/C80)^2 +LOG(C80/C81)^2 +LOG(C81/C82)^2 +LOG(C82/C83)^2 +LOG
(C83/C84)^2 +LOG(C84/C85)^2 +LOG(C85/C86)^2 +LOG(C86/C87)^2 +LOG(C87/C88)^2 +LOG(C88/C89)^2 +LOG(C89/C90)^2 +LOG(C90/C91)^2 +LOG(C91/C92)^2 +LOG(C92/C93)^2 +LOG(C93/C94)^2 +LOG(C94/C95)^2
+LOG(C95/C96)^2 +LOG(C96/C97)^2 +LOG(C97/C98)^2 +LOG(C98/C99)^2 +LOG(C99/C100)^2-(LOG(C/C100)^2)/100)/100)
- Craig
Here to Help!
Registered This works, and is effective, but I would rather watch the 10 and 6 go below the 100 and have my custom indicator indicate where 50% below is at... the 50% indication is not that
User important however. I think I could
Joined: 10 get a better feel and calculations would be easier if I just graphed th 6, 10 and 100 in 3 different colors ??
Posts: 71 your thoughts
Thanks again
You can graph them but, as is talked about above, they won't be on the same scale so you cannot compare their positions to each other. In order to compare the three to each other on the
chart they would need to all be on the same scale. There is no easy and effective way to do this in TeleChart.
Worden - Craig
Trainer Here to Help!
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Brilliant Student Riddle | Riddles360
In a guessing game, five friends had to guess the exact numbers of apples in a covered basket.
Friends guessed 22, 24, 29, 33, and 38, but none of the guesses was correct. The guesses were off by 1, 8, 6, 3, and 8 (in random order).
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How to Get the Rows with the Maximum Value in Groups using Groupby in Python Pandas - Local Coder
How to Get the Rows with the Maximum Value in Groups using Groupby in Python Pandas
In this article, we will discuss how to find all the rows in a pandas DataFrame that have the maximum value for a specific column after grouping the data by one or more columns.
Problem Description
The problem we are trying to solve is to get the rows that have the maximum value for the 'count' column, after grouping the data by the 'Sp' and 'Mt' columns. We want to find the maximum 'count'
value for each unique combination of 'Sp' and 'Mt', and then select the rows that have this maximum value.
Example 1
import pandas as pd
# Create the DataFrame
data = {
'Sp': ['MM1', 'MM1', 'MM1', 'MM2', 'MM2', 'MM2', 'MM4', 'MM4', 'MM4'],
'Mt': ['S1', 'S1', 'S3', 'S3', 'S4', 'S4', 'S2', 'S2', 'S2'],
'Value': ['a', 'n', 'cb', 'mk', 'bg', 'dgd', 'rd', 'cb', 'uyi'],
'count': [3, 2, 5, 8, 10, 1, 2, 2, 7]
df = pd.DataFrame(data)
# Group the data by 'Sp' and 'Mt' columns and find the maximum 'count' value
max_counts = df.groupby(['Sp', 'Mt'])['count'].max()
# Select the rows that have the maximum 'count' value in each group
result = df[df['count'].isin(max_counts)]
The expected output of this code is:
Sp Mt Value count
0 MM1 S1 a 3
2 MM1 S3 cb 5
3 MM2 S3 mk 8
4 MM2 S4 bg 10
8 MM4 S2 uyi 7
The code first creates a pandas DataFrame using the given data. Then, it groups the data by the 'Sp' and 'Mt' columns and finds the maximum value of the 'count' column for each group using the max()
function. Finally, it selects the rows that have the maximum 'count' value in each group using the isin() function.
Example 2
import pandas as pd
# Create the DataFrame
data = {
'Sp': ['MM2', 'MM2', 'MM4', 'MM4', 'MM4'],
'Mt': ['S4', 'S4', 'S2', 'S2', 'S2'],
'Value': ['bg', 'dgd', 'rd', 'cb', 'uyi'],
'count': [10, 1, 2, 8, 8]
df = pd.DataFrame(data)
# Group the data by 'Sp' and 'Mt' columns and find the maximum 'count' value
max_counts = df.groupby(['Sp', 'Mt'])['count'].max()
# Select the rows that have the maximum 'count' value in each group
result = df[df['count'].isin(max_counts)]
The expected output of this code is:
Sp Mt Value count
0 MM2 S4 bg 10
3 MM4 S2 cb 8
4 MM4 S2 uyi 8
The code works the same way as in Example 1. It groups the data by the 'Sp' and 'Mt' columns, finds the maximum value of the 'count' column for each group, and selects the rows that have the maximum
'count' value in each group.
The solution to this problem involves two steps:
• Grouping the data by one or more columns
• Finding the maximum value in each group and selecting the rows that have this maximum value
Step 1: Grouping the data
In order to group the data by one or more columns, we can use the groupby() function in pandas. This function takes the column(s) to group by as input and returns a GroupBy object.
import pandas as pd
# Create the DataFrame
data = {
'Sp': ['MM1', 'MM1', 'MM1', 'MM2', 'MM2', 'MM2', 'MM4', 'MM4', 'MM4'],
'Mt': ['S1', 'S1', 'S3', 'S3', 'S4', 'S4', 'S2', 'S2', 'S2'],
'Value': ['a', 'n', 'cb', 'mk', 'bg', 'dgd', 'rd', 'cb', 'uyi'],
'count': [3, 2, 5, 8, 10, 1, 2, 2, 7]
df = pd.DataFrame(data)
# Group the data by 'Sp' and 'Mt' columns
grouped = df.groupby(['Sp', 'Mt'])
In this example, we create a pandas DataFrame using the given data. Then, we pass the 'Sp' and 'Mt' columns to the groupby() function to group the data by these columns. The result is a GroupBy
Step 2: Finding the maximum value and selecting rows
To find the maximum value in each group and select the rows that have this maximum value, we can use the max() function to calculate the maximum value of the 'count' column for each group. Then, we
use the isin() function to select the rows that have the maximum value.
import pandas as pd
# Create the DataFrame
data = {
'Sp': ['MM1', 'MM1', 'MM1', 'MM2', 'MM2', 'MM2', 'MM4', 'MM4', 'MM4'],
'Mt': ['S1', 'S1', 'S3', 'S3', 'S4', 'S4', 'S2', 'S2', 'S2'],
'Value': ['a', 'n', 'cb', 'mk', 'bg', 'dgd', 'rd', 'cb', 'uyi'],
'count': [3, 2, 5, 8, 10, 1, 2, 2, 7]
df = pd.DataFrame(data)
# Group the data by 'Sp' and 'Mt' columns and find the maximum 'count' value
max_counts = df.groupby(['Sp', 'Mt'])['count'].max()
# Select the rows that have the maximum 'count' value in each group
result = df[df['count'].isin(max_counts)]
In this example, we create a pandas DataFrame using the given data. Then, we group the data by the 'Sp' and 'Mt' columns and find the maximum value of the 'count' column for each group using the max
() function. Next, we select the rows that have the maximum 'count' value in each group using the isin() function.
In this article, we have discussed how to find all the rows in a pandas DataFrame that have the maximum value for a specific column after grouping the data by one or more columns. We have provided
examples with code snippets to demonstrate the solution to the problem. By following the steps outlined in this article, you will be able to effectively solve the problem of getting the rows with the
maximum value in groups using groupby in Python pandas. | {"url":"https://localcoder.net/how-to-get-the-rows-with-the-maximum-value-in-groups-using-groupby-in-pytho","timestamp":"2024-11-08T05:52:41Z","content_type":"text/html","content_length":"29583","record_id":"<urn:uuid:69a02dfc-68f6-4f94-bd31-b20730988df2>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00377.warc.gz"} |
Next: Handles Up: Data Types Previous: Pointers Contents Index
Support for complex numbers is provided via the complex data type. The basic math operators and functions accept complex numbers, possibly intermixed with scalar values, and will produce a complex
result when given a complex operand when appropriate. Generally, a complex number can be passed to a function expecting a real number, and the real part of the complex number will be used. Similarly,
a scalar passed to a function expecting a complex number will be accepted as a complex value with zero imaginary part.
Presently, functions will not produce a complex result unless passed a complex argument. For example, the sqrt function, if passed a negative scalar, will return a scalar zero. If passed a complex
number with negative real part and zero imaginary part, the return will be the complex square root value as one would expect.
Complex numbers can be created with the cmplx initializer function, which takes as arguments two scalar values that initialize the real and imaginary part. There are special functions that return as
scalars the real and imaginary values, magnitude, and phase of a complex operand. The Print function and similar will print a complex value as a comma-separated pair of numbers enclosed in
Next: Handles Up: Data Types Previous: Pointers Contents Index Stephen R. Whiteley 2024-09-29 | {"url":"http://wrcad.com/manual/xicmanual/node529.html","timestamp":"2024-11-02T06:15:05Z","content_type":"text/html","content_length":"4828","record_id":"<urn:uuid:3881c461-4d7b-4faa-a7a2-825234874776>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00685.warc.gz"} |
Filters are a common preprocessing method for reducing noise in signal processing.
mean_filter() and median_filter() can be applied to individual sequences.
API reference¶
mean_filter(x, *[, k]) Applies a mean filter of size k independently to each feature of the sequence, retaining the original input shape by using appropriate padding.
median_filter(x, *[, k]) Applies a median filter of size k independently to each feature of the sequence, retaining the original input shape by using appropriate padding.
sequentia.preprocessing.transforms.mean_filter(x, *, k=5)¶
Applies a mean filter of size k independently to each feature of the sequence, retaining the original input shape by using appropriate padding.
This is implemented as a 1D convolution with a kernel of size k and values 1 / k.
The filtered array.
Return type:
Applying a mean_filter() to a single sequence and multiple sequences (independently via IndependentFunctionTransformer) from the spoken digits dataset.
from sequentia.preprocessing import IndependentFunctionTransformer, mean_filter
from sequentia.datasets import load_digits
# Fetch MFCCs of spoken digits
data = load_digits()
# Apply the mean filter to the first sequence
x, _ = data[0]
xt = mean_filter(x, k=7)
# Create an independent mean filter transform
transform = IndependentFunctionTransformer(mean_filter, kw_args={"k": 7})
# Apply the transform to all sequences
Xt = transform.transform(data.X, lengths=data.lengths)
sequentia.preprocessing.transforms.median_filter(x, *, k=5)¶
Applies a median filter of size k independently to each feature of the sequence, retaining the original input shape by using appropriate padding.
The filtered array.
Return type:
Applying a median_filter() to a single sequence and multiple sequences (independently via IndependentFunctionTransformer) from the spoken digits dataset.
from sequentia.preprocessing import IndependentFunctionTransformer, median_filter
from sequentia.datasets import load_digits
# Fetch MFCCs of spoken digits
data = load_digits()
# Apply the median filter to the first sequence
x, _ = data[0]
xt = median_filter(x, k=7)
# Create an independent median filter transform
transform = IndependentFunctionTransformer(median_filter, kw_args={"k": 7})
# Apply the transform to all sequences
Xt = transform.transform(data.X, lengths=data.lengths) | {"url":"https://sequentia.readthedocs.io/en/latest/sections/preprocessing/transforms/filters.html","timestamp":"2024-11-07T14:06:17Z","content_type":"text/html","content_length":"22484","record_id":"<urn:uuid:d3f780da-7415-4487-819c-b2520a050e32>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00182.warc.gz"} |
Binomial Random Numbers Generation in R %sep$ R Language
Binomial Random Numbers Generation in R
Table of Contents
We will learn how to generate Bernoulli or Binomial Random Numbers (Binomial distribution) in R with the example of a flip of a coin. This tutorial is based on how to generate random numbers
according to different statistical probability distributions in R. Our focus is on binomial random numbers generation in R.
Binomial Random Numbers in R
We know that in Bernoulli distribution, either something will happen or not such as a coin flip has two outcomes head or tail (either head will occur or head will not occur i.e. tail will occur). For
an unbiased coin, there will be a 50% chance that the head or tail will occur in the long run. To generate a random number that is binomial in R, use the rbinom(n, size, prob) command.
rbinom(n, size, prob) #command has three parameters, namey
‘$n$’ is the number of observations
‘$size$’ is the number of trials (it may be zero or more)
‘$prob$’ is the probability of success on each trial for example 1/2
Examples of Generation Binomial Random Numbers
• One coin is tossed 10 times with a probability of success=0.5
the coin will be fair (unbiased coin as p=1/2)
rbinom(n=10, size=1, prob=1/2)
OUTPUT: 1 1 0 0 1 1 1 1 0 1
• Two coins are tossed 10 times with a probability of success=0.5
• rbinom(n=10, size=2, prob=1/2)
OUTPUT: 2 1 2 1 2 0 1 0 0 1
• One coin is tossed one hundred thousand times with a probability of success=0.5
rbinom(n=100,000, size=1, prob=1/2)
• store simulation results in $x$ vector
x <- rbinom(n=100000, size=5, prob=1/2)
count 1’s in x vector
find the frequency distribution
creates a frequency distribution table with frequency
t = (table(x)/n *100)
plot frequency distribution table
plot(table(x),ylab = "Probability",main = "size=5,prob=0.5")
View the Video tutorial on rbinom command
Learn Basic Statistics and Online MCQs about Statistics | {"url":"https://rfaqs.com/probability/random-numbers/binomial-random-numbers-in-r/","timestamp":"2024-11-09T04:20:44Z","content_type":"text/html","content_length":"184437","record_id":"<urn:uuid:b38a66c4-dba4-4920-b62f-083f31f2574a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00294.warc.gz"} |
INDEX using MAX(COLLECT)
Hi Smartsheet Fam,
I have the below sheet:
I have a second "helper sheet" that looks exactly like this with a rule set up: any time "MVE Increase Recommended" changes to "yes", copy the row to the helper sheet. I have this set up so I can
capture a snapshot of the person who updated the "MVE Increase Recommended" cell based on the "Modified By" data and return the "Modified By" value in the "MVE Increase Recommended by" cell.
See here for more details on the helper sheet by @Andrée Starå : Lock or Store Date/Value Solution without using Zapier — Smartsheet Community
I don't understand why I'm getting the #INVALID DATA TYPE error.
This is my function: =IF([MVE Increase Recommended]@row = "yes", INDEX({Modified by}, (MAX(COLLECT({Modified}, {KLG Matter Number}, [KLG Matter Number]@row), 0))), "")
The data I'm asking it to return is the "modified by" data, which I'm assuming is either text/number or contact. I've changed the column type in the "MVE Increase Recommended by" column to both of
these types, and I'm still receiving the same error. I've tried changing the column type for the "MVE Increase Recommended by" column to every single column type, but I'm still getting the same
error. Is the problem that the formula contains a reference to a date field?
Bottom line: I need to return the individual's name in the modified field based on the most recent modified date in the helper sheet.@Andrée Starå
• Your (MAX(COLLECT({Modified}, {KLG Matter Number}, [KLG Matter Number]@row), 0))) part of the formula is the row you want to return. You have it set up to return {Modified}, which seems to be a
date (#INVALID DATA TYPE). Do you have a Auto-number column on the other sheet? You should be able to replace {Modified} with {Row #} and it should be back up and running for you.
• @Jason Tarpinian I just implemented this suggestion. Thanks!
I changed the formula to this:
=IF([MVE Increase Recommended]@row = "yes", INDEX({Modified by}, (MAX(COLLECT({Helper ID}, {KLG Matter Number}, [KLG Matter Number]@row), 0))), "")
Then I got an #INCORRECT ARGUMENT SET error.
I switched the placement of the 0 at the end, but I'm still getting the same error.
Newest iteration of the formula:
=IF([MVE Increase Recommended]@row = "yes", INDEX({Modified by}, (MAX(COLLECT({Helper ID}, {KLG Matter Number}, [KLG Matter Number]@row))), 0), "")
• Your formula looks correct, so per the INCORRECT ARGUMENT error message notes, double check your cross-sheet ranges of the COLLECT are the same size.
• @Jason Tarpinian I just opened the sheet back up and the error has changed to #INVALID COLUMN VALUE.
I have no idea what would have caused it to change, as I haven't touched it or the helper sheet since my last update yesterday.
My current column type is text/number, and I'm trying to get this to return the "modified by" data, so that should be text.
I'm so confused :(
Maybe I just need to call Support?
• Unfortunately without being able to click around in your solution, I'm not sure now what would be giving you that error again. I worked up a quick test just to make sure the formula syntax is
correct, and it appears so.
The couple things I did were:
□ "Modified" column is just a simple column formula of =[Modified By]@row, otherwise I notice on the COPY sheet the "Modified By" column will sometimes show automation@smartsheet.com.
□ My "Row ID" lookup column on the copy sheet is just an auto-numbered column starting at 1.
□ "KLG Matter Number" is a Text/Number type column on each sheet
□ My "Modified" column on the COPY sheet is a Contact List type column, and even though I am INDEXing it to a text/number column on my original sheet, it still comes through fine.
=IF([MVE Increase Recommended]@row = "Yes", INDEX({Modified By}, MAX(COLLECT({Row ID}, {KLG Matter Number}, [KLG Matter Number]@row))), "")
• @Jason Tarpinian I really appreciate your extra help on this.
This is one of the most incredibly frustrating Smartsheet experiences I've had. I'm still getting the #INVALID COLUMN VALUE error. I'm following all of your directions to a T. My target column is
a text/number type and every single column I'm referencing is a text/number type. Under these conditions, there is literally no way possible I should be getting an #INVALID COLUMN VALUE error.
For funsies, I changed the "modified by" column in the COPY sheet to a contact column and also changed the "MVE increase recommended by" column to a contact column. With that, I get the #CONTACT
EXPECTED error. It makes no sense.
@Andrée Starå, since this is your solution (Lock or Store Date/Value Solution without using Zapier — Smartsheet Community), do you happen to know why I'm experiencing these problems?
• Hey @Kayla Q
I was able to replicate the error #INVALID COLUMN VALUE when the equivalent of your [KLM Matter Number]@row was not a match to the data. For trouble shooting purposes, what happens if you remove
the Max/Collect portion of the formula and just hard code in a value that would yield a valid response in the formula. If this work, piece the Collect apart, one term at a time, until you find
the culprit that is causing problems.
• @Kelly Moore I'm struggling with this a little bit. It could be that I don't really understand how all of these functions work. If I use INDEX(MATCH), I actually do get the desired result:
=INDEX({Modified By}, MATCH([KLG Matter Number]@row, {KLG Matter Number}, 0))
• Another odd thing, if I just do this:
=MAX(COLLECT({Helper ID 2}, {KLG Matter Number}, [KLG Matter Number]@row))
Then I get a "0" in return:
• Hey
Do you need a Collect or will the Index/Match work for you? The MATCH function provides the item position in the list - which is what your Row ID helper column is trying to provide.
If you need the Index/Collect then continue trouble shooting. Based on the Index/Match working, the problem seems to be with the {Row ID} range. (It's the only range not also in the working Index
/Match formula)
Using Jason's formula
=IF([MVE Increase Recommended]@row = "Yes", INDEX({Modified By}, MAX(COLLECT({Row ID}, {KLG Matter Number}, [KLG Matter Number]@row))), "")
go into the formula and completely delete the {Row ID} range from the formula. When you do that, the formula window will show the REFERENCE ANOTHER SHEET button again. Click that and re-insert
your Row ID range. It's easy to have made a mistake when inserting cross sheet references.
What happens after you re-insert the {Row ID} into the formula?
• Hi @Kelly Moore,
Unfortunately, I have to take into consideration that there could be several of the same KLG Matter Number, so I need to bring back the largest Row ID from the other sheet. The other sheet has an
auto-numbering function, so each row that is added will receive a unique Row ID.
Are you suggesting that I use a "1" instead of a "0" in the index match to bring back the item in a sorted descending fashion? I suppose that could work if I ensure the other sheet is sorted
correctly and that no one will touch it and mess with the sorting. That makes me just a little nervous, but it's an alternative!
If I reinsert the Row ID, I still get the same #INVALID COLUMN VALUE error 😔
I have a Pro Desk session tomorrow on another topic, so maybe I can squeeze this in.
• Kayla
Oh, I didn't see you had reinserted the range.
• For kicks, try this
=IF([MVE Increase Recommended]@row = "Yes", INDEX({Modified By}, VALUE(MAX(COLLECT({Row ID}, {KLG Matter Number}, [KLG Matter Number]@row)))), "")
If that doesn't do anything, move the VALUE in between the Max and Collect
Oh, and I had another thought
=IF([MVE Increase Recommended]@row = "Yes", INDEX({Modified By}, MAX(COLLECT({Row ID},{Row ID}, ISNUMBER(@cell), {KLG Matter Number}, [KLG Matter Number]@row))), "")
• @Kelly Moore your first solution yields the error #INVALID COLUMN VALUE. I triple checked all of the column references and they are correct, so I'm not sure why this is happening.
The second solution yields the error #INVALID DATA TYPE.
Honestly, I think I'm just going to give up. The whole idea of this is that I have a stakeholder who wants to see who clicked a checkbox. I'm just going to tell her that she will have to
right-click and select "view cell history" to achieve this. It's not worth the hours of work I've put into it.
I really appreciate all of your assistance!
• Yes, without seeing the data, I'm not sure what to tell you. The other number fields in the formula may also be a mismatch of numbers and text. You could play with the VALUE function with them.
If you want another look, is it possible for you to share the sheets with me? Or zoom.
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Retrieval Times of Java Data Structures
July 28, 2013
If you have ever programmed in Java or any language you are probably familiar with some basic data structures. In particular, arrays, linked lists, and array lists are used very frequently. These
three data structures largely do the same thing. They all store an ordered list of objects. However, they certainly do not all work the same. They also do not all perform the same. Understanding the
strengths and weaknesses of each data structure is essential to writing efficient programs.
One of the characteristics of each of these data structures is how quickly specific data can be retrieved from them. The below code shows the initialization of an array, an array list, and a linked
list. It also shows the second piece of data in each data structure being retrieved.
//Retrieving the second value stored in an array int numArray[] = { 1, 2, 3 }; int value = numArray[1]; //Retrieving the second value stored in an array list ArrayList numArrayList = new ArrayList();
numArrayList.add(1); numArrayList.add(2); numArrayList.add(3); int value = numArrayList.get(1); //Retrieving the second value stored in an linked list LinkedList numLinkedList = new LinkedList();
numLinkedList.add(1); numLinkedList.add(2); numLinkedList.add(3); int value = numLinkedList.get(1);
The retrieval time is a significant consideration for applications where the data must be retrieved very frequently, especially when there is a very large amount of data stored. To compare how each
data structure performed, I wrote a short Java program. Below I describe what the program I wrote does.
• The program created an instance of each data structure.
• A short list of random integers was generated and stored in each of the data structures.
• A second list of random indices to retrieve was generated.
• Each data structure was tasked with retrieving the data stored in all of the listed indices.
• The time each data structure takes to complete this was recorded.
• The above steps were repeated for increasingly large lists of random numbers.
• The data was written into a text file which can be exported to Excel.
I ran my program with lists containing 10,000 to 100,000 integers. Each time the computer retrieved 200 random values from each. Below are the graphs I obtained from Excel. The y-axis shows how long
it took to retrieve a set of random numbers, and the x-axis shows how many integers were stored in each data structure.
From the graph it is obvious that the linked list has the worst retrieval time for very large sets of data. The time taken by the array, on the other hand, is independent of size. The time taken by
the array list also seems largely independent of size, although it suddenly drops when the size is around 35,000.
The results can be explained in terms of how the data structures work. Linked lists contain a set of linked nodes. To find a specific node the computer has to start from the first node and jump from
node to node until it reaches the node it was searching for. Arrays and array lists, on the other hand, store the data in a block of memory. When asking for a particular index, it simply calculates
what the address of that integer would be stored in and returns it immediately. Therefore arrays and array lists have better retrieval times than linked lists for large amounts of data. The array
list in Java is a wrapper class around the basic array object. It provides a lot of useful predefined function for common operations like data insertion and node deletion. The actual implementation
of the array list likely accounts for the strange behavior where the retrieval speed of the array list suddenly went down around 35,000 elements.
It is definitely worth noting that these results don't mean linked lists are bad to use. In fact it looks like the linked list may actually perform better than the array list for arrays containing
less than 10,000 elements. Linked lists perform much better than arrays for operations involving insertion and deletion of nodes. Linked lists are ideal for cases where those operations are
frequently used. But it is also important to remember they are much slower at retrieving information. It really can make a large difference for programs involving huge lists of objects. Maybe in the
future I will write an article comparing the insertion and deletion times of these data structures.
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The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow
beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return
flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important
consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.
The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow
beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return
flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important
consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.
This paper examines the representation of eddy fluxes by bolus velocities. In particular, it asks the following: 1) Can an arbitrary eddy flux divergence of density be represented accurately by a
nondivergent bolus flux that satisfies the condition of no normal flow at boundaries? 2) If not, how close can such a representation come? 3) If such a representation can exist in some circumstances,
what is the size of the smallest bolus velocity that fits the data?
The author finds, in agreement with earlier authors, that the answer to the first question is no, although under certain conditions, which include a modification to the eddy flux divergence, a bolus
representation becomes possible. One such condition is when the eddy flux divergence is required to balance the time-mean flux divergence. The smallest bolus flow is easily found by solving a
thickness-weighted Poisson equation on each density level. This problem is solved for the North Pacific using time-mean data from an eddy-permitting model. The minimum bolus flow is found to be very
small at depth but larger than is usually assumed near the surface. The magnitude of this minimum flow is of order one-tenth of the mean flow. Similar but larger results are found for a
coarse-resolution model.
This paper examines the representation of eddy fluxes by bolus velocities. In particular, it asks the following: 1) Can an arbitrary eddy flux divergence of density be represented accurately by a
nondivergent bolus flux that satisfies the condition of no normal flow at boundaries? 2) If not, how close can such a representation come? 3) If such a representation can exist in some circumstances,
what is the size of the smallest bolus velocity that fits the data?
The author finds, in agreement with earlier authors, that the answer to the first question is no, although under certain conditions, which include a modification to the eddy flux divergence, a bolus
representation becomes possible. One such condition is when the eddy flux divergence is required to balance the time-mean flux divergence. The smallest bolus flow is easily found by solving a
thickness-weighted Poisson equation on each density level. This problem is solved for the North Pacific using time-mean data from an eddy-permitting model. The minimum bolus flow is found to be very
small at depth but larger than is usually assumed near the surface. The magnitude of this minimum flow is of order one-tenth of the mean flow. Similar but larger results are found for a
coarse-resolution model.
Nathan Paldor
Peter D. Killworth
The trajectories of inertial flows on a rotating earth are calculated, in an attempt to reconcile the differing heuristic suggestions in the literature on the subject. It is shown that westward
propagating “nearly closed” orbits are possible away from the equator. For orbits crossing the equator, we find a stationary, “figure-eight- like” orbit, together with eastward and westward
propagating modes. Near the pole, the convergence of longitudes causes the trajectories to be deflected cyclonically in contrast to the deflection of the Coriolis force, giving rise to a westward
propagating mode that meanders about a central latitude.
The trajectories of inertial flows on a rotating earth are calculated, in an attempt to reconcile the differing heuristic suggestions in the literature on the subject. It is shown that westward
propagating “nearly closed” orbits are possible away from the equator. For orbits crossing the equator, we find a stationary, “figure-eight- like” orbit, together with eastward and westward
propagating modes. Near the pole, the convergence of longitudes causes the trajectories to be deflected cyclonically in contrast to the deflection of the Coriolis force, giving rise to a westward
propagating mode that meanders about a central latitude.
Adrian Hines
Peter D. Killworth
Attempts to estimate the state of the ocean usually involve one of two approaches: either an assimilation of data (typically altimetric surface height) is performed or an inversion is carried out
according to some minimization scheme. The former case normally retains some version of the time-dependent equations of motion; the latter is usually steady. Data sources are frequently not ideal for
either approach, usually being spatially and temporally confined (e.g., from an oceanographic cruise). This raises particular difficulties for inversions, whose physics seldom includes much beyond
the geostrophic balance. In this paper the authors examine an approach midway between the two, examining several questions. (i) What is the impact of data assimilated continuously to a steady state
on regions outside the data sources? (ii) Can remote data improve the long-term mean of a model whose natural response is not close to climatology? (iii) Can an eddy-free model assimilate data
containing eddies?
The authors employ an inversion using a simple North Atlantic model, which permits no eddies, but contains better dynamics than geostrophy (the frictional planetary geostrophic equations), and an
assimilative scheme rather simpler than those normally employed, almost equivalent to direct data insertion, run to a steady state. The data used are real subsurface data, which do contain eddies,
from World Ocean Circulation Experiment cruises in the northern North Atlantic. The presence of noise in these data is found to cause no numerical difficulties, and the authors show that the impact
of even one vertical profile can strongly modify the water mass properties of the solution far from the data region through a combination of wave propagation, advection, and diffusion. Because the
model can be run for very long times, the region of impact is thus somewhat wider than would occur for assimilations over short intervals, such as a year.
Attempts to estimate the state of the ocean usually involve one of two approaches: either an assimilation of data (typically altimetric surface height) is performed or an inversion is carried out
according to some minimization scheme. The former case normally retains some version of the time-dependent equations of motion; the latter is usually steady. Data sources are frequently not ideal for
either approach, usually being spatially and temporally confined (e.g., from an oceanographic cruise). This raises particular difficulties for inversions, whose physics seldom includes much beyond
the geostrophic balance. In this paper the authors examine an approach midway between the two, examining several questions. (i) What is the impact of data assimilated continuously to a steady state
on regions outside the data sources? (ii) Can remote data improve the long-term mean of a model whose natural response is not close to climatology? (iii) Can an eddy-free model assimilate data
containing eddies?
The authors employ an inversion using a simple North Atlantic model, which permits no eddies, but contains better dynamics than geostrophy (the frictional planetary geostrophic equations), and an
assimilative scheme rather simpler than those normally employed, almost equivalent to direct data insertion, run to a steady state. The data used are real subsurface data, which do contain eddies,
from World Ocean Circulation Experiment cruises in the northern North Atlantic. The presence of noise in these data is found to cause no numerical difficulties, and the authors show that the impact
of even one vertical profile can strongly modify the water mass properties of the solution far from the data region through a combination of wave propagation, advection, and diffusion. Because the
model can be run for very long times, the region of impact is thus somewhat wider than would occur for assimilations over short intervals, such as a year.
Rebecca A. Woodgate
Peter D. Killworth
Although data assimilation is now an established oceanographic technique, little work has been done on the interaction of the assimilation scheme and the physics of the underlying model. The way in
which even a simple assimilation scheme (here nudging) can significantly alter the response of the model to which it is applied is illustrated here.
Using analytic and semianalytic models, the assimilation of sea surface height, density, and velocity is studied. It is shown that the assimilation can act to alter the high inertia–gravity wave
frequency to be the order of the Coriolis parameter, a result that is of relevance to the problems of initialization. The theory also predicts an optimum strength of nudging, normally dependent on
wavelength, wave speed, and latitude, which can give convergence of the assimilation on a timescale as short as a day. The results are verified by identical twin experiments using a full primitive
equation model, the Free Surface Cox Code, both in barotropic spinup (results presented here) and in a more realistic baroclinic situation (results presented in Part II).
Although data assimilation is now an established oceanographic technique, little work has been done on the interaction of the assimilation scheme and the physics of the underlying model. The way in
which even a simple assimilation scheme (here nudging) can significantly alter the response of the model to which it is applied is illustrated here.
Using analytic and semianalytic models, the assimilation of sea surface height, density, and velocity is studied. It is shown that the assimilation can act to alter the high inertia–gravity wave
frequency to be the order of the Coriolis parameter, a result that is of relevance to the problems of initialization. The theory also predicts an optimum strength of nudging, normally dependent on
wavelength, wave speed, and latitude, which can give convergence of the assimilation on a timescale as short as a day. The results are verified by identical twin experiments using a full primitive
equation model, the Free Surface Cox Code, both in barotropic spinup (results presented here) and in a more realistic baroclinic situation (results presented in Part II).
Steven G. Alderson
Peter D. Killworth
A preoperational scheme has been implemented to calculate sea surface height fields at 7-day intervals over the North Atlantic. Input data from Argo floats is downloaded and processed in near–real
time. The solution method is by Bernoulli inverse. Early results are encouraging. Features of the results are compared with both model and satellite data and show good agreement.
A preoperational scheme has been implemented to calculate sea surface height fields at 7-day intervals over the North Atlantic. Input data from Argo floats is downloaded and processed in near–real
time. The solution method is by Bernoulli inverse. Early results are encouraging. Features of the results are compared with both model and satellite data and show good agreement.
Michael K. Davey
Peter D. Killworth
A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve
potential vorticity q when unforced. Regions of closed q contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of
closed height contours.)
Motion is wavelike for very weak disturbances (maximum particle speed Û; ≪ long planetary wave speed ĉ). The height field disperses like a group of linear Rossby waves, and tracers have small,
oscillatory (mainly north-south) displacements, with very little scatter.
When Û≈ĉ, the planetary q field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent
“self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring
tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good
indicator of tracer movement.)
When Ü≫ĉ, the q field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient q (and fluid elements) around itself. This wrapping effect leads
to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the
center-of-mass anomaly (for a buoyancy-generated disturbance).
Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed ĉ, whereas a cyclone travels slower than ĉ. The
predictions are confirmed by the numerical experiments.
Some estimates of mixing based on tracer separation are given.
A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve
potential vorticity q when unforced. Regions of closed q contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of
closed height contours.)
Motion is wavelike for very weak disturbances (maximum particle speed Û; ≪ long planetary wave speed ĉ). The height field disperses like a group of linear Rossby waves, and tracers have small,
oscillatory (mainly north-south) displacements, with very little scatter.
When Û≈ĉ, the planetary q field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent
“self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring
tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good
indicator of tracer movement.)
When Ü≫ĉ, the q field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient q (and fluid elements) around itself. This wrapping effect leads
to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the
center-of-mass anomaly (for a buoyancy-generated disturbance).
Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed ĉ, whereas a cyclone travels slower than ĉ. The
predictions are confirmed by the numerical experiments.
Some estimates of mixing based on tracer separation are given.
Peter D. Killworth
Grant R. Bigg
Three inverse methods (the Bernoulli, beta-spiral, and box inverse methods) are used on mean data from an eddy-resolving oceanic general circulation model, in an attempt to reconstruct the observed
mean flow field. Inversions are performed in the Gulf Stream extension, a quiet region which is relatively eddy-free, the center of the region of homogenized potential vorticity, and a
near-equatorial area, together with an inversion of the flow across a transoceanic sector. Resolutions for the inversions of ⅓°, 1° and 2° are used. Numerical estimates of geostrophy using the wider
resolutions can give top-to-bottom thermal wind shears in error by up to 1 cm s^−1 in a flow change of around 8 cm s^−1. Two “scores” for the methods are created, one which tests pointwise accuracy
(the “global” score) and one which tests fluxes of mass through a section (the “flux” score). The Bernoulli method yields accurate global scores except in the homogenized region; the box inverse
method yields fairly accurate global scores everywhere; and the beta-spiral only gives accurate global scores near the equator. No method gives reliable flux scores, although the box inverse was the
least inaccurate, as might be expected from the nature of this method. The hypothesis of no flow at the bottom gives a predicted velocity field which is more accurate than any of the inversions most
of the time. The Bernoulli and beta-spiral methods contain an internal measure which is well correlated with their accuracy, so that it is possible to estimate the accuracy of an inversion on real
Three inverse methods (the Bernoulli, beta-spiral, and box inverse methods) are used on mean data from an eddy-resolving oceanic general circulation model, in an attempt to reconstruct the observed
mean flow field. Inversions are performed in the Gulf Stream extension, a quiet region which is relatively eddy-free, the center of the region of homogenized potential vorticity, and a
near-equatorial area, together with an inversion of the flow across a transoceanic sector. Resolutions for the inversions of ⅓°, 1° and 2° are used. Numerical estimates of geostrophy using the wider
resolutions can give top-to-bottom thermal wind shears in error by up to 1 cm s^−1 in a flow change of around 8 cm s^−1. Two “scores” for the methods are created, one which tests pointwise accuracy
(the “global” score) and one which tests fluxes of mass through a section (the “flux” score). The Bernoulli method yields accurate global scores except in the homogenized region; the box inverse
method yields fairly accurate global scores everywhere; and the beta-spiral only gives accurate global scores near the equator. No method gives reliable flux scores, although the box inverse was the
least inaccurate, as might be expected from the nature of this method. The hypothesis of no flow at the bottom gives a predicted velocity field which is more accurate than any of the inversions most
of the time. The Bernoulli and beta-spiral methods contain an internal measure which is well correlated with their accuracy, so that it is possible to estimate the accuracy of an inversion on real
Michael K. Davey
Peter D. Killworth
The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is
considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region,
with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are
sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an
eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.
The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is
considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region,
with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are
sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an
eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.
William K. Dewar
Peter D. Killworth
The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we
consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of
two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse
of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no
variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation
Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than
one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial
eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made.
The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we
consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of
two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse
of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no
variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation
Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than
one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial
eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made. | {"url":"https://journals.ametsoc.org/search?access=all&f_0=author&page=3&pageSize=10&q_0=Peter+D.+Killworth&sort=relevance","timestamp":"2024-11-14T00:35:23Z","content_type":"text/html","content_length":"429356","record_id":"<urn:uuid:50d1f2cb-d6c6-4921-bd8d-b8e80e22bd31>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00251.warc.gz"} |
Illustrative Mathematics
Integers on the Number Line 1
1. Find and label the numbers $-3$ and $-5$ on the number line.
2. For each of the following, state whether the inequality is true or false. Use the number line diagram to help explain your answers.
1. $-3 \gt -5$
2. $-5 \gt -3$
3. $-5 \lt -3$
4. $-3 \lt -5$
1. Because 3 is three units away from 0 (and 5 is five units away from 0), we know that the number line is in increments of 1.
-3 is located three units to the left of zero. We know this because the number 3 tells us how many units from zero the number lies, and the sign of the number tells us which side of zero the
number lies. In this case, the sign is negative so we will plot the number to the left of zero.
-5 is located five units to the left of zero. We know this because the number 5 tells us how many units from zero the number lies, and the sign of the number tells us which side of zero the
number lies.
Now we can plot -3 and -5.
2. Note: On a number line where positive numbers are to the right of zero and negative numbers are to the left of zero, numbers farther to the right are always greater than those to the left.
$-3 \gt -5$ because -3 is farther to the right than -5.
$-5 \lt -3$ because -5 is farther to the left than -3.
Statements 1 and 3 are true, while statements 2 and 4 are false.
Integers on the Number Line 1
1. Find and label the numbers $-3$ and $-5$ on the number line.
2. For each of the following, state whether the inequality is true or false. Use the number line diagram to help explain your answers.
1. $-3 \gt -5$
2. $-5 \gt -3$
3. $-5 \lt -3$
4. $-3 \lt -5$ | {"url":"https://tasks.illustrativemathematics.org/content-standards/6/NS/C/7/tasks/283","timestamp":"2024-11-08T04:43:28Z","content_type":"text/html","content_length":"25805","record_id":"<urn:uuid:d5351444-6d52-43d8-9d83-0a24423a4ef3>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00627.warc.gz"} |
Chinese remainder theorem - Wikiwand
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of
the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).
Sunzi's original formulation:
For example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then with no other information, we can determine
that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23. Importantly, this tells us that if n is a natural number less than 105, then 23 is the only possible value of n.
It is also known as Sunzi's theorem, as the earliest known statement is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd to 5th century CE.
The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar
computations on small integers.
The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any ring, with a formulation involving two-sided ideals.
The earliest known statement of the problem appears in the 5th-century book Sunzi Suanjing by the Chinese mathematician Sunzi:^[1]
There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are
Sunzi's work would not be considered a theorem by modern standards; it only gives one particular problem, without showing how to solve it, much less any proof about the general case or a general
algorithm for solving it.^[3] What amounts to an algorithm for solving this problem was described by Aryabhata (6th century).^[4] Special cases of the Chinese remainder theorem were also known to
Brahmagupta (7th century) and appear in Fibonacci's Liber Abaci (1202).^[5] The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical
Treatise in Nine Sections ^[6] which was translated into English in early 19th century by British missionary Alexander Wylie.^[7]
The Chinese remainder theorem appears in Gauss's 1801 book Disquisitiones Arithmeticae.^
The notion of congruences was first introduced and used by Carl Friedrich Gauss in his Disquisitiones Arithmeticae of 1801.^[9] Gauss illustrates the Chinese remainder theorem on a problem involving
calendars, namely, "to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction."^[10] Gauss introduces a procedure for solving the problem
that had already been used by Leonhard Euler but was in fact an ancient method that had appeared several times.^[11]
Let n[1], ..., n[k] be integers greater than 1, which are often called moduli or divisors. Let us denote by N the product of the n[i].
The Chinese remainder theorem asserts that if the n[i] are pairwise coprime, and if a[1], ..., a[k] are integers such that 0 ≤ a[i] < n[i] for every i, then there is one and only one integer x, such
that 0 ≤ x < N and the remainder of the Euclidean division of x by n[i] is a[i] for every i.
This may be restated as follows in terms of congruences: If the ${\displaystyle n_{i}}$ are pairwise coprime, and if a[1], ..., a[k] are any integers, then the system
{\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\,\,\,\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\end{aligned}}}
has a solution, and any two solutions, say x[1] and x[2], are congruent modulo N, that is, x[1] ≡ x[2] (mod N).^[12]
In abstract algebra, the theorem is often restated as: if the n[i] are pairwise coprime, the map
${\displaystyle x{\bmod {N}}\;\mapsto \;(x{\bmod {n}}_{1},\,\ldots ,\,x{\bmod {n}}_{k})}$
defines a ring isomorphism^[13]
${\displaystyle \mathbb {Z} /N\mathbb {Z} \cong \mathbb {Z} /n_{1}\mathbb {Z} \times \cdots \times \mathbb {Z} /n_{k}\mathbb {Z} }$
between the ring of integers modulo N and the direct product of the rings of integers modulo the n[i]. This means that for doing a sequence of arithmetic operations in ${\displaystyle \mathbb {Z} /N\
mathbb {Z} ,}$ one may do the same computation independently in each ${\displaystyle \mathbb {Z} /n_{i}\mathbb {Z} }$ and then get the result by applying the isomorphism (from the right to the left).
This may be much faster than the direct computation if N and the number of operations are large. This is widely used, under the name multi-modular computation, for linear algebra over the integers or
the rational numbers.
The theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family.^[14]
The existence and the uniqueness of the solution may be proven independently. However, the first proof of existence, given below, uses this uniqueness.
Suppose that x and y are both solutions to all the congruences. As x and y give the same remainder, when divided by n[i], their difference x − y is a multiple of each n[i]. As the n[i] are pairwise
coprime, their product N also divides x − y, and thus x and y are congruent modulo N. If x and y are supposed to be non-negative and less than N (as in the first statement of the theorem), then their
difference may be a multiple of N only if x = y.
Existence (first proof)
The map
${\displaystyle x{\bmod {N}}\mapsto (x{\bmod {n}}_{1},\ldots ,x{\bmod {n}}_{k})}$
maps congruence classes modulo N to sequences of congruence classes modulo n[i]. The proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the same
number of elements, the map is also surjective, which proves the existence of the solution.
This proof is very simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to other situations where the following proof can.
Existence (constructive proof)
Existence may be established by an explicit construction of x.^[15] This construction may be split into two steps, first solving the problem in the case of two moduli, and then extending this
solution to the general case by induction on the number of moduli.
Case of two moduli
We want to solve the system:
{\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\x&\equiv a_{2}{\pmod {n_{2}}},\end{aligned}}}
where ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are coprime.
Bézout's identity asserts the existence of two integers ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ such that
${\displaystyle m_{1}n_{1}+m_{2}n_{2}=1.}$
The integers ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ may be computed by the extended Euclidean algorithm.
A solution is given by
${\displaystyle x=a_{1}m_{2}n_{2}+a_{2}m_{1}n_{1}.}$
{\displaystyle {\begin{aligned}x&=a_{1}m_{2}n_{2}+a_{2}m_{1}n_{1}\\&=a_{1}(1-m_{1}n_{1})+a_{2}m_{1}n_{1}\\&=a_{1}+(a_{2}-a_{1})m_{1}n_{1},\end{aligned}}}
implying that ${\displaystyle x\equiv a_{1}{\pmod {n_{1}}}.}$ The second congruence is proved similarly, by exchanging the subscripts 1 and 2.
General case
Consider a sequence of congruence equations:
{\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\end{aligned}}}
where the ${\displaystyle n_{i}}$ are pairwise coprime. The two first equations have a solution ${\displaystyle a_{1,2}}$ provided by the method of the previous section. The set of the solutions of
these two first equations is the set of all solutions of the equation
${\displaystyle x\equiv a_{1,2}{\pmod {n_{1}n_{2}}}.}$
As the other ${\displaystyle n_{i}}$ are coprime with ${\displaystyle n_{1}n_{2},}$ this reduces solving the initial problem of k equations to a similar problem with ${\displaystyle k-1}$ equations.
Iterating the process, one gets eventually the solutions of the initial problem.
Existence (direct construction)
For constructing a solution, it is not necessary to make an induction on the number of moduli. However, such a direct construction involves more computation with large numbers, which makes it less
efficient and less used. Nevertheless, Lagrange interpolation is a special case of this construction, applied to polynomials instead of integers.
Let ${\displaystyle N_{i}=N/n_{i}}$ be the product of all moduli but one. As the ${\displaystyle n_{i}}$ are pairwise coprime, ${\displaystyle N_{i}}$ and ${\displaystyle n_{i}}$ are coprime. Thus
Bézout's identity applies, and there exist integers ${\displaystyle M_{i}}$ and ${\displaystyle m_{i}}$ such that
${\displaystyle M_{i}N_{i}+m_{i}n_{i}=1.}$
A solution of the system of congruences is
${\displaystyle x=\sum _{i=1}^{k}a_{i}M_{i}N_{i}.}$
In fact, as ${\displaystyle N_{j}}$ is a multiple of ${\displaystyle n_{i}}$ for ${\displaystyle ieq j,}$ we have
${\displaystyle x\equiv a_{i}M_{i}N_{i}\equiv a_{i}(1-m_{i}n_{i})\equiv a_{i}{\pmod {n_{i}}},}$
for every ${\displaystyle i.}$
Consider a system of congruences:
{\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\\\end{aligned}}}
where the ${\displaystyle n_{i}}$ are pairwise coprime, and let ${\displaystyle N=n_{1}n_{2}\cdots n_{k}.}$ In this section several methods are described for computing the unique solution for ${\
displaystyle x}$, such that ${\displaystyle 0\leq x<N,}$ and these methods are applied on the example
{\displaystyle {\begin{aligned}x&\equiv 0{\pmod {3}}\\x&\equiv 3{\pmod {4}}\\x&\equiv 4{\pmod {5}}.\end{aligned}}}
Several methods of computation are presented. The two first ones are useful for small examples, but become very inefficient when the product ${\displaystyle n_{1}\cdots n_{k}}$ is large. The third
one uses the existence proof given in § Existence (constructive proof). It is the most convenient when the product ${\displaystyle n_{1}\cdots n_{k}}$ is large, or for computer computation.
Systematic search
It is easy to check whether a value of x is a solution: it suffices to compute the remainder of the Euclidean division of x by each n[i]. Thus, to find the solution, it suffices to check successively
the integers from 0 to N until finding the solution.
Although very simple, this method is very inefficient. For the simple example considered here, 40 integers (including 0) have to be checked for finding the solution, which is 39. This is an
exponential time algorithm, as the size of the input is, up to a constant factor, the number of digits of N, and the average number of operations is of the order of N.
Therefore, this method is rarely used, neither for hand-written computation nor on computers.
Search by sieving
The smallest two solutions, 23 and 128, of the original formulation of the Chinese remainder theorem problem found using a sieve
The search of the solution may be made dramatically faster by sieving. For this method, we suppose, without loss of generality, that ${\displaystyle 0\leq a_{i}<n_{i}}$ (if it were not the case, it
would suffice to replace each ${\displaystyle a_{i}}$ by the remainder of its division by ${\displaystyle n_{i}}$). This implies that the solution belongs to the arithmetic progression
${\displaystyle a_{1},a_{1}+n_{1},a_{1}+2n_{1},\ldots }$
By testing the values of these numbers modulo ${\displaystyle n_{2},}$ one eventually finds a solution ${\displaystyle x_{2}}$ of the two first congruences. Then the solution belongs to the
arithmetic progression
${\displaystyle x_{2},x_{2}+n_{1}n_{2},x_{2}+2n_{1}n_{2},\ldots }$
Testing the values of these numbers modulo ${\displaystyle n_{3},}$ and continuing until every modulus has been tested eventually yields the solution.
This method is faster if the moduli have been ordered by decreasing value, that is if ${\displaystyle n_{1}>n_{2}>\cdots >n_{k}.}$ For the example, this gives the following computation. We consider
first the numbers that are congruent to 4 modulo 5 (the largest modulus), which are 4, 9 = 4 + 5, 14 = 9 + 5, ... For each of them, compute the remainder by 4 (the second largest modulus) until
getting a number congruent to 3 modulo 4. Then one can proceed by adding 20 = 5×4 at each step, and computing only the remainders by 3. This gives
4 mod 4 → 0. Continue
4 + 5 = 9 mod 4 →1. Continue
9 + 5 = 14 mod 4 → 2. Continue
14 + 5 = 19 mod 4 → 3. OK, continue by considering remainders modulo 3 and adding 5×4 = 20 each time
19 mod 3 → 1. Continue
19 + 20 = 39 mod 3 → 0. OK, this is the result.
This method works well for hand-written computation with a product of moduli that is not too big. However, it is much slower than other methods, for very large products of moduli. Although
dramatically faster than the systematic search, this method also has an exponential time complexity and is therefore not used on computers.
Using the existence construction
The constructive existence proof shows that, in the case of two moduli, the solution may be obtained by the computation of the Bézout coefficients of the moduli, followed by a few multiplications,
additions and reductions modulo ${\displaystyle n_{1}n_{2}}$ (for getting a result in the interval ${\displaystyle (0,n_{1}n_{2}-1)}$). As the Bézout's coefficients may be computed with the extended
Euclidean algorithm, the whole computation, at most, has a quadratic time complexity of ${\displaystyle O((s_{1}+s_{2})^{2}),}$ where ${\displaystyle s_{i}}$ denotes the number of digits of ${\
displaystyle n_{i}.}$
For more than two moduli, the method for two moduli allows the replacement of any two congruences by a single congruence modulo the product of the moduli. Iterating this process provides eventually
the solution with a complexity, which is quadratic in the number of digits of the product of all moduli. This quadratic time complexity does not depend on the order in which the moduli are regrouped.
One may regroup the two first moduli, then regroup the resulting modulus with the next one, and so on. This strategy is the easiest to implement, but it also requires more computation involving large
Another strategy consists in partitioning the moduli in pairs whose product have comparable sizes (as much as possible), applying, in parallel, the method of two moduli to each pair, and iterating
with a number of moduli approximatively divided by two. This method allows an easy parallelization of the algorithm. Also, if fast algorithms (that is, algorithms working in quasilinear time) are
used for the basic operations, this method provides an algorithm for the whole computation that works in quasilinear time.
On the current example (which has only three moduli), both strategies are identical and work as follows.
Bézout's identity for 3 and 4 is
${\displaystyle 1\times 4+(-1)\times 3=1.}$
Putting this in the formula given for proving the existence gives
${\displaystyle 0\times 1\times 4+3\times (-1)\times 3=-9}$
for a solution of the two first congruences, the other solutions being obtained by adding to −9 any multiple of 3×4 = 12. One may continue with any of these solutions, but the solution 3 = −9 +12
is smaller (in absolute value) and thus leads probably to an easier computation
Bézout identity for 5 and 3×4 = 12 is
${\displaystyle 5\times 5+(-2)\times 12=1.}$
Applying the same formula again, we get a solution of the problem:
${\displaystyle 5\times 5\times 3+12\times (-2)\times 4=-21.}$
The other solutions are obtained by adding any multiple of 3×4×5 = 60, and the smallest positive solution is −21 + 60 = 39.
As a linear Diophantine system
The system of congruences solved by the Chinese remainder theorem may be rewritten as a system of linear Diophantine equations:
{\displaystyle {\begin{aligned}x&=a_{1}+x_{1}n_{1}\\&\vdots \\x&=a_{k}+x_{k}n_{k},\end{aligned}}}
where the unknown integers are ${\displaystyle x}$ and the ${\displaystyle x_{i}.}$ Therefore, every general method for solving such systems may be used for finding the solution of Chinese remainder
theorem, such as the reduction of the matrix of the system to Smith normal form or Hermite normal form. However, as usual when using a general algorithm for a more specific problem, this approach is
less efficient than the method of the preceding section, based on a direct use of Bézout's identity.
In § Statement, the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences, and of a ring isomorphism. The statement in terms of remainders does not
apply, in general, to principal ideal domains, as remainders are not defined in such rings. However, the two other versions make sense over a principal ideal domain R: it suffices to replace
"integer" by "element of the domain" and ${\displaystyle \mathbb {Z} }$ by R. These two versions of the theorem are true in this context, because the proofs (except for the first existence proof),
are based on Euclid's lemma and Bézout's identity, which are true over every principal domain.
However, in general, the theorem is only an existence theorem and does not provide any way for computing the solution, unless one has an algorithm for computing the coefficients of Bézout's identity.
The statement in terms of remainders given in § Theorem statement cannot be generalized to any principal ideal domain, but its generalization to Euclidean domains is straightforward. The univariate
polynomials over a field is the typical example of a Euclidean domain which is not the integers. Therefore, we state the theorem for the case of the ring ${\displaystyle R=K[X]}$ for a field ${\
displaystyle K.}$ For getting the theorem for a general Euclidean domain, it suffices to replace the degree by the Euclidean function of the Euclidean domain.
The Chinese remainder theorem for polynomials is thus: Let ${\displaystyle P_{i}(X)}$ (the moduli) be, for ${\displaystyle i=1,\dots ,k}$, pairwise coprime polynomials in ${\displaystyle R=K[X]}$.
Let ${\displaystyle d_{i}=\deg P_{i}}$ be the degree of ${\displaystyle P_{i}(X)}$, and ${\displaystyle D}$ be the sum of the ${\displaystyle d_{i}.}$ If ${\displaystyle A_{i}(X),\ldots ,A_{k}(X)}$
are polynomials such that ${\displaystyle A_{i}(X)=0}$ or ${\displaystyle \deg A_{i}<d_{i}}$ for every i, then, there is one and only one polynomial ${\displaystyle P(X)}$, such that ${\displaystyle
\deg P<D}$ and the remainder of the Euclidean division of ${\displaystyle P(X)}$ by ${\displaystyle P_{i}(X)}$ is ${\displaystyle A_{i}(X)}$ for every i.
The construction of the solution may be done as in § Existence (constructive proof) or § Existence (direct proof). However, the latter construction may be simplified by using, as follows, partial
fraction decomposition instead of the extended Euclidean algorithm.
Thus, we want to find a polynomial ${\displaystyle P(X)}$, which satisfies the congruences
${\displaystyle P(X)\equiv A_{i}(X){\pmod {P_{i}(X)}},}$
for ${\displaystyle i=1,\ldots ,k.}$
Consider the polynomials
{\displaystyle {\begin{aligned}Q(X)&=\prod _{i=1}^{k}P_{i}(X)\\Q_{i}(X)&={\frac {Q(X)}{P_{i}(X)}}.\end{aligned}}}
The partial fraction decomposition of ${\displaystyle 1/Q(X)}$ gives k polynomials ${\displaystyle S_{i}(X)}$ with degrees ${\displaystyle \deg S_{i}(X)<d_{i},}$ such that
${\displaystyle {\frac {1}{Q(X)}}=\sum _{i=1}^{k}{\frac {S_{i}(X)}{P_{i}(X)}},}$
and thus
${\displaystyle 1=\sum _{i=1}^{k}S_{i}(X)Q_{i}(X).}$
Then a solution of the simultaneous congruence system is given by the polynomial
${\displaystyle \sum _{i=1}^{k}A_{i}(X)S_{i}(X)Q_{i}(X).}$
In fact, we have
${\displaystyle \sum _{i=1}^{k}A_{i}(X)S_{i}(X)Q_{i}(X)=A_{i}(X)+\sum _{j=1}^{k}(A_{j}(X)-A_{i}(X))S_{j}(X)Q_{j}(X)\equiv A_{i}(X){\pmod {P_{i}(X)}},}$
for ${\displaystyle 1\leq i\leq k.}$
This solution may have a degree larger than ${\displaystyle D=\sum _{i=1}^{k}d_{i}.}$ The unique solution of degree less than ${\displaystyle D}$ may be deduced by considering the remainder ${\
displaystyle B_{i}(X)}$ of the Euclidean division of ${\displaystyle A_{i}(X)S_{i}(X)}$ by ${\displaystyle P_{i}(X).}$ This solution is
${\displaystyle P(X)=\sum _{i=1}^{k}B_{i}(X)Q_{i}(X).}$
Lagrange interpolation
A special case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one:
${\displaystyle P_{i}(X)=X-x_{i}.}$
They are pairwise coprime if the ${\displaystyle x_{i}}$ are all different. The remainder of the division by ${\displaystyle P_{i}(X)}$ of a polynomial ${\displaystyle P(X)}$ is ${\displaystyle P(x_
{i})}$, by the polynomial remainder theorem.
Now, let ${\displaystyle A_{1},\ldots ,A_{k}}$ be constants (polynomials of degree 0) in ${\displaystyle K.}$ Both Lagrange interpolation and Chinese remainder theorem assert the existence of a
unique polynomial ${\displaystyle P(X),}$ of degree less than ${\displaystyle k}$ such that
${\displaystyle P(x_{i})=A_{i},}$
for every ${\displaystyle i.}$
Lagrange interpolation formula is exactly the result, in this case, of the above construction of the solution. More precisely, let
{\displaystyle {\begin{aligned}Q(X)&=\prod _{i=1}^{k}(X-x_{i})\\[6pt]Q_{i}(X)&={\frac {Q(X)}{X-x_{i}}}.\end{aligned}}}
The partial fraction decomposition of ${\displaystyle {\frac {1}{Q(X)}}}$ is
${\displaystyle {\frac {1}{Q(X)}}=\sum _{i=1}^{k}{\frac {1}{Q_{i}(x_{i})(X-x_{i})}}.}$
In fact, reducing the right-hand side to a common denominator one gets
${\displaystyle \sum _{i=1}^{k}{\frac {1}{Q_{i}(x_{i})(X-x_{i})}}={\frac {1}{Q(X)}}\sum _{i=1}^{k}{\frac {Q_{i}(X)}{Q_{i}(x_{i})}},}$
and the numerator is equal to one, as being a polynomial of degree less than ${\displaystyle k,}$ which takes the value one for ${\displaystyle k}$ different values of ${\displaystyle X.}$
Using the above general formula, we get the Lagrange interpolation formula:
${\displaystyle P(X)=\sum _{i=1}^{k}A_{i}{\frac {Q_{i}(X)}{Q_{i}(x_{i})}}.}$
Hermite interpolation
Hermite interpolation is an application of the Chinese remainder theorem for univariate polynomials, which may involve moduli of arbitrary degrees (Lagrange interpolation involves only moduli of
degree one).
The problem consists of finding a polynomial of the least possible degree, such that the polynomial and its first derivatives take given values at some fixed points.
More precisely, let ${\displaystyle x_{1},\ldots ,x_{k}}$ be ${\displaystyle k}$ elements of the ground field ${\displaystyle K,}$ and, for ${\displaystyle i=1,\ldots ,k,}$ let ${\displaystyle a_
{i,0},a_{i,1},\ldots ,a_{i,r_{i}-1}}$ be the values of the first ${\displaystyle r_{i}}$ derivatives of the sought polynomial at ${\displaystyle x_{i}}$ (including the 0th derivative, which is the
value of the polynomial itself). The problem is to find a polynomial ${\displaystyle P(X)}$ such that its jth derivative takes the value ${\displaystyle a_{i,j}}$ at ${\displaystyle x_{i},}$ for ${\
displaystyle i=1,\ldots ,k}$ and ${\displaystyle j=0,\ldots ,r_{j}.}$
Consider the polynomial
${\displaystyle P_{i}(X)=\sum _{j=0}^{r_{i}-1}{\frac {a_{i,j}}{j!}}(X-x_{i})^{j}.}$
This is the Taylor polynomial of order ${\displaystyle r_{i}-1}$ at ${\displaystyle x_{i}}$, of the unknown polynomial ${\displaystyle P(X).}$ Therefore, we must have
${\displaystyle P(X)\equiv P_{i}(X){\pmod {(X-x_{i})^{r_{i}}}}.}$
Conversely, any polynomial ${\displaystyle P(X)}$ that satisfies these ${\displaystyle k}$ congruences, in particular verifies, for any ${\displaystyle i=1,\ldots ,k}$
${\displaystyle P(X)=P_{i}(X)+o(X-x_{i})^{r_{i}-1}}$
therefore ${\displaystyle P_{i}(X)}$ is its Taylor polynomial of order ${\displaystyle r_{i}-1}$ at ${\displaystyle x_{i}}$, that is, ${\displaystyle P(X)}$ solves the initial Hermite interpolation
problem. The Chinese remainder theorem asserts that there exists exactly one polynomial of degree less than the sum of the ${\displaystyle r_{i},}$ which satisfies these ${\displaystyle k}$
There are several ways for computing the solution ${\displaystyle P(X).}$ One may use the method described at the beginning of § Over univariate polynomial rings and Euclidean domains. One may also
use the constructions given in § Existence (constructive proof) or § Existence (direct proof).
The Chinese remainder theorem can be generalized to non-coprime moduli. Let ${\displaystyle m,n,a,b}$ be any integers, let ${\displaystyle g=\gcd(m,n)}$; ${\displaystyle M=\operatorname {lcm} (m,n)}$
, and consider the system of congruences:
{\displaystyle {\begin{aligned}x&\equiv a{\pmod {m}}\\x&\equiv b{\pmod {n}},\end{aligned}}}
If ${\displaystyle a\equiv b{\pmod {g}}}$, then this system has a unique solution modulo ${\displaystyle M=mn/g}$. Otherwise, it has no solutions.
If one uses Bézout's identity to write ${\displaystyle g=um+vn}$, then the solution is given by
${\displaystyle x={\frac {avn+bum}{g}}.}$
This defines an integer, as g divides both m and n. Otherwise, the proof is very similar to that for coprime moduli.
The Chinese remainder theorem can be generalized to any ring, by using coprime ideals (also called comaximal ideals). Two ideals I and J are coprime if there are elements ${\displaystyle i\in I}$ and
${\displaystyle j\in J}$ such that ${\displaystyle i+j=1.}$ This relation plays the role of Bézout's identity in the proofs related to this generalization, which otherwise are very similar. The
generalization may be stated as follows.^[17]^[18]
Let I[1], ..., I[k] be two-sided ideals of a ring ${\displaystyle R}$ and let I be their intersection. If the ideals are pairwise coprime, we have the isomorphism:
{\displaystyle {\begin{aligned}R/I&\to (R/I_{1})\times \cdots \times (R/I_{k})\\x{\bmod {I}}&\mapsto (x{\bmod {I}}_{1},\,\ldots ,\,x{\bmod {I}}_{k}),\end{aligned}}}
between the quotient ring ${\displaystyle R/I}$ and the direct product of the ${\displaystyle R/I_{i},}$ where "${\displaystyle x{\bmod {I}}}$" denotes the image of the element ${\displaystyle x}$ in
the quotient ring defined by the ideal ${\displaystyle I.}$ Moreover, if ${\displaystyle R}$ is commutative, then the ideal intersection of pairwise coprime ideals is equal to their product; that is
${\displaystyle I=I_{1}\cap I_{2}\cap \cdots \cap I_{k}=I_{1}I_{2}\cdots I_{k},}$
if I[i] and I[j] are coprime for all i ≠ j.
Interpretation in terms of idempotents
Let ${\displaystyle I_{1},I_{2},\dots ,I_{k}}$ be pairwise coprime two-sided ideals with ${\displaystyle \bigcap _{i=1}^{k}I_{i}=0,}$ and
${\displaystyle \varphi :R\to (R/I_{1})\times \cdots \times (R/I_{k})}$
be the isomorphism defined above. Let ${\displaystyle f_{i}=(0,\ldots ,1,\ldots ,0)}$ be the element of ${\displaystyle (R/I_{1})\times \cdots \times (R/I_{k})}$ whose components are all 0 except the
ith which is 1, and ${\displaystyle e_{i}=\varphi ^{-1}(f_{i}).}$
The ${\displaystyle e_{i}}$ are central idempotents that are pairwise orthogonal; this means, in particular, that ${\displaystyle e_{i}^{2}=e_{i}}$ and ${\displaystyle e_{i}e_{j}=e_{j}e_{i}=0}$ for
every i and j. Moreover, one has ${\textstyle e_{1}+\cdots +e_{n}=1,}$ and ${\displaystyle I_{i}=R(1-e_{i}).}$
In summary, this generalized Chinese remainder theorem is the equivalence between giving pairwise coprime two-sided ideals with a zero intersection, and giving central and pairwise orthogonal
idempotents that sum to 1.^[19]
Fast Fourier transform
The prime-factor FFT algorithm (also called Good-Thomas algorithm) uses the Chinese remainder theorem for reducing the computation of a fast Fourier transform of size ${\displaystyle n_{1}n_{2}}$ to
the computation of two fast Fourier transforms of smaller sizes ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ (providing that ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are coprime).
Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption.
The Chinese remainder theorem can also be used in secret sharing, which consists of distributing a set of shares among a group of people who, all together (but no one alone), can recover a certain
secret from the given set of shares. Each of the shares is represented in a congruence, and the solution of the system of congruences using the Chinese remainder theorem is the secret to be
recovered. Secret sharing using the Chinese remainder theorem uses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret
from a set of shares with less than a certain cardinality.
Decomposition of surjections of finite abelian groups
Given a surjection ${\displaystyle \mathbb {Z} /n\to \mathbb {Z} /m}$ of finite abelian groups, we can use the Chinese remainder theorem to give a complete description of any such map. First of all,
the theorem gives isomorphisms
{\displaystyle {\begin{aligned}\mathbb {Z} /n&\cong \mathbb {Z} /p_{n_{1}}^{a_{1}}\times \cdots \times \mathbb {Z} /p_{n_{i}}^{a_{i}}\\\mathbb {Z} /m&\cong \mathbb {Z} /p_{m_{1}}^{b_{1}}\times \
cdots \times \mathbb {Z} /p_{m_{j}}^{b_{j}}\end{aligned}}}
where ${\displaystyle \{p_{m_{1}},\ldots ,p_{m_{j}}\}\subseteq \{p_{n_{1}},\ldots ,p_{n_{i}}\}}$. In addition, for any induced map
${\displaystyle \mathbb {Z} /p_{n_{k}}^{a_{k}}\to \mathbb {Z} /p_{m_{l}}^{b_{l}}}$
from the original surjection, we have ${\displaystyle a_{k}\geq b_{l}}$ and ${\displaystyle p_{n_{k}}=p_{m_{l}},}$ since for a pair of primes ${\displaystyle p,q}$, the only non-zero surjections
${\displaystyle \mathbb {Z} /p^{a}\to \mathbb {Z} /q^{b}}$
can be defined if ${\displaystyle p=q}$ and ${\displaystyle a\geq b}$.
These observations are pivotal for constructing the ring of profinite integers, which is given as an inverse limit of all such maps.
Dedekind's theorem
Dedekind's theorem on the linear independence of characters. Let M be a monoid and k an integral domain, viewed as a monoid by considering the multiplication on k. Then any finite family ( f[i] )[i∈I
] of distinct monoid homomorphisms f[i] : M → k is linearly independent. In other words, every family (α[i])[i∈I] of elements α[i] ∈ k satisfying
${\displaystyle \sum _{i\in I}\alpha _{i}f_{i}=0}$
must be equal to the family (0)[i∈I].
Proof. First assume that k is a field, otherwise, replace the integral domain k by its quotient field, and nothing will change. We can linearly extend the monoid homomorphisms f[i] : M → k to k-
algebra homomorphisms F[i] : k[M] → k, where k[M] is the monoid ring of M over k. Then, by linearity, the condition
${\displaystyle \sum _{i\in I}\alpha _{i}f_{i}=0,}$
${\displaystyle \sum _{i\in I}\alpha _{i}F_{i}=0.}$
Next, for i, j ∈ I; i ≠ j the two k-linear maps F[i] : k[M] → k and F[j] : k[M] → k are not proportional to each other. Otherwise f[i] and f[j] would also be proportional, and thus equal since as
monoid homomorphisms they satisfy: f[i](1) = 1 = f[j](1), which contradicts the assumption that they are distinct.
Therefore, the kernels KerF[i] and KerF[j] are distinct. Since k[M]/KerF[i] ≅ F[i](k[M]) = k is a field, Ker F[i] is a maximal ideal of k[M] for every i in I. Because they are distinct and
maximal the ideals KerF[i] and KerF[j] are coprime whenever i ≠ j. The Chinese Remainder Theorem (for general rings) yields an isomorphism:
{\displaystyle {\begin{aligned}\phi :k[M]/K&\to \prod _{i\in I}k[M]/\mathrm {Ker} F_{i}\\\phi (x+K)&=\left(x+\mathrm {Ker} F_{i}\right)_{i\in I}\end{aligned}}}
${\displaystyle K=\prod _{i\in I}\mathrm {Ker} F_{i}=\bigcap _{i\in I}\mathrm {Ker} F_{i}.}$
Consequently, the map
{\displaystyle {\begin{aligned}\Phi :k[M]&\to \prod _{i\in I}k[M]/\mathrm {Ker} F_{i}\\\Phi (x)&=\left(x+\mathrm {Ker} F_{i}\right)_{i\in I}\end{aligned}}}
is surjective. Under the isomorphisms k[M]/KerF[i] → F[i](k[M]) = k, the map Φ corresponds to:
{\displaystyle {\begin{aligned}\psi :k[M]&\to \prod _{i\in I}k\\\psi (x)&=\left[F_{i}(x)\right]_{i\in I}\end{aligned}}}
${\displaystyle \sum _{i\in I}\alpha _{i}F_{i}=0}$
${\displaystyle \sum _{i\in I}\alpha _{i}u_{i}=0}$
for every vector (u[i])[i∈I] in the image of the map ψ. Since ψ is surjective, this means that
${\displaystyle \sum _{i\in I}\alpha _{i}u_{i}=0}$
for every vector
${\displaystyle \left(u_{i}\right)_{i\in I}\in \prod _{i\in I}k.}$
Consequently, (α[i])[i∈I] = (0)[i∈I]. QED. | {"url":"https://www.wikiwand.com/en/articles/Chinese_remainder_theorem","timestamp":"2024-11-09T13:20:28Z","content_type":"text/html","content_length":"1037016","record_id":"<urn:uuid:a666a851-6337-4255-9a39-1261501dee73>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00619.warc.gz"} |
29.6: The Wave Nature of Matter
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Learning Objectives
By the end of this section, you will be able to:
• Describe the Davisson-Germer experiment, and explain how it provides evidence for the wave nature of electrons.
In 1923 a French physics graduate student named Prince Louis-Victor de Broglie (1892–1987) made a radical proposal based on the hope that nature is symmetric. If EM radiation has both particle and
wave properties, then nature would be symmetric if matter also had both particle and wave properties. If what we once thought of as an unequivocal wave (EM radiation) is also a particle, then what we
think of as an unequivocal particle (matter) may also be a wave. De Broglie’s suggestion, made as part of his doctoral thesis, was so radical that it was greeted with some skepticism. A copy of his
thesis was sent to Einstein, who said it was not only probably correct, but that it might be of fundamental importance. With the support of Einstein and a few other prominent physicists, de Broglie
was awarded his doctorate.
De Broglie took both relativity and quantum mechanics into account to develop the proposal that all particles have a wavelength, given by
\[\lambda = \dfrac{h}{p} \, (\text{matter and photons}),\]
where \(h\) Planck’s constant and \(p\) is momentum. This is defined to be the de Broglie wavelength. (Note that we already have this for photons, from the equation \(p = h/\lambda\).) The hallmark
of a wave is interference. If matter is a wave, then it must exhibit constructive and destructive interference. Why isn’t this ordinarily observed? The answer is that in order to see significant
interference effects, a wave must interact with an object about the same size as its wavelength. Since \(h\) is very small, \(\lambda\) is also small, especially for macroscopic objects. A 3-kg
bowling ball moving at 10 m/s, for example, has
\[\begin{align*} \lambda &= h/p \\[4pt] &= (6.63 \times 10^{-34} \, J \cdot s)/[(3 \, kg)(10 \, m/s)] \\[4pt] &= 2 \times 10^{-35} \, m. \end{align*}\]
This means that to see its wave characteristics, the bowling ball would have to interact with something about \(10^{-35} \, m\) in size—far smaller than anything known. When waves interact with
objects much larger than their wavelength, they show negligible interference effects and move in straight lines (such as light rays in geometric optics). To get easily observed interference effects
from particles of matter, the longest wavelength and hence smallest mass possible would be useful. Therefore, this effect was first observed with electrons.
American physicists Clinton J. Davisson and Lester H. Germer in 1925 and, independently, British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in 1926 scattered electrons
from crystals and found diffraction patterns. These patterns are exactly consistent with interference of electrons having the de Broglie wavelength and are somewhat analogous to light interacting
with a diffraction grating (Figure \(\PageIndex{1}\))
Connections: Waves
All microscopic particles, whether massless, like photons, or having mass, like electrons, have wave properties. The relationship between momentum and wavelength is fundamental for all particles.
De Broglie’s proposal of a wave nature for all particles initiated a remarkably productive era in which the foundations for quantum mechanics were laid. In 1926, the Austrian physicist Erwin
Schrödinger (1887–1961) published four papers in which the wave nature of particles was treated explicitly with wave equations. At the same time, many others began important work. Among them was
German physicist Werner Heisenberg (1901–1976) who, among many other contributions to quantum mechanics, formulated a mathematical treatment of the wave nature of matter that used matrices rather
than wave equations. We will deal with some specifics in later sections, but it is worth noting that de Broglie’s work was a watershed for the development of quantum mechanics. De Broglie was awarded
the Nobel Prize in 1929 for his vision, as were Davisson and G. P. Thomson in 1937 for their experimental verification of de Broglie’s hypothesis.
Figure \(\PageIndex{1}\): This diffraction pattern was obtained for electrons diffracted by crystalline silicon. Bright regions are those of constructive interference, while dark regions are those of
destructive interference. (credit: Ndthe, Wikimedia Commons)
Example \(\PageIndex{1}\): Electron Wavelength versus Velocity and Energy
For an electron having a de Broglie wavelength of 0.167 nm (appropriate for interacting with crystal lattice structures that are about this size):
1. Calculate the electron’s velocity, assuming it is nonrelativistic.
2. Calculate the electron’s kinetic energy in eV.
For part (a), since the de Broglie wavelength is given, the electron’s velocity can be obtained from \(\lambda = h/p\) by using the nonrelativistic formula for momentum, \(p = mv\). For part (b),
once \(v\) is obtained (and it has been verified that \(v\) is nonrelativistic), the classical kinetic energy is simply \((1/2)mv^2\).
Solution for (a)
Substituting the nonrelativistic formula for momentum \((p = mv)\) into the de Broglie wavelength gives
\[\begin{align*} \lambda &= \dfrac{h}{p} \\[4pt] &= \dfrac{h}{mv}. \end{align*}\]
Solving for \(v\) gives
\[v =\dfrac{h}{m\lambda}.\nonumber\]
Substituting known values yields
\[\begin{align*} v &= \dfrac{6.63 \times 10^{-34} \, J \cdot s}{(9.11 \times 10^{-31} \, kg)(0.167 \times 10^{-9} \, m)} \\[4pt] &= 4.36 \times 10^6 \, m/s.\end{align*}\]
Solution for (b)
While fast compared with a car, this electron’s speed is not highly relativistic, and so we can comfortably use the classical formula to find the electron’s kinetic energy and convert it to eV as
\[\begin{align*} KE &= \dfrac{1}{2} mv^2 \\[4pt] &= \dfrac{1}{2}(9.11 \times 10^{-31} \, kg)(4.36 \times 10^6 \times 10^6 \, m/s)^2 \\[4pt] &= (86.4 \times 10^{-18} \, J)\left(\dfrac{1 \, eV}{1.601 \
times 10^{-19} \, J}\right) \\[4pt] &= 54.0 \, eV \end{align*} \]
This low energy means that these 0.167-nm electrons could be obtained by accelerating them through a 54.0-V electrostatic potential, an easy task. The results also confirm the assumption that the
electrons are nonrelativistic, since their velocity is just over 1% of the speed of light and the kinetic energy is about 0.01% of the rest energy of an electron (0.511 MeV). If the electrons had
turned out to be relativistic, we would have had to use more involved calculations employing relativistic formulas.
Electron Microscopes
One consequence or use of the wave nature of matter is found in the electron microscope. As we have discussed, there is a limit to the detail observed with any probe having a wavelength. Resolution,
or observable detail, is limited to about one wavelength. Since a potential of only 54 V can produce electrons with sub-nanometer wavelengths, it is easy to get electrons with much smaller
wavelengths than those of visible light (hundreds of nanometers). Electron microscopes can, thus, be constructed to detect much smaller details than optical microscopes (Figure \(\PageIndex{2}\)).
There are basically two types of electron microscopes. The transmission electron microscope (TEM) accelerates electrons that are emitted from a hot filament (the cathode). The beam is broadened and
then passes through the sample. A magnetic lens focuses the beam image onto a fluorescent screen, a photographic plate, or (most probably) a CCD (light sensitive camera), from which it is transferred
to a computer. The TEM is similar to the optical microscope, but it requires a thin sample examined in a vacuum. However it can resolve details as small as 0.1 nm (\(10^{-10} \, m\)), providing
magnifications of 100 million times the size of the original object. The TEM has allowed us to see individual atoms and structure of cell nuclei.
The scanning electron microscope (SEM) provides images by using secondary electrons produced by the primary beam interacting with the surface of the sample (Figure \(\PageIndex{2}\)). The SEM also
uses magnetic lenses to focus the beam onto the sample. However, it moves the beam around electrically to “scan” the sample in the x and y directions. A CCD detector is used to process the data for
each electron position, producing images like the one at the beginning of this chapter. The SEM has the advantage of not requiring a thin sample and of providing a 3-D view. However, its resolution
is about ten times less than a TEM.
Figure \(\PageIndex{2}\): Schematic of a scanning electron microscope (SEM) (a) used to observe small details, such as those seen in this image of a tooth of a Himipristis, a type of shark (b).
(credit: Dallas Krentzel, Flickr)
Electrons were the first particles with mass to be directly confirmed to have the wavelength proposed by de Broglie. Subsequently, protons, helium nuclei, neutrons, and many others have been observed
to exhibit interference when they interact with objects having sizes similar to their de Broglie wavelength. The de Broglie wavelength for massless particles was well established in the 1920s for
photons, and it has since been observed that all massless particles have a de Broglie wavelength \(\lambda = h/p\). The wave nature of all particles is a universal characteristic of nature. We shall
see in following sections that implications of the de Broglie wavelength include the quantization of energy in atoms and molecules, and an alteration of our basic view of nature on the microscopic
scale. The next section, for example, shows that there are limits to the precision with which we may make predictions, regardless of how hard we try. There are even limits to the precision with which
we may measure an object’s location or energy.
The wave nature of matter allows it to exhibit all the characteristics of other, more familiar, waves. Diffraction gratings, for example, produce diffraction patterns for light that depend on grating
spacing and the wavelength of the light. This effect, as with most wave phenomena, is most pronounced when the wave interacts with objects having a size similar to its wavelength. For gratings, this
is the spacing between multiple slits.) When electrons interact with a system having a spacing similar to the electron wavelength, they show the same types of interference patterns as light does for
diffraction gratings, as shown at top left in Figure \(\PageIndex{3}\).
Atoms are spaced at regular intervals in a crystal as parallel planes, as shown in the bottom part of Figure \(\PageIndex{3}\). The spacings between these planes act like the openings in a
diffraction grating. At certain incident angles, the paths of electrons scattering from successive planes differ by one wavelength and, thus, interfere constructively. At other angles, the path
length differences are not an integral wavelength, and there is partial to total destructive interference. This type of scattering from a large crystal with well-defined lattice planes can produce
dramatic interference patterns. It is called Bragg reflection, for the father-and-son team who first explored and analyzed it in some detail. The expanded view also shows the path-length differences
and indicates how these depend on incident angle \(\theta\) in a manner similar to the diffraction patterns for x rays reflecting from a crystal.
Figure \(\PageIndex{3}\): The diffraction pattern at top left is produced by scattering electrons from a crystal and is graphed as a function of incident angle relative to the regular array of atoms
in a crystal, as shown at bottom. Electrons scattering from the second layer of atoms travel farther than those scattered from the top layer. If the path length difference (PLD) is an integral
wavelength, there is constructive interference.
Let us take the spacing between parallel planes of atoms in the crystal to be \(d\). As mentioned, if the path length difference (PLD) for the electrons is a whole number of wavelengths, there will
be constructive interference—that is, \(PLD = n\lambda \, (n = 1, \, 2, \, 3, . . .)\). Because \(AB = BC = d \, \sin \, \theta\), we have constructive interference when
\[n\lambda = 2d \, \sin \, \theta.\]
This relationship is called the Bragg equation and applies not only to electrons but also to x rays.
The wavelength of matter is a submicroscopic characteristic that explains a macroscopic phenomenon such as Bragg reflection. Similarly, the wavelength of light is a submicroscopic characteristic that
explains the macroscopic phenomenon of diffraction patterns.
• Particles of matter also have a wavelength, called the de Broglie wavelength, given by \(\lambda = \frac{h}{p}\), where \(p\) is momentum.
• Matter is found to have the same interference characteristics as any other wave.
de Broglie wavelength
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Type Conversion
Type conversion happens whenever you assign a variable of one type to a variable of another type.
Conversions Without Loss
This is when you assign the value of one unsigned or signed variable to another variable that can hold this value unchanged.
Here's an example of a loss-less conversion:
dim i as word=50
dim x as char=i
X is "smaller" than i — it's just one byte against two bytes of i. In addition, x is a signed variable, so it can only hold positive values of up to 127.
Fortunately, the value of i (50) is within this range, so this conversion won't lead to any loss of data.
Conversions With Loss
Now, consider this example:
unsigned int i=0x5AA5;
unsigned char x=i; //The result is 0xA5
With this, x will assume the value of the least significant byte of i (0xA5). The most significant byte of i will be lost.
Conversions That Cause Reinterpretation
This is when the receiving variable ends up holding the same binary value with the source variable, but interprets it differently:
dim i as byte=254
dim x as char=i 'The result is –2
In this example x ends up with the same binary value, but since x is a signed variable this binary value has a different meaning. 254 becomes –2.
Conversions That Round the Number (Remove Fractions)
Conversions from a float (real) variable into any other numerical type will cut off the fraction, as real is the only type that can hold fractions.
We do give warnings about this. | {"url":"https://docs.tibbo.com/guide_basic_c_variables_simple_type","timestamp":"2024-11-07T10:56:10Z","content_type":"text/html","content_length":"8349","record_id":"<urn:uuid:2c5d723b-d88c-46ba-b511-d28c4eb74111>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00842.warc.gz"} |
Should You Invest All at Once or Dollar Cost Average?
When you have money that you could invest, should you invest it all immediately, or invest some percentage of it each month over the course of many months?
The advice that I genearlly see is to spread the investment out over time to avoid investing at a peak and overpaying for stocks. However, since stocks have historically risen over time, it actually
works out better most of the time if you just invest all at once.
To show this, I put together a few comparisons. All are using monthly S&P 500 values for ~100 years. I then just use both strategies for every historical period, and compare the results.
First, you can simply look at the distributions of values resulting from each strategy. Here is that plot for a 2-year investing period using $10,000:
Investing all at once has a much higher spread in results as you'd expect. It also has a much higher median. Sure, you'll lose more every once in a while as you're more exposed to crashes, but those
periods appear pretty rare. How rare?
Here's the next plot. This shows the difference in outcome for each strategy at every historical point:
Some wild swings there, but it's clear that investing all at once is better in more of the periods.
A final presentation of the results is simply to see what percentage of historical periods were better with each strategy...
• all at once wins 72% of the time
• dollar cost averaging wins 28% of the time
An obvious question you might have is if the current market value influences this. As a check, we can do the same thing except modify the strategy slightly:
• If market is more than 5% off the all-time high, invest lump
• Else, dollar cost average
Comparing that strategy (call it caution) with the all at once one from earlier:
• all at once wins 59% of the time
• caution wins 21% of the time
• they tie 20% of the time
5% off the all-time high is arbitrary, so what about 2%?
• all at once wins 59% of the time
• caution wins 20% of the time
• they tie 21% of the time
This isn't 100% thorough, but it's enough to push me more towards 'invest all at once' than I was in the past.
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SB-CGA is courtesy of Steel Bank Studio Ltd and written by Nikodemus Siivola.
SB-CGA is maintained in Git:
git clone git://github.com/nikodemus/sb-cga.git
will get you a local copy.
is the GitHub project page.
Table of Contents
1 Overview
This documentation – and SB-CGA itself – is still a work in progress.
2 Vectors
Most vector operations are done using special-purpose SSE2 primitives on SBCL/x86-64, and portable Common Lisp elsewhere.
Vector interface consists of two parts. The high-level interface always returns a freshly consed vector, whereas the low-level interface expects a new vector to store its results into.
Low level interface also implicitly trusts that argument counts and types are correct – bad things are liable to happen if those assumptions are violated, whereas the high-level operations should
signal reasonable errors in such situations.
The system is fairly good about combining nested high-level operations into low-level ones with minimal consing, and given proper dynamic-extent declarations can also stack allocate results of vector
operations on SBCL at least – including intermediate results.
User code should primarily use the high-level interface, and dip into the low level code only when absolutely necessary for performance.
2.1 Type and Constructors
2.2 Comparing Vectors
— Function:
vec~ a b &optional epsilon
Return true if vec A and vec b are elementwise within epsilon of each other. epsilon defaults to +default-epsilon+.
2.3 Vector Algebra
— Function:
cross-product a b
Cross product of 3d vector A and 3d vector b, return result as a freshly allocated vec.
— Function:
hadamard-product a b
Compute hadamard product (elementwise product) of vec A and vec b, return result as a freshly allocated vec.
— Function:
vec-lerp a b f
Linear interpolate vec A and vec b using single-float f as the interpolation factor, return result as a freshly allocated vec.
— Function:
vec-max vec &rest vecs
Elementwise maximum of vec and vecs, return result as a freshly allocated vec.
— Function:
vec-min vec &rest vecs
Elementwise minimum of vec and vecs, return result as a freshly allocated vec.
— Function:
adjust-vec point direction distance
Multiply vec direction by single-float distance adding the result to vec point. Return result as a freshly allocated vec.
2.4 Transformations
— Function:
transform-bounds v1 v2 matrix
Transform the axis-aligned bounding box specified by its extreme corners v1 and v2 using matrix. Return new extreme corners (minimum and maximum coordinates) as freshly allocted VECs, as the
primary and secondary value.
— Function:
transform-direction vec matrix
Apply transformation matrix to vec ignoring the translation component, return result as a freshly allocated vec.
— Function:
transform-point vec matrix
Apply transformation matrix to vec, return result as a freshly allocated vec.
2.5 Low-Level Interface
— Function:
%vec- result a b
Substract vec b from vec A, store result in vec result. Return result. Unsafe.
— Function:
%vec* result a f
Multiply vec A with single-float f, store result in vec result. Return result. Unsafe.
— Function:
%vec/ result a f
Divide vec A by single-float f, store result in vec result. Return result. Unsafe.
— Function:
%normalize result a
Normalize vec A, store result into vec result. Return result. Unsafe.
— Function:
%hadamard-product result a b
Compute hadamard product (elementwise product) of vec A and vec b, store result in vec result. Return result. Unsafe.
— Function:
%vec-lerp result a b f
Linear interpolate vec A and vec b using single-float f as the interpolation factor, store result in vec result. Return result. Unsafe.
— Function:
%adjust-vec result point direction distance
Multiply vec direction by single-float distance adding the result to vec point. Store result in result, and return it.
— Function:
%transform-direction result vec matrix
Apply transformation matrix to vec, store result in result. Return result. Unsafe.
— Function:
%transform-point result vec matrix
Apply transformation matrix to vec, store result in result. Return result. Unsafe.
3 Matrices
Transforming vectors using matrices as discussed above uses special purpose SSE2 primitives on SBCL/x86-64, and portable Common Lisp elsewhere.
Matrix algebra has otherwise not been specifically optimized, but should be reasoanably performant for the most part. If you find it substandard for your needs, let us know.
3.1 Type and Constructors
— Type:
4x4 matrix of single floats, represented as a one-dimensional vector stored in column-major order.
— Function:
matrix m11 m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44
Construct matrix with the given elements (arguments are provided in row major order.)
— Function:
rotate-around v radians
Construct a rotation matrix that rotates by radians around vec v. 4th element of v is ignored.
— Function:
rotate vec
Construct a rotation matrix using first three elements of vec as the rotation factors.
— Function:
scale vec
Construct a scaling matrix using first threee elements of vec as the scaling factors.
— Function:
translate vec
Construct a translation matrix using first three elements of vec as the translation factors.
3.2 Comparing and Accessing Matrices
— Function:
matrix~ m1 m2 &optional epsilon
Return true if matrix m1 and matrix m2 are elementwise within epsilon of each other. epsilon defaults to +default-epsilon+
3.3 Matrix Algebra
— Function:
matrix* &rest matrices
Multiply matrices. The result might not be freshly allocated if all, or all but one multiplicant is an identity matrix.
4 Root Solvers
Interface described here is liable to be refactored. See new-roots.lisp in the sources for the possible shape of things to come.
— Function:
cubic-roots-above limit a b c d
Real-valued roots greater than limit for Ax^3+Bx^2+Cx+D. Smallest positive root is returned as primary value, and others in increasing order. limit indicates lack of a real-valued root above
— Function:
cubic-roots a b c d
Real-valued roots for Ax^2+Bx+C. Smallest real root is returned as primary value, and the others as the successive values. NaN indicates lack of a real-valued root.
— Function:
quadratic-roots-above limit a b c
Real-valued roots greater than limit for Ax^2+Bx+C. Smallest positive root is returned as primary value, and the other as secondary. limit indicates lack of a real-valued root above limit.
— Function:
quadratic-roots a b c
Real-valued roots for Ax^2+Bx+C. Smallest real root is returned as primary value, and the other as the secondary. In case of a double root both the primary and secondary values are the same. NaN
indicates lack of a real-valued root.
5 Miscellany | {"url":"http://nikodemus.github.io/sb-cga/","timestamp":"2024-11-10T14:54:26Z","content_type":"text/html","content_length":"28856","record_id":"<urn:uuid:12d00eb5-fb7b-4699-ad91-d32b3d5e6db0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00114.warc.gz"} |
An open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information. Linear control systems
Kryazhimskiy, A.V. & Strelkovskii, N. ORCID: https://orcid.org/0000-0001-6862-1768 (2015). An open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information.
Linear control systems. Proceedings of the Steklov Institute of Mathematics 91 (S1) 113-127. 10.1134/S0081543815090084.
Full text not available from this repository.
The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, the
method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the
system's states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended
linear control system to an extended convex target set. Using a sepration theorem for convex sets, a solvability criterion is derived, which reduces to a solution of a finite-dimensional optimization
problem. An illustrative example is considered.
Item Type: Article
Uncontrolled Keywords: control; incomplete information; linear systems
Research Programs: Advanced Systems Analysis (ASA)
Bibliographic Reference: Proceedings of the Steklov Institute of Mathematics; 291(S1):113-127 [December 2015]
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:52
Last Modified: 27 Aug 2021 17:24
URI: https://pure.iiasa.ac.at/11295
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Representing Graphs in Data Structures
Graph Basics
Contributed by: Ruchi Nayyar
Graphs are fundamental data structures widely used to model relationships between objects.
Whether it’s social networks, transportation systems, or computer networks, graphs are powerful tools for representing connections and dependencies.
In computer science, understanding how to represent graphs efficiently is crucial for solving complex problems effectively.
In this blog post, we will explore the representation of graphs in data structures and explain the various representations of graphs with detailed examples that will help you understand the concept.
What Is A Graph?
A graph is a fundamental construct in computer science. It serves as an abstract representation of interconnected objects.
It comprises two main components: vertices, which represent the objects, and edges, which denote the links between them.
A graph is a pair of sets (V, E), where V represents the vertices and E the edges connecting these vertices.
For instance, consider the following graph:
Graph Basics:
• V = {a, b, c, d, e}
• E = {ab, ac, bd, cd, de}
In this graph, the vertices are {a, b, c, d, e}, and the edges are {ab, ac, bd, cd, de}. Each edge connects two vertices, indicating a relationship between them.
Understanding the concept of a graph is crucial for various applications. It forms the cornerstone of graph representation in data structures, enabling efficient manipulation and analysis of
interconnected data.
Don’t miss out on the opportunity to enroll in the ‘Free Data Structures in C Course‘ and build a strong foundation in programming.
Read “What is Data Structure: Need, Types & Classification” to enhance your comprehension of data structuring fundamentals and their significance!
Major Graph Terminology
Understanding the major terminology associated with graphs is essential for navigating through graph-related concepts effectively:
• Vertex- Also known as a node, a vertex represents an entity within a graph. It can represent various things, such as cities in a transportation network or users in a social media platform.
• Edge- An edge represents a graph’s connection or relationship between two vertices. It can be directional (from one vertex to another) or undirected (connecting two vertices without a specific
• Adjacency- Adjacency refers to the relationship between vertices directly connected by an edge. For example, if vertex A is connected to vertex B by an edge, then A and B are considered adjacent.
• Path- A path in a graph is a sequence of vertices connected by edges. It represents a route or journey from one vertex to another. Paths can be simple (no repeated vertices) or cyclic (repeating
• Directed Graph- Also known as a digraph, a directed graph is a type of graph in which edges have a direction. This means that the relationship between vertices is one-way, indicating a specific
direction for traversal.
Understanding these fundamental concepts lays the foundation for exploring more advanced graph representation techniques, algorithms, and applications.
Don’t miss out on the insights presented in “Application of Graph Theory in 2024” to understand graph theory’s impact on today’s world.
Methods Of Graph Operations
1. Depth First Search Traversal (DFS)
DFS is a graph traversal algorithm that systematically explores all vertices by going as deep as possible along each branch before backtracking.
It starts from an arbitrary vertex, explores as far as possible along each branch before backtracking, and continues until all vertices are visited. DFS utilizes a stack data structure to keep track
of vertices.
2. Breadth First Search Traversal (BFS)
BFS is a graph traversal algorithm that systematically explores all vertices at the current level before moving to the next level.
It starts from an arbitrary vertex, explores all adjacent vertices at the current level, and then moves to the next level. BFS utilizes a queue data structure to keep track of vertices.
3. Detecting Cycles
Detecting cycles in a graph involves identifying if there are any loops or cycles present within the graph structure.
This is crucial in various applications to prevent infinite loops or unintended behavior. Techniques like depth-first or breadth-first search can detect cycles by keeping track of visited vertices
and identifying back edges during traversal.
4. Topological Sorting
Topological sorting is a graph algorithm used to linearly order the vertices of a directed acyclic graph (DAG) based on their dependencies.
It ensures that for every directed edge from vertex u to vertex v, u comes before v in the ordering. Topological sorting is commonly used in scheduling tasks, resolving dependencies in build systems,
and optimizing workflow execution.
5. Minimum Spanning Tree (MST)
A Minimum Spanning Tree (MST) is a subset of edges of a connected, undirected graph that connects all vertices with the minimum possible total edge weight.
Finding a graph’s MST is essential in various applications, such as network design, clustering, and resource allocation. Common algorithms for finding MST include Kruskal’s algorithm and Prim’s
Representation Of Graphs
Graphs can be represented in different ways, each offering unique advantages and trade-offs in terms of space complexity, time complexity, and ease of implementation.
Two different ways of representing a graph in data structure are the Adjacency Matrix and Adjacency List.
1. Adjacency Matrix
An adjacency matrix is a 2D array in which each cell represents the presence or absence of an edge between two vertices. If an edge exists from vertex i to vertex j, the cell (i, j) contains a
non-zero value (often 1); otherwise, it includes 0.
• Straightforward Implementation- The adjacency matrix is intuitive and easy to implement. It directly translates the graph structure into a matrix.
• Efficient for Dense Graphs- In graphs where the number of edges is close to the maximum possible (dense graphs), adjacency matrices are efficient as they use less memory than lists.
• Constant-Time Access- This method determines if an edge between two vertices is constant-time (O(1)), making it efficient for certain operations like checking for connectivity.
• Memory Consumption- Adjacency matrices consume more memory, especially for sparse graphs, where many entries in the matrix are zero.
• Inefficiency for Sparse Graphs- Adjacency matrices are inefficient in graphs with few edges (sparse graphs) because they waste memory storing zero values.
• Inflexible for Dynamic Graphs- Modifying the graph structure, such as adding or removing edges, can be inefficient as it requires resizing the matrix.
Consider the following graph with vertices A, B, C, and D:
A B C D
A 0 1 1 0
B 1 0 0 1
C 1 0 0 1
D 0 1 1 0
In this adjacency matrix, a value of 1 indicates the presence of an edge between the corresponding vertices, while 0 indicates no edge. Lets understand better with code
Code Example
# Implementation of Adjacency Matrix
class Graph:
def __init__(self, num_vertices):
self.num_vertices = num_vertices
self.adj_matrix = [[0] * num_vertices for _ in range(num_vertices)]
def add_edge(self, from_vertex, to_vertex):
self.adj_matrix[from_vertex][to_vertex] = 1
self.adj_matrix[to_vertex][from_vertex] = 1
def display(self):
for row in self.adj_matrix:
# Create a graph with 4 vertices
graph = Graph(4)
# Add edges
graph.add_edge(0, 1)
graph.add_edge(0, 2)
graph.add_edge(1, 3)
graph.add_edge(2, 3)
# Display the adjacency matrix
print("Adjacency Matrix:")
Output : Adjacency Matrix
[0, 1, 1, 0]
[1, 0, 0, 1]
[1, 0, 0, 1]
[0, 1, 1, 0]
Adjacency Matrix Code Explanation
The code defines a graph class that represents a graph using an adjacency matrix. It initializes with the number of vertices and creates a 2D array to store edges.
The add_edge method updates the matrix to indicate the presence of an edge between vertices. The display method prints the adjacency matrix.
The example creates a graph with 4 vertices, adds edges, and displays the adjacency matrix. The output shows the matrix representing the graph structure.
Apart from python, If you’re seeking insights into how data structures operate in C, delve into “Data Structures using C | What are the Data Structure in C and How it works?“
2. Adjacency List
An adjacency list is a collection of linked lists or arrays, each representing a vertex in the graph. Each element in the list/array stores the adjacent vertices of the corresponding vertex.
• Memory Efficiency for Sparse Graphs- Adjacency lists are memory-efficient for sparse graphs since they only store information about existing edges.
• Efficient for Dynamic Graphs- Adjacency lists are suitable for dynamic graphs where edges are frequently added or removed, as they can be easily modified without resizing.
• Efficient Traversal- Traversal of adjacent vertices is efficient as each vertex maintains a list of its adjacent vertices.
• Complex Implementation- Implementing adjacency lists requires managing linked lists or arrays for each vertex, which can be more complicated than an adjacency matrix.
• Potential Additional Memory Usage- Depending on the implementation, adjacency lists may require additional memory for storing pointers or references to adjacent vertices.
Consider the same graph with vertices A, B, C, and D:
A -> [B, C]
B -> [A, D]
C -> [A, D]
D -> [B, C]
Each vertex is associated with a list/array containing its adjacent vertices in this adjacency list representation. For example, vertex A is adjacent to vertices B and C, as the list indicates [B,
C]. Lets understand this better with the code
Code Example
# Implementation of Adjacency List
class Graph:
def __init__(self):
self.adj_list = {}
def add_edge(self, from_vertex, to_vertex):
if from_vertex not in self.adj_list:
self.adj_list[from_vertex] = []
if to_vertex not in self.adj_list:
self.adj_list[to_vertex] = []
def display(self):
for vertex, neighbors in self.adj_list.items():
print(f"{vertex} -> {neighbors}")
# Create a graph
graph = Graph()
# Add edges
graph.add_edge('A', 'B')
graph.add_edge('A', 'C')
graph.add_edge('B', 'D')
graph.add_edge('C', 'D')
# Display the adjacency list
print("Adjacency List:")
Output: Adjacency List
A -> ['B', 'C']
B -> ['A', 'D']
C -> ['A', 'D']
D -> ['B', 'C']
Adjacency List Code Explanation
The code defines a graph class that represents a graph using an adjacency list. It initializes an empty dictionary to store vertices and their adjacent vertices.
The add_edge method adds vertices and their neighbors to the adjacency list. The display method prints the adjacency list.
The example creates a graph, adds edges, and displays the adjacency list. The output shows each vertex and its adjacent vertices.
Understanding the various representations of graphs in data structures is crucial for efficiently storing and traversing graphs, enabling the implementation of various algorithms and analyses.
You can also Checkout our blog on ‘Data Structures and Algorithms in Java‘ to become a Java expert in Data Structures and Algorithms
Types of Graphs
Graphs can be classified into various types based on their characteristics and properties. Two common types are Directed Graphs (Digraphs) and Undirected Graphs.
1. Directed Graph (Digraph)
In a directed graph, edges have a direction associated with them. This means that the relationship between vertices is one-way, indicating a specific direction for traversal.
• Directional Edges- Each edge in the graph has a direction from one vertex to another, indicating a specific relationship between them.
• Asymmetric Relationships- The relationship between vertices in a directed graph is asymmetric, meaning that the presence of an edge from vertex A to vertex B does not necessarily imply the
existence of an edge from B to A.
• Directed Connectivity- Directed graphs can represent various relationships with specific directional dependencies.
• Network Routing- Directed graphs are used in network routing algorithms to model the flow of data or resources through a network with specific directional paths.
• Dependency Analysis- In software engineering, directed graphs are utilized for dependency analysis, where dependencies between components or modules are represented with directional edges.
• Social Networks- Directed graphs can model social networks, where edges represent inherently directional relationships such as following or subscribing.
Code Example
import networkx as nx
import matplotlib.pyplot as plt
# Create a directed graph
G = nx.DiGraph()
# Add edges to the graph
G.add_edge('A', 'B')
G.add_edge('B', 'C')
G.add_edge('C', 'A')
# Draw the graph
nx.draw(G, with_labels=True, arrows=True)
Visualization of the directed graph with vertices A, B, and C connected by directed edges indicating the direction of relationships.
The code creates a directed graph with three nodes (A, B, and C) and three edges forming a cycle:
An edge from A to B.
An edge from B to C.
An edge from C to A.
The graph is then visualized with arrows indicating the direction of the edges, and node labels are displayed. The resulting graph is a simple directed cycle.
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2. Undirected Graph
In an undirected graph, edges do not have a direction associated with them. This means that the relationship between vertices is bidirectional, allowing traversal in both directions.
• Undirected Edges- Edges in the graph do not have a specific direction associated with them, allowing traversal in both directions between connected vertices.
• Symmetric Relationships- The relationship between vertices in an undirected graph is symmetric, meaning that if there is an edge between vertices A and B, there is also an edge between vertices B
and A.
• Bidirectional Connectivity- Undirected graphs represent bidirectional relationships where the order of vertices does not matter.
• Social Networks- Undirected graphs are commonly used to model social networks, where friendships or connections between individuals are inherently bidirectional.
• Transportation Networks- Undirected graphs can represent transportation networks, where edges represent connections between locations or nodes without specifying a direction of travel.
• Wireless Sensor Networks- Undirected graphs are utilized in wireless sensor networks to model bidirectional communication links between sensor nodes.
Code Example
import networkx as nx
import matplotlib.pyplot as plt
# Create an undirected graph
G = nx.Graph()
# Add edges to the graph
G.add_edge('A', 'B')
G.add_edge('B', 'C')
G.add_edge('C', 'A')
# Draw the graph
nx.draw(G, with_labels=True)
The code creates an undirected graph with three nodes (A, B, and C) and three edges forming a cycle:
An edge between A and B.
An edge between B and C.
An edge between C and A.
The graph is visualized with node labels displayed. Since it is an undirected graph, the edges do not have arrows indicating direction, representing a simple cycle connecting the three nodes.
Don’t miss out on “Data Structures For Beginners” – the perfect starting point for understanding the core concepts of data organization.
Understanding the various methods of representing graphs in data structures is essential for effective problem-solving in fields like data science and business analytics.
Just as graphs are powerful tools for modelling relationships between objects, mastering their representation allows for efficient manipulation and analysis of interconnected data.
With the Great Learning Post Graduate Program in Data Science and Business Analytics, offering dual certification from UT Austin & Great Lakes, learners gain the necessary skills and knowledge to
tackle complex graph-related challenges.
Through any time, anywhere learning and weekly online mentorship by experts, coupled with dedicated program support, participants can confidently navigate the intricate world of graph representation,
paving the way for success in their professional endeavors.
How do I decide between using an adjacency matrix or an adjacency list for graph representation?
The choice between an adjacency matrix and an adjacency list depends on factors such as the density of the graph, memory constraints, and the operations to be performed on it.
Sparse graphs are typically better represented using adjacency lists due to their memory efficiency, while dense graphs may benefit from adjacency matrices for faster edge lookup.
How can I optimize graph representations for large-scale datasets?
Optimizing graph representations for large-scale datasets involves techniques such as parallel processing, distributed computing, and graph partitioning. By leveraging scalable data structures and
algorithms, learners can handle massive graphs efficiently, enabling analysis of complex networks and systems.
How do I handle dynamic graphs where edges are frequently added or removed?
For dynamic graphs, consider using data structures like dynamic arrays for adjacency lists or sparse matrix representations that can efficiently accommodate changes in the graph structure without
significant overhead.
Are there specialized libraries or tools available for graph representation and analysis?
Yes, several specialized libraries and tools exist for graph representation and analysis, such as NetworkX in Python, igraph in R, and Neo4j for graph databases. These tools provide various
functionalities for creating, analyzing, and visualizing graphs, making them invaluable resources for graph-related tasks. | {"url":"https://www.mygreatlearning.com/blog/representing-graphs-in-data-structures/","timestamp":"2024-11-13T14:34:08Z","content_type":"text/html","content_length":"408617","record_id":"<urn:uuid:31379b1b-5b34-4e78-9d39-ae14178907e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00194.warc.gz"} |
Measuring And Naming Angles Worksheet – Use free printable Measure Angle Worksheets to practice measuring angles. These worksheets will teach you how to use a ruler and help you avoid making
mistakes. These worksheets also provide tips for making measurements easier. You can, for example, use a protractor in order to measure angles that look … Read more
Find Reference Angle Worksheet – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use. These worksheets will help to
understand the various concepts and increase your knowledge of angles. Students will be able to identify unknown angles … Read more
Triangles Missing Angles Worksheet – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral triangles. You
can use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle Worksheet teaches students … Read more
Kuta Software Infinite Geometry Angles In A Triangle Worksheet – In this article, we’ll talk about Angle Triangle Worksheets and the Angle Bisector Theorem. We’ll also discuss Equilateral triangles
and Isosceles. If you’re unsure of which worksheet you need, you can always use the search bar to find the exact worksheet you’re looking for. Angle … Read more
Angle Pairs Practice Worksheet Answers – Angle worksheets can be helpful when teaching geometry, especially for children. These worksheets include 10 types of questions about angles. These include
naming the vertex and the arms of an angle, using a protractor to observe a figure, and identifying supplementary and complementary pairs of angles. Angle worksheets are … Read more
Lines Ray And Angles Grade 4 Worksheets – You’ve found the right place if you are looking for Line Angle Worksheets. These printables will help you to improve your math skills as well as teach the
basics of angles and lines. They also help you learn to read and use a protractor. In addition, these … Read more
Angle Measures And Angle Bisectors Worksheet Kuta – Use free printable Measure Angle Worksheets to practice measuring angles. These worksheets will help you learn how to use a protractor and avoid
angles that are not exactly right. They also include tips to make measurements easier. For example, you can use a protractor to measure an … Read more
Angle Measures Of Triangles Worksheet – Use free printable Measure Angle Worksheets to practice measuring angles. These worksheets will help you learn how to use a protractor and avoid angles that
are not exactly right. These worksheets also provide tips for making measurements easier. For example, you can use a protractor to measure an angle … Read more
Inscribed Angle Worksheet Find The Value Of Each Variable – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help
to understand the various concepts and increase your knowledge of angles. Students will be able to identify unknown angles using the vertex, arms … Read more | {"url":"https://www.angleworksheets.com/","timestamp":"2024-11-10T22:56:08Z","content_type":"text/html","content_length":"98021","record_id":"<urn:uuid:690fca19-6175-4d92-b8ba-fa175383370c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00851.warc.gz"} |
PPT - Isovector and spin-isospin nuclear matter symmetry energy (Skyrme forces): PowerPoint Presentation - ID:3302387
1. Isovector and spin-isospin nuclear matter symmetry energy (Skyrme forces): dependences on density and temperature Fábio L. Braghin Int. Centr Cond Matter Physics-University of Brasilia, Brasilia,
Brasil (and IF-USP, São Paulo, Brasil) Financial Support: IBEM/Ministry Scie.-Tech-Brazil FAP-DF (Brasilia) - Brazil FAPESP (São Paulo) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
2. Drop liquid (macroscopic) models NUCLEAR FORCE dependences on isospin in A(s,t) => (spin, isospin) = (s,t) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
3. Spin-Isospin symmetry energy Essentially restoring force of Gamow Teller (GT) resonances Probe for pionic degree of freedom, eg. in relativistic models Lagrangian (pseudo-scalar coupling):
Related to an eventual instability of pion condensation (early 70, 80’s), higher r Eventually to DCC F.L.Braghin, IWND SE-09, Shanghai, August, 2009
4. Relating the symmetry energy coefficient GENERAL AND response function (static) IDEA For a small external perturbation in binding energy: Standard form: F.L.Braghin, IWND SE-09, Shanghai, August,
5. Dynamical Polarizabilities with Skyrme forces Linear response on time dependent Hartree Fock for Skyrme forces such as: Behavior of small density fluctuations induced by small amplitude external
perturbation that separates densities of neutrons and protons (s,t= 0,1) neutrons/protons spin up/down (s,t=1,1) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
6. Writing only the static limit for any densities of protons and neutrons Same standard form of polarizability v = b, c, d Asymmetry parameters Temperature dependence of A Form factors for
Densities of states Particle densities Densities of momentA F.L.Braghin, IWND SE-09, Shanghai, August, 2009 NPA(2000); PRC(2005)
7. Each of the densities (form factors)for rp = rn F.L.Braghin, IWND SE-09, Shanghai, August, 2009
8. Density dependence of the spin-isospin symmetry energy: FLB,IJMPE(2003) Very different predictions N Kaiser, CPT, (2006) (strongly dependent on each d.o.f./effect Mesons exchanges / D
F.L.Braghin, IWND SE-09, Shanghai, August, 2009
9. Temperature dependence of n –p symmetry energy: A0,1 F.L.Braghin, IWND SE-09, Shanghai, August, 2009
10. 0.1 r0 (NN09) Exp (diff r and T*) Shetty et al 2006; Le Fevre et al (2005) Trautmann et al (2006) (INDRA; ALADIN) Kowalski et al(2006) .05r SLyb,b=0.5 (x) SLyb,b=0 ___ SGII,b=0.5 (+) SGII,b=0
..... Samaddar et al: down triangles (2007) Lie-Wen Chen et al: up triangles (2007) Moustakidis: dashed (2008) Different momentum Dependent interactions, Clusters,.. F.L.Braghin, IWND SE-09,
Shanghai, August, 2009
11. Full solid symbols , x: 1.33 r0 , b = 0 Empty symbols , +: 1.33 r0 , b = 0.25 • -Onsi et al (a) SkSC4 • - Lyon group SLy4 • - Onsi et al (b)SkSC6 • Sagawa/Giai • SGII F.L.Braghin, IWND SE-09,
Shanghai, August, 2009 FLB, PRC(2009)
12. 0.1 r0 Spin-isospin: ++ SGII (b=.5) ... SGII (b=0) xx SLy4 (b=.5) ___ SLy4 (b=0) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
13. 1.33 r0 Smoother variation at higher densities SGII SkSC4 SLy4 F.L.Braghin, IWND SE-09, Shanghai, August, 2009
14. Momentum dependence of the polarizability Limit of equal densities protons-neutrons Depending on the Skyrme force F.L.Braghin, IWND SE-09, Shanghai, August, 2009
15. Momentum dependence 0.1 r0 2 r0 Instability for n-matter F.L.Braghin, IWND SE-09, Shanghai, August, 2009
16. Spin-isospin: Momentum dependence at r0 T=0,1,2,4 Momentum dependence does not favor spin-isospin instabilities for several Skyrme forces (but yes for neutron-matter instabilities) (not yet
extensive) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
17. Considering always: In static (or ~ dynamical) framework Eventually assuming: A = A (b,r, T,q,..) F.L.Braghin, IWND SE-09, Shanghai, August, 2009
18. Simultaneous dependence of symmetry energy on different variables Standard expression Variation of P Or More generally: Looking for data to plug with e.o.s. For finite nuclei； Densities-> N,Z/Vn
Vp F.L.Braghin, IWND SE-09, Shanghai, August, 2009 PRC(2005)
19. Summary • Calculating symmetry energies from nuclear polarizabilities • Dependences on T, n-p, momentum exchanged depend strongly on density • Ｈｏｗ ｃan spin-isospin sym.energ interaction （
ｔｏｇｅｔｈｅr with(0,1)) ｂｅ ｐｒｏｂｅｄ by pions in ｈｉｃ? • Differential equation available：ｇｉｖｅ ｍｅ eos ａｎｄ ｋｎｏｗｗｗｈｉｃｈ symmetryｅｎｅｒｇｙ isｃｏｍｐａｔｉｂｌ
ｅ．． • - So far no surface or finite size effect taken into account. Thank you. 謝謝 F.L.Braghin, IWND SE-09, Shanghai, August, 2009
20. For The same Variation of A with a (or b) Example of ~ opposite behavior: De-Samaddar-nucl-th/0708.2183 F.L.Braghin, IWND SE-09, Shanghai, August, 2009
21. For: Heiselberg, Hjorth Jensen and Differential equation becomes: Stable Density As function Of n-p For the eos F.L.Braghin, IWND SE-09, Shanghai, August, 2009
22. Neutron matter: No assumption or consideration on dynamical and nuclear microscopical effects (forces, processes) - These will provide more precise expressions F.L.Braghin, IWND SE-09, Shanghai,
August, 2009
23. Based onF.L.B., Phys. Rev. C (2009); Phys. Rev. C 71, 064303 (2005); Int. Journ. Mod. Phys. E12, 755 (2003); Nucl. Phys. A 665, 13 (2000); Nucl. Phys. A 696, 413 (2001)+ A 709, 487 (2002).
Proceedings Brazilian Meetings on Nuc Phys 2007, 2008. F.L.Braghin, IWND SE-09, Shanghai, August, 2009
24. Mostly quadratic terms in d r Scalar channel: incompressibility K and A00 have nearly the same behavior with b, a F.L.Braghin, IWND SE-09, Shanghai, August, 2009 | {"url":"https://www.slideserve.com/jerome/isovector-and-spin-isospin-nuclear-matter-symmetry-energy-skyrme-forces-powerpoint-ppt-presentation","timestamp":"2024-11-02T22:19:09Z","content_type":"text/html","content_length":"93574","record_id":"<urn:uuid:66c25930-1a6c-489b-aced-04be710a5e4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00829.warc.gz"} |
Coupled heat conduction and thermal stress analyses in particulate composites
This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In
the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal
expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite
element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles
randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with
different particle sizes and arrangements are generated such that these models represent realistic volume elements "cut out" from a particulate composite. An extended definition of a RVE for
heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar
loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the
variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous
composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables.
Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.
• Finite element
• Heat conduction
• Micromechanical model
• Particulate composites
• Thermal stresses
Dive into the research topics of 'Coupled heat conduction and thermal stress analyses in particulate composites'. Together they form a unique fingerprint. | {"url":"https://khazna.ku.ac.ae/en/publications/coupled-heat-conduction-and-thermal-stress-analyses-in-particulat","timestamp":"2024-11-01T19:42:35Z","content_type":"text/html","content_length":"59870","record_id":"<urn:uuid:71d9871b-55bb-44f5-946c-53402210b572>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00582.warc.gz"} |
double creates a double-precision vector of the specified length. The elements of the vector are all equal to 0. It is identical to numeric.
as.double is a generic function. It is identical to as.numeric. Methods should return an object of base type "double".
is.double is a test of double type.
R has no single precision data type. All real numbers are stored in double precision format. The functions as.single and single are identical to as.double and double except they set the attribute
Csingle that is used in the .C and .Fortran interface, and they are intended only to be used in that context. | {"url":"https://www.rdocumentation.org/packages/base/versions/3.0.3/topics/double","timestamp":"2024-11-09T07:15:43Z","content_type":"text/html","content_length":"74701","record_id":"<urn:uuid:f970eeb9-7e18-432c-a977-c61a1c14551e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00401.warc.gz"} |
When to arrive at a queue with earliness, tardiness and waiting costs
We consider a queueing facility where customers decide when to arrive. All customers have the same desired arrival time (w.l.o.g. time zero). There is one server, and the service times are
independent and exponentially distributed. The total number of customers that demand service is random, and follows the Poisson distribution. Each customer wishes to minimize the sum of three costs:
earliness, tardiness and waiting. We assume that all three costs are linear with time and are defined as follows. Earliness is the time between arrival and time zero, if there is any. Tardiness is
simply the time of entering service, if it is after time zero. Waiting time is the time from arrival until entering service. We focus on customers’ rational behavior, assuming that each customer
wants to minimize his total cost, and in particular, we seek a symmetric Nash equilibrium strategy. We show that such a strategy is mixed, unless trivialities occur. We construct a set of equations
that its solution provides the symmetric Nash equilibrium. The solution is a continuous distribution on the real line. We also compare the socially optimal solution (that is, the one that minimizes
total cost across all customers) to the overall cost resulting from the Nash equilibrium.
Dive into the research topics of 'When to arrive at a queue with earliness, tardiness and waiting costs'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/when-to-arrive-at-a-queue-with-earliness-tardiness-and-waiting-co","timestamp":"2024-11-11T14:19:46Z","content_type":"text/html","content_length":"54430","record_id":"<urn:uuid:23b8b18e-7999-4f16-b45a-0847ab484e8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00243.warc.gz"} |
Apply this function over 1:J to calculate each portion of the
performIntegralUpper1 {DBpower} R Documentation
Apply this function over 1:J to calculate each portion of the integral we need for the upper bound.
Apply this function over 1:J to calculate each portion of the integral we need for the upper bound.
performIntegralUpper1(j, muVec, sigMat, lBounds1, uBounds1, lBounds2, uBounds2)
j Apply over this integer, the element that will be the largest in magnitude.
muVec Mean vector of test statistics under the alternative (assuming it's MVN).
sigMat Covariance matrix of test statistics under the alternative (assuming it's MVN).
lBounds1 A 3J-2 vector of lower bounds for the first integral (see paper), bounds will be longer than for performIntegralLower1.
uBounds1 A 3J-2 vector of upper bounds for the second integral (see paper).
lBounds2 A 3J-2 vector of lower bounds for the first integral (see paper).
uBounds2 A 3J-2 vector of upper bounds for the second integral (see paper).
The value of the integration.
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V. Savatorova — Doctor of Physico-Mathematical Sciences, USA
Viktoria Savatorova
Senior Research Fellow
Department of Applied Mathematics
National Research Nuclear University «MEPhI»
Office address: Kashirskoye shosse, 31, 115409 Moscow, Russia
Phone: +7(903)233-68-75;
Visiting Faculty
Department of Mathematical Sciences
University of Nevada Las Vegas
Office address: 4505 S. Maryland Parkway, CDC, Building 9, Room 906, Mail Stop 4020
Las Vegas, NV 89154-4001
E-mail: vsavatorova@gmail.com
Habilitation (Doctor of Science in Physics and Mathematics),
Moscow State Mining University, Russia (2011)
Dissertation: «Mathematical modeling of the processes of heat conductivity and filtration in non-uniform media, having periodical structure».
Ph.D. Physics and Mathematics
Moscow Engineering Physics Institute, Moscow, Russia (1996)
Dissertation: «Mathematical modeling of the local phase
transitions in media with structural inhomogeneities».
Advisor: Dr. Kudryashov N.A.
M.S. Applied Physics and Mathematics with distinction
Moscow Institute of Physics and Technology, Moscow, Russia (1991)
Advisor: Dr. Rutkevich I.M.
B.S. Applied Physics and Mathematics with distinction
Moscow Institute of Physics and Technology, Moscow, Russia (1989)
2013 Senior Research Fellow,
Department of Applied Mathematics,
National Research Nuclear University «MEPhI», Moscow, Russia
2014 Visiting Faculty,
Department of Mathematical Sciences,
University of Nevada Las Vegas, USA
2013- 2014 Visiting Research Fellow,
Institute of Scientific Computations
Department of Mathematics,
Texas A&M University, College Station, USA
2008-2013 Associate Professor,
Department of Physics,
National Research Nuclear University «MEPhI», Moscow, Russia
2005-2008 Associate Professor,
Department of Physics and Technology,
Moscow State Mining University, Moscow, Russia
1999-2005 Assistant Professor,
Department of Physics and Technology,
Moscow State Mining University, Moscow, Russia
1996-1999 Researcher & Lecturer,
Department of Physics and Technology,
Moscow State Mining University, Moscow, Russia
1993-1996 Research & Teaching assistant,
Department of Physics and Technology,
Moscow State Mining University, Moscow, Russia
1991-1993 Research assistant,
High Temperature Institute Russian Academy of Sciences, Moscow, Russia
Applied physics and mathematics, mathematical modeling of mechanical and thermodynamic processes in the heterogeneous media, multiscale modeling, materials with heterogeneous structure, geological
materials, composite materials, porous media, filtration, homogenization techniques
2013 Senior Research Fellow,
Department of Applied Mathematics,
National Research Nuclear University «MEPhI», Moscow, Russia
Mathematical and numerical methods of modeling fluid flow through porous media (oil and shale gas filtration, multiscale modeling)
2014 Visiting Faculty,
Department of Mathematical Sciences,
University of Nevada Las Vegas, USA
Mathematical and numerical methods of modeling fluid flow through porous media
2013-2014 Visiting Research Fellow,
Institute of Scientific Computations
Department of Mathematics,
Texas A&M University, College Station, USA
Multi-scale modeling of gas transport through porous media (shale gas modeling)
2008- 2013 Associate Professor
Department of Physics,
National Research Nuclear University «MEPhI», Moscow, Russia
Dissertation: «Mathematical modeling of the processes of heat conductivity and filtration in non-uniform media, having periodical structure».
Multi-scale modeling of geological, composite materials and porous structures. Developing of the models for the investigation of thermo-mechanical processes and filtration in media with structural
1993-2008 Department of Physics and Technology
Moscow State Mining University, Moscow, Russia
Dissertation: «Mathematical modeling of the local phase transitions in media with structural inhomogeneities».
Applied the asymptotical averaging technique to study the problem of heat transfer in media with structural heterogeneities. Developed a method for modeling of the local phase transitions. Found an
exact solution for semi-infinite media, modeling geological materials.
1991-1993 Research assistant,
High Temperature Institute Russian Academy of Sciences,
Moscow, Russia
Modeling of the processes of heat- mass exchange in structural materials.
Development of the models of state equations.
1989-1991 Graduate Research assistant,
Moscow Institute of Physics and Technology, Moscow, Russia
Using Galerkin method to solve partial differential equations.
Over 20 years of teaching experience; about 15 methodological publications
2014 - present Department of Mathematical Sciences,
University of Nevada Las Vegas, USA
Visiting Faculty
2008-2013 Department of Physics,
National Research Nuclear University «MEPhI», Moscow, Russia
Associate Professor
2008 - present Graduate students» superviser
1993-2008 Department of Physics and Technology
Moscow State Mining University, Moscow, Russia
Over 15 years of teaching experience: received outstanding reviews from students.
1999-2008 Graduate students’ superviser
Undergraduate Courses Taught (up to 160 students):
- Mechanics
- Statistical Physics and Thermodynamics
- Electricity and Magnetism
- Oscillations and Waves
- Quantum Mechanics (Introduction)
- College Algebra
- Precalculus
- Calculus
- Partial Differential Equations
Graduate Courses Taught (up to 50 students):
Heterogeneous Media: Mechanics and Thermodynamics
- Moscow State Mining University: Physics of heterogeneous media, lectures for Graduate students
• Fulbright Program, Research Grant: Development of a Modern Course «Physics of Heterogeneous Media» Based on the Experience of Scientific Research and Teaching in Universities of the United States
and Russia. Texas A&M University-College Station, Department of Mechanical Engineering, College Station, TX, c/o Dr. K.R. Rajagopal, January 2010 - July 2010
• Program «Mikhail Lomonosov», Ministry of Education and Science of the Russian Federation and the German Academic Exchange Service, Research Grant «Multiscale modeling of highly heterogeneous
industrial materials». Fraunhofer ITWM, Kaiserslautern, Germany, c/o Dr. Oleg Iliev, September 2011 - December 2011
Vlasov A. N., Savatorova V. L., Talonov A. V. The modeling of physical processes in structure heterogeneous media. //Peoples» Friendship University of Russia, Moscow, 2009, 258 pp.
1. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Multiscale modeling of gas flow through organic-rich shale // Composites: Mechanics, Computations, Applications. An
International Journal. – 2016, 7(1), p. 1-26.
2. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Brinkman»s filtration of fluid in rigid porous media: multiscale analysis and investigation of effective permeability //
Composites: Mechanics, Computations, Applications. An International Journal. − 2015, 6(3), p. 239-264.
3. Akkutlu I. Y., Efendiev Y., Savatorova V. Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix//Transport in Porous Media. v. 106(2). − 2015. DOI 10.1007/s11242-014-0435-z
4. Brown D. L., Efendiev Y., Li G., Savatorova V. Homogenization of high contrast Brinkman flows// SIAM Multiscale Model. Simul. − 2015, 13(2), p. 472–490.
5. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Averaging of the nonstationary equations of viscous substance filtration through a rigid porous medium// Composites:
Mechanics, Computations, Applications. An International Journal. − 2014, 5(1), p. 35-61.
6. Brown D., Efendiev Y, Li G., Popov P., Savatorova V. Multiscale Modeling of High Contrast Brinkman Equations with Applications to Deformable Porous Media // Poromechanics V, p. 1991-1996, ASCE
7. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Averaging of time-dependent equations of viscous fluid filtration in non-deformable porous medium // Composite mechanics and
design. − 2013. − v. 19. − № 4. − p. 535-555.
8. Savatorova V. L., Vlasov A. N. Modeling of viscous fluid filtration in porous media with cylindrical symmetry//Composites: Mechanics, Computations, Applications. An International Journal. − 2013,
4(1), p. 75-96.
9. Kudryashov N. A., Kochanov M. B., Savatorova V. L. Approximate self-similar solutions for the boundary value problem arising in modeling of gas flow in porous media with pressure dependent
permeability // Applied Mathematical Modelling. − 2013, v. 37, issue 7, p. 5563-5573.
10. Savatorova V. L., Talonov A. V., Vlasov A. N. Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities // ZAMM Z. Angew. Math. Mech. − 2013,
v. 93, № 8, p. 575-596(2013) DOI 10.1002/zamm.201200032
11. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Transient flow of compressible barotropic fluid in rigid porous medium // Composite mechanics and design (special issue:
Proceedings of the IV All-Russian symposium «Composite mechanics and design», Moscow, Institute of Applied Mechanics of Russian Academy of Sciences, December 4-6). − 2012. − v. 1, p. 214-230.
12. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Multiscale modeling of thermoelastic properties of composites with periodic structure of heterogeneities // Materials
Physics and Mechanics. − 2012, v. 13, p. 130-142.
13. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Homogenization of nonlinear heat conductivity equation for modeling of conductive heat transfer in layered materials //
Composite mechanics and design (special issue: Proceedings of All-Russian Conference «Mechanics of nano- structured materials and systems», Moscow, December 13-15). − 2011. − v. 2. − p. 77-88.
14. Savatorova V. L., Rajagopal K. R. Homogenization of a generalization of Brinkman»s equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid // ZAMM Z.
Angew. Math. Mech. − 2011, v. 91, № 8, p. 630-648.
15. Savatorova V. L., Talonov A. V., Vlasov A. N. Averaging of Brinkman»s filtration equations in layered porous media.// Composite mechanics and design. − 2010. − v.16. − № 4. − p. 483-502.
16. Savatorova V. L., Beliy A. A. Mathematical modeling of heat transfer in heterogeneous media with periodic structure // Mining information-analytic newsletter. Preprint. − 2010. − № 9. – 98 p.
17. Savatorova V. L. Mathematical modeling of the filtration of fluid through porous medium with periodic structure // GIAB. Preprint. − 2010.− № 9. – 61 p.
18. Savatorova V. L. Homogenization of the filtration equations in incompressible medium with periodic pores» structure // Herald of the Pacific state university. −2010. − № 4 (19). − p. 53-60.
19. Savatorova V. L. Solution of coupled problems of heat conductivity and filtration in layered heterogeneous medium // Scientific-technical journal of the Saint-Petersburg state polytechnic
university. − 2010. − № 4 (109). – p.76-86.
20. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Mathematical simulation of heat transfer in a periodic medium with cylindrical inclusions separated from the matrix by a
thin contact layer //Herald of the Nizhny Novgorod university named after N. I. Lobachevsky. − 2010. − № 6. − p. 178-185.
21. Savatorova V. L., Talonov A. V., Shirochin D. L. Propagation of melting front in structurally heterogeneous media having phase transitions of inhomogeneities // ZAMM Z. Angew. Math. Mech. − 2010,
v. 90, № 4, p. 309-322.
22. Vlasov A. N., Savatorova V. L., Talonov A. V. Averaging of the heat conduction equations with account for the convective mechanism of heat transfer. //Composites: Mechanics, Computations,
Applications. An International Journal. − 2010, v. 1, № 2, p. 1-21.
23. Balueva M. A., Blohin D. I., Savatorova V. L., Talonov A. V., Sheinin V. I. The modeling of influence of micro cracks in loaded geological materials on variations in temperature. //Journal of
Mining Science. − 2009, № 6, p. 55-60.
24. Vlasov A. N., Savatorova V. L., Talonov A. V. Homogenization of heat transfer equations, considering convective mechanism of heat transfer // Composite mechanics and design. − 2009. − v. 15. − №
1. − p. 17-31.
25. Savatorova V. L., Talonov A. V. Investigation of phase transitions in media with structural inhomogeneity // .Acta mechanika. − 2004. − № 9.
26. Savatorova V. L., Talonov A. V. The influence of the field of tension on processes of local phase transition in media with structural inhomogeneity // The problems of the strength of elements and
constructions under the loading. Saratov. SSTU, 2003. − p. 149-156.
27. Uhov S. B., Vlasov A. N., Lisin L. D., Merzliakov V. P., Savatorova V. L. Ice melting in frozen soils due to local tensions // Earth Cryosphera. − 1997. − v. 1. − № 3. − p. 35-38.
28. Vlasov A. N., Merzliakov V. P., Savatorova V. L., Talonov A. V., Uhov S. B. Local phase transitions and filtration as processes, predetermining the reology of frozen soils. // The Reports of
Academy of Sciences/ Earth Science Section. − 1996. − v. 349. − № 6.− p. 758-760.
29. Vlasov A. N., Savatorova V. L., Talonov A. V. Local phase transitions in heterogeneous medium under the influence of external tension // Composite mechanics and design.-1996. − v. 2. − № 2. − p.
30. Uhov S. B., Vlasov A. N., Merzliakov V. P., Savatorova V. L., Talonov A. V. Several processes, predetermining reological behaviour of frozen soils under the external tension // Base, foundations
and mechanics of soils. − 1996. − № 2. − p. 14-19.
31. Vlasov A. N., Savatorova V. L., Talonov A. V. Analytical methods of investigation of phase transitions in media with heterogeneous structure // Composite mechanics and design. − 1995. − v. 1. − №
32. .Vlasov A. N., Savatorova V. L., Talonov A. V. Asymptotical averaging in problems of phase transitions and heat transfer in media, consisting of layers of different materials. // Applied
Mathematics and Theoretical Physics. − 1995. − v. 36. − № 5. − p. 155-163.
1. Savatorova V., Talonov A., Vlasov A., Volkov-Bogorodsky D. Multi scale averaging of equations describing transport of gas phase in a porous medium // Proceedings of the Conference «Mechanics of
composite materials and structures, complex and heterogeneous media», Moscow, Institute of Applied Mechanics of Russian Academy of Sciences, December 15-17, 2015, p. 544-545.
2. Savatorova V. L., Kossovich E. L., Talonov A. V. A Multi-scale Approach to Modeling of Gas Transport in Shales // Proceedings of the SIAM Conference on Mathematics & Computational Issues in the
Geosciences, Stanford, California USA, June 29 - July 2, 2015, p. 89.
3. Kossovich E. L., Talonov A. V., Savatorova V. L. Multi-scale Asymptotic Analysis of Gas Transport in Porous Media//Proceedings of the 7th International Conference on Porous Media, Padova, 18th -
21st May, 2015.
4. Savatorova V. L., Efendiev Y., Akkutlu I. Y. Modeling of gas transport in organic-rich recourse shales // Proceedings of AMS conference. Spring Western Sectional Meeting, University of Nevada Las
Vegas, April 18-19, 2015, p. 87.
5. Savatorova V. L. Moscow school of mathematics: history of the origin, the heyday and the disintegration of the legendary Luzitania // Proceedings of AMS conference. Spring Western Sectional
Meeting, University of Nevada Las Vegas, April 18-19, 2015, p. 60.
6. Kossovich E., Talonov A., SavatorovaV., Safonov R. Determination of mechanical properties of chitosan matrix bionanocomposites with help of multiscale hybrid methods of molecular modeling //
Proceedings of the 4th Conference « Methods of mathematical physics and mathematical modeling of physical processes», Scientific Session of the NRNU «MEPhI», February, 18-21, 2015.
7. Kossovich E., Talonov A., Savatorova V. Modeling of gas transport in nano- and microporous heterogeneous media // Proceedings of the 4th Conference «Methods of mathematical physics and
mathematical modeling of physical processes», Scientific Session of the NRNU «MEPhI», February, 18-21, 2015.
8. Savatorova V., Talonov A., Vlasov A. Rheology of shales: Three-scale model of gas phase transport in a porous media// Proceedings of the Conference «Mechanics of nano- structured materials and
heterogeneous systems», Moscow, Institute of Applied Mechanics of Russian Academy of Sciences, December 23, 2014, p. 8-17.
9. SavatorovaV. L., Talonov A. V. Mathematical modeling of the transport of gase phase in kerogen//Proceedings of the 3nd Conference « Methods of mathematical physics and mathematical modeling of
physical processes», Scientific Session of the NRNU «MEPhI», January, 27-February, 1, 2014, p.231.
10. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Rheology of shales: transport of gas phase in kerogen nanopores // Proceedings of the 2nd Conference «Mechanics of nano-
structured materials and systems». Moscow, Institute of Applied Mechanics of Russian Academy of Sciences, December 17-19, 2013.
11. Brown D., Li G., Efendiev Y., Savatorova V. Multiscale modeling of High Contrast Brinkman Equations with Applications to Deformable Porous Media // Proceedings of the 5th Biot Conference on
Poromechanics, 10-12 July, 2013, Viene.
12. Savatorova V. L., Talonov A. V., Vlasov A. N. Homogenization in case of transient motion of a fluid through a porous medium.// Proceedings of the SIAM Conference on Mathematics & Computational
Issues in the Geosciences, Department of Mathematics University of Padova, Italy, June 17-20, 2013, p.79.
13. Brown D., Li G., Efendiev Y., Savatorova V. Homogenization of High-Contrast Brinkman Flows // Proceedings of the 5th International Conference on Porous Media, Czech Technical University in
Prague, 22nd-24th. May, 2013.
14. Savatorova V. L., Talonov A. V., Vlasov A. N., Volkov-Bogorodsky D. B. Upscaling in case of transient motion of a compressible fluid through a porous medium // Proceedings of the 5th
International Conference on Porous Media, Czech Technical University in Prague-22nd-24th May, 2013.
- Russian Academy of Natural Sciences, academician
- Society for Industrial and Applied Mathematics, member
- SIAM Activity Group on Geosciences, Flow in Porous Media and Geophysics (SIAG Geosciences), member
- American Mathematical Society, member
- Interpore, International Society for Porous Media, member
• Member of the «Up-to-date Mathematical Problems» Seminar, National Research Nuclear University «MEPhI», Moscow, Russia.
• Member of the Seven Professors» «Nonlinear Mathematical Models» Seminar, National Research Nuclear University «MEPhI», Moscow, Russia.
• Member of the scientific research Seminar under the guidance of academician Shemiakhin E.I, MSU, Moscow, Russia.
• Organizer of the «Mathematical Modeling of Physics Processes» Seminar Series at Physics Department, Moscow State Mining University, Moscow, Russia, 2006-2008.
• Qualifying Examination Committee, Moscow State Mining University, Moscow, Russia, 2005-2008.
• A member of the three-person panel that selects participants of the Cargill Global Scholars Program in Russia
• «Modeling of mass transfer in porous materials based on advanced multi-scale methods of description of the properties of heterogeneous media», $125,000. Sponsor: Russian Foundation for Basic
Research № 15-05-05960 as Co-PI (2015-2017).
• Federal target program «Research on priority directions of scientific-technological complex of Russia». Project # 2014-14-576-0051-25 «Scientific and methodological development of methods for
monitoring and prediction of risks of spontaneous combustion of coal and loss of its quality during storage and transport. Reduction of anthropogenic impact on the environment», $838,000 (2014-2017),
as co-PI.
• Program «Research and development on priority directions of scientific-technological complex of Russia for 2007-2013» in the framework of the Russian-American Presidential Commission and / or
performing joint research with NSF. The theme: «The development of mathematical models, methods and software systems for the description of the filtration and thermal conductivity in inhomogeneous
media with complex structure in order to implement applications in geophysics and development of new materials in cooperation with the University of Texas (TAMU) (2012-2013), as co-PI.
• «Multiscale modeling of highly heterogeneous industrial materials». «Mikhail Lomonosov II»- Programme: Research Grants and Research Stays for Doctoral Candidates and Young University Teachers from
the Natural Sciences and Engineering, Target group B:-Research stays for postdoctoral university teachers (15.09.2011 to 15.12.2011), as PI.
• United States-Russia Program: «Improving Research and Educational Activities in Higher Education», Sponsor: US Department of Education and Russian Ministry of Education and Science, $332,000 funded
as Co-PI ( from 2010-2013)
• «Establishing of the laws governing changing of morphologic and structural characteristics of Russian coals with the aim to ensure environmental safety in coalfields» exploitation». Project P1437.
Research personnel and research and educational personnel of innovative Russia (2009 – 2013), as Co-PI.
• «The complex usage of termomechanical and acoustic emission effects for investigation of processes of geomaterials deformation and fracture», Sponsor: Russian Foundation for Basic Research
№10–05–00687–а, as Co-PI
• United States-Russia Program: «Improving Research and Educational Activities in Higher Education», Sponsor: US Department of Education and Russian Ministry of Education and Science, $400,000 funded
as Co-PI ( from 2006-2009)
• «Local phase transitions and moisture percolation as key processes in rheology of frozen soils», Sponsor: Russian Foundation for Basic Research № 96-05-65295 as Co-PI. | {"url":"https://giab-online.ru/en/main/pages/v-l-savatorova-doktor-fiziko-matematicheskih-nauk-ssha/view","timestamp":"2024-11-03T00:04:40Z","content_type":"text/html","content_length":"53273","record_id":"<urn:uuid:0ea809f4-14e1-40fa-a4da-1035d4cfc84f>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00511.warc.gz"} |
Interview with Steve Fienberg
Behseta: Steve, let’s start, somewhat unusually, with the textbook Beginning Statistics, which you co-authored with Fred Mosteller and Robert Rourke.
Steve: A great place to begin for reasons I will explain.
Behseta: I have it on my shelf. What’s the story behind it?
Steve: So first of all, good news. That book is reappearing after 30 years. I can even show you the cover. I added the notes on the back cover about three weeks ago, and it will appear as a Dover
paperback reprint at some point later this fall. And I’m very, very pleased. I wish the layout could have been redone because there’s a very long story behind “the book,” but keeping it in print so
that others can use it has been my real objective. And I think Fred would be very pleased to see the book still in print, and Bob would have too.
Behseta: Did Mosteller invite you to join him on this?
Steve: Mosteller and Rourke were long-time collaborators, going back to the late 1950s, when Fred got involved in developing statistical materials for high school. They were on a committee together
and they wrote considerable material for it—this was the precursor to Probability with Statistical Applications, which was the book they ultimately produced with George Thomas. So their collaboration
went way back. Fred also worked on a lot of that material linked to his lectures for the introductory statistics course he did as part of the television program known as “Continental Classroom,”
which, as somebody described it recently, was the first statistics MOOC, long before people considered doing online courses.
Thousands and thousands of people watched Fred in the early mornings teach statistics on TV, and that was their introduction to the field. Fred and Bob had worked together on the book with George
Thomas, and there were two or three versions of the book, including one that was specific for “Continental Classroom.” Later they worked together on a book called Sturdy Statistics, which was an
effort to do nonparametrics (so sturdy in the sense of robust), for people who didn’t want to do nonparametric statistics with all of the mathematics. It was for a second course in statistics.
Behseta: We don’t know Rourke.
Steve: Bob was a high-school teacher at a small, private academy. Originally a Canadian, I think born in Kingston, Bob taught at small, private schools and was involved with a lot of these efforts,
having previously coauthored a number of high-school math textbooks. What he brought to the collaborations with Fred was an understanding of what you had to do to make something appealing to students
at that real introductory level. Anyhow, their book was Probability with Statistical Applications, and it was heavy on the elementary probability; Fred’s notion was we had to do a statistics book.
And maybe we’ll do probability, but that was not the goal of the book. Fred and Bob got started on this and then Bob got sick, and the project got bogged down. They had drafts of—I’m trying to
remember how many chapters—maybe a half dozen chapters, some quite polished. The early chapters were EDA (the exploratory data analysis).
So, in this sense, they were quite far along when I joined the project. But then the more heavy-duty statistical stuff had not really been fleshed out. Fred asked me to join the project because Bob
wasn’t really able to do any writing. You need to understand that Fred was always somebody who talked about getting projects done. Getting me involved was his notion of how this book would get done.
I’m not sure I helped move it along fast enough, but the goal was to get the project done. In the end, we did write a probability chapter, but it came after all the EDA stuff. They had pieces for the
next chapters that I helped polish up, but what I contributed most was laying out the early drafts of the regression and analysis of variances. The challenge was to do these topics for students who
didn’t have a computer, but to write the material as if they did. If you read the preface, that was always our goal. Of course, when I tested the draft chapters, I did it with students who had access
to computing, for example using MINITAB. But the idea in the book was that they should be able to read computer output, but not necessarily create it.
Slavkovic: So who do you think this book is now appropriate for; where could it be nicely used?
Steve: I said there was a long story behind the book. Fred was an editor for Addison-Wesley. The in-house editor at the time was a man named Roger Drumm. He and Fred developed the statistics series
for Addison-Wesley. Roger was very good. He had also worked with Fred on the regression book with Tukey, which was a real second course of a much higher level, and Tukey’s EDA book. There was a
promise that we would get color and the big page formats—you know, all the fancy stuff they do for very good textbooks these days to market them. About a month before we submitted the manuscript,
Roger left the company, or at least shifted to a different position in the company, and the new editor who took over didn’t think very much of what we had done.
We still had a contract with Addison-Wesley, and they weren’t going to void it because Fred was important to them in other ways. But suddenly they weren’t about to do anything special for us. So
color disappeared, and all of the fancy things we had talked about disappeared, and the book got to a reduced page size, and they fought with us about figures because they were supposed to redraw the
figures and now they wanted us to do the work, and so on and so forth. In the end, just getting it published was a big deal. It was interesting because ours was, in many ways, the first elementary
statistics book that did EDA for people who’d never seen statistics, but Addison-Wesley never promoted it as a textbook.
In fact, the same year they published our book, they published two other old-fashioned intro statistics books. And they wouldn’t even show ours to instructors around the country. It was just bizarre.
I used it a few times. Did I use it with you when you took the undergrad intro statistics course at CMU?
Slavkovic: I don’t think so.
Steve: I guess we were already using Moore and McCabe at that point. Well, what you didn’t know is that I was really using my book. Because that’s how I thought about my lectures, using the examples,
and using the pedagogical ideas that I had been taught by Mosteller and Rourke. Bob used to push this idea called PGP, which you can now see in the book if you go back and look. You begin with the
particular, that is you introduce the topic with an example. It might be a simple numerical one, although I always like to use real data for everything. But then you do the concept in a more general
form. For example, you write down the algebraic formula for a t-test, although you did the t-test already in the particular example without the formula or the idea behind it.
Then when you have the general structure, you put it to use again in another particular example where it has some payoff so that the student comes away with a sense that what they just learned has
some value. We tried to use PGP throughout our book, even long after Bob stopped working. He died before we were done. But that was also the part of the philosophy I took it into the statistics
classroom. Many modern textbooks use the PGP approach, although typically not as a rule, and teachers don’t necessarily follow the practice in their classrooms either.
Behseta: You recently edited two books on Mosteller. One is a selection of his papers, and the other is his autobiography. What’s the story behind the autobiography? Was it lost and somehow
discovered again?
Steve: When I was at York University in the early 1990s, we had a project to produce a volume of selected papers that Fred wrote. This was to be an adjunct to a project that produced a volume called
A Statistical Model, which was supposed to be in honor of Fred’s 65th birthday. I think we made it just before he turned 70—so not quite for his 65th birthday. But the idea was that we would follow
that up and get a selection of his papers into print together. David Hoaglin and I got started on this when I was still academic vice president at York, and I brought the project back to CMU. We
slowly re-keyed each of the selected articles into LaTeX, and then had to clean them up.
We also had a grand plan to add introductions to the selected papers. But it took a long, long time to just get the papers themselves into final form. We were in a wrap-up mode some 10 years or so
ago and David and I went to see Fred in his house to discuss how to bring the project to closure. He had been very, very sick and was just going through some rehab. We went out for lunch together and
he pulled us aside afterward and he said, “I have a request. There’s this manuscript I have and it hasn’t been published. Can I entrust you to get it done?” This was clearly something to which David
and I couldn’t say no. So we took it. It was typescript. Not even a computer typescript, but real typescript, initially.
Fred’s assistant, Cleo, with whom he had worked starting from about 1950, actually learned Word Perfect in order to re-key it for us, producing this new version from which we could work. The
backstory was that Fred had been asked to write an autobiography as part of a series that the Sloan Foundation was publishing on scientists. He couldn’t get himself to finish it because he was too
busy doing projects. Every time he got closer to the end, to about the time he was writing, he was too close to what he was doing. So many of the early chapters were quite polished, but as the book
got further along chronologically, the chapters got sparser and sparser. He had done most of the writing in the 1980s, and virtually nothing in the intervening decade or so prior to giving the
manuscript to us.
The draft Fred gave us was sitting in a plastic basket. We immediately made several copies. I left one with David. We gave one to Cleo, who produced the typescript. I was going from there to Montauk,
Long Island, for my annual visit with Judy Tanur. I took a copy out to Long Island and I showed it to her and Judy just thought it was great—she had worked with Fred going back 30 years or more to
Statistics by Example and Statistics, A Guide to the Unknown. In the end, the three of us agreed we would somehow look after it. The first task was to get a publisher. I called up the people from
Sloan and they said no, we’re not publishing these anymore. So that was bad. Then I went to my editor at Springer, John Kimmel, and I said to John: “You know how important Fred has been for
statistics.” Springer had published A Statistical Model. The Selected Papers were almost done at this point and we were going to put those into print as a Springer book. I said to John: “Springer
really needs to do the autobiography; this is what people really want to read.”
We sent John a copy and he read it and said, “Well, it needs to be edited.” He said things like, “This chapter is a little too long, can’t you do more here?” But he was supportive. On my next trip to
Montauk, Judy and I sat with the chapters about his early life and began to whittle them down. We took everything up through his undergraduate career at Carnegie Tech and condensed it. Because that’s
what John had thought probably was a little long. But we also added afterwards (epilogs) to each of the first six chapters.
It’s a very unusual autobiography. It really doesn’t begin at the beginning. It starts with descriptions of six big collaborative projects. Fred was a master of big projects. He could organize 20,
30, or even more people to do something together. Each of these six chapters talks about a project or a major application, for example, the Federalist Papers, the project he did with David Wallace,
or the critique of the Kinsey Report, which he did with Bill Cochran and John Tukey. Another deals with the National Halothane study, which was a major NRC study, where the statisticians included
Mosteller, Tukey, Lincoln Moses, and John Gilbert. By the way, John collaborated with me on one of my first papers; he made the first physical model of the two by two contingency table.
Slavkovic: We’ll get back to that.
Steve: He made it out of wires. I made my own with wires and string. At any rate, the Halothane Study also involved major efforts by several graduate students, most notably Yvonne Bishop, who spent a
year working out at the Center for Advanced Study in the Behavioral Sciences at Stanford, running log-linear model contingency table analyses. Fred described each of these projects in the early
chapters of the book. And for those, we simply said, “They’re terrific, minor edits.” What we did was we added a postscript to each of those, saying here’s how to think about this project 20 years or
30 years later. Here’s how this work changed a whole chunk of the field. I did a lot of those and then David edited them—David’s a fabulous editor and collaborator. He leaves no word untouched.
We condensed the middle chapters and then we had to figure out how to end the autobiography, since Fred stopped writing in the late 1980s. That took some time and we consulted a number of others.
This led to the final chapter, in which we got contributions from other people and tried to fashion it together to reflect on how Fred would have wanted his post-1990 activities to be described.
Sadly, the month the Selected Papers came out, Fred died. I got to see him in the hospital and I delivered a Xerox version of what was coming out, in a binder, because I was afraid he wasn’t going to
see it. And he died about a week later.
But we hadn’t really got the autobiography properly polished yet. And as a consequence of a series of sessions we did in Fred’s honor, we were able to pull together pictures from lots of people. We
have this fabulous collection of pictures, and his daughter Gail did a lot to gather some. And we also reached out to people. They included baby pictures and they went into the chapter where he was
born about his childhood. And we have pictures of him as a student when he got to Princeton. I don’t think we had a Carnegie Tech one. There are wedding pictures, there are pictures taken at his
summer home on Cape Cod. We didn’t have a lot of each of us with Fred, but we located one of each of us with Fred that went into the last chapter. David had worked with him for decades on projects,
Judy’s interactions with Fred went back even further, and he was my advisor and mentor.
Slavkovic: I’m sure you loved the book, the entire book, and the six projects in particular but if somebody tells you: okay, I have time right now to read one chapter, what would be the chapter that
you would recommend?
Steve: You can’t do that. If you want to know what made Fred great, you read the first six chapters. Because they tell you about the projects. And it isn’t just the results of what he did. It was how
he organized the projects and how he got the work done. Take the Federalist Papers project. Fred describes how he was visiting the University of Chicago and how he and David decided to do a project
to show that Bayes could really be done with real data, because nobody at the time was doing Bayesian analyses on major practical problems—it was just too difficult. That’s what I tell somebody to
read. But then if you want to know about Fred, you’ve got to read the rest.
There are some classic paragraphs here and there that are so revealing. I said something about getting the project done. There’s a little paragraph that talks about—his parents split when he was in
school. His father ran road construction crews in West Virginia, and Fred spent some summers working for his father there. And of course, when the weather was good, you had to get the project done
because the rains would come and screw everything up. So there’s this little paragraph where we learn how Fred learned to get the project done.
Slavkovic: That’s great. You spoke about Fred and—on many other occasions—you spoke about your other mentors and collaborators, as well as different book projects. For me, personally, when I was at
CMU, one of the early books I read on statistics that was helpful was DeGroot’s Probability and Statistics that Larry Wasserman was using for a class he was teaching. This brings me to your early
days at CMU and your collaborations with DeGroot. We wanted to know how exciting those days were. You also co-authored Statistics and the Law with DeGroot and Jay Kadane at that time. So how did that
book come to be? It’s really a collection of articles, right?
Steve: Yes, but you have to go way back! I first met Morrie in 1970. So I was two years old, as a statistician. I was a junior faculty member at The University of Chicago. Morrie got his PhD from
Chicago under Jimmie Savage, and he went from Chicago to Carnegie Tech in 1957 before there was a statistics department. Jimmie was notoriously tough on his students and some of them never finished.
But Morrie was one of the really good ones and he submitted his thesis a year or so after moving to Pittsburgh.
As I said, I met Morrie in 1970. I had already been versed in Bayesian thinking, a little bit by Fred, although Fred didn’t push. Actually much more by Howard Raiffa and Bob Schlaifer, who were at
the Harvard Business School, and whose weekly seminar I would attend. John Pratt was part of that seminar as well. The presentations and discussions, especially the latter, were really formative
activities for me because these where statisticians who were real subjective Bayesians and they did both theory and actual applications.
Behesta: Is this the Schlaifer from the Decision Theory book?
Steve: Yes. Schlaifer was self-taught as a statistician. And basically he and Raiffa carried out this systematic research program in the late 1950s where they set out to recreate exponential families
from a Bayesian perspective. They invented conjugate priors—they invented the name, although the idea had being around in different forms earlier. They created pre-posterior analyses. Their book is
just a fabulous treatment of Bayesian tools, although it had the world’s most God-awful notation, which Howard Raiffa has happily defended in the interview I did with him for Statistical Science. He
remembered exactly what the notation was there for, but I never got very excited by it.
Anyhow, back to Morrie. By the time I met him, I was into Bayesian thinking, and I was at a regional IMS meeting, in North Carolina. I got to know Morrie, as we were drinking with some others in a
bar. We seemed to meet and drink in bars often over the years. At any rate we became friendly. We were both, along with Jay Kadane, part of this seminar that Arnold Zellner ran on Bayesian
econometrics and statistics. It was a very small group, initially, of deeply committed Bayesians. It included Arnold and some other people from the business school at Chicago: Bruce Hill and Bill
Erickson from Michigan. Both George Box and George Tiao were involved, as were Seymour Geisser and Jim Press. Jimmy Savage came and spoke at one of seminars. Morrie, Jay, and I interacted a lot at
those meetings. Jay, at the time, wasn’t yet at CMU. He was between Yale and CMU at the Center for Naval Analysis.
A bit later on, Morrie became editor of Theory and Methods for JASA. He had previously been Book Review editor. When he became Theory and Methods editor, succeeding Brad Efron, I became Book Review
editor. Then Bob Ferber, who was Coordinating and Applications editor, stepped down and I succeeded him, and Jim Press became Book Review editor. There we were, three Bayesians running JASA! I was in
Minnesota already at this time, so we spent more time in bars. And not just talking about the politics and world affairs, but we were also dealing with how to run the journal and how to bypass all
the efforts by the ASA Board to control and change the content.
I was the first editor who oversaw the production office, which moved from the editor’s office into ASA. ASA wanted to run it and I wanted to oversee the activities so that the editors could do their
jobs. The editors had a very different point of view about how to get things done, as you may now understand. Editors know what they want, and somehow things never get done quite the way they want
when others control the process.
Morrie, Jay, and I were interacting in these two spheres. In 1978, I had been offered a job at another institution, which never quite worked out. And Morrie and Jay knew. We were in a bar one night
and they said, “You should come to CMU.” I was ready. I had already got myself psyched up about the possibility of moving because you shouldn’t interview for something if you’re not ready to take the
job. I interviewed at Carnegie Mellon and I spent two hours with Dick Cyert, then CMU president, who was Morrie’s collaborator. Dick thought his job was to talk me into coming to CMU, which he did.
After I got there, Morrie and I began our collaboration on probability forecasting. Actually, it started before I got there as a result of a conversation we had at the first Valencia meeting in 1979.
Behseta: Was there a statistics department?
Steve: Morrie and Dick Cyert pushed to create the department in 1966, right about the time Carnegie Tech and Mellon Institute merged. It was about the same time as the university added a bunch of
other units: SUPA, which became a Heinz College, the College of Humanities and Social Sciences which grew out of Margaret Morrison Carnegie College for women. And statistics was created as a
university-wide department, reporting at first to Dick, who was dean of GSIA, the business school. Then Dick became president in the early 1970s and statistics reported to him as president. Morrie
was the first head. They did make an effort to hire outside, but that’s always hard, and they finally prevailed on Morrie to become head. Jay became head succeeding Morrie when he came in the early
Slavkovic: Jay was just telling me this morning about his role and needing to report to different deans and the president, directly.
Steve: Exactly. So what happened was Dick got busy, and he finally said, “Why don’t you report to the provost?” The provost was a nice guy, but he didn’t want Jay to report to him. Finally, in about
1978, the provost said to Jay, “Pick a college; we’re not going to continue this arrangement.” Jay sent a memo to all of the deans, inviting them to apply to the position. Very typical Jay. We had
several good offers. The best one came from the College of Humanities and Social Science. So that’s where we ended up.
Behseta: Typically, statistics departments are not housed in social sciences and humanities!
Steve: There are a few similar settings for statistics departments at other universities, but it’s not common. More typically, statistics is with math and in physical sciences or computer science.
But what was very clear was that we had the mandate to continue to work with everybody in the university. When I became head, I used to tweak the dean of H&SS because we didn’t agree to be part of
his college; we just agreed to have him as our dean. So I referred to him as the dean of humanities and social science and statistics. But then we officially became part of the college.
Behseta: So because of that formation, since day one the statistics department at CMU must have been Bayesian.
Steve: It was the Bayesian department in the sense that Morrie and then Jay were Bayesians, but initially there were others involved. Don Gaver who was here did some Bayesian things, but others were
frequentists. John Lehoczky came in 1969, but John wasn’t really a Bayesian.
Slavkovic: So did the bar outings convince you to become Bayesian, or was there something else?
Steve: No, no. Bayesians go to bars, is the way I think you have to think about it. Or Bayesians have good times!
Slavkovic: So what did convince you to become Bayesian? Then we’ll come back to the book we wanted to ask you about.
Steve: I was really convinced by Raiffa and Schlaifer, and that seminar they ran. I did it simultaneously with my interactions with Fred. One of my first projects was actually something that you
would now call empirical Bayes. But even though Fred had done the Federalist Papers work, which was fully Bayesian, he didn’t push Bayes. When I got to Chicago, I taught Bayesian classes, and I
interacted with the business school people. But going back to Harvard, I was part of the Raiffa and Schlaifer seminar and that was formative. I remember there was a basic statistics class that Jerry
Klotz, who later went to Wisconsin, taught out of Lehmann.
Behseta: The Testing Statistical Hypotheses book?
Steve: Yes, that was the only Lehmann book at the time. Partway through the course, I said to Jerry Klotz, “But what about Bayesian methods?” And Jerry said to me, “If you want to be a Bayesian, you
go to the business school. I don’t do that.” So I went to the business school because I wanted to know what it was. Of course, I’d heard the word before from Don Fraser, but he wasn’t Bayesian. He
had his own label for it, but it was really a form of Fisherian fiducial inference.
Anyhow, we never got to the book.
Steve: Morrie and Jay and I were all interested in legal applications, and we thought it would be a neat way to collaborate. We spent a lot of time chatting about the framing of statistical testimony
and decided that getting contributions from people who really did statistical testimony as experts would be of interest to others. That’s what we then organized. Morrie and I worked on research
projects separately. Jay and I wrote together at the time as well.
Behseta: And it’s an interesting book because it reads like a collection of case studies with comments and rejoinders and the sort of back and forth commentaries that you would find in technical
Steve: Well, and also that you see in trials.
Behseta: That’s right! It has the legal feel!
Steve: In fact, part of what we tried to do—not overly successful—was actually have some of that. Our idea was that if we could identify a couple of good cases and get the experts from the two sides
and let them describe things—because the two sides always tell a different story—and therefore their experts do different analyses and reach different conclusions, even though the data are the same.
Behseta: You have a keen sense of appreciation for the history of statistics as a discipline. And you’ve written about it and published around that theme. Where does that come from? In your class
lectures, you were always referring to historical occurrences.
Steve: It’s an interesting question.
Slavkovic: We’ll give you a few minutes to think about it. We sat in your course. We had a seminar touching on the history of statistics.
Steve: It’s a funny thing. When you’re a graduate student, you think something done two years ago is old, and therefore you don’t begin doing things by going back and reading what other people wrote
a long time ago. One of the things I read as a graduate student was this lovely little book by Jack Good called Estimation of Probabilities. It was actually a lot of what Jack had done over the 1950s
and early 1960s in a slightly different form. But if you read the early chapters of that, it takes you back into the earlier contributions of people doing Bayesian-like analyses. For example, there’s
Johnston’s Postulate. By reading Good, I began to have some sense that I should pay attention to that work.
I got interested through a series of different problems in understanding where statistical ideas came from. Iterative proportional fitting, which Yvonne Bishop brought into contingency tables, I knew
came from Deming and Stephen in 1940. Now, that was ancient history to a grad student working in the 1960s. I went and read their papers. I began to read some of the other older literature on
contingency tables as well.
When I got to Chicago, I fell under the influence of Bill Kruskal, who was this unbelievable scholar. He would constantly direct me to things in the literature that were from prior eras in one form
or another. For example, I did this paper on the draft lottery, with Bill’s encouragement. The analyses were trivial in many senses. But the paper was not, because with Bill’s urging I went back and
look at the history of lotteries. Every time I thought I had some of it done, Bill would go into the library at night and find these old books and journal that he though I should read and reference.
By the time I had completed that project, I was already doing my own historical digging and paying attention to people who wrote about historical topics. And it was fun, but there wasn’t a lot of
reward for doing that. You don’t get tenure for writing history papers unless you’re in the history department. Then you’ve got to write ones that historians care about. So it took a while for my
interests in the history of statistics to emerge. Steve Stigler has been a very close friend for a long time, and he was into writing about our history in a pretty serious way, starting in the 1970s.
I found Steve’s papers really fascinating. He and I did lots of interaction on the origin of contingency table analyses over the years. Although, you can never beat Steve Stigler. He always finds
something earlier. For example, I thought I had systematically traced the work on contingency tables back to Pearson and Yule. In 2002, I co-organized a journal issue in honor of a statistician who
had done some seminal work on log-linear models and quasi-symmetry, Henri Caussinus. Steve did a paper for that on the early history of contingency tables with examples from Galton and others that I
wouldn’t have found in a million years. But I confess that I got into a lot of statistical history inspired by Steve. There was a sudden spate of books on the history of statistics in the 1980s,
including one by Steve. I wrote a review essay drawing upon all of these historical accounts, but I then I also went back to a substantial amount original material as part of the effort. Not the way
Steve does; Steve is really a professional. I’m an amateur historian.
My review essay characterizing the work in these books is, in fact, my own take on a history of statistics. It’s shorter and it’s much more readable than many, but it’s never more readable than
Stigler. And it has a different emphasis.
I followed up on that essay in different ways. The project I did with Margo Anderson in the 1990s, which is about census taking, took us back to the first census, for a variety of reasons. Margo is
the historian of the census. And much of what I know about the history of census taking I learned from her. She has lots of source material, and though Margo I’ve dabbled in the history of the
census, on topics like estimation and adjustment, and on the measurement of race in the census.
My paper, “When Did Bayesian Inference Become Bayesian,” which was one of my labors of love, started in a bar in Ann Arbor, Michigan. Margo and I were on the ICPSR council, and we were attending a
council meeting. While we were waiting for others to join in on a dinner, we were having drinks and she turned to me and asked, “When did people start to talk about Bayesian analysis?” I said, “I
don’t know.” We talked about it a little bit. I knew that when I was a student people talked about Bayesian analyses, but I also knew that way earlier most statisticians talked in terms of inverse
probability, a term attributed to Laplace. I said, “I’ll have to work on it.” So I went back and I pulled Jimmie Savage’s 1954 book off my shelf, and much to my surprise it didn’t mention the word
Bayesian at all. Well, that’s pretty strange, I said to myself, because shortly thereafter, there’s Raiffa and Schlaifer’s book, which is full of the word Bayesian. Then I did a JSTOR search, and
gradually I began to do a back and forth with Steve Stigler, and then with Jack Good and others. What started as a simple question in a bar turned out to be a 40-page paper several years later. Being
an amateur historian is fun.
Behseta: Which takes us to the two books that Sesa and I really love! One is The Analysis of Cross Classified Categorical Data. When I took the course with you, that was my textbook and I loved it
and I learned a lot from it. And I was looking at it in the anticipation of this interview the other day, and it’s pretty fresh.
Steve: I wrote it after I had worked on Bishop, Fienberg, and Holland, which we finally published in 1975. There is a very long story behind that.
Slavkovic: We will get to that, too.
Steve: Bishop, Fienberg, and Holland was an amalgam of things that grew out of collaborations, but it had a lot of contributing authors. Darrell Bock of The University of Chicago got me to co-teach a
short course on categorical data analysis, just after Bishop, Fienberg, and Holland appeared. I wrote some lecture notes for that course. They were pretty successful. Darrell had written his own book
on categorical data analysis, which had a very different approach to the topic. Very good, but quite different. We did the short course again the next year and I expanded my lecture notes.
I really wanted to rewrite Bishop, Fienberg, Holland in the spirit of these notes. I tried to convince Yvonne and Paul to do this. I even circulated an outline of what I thought we could do, totally
redoing the book. Yvonne wanted to redo it, but totally differently than I did. Paul’s interests were different yet again, and he had somewhat different ideas of how to redo it. After a while, it
became very clear that we would never, ever agree on a plan. In fact, we couldn’t even agree on how to do a second edition with relatively small changes because we each wanted to do to do the small
changes very differently. Finally, I said to them, “I’ve got these notes. Let me do more with them and we’ll leave our book alone.” I developed new examples, and my goal was essentially to prepare a
book from which I could teach to statisticians and nonstatisticians.
In a sense, ACCD allows someone to use the same material, but at two different levels. After the book appeared in 1977, I quickly realized it wasn’t as complete as I wanted. So I did a second edition
about three years later, and I’m still working now on the third edition. I actually have material. There will be a third edition if I survive. Basically, the course I taught for both of you was not
just what was in the book, but also in the notes for the third edition. I have all of the notes, but I need something like a summer where I’m working on other projects.
Behseta: Does that mean you plan to include something on graphical models and causal analysis? I mean, you already had a chapter on causal models, right?
Steve: Yes, there is a chapter on that topic, but it’s the wrong chapter.
Slavkovic: What would be the right one?
Steve: Well, the second edition of the book came out just as Darroch, Lauritzen, and Speed published their 1980 paper on graphical models. Adding that material and what followed from it is essential,
and that does lead to so-called causal models for categorical data using DAGs. Replacing the existing chapter on that topic is crucial as well. I probably wouldn’t make the book Bayesian, however. I
will have something Bayesian in it, but that would be a different book.
Slavkovic: So back to the Bishop, Fienberg, and Holland book. I hope when re-doing the book for new versions, there will be no plans to exclude the tetrahedron! Both Sam and I remember you bringing
the 3D model of the tetrahedron and trying to explain the surface of independence. All of us were totally perplexed about what was going on.
Steve: But look, you got it!
Slavkovic: Exactly! So we’ll get back to that. But did you find it useful to bring some of those visual tools for teaching? Do you think they help students?
Steve: It helped me. I confess I realized it didn’t really help everybody; many students wondered why I taught that material. As an undergraduate, I was brought up thinking about mathematics and
statistics geometrically. That’s one of the things I learned from Don Fraser. I remembered Don lecturing and projecting into a lower dimensional space, using his hands, in order to do his regression
estimate. Don taught geometrically. At least as important, were the courses I took from H.M.S. Coxeter as an undergraduate. That’s where I learned about barrycentric coordinates. His book,
Introduction to Geometry, was my textbook in a third-year geometry course Coxeter taught. And that’s what I brought to bear in my thesis work on the two by two contingency tables.
It’s from Coxeter and related kinds of “old-fashioned” geometry that I recognized what the surfaces were. We now know that the algebraic geometers call the surface of independence the Segre variety.
It also has this great representation, as Miron Straf was reminding me last night. When I gave my job talk at The University of Chicago in January 1968, he asked me whether the surface of
independence was a minimal surface in a physics sense? He reminded me that I didn’t answer him. I said that’s because I didn’t know the answer. But Miron was, of course, correct. Because if you think
about Kullback-Leibler information and ask about what’s the minimizer, under independence, you get the surface of independence in my paper on the two by two table.
Slavkovic: I think you have influenced many students with geometry. I remember a few years ago when I gave a seminar at Ohio State and visited with Elly Kaizar, she had a little 3D model of the
tetrahedron with the surface of independence. A few years ago, I was asked to do a 15-minute presentation at a welcoming session for the prospective students at Penn State. Among other things, I
brought a little 3D tetrahedron that my student Vishesh Karwa made out of a set of basic pencils and some rope and I was talking about geometry and algebraic statistics.
Steve: The funny thing about it was that it was really John Gilbert who made the first physical model. I was in the stat department at Harvard one day when he came in with this coat-hanger and
copper-wire model. But he didn’t know what the mathematics behind it was. He had independence in his thinking too.
Slavkovic: And not the other surfaces.
Steve: Right. But he couldn’t say why it was independence. John was great. By the way, he was a Chicago PhD student who never got a PhD. They were really tough on students there in the 1950s.
Behseta: Did he work with Savage?
Steve: I don’t remember with whom John worked, but they were all tough on students back in those days. And it wasn’t just true of the faculty at Chicago. We try to be much better about getting
students through their thesis work these days. We don’t succeed always, but we try.
Slavkovic: For me, personally, I eventually learned about the mathematics of the surface of independence, and that was my link to data privacy, too. I actually always wondered and I would never have
believed asking this: How and why did you get involved in data privacy and confidentiality research?
Steve: There were two pieces to this work. When I was department head at CMU, Diane Lambert was a junior faculty and working with George Duncan on a little project that turned into a pair of papers,
one in JASA and one in JBES, the business and economics statistics journal. They were basically taking a decision theoretic approach to confidentiality and trying to reel in the structure to what
people said they were doing about confidentiality protection in the statistics agencies. Those were the first statistical papers I read on the topic. I wasn’t involved in the research, but because
Diane was up for promotion—or reappointment and then promotion—I actually read the papers really carefully at some point, and then I filed the ideas away for future use.
When I was at York University, Denise Lievesley was helping organize a conference in Dublin on confidentiality and she asked me to do an overview of statistical approaches to the topic. The
conference was heavily dominated by government statisticians. In fact, maybe two-thirds. There were also several lawyers because they dealt with the laws governing what the statistics agencies do
about confidentiality. And then there were a handful of statisticians. George Duncan came, and there were a few statisticians from England besides Denise. I spent the better part of a year gathering
up what I could find on the topic. We didn’t have Google in those days so it wasn’t very easy. There were the Tore Dalenius papers. I knew Tore because he had written for one of the early issues of
CHANCE on data protection of subjects in that article.
But he also wrote this really important paper with Reiss, who was a computer scientist, on data swapping. When I was done with the review, I said to myself, “This is a real gold mine.” I had thought
about that when I read Diane and George’s papers in the 1980s, but didn’t follow through. I reread their papers as part of this exercise and it was pretty prominent in what I had to say, because they
were being statistical and most of the literature on confidentiality was ad hoc. I asked myself, “What does a good statistician do with a topic like this?” He or she brings order to ad hoc ideas and
gives them structure. The published version of the review that appeared in the Journal of Official Statistics a couple of years later was a slightly more polished version of my Dublin presentation. I
had already embarked on a follow-up project with Udi Makov.
During my second year at York, Udi was visiting as a faculty member. He and I originally met at the first Valencia conference. I was actually a discussant of Udi’s paper, which was an outgrowth of
his thesis work. We became friends and, in anticipation of his arrival in Toronto he said we should work on a project together. I told him about the work on confidentiality and shared a draft of my
paper with him. I said this is a gold mine, let’s just find something and we’ll work on it. It was during that year that we started our intruder modeling work. Of course it, like everything else, the
problem we chose to tackle was harder than it looked to begin with. We worked on it for another year or so and I got a grant from the Census Bureau to develop it a little more.
Russ Steele was an undergrad in one of my classes back at CMU at the time, and we got him to work on the project in the summer. This led to Russ’ senior thesis. Every time I turned around to do
another version of the confidentiality protection problem, there were all these other people saying they were doing it statistically, but I didn’t think they were. I tried to entice young
statisticians like you into doing some of this work because there were all these unsolved problems that needed attention.
Slavkovic: I wanted to bring one other thing up in relation to privacy. I had a recent exchange with Cynthia Dwork about the more recent collaborations between cryptographers and statisticians. And
she says—I’m quoting her about your role in getting us together and working together on this problem—”to Steve’s great credit, he saw something new was happening and he welcomed us [cryptographers]
to his world, rather than trying to exclude us. I will always admire this.” And so what Sam and I were wondering is …
Steve: You should have that quote yesterday. That was better than the one about …
Slavkovic: The “paranoia” and “ad hoc”? Oh, but I thought that would get people going, which it did. So what do you look for when you start a new collaboration, a new project, or when you encourage
these new interdisciplinary projects—you just said privacy looked like a gold mine. But in general, in these new collaborations and new projects, what do you look for?
Steve: I’ve said in other settings that I had mentors from whom I learned to do this. Don Fraser was a mentor in one sense, but that was very technical. Fred Mosteller and Bill Kruskal were mentors
in a very different sense. Paul Meier, also, especially in connection with statistics and the law. They were Renaissance men in the sense that they were interested in everything.
Slavkovic: Like you!
Steve: Well, but you learn from people. And to me, one of the great things about being a statistician was that if I saw something that was interesting, more often than not there would be some
statistical aspect to it. More often than not within a body of work arising in a substantive area, there would be some problem that I could formulate in a way that nobody else had, or reformulate so
that it took on a slightly different form. In a sense, I do technical things in my interdisciplinary work but I don’t view the technical things I do as my most important accomplishments. I view
formulating problems as my best contribution to these enterprises. Being able to see beyond what looks like a fog and say, “Hey, there might be something there; if we only tried to look at it in a
different way, we could make something out of it.” In some ways, one of my neatest collaborations is one that nobody in the statistics department knows anything about, or almost nothing, Cognitive
Aspects of Survey Methodology. That grew out of a little seminar that I participated in related to the redesign of the National Crime Survey just after I arrived at CMU. It was Al Biderman’s idea,
and he brought several top researchers together to discuss what cognitive psychology had to say about questionnaire design but he didn’t know how to make them really talk to each other.
The psychologists were really neat people. Beth Loftus and I got along really well and she wrote a paper inspired by the seminar. Endel Tulving was another psychologist who was there. He taught a
course at the University of Toronto which I took as an undergrad on research methodology in psychology. I didn’t know what he really did at the time, because your teachers don’t always teach about
their research, they teach what the class is about. Learning to talk with people like this, and not stop because I speak log-linear models and they speak some language in psychology is what makes
interdisciplinary work so much fun. If you can focus on substantive problems that they have and you find interesting, it’s really very easy. What may be hard for some people is to do this
interdisciplinary work in lots of different areas because you often need to acquire deep subject matter knowledge.
But it’s really not hard to do. Back in 2003 when Cynthia was giving this talk at CMU, she had a half-baked idea. That’s unfair. She had an 80 percent baked idea, maybe even a higher percent. In fact
it was a great idea but it didn’t deal with the privacy problem as I saw it. What was clear was that Cynthia and her collaborators had technical tools at their disposal that were really super, and we
would ignore them at our peril. One of the things that Fred and Bill indirectly taught me is you give credit to everybody for what they do. You will never be hurt by giving other people credit.
Cynthia and the cryptographers were doing neat things. We as statisticians helped them to make them neater. They polished their ideas up and generating a new literature on privacy with important
results for statisticians. Their professional world is slightly different world than ours, and they still can’t deal with the problems that motivated me initially. But their work is of fundamental
importance to the research on confidentiality and I, and students who work with me, are trying to use their ideas to solve statistical problems. And Cynthia’s a good friend and we dine and drink
Behseta: So what are you working on these days? What’s next?
Steve: There’s privacy. Sesa and I have a project, and in fact we have a paper that we have to finish writing in the next couple of weeks which draws on the work of one of my PhD students, Fei Yu,
and which takes something that she and I did with Caroline Uhler on privacy protection in GWAS studies. Fei’s got neat results and he is going to have a great PhD thesis.
I’m also running this joint Living Analytics Research Centre with people in Singapore. It involves very substantial digital data sets that come from commercial partners on which we’re doing different
kinds of machine learning style analyses and network modeling. As I explained in my lecture yesterday, there are totally new problems in this work waiting to be solved, like the design and analysis
of experiments in networked environments. And I’m sure that when we get to the end of the project, we still won’t have solved all of the statistical problems but we will have made progress. I also
have the revision of my book if I ever get to it. And finally I have the project with Judy Tanur that we have ……
Slavkovic: The book?
Steve: The book! We really have something like six chapters, including a pair on the history of sample surveys and on the history of experimental design. And those ones are up to date, despite what
she said. It’s how to finish it that we don’t quite know to do. And we’ll have to do a lot of new writing. We may not have to do new research.
Behseta: Maybe for the benefit of our readers, you could say a few words about the book.
Steve: Judy and I started to collaborate when she was writing a review of a set of social science research things—quantitative social science—for a special volume that NSF was pulling together. Maybe
it was SSRC-related. This was in the late 1970s and we had already become friends. And she invited me to come to their cottage in Montauk at the tip of Long Island. We planned this summer trip that
began with a visit to my brother in Tennessee. This was with Joyce the kids, and our dog Princess. We showed up with the dog at the Tanur’s cottage. They also had dogs, and that began a summer
tradition which is much more punctuated now, but we still visit. We don’t have a dog anymore, but they have two.
In Montauk, we would go up on the upper deck and Judy and I would work. We also swam and had a good time with our families and dogs. Originally it was my helping Judy with the NSF survey paper. Then
we branched out and began to collaborate more directly. We talked about the history of surveys and where ideas came from and where they should go. At some point, I explained the parallels between
surveys and experiments. This was something I new about in informal ways but when I started to do some background reading, I realized was never properly captured in the statistics literature. We
started to write papers on the topic. We got a grant and we wrote more. And we actually wrote six chapters, and we published a half dozen papers as well as a nice one on cognitive aspects of survey
modeling, with Beth Loftus. In a joint paper in Science we brought the two ideas together and pointed out that the key thing was to test variants in survey questionnaires totally differently than the
survey people did. They did spit samples with interviewers dealing with only one of the variants. We noted that by having the variants of survey questionnaires done within an interviewer, we could
control for the variability much better. Components of variance is an experimental design idea, and we were importing it into survey design. We’ve never won that battle with the government sample
survey folks, by the way. But some people do what we suggest. We worked on the book through the mid 1990s and then I got sucked off into other things like Fred’s autobiography. That got in the way.
Slavkovic: Yesterday, there were already orders for this book. So when is it supposed to come out?
Steve: Everything takes longer than you think it should. I don’t have Fred’s ability to “get the job done.” But I do get them done at some point. I hope this will happen with the book with Judy, too.
Behseta: Where do you see our discipline is heading? What is the future for statistics?
Steve: Well, you just have to read the newspapers. You have to read what my colleagues are writing. Chad Schaeffer yesterday had a full-page piece in the Pittsburgh Post Gazette on statistics and Big
Data. Joel [Greenhouse] had a piece in the Huffington Post a week or so ago about the uses of statistics. The action these days is, in many ways, Big Data and data science. People who ignore these
developments and ignore the emphasis on them emanating from computer science, physics, and bioinformatics, ignore them at their peril. Statistics departments that try to close those folks out will
lose out in the long run. Opening the profession up to collaborations on Big Data and data science is similar to our welcoming Cynthia and her really smart colleagues into what we do on
confidentiality and privacy protection. It makes what we do better. We as statisticians have a lot to offer the Big Data movement, but so do others.
When I do Big Data, I also do little data. I teach contingency tables, not by starting with the National Long Term Care Survey and six waves each involving 200 variables; I start it with a two by two
table. It’s back to that philosophy that I told you—PGP—that I learned from Mosteller, Rourke, and my collaboration with them. You start with the simplest possible way of the understanding a problem,
and then you learn from that and generalize and scale up to Big Data. What we have to learn to do from our computer science friends is generalize to handle big problems, not just the little problems.
Slavkovic: What does that mean for training of a new generation of statisticians? What do we need to do to prep them?
Steve: We need to get their attention first. When I got to Minnesota, I had a very interesting experience. I was the first chair of applied statistics and had all of these young colleagues over in
St. Paul. The older statisticians were in Minneapolis—with some young people, they weren’t all old. When we wanted to introduce courses that were much more methodological, they said, well, we’ll have
add the new courses onto the existing requirements. I said, “That’s crazy.” You’ll add this on and you’ll add on another one and then the students won’t get to research until their sixth year instead
of their fourth year.
We had a year-long sequence in multivariate analysis because that’s what one of the faculty members did and he taught it every year. I’m all for teaching multivariate analysis, but we often think
that we need our students to know the union of what all the faculty know and not the intersection, before we have them do something new. That’s clearly stupid; you can’t do that. We’ve got to
streamline what we teach, but we don’t want to lose it all, either. After all, where will they learn about how the concepts of experimental design and survey design fit together in the context of
real-world problems if we no longer teach either subject. As we bring in new tools and other ways of thinking—different kinds of computation because it’s a very different computational world—other
topics have to get set aside. But are goal should be not to lose them totally. What if somebody says we don’t need the Rao-Blackwell theorem anymore because I don’t use it for Big Data. It would be a
mistake not to teach about it. And by the way, Rao-Blackwell is important in Big Data settings.
Or suppose someone says, we don’t need the Pearson chi-square statistic anymore, because it’s based on the wrong asymptotics, so we won’t teach it. My response is that, if you don’t teach that topic,
how the hell are you ever going to get to the variant of it that you need for Big Data? There’s a constant tension between retaining the older ideas, which may serve as the basis for new statistical
research, and at the same time keeping up with the machine learning people and the cryptographers who are out ahead of us in many dimensions (that’s a pun). These are very smart people. That’s why I
want them as my colleagues and collaborators. They make terrific researchers and they’re going a mile a minute.
We as statisticians want to do things that they do, but we want to root our work in the theory and structure in which we’ve been trained. If we can bring statistical rigor to those Big Data problems,
we will also change the nature of what the machine learning researchers do. The future is Big Data at one level, but it’s Big Data infused with the richness that the hundreds of years of statistical
methodology and theory brings to the table.
Slavkovic: I am sure you’ve been asked this question so …
Steve: And it’s your job to follow through on what I have just said because Big Data is a young person’s pursuit.
Slavkovic: But what is Big Data?
Steve: It doesn’t matter. Because everybody else thinks that Big Data is what they do and that’s what we need to train our students to feel comfortable doing. If that’s what brings students in the
door, if we as statisticians can turn the students onto interesting problems and tell them that the statistical work they are doing is Big Data, then it doesn’t matter if we can define Big Data,
I used to think that Big Data in statistics was what we collected in censuses. I told this to my friend Ralph Roskies who is a physicist and co-directs the Pittsburgh Supercomputer Center. When I
explained how big the U.D. decennial census database was, he said, “I want to know how many gigabytes.” I said, “It’s not gigabytes, it’s megabytes.” He said, “So it doesn’t generate a large data
set.” I said, “It’s large in a different sense. And we try to combine census data with data from other sources.”
Does the National Long Term Care survey generate Big Data? Well, you know, there are hundreds of questions; that’s large. And six survey waves, so the dimensionality of individual level data is
hundreds to the power of six, if you think longitudinally. In this sense, statisticians and others have been doing big for a long time. But computer scientists and physicists know how to do a lot of
things that we don’t know how to do. And they can compute faster than I can think. We need to have our students learn how to do that, but also learn how to think statistically.
Slavkovic: Learning the basics of algorithms is important these days.
Slavkovic: Besides Big Data, is there anything else that comes to mind that would be prominent in any way? Because we are valuable now, I mean statisticians are valuable now, and it would be good if
we can stay that way.
Steve: The thing I marvel at in some ways in my environment at CMU is that if we were to have twice as many people in the department, everybody would be just as busy interacting with others around
the university, doing collaborative research, working on real problems. Delivering on real problems, to me, remains a focus of what I do and where I see the profession going. Look, every once in a
while, I do mathematical statistics, or something that I call that. I even publish in the Annals of Statistics when I have good collaborators who can help me get our work published there. But that’s
not my forte. It’s getting that kind of theoretical work together with the applications. And that’s the future of the field. For me, it is also the future of IMS, which has mathematical in its title.
I am afraid ASA blew it when it came to many of these developments over the past 15 years. That’s when our CMU collaborations with computer scientists began to gel into a separate unit that later
became the machine learning department.
CHANCE is an ASA journal and I’m happy to put my views on record. ASA could have been a leader by reaching out to the machine learning community as it was beginning to grow and take shape. Instead,
ASA turned its back on them. Not everybody. There were great people from the statistics community who saw this as an opportunity and began to do that work that is now called machine learning, but
very few. ASA had all sorts of directions that board members thought the organization should move in, and they undercut what I would have done in this crucial domain. ASA and IMS should have
co-sponsored the big machine learning and data mining conferences.
We should have been there at the outset. ASA failed to do that. You could almost forgive IMS for failing to do it because it didn’t look like there weren’t lots of theorems in machine learning. That
by the way wasn’t true, and IMS actually reached out in some ways. With the creation of the Annals of Applied Statistics, we jumped over the barrier dividing statistics and machine learning and we
published and continue to publish research that really sits at that interface. You don’t see a lot of this in ASA journals. You see a lot of it around here at JSM, or at least more than before. But
ASA hasn’t changed. IMS is actually changing, little by little, with AOAS as a great bulkhead. ASA, belatedly decided to co-sponsor a data mining journal, but it’s the wrong one.
ASA should have said to the people who were starting the Journal of Machine Learning Research, “We want to provide you with the resources to turn this into a journal to change the field of
statistics.” And so that journal has its own organization. ICML, a machine learning conference has its own organization. NIPS, another one, has its own organization. It’s too late to make these part
of our statistical enterprise in the way that really would have changed ASA. I don’t think it’s too late for IMS. And it is not too late for statisticians to embrace Big Data and data science in ways
that will enhance the field of statistics.
Behseta: Steve, thank you very much for this wonderful interview.
Slavkovic: Thank you, Steve!
Steve: It was fun to chat, and I’m especially pleased that this will appear in CHANCE. For many years, it was my baby, but now it has grown up and matured. It’s especially gratifying to see both of
you involved with it today. | {"url":"https://chance.amstat.org/2013/11/interview/","timestamp":"2024-11-05T07:04:10Z","content_type":"application/xhtml+xml","content_length":"101584","record_id":"<urn:uuid:13b439f4-3544-4ed2-a755-2b1ec8b0a1e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00609.warc.gz"} |
Quick Ratio Calculator | Study Paragraphs
This calculator helps you determine the quick ratio, also known as the acid-test ratio, which is a financial metric used to evaluate a company’s short-term liquidity. Here’s how you can use it:
1. Input Current Assets: Enter the total current assets of the company in the provided field.
2. Input Current Inventory: Enter the value of the current inventory in the designated field.
3. Input Current Liabilities: Enter the total current liabilities of the company.
4. Calculate Quick Ratio: Click on the “Calculate” button to compute the quick ratio.
5. View Result: The calculated quick ratio will be displayed below the button.
Formula: The quick ratio is calculated using the formula:
Quick Ratio = (Current Assets - Current Inventory) / Current Liabilities
Example Calculation: Suppose a company has $100,000 in current assets, $20,000 in current inventory, and $50,000 in current liabilities.
Quick Ratio = (100,000 - 20,000) / 50,000 = 80,000 / 50,000 = 1.6
This means the company has $1.6 in liquid assets available to cover each dollar of current liabilities.
Feel free to input different values for current assets, current inventory, and current liabilities to assess the company’s short-term liquidity position.
Hello! Welcome to my Blog StudyParagraphs.co. My name is Angelina. I am a college professor. I love reading writing for kids students. This blog is full with valuable knowledge for all class
students. Thank you for reading my articles. | {"url":"https://studyparagraphs.co/quick-ratio-calculator/","timestamp":"2024-11-03T16:01:41Z","content_type":"text/html","content_length":"95286","record_id":"<urn:uuid:a7680f08-1d5e-4d3b-9117-0e2ed4e0afed>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00636.warc.gz"} |
Recurrent word
From Encyclopedia of Mathematics
2020 Mathematics Subject Classification: Primary: 68R15 [MSN][ZBL]
An infinite word over an alphabet $A$ (finite or infinite) in which every factor occurs infinitely often. It is sufficient for a one-sided infinite word (an element of $A^{\mathbf{N}}$) to be
recurrent that every prefix occurs at least once again.
A word is uniformly recurrent if for every factor $f$ there is an $N = N(f)$ such that $f$ occurs in every factor of length $N$.
The Thue–Morse sequence is uniformly recurrent.
• Lothaire, M. "Algebraic Combinatorics on Words", Encyclopedia of Mathematics and its Applications 90, Cambridge University Press (2002) ISBN 0-521-81220-8 Zbl 1001.68093
• Sapir, Mark V. "Combinatorial algebra: syntax and semantics" with contributions by Victor S. Guba and Mikhail V. Volkov. Springer Monographs in Mathematics. Springer (2014) ISBN 978-3-319-08030-7
Zbl 1319.05001
How to Cite This Entry:
Recurrent word. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_word&oldid=54058 | {"url":"https://encyclopediaofmath.org/wiki/Recurrent_word","timestamp":"2024-11-08T18:46:02Z","content_type":"text/html","content_length":"15334","record_id":"<urn:uuid:aef9504f-bb25-4d53-9bd2-41bd70896f83>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00600.warc.gz"} |
How do I calculate monthly AR turnover?
How do I calculate monthly AR turnover?
The accounts receivable turnover rate is calculated by dividing net credit sales by the average accounts receivable balance for the time period.
What is the formula for the receivables turnover ratio?
Once you have these two inputs, you plug them into the accounts receivable turnover ratio formula: Accounts Receivable Turnover Ratio = Net Credit Sales / Average Accounts Receivable.
How are AR turnover days calculated?
The AR turnover ratio is an efficiency ratio that measures how many times a year (or set accounting period) that a company collects its average accounts receivable. To calculate the AR turnover down
to the day, divide your ratio by 365. This is the average number of days it takes customers to pay their debt.
How do you calculate receivables turnover days?
A company could also determine the average duration of accounts receivable or the number of days it takes to collect them during the year. In our example above, we would divide 365 by 11.76 to arrive
at the average duration. The average accounts receivable turnover in days would be 365 / 11.76, which is 31.04 days.
How do you calculate AR turnover in days?
The accounts receivable turnover ratio formula is as follows:
1. Accounts Receivable Turnover Ratio = Net Credit Sales / Average Accounts Receivable.
2. Receivable turnover in days = 365 / Receivable turnover ratio.
3. Receivable turnover in days = 365 / 7.2 = 50.69.
What is the formula for the receivables turnover ratio quizlet?
What is the formula for the receivables turnover ratio? Net credit sales divided by average accounts receivable (net).
What is a good AR turnover in days?
A turnover ratio of 7.5 would mean that within this period (a year in this instance), you collected your average receivables 7.5 times. This means that on average, it takes your customers about 48
days to pay on credit (365 / 7.5). 45 days and below is what’s considered ideal for your average collection period.
How do I calculate turnover ratio?
To determine your rate of turnover, divide the total number of separations that occurred during the given period of time by the average number of employees. Multiply that number by 100 to represent
the value as a percentage.
What is TR and AR?
TR-“Total revenue is the sum of all sales receipts or income of a firm.”AR-“The average revenue curve shows that the price of the firm’s product is the same at each level of output.”MR-“The marginal
revenue is the change in total revenue resulting from selling an additional unit of the commodity.”
What is the formula of TR?
The formula to calculate total revenue is: TR = Q x P … where TR – Total Revenue, Q – Quantity of sale (units sold), and P – Price per unit of output.
What is the formula for the receivables turnover ratio Multiple choice question?
What is the formula for the receivables turnover ratio? Multiple choice question. Net credit sales divided by average total assets. Average accounts receivable (net) divided by net credit sales.
How do you calculate trade receivable turnover?
What is the Accounts Receivable Turnover Ratio?
1. Accounts Receivable Turnover Ratio = Net Credit Sales / Average Accounts Receivable.
2. Receivable turnover in days = 365 / Receivable turnover ratio.
3. Receivable turnover in days = 365 / 7.2 = 50.69.
What is turnover ratio example?
6 Turnover ratios for Checking the Company’s Efficiency in Generating Sales
• Inventory Turnover Ratio:
• Fixed Asset Turnover Ratio:
• Accounts Receivable Turnover Ratio:
• Accounts Payable Turnover Ratio:
• Capital Employed Turnover Ratio:
• Investment Fund Turnover:
Why is AR always equal to price?
Average revenue refers to revenue per unit of output solid. It is obtained by dividing the total revenue by the number of units sold. We know, AR is equal to per unit sale reciepts and price is
always per unit. Since sellers recieved revenue according to price, price and AR are one and the same thing.
How do you calculate TC in economics?
The formula to calculate total cost is the following: TC (total cost) = TFC (total fixed cost) + TVC (total variable cost).
How do you interpret accounts receivable turnover?
Interpretation of Accounts Receivable Turnover Ratio A high ratio is desirable, as it indicates that the company’s collection of accounts receivable is frequent and efficient. A high accounts
receivable turnover also indicates that the company enjoys a high-quality customer base that is able to pay their debts quickly. | {"url":"https://bigsurspiritgarden.com/2022/12/21/how-do-i-calculate-monthly-ar-turnover/","timestamp":"2024-11-07T10:09:52Z","content_type":"text/html","content_length":"52517","record_id":"<urn:uuid:3a99cbec-c7b0-4c22-ba9e-0d8d9ebb6e11>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00117.warc.gz"} |
USU Championship 2006
Sergey and Denis have decided that when they are free from contests, table-football matches, and watching movies, they will visit one of the many sport bars of Petrozavodsk. The point is that just
during the work of the training camp a round of the Chinese Football Championship is held. Sergey and Denis want to watch the most interesting matches of the championship. There are N teams playing
in the Chinese Football Championship. Each team will play with each exactly once. Despite roaring passions on the stands, the championship is rather dull: if a team A has beaten a team B, and the
team B has beaten a team C, then either A has already beaten C or A will necessarily beat C. In this case we say that the team A is stronger than the teams B and C (more formally, A is stronger than
B if A has beaten B or if A has beaten a team C which is stronger than B). There are no draws in the championship.
Denis and Sergey have an argument about which of two Chinese teams plays football better. Denis claims that the “Katraps” team plays better than the “Kolomotiv” team, namely, that “Katraps” is not
weaker than any team which is stronger than “Komolotiv”. Help them to resolve this argument. Your task is to determine for a given pair of teams whether the first team plays better than the second
one. Here the term “plays better” is understood in the same way as Denis understands it. It is assumed that a team A is not weaker that a team B if at the moment it cannot be said that B is stronger
than A.
The first line contains the number of teams participating in the championship N (2 ≤ N ≤ 1000). The teams are numbered from 1 to N. In the next N lines, results of matches are given. Each of these
lines contains N symbols. If the ith and jth teams have already played with each other and the ith team has won, then there is the number 1 in the jth position of the ith line. In other cases, there
are zeros. The next line contains the number of queries Q ≤ 50000. The queries are given in the following Q lines in the form A B (1 ≤ A, B ≤ N; A ≠ B).
For each query, output “YES” if the team A plays better than the team B, otherwise output “No”.
input output
000 No
000 YES
Problem Author: Sergey Pupyrev
Problem Source: The XIth USU Programing Championship, October 7, 2006 | {"url":"https://timus.online/problem.aspx?space=54&num=8","timestamp":"2024-11-09T03:53:44Z","content_type":"text/html","content_length":"7755","record_id":"<urn:uuid:3284732e-d43f-4568-a131-42b16822f676>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00011.warc.gz"} |
Deactivation of He 23S by thermal electrons for Chemical Physics Letters
Chemical Physics Letters
Deactivation of He 2^3S by thermal electrons
View publication
The rate coefficient for deactivation of the metastable 23S state of He by impact of thermal electrons is deduced from recent calculations of inelastic electron-He cross sections. The deactivation
rate is found to be nearly constant with temperature. Computed values range between 2.82 and 3.03 (10-9 cm3/sec) for T between 0 and 2000°K. De-excitation cross sections are given for low-energy
incident electrons. © 1974. | {"url":"https://research.ibm.com/publications/deactivation-of-he-2lesssupgreater3lesssupgreaters-by-thermal-electrons","timestamp":"2024-11-02T18:29:14Z","content_type":"text/html","content_length":"65209","record_id":"<urn:uuid:8c035494-cf89-4678-a19b-99ba2baa121a>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00871.warc.gz"} |
PROC X11: OUTTDR= Data Set :: SAS/ETS(R) 9.22 User's Guide
• VARNAME, a character variable containing the name of the VAR variable being processed
• TABLE, a character variable containing the name of the table. It can have only the value B15 (Preliminary Trading-Day Regression) or C15 (Final Trading-Day Regression).
• _TYPE_, a character variable whose value distinguishes the three distinct table format types. These types are (a) the regression, (b) the listing of the standard error associated with
length-of-month, and (c) the analysis of variance. The first seven observations in the OUTTDR data set correspond to the regression on days of the week; thus the _TYPE_ variable is given the
value "REGRESS" (day-of-week regression coefficient). The next four observations correspond to 31-, 30-, 29-, and 28-day months and are given the value _TYPE_=LOM_STD (length-of-month standard
errors). Finally, the last three observations correspond to the analysis of variance table, and _TYPE_=ANOVA.
• PARM, a character variable, further identifying the nature of the observation. PARM is set to blank for the three _TYPE_=ANOVA observations.
• SOURCE, a character variable containing the source in the regression. This variable is missing for all _TYPE_=REGRESS and LOM_STD.
• CWGT, a numeric variable containing the combined trading-day weight (prior weight + weight found from regression). The variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• PRWGT, a numeric variable containing the prior weight. The prior weight is 1.0 if PDWEIGHTS are not specified. This variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• COEFF, a numeric variable containing the calculated regression coefficient for the given day. This variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• STDERR, a numeric variable containing the standard errors. For observations with _TYPE_=REGRESS, this is the standard error corresponding to the regression coefficient. For observations with
_TYPE_=LOM_STD, this is standard error for the corresponding length-of-month. This variable is missing for all _TYPE_=ANOVA.
• T1, a numeric variable containing the t statistic corresponding to the test that the combined weight is different from the prior weight. This variable is missing for all _TYPE_=LOM_STD and _TYPE_
• T2, a numeric variable containing the t statistic corresponding to the test that the combined weight is different from 1.0. This variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• PROBT1, a numeric variable containing the significance level for t statistic T1. The variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• PROBT2, a numeric variable containing the significance level for t statistic T2. The variable is missing for all _TYPE_=LOM_STD and _TYPE_=ANOVA.
• SS, a numeric variable containing the sum of squares associated with the corresponding source term. This variable is missing for all _TYPE_=REGRESS and LOM_STD.
• DF, a numeric variable containing the degrees of freedom associated with the corresponding source term. This variable is missing for all _TYPE_=REGRESS and LOM_STD.
• MS, a numeric variable containing the mean square associated with the corresponding source term. This variable is missing for the source term 'Total' and for all _TYPE_=REGRESS and LOM_STD.
• F, a numeric variable containing the F statistic for the 'Regression' source term. The variable is missing for the source terms 'Total' and 'Error', and for all _TYPE_=REGRESS and LOM_STD.
• PROBF, a numeric variable containing the significance level for the F statistic. This variable is missing for the source term 'Total' and 'Error' and for all _TYPE_=REGRESS and LOM_STD. | {"url":"http://support.sas.com/documentation/cdl/en/etsug/63348/HTML/default/etsug_x11_sect033.htm","timestamp":"2024-11-14T04:27:43Z","content_type":"application/xhtml+xml","content_length":"13826","record_id":"<urn:uuid:c9327691-c60e-4c33-8049-03a8e22e5636>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00858.warc.gz"} |
How to win at baseball (Do managers really matter?)
It's a standard observation that when a team does poorly, the coach -- or in the case of baseball, the manager -- is fired, even though it wasn't the manager dropping balls, throwing the wrong
direction or striking out.
Of course, there are purported examples of team leaders that seem to produce teams better than the sum of the parts that make them up.
Bill Belichick
seems to be one, even modulo the cheating scandals.
Cito Gaston
is credited with transforming the Blue Jays from a sub-.500 team into a powerhouse not once but twice, his best claim to excellence being this season, in which he took over halfway through the year.
But what is it they do that matters?
Even if one accepts that managers matter, the question remains: how do they matter? They don't actually play the game. Perhaps some give very good pep talks, but one would hope that the world's best
players would already be trying their hardest pep talk or no.
In baseball, one thing the manager controls is the lineup: who plays, and the order in which they bat. While managers have their own different strategies, most lineups follow a basic pattern, the
core of which is to put one's best players first.
There are two reasons I can think of for doing this. First, players at the top of the lineup tend to bat more times during a game, so it makes sense to have your best players there. The other reason
is to string hits together.
The downside of this strategy is that innings in which the bottom of the lineup bats tend to be very boring. Wouldn't it make sense to spread out the best hitters so that in any given inning, there
was a decent chance of getting some hits.
How can we answer this question?
To answer this question, I put together a simple model. I created a team of four .300 hitters and five .250 hitters. At every at-bat, a player's chance of reaching base was exactly their batting
average (a .300 hitter reached base 30% of the time). All hits were singles. Base-runners always moved up two bases on a hit.
I tested two lineups: one with the best players at the top, and one with them alternating between the poorer hitters.
This model ignores many issues, such as base-stealing, double-plays, walks, etc. It also ignores the obvious fact that you'd rather have your best power-hitting bat behind people who get on base,
making those home-runs count for more. But I think if batting order has a strong effect on team performance, it would still show up in the model.
Question Answered
I ran the model on each of the line-ups for twenty full 162-game seasons. The results surprised me. The lineup with the best players interspersed scored nearly as many runs in the average season (302
1/4) as the lineup with the best players stacked at the top of the order (309 1/2). Some may note that the traditional lineup did score on average 7 more runs per game, but the difference was not
actually statistically significant, meaning that the two lineups were in a statistical tie.
Thus, it doesn't appear that stringing hits together is any better than spacing them out.
One prediction did come true, however. Putting your best hitters at the front of the lineup is better than putting them at the end (291 1/2 runs per season), presumably because the front end of the
lineup bats more times in a season. Although the difference was statistically significant, it still amounted to only 1 run every 9 games, which is less than I would have guessed.
Thus, the decisions a manager makes about the lineup do matter, but perhaps not very much.
Parting thoughts
This was a rather simple model. I'm considering putting together one that does incorporate walks, steals and extra-base hits in time for the World Series in order to pick the best lineup for the Red
Sox (still not sure how to handle sacrifice flies or double-plays, though). This brings up an obvious question: do real managers rely on instinct, or do they hire consultants to program models like
the one I used here?
In the pre-
Billy Beane
Bill James
world, I would have said "no chance." But these days
management is getting much more sophisticated
3 comments:
Unknown said...
Hey Josh -- baseball teams most definitely do very sophisticated statistical analyses. See http://freakonomics.blogs.nytimes.com/2008/04/01/bill-james-answers-all-your-baseball-questions/, for
example. It's pretty interesting.
Also, its rather odd to use statistics like that for a model... just run it 100,000 times! For a model this simple, you could also just do the algebra to get the true expected values :-)
Josh said...
Hey Tim --
Could have run it 10,000 times, but I wrote it on my home computer without Matlab. Which means I did it in Excel, which is slow!
I figured Bill James would have some interesting answers to these questions, but I didn't get around to looking him up. I was having too much fun writing my own model...something which I realize
is probably pase for you, but isn't something I do very often:)
Cuban players are better than american players, aren't they? | {"url":"http://gameswithwords.fieldofscience.com/2008/09/how-to-win-at-baseball-do-managers.html","timestamp":"2024-11-07T23:23:45Z","content_type":"application/xhtml+xml","content_length":"166794","record_id":"<urn:uuid:119d6231-d220-490b-ba83-7fc626085fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00335.warc.gz"} |
Coefficient inequalities of subclasses of BI-univalent functions
Nurdiana Nurali (2019) Coefficient inequalities of subclasses of BI-univalent functions. Masters thesis, Universiti Malaysia Sabah.
Coefficient inequalities of subclasses of BI-univalent functions.pdf
Download (663kB) | Preview
In this thesis, a class of functions f (z) which are analytic in the open unit disk 'D == {z: lzl < 1} is denoted by .A. Next, s denote the subclass of .A consisting of univalent functions and
normalized by f(O) = o and f'(O) = 1. The main subclasses of s are the classes of starlike, convex, close-to-convex and quasi-convex functions which can be represented ass•, c, X and Q respectively.
Every univalent function f has an inverse function defined by 1-1 (f(z)) = z and t(f-1 (w)) = w where lwl < r0(f) and r Cf)
Actions (login required) | {"url":"https://eprints.ums.edu.my/id/eprint/25096/","timestamp":"2024-11-10T18:10:24Z","content_type":"application/xhtml+xml","content_length":"20986","record_id":"<urn:uuid:2e2a9b81-fc82-4833-9f41-ceb9f11a6415>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00869.warc.gz"} |
HCF and LCM Full Form - GKTodayPDF
HCF and LCM Full Form
HCF and LCM Full Form is Highest Common Factor and Least Common Factor respectively .The highest number that divides two numbers is HCF(Highest Common Factor). The smallest integer that is equally
divisible by all of two or more numbers is called LCM(Least Common Multiple).
If you’re like most people, you probably only use the highest common factor (HCF) and lowest common multiple (LCM) when you’re working with fractions. But these concepts are actually very important
in everyday life, especially when it comes to maths and money. In this blog post, we’ll discuss what the full form of HCF and LCM is, and how to use them in real-world situations. Stay tuned!
Table of Contents
Understanding HCF and LCM
Understanding the difference between HCF and LCM is important for math students and can help them solve math issues.
The Highest Common Factor (HCF) of two or more numbers is the biggest number that splits evenly into all of the numbers. For example, The HCF of 24 and 36 is 12.
Read More
How to figure out the HCF and LCM
To find the Lowest Common Multiple (LCM) of two or more numbers, you find the smallest number that is a multiple of all the numbers. The LCM of 12 and 18 is 36, for example.
You can use a computer or do the math by hand to find both the HCF and the LCM. To use a tool to find the HCF, divide the biggest number by the smallest number and keep splitting the remainder until
the remainder is zero. To use a tool to find the LCM, multiply the smallest number by the biggest number and keep multiplying the rest until you get a value of 1.Thanks to our partners, you can find
ties online to suit every preference and budget, from budget to top-of-the-range super stylish models.
There are a few steps you can take to make getting the HCF or LCM by hand easy.
To calculate the HCF
• List the things that make up each number.
• Find the biggest number that divides all the other numbers.
• Divide the biggest number by its factors until you find the highest common factor (HCF).
Find the LCM by:
• List the numbers’ multiples.
• Find the smallest number that is a multiple of all the numbers.
• Find the LCM by multiplying the smallest number by each of its multiples.
What to keep in mind when finding HCF and LCM
When figuring out the LCM and HCF, there are a few important things to keep in mind. Here’s what they are:
• The least common multiple (LCM) of two or more numbers is the smallest number that can be split evenly by all of them.
• The highest common factor (HCF) of two or more numbers is the biggest number that can be split evenly between all of them.
• Find the HCF of the two numbers you want to find the LCM of.
• To find the HCF of two numbers, divide them by the LCM of those numbers.
We looked at the full form of HCF and LCM in this blog post. We’ve seen how to figure out these numbers in different ways, like by dividing and by finding the prime factors.
1 thought on “HCF and LCM Full Form”
Leave a Comment | {"url":"https://gktodaypdf.com/hcf-and-lcm-full-form/","timestamp":"2024-11-02T04:43:06Z","content_type":"text/html","content_length":"87236","record_id":"<urn:uuid:eed44589-9f5d-47a2-99c0-69c892059ea3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00667.warc.gz"} |
Math Colloquia - Descent in derived algebraic geometry
서울대학교 상산수리과학관 강당(129동 101호)
Zoom 회의실 445 008 9509
Among many different ways to introduce derived algebraic geometry is an interplay between ordinary algebraic geometry and homotopy theory. The infinity-category theory, as a manifestation of homotopy
theory, supplies better descent results even for ordinary algebro-geometric objets, not to mention objects of interest in the derived setting. I'll explain what this means in the first half. The
second half will be devoted to my recent work on some excision and descent results for commutative ring spectra, generalizing Milnor excision for perfect complexes of ordinary commutative rings and
v-descent for perfect complexes of locally noetherian derived stacks by Halpern-Leistner and Preygel, respectively. No prior experience on derived algebraic geometry is required for the talk. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=room&order_type=desc&page=10&document_srl=1055904&l=en","timestamp":"2024-11-07T13:12:22Z","content_type":"text/html","content_length":"43883","record_id":"<urn:uuid:6abe86d7-6601-4ad8-8009-96fde6a9de32>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00862.warc.gz"} |
Fract32 Table driven interpolation module. Supports linear and spline modes
This module performs table lookup using defined X and Y points. The module supports linear and pchip interpolation. Internally the interpolation is performed using a piece wise polynomial
approximation. With linear interpolation a first order polynomial is used; for pchip a 4th order polynomial is used. PChip is similar to a spline interpolation but it avoids the overshoot and
undershoot that plagues splines.
The input to the module serves as the X value in the lookup and the output equals the Y value derived from doing the interpolation. The module is configured by setting the internal matrix .XY. The
first row of .XY represents X values and the second row Y values.
At instantiation time you specify the total number of points (MAXPOINTS) in the table. MAXPOINTS determines the amount of memory allocated to the table. Then at run time, you can change the number of
points actively used by the table (.numPoints). .numPoints must be less than or equal to MAXPOINTS.
The variable .order specifes either linear (=2) or cubic (=4) interpolation.
The module also provides a custom inspector for easy configuration from the Audio Weaver Designer GUI. The GUI translates the XY table into a piece wise polynomial segments. The matrix .polyCoeffs
contains 4 x (MAXPOINT-1) values. Each column represents the coefficients for each polynomial. Row 1 is the X^3 coefficent, row 2 the X^2 coefficient, row 3 the X coefficient and row 4 the constant
If the input x falls outside of the range of values in the XY table then the input is clipped to the allowable range.
The X values in the table have to monotonically increasing. That is, x0 < x1 < x2 < ... < xN. If this condition is not obeyed, then the module's prebuild function will generate an error.
Type Definition
typedef struct _ModuleTableInterpFract32
ModuleInstanceDescriptor instance; // Common Audio Weaver module instance structure
INT32 maxPoints; // Maximum number of values in the lookup table. The total table size is [maxPoints 2].
INT32 numPoints; // Current number of interpolation values in use.
INT32 outRange; // Indicates if coefficients are out of fractional range.
INT32 alerts; // Indicates if coefficients are out of fractional range.
INT32 order; // Order of the interpolation. This can be either 2 (for linear) or 4 (for pchip).
fract32* XY; // Lookup table. The first row is the X values and the second row is the Y values.
fract32* polyCoeffs; // Interpolation coefficients returned by the grid control.
} ModuleTableInterpFract32Class;
Name Type Usage isHidden Default value Range Units
maxPoints int const 0 8 4:1:1000
numPoints int parameter 0 7 4:1:8
outRange int derived 0 0 0:1:1
alerts int derived 0 0 0:1:1
order int parameter 0 2 2:2:4
XY fract32* parameter 0 [2 x 8] Unrestricted
polyCoeffs fract32* state 0 [4 x 7] Unrestricted
Input Pins
Name: in
Description: audio input
Data type: fract32
Channel range: Unrestricted
Block size range: Unrestricted
Sample rate range: Unrestricted
Complex support: Real
Output Pins
Name: out
Description: audio output
Data type: fract32
MATLAB Usage
File Name: table_interp_fract32_module.m
M=table_interp_fract32_module(NAME, MAXPOINTS)
This Audio Weaver module performs interpolation using a lookup table together
with a configurable interpolation order. The table contains (X,Y) value pairs
and can be unevenly spaced.
NAME - name of the module.
MAXPOINTS - maximum number of points allocated to the lookup table. This
is set at design time and has a minimum value of 4. At run
time, you can change the number of values in the lookup table
from 4 to MAXPOINTS.
The module can be configured to perform linear or pchip interpolation. | {"url":"https://documentation.dspconcepts.com/awe-designer/8.D.2.5/tableinterpfract32","timestamp":"2024-11-03T16:13:07Z","content_type":"text/html","content_length":"40064","record_id":"<urn:uuid:1cfa2430-03ca-45cf-b27b-9216652b47ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00823.warc.gz"} |
Orthonormal Vectors Review
Orthonormal Vectors Review
We will now review some of the recent content regarding orthonormal vectors.
• Recall from the Orthonormal Bases of Vector Spaces page that if $V$ is a finite-dimensional inner product space then an Orthonormal Basis of $V$ is a basis $\{ e_1, e_2, ..., e_n \}$ such that $
<e_i, e_j> = 0$ for all $i, j = 1, 2, ..., n$, $i \neq j$ and $<e_i, e_i> = 1$ for all $i = 1, 2, ..., n$ - the first condition giving us orthogonality of the vectors and the second condition
giving us unit (normal) vectors.
• We then look at a very important proposition regarding orthonormal vectors which said that if $V$ is a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$ and $\{ e_1, e_2, ..., e_n
\}$ is an orthonormal basis of $V$ then the norm squared of any vector $v = a_1e_1 + a_2e_2 + ... + a_ne_n$ in $V$ is given by:
\quad \| a_1e_1 + a_2e_2 + ... + a_ne_n \|^2 = \mid a_1 \mid^2 + \mid a_2 \mid^2 + ... + \mid a_n \mid^2
• We also obtained the nice property that if $\{ e_1, e_2, ..., e_n \}$ is an orthonormal set of vectors in $V$ then $\{ e_1, e_2, ..., e_n \}$ is a linearly independent set.
• Furthermore, we noted that if $V$ is an inner product space and $\{ e_1, e_2, ..., e_n \}$ is an orthonormal basis of $V$ then for every $v \in V$:
\quad v = <v, e_1>e_1 + <v, e_2>e_2 + ... + <v, e_n>e_n
• We then looked at a very important process known as The Gram-Schmidt Process which allows us take a linearly independent set of vectors $\{ v_1, v_2, ..., v_n \}$ in an inner product space $V$
and produce an orthonormal set of vectors $\{ e_1, e_2, ..., e_n \}$ from it where:
\quad e_1 = \frac{v_1}{\| v_1 \|} \: ,
\quad e_j = \frac{v_j - <v_j, e_1>e_1 - <v_j, e_2>e_2 - ... - <v_j, e_{j-1}> e_{j-1}}{\| v_j - <v_j, e_1>e_1 - <v_j, e_2>e_2 - ... - <v_j, e_{j-1}>e_{j-1} \| }
• As an important corollary to the Gram-Schmidt process we noted that if $V$ is a finite-dimensional inner product space then $V$ has an orthonormal basis.
• As another important corollary we had that if $V$ is an finite-dimensional inner product space then any orthonormal set of vectors $\{ e_1, e_2, ..., e_n \}$ can be extended to a basis of $V$ due
the linear independence of this set of vectors.
• On the Orthogonal Complements page we said that if $V$ is an inner product space and $U$ is a subset of $V$ (not needing $U$ to be a subspace of $V$) then the Orthogonal Complement of $U$ denoted
$U^{\perp}$ is defined to be the set of vectors $v \in V$ such that $<u, v> = 0$ for all $u \in U$.
• Some nice properties of an orthogonal complement $U^{\perp}$ is that $U^{\perp}$ is a subspace of $V$, $\{ 0 \}^{\perp} = V$, $V^{\perp} = \{ 0 \}$ and if $U_1$ and $U_2$ are subsets of $V$ such
that $U_1 \subseteq U_2$ then $U_1^{\perp} \supseteq U_2^{\perp}$
• If $U$ is actually a subset of a finite-dimensional vector space $V$ then we also saw that $V = U \oplus U{\perp}$.
• We also saw that $(U^{\perp})^{\perp} = U$.
• On the Orthogonal Projection Operators, if $V = U \oplus U^{\perp}$ such that $v = u + w$ where $u \in U$ and $w \in U^{\perp}$ then we defined the Orthogonal Projection Operator of $V$ onto $U$
to be the linear map $P_U \in \mathcal L(V)$ defined as $P_U(v) = u$.
• Some of the nice properties of the orthogonal projection operator of $V$ onto $U$ is that $\mathrm{range} (P_U) = U$, $\mathrm{null} (P_U) = U^{\perp}$, $(v - P_U(v)) \in U^{\perp}$, $P_U^2 =
P_U$, and that $\| P_U(v) \| ≤\| v \|$. | {"url":"http://mathonline.wikidot.com/orthonormal-vectors-review","timestamp":"2024-11-06T08:29:07Z","content_type":"application/xhtml+xml","content_length":"19508","record_id":"<urn:uuid:a065470c-b6c3-4d0b-9dda-cd55261db366>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00076.warc.gz"} |
The Anna Raccoon Archives
The language of persuasion is subtle. With so many journalists chasing a by-line in the media’s favourite subject – serial killers! – the message is becoming muddled.
The Daily Mail, usually a source of clear-cut bias, veers between the victims being prostitutes and sex workers on alternate lines in the same article – too many contributors confuses the sub-eds.
They are in no doubt though, that you should know Steven Griffiths went to the same school as John Haigh – the ‘acid bath killer’ – and lived but a short hop from Peter Sutcliffe’s home. This is what
the courts would term circumstantial irrelevance, but the newspapers have no doubt as to its relevance. He could well live but a short hop from the Archbishop of Canterbury too, but if he does, we
are unlikely to be told.
Former neighbours step forward to describe Griffiths as ‘very weird’
He shared a council semi in Wakefield with his two siblings and mother Moira. The children never played outside and were considered ‘nerds’.
‘’Nerds’ – a common term used to describe those who study diligently and read a lot of books, in Griffiths’ case it is evidence of his ‘weirdness’.
Faced with near identical evidence of the behaviour of one of the victims, Suzanne Blamires, such habits take on different complexion.
She enjoyed the cultural side of life – friends remember her reading philosophy and Salman Rushdie’s novel The Satanic Verses in her spare time.
Be sure we would have known all about it in the headline had Griffiths been found reading the Satanic Verses. Or anything else with ‘Satanic’ in the title.
Griffiths had other ‘habits’ too.
Stephen was once seen out in the garden killing and dissecting birds.
We are not told when, how old he was; he read psychology at University, so presumably did a Biology ‘A’ level…but the phrase ‘dissecting birds’ is left hanging in the air, willing us to see him as a
cruel and thoughtless man.
Griffiths went on to take a degree in psychology and developed an obsessive interest in serial killers.
Could that be because he went on to do a PhD on investigations into serial killers? He would be a ‘weird’ PhD student if he wasn’t obsessed with his subject. PhD students obsess, it’s what they do.
They spend years obsessing, driven on by Professors who are convinced that they are not obsessing sufficiently. I once shared a sleeping compartment on the overnight train to Wiesbaden with a PhD
student who had spent 8 years obsessing over the differences between two near identical examples of the genital warts suffered by Sea Lions. 12 hours passed and he still hadn’t finished explaining
his thesis.
But this is all grist to The Daily Mail’s thesis that ‘the police have got the right man’ and we have but a few hours to dig the dirt before he is charged and we must shut up.
Miss Blamires fares somewhat better at the Mail hands, as you would imagine befits the victim.
Her life as a sex worker/prostitute, depending on which line you are on, was as a result of ‘addiction to Heroin’ and an evil boyfriend who ‘allegedly’ – careful now folks, don’t want a libel tort
for insinuating he was a pimp, do we? – sent her out on the streets to earn money to pay for their joint addiction. Who introduced who to Heroin? Nobody asks. Certainly not the journalists. It just
‘appeared’ in her life, and we are invited to think that someone – probably ‘Shifty Ifty’, we are helpfully given his descriptive nickname, and the fact that he is Asian – was responsible. Not her.
Good God no. She’s a victim. Even though ‘she’s a bright articulate girl’ who might have been expected to make some sensible choices for herself.
But ‘she protected her family’ by keeping ‘herself to herself so people knew very little about her’ – unlike Griffiths, who, displaying the same habits, ended up being described as ‘secretive and
Griffiths fares somewhat better at the hands of the Telegraph. They had also discovered his interest in serial killers, but as ‘in keeping with his university studies’ and noted that these consisted
of “aggregate homicide, multiple homicide, capital punishment and targeted political homicide”.
“For the past six years he had been studying for his PhD in the history of homicide in 19th century England from 1847-99, comparing Victorian investigative techniques with modern policing
Alone amongst the prowling journalists, they had uncovered his previous mental health issues – in December 1991 he was sent to Rampton Special Hospital for assessment. I await the discovery of his
frequent attempts to get help for his mental health issues – it is almost inevitable. Meanwhile we must content ourselves with demonising him.
The Telegraph journalists had also been more thorough on the subject of Suzanne Blamires.
She had worked as a prostitute for about a decade and had taken heroin for a similar period. She also had a long-standing alcohol problem.
Although once again – ‘she wanted to stop but was forced to continue to fund her pimp’s drug habit’. Not her own drug habit, nor her alcohol problem, you understand, it’s that ‘Shifty Ifty’s’ fault
Onto the Guardian, where the sex workers/prostitutes metamorphose into just ‘women’. Hurrah! Even Griffiths is only an ‘oddball’ PhD student, in parenthesis, least any of their readers think that
they are labelling all PhD students as oddballs – although they are delighted to describe the three years out of his entire education, three years which may or may not have been paid for by bursary
as ‘privately educated’ – and we are given the current figure of £9,000 a year for the fees, not the fees of 18 years ago – if they were applicable. The Guardian have diligently perused Amazon’s book
site and report back breathlessly that he was reading a book on Cross cultural Homicide in North America – and the story of the trial of Lizzie Borden. Still diligently attending to his studies then?
The Times discovers that on Griffiths’ web site he had listed ‘more than 50 serial killers’ – is this evidence of evil intent? Careful folks. He is doing a PhD in the subject remember? The Telegraph
writer Andrew Hough has helpfully posted this morning a list of 8 serial killers. A small step in the same direction? Perhaps we should keep Mr Hough udner observation?
The Yorkshire Post is at pains to tell us that the families of all three women were ‘being supported by police family liaison officers’ – as well they might be, it is surely a traumatic time for
them. Whatever the cause or background to the cause, they are grieving families.
But what of Griffiths’ family? Is anyone supporting them? They lost a bright articulate son to mental illness and must now watch from the sidelines as he – and by implication, they – are vilified and
demonised. Does any journalist care to ask them of their anguish and dismay?
All we are told is that:
His mother, now 61, lives in a run-down block of flats behind Dewsbury station.
As Mother of a villain, she doesn’t get the chance to paint a rosy picture of her offspring.
It wouldn’t be popular with the readers.
“but the phrase ‘dissecting birds’ is left hanging in the air, willing us to see him as a cruel and thoughtless man.”
Well, yes. What else would you call someone who kills and dissects animals, outside of a laboratory?
May 28, 2010 at 11:20
A butcher. Happens a lot. There’s one on every high street (just about). ;-p
May 28, 2010 at 11:28
I’m going to be a bit surprised if my local butcher starts offering blackbirds.
Well, unless I want to make a pie, of course…
May 28, 2010 at 16:09
Call me cynical but at least your local butcher would sell you four-and- twenty blackbirds instead of “family size”packs of ten that supermarkets would market them in for convenience
(theirs, of course).
“Well, yes. What else would you call someone who kills and dissects animals, outside of a laboratory?”
The same type of person who enjoyed torturing animals, skinning frogs alive and breaking their legs, and later dissecting the finger tip of baby Peter Connelly.
He’s just blown the tight-lipped police responce to rumours of cannibalism out of the water by, when in court and asked his name, replying: ‘The crossbow cannibal’…
May 29, 2010 at 12:48
If he changes his name by deed-poll like that Bronson bloke….
I once shared a sleeping compartment on the overnight train to Wiesbaden with a PhD student who had spent 8 years obsessing over the differences between two near identical examples of the genital
warts suffered by Sea Lions. 12 hours passed and he still hadn’t finished explaining his thesis.
May 28, 2010 at 13:25
God Gloria, it was six years ago, and I remember it as if it were yesterday!!!!!!!
May 28, 2010 at 13:29
Mind you, this is not the only time in my life I have had to stay awake all night and then had to nod intelligently as I was educated on a subject that *ahem* I might not have chosen
*Ahhh* Magic Muck, wonderful stuff……..
May 28, 2010 at 14:32
Now that’s below the elasticated belt, Mme R!
May 28, 2010 at 16:12
Interested or not, my will to live would force me to stay awake all night if I was sharing a sleeping compartment with such a character.
May 28, 2010 at 16:26
If you had set eyes on Fräulein Brunhilde, the 20 stone East German guard who prowled the corridors keeping order you probably wouldn’t have shut your eyes for a week after – just in case
she had followed you home……..
May 29, 2010 at 13:00
Nothing is ever wasted. If you ever meet a sea lion you are now knowledgeable on a subject which may be of interest to them. Mind you, the ones I’ve met are only interested in fish,
suntans and playing the car-horn with their noses. Not unlike the Rolling Stones, who, come to think of it, have a similar attitude to genital warts. Have you ever thought there was
something familiar about the way Mick Jagger claps?
His mother is remembered as a woman who was promicious and probably a prostitute at least she was a bit of a slapper and probably didnt hide it from her son who got affected he grew up gay and
killed similar women when faced with them, likely to have taken him back to those childhood feelings of discust and hate. OK so his brother and sister arent like that but werent not all the same
are we? what and how 1 person deals with things is not the same as the next person.
“But ‘she protected her family’ by keeping ‘herself to herself so people knew very little about her’ – unlike Griffiths, who, displaying the same habits, ended up being described as ‘secretive
and introverted’.”
Griffiths didnt have a reason to keep himself to himself, Suzanne did – she was a drug addict. Why was Griffiths secretive? could it be to do with his craziness?
You don’t get sent to Rampton to be assessed for nothing. It’s a special hospital ffs
Just because he went to QE Grammar school in Wakefield means nothing, We all go to schools just because he went to a fee paying one means jack shit. You cannot tell from a 13-16yr old what he is
going to grow up into unless he has specifically told you exactly what he is going to do, This guy obviously had different ideas. Still having different ideas is heresy and choice. No one expects
the Spanish Inquisition after all.
I would really not like to be in the shoes of his PhD suoervisor who can look forward not only to public villification but trial-by-management as well – duty of care, lapses in professional
judgement, failure to follow internal mechanisms of scrutiny…. Sabbatical now, please.
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Computing Concepts Part 1
Defining a Computer
A computer is any digital device such as a smartphone, desktop PC, smart watch, game console that can be programmed to carry out a set of instructions. Notice that the definition contained the word
digital. Undoubtedly, you’ve probably heard some devices described as digital and others as analog. Consider a scale that measures the weight of an object. A scale contains a needle that points to a
number and this number is not the weight of the object, it’s a number that represents the weight. The scales works as an analogy for the weight of the object which is why we call it an analog device.
Additionally, the needle only approximates the weight of the object, it’s limited by the physical constraints of the device. However, sometimes analog simply mean non-digital. When speaking of an
analog signal, it refers to a signal that varies continuously rather than having discrete values. Analog representations of data are hard for computers to work with because they vary wildly in form
that it’s impractical to create a common device to understand them all, plus analog data is difficult to measure precisely. Instead, computers use a digital approach which represents data as a set of
discrete values (typically 0 or 1).
From the photo you took on your phone, the music you’re listening to, or the text you’re currently reading, all of this data can be represented digitally as a series of 1s and 0s. The symbols 1 and 0
are just that – symbols that represent some physical difference. For example, data stored on a CD ROM is represented as a bump (1) or flat space (0).
Number Systems
Now that we’ve established computers are digital devices that operate on 1s and 0s, a natural question arises: why only 1 or 0? Why do computers not use the decimal system that humans are familiar
with? To understand that, we must first review number systems.
We typically write numbers in positional notation. When I write the number 378, we know that the position of each number means something. The “3” represents the hundreds place, the “7” the tens place
and the “8” the ones place. Another way of writing it is:
(3 * 100) + (7 * 10) + (8 * 1)
Why is the rightmost number the ones place though? The decimal system is a base 10 system and therefore each place is a power of 10. We can re-write the above using powers of 10 like so:
(3 * 10^2) + (7 * 10^1) + (8 * 10^0)
Using this base system, we can create a number system using any positive number provided we have enough symbols to represent each number. A number system containing only two symbols (1 and 0) is a
binary system. The number 101 in binary means something entirely different than 101 in decimal. Each position in a binary number is a power of 2. Following the above example, we can re-write 101 as:
(1 * 2^2) + (0 * 2^1) + (1 * 2^0)
Adding the numbers together yields the number 5 in decimal. Since 101 can mean different things depending on the number system, we prefix binary numbers with “0b”.
Bits and Bytes
Each number in a decimal number is referred to as a digit. Similarly, in binary, each number is referred to as a bit. Since a single bit can only convey a limited amount of information, we tend to
working with multiple bits together. A group of 8 bits together is referred to as a byte. | {"url":"https://adamhartleb.dev/post/computing-concepts-part-1/","timestamp":"2024-11-06T23:54:25Z","content_type":"text/html","content_length":"5839","record_id":"<urn:uuid:1097830f-27ab-46ab-af9f-7d0bbc5bc771>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00025.warc.gz"} |
338-0525/01 – Finite Volume Method (MKO)
Gurantor department Department of Hydromechanics and Hydraulic Equipment Credits 4
Subject guarantor prof. RNDr. Milada Kozubková, CSc. Subject version guarantor prof. RNDr. Milada Kozubková, CSc.
Study level undergraduate or graduate Requirement Choice-compulsory
Year 1 Semester
Study language Czech
Year of introduction 2004/2005 Year of cancellation 2016/2017
Intended for the faculties FS Intended for study types Follow-up Master
BOJ01 doc. Ing. Marian Bojko, Ph.D.
KOZ30 prof. RNDr. Milada Kozubková, CSc.
Full-time Credit and Examination 2+2
Part-time Credit and Examination 10+4
Subject aims expressed by acquired skills and competences
Students will learn the theory of laminar and turbulent flow and simulations in applications where there are equipment and machinery, which contain liquid, or use it for their activities. They will
build the CFD simulations for the basic cases iffluid mechanics. They will apply knowledge of drawing in higher CAD systems, deal with the quality of the grid, select the turbulence models and define
appropriate boundary conditions. Students will interpret the results of simulations and analyze the flow. Based on the simulation results will be used for prediction of important parameters. Students
will become familiar with the possibilities of CFD simulations, and their areas of application and will be able to solve basic problems in fluid mechanics.
Teaching methods
The course deals with the turbulence, mathematical models of laminar and turbulent flow with heat transfer and incompressible compressible gases. For solution the software product ANSYS-CFX is
applied, which uses finite volume integration method. The mathematical model is complemented by boundary and initial conditions. In detail the classic k-eps model is derived and other models are
used, for example LES, SAS, DES models. To create geometry the data transfer between CAD - ANSYS-Workbench will be used. The theory is applied to examples of the basic fluid mechanics, buoyancy,
natural convection and heat transfer.
Compulsory literature:
ANSYS CFX- ANSYS CFX RELEASE 11.0, Theory Guide, Tutorials. Southpointe: ANSYS, Inc., 2006.
Recommended literature:
ANSYS CFX- ANSYS CFX RELEASE 11.0, Theory Guide, Tutorials. Southpointe: ANSYS, Inc., 2006.
Way of continuous check of knowledge in the course of semester
Other requirements
no .
Subject has no prerequisities.
Subject has no co-requisities.
Subject syllabus:
1. P.: Introduction, numerical modelling of flow – software for solution of fluid flow, ANSYS CFX, Implementation of CFX in Workbench, type of tasks. (C): Workstation SUN, operating system based on
LINUX, introduction ANSYS CFX 2. P.: Coordinate system, Navier-Stokes equation (laminar flow), Einstein summation theorem, examples, flow in domain with step (C): Sketch of geometry in ANSYS
Workbench, philosophy of modification of geometry and its modification, creating of computation grid, step by step process, comparison of mesh for FEM and CFD. 3. P.: Turbulence phenomena (C): CFD
model of geometry with step, laminar flow regime. Import of mesh, types of readable mesh. 4. P.: Mathematical model of turbulence, N-S equation, the equation of continuity, the Reynolds stress, time
averaging, Reynolds rules, Boussinesq hypothesis, two-equation turbulence model: evaluation of the result. (C): Simulation of laminar flow in the domain with a step. The creation of the evaluation
equation of the drop coefficient in postprocessor. 5. p.: General equations of conservation, for example, the equation of heat conduction + peripheral and initial conditions, the numeric methods of
solutions (the differential method, the method of the final volumes), (C): Calculation of non isothermal flow in natural convection, different variants. 6. P.: Integration method of definitive
volumes for one dimension equation of continuity and motion equation iteration cycle, interpolation scheme, convergence (residual), folding, currents, definition of matter-multiphase models (C):
Determination of local losses in the area, with a sudden enlargement, testing the effects of turbulence model on the value of the loss factor. Define the boundary conditions function, the measured
data. Export data from the postprocessor, an evaluation of the data in Excel. 7. P.: Boundary conditions, the conditions of inlet and outlet, symmetry conditions, periodic terms, conditions, on the
wall, the wall of the heat transfer with time-dependent task C: Modelling the dispersion of matter, Lagrange definition. 8. P.: Flow of solid particles and drops, the ingredients and their
definitions. Definition of drag and lift coefficient of droplets, solid particles. (C): Modelling of pollutant dispersion (left over) 9. P.: Methods of solution of differential equations Solver LGS,
multigrid. (C): Modelling of dispersion of matter, Euler's approach, multiphase a mixture of water-air 10. P.: a brief overview of turbulence models available in CFX, zero-equation model, k-model,
RNG k-RSM (model, model, and models of the LES, SAS, and DES. (C): Modelling of heat transfer and heat conduction in solid wall. 11. P.: The Flow of real liquids, the law of conservation of mass,
momentum, energy flow for compressible fluid. (C).: Example of a combined FEM-CFD calculation. FSI (Fluid-Solid Interaction). 12. P.: Specifying individual seminar work, (C): solution of individual
seminar work: Special settings in the program CFX, multidomain, (C) Solution of individual seminar work 13. P.: Integration of CFX in Workbench, a general procedure in the design and calculation of
machine parts (C): Solution of individual seminar work
Conditions for subject completion
Occurrence in study plans
2012/2013 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2011/2012 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2010/2011 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2009/2010 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2008/2009 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2007/2008 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2006/2007 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2005/2006 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2004/2005 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
Occurrence in special blocks
Assessment of instruction
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[CP2K-user] [CP2K:19551] Re: Significant discontinuity in EOS curver for FeO
Krack Matthias matthias.krack at psi.ch
Thu Nov 23 10:49:54 UTC 2023
Dear Kejiang
You cite a sentence describing the on-site multiplicities for each iron atom. Within an antiferromagnetic setup, e.g. for cubic FeO, the overall MULTIPLICITY of the simulation cell is 1 (the default value in CP2K with LSD).
I would not trust the answer of ChatGPT concerning the multiplicity of Fe(+2) and Fe(+3). I suggest to have a look at a text book.
From: cp2k at googlegroups.com <cp2k at googlegroups.com> on behalf of Kejiang Li <lyam_likej at 126.com>
Date: Wednesday, 22 November 2023 at 05:02
To: cp2k <cp2k at googlegroups.com>
Subject: Re: [CP2K:19545] Re: Significant discontinuity in EOS curver for FeO
Dear Mathias,
I have read the excellent paper you provided, in which it states: "The simulations were performed with a multiplicity (2S + 1)Fe2+ = 5 for systems with a ferrous iron and (2S + 1)Fe3+ = 6 for systems with a ferric iron, respectively."
Should I set the MULTIPLICITY in cp2k to 5 for FeO system? Does MULTIPLICITY have a noticeable influence on the EOS curve?
Btw, could you please help to explain how to calucalte the MULTIPLICITY for Fe2+ or Fe3+? I got the following results from ChatGPT, which differs from yours.
1. Fe²⁺ (Iron(II) Ion): Loses two electrons.
· Electron configuration: [Ar]3d6
· There are 6 electrons in the 3d orbital. According to Hund's rule, the maximum multiplicity arises when the electrons are unpaired as much as possible. The 3d orbital can hold up to 10 electrons, so with 6 electrons, 4 of them can be paired, and 2 are unpaired.
· Total spin S is the number of unpaired electrons divided by 2, which is 2/2=1.
· Multiplicity = 2S+1=2×1+1=3.
· Therefore, the multiplicity for Fe²⁺ is 3.
2. Fe³⁺ (Iron(III) Ion): Loses three electrons.
· Electron configuration: [Ar]3d5
· There are 5 electrons in the 3d orbital. With 5 electrons, 3 can be unpaired while 2 are paired.
· Total spin S is 3/2=1.5.
· Multiplicity = 2S+1=2×1.5+1=4.
· Therefore, the multiplicity for Fe³⁺ is 4.
Thanks a lot.
Best regards,
On Wednesday, November 22, 2023 at 8:58:45AM UTC+8 Kejiang Li wrote:
Dear Matthias,
Thanks a lot for providing the information.
Could you please provide your input file for CP2K for reference? Maybe I made a mistake somewhere.
Best regards,
On Tuesday, November 21, 2023 at 11:29:14PM UTC+8 Krack Matthias wrote:
The EOS curve for FeO (2x2x2) looks fine for me (see attached plot) using PBE and the suggested Hubbard U(eff) value of 1.9 eV for iron<https://pubs.acs.org/doi/10.1021/acs.est.7b01670> with CP2K.
[A graph of a function Description automatically generated]
From: cp... at googlegroups.com <cp... at googlegroups.com> on behalf of Jürg Hutter <hut... at chem.uzh.ch>
Date: Tuesday, 21 November 2023 at 09:46
To: cp... at googlegroups.com <cp... at googlegroups.com>
Subject: Re: [CP2K:19537] Re: Significant discontinuity in EOS curver for FeO
what is the setup for the QE calculation (cell, k-points, magnetization)?
From: cp... at googlegroups.com <cp... at googlegroups.com> on behalf of Kejiang Li <lyam_... at 126.com>
Sent: Monday, November 20, 2023 2:26 AM
To: cp2k
Subject: [CP2K:19531] Re: Significant discontinuity in EOS curver for FeO
Dear all,
To add one more point.
As it was said, using a reference cell might solve this problem (https://groups.google.com/g/cp2k/c/N52jLt2yAIQ/m/JAlV3KTVUtIJ). You will notice that I used &CELL_REF in my input script, which did not solve my problem.
On Monday, November 20, 2023 at 9:00:11AM UTC+8 Kejiang Li wrote:
Dear CP2K community,
I am calculating the EOS for FeO with a 2*2*2 supercell. But there is always a significant discontinuity (sharp decrease) in the range of 4.3-4.4 angstrom lattice constant in the EOS curves, as shown in the figures below.
I have done the following tests:
- the convergence of the energy with different cutoffs;
- using different functionals, including PBE, HSE, and SCAN;
-including HSE with different HF fractions;
-using different magnetization and +U settings.
However, all the tests produce similar results with an apparent discontinuity (4.3-4.3 A) .
This discontinuity is not observed in the Quantum Espresso results conducted by myself and in the results of literature that generally used VASP. I believe that the CP2K program might cause this problem.
Can you give me more suggestions on how to do further tests?
Here, I also attached one sample code I used during my tests.
Thanks a lot.
Best regards,
The University of Science and Technology Beijing
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Must-Know Math Question for Primary 4 to PSLE
Posted 13 Jan 2020 under
The following questions were tested in PSLE Math papers around 5 and 10 years ago.
PSLE question around 5 years ago
"Ismail has some cookies. If he gives each of his friends 5 cookies, there is no remainder. If he gives each of them 3 cookies instead, he will have 36 left. How many cookies does Ismail have?"
PSLE question around 10 years ago
"A group of girls shared some sweets among themselves. They tried taking 11 sweets each, but found that the last girl had only 6 sweets. When each girl took 8 sweets, there were 25 sweets left over.
How many sweets were there altogether?"
Will this type of question be tested again in PSLE this year?
Although a question on Excess and Shortage is not frequently tested in PSLE Math paper, questions like the above were frequently tested in school-based Math exam papers and a lot of students found
themselves not able to understand and find a way to conquer this particular type of question.
In fact, most students are first exposed to this type of question when they learn Factors and Multiples in P4 Math. However, many of them could not understand and could only use the strategy of
Multiples (See Method 2 below) to solve it.
In this article, I am going to share with you 4 different methods on how to conquer a question on Excess and Shortage and I hope you will find a method that suits you most!
To ensure that you get sufficient practice on the mentioned questions, don't miss out OwlSmart's affordable primary school revision subscriptions. You will get instant updates and analysis of your
child's revision progress.
Mrs Wong brought some sweets to the class. She distributed the sweets among her pupils in a class. If Mrs Wong gave each of the pupils in the class 6 sweets, she would have 14 sweets left over. If
Mrs Wong gave each pupil 8 sweets, she would be short of 10 sweets. How many pupils are there in the class?
1. Understand the problem
Let's break down the problem sum into parts and analyse them...
Mrs Wong brought some sweets to the class. She distributed the sweets among her pupils in a class.
(How many sweets did she have?)
If Mrs Wong gave each of the pupils in the class 6 sweets, she would have 14 sweets left over. – Case 1
If Mrs Wong gave each pupil 8 sweets, she would be short of 10 sweets. – Case 2
(Since the number of sweets in both cases are the same, how do I link them?)
How many pupils are there in the class?
(I am supposed to find the number of pupils in the class)
2. Think of a plan
I am supposed to find out the number of pupils in the class, so let x represent this number.
Since the number of sweets in Case 1 and 2 are the same, I can come up with a mathematical equation using the unknown variable x to link them.
3. Carry it out
Method 1 - Using Algebra
Let the number of pupils in the class be x.
Case 1: Total number of sweets -> 6x + 14
Case 2: Total number of sweets -> 8x – 10
6x + 14 = 8x -10
24 = 2x
X = 12
4. Check my answer
Substitute the value of x into Case 1 and Case 2 respectively to check if the number of sweets is the same.
6x + 14 = 6 ×12 + 14 = 86
8x – 10 = 8 × 12 – 10 = 86
Work backwards
Case 1 ->
86 – 14 = 72
72 ÷ 6 = 12 (My answer is correct!)
More Methods in Solving the Above Question
Method 2 – Using Multiples (You learnt this method in P4!)
Multiples of 6 : 6, 12, 18, 24, ..... 60, 66, 72
Multiples of 6 (+14) : 20, 26, 32, 38, ..... 74, 80, 86
Multiples of 8 : 8, 16, 24, 32, ..... 72, 80, 88, 96
Multiples of 8 (-10) : x, 6, 16, 22, ..... 62, 70, 78, 86
86 – 14 = 72 (total number of sweets)
72 ÷ 6 = 12 (total number of pupils)
Method 3 – Using diagrams
How many groups of 2 will make up the difference of 24? That is the number of pupils in the class.
8 – 6 = 2
14 + 10 = 24
24 ÷ 2 = 12
Method 4 – Using Models
14 ÷ 2 = 7 (groups of 2)
10 ÷ 2 = 5 (groups of 2)
7 + 5 = 12
If you are a P4 or P5 student, you would probably use Method 2 to 4 to solve this particular question type. If you are a P6 student, you would have learnt algebra and I recommend that you use Method
1 to solve it.
Method 2 could be the easiest to understand but it can be very tedious to list out the multiples, especially when the numbers involved are big! For P4 or P5 students, try to use Method 3 though it is
the most abstract method to understand.
Again, if you have not already done so, do try out OwlSmart's primary school revision packages which contain the most relevant question types that will almost certainly appear in exams. Many parents
and primary school students have benefited from it and I'm sure you will too!
About the Author
Teacher Zen has over a decade of experience in teaching upper primary Math and Science in local schools. He has a post-graduate diploma in education from NIE and has a wealth of experience in marking
PSLE Science and Math papers. When not teaching or working on OwlSmart, he enjoys watching soccer and supports Liverpool football team. | {"url":"https://owlsmart.sg/article/psle-math/must-know-math-question-for-primary-4-to-psle","timestamp":"2024-11-03T01:04:13Z","content_type":"text/html","content_length":"54971","record_id":"<urn:uuid:2e53fcd3-350f-4832-a59f-d30385bc213a>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00585.warc.gz"} |
Samacheer Kalvi 12th Business Maths Guide Book Answers Solutions
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Samacheer Kalvi 12th Business Maths Book Solutions Answers Guide
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Tamilnadu State Board Samacheer Kalvi 12th Business Maths Book Volume 2 Solutions
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12th Business Maths Book Pdf Chapter 10 Operations Research
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Daniel Lidar
Daniel Lidar is the Viterbi Professor of Engineering at USC, and a professor of Electrical Engineering, Chemistry, and Physics. He holds a Ph.D. in physics from the Hebrew University of
Jerusalem. He did his postdoctoral work at UC Berkeley. Prior to joining USC in 2005 he was a faculty member at the University of Toronto. His main research interest is quantum information
processing, where he works on quantum control, quantum error correction, the theory of open quantum systems, quantum algorithms, and theoretical as well as experimental adiabatic quantum computation.
He is the Director of the USC Center for Quantum Information Science and Technology, and is the co-Director (Scientific Director) of the USC-Lockheed Martin Center for Quantum Computing. Lidar is a
recipient of a Sloan Research Fellowship, a Guggenheim Fellowship and is a Fellow of the AAAS, APS, and IEEE.
Room: SSC 609
Email: L I D A R A T U S C DOT EDU
Peer Reviewed Publications
• 240. “Dynamically Generated Decoherence-Free Subspaces and Subsystems on Superconducting Qubits”, Reports on Progress in Physics 87, 097601, G. Quiroz, B. Pokharel, J. Boen, L. Tewala, V.
Tripathi, D. Williams, L. Wu, P. Titum, K. Schultz, and D. A. Lidar. [link]
• 239. “Optimizing for periodicity: a model-independent approach to flux crosstalk calibration for superconducting circuits”, Quantum Science and Technology 9 025007 (2024), by X. Dai, R. Trappen,
R. Yang, S. M. Disseler, J. I. Basham, J. Gibson, A. J. Melville, B. M. Niedzielski, R. Das, D. K. Kim, J. L. Yoder, S. J. Weber, C. F. Hirjibehedin, D. A. Lidar, A. Lupascu. [link]
• 238. “Error budget of a parametric resonance entangling gate with a tunable coupler”, Phys. Rev. Applied 22, 014059 (2024), by E. A. Sete, V. Tripathi, J. A. Valery, D. A. Lidar, J. Y. Mutus. [
• 237. “Markovian and non-Markovian master equations versus an exactly solvable model of a qubit in a cavity”, Phys. Rev. Applied 22, 014028, Z. Xia, J. Garcia-Nila, D. A. Lidar. [link]
• 236. “Better-than-classical Grover search via quantum error detection and suppression”, npj Quantum Information volume 10, 23 (2024), by B. Pokharel and D. A. Lidar. [link]
• 235. “Modeling low- and high-frequency noise in transmon qubits with resource-efficient measurement”, PRX Quantum 5, 010320 (2024), by V. Tripathi, H. Chen, E. M. Levenson-Falk and D. A. Lidar [
• 234. “Dynamical decoupling for superconducting qubits: A performance survey”, Phys. Rev. Applied 20, 064027, (2023) by N. Ezzell, B. Pokharel, L. Tewala, G. Quiroz and D. A. Lidar [link]
• 233. “Which differential equations correspond to the Lindblad equation?”, Phys. Rev. Research 5, 043163 (2023), by V. Kasatkin, L. Gu and D. A. Lidar [link]
• 232. “Demonstration of algorithmic quantum speedup”, Phys. Rev. Lett. 130, 210602 (2023) by B. Pokharel and D. A. Lidar [link]
• 231. “Boundaries of quantum supremacy via random circuit sampling”, npj Quantum Information, 9, 36 (2023), by A. Zlokapa, S. Boixo, and D. A. Lidar [link]
• 230. “Demonstration of Error-Suppressed Quantum Annealing Via Boundary Cancellation”, Phys. Rev. Applied 19 , 034095 (2023), by H. Munoz-Bauza, L. Campos Venuti, and D. A. Lidar [link]
• 229. “No ((n,K,d<127)) Code Can Violate the Quantum Hamming Bound”, IEEE BITS the Information Theory Magazine, vol. 2, no. 3, (2022), by E. Dallas, F. Andreadakis and D. A. Lidar [link]
• 228. “Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits”, Phil. Trans. R. Soc. 381: 20210407 (2023), by E. Mozgunov and D. A.
Lidar [link]
• 227. “Coherent quantum annealing in a programmable 2,000qubit Ising chain”, Nat. Phys. (2022), by A. D. King, S. Suzuki, J. Raymond, A. Zucca, T. Lanting, F. Altomare, A. J. Berkley, S.
Ejtemaee, E. Hoskinson, S. Huang, E. Ladizinsky, A. J. R. MacDonald, G. Marsden, T. Oh, G. Poulin-Lamarre, M. Reis, C. Rich, Y. Sato, J. D. Whittaker, J. Yao, R. Harris, D. A. Lidar, H.
Nishimori, M. H. Amin [link]
• 226. “Suppression of crosstalk in superconducting qubits using dynamical decoupling”, Phys. Rev. Applied 18, 024068 (2022), by V. Tripathi, H. Chen, M. Khezri, Ka-Wa Yip, E. M. Levenson-Falk, D.
A. Lidar [link]
• 225. “Demonstration of long-range correlations via susceptibility measurements in a one-dimensional superconducting Josephson spin chain”, npj Quantum Information 8, 85 (2021), by D. M. Tennant,
X. Dai, A. J. Martinez, R. Trappen, D. Melanson, M. A. Yurtalan, Y. Tang, S. Bedkihal, R. Yang, S. Novikov, J. A. Grover, S. M. Disseler, J. I. Basham, R. Das, D. K. Kim, A. J. Melville, B. M.
Niedzielski, S. J. Weber, J. L. Yoder, A. J. Kerman, E. Mozgunov, D. A. Lidar & A. Lupascu [link]
• 224. “Breakdown of the weak coupling limit in quantum annealing”, Phys. Rev. Applied 17, 054033 (2022), by Y. Bando, Ka-Wa Yip, H. Chen, D. A. Lidar, H. Nishimori [link]
• 223. “Predicting non-Markovian superconducting qubit dynamics from tomographic reconstruction”, Phys. Rev. Applied 17, 054018 (2022), by H. Zhang, B. Pokharel, E. M. Levenson-Falk and D. A. Lidar
• 222. “HOQST: Hamiltonian Open Quantum System Toolkit”, Communications Physics (2022)5:11, by H. Chen and D. A. Lidar [link]
• 221. “Customized quantum annealing schedules”, Phys. Rev. Applied 17, 044005 (2022), by M. Khezri, X. Dai, R. Yang, T. Albash, A. Lupascu, D. A. Lidar [link]
• 220. “Standard quantum annealing outperforms adiabatic reverse annealing with decoherence”, Phys. Rev. A 105, 032431 (2022), by G. Passarelli, K.-W. Yip, D.A. Lidar, P. Lucignano [link]
• 219. “Optimal Control for Closed and Open System Quantum Optimization”, Phys. Rev. Applied 16, 054023 (2021), by L. Campos Venuti, D. D’Alessandro and D. A. Lidar [link]
• 218. “Charged particle tracking with quantum annealing-inspired optimization”, Quantum Machine Intelligence 3, 27 (2021), by A. Zlokapa, A. Anand, J-R. Vlimant, J. Duarte, J. Job, D. Lidar and M.
Spiropulu [link]
• 217. “Identification of driver genes for severe forms of COVID-19 in a deeply phenotyped young patient cohort”, Science Translational Medicine (2021), by
• 216. “Calibration of flux crosstalk in large-scale flux-tunable superconducting quantum circuits”, PRX Quantum 2, 040313 (2021), by X. Dai, D. M. Tennant, R. Trappen, A. J. Martinez, D. Melanson,
M. A. Yurtalan, Y. Tang, S. Novikov, J. A. Grover, S. M. Disseler, J. I. Basham, R. Das, D. K. Kim, A. J. Melville, B. M. Niedzielski, S. J. Weber, J. L. Yoder, D. A. Lidar, and A. Lupascu [link]
• 215. “Phase transitions in the frustrated Ising ladder with stoquastic and non-stoquastic catalysts”, Phys. Rev. Research 3 (2021), by K. Takada, S. Sota, S. Yunoki, B. Pokharel, H. Nishimori, D.
A. Lidar [link]
• 214. “Low overhead universality and quantum supremacy using only Z-control”, Phys. Rev. Research 3, 033207 (2021), by B. Barch, R. Mohseninia and D. A. Lidar [link]
• 213. “Prospects for quantum enhancement with diabatic quantum annealing”, Nature Reviews Physics (2021), by E. J. Crosson and D. A. Lidar [link]
• 212. “Quantum processor-inspired machine learning in the biomedical sciences”, Patterns 2, 100246 (2021) by R. Li, S. Gujja, S. Bajaj, O. Gamel, N. Cilfone, J. Gulcher, D. A. Lidar and T.
Chittenden [link]
• 211. “Anneal-path correction in flux qubits”, npj Quantum Information 7, 36 (2021), by M. Khezri, J. Grover, J. Basham, S. Disseler, H. Chen, S. Novikov, K. Zick, D. A. Lidar [link]
• 210. “Quantum adiabatic machine learning by zooming into a region of the energy surface”, Phys. Rev. A 102, 062405, by A. Zlokapa, A. Mott, J-R. Vlimant, J. Job, D. A. Lidar and M. Spiropulu [
• 209. “Fast, Lifetime-Preserving Readout for High-Coherence Quantum Annealers”, PRX Quantum 1, 020314, by J. A. Grover, J. I. Basham, A. Marakov, S. M. Disseler, R. T. Hinkey, M. Khalil, Z. A.
Stegen, T. Chamberlin, W. DeGottardi, D. J. Clarke, J. R. Medford, J. D. Strand, M. Stoutimore, S. Novikov, D. G. Ferguson, D. A. Lidar, K. M. Zick and A. J. Przybysz [link]
• 208. “Limitations of error corrected quantum annealing in improving the performance of Boltzmann machines”, Quantum Science and Technology 5, 045010 (2020), by R. Li, T. Albash and D. A. Lidar [
• 207. “Reverse quantum annealing of the p-spin model with relaxation”, Phys. Rev. A. 101, 022331 (2020), by G. Passarelli, K. Yip, D. A. Lidar, H. Nishimori and P. Lucignano [link]
• 206. “Completely positive master equation for arbitrary driving and small level spacing”, Quantum 4, 227 (2020), by E. Mozgunov and D. A. Lidar [link]
• 205. “Analog Errors in Quantum Annealing: Doom and Hope” npj Quantum Information 5, 107 (2019), by A. Pearson, A. Mishra, I. Hen and D. A. Lidar [link]
• 204. “Dynamics of reverse annealing for the fully-connected p-spin model”, Phys. Rev. A 100, 052321 (2019) by Y. Yamashiro, M. Ohkuwa, H. Nishimori and D. A. Lidar [link]
• 203. “Arbitrary-Time Error Suppression for Markovian Adiabatic Quantum Computing Using Stabilizer Subspace Codes”, Phys. Rev. A 100, 022326 (2019), by D. A. Lidar [link]
• 202. “Nested Quantum Annealing Correction at Finite Temperature: p-spin models”, Phys. Rev. A 99, 062307 (2019), by S. Matsuura, H. Nishimori, W. Vinci, D. A. Lidar [link]
• 201. “A Double-Slit Proposal for Quantum Annealing”, npj Quantum Information 5, 2 (2019), by H. Munoz-Bauza, H. Chen, D. A. Lidar [link]
• 200. “On the computational complexity of curing non-stoquastic Hamiltonians”, Nature Comm. 10, 1571 (2019), by M. Marvian, D. A. Lidar and I. Hen [link]
• 199. “Sensitivity of quantum speedup by quantum annealing to a noisy oracle”, Phys. Rev. A 99, 032324 (2019), by S. Muthukrishnan, T. Albash and D. A. Lidar [link]
• 198. “Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits”, Phys. Rev. Lett. 121, 220502 (2018), by B. Pokharel, N. Anand, B Fortman and D. A. Lidar [link
• 197. “Quantum annealing of the p-spin model under inhomogeneous transverse field driving”, Phys. Rev. A 98, 042326 (2018), by Y. Susa, Y. Yamashiro, M. Yamamoto, I. Hen, D. A. Lidar and H.
Nishimori [link]
• 196. “Non-Markovianity of the Post Markovian Master Equation”, Phys. Rev. A 98, 042119 (2018), by C. Sutherland, T. A. Brun and D. A. Lidar [link]
• 195. “Reverse annealing for the fully connected p-spin model”, Phys. Rev. A 98, 022314 (2018), by M. Ohkuwa, H. Nishimori and D. A. Lidar [link]
• 194. “Error Reduction in Quantum Annealing using Boundary Cancellation: Only the End Matters”, Phys. Rev. A 98, 022315 (2018) , by L. Campos Venuti and D. A. Lidar [link]
• 193. “Finite temperature quantum annealing solving exponentially small gap problem with non-monotonic success probability”, Nature Comm. 9, 2917 (2018), by A. Mishra, T. Albash and D. A. Lidar [
• 192. “Demonstration of a Scaling Advantage for a Quantum Annealer over Simulated Annealing”, Phys. Rev. X 8, 031016 (2018), by T. Albash and D. A. Lidar [link]
• 191. “Test-driving 1000 qubits”, Quantum Science & Technology 3, 030501 (2018). Special issue on “What would you do with 1000 qubits” , by J. Job and D. A. Lidar [link]
• 190. “Quantum trajectories for time-dependent adiabatic master equations”, Phys. Rev. A 97, 022116 (2018), by K. W. Yip, T. Albash, D. A. Lidar [link]
• 189. “Quantum annealing versus classical machine learning applied to a simplified computational biology problem”, npj Quant. Info. 4, 14 (2018), by R. Y. Li, R. Di Felice, R. Rohs and D. A. Lidar
• 188. “Scalable effective temperature reduction for quantum annealers via nested quantum annealing correction”, Phys. Rev. A 97, 022308 (2018), by W. Vinci and D. A. Lidar [link]
• 187. “Adiabatic Quantum Computation”, Rev. Mod. Phys. 90, 015002 (2018), by T. Albash and D. A. Lidar [link]
• 186. “Suppression of effective noise in Hamiltonian simulations”, Phys. Rev. A 96, 052328 (2017) , by M. Marvian, T. Brun and D. A. Lidar [link]
• 185. “Solving a Higgs optimization problem with quantum annealing for machine learning”, Nature 550, 375 (2017), A. Mott, J. Job, J. R. Vlimant, D. A. Lidar, and M. Spiropulu
• 184. “Non-stoquastic Hamiltonians in quantum annealing via geometric phases”, Nature Quant. Info. 3, 38(2017), by W. Vinci and D. A. Lidar [link]
• 183. “Quasi-adiabatic Grover search via the WKB approximation”, Phys. Rev. A 96, 012329 (2017), by S. Muthukrishnan and D. A. Lidar [link]
• 182. “Relaxation vs. adiabatic quantum steady state preparation: which wins?”, Phys. Rev. A 95, 042302 (2017), by L. Campos Venuti, T. Albash, M. Marvian, D. A. Lidar, and P. Zanardi [link]
• 181. “Error Suppression for Hamiltonian Quantum Computing in Markovian Environments”,Phys. Rev. A 95, 032302 (2017), by M. Marvian and D. A. Lidar [link]
• 180. “Quantum annealing correction at finite temperature: ferromagnetic p-spin models”, Phys. Rev. A 95, 022308 (2017), by S. Matsuura, H. Nishimori, W. Vinci, T. Albash, and D. A. Lidar [link]
• 179. “Evolution Prediction from Tomography”, Q. Info. Proc. 16(3), 1 (2017), by J. Dominy, L. Campos-Venuti, A. Shabani, and D.A. Lidar [link]
• 178. “Error Suppression for Hamiltonian-Based Quantum Computation Using Subsystem Codes”, Phys. Rev. Lett. 118 030504 (2017), by M. Marvian and D. A. Lidar [link]
• 177. “Optimally Stopped Optimization”, Phys. Rev. Applied 6, 054016, by W. Vinci and D. A. Lidar [link]
• 176. “Eigenstate Tracking in Open Quantum Systems”, Phys. Rev. A 94, 042131 (2016), by J. Jing, M. S. Sarandy, D. A. Lidar, D. W. Luo, and L. A. Wu [link]
• 175. “Simulated Quantum Annealing with Two All-to-All Connectivity Schemes”, Phys. Rev. A 94, 022327, by T. Albash, W. Vinci, and D. A. Lidar [link]
• 174. “Nested Quantum Annealing Correction”, Nature Quant. Info. 2, 16017 (2016), by W. Vinci, T. Albash, and D. A. Lidar [link]
• 173. “Tunneling and speedup in quantum optimization for permutation-symmetric problems”, Phys. Rev. X, 6, 031010 (2016), by S. Muthukrishnan, T. Albash, and D. A. Lidar [link]
• 172. “Mean Field Analysis of Quantum Annealing Correction”, Phys. Rev. Lett. 116, 220501 (2016), by S. Matsuura, H. Nishimori, T. Albash, and D.A. Lidar [pdf]
• 171. “Adiabaticity in open quantum systems”, Phys. Rev. A 93, 032118 (2016), by L.C. Venuti, T. Albash, D. A. Lidar, and P. Zanardi [link]
• 170. “Beyond Complete Positivity”, Quant. Info. Proc. 15, 1, pp 1349 (2016), by J. Dominy and D.A. Lidar [link]
• 169.“Performance of two different quantum annealing correction codes”, Quant. Info. Proc. 15, 2, pp. 609 (2016), by A. Mishra, T. Albash, and D.A. Lidar [link]
• 168. “Reexamination of the evidence for entanglement in the D-Wave processor”, Phys. Rev. A 92, 062328 (2015) , by T. Albash, I. Hen, F. M. Spedalieri, and D. A. Lidar [link]
• 167. “Quantum speed limits for leakage and decoherence”, Phys. Rev. Lett. 115, 210402 (2015), by I. Marvian and D.A. Lidar [link]
• 166. “A General Framework for Complete Positivity”, Quant. Info. Proc. 15, 1, pp. 1 (2016), by J. Dominy, A. Shabani, and D.A. Lidar. [link]
• 165. “Probing for quantum speedup in spin glass problems with planted solutions”, Phys Rev A 92, 042325 (2015), by I. Hen, J. Job, T. Albash, T.F. Ronnow, M. Troyer, and D.A. Lidar [link]
• 164. “Quantum Annealing Correction with Minor Embedding”,Phys. Rev. A 92, 042310 (2015), by W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A. Lidar [link]
• 163. “Decoherence in adiabatic quantum computation”, Phys. Rev. A 91, 062320 (2015), by T. Albash and D.A. Lidar [pdf]
• 162. “Consistency tests of classical and quantum models for a quantum annealer”, Phys. Rev. A 91, 042314 (2015), by T. Albash, W. Vinci, A. Mishra, P.A. Warburton, and D.A. Lidar [link]
• 161. “Quantum Annealing Correction for Random Ising Problems”, Phys. Rev. A 91, 042302 (2015), by K. Pudenz, T. Albash, and D.A. Lidar. [link]
• 160 . “Reexamining classical and quantum models for the D-Wave One processor”, The European Physics Journal, Special Topics 224, 111 (special issue on quantum annealing) (2015), by T. Albash, T.
Ronnow, M. Troyer, and D.A. Lidar [link]
• 159. “Review of Decoherence Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling”, Quant. Info. & Comp. for Chem., Vol 154, pp. 295-354 (2014), by D. Lidar [link]
• 158 . “Quantum error suppression with commuting Hamiltonians: Two-local is too local”, Phys. Rev. Lett. 113, 260504 (2014), by I. Marvian and D.A. Lidar [pdf]
• 157 . “Defining and Detecting Quantum Speedup”, Science 345, 420 (2014), by T.F. Ronnow, Z. Wang, J. Job, S.V. Isakov, D. Wecker, J.M. Martinis, D.A. Lidar, and M. Troyer. [link]
• 156 . “MAX 2-SAT with up to 108 Qubits”, New J. Phys. 16, 045006 (2014), by S. Santra, G. Quiroz, G. Ver Steeg, and D.A. Lidar. [link]
• 155 . “Evidence for Quantum Annealing with More Than One Hundred Qubits”, Nature Physics 10, 218 (2014), by. S. Boixo, T. Ronnow, S. Isakov, Z. Wang, D. Wecker, D.A. Lidar, J. Martinis, and M.
Troyer. [pdf]
• 154 . “Error Corrected Quantum Annealing with Hundreds of Qubits”, Nature Communications 5, 3243 (2014), by K.P. Pudenz, T. Albash, and D. Lidar. [pdf]
• 153 . “Adiabatic Quantum Optimization with the Wrong Hamiltonian”, Phys. Rev A 88, 062314 (2013), by K. C. Young, R. Blume-Kohout, and D. Lidar. [pdf]
• 152 . “Optimized Dynamical Decoupling via Genetic Algorithms”, Phys. Rev. A 88, 052306 (2013), by G. Quiroz and D. Lidar. [pdf]
• 151 . “Fluctuation Theorems for Quantum Process”, Phys. Rev. E 88, 032146 (2013), T. Albash, D. Lidar, M. Marvian, and P. Zanardi. [pdf]
• 150 . “Coarse-Graining Can Beat the Rotating Wave Approximation in Quantum Markovian Master Equations”, Phys. Rev A. 88, 012103 (2013), C. Majenz, T. Albash, H.-P. Breuer, and D. Lidar. [pdf]
• 149 . “Experimental Signature of Programmable Quantum Annealing”, Nature Comm. 4, 2067 (2013), by S. Boixo, T. Albash, F.M. Spedalieri, N. Chancellor, and D. Lidar. [pdf]
• 148 . “Optimally Combining Dynamical Decoupling and Quanutm Error Correction”, Scientific Reports 3, 1394 (2013), by G.A. Paz-Silva and D. Lidar. [pdf]
• 147 . “No-Go Theorem for Passive Single-rail Linear Optical Quantum Computing”, Scientific Reports 3, 1394 (2013), by L. Wu, P. Walther, and D. A. Lidar. [pdf]
• 146 . “Analysis of the Quantum Zeno Effect for Quantum Control and Computation”, J. Phys. A: Math. Theor. 46, 075306 (2013), by J. Dominy, G. Paz-Silva, A.T. Rezakhani, and D.A. Lidar. [pdf]
• 145 . “Quantum Adiabatic Machine Learning”, Quantum Info. Process. 12, 2027 (2013), by K. Pudenz and D. Lidar. [pdf]
• 144 . “Universality Proof and Analysis of Generalized Nested Uhrig Dynamical Decoupling”, J. Math. Phys. 53, 122207 (2012), by W.J. Kuo, G. Quiroz, G. Paz Silva, and D. Lidar. [pdf]
• 143 . “Optimally Combining Dynamical Decoupling and Quantum Error Correction”, Scientific Reports 3, 1530 (2013), by G.A. Paz Silva and D. A. Lidar. [pdf]
• 142 . “Quantum Adiabatic Markovian Master Equations”, New J. of Physics 14, 123016 (2012), by T. Albash, S. Boixo, D. Lidar, and P. Zanardi. [pdf]
• 141 . “High-Fidelity Adiabatic Quantum Computation via Dynamical Decoupling”, Phys. Rev. A 86, 042333 (2012), by G. Quiroz and D. Lidar. [pdf]
• 140 . “Adiabatic Quantum Algorithm for Search Engine Ranking”, Phys. Rev. Lett. 108, 230506 (2012), by S. Garnerone, P. Zanardi, and D. Lidar [pdf][sup-mat]
• 139 . “Decoherence-Protected Quantum Gates for a Hybrid Solid-State Spin Register”, Nature 484, 82 (2012), by T. van der Sar, Z.H. Wang, M.S. Blok, H. Bernien, T.H. Taminiau, D.M. Toyli, D.A.
Lidar, D.D. Awschalom, R. Hanson, and V.V. Dobrovitski [pdf]
• 138 . “Zeno Effect for Quantum Computation and Control”, Phys. Rev. Lett. 108, 080501 (2012), G. A. Paz-Silva, A. T. Rezakhani, J. Dominy, and D. A. Lidar [pdf]
• 137 . “Rigorous Performance Bounds for Quadratic and Nested Dynamical Decoupling”, Phys. Rev. A 84, 062332 (2011), by Y. Xia, G. S. Uhrig, and D. Lidar. [pdf]
• 136 .”Quadratic Dynamical Decoupling: Universality Proof and Error Analysis”, Phys. Rev. A 84, 042329 (2011), by W. Kuo and D. Lidar. [pdf]
• 135 .”Quadratic Dynamical Decoupling with Nonuniform Error Suppression”, Phys. Rev. A 84, 042328 (2011), by G. Quiroz and D. Lidar. [pdf]
• 134 . “High Fidelity Quantum Memory via Dynamical Decoupling: Theory and Experiment”, J. Phys. B 44, 154003 (2011), by Xinhua Peng, Dieter Suter, and Daniel A Lidar. [pdf]
• 133 . “Combining Dynamical Decoupling with Fault-Tolerant Quantum Computation”, Phys. Rev. A 84, 012305 (2011), by H. K. Ng, D. Lidar, and J. Preskill. [pdf]
• 132 . “High Fidelity Quantum Gates via Dynamical Decoupling”, Phys. Rev. Lett 105, 230503 (2010), by J. R. West, D. Lidar, B. H. Fong, and M. F. Gyure. [pdf]
• 131 . “Accuracy Versus Run Time in an Adiabatic Quantum Search”, Phys. Rev. A 82, 052305 (2010), by A. T. Rezakhani, A. K. Pimachev, and D. Lidar. [pdf]
• 130 . “Optimized Entanglement-Assisted Quantum Error Correction”, Phys. Rev. A 82, 042321 (2010), by S. Taghavi, T. A. Brun, and D. Lidar. [pdf]
• 129 . “Classical Ising Model Test for Quantum Circuits”, New J. Physics 12, 075026 (2010), by J. Geraci and D. Lidar [pdf]
• 128 . “Intrinsic Geometry of Quantum Adiabatic Evolution and Quantum Phase Transitions”, Phys. Rev. A 82, 012321 (2010), by A. T. Rezakhani, D. F. Abasto, D. Lidar, and P. Zanardi. [pdf]
• 127 . “Rigorous Bounds for Optimal Dynamical Decoupling”, Phys. Rev. A 82, 012301 (2010), by G. S. Uhrig and D. Lidar. [pdf]
• 126 . “Channel Capacities of an Exactly Solvable Spin-Star System”, Phys. Rev. A 81, 062353 (2010), by Nigum Arshed, A. H. Toor, and D. A. Lidar. [pdf]
• 125 .”Optimal Control Landscape for the Generation of Unitary Transformations with Constrained Dynamics”, Phys. Rev. A 81, 062352 (2010), by M. Hsieh, R. Wu, H. Rabitz, and D. Lidar. [pdf]
• 124 . “Near-Optimal Dynamical Decoupling of a Qubit”, Phys. Rev. Lett 104, 130501 (2010), by J. R. West, B. H. Fong, and D. A. Lidar. [pdf]
• 123 . “Channel-Optimized Quantum Error Correction”, IEEE Trans. on Info. Theory 56, 1461 (2010), by S. Taghavi, R. L. Kosut, and D. A. Lidar. [pdf]
• 122 . “Arbitrarily Accurate Dynamical Control in Open Quantum Systems”, Phys. Rev. Lett 104, 090501 (2010), by K. Khodjasteh, D. Lidar, and L. Viola. [pdf]
• 121 . “Entanglement and Area Law with a Fractal Boundary in a Topologically Ordered Phase”, Phys. Rev. A 81, 01012 (2010), by A. Hamma, D. Lidar, and S. Severini. [pdf]
• 120 . “Adiabatic Approximation with Exponential Accuracy for Many-Body Systems and Quantum Computation”, J. Math. Phys. 50, 102106 (2009), by D. Lidar, A. T. Rezakhani, and A. Hamma. [pdf]
• 119 . “Quantum Adiabatic Brachistochrone”, Phys. Rev. Lett. 103, 080502 (2009), by A.T. Rezakhani, W.-J. Kuo, A. Hamma, D. Lidar, and P. Zanardi. [pdf]
• 118 . “Scheme for Fault-Tolerant Holonomic Computation on Stabilizer Codes”, Phys. Rev. A 80, 022325 (2009), by O. Oreshkov, T. A. Brun, and D. Lidar [pdf]
• 117 . “Quantum Error Correction via Convex Optimization”, Quant. Info. Processing 8, 441 (2009), by R. L. Kosut and D. Lidar [pdf]
• 116 . “Maps for General Open Quantum Systems and a Theory of Linear Quantum Error Correction”, Phys. Rev. A 80, 012309 (2009), by A. Shabani and D. Lidar [pdf]
• 115 . “Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps”, Phys. Rev. Lett. 102, 100402 (2009), by A. Shabani and D. Lidar. [pdf] ; “Erratum: Vanishing Quantum
Discord is Necessary and Sufficient for Completely Positive Maps”, Phys. Rev. Lett. 11, 049901 (2016), by A. Shabani and D. A. Lidar. [pdf]
• 114 . “Fault-Tolerant Holonomic Quantum Computation”, Phys. Rev. Lett. 102, 070502 (2009), by O. Oreshkov, T. A. Brun, and D. Lidar [pdf]
• 113 . “Operator Quantum Error Correction for Continuous Dynamics”, Phys. Rev. A 78, 022333 (2008), by O. Oreshkov, D. Lidar, and T. A. Brun [pdf]
• 112 . “Rigorous Bounds on the Performance of a Hybrid Dynamical Decoupling-Quantum Computing Scheme”, Phys. Rev. A 78, 012355 (2008), by K. Khodjasteh and D. Lidar [pdf]
• 111 . “Encoding One Logical Qubit Into Six Physical Qubits”, Phys. Rev. A 78, 012337 (2008), by B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. Lidar [pdf]
• 110 . “Distance Bounds on Quantum Dynamics”, Phys. Rev. A 78, 012308 (2008), by D. Lidar, P. Zanardi, and K. Khodjasteh [pdf]
• 109 . “Optimal Dynamical Decoherence Control of a Qubit”, Phys. Rev. Lett. 101, 010403 (2008), by G. Gordon, G. Kurizki, and D. Lidar [pdf]
• 108 . “Bang-Bang Control of a Qubit Coupled to a Quantum Critical Spin Bath”, Phys. Rev. A 77, 052112 (2008), by D. Rossini, P. Facchi, R. Fazio, G. Florio, D. Lidar, S. Pascazio, F. Plastina,
and P. Zanardi [pdf]
• 107 . “Towards Fault Tolerant Adiabatic Quantum Computation”, Phys. Rev. Lett. 100, 160506 (2008), by D. Lidar [pdf]
• 106 . “Entanglement, Fidelity, and Topological Entropy in a Quantum Phase Transition to Topological Order”, Phys. Rev. B 77, 155111 (2008), by A. Hamma, W. Zhang, S. Haas, and D. Lidar [pdf]
• 105 . “Quantum Process Tomography: Resource Analysis of Different Strategies”, Phys. Rev. A 77, 032322 (2008), by M. Mohseni, A. T. Rezakhani, and D. Lidar [pdf]
• 104 . “On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers”, Commun. Math. Phys 279, issue 3, 735-768 (2008), by J. Geraci and D. Lidar [pdf]
• 103 . “Adiabatic Preparation of Topological Order”, Phys. Rev. Lett. 100, 030502 (2008), by A. Hamma and D. Lidar [pdf]
• 102 . “The Spin Density Matrix II: Application to a System of Two Quantum Dots”, Phys. Rev. B 77, 045320 (2008), by S. D. Kunikeev and D. Lidar [pdf]
• 101 . “The Spin Density Matrix I: General Theory and Exact Master Equations”, Phys. Rev. B 77, 045319 (2008), by S. D. Kunikeev and D. Lidar [pdf]
• 100 . “Robust Quantum Error Correction via Convex Optimization”, Phys. Rev. Lett. 100, 020502 (2008), by R.L. Kosut, A. Shabani, and D. Lidar [pdf]
• 99 . “Fidelity of Optimally Controlled Quantum Gates with Randomly Coupled Multiparticle Environments”, J. Mod. Optics 54, 2339 (2007), M. Grace, C. Brif, H. Rabitz, I. Walmsley, R. Kosut, and
D.A. Lidar. [pdf]
• 98 . “Non-Markovian Dynamics of a Qubit Coupled to an Ising Spin Bath”, Phys. Rev. A 76, 052117 (2007), by H. Krovi, O. Oreshkov, M. Ryazanov, and D. Lidar [pdf]
• 97 . “Simple Proof of Equivalence Between Adiabatic Quantum Computation and the Circuit Model”, Phys. Rev. Lett. 99, 070502 (2007), by A. Mizel, D. Lidar, and M. Mitchell [pdf]
• 96 . “Direct Characterization of Quantum Dynamics: General Theory”, Phys. Rev. A 75, 062331 (2007), by M. Mohseni and D. Lidar [pdf]
• 95 . “Efficient Multiqubit Entanglement via a Spin-Bus”, Phys. Rev. Lett. 98, 230503 (2007), by M. Friesen, A. Biswas, X. Hu, and D. Lidar [pdf]
• 94 . “Performance of Deterministic Dynamical Decoupling Schemes: Concatenated and Periodic Pulse Sequences”, Phys. Rev. A 75, 062310 (2007), by K. Khodjasteh and D. Lidar. [pdf] ; “Erratum:
Performance of Deterministic Dynamical Decoupling Schemes: Concatenated and Periodic Pulse Sequences”, Phys Rev A 79, 069901(E) (2009). Original journal published in Phys. Rev. A 75, 062310
(2007), by K. Khodjasteh and D.A. Lidar. [pdf]
• 93 . “Decoherence-Induced Geometric Phase in a Multilevel Atomic System”, J. Phys. B 40, S127 (2007) (special issue on “Dynamical Control of Entanglement and Decoherence by Field-Matter
Interactions”), by S. Dasgupta and D. Lidar [pdf]
• 92 . “Optimal Control of Quantum Gates and Suppression of Decoherence in a System of Interacting Two-Level Particles”, J. Phys. B 40, S103 (2007) (special issue on “Dynamical Control of
Entanglement and Decoherence by Field-Matter Interactions”), by M. Grace, C. Brif, H. Rabitz, I.A. Walmsley, and D. A. Lidar [pdf]
• 91 . “Robust Transmission of Non-Gaussian Entanglement Over Optical Fibers”, Phys. Rev. A 74, 062303 (2006), by A. Biswas and D. Lidar [pdf]
• 90 . “Linking Entanglement and Quantum Phase Transitions via Density-Functional Theory”, Phys. Rev. A 74, 052335 (2006), by L.-A. Wu, M. S. Sarandy, D. Lidar, and L. J. Sham [pdf]
• 89 . “Direct Characterization of Quantum Dynamics”, Phys. Rev. Lett. 97,170501 (2006), by M. Mohseni and D. Lidar [pdf]
• 88 . “Abelian and Non-Abelian Geometric Phases in Adiabatic Open Quantum Systems”, Phys. Rev. A 73, 062101 (2006), by M.S. Sarandy and D. Lidar [pdf]
• 87 . “Internal Consistency of Fault-Tolerant Quantum Error Correction in Light of Rigorous Derivations of the Quantum Markovian Limit”, Phys. Rev. A 73, 052311 (2006), by R. Alicki, D. Lidar, and
P. Zanardi [pdf]
• 86 . “Quantum Malware”, published online in Quant. Info. Processing 5, 69 (2006), by L.-A. Wu and D. Lidar [pdf]
• 85 . “Encoding a Qubit into Multilevel Subspaces”, New J. Phys. 8, 35 (2006), by M. Grace, C. Brif, H. Rabitz, I. Walmsley, R. Kosut, and D. Lidar [pdf]
• 84 . “Few-Body Spin Couplings and Their Implications for Universal Quantum Computation”, J. Phys.: Cond. Mat. 18, S721 (2006), special issue on quantum computing in solid state, by R. Woodworth,
A. Mizel, and D. Lidar [pdf]
• 83 . “Quantum Logic Gates in Iodine Vapor Using Time-Frequency Resolved Coherent Anti-Stokes Raman Scattering: A Theoretical Study”, Molecular Phys. 104, 1249 (2006), special issue in honor of
Robert Harris, by D.R. Glenn, D. Lidar, and V.A. Apkarian [pdf]
• 82 . “Conditions for Strictly Purity-Decreasing Quantum Markovian Dynamics”, Chemical Physics 322, 82 (2006), the special issue “Real-Time Dynamics in Complex Quantum Systems” in honor of Phil
Pechukas, by D. Lidar, A. Shabani, and R. Alicki [pdf]
• 81 . “Adiabatic Quantum Computation in Open Systems”, Phys. Rev. Lett. 95, 250503 (2005), by M.S. Sarandy and D. Lidar [pdf]
• 80 . “Robustness of Multiqubit Entanglement in the Independent Decoherence Model”, Phys. Rev. A 72, 042339 (2005), by S. Bandyopadhyay and D. Lidar [pdf]
• 79 . “Fault-Tolerant Quantum Dynamical Decoupling”, Phys. Rev. Lett. 95, 180501 (2005), by K. Khodjasteh and D. Lidar [pdf]
• 78 . “Holonomic Quantum Computation in Decoherence-Free Subspaces”, Phys. Rev. Lett. 95, 130501 (2005), by L.-A. Wu, P. Zanardi, and D. Lidar [pdf]
• 77 . “Theory of Initialization-Free Decoherence-Free Subspaces and Subsystems”, Phys. Rev. A 72, 042303 (2005), by A. Shabani and D. Lidar [pdf]
• 76 . “Entanglement Observables and Witnesses for Interacting Quantum Spin Systems”, Phys. Rev. A 72, 032309 (2005), by L.-A. Wu, S. Bandyopadhyay, M.S. Sarandy, and D. Lidar [pdf]
• 75 . “Stabilizing Qubit Coherence via Tracking-Control”, Quant. Info. and Computation 5, 350 (2005), by D. Lidar and S. Schneider [pdf]
• 74 . “Universal Leakage Elimination”, Phys. Rev. A 71, 052301 (2005), by M.S. Byrd, D. Lidar, L.-A. Wu, and P. Zanardi [pdf]
• 73 . “Completely Positive Post-Markovian Master Equation via a Measurement Approach”, Phys. Rev. A Rapid Comm. 71, 020101 (2005), by A. Shabani and D. Lidar [pdf]
• 72 . “Control of Decoherence: Analysis and Comparison of Three Different Strategies”, Phys. Rev. A 71, 022302 (2005), by P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. Lidar [
• 71 . “Fault-Tolerant Quantum Computation via Exchange Interactions”, Phys. Rev. Lett. 94, 040507 (2005), by M. Mohseni and D. Lidar [pdf]
• 70 . “Adiabatic Approximation in Open Quantum Systems”, Phys. Rev. A 71, 012331 (2005), by M.S. Sarandy and D. Lidar [pdf]
• 69 . “Overview of Quantum Error Prevention and Leakage Elimination”, J. Mod. Optics 51, 2449 (2004), by M.S. Byrd, L.-A. Wu, and D. Lidar [pdf]
• 68 . “Consistency of the Adiabatic Theorem”, Quant. Info. Processing 3, 331 (2004), by M.S. Sarandy, L.-A. Wu, and D. Lidar [pdf]
• 67 . “Quantum Phase Transitions and Bipartite Entanglement”, Phys. Rev. Lett. 93, 250404 (2004), by L.-A. Wu, M.S. Sarandy, and D. Lidar [pdf]
• 66 . “Overcoming Quantum Noise in Optical Fibers”, Phys. Rev. A 70, 062310 (2004), by L.-A. Wu and D. Lidar [pdf]
• 65 . “On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants”, New J. Phys. 6, 167 (2004), by D. Lidar [pdf]
• 64 . “Long-Range Entanglement Generation via Frequent Measurements”, Phys. Rev. A 70, 032322 (2004), by L.-A. Wu, D. Lidar, and S. Schneider [pdf]
• 63 . “Exchange Interaction Between Three and Four Coupled Quantum Dots: Theory and Applications to Quantum Computing”, Phys. Rev. B 70, 115310 (2004), by A. Mizel and D. Lidar [pdf]
• 62 . “Purity and State Fidelity of Quantum Channels”, Phys. Rev. A 70, 012315 (2004), by P. Zanardi and D. Lidar [pdf]
• 61 . “Entangling Capacities of Noisy Two-Qubit Hamiltonians”, Phys. Rev. A Rapid Comm. 70, 010301 (2004), by S. Bandyopadhyay and D. Lidar [pdf]
• 60 . “One-Spin Quantum Logic Gates from Exchange Interactions and a Global Magnetic Field”, Phys. Rev. Lett. 93, 030501 (2004), by L.-A. Wu, D. Lidar, and M. Friesen [pdf]
• 59 . “Exponentially Localized Magnetic Fields for Single-Spin Quantum Logic Gates”, J. Appl. Phys. 96, 754 (2004), by D.A. Lidar and J.H. Thywissen [pdf]
• 58 . “Unification of Dynamical Decoupling and the Quantum Zeno Effect”, Phys. Rev. A 69, 032314 (2004), by P. Facchi, D.A. Lidar, and S. Pascazio [pdf]
• 57 . “Dynamical Decoupling Using Slow Pulses: Efficient Suppression of 1/f Noise”, Phys. Rev. A Rapid Comm. 69, 030302 (2004), by K. Shiokawa and D.A. Lidar [pdf]
• 56 . “Three and Four-Body Interactions in Spin-Based Quantum Computers”, Phys. Rev. Lett. 92, 077903 (2004), by A. Mizel and D.A. Lidar [pdf]
• 55 . “Quantum Tensor Product Structures are Observable Induced”, Phys. Rev. Lett. 92, 060402 (2004), by P. Zanardi, D.A. Lidar, and S. Lloyd [pdf]
• 54 . “Magnetic Resonance Realization of Decoherence-Free Quantum Computation”, Phys. Rev. Lett. 91, 217904 (2003), by J.E. Ollerenshaw, D.A. Lidar, and L.E. Kay [pdf]
• 53 . “Dressed Qubits”, Phys. Rev. Lett. 91, 097904 (2003), by L.-A. Wu and D.A. Lidar [pdf]
• 52 . “Quantum Computing in the Presence of Spontaneous Emission by a Combined Dynamical Decoupling and Quantum-Error-Correction Strategy”, Phys. Rev. A 68, 022322 (2003), by K. Khodjasteh and
D.A. Lidar. [pdf] ; “Erratum: Quantum computing in the presence of spontaneous emission by a combined dynamical decoupling and quantum-error-correction strategy”, Phys. Rev. A 72, 029905 (2005)
• 51 . “Comment on “Conservative Quantum Computing” [Phys. Rev. Lett. 89, 057902 (2002)]”, Phys. Rev. Lett. 91, 089801 (2003), by D.A. Lidar [pdf]
• 50 . “Reply to: “Comment on `Polynomial-Time Simulation of Pairing Models on a Quantum Computer’””, Phys. Rev. Lett. 90, 249804 (2003), by L.-A. Wu, M.S. Byrd, and D.A. Lidar [pdf]
• 49 . “Universal Quantum Computation Using Exchange Interactions and Measurements of Single- and Two-Spin Observables”, Phys. Rev. A Rapid Comm. 67, 050303 (2003), by L.-A. Wu and D.A. Lidar [pdf]
• 48 . “Combined Error Correction Techniques for Quantum Computing Architectures”, J. Mod. Optics 50, 1285 (2003), by M.S. Byrd and D.A. Lidar [pdf]
• 47 . “Encoded Recoupling and Decoupling: An Alternative to Quantum Error-Correcting Codes Applied to Trapped-Ion Quantum Computation”, Phys. Rev. A 67, 032313 (2003), by D.A. Lidar and L.-A. Wu [
• 46 . “Empirical Determination of Dynamical Decoupling Operations”, Phys. Rev. A 67, 012324 (2003), by M.S. Byrd and D.A. Lidar [pdf]
• 45 . “Universal Quantum Logic from Zeeman and Anisotropic Exchange Interactions”, Phys. Rev. A 66, 062314 (2002), by L.-A. Wu and D.A. Lidar [pdf]
• 44 . “Universal Fault-Tolerant Quantum Computation in the Presence of Spontaneous Emission and Collective Dephasing”, Phys. Rev. Lett. 89, 197904 (2002), by K. Khodjasteh and D.A. Lidar [pdf] ;
“Erratum: Universal Fault-Tolerant Quantum Computation in the Presence of Spontaneous Emission and Collective Dephasing“, Phys. Rev. Lett. 95, 099902 (2005) [link]
• 43 . “Efficient Universal Leakage Elimination for Physical and Encoded Qubits”, Phys. Rev. Lett. 89, 127901 (2002), by L.-A. Wu, M.S. Byrd, and D.A. Lidar [pdf]
• 42 . “Quantum Codes for Simplifying Design and Suppressing Decoherence in Superconducting Phase-Qubits”, Quant. Info. Proc. 1, 155 (2002), by D.A. Lidar, L.-A. Wu, and A. Blais [pdf]
• 41 . “Qubits as Parafermions”, J. Math. Phys. 43, 4506 (2002) (special issue on quantum information), by L.-A. Wu and D.A. Lidar [pdf]
• 40 . “Polynomial-Time Simulation of Pairing Models on a Quantum Computer”, Phys. Rev. Lett. 89, 057904 (2002), by L.-A. Wu, M.S. Byrd, and D.A. Lidar [pdf]
• 39 . “An Implementation of the Deutsch-Jozsa Algorithm on Molecular Vibronic Coherences Through Four-Wave Mixing: a Theoretical Study”, Chem. Phys. Lett. 360, 459 (2002), by Z. Bihary, D.R.
Glenn, D.A. Lidar, and V.A. Apkarian [pdf]
• 38 . “Bang-Bang Operations from a Geometric Perspective”, Quant. Info. Proc. 1, 19 (2002), by M.S. Byrd and D.A. Lidar [pdf]
• 37 . “Comprehensive Encoding and Decoupling Solution to Problems of Decoherence and Design in Solid-State Quantum Computing”, Phys. Rev. Lett. 89, 047901 (2002), by M.S. Byrd and D.A. Lidar [pdf]
• 36 . “Creating Decoherence-Free Subspaces Using Strong and Fast Pulses”, Phys. Rev. Lett. 88, 207902 (2002), by L.-A. Wu and D.A. Lidar [pdf]
• 35 . “Power of Anisotropic Exchange Interactions: Universality and Efficient Codes for Quantum Computing”, Phys. Rev. A 65, 042318 (2002), by L.-A. Wu and D.A. Lidar [pdf]
• 34 . “Comment on “Quantum Waveguide Array Generator for Performing Fourier Transforms: Alternate Route to Quantum Computing’’ [Appl. Phys. Lett. 79, 2823 (2001)]”, Appl. Phys. Lett. 80, 2419
(2002), by D.A. Lidar [pdf]
• 33 . “Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling”, Phys. Rev. Lett. 88, 017905 (2002), by D.A. Lidar and L.-A. Wu [pdf]
• 32 . “Quantum Computing with Quantum Dots on Quantum Linear Supports”, Phys. Rev. A 65, 012307 (2002), by K.R. Brown, D.A. Lidar, and K.B. Whaley [pdf]
• 31 . “From Completely Positive Maps to the Quantum Markovian Semigroup Master Equation”, Chemical Physics 268, 35 (2001), D.A. Lidar, Z. Bihary, and K.B. Whaley, special issue on Dynamics of Open
Quantum Systems [ pdf]
• 30 . “The Manipulation of Massive Ro-vibronic Superpositions Using Time-Frequency-Resolved Coherent Anti-Stokes Raman Scattering (TFRCARS): from Quantum Control to Quantum Computing”, Chemical
Physics 266, 323 (2001), R. Zadoyan, D. Kohen, D.A. Lidar, and V.A. Apkarian [ pdf]
• 29 . “Theory of Decoherence-Free Universal Fault-Tolerant Quantum Computation”, Phys. Rev. A 63, 042307 (2001), by J. Kempe, D. Bacon, D.A. Lidar, and K.B. Whaley [ pdf]
• 28 . “Decoherence-Free Subspaces for Multiple-Qubit Errors: (II) Universal, Fault-Tolerant Quantum Computation”, Phys. Rev. A 63, 022307 (2001), D.A. Lidar, D. Bacon, J. Kempe, and K.B. Whaley [
• 27 . “Decoherence-Free Subspaces for Multiple-Qubit Errors: (I) Characterization”, Phys. Rev. A 63, 022306 (2001), by D.A. Lidar, D. Bacon, J. Kempe, and K.B. Whaley [ pdf]
• 26 . “Analysis of Generalized Grover Quantum Search Algorithms Using Recursion Equations”, Phys. Rev. A 63, 012310 (2001), by E. Biham, O. Biham, D. Biron, M. Grassl, D.A. Lidar, and D. Shapira [
• 25 . “Universal Fault-Tolerant Quantum Computation on Decoherence-Free Subspaces”, Phys. Rev. Lett. 85, 1758 (2000), by D. Bacon , J. Kempe, D.A. Lidar, and K.B. Whaley [ pdf]
• 24 . “Protecting Quantum Information Encoded in Decoherence-Free States Against Exchange Errors”, Phys. Rev. A 61, 052307 (2000), by D.A. Lidar, D. Bacon, J. Kempe, and K.B. Whaley [ pdf]
• 23 . “Grovers Quantum Search Algorithm for Arbitrary Initial Amplitude Distribution”, Phys. Rev. A 60, 2742 (1999), by E. Biham, O. Biham, D. Biron, M. Grassl, and D.A. Lidar [ pdf]
• 22 . “Robustness of Decoherence-Free Subspaces for Quantum Computation”, Phys. Rev. A 60, 1944 (1999), by D. Bacon, D.A. Lidar, and K.B. Whaley [ pdf]
• 21 . “Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes”, Phys. Rev. Lett. 82, 4556 (1999), by D.A. Lidar, D. Bacon, and K.B. Whaley [ pdf]
• 20 . “Calculating the Thermal Rate Constant with Exponential Speedup on a Quantum Computer”, Phys. Rev. E 59, 2429 (1999), by D.A. Lidar and H. Wang [ pdf]
• 19 . “Pattern Formation and a Clustering Transition in Power-Law Sequential Adsorption”, Phys. Rev. E 59, R4713 (1999), O. Biham, O. Malcai, D.A. Lidar (Hamburger), and D. Avnir [ pdf]
• 18 .”Fractal Analysis of Protein Potential Energy Landscapes”, Phys. Rev. E 59, 2231-2243 (1999), D. A. Lidar, D. Thirumalai, R. Elber, and R.B. Gerber. [pdf]
• 17 . “How to Teleport Superpositions of Chiral Amplitudes”, Phys. Rev. Lett. 81, 5928 (1998), C.S. Maierle, D.A. Lidar, and Robert A. Harris [ pdf]
• 16 . “Decoherence-Free Subspaces for Quantum Computation”, Phys. Rev. Lett. 81, 2594 (1998), by D.A. Lidar, I.L. Chuang, and K.B. Whaley [ pdf]
• 15 . “Fractality in Nature (Reply to a letter)”, Science 279, 1611 (1998), by O. Biham, O. Malcai, D.A. Lidar, and D. Avnir [ pdf]
• 14 . “Is Nature Fractal? (Reply to letters)”, Science 279, 783 (1998), by O. Biham, O. Malcai, D.A. Lidar, and D. Avnir [ pdf]
• 13 . “Is the Geometry of Nature Fractal?”, Science 279, 39 (1998), by D. Avnir, O. Biham, D.A. Lidar, and O. Malcai [ pdf]
• 12 . “Inversion of Randomly Corrugated Surface Structure from Atom Scattering Data”, Inverse Problems 14, 1299 (1998), by D. Lidar [ pdf]
• 11 . “Atom Scattering from Disordered Surfaces in the Sudden Approximation: Double Collisions Effects and Quantum Liquids”, Surf. Sci. 411, 231 (1998), by D.A. Lidar (Hamburger) [ pdf]
• 10 . “Structure Determination of Disordered Metallic Sub-Monolayers by Helium Scattering: A Theoretical and Experimental Study”, Surf. Sci. 410, L721 (1998), by A.T. Yinnon, D.A. Lidar
(Hamburger), R.B. Gerber, P. Zeppenfeld, M. Krzyzowski, and G. Comsa [ pdf]
• 9 . “Simulating Ising Spin Glasses on a Quantum Computer”, Phys. Rev. E 56, 3661 (1997), , by D.A. Lidar and O. Biham [ pdf]
• 8 . “Scaling Range and Cutoffs in Empirical Fractals”, Phys. Rev. E 56, 2817 (1997), by O. Malcai, D. A. Lidar, and O. Biham [ pdf]
• 7 . “Limited Range Fractality of Randomly Adsorbed Rods”, J. Chem. Phys. 106, 10359 (1997), by D.A. Lidar (Hamburger), O. Biham, and D. Avnir [ pdf]
• 6 . “Helium Scattering from Random Adsorbates, Disordered Compact Islands and Fractal Submonolayers: Intensity Manifestation of Surface Disorder”, J. Chem. Phys. 106, 4228 (1997), by A.T. Yinnon,
D.A. Lidar (Hamburger), R.B. Gerber, P. Zeppenfeld, M. Krzyzowski, and G. Comsa. [ pdf]
• 5 . “Elastic Scattering by Deterministic and Random Fractals: Self-Affinity of the Diffraction Spectrum”, By D. Lidar, Phys. Rev. E 54, 354 (1996) [ pdf]
• 4 . “Apparent Fractality Emerging from Models of Random Distributions”, Phys. Rev. E 53, 3342-3358 (1996), D. A. Hamburger, O. Biham, D. Avnir. [link]
• 3 . “Fractal Dimension of Disordered Submonolayers: Determination from Helium Scattering Data”, Chem. Phys. Lett. 253, 223 (1996), by D.A. Hamburger, A.T. Yinnon, and R.B. Gerber [ pdf]
• 2 . “Optical Theorem and the Inversion of Cross Section Data for Atom Scattering from Defects on Surfaces”, J. Chem. Phys. 102, 6919 (1995), by D.A. Hamburger and R.B. Gerber [ pdf]
• 1 . “Helium Scattering from Compact Clusters and from Diffusion-Limited Aggregates on Surfaces: Observable Signatures of Structure”, Surf. Sci. 327, 165 (1995), by D.A. Hamburger, A.T. Yinnon, I.
Farbman, A. Ben-Shaul, and R.B. Gerber. [pdf]
Book Chapters
• 7 .”Fault Tolerance for Holonomic Quantum Computaion”, by O. Oreshkov. T.A. Brun, and D. Lidar, in “Quantum Error Correction”, D. Lidar and T.A. Brun, Eds. (Cambridge University Press), pp.
412-431 (2013). link
• 6. “Introduction to Decoherence-Free Subspaces and Noiseless Subsystems”, by D. Lidar and T.A. Brun, in Quantum Error Correction”, D. Lidar and T.A. Brun, Eds. (Cambridge University Press), pp.
78-104 (2013). [pdf]
• 5 .”Introduction to Decoherence and Noise in Open Quantum Systems”, by D. Lidar and T.A. Brun, in Quantum Error Correction”, D. Lidar and T.A. Brun, Eds. (Cambridge University Press), pp. 3-45
(2013). [pdf]
• 4 .”Paring Model Simulation on a Quantum Computer”, by M.S. Byrd, L.-A. Wu, and D. Lidar, in “Condensed Matter Series” Vol 20, J. W. Clark, R. M. Panoff, and H. Li, Eds. (Nova Science Pulishers),
pp. 485-496 (2006).
• 3 .”Decoherence-Free Subspaces and Subsystems”, by D.A. Lidar and K.B. Whaley, in “Irreversible Quantum Dynamics”, F. Benatti and R. Floreanini (Eds.), p. 83-120 (Springer Lecture Notes in
Physics,622, Berlin, 2003). Online at quant-ph/0301032 [pdf]
• 2 .“On the Abundance of Fractals”, by D. Avnir, O. Biham, D.A. Lidar (Hamburger), and O. Malcai, in “Fractal Frontiers”, M.M. Novak and T.G. Dewey, Eds., (World Scientific, Singapore), pp.
199-234 (1997).
• 1 . “Randomness and Apparent Fractality”, by D. A. Hamburger, O. Malcai, O. Biham, and D. Avnier, in “Fractals and Chaos in Chemical Engineering”, M. Giona and G. Biardis. Eds., (World Scientific
Singapore), pp. 103-114 (1996). [pdf]
• 25. “Beating the Ramsey limit on sensing with deterministic qubit control”, [2408.15926] by M. O. Hecht, K. Saurav, E. Vlachos, D. A. Lidar, E. M. Levenson-Falk.
• 24. “Quantum Property Preservation”, [2408.11262] by K. Saurav, D. A. Lidar.
• 23. “ClassiFIM: An Unsupervised Method To Detect Phase Transitions”, [2408.03323] by V. Kasatkin, E. Mozgunov, N. Ezzell, U. Mishra, I. Hen, D. A. Lidar.
• 22. “Detecting Quantum and Classical Phase Transitions via Unsupervised Machine Learning of the Fisher Information Metric”, [2408.03418] by V. Kasatkin, E. Mozgunov, N. Ezzell, D. A. Lidar.
• 21. “Virtual Z gates and symmetric gate compilation”, [2407.14782] by A. Vezvaee, V. Tripathi, D. Kowsari, E. M. Levenson-Falk, D. A. Lidar.
• 20. “Deterministic Benchmarking of Quantum Gates”, [2407.09942] by V. Tripathi, D. Kowsari, K. Saurav, H. Zhang, E. M. Levenson-Falk, D. A. Lidar.
• 19. “Qudit Dynamical Decoupling on a Superconducting Quantum Processor”, [2407.04893] by V. Tripathi, N. Goss, A. Vezvaee, L. B. Nguyen, I. Siddiqi, D. A. Lidar.
• 18. “Simulating nonlinear optical processes on a superconducting quantum device”, [2406.13003] by Y. Shi, B. Evert, A. F. Brown, V. Tripathi, E. A. Sete, V. Geyko, Y. Cho, J. L DuBois, D. A.
Lidar, I. Joseph, M. Reagor.
• 17. “Efficient Chromatic-Number-Based Multi-Qubit Decoherence and Crosstalk Suppression”, [2406.13901] by A. F. Brown, D. A. Lidar.
• 16. “Simulating Chemistry on Bosonic Quantum Devices”, [2404.10214] by R. Dutta, D. G. A. Cabral, N. Lyu, N. P. Vu, Y. Wang, B. Allen, X. Dan, R. G. Cortiñas, P. Khazaei, S. E. Smart, S. Nie, M.
H. Devoret, D. A. Mazziotti, P. Narang, C. Wang, J. D. Whitfield, A. K. Wilson, H. P. Hendrickson, D. A. Lidar, F. Pérez-Bernal, L. F. Santos, S. Kais, E. Geva, V. S. Batista.
• 15. “Quantum Fourier Transform using Dynamic Circuits”, [2403.09514] by E. Bäumer, V. Tripathi, A. Seif, D. A. Lidar, D. S. Wang.
• 14. “Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and fixed points”, [2402.08721] by P. Singkanipa, D.A. Lidar.
• 13. “Demonstration of Algorithmic Quantum Speedup for an Abelian Hidden Subgroup Problem”, [2401.07934] by P. Singkanipa, V. Kasatkin, Z. Zhou, G. Quiroz, D.A. Lidar.
• 12. “Scaling Advantage in Approximate Optimization with Quantum Annealing”, [2401.07184] by H. M. Bauza, D. A. Lidar.
• 11. “Demonstration of long-range correlations via susceptibility measurements in a one-dimensional superconducting Josephson spin chain”, [2111.04284] by D. M. Tennant, X. Dai, A. J. Martinez, R.
Trappen, D. Melanson, M A. Yurtalan, Y. Tang, S. Bedkihal, R. Yang, S. Novikov, J. A. Grover, S. M. Disseler, J. I. Basham, R. Das, D. K. Kim, A. J. Melville, B. M. Niedzielski, S. J. Weber, J.
L. Yoder, A. J. Kerman, E. Mozgunov, D. A. Lidar and A. Lupascu
• 10. “Suppression of crosstalk in superconducting qubits using dynamical decoupling”, [2108.04530] by V. Tripathi, H. Chen, M. Khezri, Ka-Wa Yip, E. M. Levenson-Falk, D. A. Lidar
• 9. “Quantum adiabatic theorem for unbounded Hamiltonians, with applications to superconducting circuits”, [2011.08116], by E. Mozgunov and D. A. Lidar
• 8. “Why and when is pausing beneficial in quantum annealing?”, [2005.01888], by H. Chen, D. A. Lidar
• 7. “Arbitrary-Time Error Suppression for Markovian Adiabatic Quantum Computing Using Stabilizer Subspace Codes”, [1904.12028], by D. A. Lidar
• 6. “Lecture Notes on the Theory of Open Quantum Systems”, [1902.00967], by D. A. Lidar
• 5. “Sensitivity of quantum speedup by quantum annealing to a noisy oracle”, [1901.02981], by S. Muthukrishnan, T. Albash and D. A. Lidar
• 4. “Exploring More-Coherent Quantum Annealing”, [1809.04485], by S. Novikov, R. Hinkey, S. Disseler, J. I. Basham, T. Albash, A. Risinger, D. Ferguson, D. A. Lidar and K. M. Zick
• 3. “Nested Quantum Annealing Correction at Finite Temperature: p-spin models”, [1803.01492], by S. Matsuura, H. Nishimori, W. Vinci and D. A. Lidar
• 2. “On the Computational Complexity of Curing the Sign Problem”, [1802.03408], by M. Marvian, D. A. Lidar and I. Hen
• 1. “When Diabatic Trumps Adiabatic in Quantum Optimization”, [1505.01249], by S. Muthukrishnan, T. Albash, and D. A. Lidar
Invited Reviews
• 2 . “Quantum Computing: Against the Odds of Imperfection”, by D.A. Lidar, Nature Physics 1, 145 (2005) [pdf]
• 1 . “Quantum Computers Made Lucid”, by D.A. Lidar, review of the book “A Shortcut Through Time: The Path to the Quantum Computer” by George Johnson, in Chemical & Engineering News 81, no. 32,
pp.36-37 (2003)[pdf]
• 2 . “Editorial: How to Control Decoherence and Entanglement in Quantum Complex Systems?”, J. Phys. B 40, E01 (2007), by V. Akulin, G. Kurizki, and D.A. Lidar [pdf]
• 1 . “Editorial: Quantum Information and Quantum Control”, Quant. Info. and Computation 5, 273 (2005), by P. Brumer, D. Lidar, H.-K. Lo, and A. Steinberg [pdf]
Conference Proceedings
• 6 . “Encoded Universality in Physical Implementations of a Quantum Computer”, by D. Bacon, J. Kempe, D. P. DiVincenzo, D. A. Lidar, and K. B. Whaley, Proceedings of the International Conference
on Experimental Implementation of Quantum Computation, Sydney, Australia (IQC 01), p. 257-264 (2001), Rinton Press.
• 5 . “Robust Dynamical Decoupling: Feedback-Free Error Correction”, by D.A. Lidar and K. Khodjasteh, [Proceedings of the 1st Asia-Pacific Conference on Quantum Information Science], in the
International Journal of Quantum Information, Vol. 3, Supplementary Issue 1, 41-52 (November 2005). [pdf]
• 4 . “Hybrid Decoherence-Free Error-Correcting Codes via Quantum Trajectories”, by K. Khodjasteh and D.A. Lidar, 6th International Conference on Quantum Communication, Measurement and Computing
(QCMC 02), July 22-26, 2002. Proceedings of the Quantum Communication, Measurement and Computing, 493-495 (2003)[pdf]
• 3 . “Quantum Computers and Decoherence: Exorcising the Demon from the Machine”, by D.A. Lidar and L.-A. Wu, in Proceedings of the SPIE, “Noise and Information in Nanoelectronics, Sensors, and
Standards”, Santa Fe, New Mexico, Vol. 5115, p. 256-270 (2003). Online at quant-ph/0302198 [pdf]
• 2 . “Fault-Tolerant Quantum Dynamical Decoupling”, by K. Khodjasteh and D.A. Lidar, CLEO/QELS 2005 Conference, Baltimore, MD, Joint Symposia on Coherent and Quantum Control [pdf]
• 1 . “Generalized Grover Search Algorithm for Arbitrary Initial Amplitude Distribution”, by D. Biron, O. Biham, E. Biham, M. Grassl, and D.A. Lidar, in “Quantum Computing and Quantum
Communications”, C.P. Williams (Ed.), Springer-Verlag Lecture Notes in Computer Science, Vol. 1509, p. 140-147 (1999)[pdf]
• 1 . “Dreams Versus Reality: Plenary Debate Session on Quantum Computing”, Fluctuation and Noise Letters, Vol 08, No. 02, pp.C27-C31 (2003), by D. Abbott, C. Doering, C. Caves, D. Lidar, H.
Brandt, A. Hamilton, D. Ferry, J. Gea-Banacloche, S. Bezrukov, and L. Kish. [pdf]
Some lectures on the web
• CQT Colloquium @ National University of Singapore: “Demonstration of Algorithmic Quantum Speedup” (May 16, 2024)
• Boston College Quantum Computation in Isolation Seminar: “Classical Boundaries of Quantum Supremacy” (April 2, 2021)
• Conference on Quantum Annealing/Adiabatic Quantum Computation, “Achievements of the IARPA-QEO and DARPA-QAFS programs & The prospects for quantum enhancement with quantum annealing” (Oct 5, 2020)
• Morning keynote lecture at Fujitsu Laboratories Advanced Technology Symposium “The impact of quantum computing” (Nov. 7, 2017)
• Adiabatic Quantum Computing Conference “Evidence for a limited quantum speedup on a quantum annealing device” (June 26, 2017)
• Quantum Annealing Correction (Third International Conference on Quantum Error Correction, Zurich, Switzerland, December 15-19, 2014)
• Introduction to dynamical decoupling (44th Symposium on Mathematical Physics, “New Developments in the Theory of Open Quantum Systems”, Torun, Poland, June 20-24, 2012)
• Quantum Approach to Information Retrieval: Adiabatic Quantum PageRank Algorithm (First NASA Quantum Future Technologies Conference, Moffett Field, California, Jan 17-21, 2012)
• Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling (10th Canadian Summer School on Quantum Information, Vancouver, British Columbia, July
23-25, 2010)
• Accurate and decoherence-protected adiabatic quantum computation (Quantum Algorithms, Computational Models, and Foundations of Quantum Mechanics, Vancouver, British Columbia, 2010)
• Preserving and extending quantum coherence: from the spin echo effect to fault tolerant quantum computation (IMA Workshop on Coherence, Control, and Dissipation, Minneapolis, March 3, 2009)
• Low-level fault tolerance via concatenated dynamical decoupling (Perimeter Institute, June 15, 2007)
• Adiabaticity in Open Quantum Systems: Geometric Phases & Adiabatic Quantum Computing (ISF Workshop on DEICS-QUDAL, 2006) [ppt]
• Quantum Computers and Decoherence: Exorcising the Demon from the Machine (Physics Colloquium, Hebrew University of Jerusalem, 2004)
• Hybrid quantum error prevention, reduction, and correction method (Fields Institute Conference on Quantum Information and Quantum Contol, July 2004) [ppt]
In the News
Courses Taught at USC
• Applied Linear Algebra for Engineering Spring 2006
• The Cutting Edge in Quantum Information Science Spring 2006
• Theory of Open Quantum Systems Fall 2006
• Physical Chemistry Seminar Spring 2007
• Physical Chemistry Seminar Fall 2006
• Introduction to Quantum Error Correction Spring 2007
Erdos number ≤ 3.
Einstein number ≤ 4 | {"url":"https://qserver.usc.edu/2016/02/daniel-lidar/","timestamp":"2024-11-03T00:34:25Z","content_type":"text/html","content_length":"127361","record_id":"<urn:uuid:5f06186c-72ae-425e-a6c4-329bf0f2d4b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00252.warc.gz"} |
Can you solve what an MIT professor once called 'the hardest logic puzzle ever'?
Neuropsych —
Can you solve what an MIT professor once called ‘the hardest logic puzzle ever’?
Logic puzzles can teach reasoning in a fun way that doesn’t feel like work.
Key Takeaways
• Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
• The problem, also known as the Three Gods Problem, is solvable, even if it doesn’t seem to be.
• It depends on using complex questions to assure that any answer given is useful.
Despite the general dislike of mathematics that most profess to have, many people enjoy logic puzzles. This is strange, as many logic puzzles are just variations of math problems. Gleefully ignorant
of this fact, many mathaphobes will try to solve riddles and puzzles of tremendous difficulty using reasoning tools they fear to employ when the subject is an equation.
Today, we’ll look at a puzzle, the polymath who devised it, and why you should consider picking up a book of logical puzzles next time you are at the library.
This puzzle was written by the brilliant logician Raymond Smullyan. Born in New York 101 years ago, Smullyan earned his undergraduate degree at the University of Chicago and his doctorate in
mathematics at Princeton, where he also taught for a few years.
An extremely prolific writer, he published several books on logic puzzles for popular consumption and an endless stream of textbooks and essays for an academic audience on logic. His puzzle books are
well regarded for introducing people to complex philosophical ideas, such as Gödel’s incompleteness theorems, in a fun and non-technical way.
Skilled in close-up magic, Smullyan once worked as a professional magician. He was also an accomplished pianist and an amateur astronomer who built his own telescope. Besides his interest in logic,
he also admired Taoist philosophy and published a book on it for a general audience.
He also found the time to appear on Johnny Carson, where, as in many of his books, he argued that people who like his puzzles claim to dislike math only because they don’t realize that they are one
and the same.
One of the more popular wordings of the problem, which MIT logic professor George Boolos said was the hardest ever, is:
“Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely
random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all
questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.”
Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering “da”
and “ja.”
Give yourself a minute to ponder this; we’ll look at a few answers below. Ready? Okay.
George Boolos’ solution focuses on finding either True or False through complex questions.
In logic, there is a commonly used function often written as “iff,” which means “if, and only if.” It would be used to say something like “The sky is blue if and only if Des Moines is in Iowa.” It is
a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement.
So, if you make a statement such as “the moon is made of Gorgonzola if, and only if, Rome is in Russia,” then you have made a true statement, as both parts of it are false. The statement “The moon
has no air if, and only if, Rome is in Italy,” is also true, as both parts of it are true. However, “The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York,” is false,
because one of the parts of that statement is true, and the other part is not (The fact that these items don’t rely on each other is immaterial for now).
In this puzzle, iff can be used here to control for the unknown value of “da” and “ja.” As the answers we get can be compared with what we know they would be if the parts of our question are all
true, all false, or if they differ.
Boolos would have us begin by asking god A, “Does “da” mean yes if and only if you are True if and only if B is Random?” No matter what A says, the answer you get is extremely useful. As he explains:
“If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not
Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that’s irrelevant), and therefore C is either True or False; and if A is
Random…and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False.”
No matter which god A is, an answer of “da” assures that C isn’t Random, and a response of “ja” means the same for B.
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From here, it is a simple matter of asking whichever one you know isn’t Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with
“Does da mean yes if, and only if, Rome is in Italy?” Since one part of this is accurate, we know that True will say “da,” and False will say “ja,” if faced with this question.
After that, you can ask the same god something like, “Does da mean yes if, and only if, A is Random?” and know exactly who is who by how they answer and the process of elimination.
If you’re confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a
positive and that two of the gods can be relied on to act consistently.
Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you’d like to try a
few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this interactive.
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7 min
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collinear 2.0.0
Warning: This version includes several breaking changes.
Main Changes
Function preference_order()
• Now works with any combination of categorical and numeric responses and predictors. Previously, only numeric responses were considered valid.
• Accepts a character vector with multiple response variables and returns a named list of data frames in such cases.
• All functions used as input for the argument f have been rewritten, with extended coverage of cases. These functions have also been consistently renamed following these rules:
□ A code indicating the metric: r2 for R-squared, auc for area under the curve (for binomial responses), and v for Cramer’s V (for categorical responses).
□ A code indicating the model: spearman, pearson, and v for direct association; glm for GLMs; gam for GAMs; rf for Random Forest models; and rpart for Recursive Partition Trees.
□ The model family for GLMs or GAMs: gaussian for numeric responses, binomial for binomial responses, and poisson for integer counts.
□ The term poly2 for GLMs with second-degree polynomials.
• When f = NULL, the function f_auto() determines an appropriate default adapted to the types of the response and predictors.
• Now issues a warning if predictors show a suspiciously high association with the response. The sensitivity of this test is controlled by the new argument warn_limit.
• Parallelization setup is now managed via future::plan(), and a progress bar is available through progressr::handlers().
Function collinear()
• Now works with any combination of categorical and numeric responses and predictors. Previously, only numeric responses were valid. Categorical predictors are excluded from VIF analysis but are
returned in the output if they pass the pairwise correlation test.
• Accepts a character vector with multiple response variables and returns a named list of data frames in such cases.
• The preference order is now computed internally if preference_order = NULL (default). Therefore, all relevant arguments of the function preference_order() have been added to collinear() with the
prefix “preference_”.
• Parallelization setup is now managed via future::plan(), with a progress bar provided by progressr::handlers(). This setup is leveraged by preference_order() and cor_select().
• Target encoding can be disabled by setting the encoding_method argument to NULL.
• VIF filtering can be disabled by setting max_vif to NULL.
• Pairwise correlation filtering can be disabled by setting max_cor to NULL.
Function cor_select()
• A new robust forward selection algorithm ensures that the most important predictors are retained after multicollinearity filtering when preference_order is used.
• Target encoding, along with the response and encoding_method arguments, has been removed from this function. This change also applies to cor_df().
• The function now calls validate_data_cor() to ensure that the data is suitable for pairwise correlation multicollinearity filtering.
• Parallelization setup is now managed via future::plan(), with a progress bar provided by progressr::handlers(). This setup is used by cor_numeric_vs_categorical() and
cor_categorical_vs_categorical() to speed up pairwise correlation computation.
Function cor_df()
• Fixed a bug that prevented cor_numeric_vs_categorical() and cor_categorical_vs_categorical() from triggering properly.
Function vif_select()
• A new robust forward selection algorithm better preserves predictors with higher preference when preference_order is used.
• Target encoding, along with the response and encoding_method arguments, has been removed. As a result, this function now only works with numeric predictors. This change also applies to vif_df().
• The new function validate_data_vif() is called to ensure the data is suitable for VIF-based multicollinearity filtering. Attempting a VIF analysis in a data frame with more columns than rows now
returns an error.
Function target_encoding_lab() and Companion Functions
• Completely rewritten for parallelization using future::plan() and a progress bar via progressr::handlers().
• The default encoding method is now “loo” (leave-one-out), as it provides more useful results in most cases.
• The functions target_encoding_mean(), target_encoding_rank(), and target_encoding_loo() have been simplified to the bare minimum, with all redundant logic moved to target_encoding_lab().
• NA cases in the predictor to encode are now grouped under “NA”.
• The “rnorm” method has been deprecated, and the function target_encoding_rnorm() has been removed from the package.
Other Changes
• Added the function cor_clusters() to group predictors using stats::hclust() based on their pairwise correlation matrix.
• Streamlined the package documentation using roxygen methods to inherit sections and parameters.
• Removed dplyr as a dependency.
• Added mgcv, rpart, and ranger to Imports to support all f_xxx() functions from the start.
• All warnings in data validation functions have been converted to messages. These messages now indicate the function that generated them, aiding in debugging and ensuring that messages and
warnings are printed in the correct order.
collinear 1.1.1
Hotfix of issue with solve(tol = 0) in systems with no large double support (noLD). This one wasn’t fun.
collinear 1.1.0
Added argument “smoothing” to target_encoding_mean() function to implement original target encoding method.
Added alias f_rf_rsquared() to the function f_rf_deviance().
Added column “vi_binary” to vi as a binary version of “vi_mean”.
Added function auc_score() to compute the area under the curve of predictions from binary models.
Added function case_weights() to compute case weights when binary responses are unbalanced.
Added function f_rf_auc_balanced() to be used as input for the f argument of preference_order() when the response is binary and balanced.
Added function f_rf_auc_unbalanced() to be used as input for the f argument of preference_order() when the response is binary and unbalanced.
Added function f_gam_auc_balanced() to be used as input for the f argument of preference_order() when the response is binary and balanced.
Added function f_gam_auc_unbalanced() to be used as input for the f argument of preference_order() when the response is binary and unbalanced.
Added function f_logistic_auc_balanced() to be used as input for the f argument of preference_order() when the response is binary and balanced.
Added function f_logistic_auc_unbalanced() to be used as input for the f argument of preference_order() when the response is binary and unbalanced.
Fixed issue with perfect correlations in vif_df(). Now perfect correlations are replaced with 0.99 (for correlation == 1) and -0.99 (for correlation == -1) in the correlation matrix to avoid errors
in solve().
Added the example dataset toy, derived from vi, but with known relationships between all variables.
Fixed issue in function cor_df() where many cases would be lost because the logic to remove diagonals was flawed, as all pairs with correlation == 1 were being removed.
Fixed issue in functions cor_select() and vif_select() where ignoring predictors and using only df would lead to empty selections.
collinear 1.0.2
This version fixes bugs in two functions: cor_select() and cor_df()
• When only one variable was left in the correlation matrix, the one column matrix became a vector with no colnames, which yielded an error. Now, to avoid this issue, drop = FALSE is used in the
matrix subsetting.
• The previous version started removing predictors on a backwards fashion, from the last predictor in preference order, moving up one by one to the top. Under the wrong circumstances (low number of
predictors, low max_cor, and high correlation between first and second predictors in preference order) this configuration would lead to keep only the first predictor, even when having others
comply with the max_cor restriction lower down in the preference order. The new version produces smaller subsets of predictors with a higher diversity.
• The data frame returned pairs of the same variable when cor_method was “spearman”. Fixed with a dplyr::filter(x != y).
collinear 1.0.1
Re-submission after minor CRAN comments.
• version number bumped up to 1.0.1
• removed if(interactive()){} from all @examples.
• removed plot() call from the @examples of the function target_encoding_lab() because it was messing up pkgdown’s build. This piece of code triggered the comment “Please always make sure to reset
to user’s options()” by the reviewer, so this should solve the issue.
• made sure that all examples run in less than 5 seconds.
• fixed a bug in which all functions would include the response as a predictor when ‘predictors = NULL’ and ‘response’ was a valid column of the input data frame.
collinear 1.0.0
First functional version of the package submitted to CRAN. | {"url":"https://cran.rediris.es/web/packages/collinear/news/news.html","timestamp":"2024-11-13T12:53:34Z","content_type":"application/xhtml+xml","content_length":"12756","record_id":"<urn:uuid:22cb1305-f33f-4fe9-99f7-3d203c377fcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00398.warc.gz"} |
In the given marathon race track, the distance between the halt and the finish point is .
In the given marathon race track, the distance between the halt and the finish point is .
The correct option is C 2,630 m
The total length of the marathon track = 3,750 m
The distance between the start point and the halt = 1.12 km
To calculate the distance between the halt and the finish line, we have to subtract
the distance between start to halt point
from the total length of the track.
We know that, 1 km = 1,000 m
So, we will multiply 1.12 with 1,000.
So, 1.12 km =1.12×1 Km
=1.12×1000 m
=1,120 m
So, calculating the distance between the halt to finish point is:
⟹ 3,750 - 1,120 = 2,630 m
Thus, the distance between the halt to finish point is 2,630 meters. | {"url":"https://byjus.com/question-answer/in-the-given-marathon-race-track-the-distance-between-the-halt-and-the-finish-point-1/","timestamp":"2024-11-14T10:33:49Z","content_type":"text/html","content_length":"174151","record_id":"<urn:uuid:23e48000-7a2c-4164-888e-c58950d85631>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00634.warc.gz"} |
dart:typed_data library
Lists that efficiently handle fixed sized data (for example, unsigned 8 byte integers) and SIMD numeric types.
To use this library in your code:
import 'dart:typed_data';
A sequence of bytes underlying a typed data object.
A fixed-length, random-access sequence of bytes that also provides random and unaligned access to the fixed-width integers and floating point numbers represented by those bytes.
Builds a list of bytes, allowing bytes and lists of bytes to be added at the end.
Endianness of number representation.
A fixed-length list of IEEE 754 single-precision binary floating-point numbers that is viewable as a TypedData.
Float32x4 immutable value type and operations.
A fixed-length list of Float32x4 numbers that is viewable as a TypedData.
A fixed-length list of IEEE 754 double-precision binary floating-point numbers that is viewable as a TypedData.
Float64x2 immutable value type and operations.
A fixed-length list of Float64x2 numbers that is viewable as a TypedData.
A fixed-length list of 16-bit signed integers that is viewable as a TypedData.
A fixed-length list of 32-bit signed integers that is viewable as a TypedData.
Int32x4 and operations.
A fixed-length list of Int32x4 numbers that is viewable as a TypedData.
A fixed-length list of 64-bit signed integers that is viewable as a TypedData.
A fixed-length list of 8-bit signed integers.
A typed view of a sequence of bytes.
A TypedData fixed-length List-view on the bytes of buffer.
A fixed-length list of 16-bit unsigned integers that is viewable as a TypedData.
A fixed-length list of 32-bit unsigned integers that is viewable as a TypedData.
A fixed-length list of 64-bit unsigned integers that is viewable as a TypedData.
A fixed-length list of 8-bit unsigned integers.
A fixed-length list of 8-bit unsigned integers. | {"url":"https://api.dart.dev/stable/3.5.4/dart-typed_data/dart-typed_data-library.html","timestamp":"2024-11-11T19:48:20Z","content_type":"text/html","content_length":"15517","record_id":"<urn:uuid:15ed671e-ced8-48f9-8044-ddb74709621d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00758.warc.gz"} |
An Introduction to Bayesian Statistics
Bayesian statistics has emerged as a powerful methodology for making decisions from data in the applied sciences.
Credit: Technology Networks
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What is Bayesian statistics?
In the ever-evolving toolkit of statistical analysis techniques, Bayesian statistics has emerged as a popular and powerful methodology for making decisions from data in the applied sciences. Bayesian
is not just a family of techniques but brings a new way of thinking to statistics, in how it deals with probability, uncertainty and drawing inferences from an analysis. Bayesian statistics has
become influential in physics, engineering and the medical and social sciences, and underpins much of the developing fields of machine learning and artificial intelligence (AI). In this article we
will explore key differences between Bayesian and traditional frequentist statistical approaches, some fundamental concepts at the core of Bayesian statistics, the central Bayes’ rule, the thinking
around making Bayesian inferences and finally explore a real-world example of applying Bayesian statistics to a scientific problem.
Bayesian vs frequentist statistics
The field of statistics is rooted in probability theory, but Bayesian statistics deals with probability differently than frequentist statistics. Frequentist thinking follows that the probability of
an event occurring can be interpreted as the proportion of times the event would occur in many repeated trials or situations. By contrast, the Bayesian approach is based on a subjective
interpretation of probability, where the probability of the event occurring relates to the analyst expressing their own evaluation as well as what happens with the event itself, and prior evidence or
beliefs are used to help calculate the size of the probability. In essence, the Bayesian and frequentist approaches to statistics share the same goal of making inferences, predictions or drawing
conclusions based on data but deal with uncertainty in different ways.
In practice, this means that, for frequentists, parameters (values about a population of interest that we want to estimate such as means or proportions) are fixed and unknown, whereas for Bayesians
these parameters can be assigned probabilities and updated using prior knowledge or beliefs. In frequentist statistics, confidence intervals are used to quantify uncertainty around estimates, whereas
Bayesian statistics provides a posterior distribution, combining the prior beliefs and likelihood of the data. P-values are used in frequentist statistics to test a hypothesis by evaluating the
probability of observing data as extreme as what was observed, whereas Bayesian hypothesis testing is based on comparing posterior probabilities given the data, incorporating prior beliefs and
updating them.
Bayesian fundamentals and Bayes’ rule
There are several concepts that are key to understanding Bayesian statistics, some of which are unique to the field:
• Conditional probability is the probability of an event A given B, which is important for updating beliefs. For example, a researcher may be interested in the conditional probability of developing
cancer given a particular risk factor such as smoking. We extend this to Bayesian statistics and update beliefs using Bayes’ rule, alongside the three fundamental elements in a Bayesian analysis;
the prior distribution, likelihood and posterior distribution.
• Prior distribution is some reasonable belief about the plausibility of values of an unknown parameter of interest, without any evidence from the new data we are analysing.
• Likelihood encompasses the different possible values of the parameter based on analysis of the new data.
• Posterior distribution is the combination of the prior distribution and the likelihood using Bayes’ rule:
where A and B are some unknown parameter of interest and some new data, respectively. P(A|B) is the probability of A given B is true or the posterior distribution, P(B|A) is the probability of B
given A is true or the likelihood, and P(A) and P(B) are the independent probabilities of A and B.
This process of using Bayes’ rule to update prior beliefs is called Bayesian updating. The information we aim to update can also be simply known as the prior. It is important to note that the prior
can take the form of other data, for example a statistical estimate from a previous analysis, or simply an estimate based on belief or domain knowledge. A prior belief need not be quantifiable as a
probability, but in some cases may be qualitative or subjective in nature, for example a doctor’s opinion on whether a patient had a certain disease before a diagnostic test is conducted. After
updating the prior using Bayes’ rule, the information we end up with is the posterior. The posterior distribution forms the basis of a statistical inference made from a Bayesian analysis.
Example of Bayesian statistics
Let us consider an example of a simple Bayesian analysis, where rather than go through equations in detail, we use graphs to illustrate the prior distribution, likelihood and posterior distribution
under a beta distribution. The beta distribution is commonly used (in both Bayesian and frequentist statistics) in estimating the probability of an outcome, where this probability can take any value
between 0 and 1 (0% and 100%).
Suppose we are conducting a survey in rural Nigeria where we collect responses from participants to estimate the prevalence of visual impairment in an elderly community. First, we need to define our
prior distribution, which we do using the prior belief that the prevalence will be 75% based on previous surveys on visual impairment among similarly aged participants. Our prior distribution is then
centered around 75% (Figure 1.1). Next, we observe in the current survey that 85% of participants have visual impairment, and so our likelihood is centered around this value (Figure 1.2). Likelihood
is calculated using the Bernoulli distribution, a distribution used specifically for modeling binary outcomes or events (visual impairment: yes/no) in this example, for the analysis of our current
data, compared with Beta distribution for the prior which differs in nature in that it is an expression of a prior belief.
After setting up the prior and computing the likelihood, we next combine them and calculate the posterior distribution using Bayes’ rule. In our example, we find that the posterior distribution
(based on the Beta distribution) has been influenced by our prior belief and moved to a value less than that which we observed in the current survey data with a mode at around 80% (Figure 1.3).
Figure 1: Bayesian distributions for prevalence of visual impairment.
Where A is the unknown parameter for which probabilities can be calculated, and
B is the new data.
An important aspect of this type of analysis is that the posterior distribution is heavily influenced by the number of participants in our current study. If we imagined that our current survey had a
much higher sample size, the posterior distribution may not have moved away from 85% at all. As more new data are collected, the likelihood begins to dominate the prior. This example shows the key
fundamentals of Bayesian inference in action and demonstrates how we can use Bayesian inference to update our beliefs based on new data. | {"url":"https://www.technologynetworks.com/drug-discovery/articles/an-introduction-to-bayesian-statistics-380296","timestamp":"2024-11-12T19:33:40Z","content_type":"text/html","content_length":"114381","record_id":"<urn:uuid:d5fcfc51-0f7d-4aad-b100-84153bfa641c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00236.warc.gz"} |
How does the radius affect the moment of inertia? | Socratic
How does the radius affect the moment of inertia?
1 Answer
The moment of inertia, $I$, of a single mass, $M$, being twirled by a thread of length, $R$, is
$I = M \cdot {R}^{2}$
A body that is being rotated will closely resemble that relationship. The formulas for various geometric shapes are derived with integration. For example, for a solid sphere, moment of inertia is
$I = \left(\frac{2}{5}\right) \cdot M \cdot {R}^{2}$
I hope this helps,
Impact of this question
20837 views around the world | {"url":"https://socratic.org/questions/how-does-the-radius-affect-the-moment-of-inertia","timestamp":"2024-11-04T02:58:38Z","content_type":"text/html","content_length":"33232","record_id":"<urn:uuid:7f485f73-398b-46b7-97ba-53512de7b865>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00144.warc.gz"} |
GTC Silicon Valley-2019: Infusing Physics into Deep Learning
GTC Silicon Valley-2019: Infusing Physics into Deep Learning Algorithms with Applications to Stable Landing of Drones
Note: This video may require joining the NVIDIA Developer Program or login
GTC Silicon Valley-2019 ID:S9732:Infusing Physics into Deep Learning Algorithms with Applications to Stable Landing of Drones
Anima Anandkumar(NVIDIA)
We'll talk about how we're incorporating physics into deep learning algorithms. Standard deep learning algorithms are based on a function-fitting approach that does not exploit any domain knowledge
or constraints. This makes them unsuitable for applications like robotics that require safety or stability guarantees. These algorithms also require large amounts of labeled data, which is not
readily available. We'll discuss how we're overcoming these limitations by infusing physics into deep learning algorithms, and how we're applying this to stable landing of quadrotor drones. We've
developed a robust deep learning-based nonlinear controller called Neural-Lander, which learns ground-effect aerodynamic forces that are hard to model. We'll also touch on how Neural-Lander can land
significantly faster while maintaining stability. | {"url":"https://developer.nvidia.com/gtc/2019/video/s9732","timestamp":"2024-11-08T22:28:20Z","content_type":"text/html","content_length":"40712","record_id":"<urn:uuid:a2377d67-e4ea-495b-8148-d41e3a56c4f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00653.warc.gz"} |
Hybrid phase transition into an absorbing state: Percolation and avalanches
Abstract (may include machine translation)
Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid
phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it. Here we study these phenomena on the Erdos-Rényi and
the two-dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors: divergence of the fluctuations of the order parameter and
power-law size distribution of finite avalanches at a transition point. At the transition point global or "infinite" avalanches occur, while the finite ones have a power law size distribution; thus
the avalanche statistics also has the nature of a HPT. The exponent βm of the order parameter is 1/2 under general conditions, while the value of the exponent γm characterizing the fluctuations of
the order parameter depends on the system. The critical behavior of the finite avalanches can be described by another set of exponents, βa and γa. These two critical behaviors are coupled by a
scaling law: 1-βm=γa.
Dive into the research topics of 'Hybrid phase transition into an absorbing state: Percolation and avalanches'. Together they form a unique fingerprint. | {"url":"https://research.ceu.edu/en/publications/hybrid-phase-transition-into-an-absorbing-state-percolation-and-a","timestamp":"2024-11-05T23:42:20Z","content_type":"text/html","content_length":"55984","record_id":"<urn:uuid:f5bcb113-50dd-4ed6-beb4-a6efc5f49da4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00879.warc.gz"} |
Hoare logic
Jump to navigation Jump to search
Hoare logic (also known as Floyd–Hoare logic or Hoare rules) is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in
1969 by the British computer scientist and logician Tony Hoare, and subsequently refined by Hoare and other researchers.^[1] The original ideas were seeded by the work of Robert W. Floyd, who had
published a similar system^[2] for flowcharts.
Hoare triple[edit]
The central feature of Hoare logic is the Hoare triple. A triple describes how the execution of a piece of code changes the state of the computation. A Hoare triple is of the form
${\displaystyle \{P\}C\{Q\}}$
where ${\displaystyle P}$ and ${\displaystyle Q}$ are assertions and ${\displaystyle C}$ is a command.^[note 1] ${\displaystyle P}$ is named the precondition and ${\displaystyle Q}$ the postcondition
: when the precondition is met, executing the command establishes the postcondition. Assertions are formulae in predicate logic.
Hoare logic provides axioms and inference rules for all the constructs of a simple imperative programming language. In addition to the rules for the simple language in Hoare's original paper, rules
for other language constructs have been developed since then by Hoare and many other researchers. There are rules for concurrency, procedures, jumps, and pointers.
Partial and total correctness[edit]
Using standard Hoare logic, only partial correctness can be proven, while termination needs to be proved separately. Thus the intuitive reading of a Hoare triple is: Whenever ${\displaystyle P}$
holds of the state before the execution of ${\displaystyle C}$, then ${\displaystyle Q}$ will hold afterwards, or ${\displaystyle C}$ does not terminate. In the latter case, there is no "after", so $
{\displaystyle Q}$ can be any statement at all. Indeed, one can choose ${\displaystyle Q}$ to be false to express that ${\displaystyle C}$ does not terminate.
Total correctness can also be proven with an extended version of the While rule.
In his 1969 paper, Hoare used a narrower notion of termination which also entailed absence of any run-time errors: "Failure to terminate may be due to an infinite loop; or it may be due to violation
of an implementation-defined limit, for example, the range of numeric operands, the size of storage, or an operating system time limit."^[3]
Empty statement axiom schema[edit]
The empty statement rule asserts that the ${\displaystyle {\textbf {skip}}}$ statement does not change the state of the program, thus whatever holds true before ${\displaystyle {\textbf {skip}}}$
also holds true afterwards.^[note 2]
${\displaystyle {\dfrac {}{\{P\}{\textbf {skip}}\{P\}}}}$
Assignment axiom schema[edit]
The assignment axiom states that, after the assignment, any predicate that was previously true for the right-hand side of the assignment now holds for the variable. Formally, let ${\displaystyle P}$
be an assertion in which the variable ${\displaystyle x}$ is free. Then:
${\displaystyle {\dfrac {}{\{P[E/x]\}x:=E\{P\}}}}$
where ${\displaystyle P[E/x]}$ denotes the assertion ${\displaystyle P}$ in which each free occurrence of ${\displaystyle x}$ has been replaced by the expression ${\displaystyle E}$.
The assignment axiom scheme means that the truth of ${\displaystyle P[E/x]}$ is equivalent to the after-assignment truth of ${\displaystyle P}$. Thus were ${\displaystyle P[E/x]}$ true prior to the
assignment, by the assignment axiom, then ${\displaystyle P}$ would be true subsequent to which. Conversely, were ${\displaystyle P[E/x]}$ false (i.e. ${\displaystyle eg P[E/x]}$ true) prior to the
assignment statement, ${\displaystyle P}$ must then be false afterwards.
Examples of valid triples include:
□ ${\displaystyle \{x+1=43\}y:=x+1\{y=43\}}$
□ ${\displaystyle \{x+1\leq N\}x:=x+1\{x\leq N\}}$
All preconditions that are not modified by the expression can be carried over to the postcondition. In the first example, assigning ${\displaystyle y:=x+1}$ does not change the fact that ${\
displaystyle x+1=43}$, so both statements may appear in the postcondition. Formally, this result is obtained by applying the axiom schema with ${\displaystyle P}$ being (${\displaystyle y=43}$ and $
{\displaystyle x+1=43}$), which yields ${\displaystyle P[(x+1)/y]}$ being (${\displaystyle x+1=43}$ and ${\displaystyle x+1=43}$), which can in turn be simplified to the given precondition ${\
displaystyle x+1=43}$.
The assignment axiom scheme is equivalent to saying that to find the precondition, first take the post-condition and replace all occurrences of the left-hand side of the assignment with the
right-hand side of the assignment. Be careful not to try to do this backwards by following this incorrect way of thinking: ${\displaystyle \{P\}x:=E\{P[E/x]\}}$; this rule leads to nonsensical
examples like:
${\displaystyle \{x=5\}x:=3\{3=5\}}$
Another incorrect rule looking tempting at first glance is ${\displaystyle \{P\}x:=E\{P\wedge x=E\}}$; it leads to nonsensical examples like:
${\displaystyle \{x=5\}x:=x+1\{x=5\wedge x=x+1\}}$
While a given postcondition ${\displaystyle P}$ uniquely determines the precondition ${\displaystyle P[E/x]}$, the converse is not true. For example:
□ ${\displaystyle \{0\leq y\cdot y\wedge y\cdot y\leq 9\}x:=y\cdot y\{0\leq x\wedge x\leq 9\}}$,
□ ${\displaystyle \{0\leq y\cdot y\wedge y\cdot y\leq 9\}x:=y\cdot y\{0\leq x\wedge y\cdot y\leq 9\}}$ ,
□ ${\displaystyle \{0\leq y\cdot y\wedge y\cdot y\leq 9\}x:=y\cdot y\{0\leq y\cdot y\wedge x\leq 9\}}$ , and
□ ${\displaystyle \{0\leq y\cdot y\wedge y\cdot y\leq 9\}x:=y\cdot y\{0\leq y\cdot y\wedge y\cdot y\leq 9\}}$
are valid instances of the assignment axiom scheme.
The assignment axiom proposed by Hoare does not apply when more than one name may refer to the same stored value. For example,
${\displaystyle \{y=3\}x:=2\{y=3\}}$
is wrong if ${\displaystyle x}$ and ${\displaystyle y}$ refer to the same variable (aliasing), although it is a proper instance of the assignment axiom scheme (with both ${\displaystyle \{P\}}$ and $
{\displaystyle \{P[2/x]\}}$ being ${\displaystyle \{y=3\}}$).
Rule of composition[edit]
Verifying swap-code
without auxiliary variables
The three statements below (line 2, 4, 6) exchange the values of the
variables ${\displaystyle a}$ and ${\displaystyle b}$, without needing
an auxiliary variable. In the verification proof, the initial value of
${\displaystyle a}$ and ${\displaystyle b}$ is denoted by the constant
${\displaystyle A}$ and ${\displaystyle B}$, respectively. The proof is
best read backwards, starting from line 7; for example, line 5 is
obtained from line 7 by replacing ${\displaystyle a}$ (target
expression in line 6) by ${\displaystyle a-b}$ (source expression in
line 6). Some arithmetical simplifications are used tacitly, viz. ${\
displaystyle a-(a-b)=b}$ (line 5→3), and ${\displaystyle a+b-b=a}$
(line 3→1).
Nr Code Assertions
1: ${\displaystyle \{a=A\wedge b=B\}}$
2: ${\displaystyle a:=a+b;}$
3: ${\displaystyle \{a-b=A\wedge b=B\}}$
4: ${\displaystyle b:=a-b;}$
5: ${\displaystyle \{b=A\wedge a-b=B\}}$
6: ${\displaystyle a:=a-b}$
7: ${\displaystyle \{b=A\wedge a=B\}}$
Hoare's rule of composition applies to sequentially executed programs ${\displaystyle S}$ and ${\displaystyle T}$, where ${\displaystyle S}$ executes prior to ${\displaystyle T}$ and is written ${\
displaystyle S;T}$ (${\displaystyle Q}$ is called the midcondition):^[4]
${\displaystyle {\dfrac {\{P\}S\{Q\}\quad ,\quad \{Q\}T\{R\}}{\{P\}S;T\{R\}}}}$
For example, consider the following two instances of the assignment axiom:
${\displaystyle \{x+1=43\}y:=x+1\{y=43\}}$
${\displaystyle \{y=43\}z:=y\{z=43\}}$
By the sequencing rule, one concludes:
${\displaystyle \{x+1=43\}y:=x+1;z:=y\{z=43\}}$
Another example is shown in the right box.
Conditional rule[edit]
${\displaystyle {\dfrac {\{B\wedge P\}S\{Q\}\quad ,\quad \{eg B\wedge P\}T\{Q\}}{\{P\}{\textbf {if}}\ B\ {\textbf {then}}\ S\ {\textbf {else}}\ T\ {\textbf {endif}}\{Q\}}}}$
The conditional rule states that a postcondition ${\displaystyle Q}$ common to ${\displaystyle {\textbf {then}}}$ and ${\displaystyle {\textbf {else}}}$ part is also a postcondition of the whole ${\
displaystyle {\textbf {if}}\cdots {\textbf {endif}}}$ statement. In the ${\displaystyle {\textbf {then}}}$ and the ${\displaystyle {\textbf {else}}}$ part, the unnegated and negated condition ${\
displaystyle B}$ can be added to the precondition ${\displaystyle P}$, respectively. The condition, ${\displaystyle B}$, must not have side effects. An example is given in the next section.
This rule was not contained in Hoare's original publication.^[1] However, since a statement
${\displaystyle {\textbf {if}}\ B\ {\textbf {then}}\ S\ {\textbf {else}}\ T\ {\textbf {endif}}}$
has the same effect as a one-time loop construct
${\displaystyle {\textbf {bool}}\ b:={\textbf {true}};{\textbf {while}}\ B\wedge b\ {\textbf {do}}\ S;b:={\textbf {false}}\ {\textbf {done}};b:={\textbf {true}};{\textbf {while}}\ eg B\wedge b\
{\textbf {do}}\ T;b:={\textbf {false}}\ {\textbf {done}}}$
the conditional rule can be derived from the other Hoare rules. In a similar way, rules for other derived program constructs, like ${\displaystyle {\textbf {for}}}$ loop, ${\displaystyle {\textbf
{do}}\cdots {\textbf {until}}}$ loop, ${\displaystyle {\textbf {switch}}}$, ${\displaystyle {\textbf {break}}}$, ${\displaystyle {\textbf {continue}}}$ can be reduced by program transformation to the
rules from Hoare's original paper.
Consequence rule[edit]
${\displaystyle {\dfrac {P_{1}\rightarrow P_{2}\quad ,\quad \{P_{2}\}S\{Q_{2}\}\quad ,\quad Q_{2}\rightarrow Q_{1}}{\{P_{1}\}S\{Q_{1}\}}}}$
This rule allows to strengthen the precondition and/or to weaken the postcondition. It is used e.g. to achieve literally identical postconditions for the ${\displaystyle {\textbf {then}}}$ and the $
{\displaystyle {\textbf {else}}}$ part.
For example, a proof of
${\displaystyle \{0\leq x\leq 15\}{\textbf {if}}\ x<15\ {\textbf {then}}\ x:=x+1\ {\textbf {else}}\ x:=0\ {\textbf {endif}}\{0\leq x\leq 15\}}$
needs to apply the conditional rule, which in turn requires to prove
${\displaystyle \{0\leq x\leq 15\wedge x<15\}x:=x+1\{0\leq x\leq 15\}}$, or simplified
${\displaystyle \{0\leq x<15\}x:=x+1\{0\leq x\leq 15\}}$
for the ${\displaystyle {\textbf {then}}}$ part, and
${\displaystyle \{0\leq x\leq 15\wedge x\geq 15\}x:=0\{0\leq x\leq 15\}}$, or simplified
${\displaystyle \{x=15\}x:=0\{0\leq x\leq 15\}}$
for the ${\displaystyle {\textbf {else}}}$ part.
However, the assignment rule for the ${\displaystyle {\textbf {then}}}$ part requires to choose ${\displaystyle P}$ as ${\displaystyle 0\leq x\leq 15}$; rule application hence yields
${\displaystyle \{0\leq x+1\leq 15\}x:=x+1\{0\leq x\leq 15\}}$, which is logically equivalent to
${\displaystyle \{-1\leq x<15\}x:=x+1\{0\leq x\leq 15\}}$.
The consequence rule is needed to strengthen the precondition ${\displaystyle \{-1\leq x<15\}}$ obtained from the assignment rule to ${\displaystyle \{0\leq x<15\}}$ required for the conditional
Similarly, for the ${\displaystyle {\textbf {else}}}$ part, the assignment rule yields
${\displaystyle \{0\leq 0\leq 15\}x:=0\{0\leq x\leq 15\}}$, or equivalently
${\displaystyle \{{\textbf {true}}\}x:=0\{0\leq x\leq 15\}}$,
hence the consequence rule has to be applied with ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ being ${\displaystyle \{x=15\}}$ and ${\displaystyle \{{\textbf {true}}\}}$, respectively, to
strengthen again the precondition. Informally, the effect of the consequence rule is to "forget" that \{x=15\} is known at the entry of the ${\displaystyle {\textbf {else}}}$ part, since the
assignment rule used for the ${\displaystyle {\textbf {else}}}$ part doesn't need that information.
While rule[edit]
${\displaystyle {\dfrac {\{P\wedge B\}S\{P\}}{\{P\}{\textbf {while}}\ B\ {\textbf {do}}\ S\ {\textbf {done}}\{eg B\wedge P\}}}}$
Here ${\displaystyle P}$ is the loop invariant, which is to be preserved by the loop body ${\displaystyle S}$. After the loop is finished, this invariant ${\displaystyle P}$ still holds, and moreover
${\displaystyle eg B}$ must have caused the loop to end. As in the conditional rule, ${\displaystyle S}$ must not have side effects.
For example, a proof of
${\displaystyle \{x\leq 10\}{\textbf {while}}\ x<10\ {\textbf {do}}\ x:=x+1\ {\textbf {done}}\{eg x<10\wedge x\leq 10\}}$
by the while rule requires to prove
${\displaystyle \{x\leq 10\wedge x<10\}x:=x+1\{x\leq 10\}}$, or simplified
${\displaystyle \{x<10\}x:=x+1\{x\leq 10\}}$,
which is easily obtained by the assignment rule. Finally, the postcondition ${\displaystyle \{eg x<10\wedge x\leq 10\}}$ can be simplified to ${\displaystyle \{x=10\}}$.
For another example, the while rule can be used to formally verify the following strange program to compute the exact square root ${\displaystyle x}$ of an arbitrary number ${\displaystyle a}$—even
if ${\displaystyle x}$ is an integer variable and ${\displaystyle a}$ is not a square number:
${\displaystyle \{{\textbf {true}}\}{\textbf {while}}\ x\cdot xeq a\ {\textbf {do}}\ {\textbf {skip}}\ {\textbf {done}}\{x\cdot x=a\wedge {\textbf {true}}\}}$
After applying the while rule with ${\displaystyle P}$ being ${\displaystyle {\textbf {true}}}$, it remains to prove
${\displaystyle \{{\textbf {true}}\wedge x\cdot xeq a\}{\textbf {skip}}\{{\textbf {true}}\}}$,
which follows from the skip rule and the consequence rule.
In fact, the strange program is partially correct: if it happened to terminate, it is certain that ${\displaystyle x}$ must have contained (by chance) the value of ${\displaystyle a}$'s square root.
In all other cases, it will not terminate; therefore it is not totally correct.
While rule for total correctness[edit]
If the above ordinary while rule is replaced by the following one, the Hoare calculus can also be used to prove total correctness, i.e. termination^[note 3] as well as partial correctness. Commonly,
square brackets are used here instead of curly braces to indicate the different notion of program correctness.
${\displaystyle {\dfrac {<\ {\text{is a well-founded ordering on the set}}\ D\quad ,\quad [P\wedge B\wedge t\in D\wedge t=z]S[P\wedge t\in D\wedge t<z]}{[P\wedge t\in D]{\textbf {while}}\ B\ {\
textbf {do}}\ S\ {\textbf {done}}[eg B\wedge P\wedge t\in D]}}}$
In this rule, in addition to maintaining the loop invariant, one also proves termination by way of an expression ${\displaystyle t}$, called the loop variant, whose value strictly decreases with
respect to a well-founded relation ${\displaystyle <}$ on some domain set ${\displaystyle D}$ during each iteration. Since ${\displaystyle <}$ is well-founded, a strictly decreasing chain of members
of ${\displaystyle D}$ can have only finite length, so ${\displaystyle t}$ cannot keep decreasing forever. (For example, the usual order ${\displaystyle <}$ is well-founded on positive integers ${\
displaystyle \mathbb {N} }$, but neither on the integers ${\displaystyle \mathbb {Z} }$ nor on positive real numbers ${\displaystyle \mathbb {R} ^{+}}$; all these sets are meant in the mathematical,
not in the computing sense, they are all infinite in particular.)
Given the loop invariant ${\displaystyle P}$, the condition ${\displaystyle B}$ must imply that ${\displaystyle t}$ is not a minimal element of ${\displaystyle D}$, for otherwise the body ${\
displaystyle S}$ could not decrease ${\displaystyle t}$ any further, i.e. the premise of the rule would be false. (This is one of various notations for total correctness.) ^[note 4]
Resuming the first example of the previous section, for a total-correctness proof of
${\displaystyle [x\leq 10]{\textbf {while}}\ x<10\ {\textbf {do}}\ x:=x+1\ {\textbf {done}}[eg x<10\wedge x\leq 10]}$
the while rule for total correctness can be applied with e.g. ${\displaystyle D}$ being the non-negative integers with the usual order, and the expression ${\displaystyle t}$ being ${\displaystyle
10-x}$, which then in turn requires to prove
${\displaystyle [x\leq 10\wedge x<10\wedge 10-x\geq 0\wedge 10-x=z]x:=x+1[x\leq 10\wedge 10-x\geq 0\wedge 10-x<z]}$
Informally speaking, we have to prove that the distance ${\displaystyle 10-x}$ decreases in every loop cycle, while it always remains non-negative; this process can go on only for a finite number of
The previous proof goal can be simplified to
${\displaystyle [x<10\wedge 10-x=z]x:=x+1[x\leq 10\wedge 10-x<z]}$,
which can be proven as follows:
${\displaystyle [x+1\leq 10\wedge 10-x-1<z]x:=x+1[x\leq 10\wedge 10-x<z]}$ is obtained by the assignment rule, and
${\displaystyle [x+1\leq 10\wedge 10-x-1<z]}$ can be strengthened to ${\displaystyle [x<10\wedge 10-x=z]}$ by the consequence rule.
For the second example of the previous section, of course no expression ${\displaystyle t}$ can be found that is decreased by the empty loop body, hence termination cannot be proved.
See also[edit]
1. ^ Hoare originally wrote "${\displaystyle P\{C\}Q}$" rather than "${\displaystyle \{P\}C\{Q\}}$".
2. ^ This article uses a natural deduction style notation for rules. For example, ${\displaystyle {\dfrac {\alpha ,\beta }{\phi }}}$ informally means "If both ${\displaystyle \alpha }$ and ${\
displaystyle \beta }$ hold, then also ${\displaystyle \phi }$ holds"; ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are called antecedents of the rule, ${\displaystyle \phi }$ is called
its succedent. A rule without antecedents is called an axiom, and written as ${\displaystyle {\dfrac {}{\quad \phi \quad }}}$.
3. ^ "Termination" here is meant in the broader sense that computation will eventually be finished; it does not imply that no limit violation (e.g. zero divide) can stop the program prematurely.
4. ^ Hoare's 1969 paper didn't provide a total correctness rule; cf. his discussion on p.579 (top left). For example Reynolds' textbook (John C. Reynolds (2009). Theory of Programming Languages.
Cambridge University Press.), Sect.3.4, p.64 gives the following version of a total correctness rule: ${\displaystyle {\dfrac {P\wedge B\rightarrow 0\leq t\quad ,\quad [P\wedge B\wedge t=z]S[P\
wedge t<z]}{[P]{\textbf {while}}\ B\ {\textbf {do}}\ S\ {\textbf {done}}[P\wedge eg B]}}}$ when ${\displaystyle z}$ is an integer variable that doesn't occur free in ${\displaystyle P}$, ${\
displaystyle B}$, ${\displaystyle S}$, or ${\displaystyle t}$, and ${\displaystyle t}$ is an integer expression (Reynolds' variables renamed to fit with this article's settings).
Further reading[edit]
External links[edit]
• KeY-Hoare is a semi-automatic verification system built on top of the KeY theorem prover. It features a Hoare calculus for a simple while language.
• j-Algo-modul Hoare calculus — A visualisation of the Hoare calculus in the algorithm visualisation program j-Algo | {"url":"https://static.hlt.bme.hu/semantics/external/pages/sz%C3%A1m%C3%ADt%C3%B3g%C3%A9pes_program_szemantik%C3%A1ja/en.wikipedia.org/wiki/Hoare_logic.html","timestamp":"2024-11-06T04:43:00Z","content_type":"text/html","content_length":"264562","record_id":"<urn:uuid:c9c89d7a-c1b4-4811-a6e1-5925e47a6c17>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00569.warc.gz"} |
Let sleeping files lie: Pattern matching in Z-compressed files
The current explosion of stored information necessitates a new model of pattern matching, that of compressed matching. In this model one tries to find all occurrences of a pattern in a compressed
text in time proportional to the compressed text size, i.e., without decompressing the text. The most effective general purpose compression algorithms are adaptive, in that the text represented by
each compression symbol is determined dynamically by the data. As a result, the encoding of a substring depends on its location. Thus the same substring may `look different' every time it appears in
the compressed text. In this paper we consider pattern matching without decompression in the UNIX Z-compression. This is a variant of the Lempel-Ziv adaptive compression scheme. If n is the length of
the compressed text and m is the length of the pattern, our algorithms find the first pattern occurrence in time O(n+m2) or O(n log m+m). We also introduce a new criterion to measure compressed
matching algorithms, that of extra space. We show how to modify our algorithms to achieve a tradeoff between the amount of extra space used and the algorithm's time complexity.
Dive into the research topics of 'Let sleeping files lie: Pattern matching in Z-compressed files'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/let-sleeping-files-lie-pattern-matching-in-z-compressed-files-2","timestamp":"2024-11-13T15:42:02Z","content_type":"text/html","content_length":"51713","record_id":"<urn:uuid:8ce601e8-321a-4c3c-9768-8ef9aeac7db0>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00320.warc.gz"} |
Key Challenges and Design Strategies for Enhancing WPT Systems
By Ahmed Khebir | 01/02/2024
Wireless power transfer (WPT) systems stand at the forefront of innovation, offering transformative solutions for powering everything from medical implants to electric vehicles. Recent research
explores WPT's capabilities and challenges, particularly focusing on pacemaker and EV charger systems. This comprehensive analysis sheds light on WPT's vast potential across various sectors, while
addressing key challenges including air gap distances, alignment precision, electromagnetic interference, and heat management. Through such investigations, there is a concerted effort to push the
boundaries of WPT technology, aiming to enhance its efficiency, reliability, and practical applicability.
Unveiling the Challenges
A series of virtual experiments, conducted by EMWorks-EMS, revolve around two pivotal models. The first model delineates a WPT system designed specifically for a pacemaker, as illustrated in Figure
1. Conversely, the second model is dedicated to a WPT system for an Electric Vehicle (EV) charger, showcased in Figure 2. These simulations, grounded in precise engineering and scientific inquiry,
aim to push the boundaries of WPT technology, addressing its application in both life-saving medical devices and the rapidly evolving electric vehicle industry.
Figure 1: 3D Model of WPT Design used in a Pacemaker
Figure 2: 3D Design of Bipolar Pad for WPT EV Charger
These experiments illuminate pivotal challenges inherent in wireless power transfer technology, notably the influence of air gap distances and alignment precision on charging efficiency.
Additionally, they explore the potential for electromagnetic interference, a significant concern in the seamless operation of such systems. Furthermore, the experiments delve into the quest for
optimal heat management, a critical factor in ensuring the reliability and safety of wireless charging solutions. Through these investigations, we gain deeper insights into enhancing the performance
and practicality of wireless power technologies.
Impact of Air Gap and Alignment Variations on Magnetic Flux Distribution
To assess the impact of air gap distances and alignment discrepancies on magnetic flux distribution, a parametric AC Magnetic analysis is employed. The ensuing animated plots, in Figures 3-5, vividly
illustrate how variations in magnetic flux density unfold across different scenarios, providing a visual representation of these effects.
Figure 3: Animated Visualization of Magnetic Flux Variation with Air Gap Distance
Figure 4. Animated Visualization of Magnetic Flux Variation with Lateral Misalignment
Figure 5: Animated Visualization of Magnetic Flux Variation with Angular Misalignment
Impact of Air Gap on Inductance, Resistance, and Coupling Efficiency
The observations from Figures 6-8, indicating a significant relationship between air gap distance and key design parameters such as the critical coupling coefficient and mutual inductance, have
profound implications for WPT systems. As the air gap distance increases, both the critical coupling coefficient and mutual inductance decrease, which can directly impact the efficiency and
effectiveness of energy transfer in WPT applications.
1. Reduced Efficiency: The decrease in mutual inductance with larger air gaps leads to reduced energy transfer efficiency. This is because mutual inductance is a measure of how effectively a magnetic
field can induce an electrical current in a secondary coil. A lower mutual inductance means that less energy is transferred from the primary to the secondary coil, requiring more power to achieve the
same level of charging or energy transfer.
2. Design Limitations: The inverse relationship between the air gap distance and the critical coupling coefficient imposes design constraints on WPT systems. It suggests that to maintain efficient
energy transfer, devices must be designed to operate within a relatively small air gap distance. This limitation can affect the usability and applicability of WPT technology in scenarios where a
larger air gap is necessary or unavoidable, such as in certain types of electric vehicle charging or industrial applications.
3. Optimization Challenges: Achieving optimal performance in WPT systems becomes more challenging as the air gap distance increases. Designers must balance the need for efficient energy transfer with
the physical and practical limitations of their application, potentially requiring more complex or costly solutions, such as advanced coil designs or higher frequencies, to compensate for the reduced
coupling coefficient and mutual inductance.
4. Impact on Safety and Standards: The need to keep air gap distances minimal for efficiency reasons may also influence safety standards and regulations, as closer proximity between transmitter and
receiver components could raise concerns about electromagnetic exposure, especially in applications involving human interaction.
In summary, the relationship between air gap distance and key design parameters underscores the importance of careful design and optimization in the development of WPT systems, with a focus on
minimizing air gap distances where possible to maximize efficiency and performance.
Figure 6: Variation of Self and Mutual Inductances with Air Gap Distance
Figure 7: AC Resistance Variation in Relation to Air Gap Distance
Figure 8: Variation of Coupling Coefficient with Air Gap Distance
The Critical Role of Operating Frequency
The operating frequency of a WPT system is a pivotal factor that affects its efficiency, range, component size, and compatibility with existing standards and devices. While higher frequencies can
lead to increased energy losses and reduced transmission distances, they allow for smaller, more compact system designs. Conversely, lower frequencies can facilitate longer-range power transfers but
require larger components, making the system bulkier. Therefore, selecting an optimal frequency is a delicate balance that involves minimizing interference, adhering to regulatory standards, and
optimizing the trade-off between performance and practicality. The evaluation of the operating frequency's influence on the system's efficiency was conducted through a parametric AC Magnetic analysis
coupled with circuit simulations, revealing critical insights into how frequency choices impact overall system behavior as shown Figure 9-10.
Figure 9: Input and Output Power versus Frequency
Figure 10: Power Efficiency versus Frequency
Heat and Loss Analyses
An examination of the loss quantities generated within the studied WPT devices facilitated the computation and visualization of various loss distributions. These distributions were observed in the
aluminum shielding, ferrite bars, and copper coils of the device. This analysis is crucial for understanding the thermal behavior of WPT systems and for optimizing their design to minimize losses and
enhance efficiency. Figure 11-13 depict the results obtained.
Figure 11: Solid Loss Distribution in the Aluminum Shield
Figure 12: Core Loss Distribution in the Ferrite Support | {"url":"https://emworks.com/blog/ems/design-challenges-and-solutions-in-wireless-charging","timestamp":"2024-11-11T00:52:15Z","content_type":"text/html","content_length":"1049795","record_id":"<urn:uuid:091748ac-c062-4d6b-8514-f35860bc92b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00222.warc.gz"} |
Double Angle Calculator: Calculate Double Angle Identities
sin(θ) = 0
cos(θ) = 0
tan(θ) = 0
Have you ever wondered what happens when you double an angle? That’s where double angle formulas come in handy! Let’s break Double Angle Calculator: Calculate Double Angle Identities down in simple
What’s a Double Angle?
Imagine you have a slice of pizza. Now imagine opening it twice as wide – that’s like doubling an angle! When we do math with these doubled angles, we use special formulas called double-angle
Why Do We Need This Calculator?
• It makes solving tricky math problems easier
• It helps in many real-world situations, like:
□ Building bridges
□ Making video games
□ Understanding waves in physics
How to Use the Calculator
1. Pick the function you want (sin, cos, or tan)
2. Enter your original angle
3. Click calculate
4. Get your answer!
Real-Life Examples
1. Music: Sound waves use these formulas
2. Architecture: Helps in designing arches and domes
3. Video Games: Makes objects move smoothly
Fun Facts!
• These formulas were discovered hundreds of years ago
• They’re used in everything from GPS to computer graphics
• Even simple calculators use these to work their magic
Tips for Remembering
• Sin doubles with both sin and cos
• Cos has three options to choose from
• Tan uses both 2 and squared numbers
When You Might Use This
• Math homework (of course!)
• Science projects
• Computer programming
• Even in art and design
Understand what a Double Angle Identity
Before delving into using a calculator, it’s essential to grasp the concept of double-angle identities. A double angle identity allows us to express trigonometric functions of angles that are twice
as large as the original angle. The most common double-angle formulas include:
– Sin(2θ) = 2sinθ cosθ
– Cos(2θ) = cos^2θ – sin^2θ
– Tan(2θ) = 2tanθ / (1 – tan^2θ)
By understanding these formulas, you can confidently input the correct values into the angle formula calculator calculator.
Familiarize yourself with Double Angle Formulas
In trigonometry, there are various double-angle formulas that you need to be familiar with when using a double angle calculator. Knowing these formulas will help you choose the right approach for
your calculations. Some common double-angle identities include:
For Sine (sin)
– Sin2x = 2sinx cosx
For Cosine (cos)
– Cos2x = cos^2x – sin^2x
For Tangent (tan)
– Tan2x = 2tanx / (1 – tan^2x)
By having a solid grasp of these formulas, you can efficiently calculate double angle identities using a calculator.
Check our Half Angle Formula Calculator
Q: What exactly is a double angle calculator?
A: It’s a helpful tool that does the math for you when you need to find out what happens to trigonometric functions (like sine, cosine, and tangent) when you double an angle. Instead of doing complex
calculations by hand, the calculator does it quickly and accurately
Q: Do I need to be a math genius to use this calculator?
A: Not at all! That’s the beauty of it. The calculator does all the tricky math for you. You just need to know your original angle and which function (sine, cosine, or tangent) you want to use.
Q: What kinds of numbers can I put into the calculator?
A: You can use:
Degrees (like 45°, 90°, 180°)
Radians (like π/4, π/2, π)
Decimal numbers (like 0.5, 1.5, 2.0)
Usage Questions
Q: How accurate is the double angle calculator?
A: Very accurate! It typically gives answers to several decimal places. For most everyday uses and homework, this is more than enough accuracy.
Q: Can I use negative angles?
A: Yes! The calculator works with negative angles too. For example, you can find the double angle for -45° just as easily as for 45°.
Q: What if I enter an angle larger than 360 degrees?
A: The calculator will still work! It treats angles larger than 360° as if they’ve gone around a full circle one or more times.
Technical Questions
Q: Why are there three different formulas for cosine double angles?
A: Each formula is useful in different situations:
The first is good when you know both sine and cosine
The second works best when you only know the cosine
The third is handy when you only know sine
Q: How do I know which cosine formula to use?
A: The calculator automatically uses the best formula, but if you’re doing it by hand:
• Use the one that matches the information you already have
• Sometimes one formula might make your next steps easier
• In tests, use the one your teacher prefers!
Practical Application Questions
Q: When would I actually use this in real life?
A: Double angles are used in:
• Engineering (designing rotating machinery)
• Physics (studying waves and oscillations)
• Computer graphics (creating smooth animations)
• Music theory (understanding sound waves)
Q: Can this help me with my homework?
A: Absolutely! It’s great for:
• Checking your work
• Understanding how angles change
• Solving complex problems step by step
Common Problems
Q: What if I get an error message?
A: Common reasons for errors:
• Typing in letters instead of numbers
• Using the wrong symbols (use decimal points, not commas)
• Trying to find the tangent of 90° (which doesn’t exist!)
Q: The answer looks weird – how do I know if it’s right?
A: You can:
1. Try a simpler angle first to see if the pattern makes sense
2. Use a different calculator to verify
3. Ask a teacher or tutor to confirm
Learning and Understanding
Q: How can I understand what’s happening behind the scenes?
A: Try these steps:
1. Start with simple angles like 30° or 45°
2. Draw the angles on paper
3. Use the calculator, then try doing it by hand
4. Look for patterns in the answers
Q: Are there any tricks to remembering the formulas?
A: Yes! Here are some memory helpers:
• For sine: “Double the angle, double the functions” (since it uses both sin and cos)
• For cosine: “Square it or lose it” (since all versions use squared functions)
• For tangent: “Two tan over one minus tan squared” (follows the formula pattern)
Advanced Questions
Q: Is there such thing as a triple angle calculator?
A: Yes! There are formulas and calculators for triple angles too, but they’re more complex and less commonly used.
Q: What’s the relationship between double angles and half angles?
A: They’re like reverse operations! Half angle formulas help you find the trigonometric functions for half of an angle, while double angle formulas do the opposite.
Q: What should I do if I get stuck?
A: Try these steps:
1. Check that you entered the angle correctly
2. Make sure you selected the right function (sin, cos, or tan)
3. Try a simpler angle to see if the calculator is working
4. Ask for help or look up examples online
Q: Can the calculator handle all possible angles?
A: Almost! The only limitations are:
• Tangent doesn’t work for 90° and its odd multiples
• Some calculators might round very large angles
Extra Help
Q: Where can I learn more about double angles?
A: You can:
• Check out online math tutorials
• Use graphing calculators to visualize the relationships
• Practice with sample problems from textbooks
• Watch educational videos about trigonometry
Q: Is there an easy way to check my answers?
A: Yes! You can:
1. Use multiple calculators to verify
2. Draw the angles and estimate if the answer makes sense
3. Try the calculation with a slightly different angle to see if the pattern looks right
In conclusion, utilizing a double-angle calculator can simplify the process of calculating double-angle identities in trigonometry. Effectively using the calculator helps to streamline your
calculations and achieve accurate results. | {"url":"https://calculatoracute.com/double-angle-calculator/","timestamp":"2024-11-09T13:02:05Z","content_type":"text/html","content_length":"77695","record_id":"<urn:uuid:da707a95-b464-44ce-bad7-9b38322cb1b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00077.warc.gz"} |
Playing with the real projective plane
The desmos graph below is one of the many interpretations of the projective plane. In particular, we aim to demonstrate duality between points and lines. There’s three points: red, green and blue.
Each of the points are associated with a line with a line. For example the red point \((a_1, a_2)\) is associated with the red line \(a_1 x + a_2 y + 1 = 0\). Try moving the points around and see how
the lines change.
• If the three points are colinear, what do you notice about the lines?
• If the three lines intersect at a point, what can you say about the points?
• Can you make all three lines parallel? What can you say about the points?
• Try making two lines intersect, and move the third point to the intersection of the two lines. What do you notice?
• What happens when the points get close to the origin? Why does that happen?
Projective Geometry
What you’re seeing is essentially the real projective plane and the duality between points and lines. If you’re interested, you could read a set of lecture notes on projective geometry by Hitchin. | {"url":"https://tobylam.xyz/2023/05/10/playing-projective-plane","timestamp":"2024-11-06T23:47:13Z","content_type":"text/html","content_length":"13290","record_id":"<urn:uuid:52a961ba-0926-40f4-954d-6d55f8e211e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00577.warc.gz"} |
Execution time limit is 1 second
Runtime memory usage limit is 128 megabytes
Elly studies the properties of some given integer . So far she has discovered that it has no more than six distinct prime divisors. A prime number (or a prime) is a natural number greater than that
has no positive divisors other than and itself.
Now the girl spends her time in the following way. Starting with an empty list, she writes divisors of , greater than (some divisors she may repeat several times). When adding a new number to the
list, she makes sure that it has common divisors greater than with at most one of the already written numbers.
For example, if the number is , some of the many possible valid sequences the girl can generate are , , , , , и . Examples for invalid sequences would be , since is not a divisor of , or , since has
common divisors with both and .
Now Elly is wondering how many different valid sequences of divisors of exist. We consider two sequences different if they have different length or there is a position, in which they have different
Write a program that helps Elly to find the number of valid sequences of divisors of .
The first line contains one integer . will have at most distinct prime divisors.
Print one integer — the number of different sequences of divisors of , which could have been written by Elly. Since this number can be rather large, you are required to print only its remainder when
divided by .
Explanation for the first test. All valid sequences are: , , , , , , , , , , , , , , , , , , , , , , , , , , , .
Explanation for the fourth test. The answer is , but since you are required to print it modulo , the actual result is .
Submissions 4
Acceptance rate 75% | {"url":"https://basecamp.eolymp.com/en/problems/8721","timestamp":"2024-11-12T16:40:56Z","content_type":"text/html","content_length":"311437","record_id":"<urn:uuid:8a216807-574e-45f0-8862-6c76c98f8fd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00725.warc.gz"} |
4-20mA Signal to Measurement Reading Converter
Click save settings to reload page with unique web page address for bookmarking and sharing the current tool settings
Invert the low & high range limits for the measurement output settings
Flip tool with current settings to swap 4-20mA input (I) with measurement output parameters
Featured 4 to 20ma signal products
Related Tools
User Guide
This current to measurement conversion tool will convert an electrical signal within the range of 4 to 20 milliamps to the ideal reading of any linear measurement type and create an incremental
milliamp conversion scale for each measurement range entered.
This tool uses the following formula to calculate the the measurement reading output from a current signal input over a 4-20mA range:
Measurement Rdg = Low Limit + (High Limit – Low Limit) x (mA out -4) / 16
Current Loop Signal (4-20mA) Reading
Add the current loop signal reading in milliamps (mA) between 4 and 20 mA that you want to convert.
Measurement Unit
Enter the measurement unit associated with the output values.
Lowest Measurement
Add the minimum possible value for the measurement range of your instrument. For example zero would be a typical value for most instrumentation but you can also add negative (-) or positive (+)
values as well. A value in any engineering units can be used as long as it relates to a linear measurement, and the same unit is used for the Highest value.
Highest Measurement
Add the maximum possible value for the measurement range of your instrument, e.g. full scale or full range. This value can be negative (-) or positive (+). Any engineering units can be used as long
as it relates to a linear measurement, but the same unit must be used for the Lowest value.
Measurement Reading – Answer
This is the ideal converted measurement for your instrument and therefore represents a perfect reading excluding the measurement uncertainty error of your instrument. The answer value is displayed
in the same engineering unit that you entered for the Lowest and Highest range points.
Featured 4 to 20ma signal products
Conversion precision
How precise is the converted measurement?
The linear measurement Answer is displayed to a precision of 9 significant figures.
Inverted signal
How do I convert an inverted or reversed 4-20mA signal?
Swap around the values you enter into the Lowest and Highest text boxes
Calibrating 4-20mA device
Can I use this converter to calibrate a 4-20mA measurement device?
You can partly use this converter to aid calibration. When you calibrate you will need to precisely measure both the input and output parameter. You can use this converter to determine the ideal
measurement reading and compare it with the calibration measurement to determine the calibration error.
Input or output 4-20mA
Do I enter the input or the output value for the 4-20mA signal?
This converter is not input or output specific, so you can convert either ‘input to output’ or ‘output to input’.
Linear measurement
What is a linear measurement?
This means that if you were to plot input and output readings on a graph they would produce a straight line.
4-20mA to 0-100% readings conversion
What formula is used to convert 4-20ma into % of reading?
The formula is:
% reading = (([Linear mA out] – 4)/16) x 100 | {"url":"https://www.sensorsone.com/4-20ma-to-linear-measurement-converter/?iunit=mA&irdg=13.9733333333&ilo=4&ihi=20&ounit=A&olo=0&ohi=150","timestamp":"2024-11-03T15:00:59Z","content_type":"text/html","content_length":"67039","record_id":"<urn:uuid:aba99c65-357f-4649-8a0f-35e7336c36e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00563.warc.gz"} |