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If you are a registered user and signed in, you can here copy this script and its stimuli to your own account, where you can edit it and change it in any way you want.
It is absolutely free to register (no credit card info asked!). You can then instantly copy this experiment with one click and edit it, change its accompanying texts, its landing page, stimuli, etc.
Invite your colleagues, friends, or students to check out your experiment. There is no limit on how many people can do your experiment, even with a free account.
The catch? There is no catch! Just keep in mind that with a free account, you cannot collect data. For teaching that is usually not a problem. For research, prepaid data collection (unlimited
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// Generate positive set of length 'size' with a probe
// yes/no is not counter-balanced but fully randomized to about 50:50
// E.g., with size = 1, there is still a 50:50 chance of the probe being 1 or not
function getstimulus(size)
var row = shuffle(range(1,10)), // Array of digits 1 to 9 in random order
pos = row.slice(0,size), // First 'size' digits (also store in pos)
neg = row.slice(size-9), // Last digits not in positive_set (also store in neg)
inpositive = Math.random() < 0.5; // 50% chance of probe in positive set
return {
set: pos, // Positive set
inpositive: inpositive, // true if probe is in positive set, else false
probe: inpositive ? pos[randint(size-1)] : neg[randint(9-size-1)] // Keep yes/no ratio about 50:50
// Returns a completely randomized design of lengths 1 to 6, where trials for each length are specified
function gettrials(trials_per_length)
var trials = [], i, j;
for (i = 0; i < trials_per_length; i++)
for (j = 1; j <= 6; j++)
return shuffle(trials);
var middle = main.addblock('center',10,100,60)
.style("border","thin solid lightgrey")
.text("<h2>Instruction</h2>You will see one to six digits appear for a short period."
+ "Try to remember these. Then they will disappear and a 'probe' digit will appear. "
+ "Type 'z' for Yes: the probe digit was in the list, "
+ "or '/' for No: it was not in the list. "
+ "Be as fast and accurate as you can.<br>"
+ "<i>Click the Start button or press the Enter key to start the experiment.</i>");
var left = main.addblock('left','bottom',25,10)
.text("<b>z</b> key = <b>Yes</b>");
var right = main.addblock('right','bottom',25,10)
.text("<b>/</b> key = <b>No</b>");
var ok = main.addblock('center',70,20,20).button('Start');
trials = gettrials(1);
var stimulus, positive_set, probe, inpositive, trialno, length, t1, e, score;
for (t = 0; t < trials.length; t++)
stimulus = getstimulus(trials[t]);
positive_set = stimulus.set.join(' '); // turn array into a space-delimited string with join()
probe = stimulus.probe;
inpositive = stimulus.inpositive;
trialno = t;
length = trials[t]; // trials[t] holds the number of items in positive set
t1 = now();
e = await("keypress");
rt = now() - t1;
// storetrial("RT{stimulus}","{RT}","RT");
score = 0;
if (e.key.toLowerCase() === 'z' && inpositive)
score = 1;
if ((e.key === 'z' || e.key === '?') && !inpositive)
score = 1;
if (score)
middle.text("Not correct");
if (e.key.toLowerCase() !== 'z' && e.key !== '/' && e.key !== '?')
middle.text("Please, press <b>z</b> or <b>/</b>");
score = -1;
//log("RT: RT, Correct: score, length: length, trialno: trialno");
middle.text("Finished!<br>Thank you for participating!");
You can download the files as follows: Click on the file (link) and then right-click and choose Save as... from the menu. Some media files (e.g., sound) will have a download button for this purpose.
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Parity check matrices and product representations of squares
Let N[F] (n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field N[F] in which each column has at most r nonzero entries and every k columns are linearly
independent over N [F]. We obtain near-optimal upper bounds for N[F] (n,k,r) in the case k > r. Namely, we show that N[F] (n,k,r) ≫ n ^r/2 + cr/k where c ≈ 4/3 for large k. Our method is based on a
novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a
problem on hypergraphs and a problem in combinatorial number theory.
ASJC Scopus subject areas
• Discrete Mathematics and Combinatorics
• Computational Mathematics
Dive into the research topics of 'Parity check matrices and product representations of squares'. Together they form a unique fingerprint.
|
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|
This page displays the schedule of Bryn Mawr courses in this department for this academic year. It also displays descriptions of courses offered by the department during the last four academic years.
For information about courses offered by other Bryn Mawr departments and programs or about courses offered by Haverford and Swarthmore Colleges, please consult the Course Guides page.
For information about the Academic Calendar, including the dates of first and second quarter courses, please visit the College's calendars page.
Fall 2024 MATH
Course Title Schedule/Units Meeting Type Times/Days Location Instr(s)
MATH B101-001 Calculus I Semester / 1 Lecture: 10:10 AM-11:00 AM MWF Park 300 Sudparid,D.
MATH B101-002 Calculus I Semester / 1 Lecture: 12:10 PM-1:00 PM MWF Carpenter Library 21 Sudparid,D.
MATH B101L-099 Calculus I lab Semester / 0.5 Laboratory: 2:10 PM-3:30 PM W Park 363C Sudparid,D.
MATH B102-001 Calculus II Semester / 1 Lecture: 10:10 AM-11:00 AM MWF Carpenter Library 21 Vien,D.
MATH B102-002 Calculus II Semester / 1 Lecture: 11:10 AM-12:00 PM MWF Park 300 Myers,A.
MATH B104-001 Basic Probability and Statistics Semester / 1 Lecture: 2:10 PM-3:00 PM MWF Park 300 Kasius,P.
MATH B104-002 Basic Probability and Statistics Semester / 1 Lecture: 3:10 PM-4:00 PM MWF Park 300 Kasius,P.
MATH B201-002 Multivariable Calculus Semester / 1 Lecture: 12:10 PM-1:00 PM MWF Park 300 Traynor,L.
MATH B201-003 Multivariable Calculus Semester / 1 LEC: 10:10 AM-11:30 AM TTH Park 243 Kara,S.
MATH B206-001 Transition to Higher Mathematics Semester / 1 Lecture: 2:40 PM-4:00 PM MW Park 245 Myers,A.
MATH B206-002 Transition to Higher Mathematics Semester / 1 LEC: 1:10 PM-2:30 PM MW Park 245 Myers,A.
MATH B210-001 Differential Equations with Applications Semester / 1 Lecture: 8:40 AM-10:00 AM MW Carpenter Library 21 Cheng,L.
MATH B295-001 Select Topics in Mathematics: Evolutionary Game Theory Semester / 1 LEC: 12:10 PM-1:00 PM MWF Park 264 Chu,O.
MATH B301-001 Real Analysis I Semester / 1 Lecture: 10:10 AM-11:30 AM TTH Park 245 Cheng,L.
MATH B301-002 Real Analysis I Semester / 1 Lecture: 1:10 PM-2:30 PM TTH Park 245 Cheng,L.
MATH B303-001 Abstract Algebra I Semester / 1 Lecture: 2:40 PM-4:00 PM TTH Park 245 Kara,S.
MATH B312-001 Topology Semester / 1 LEC: 1:10 PM-2:30 PM MW Park 328 Traynor,L.
MATH B398-001 Senior Conference Semester / 1 Lecture: 10:10 AM-11:30 AM MW Park 328 Dept. staff, TBA
MATH B400-001 Senior Thesis Semester / 1 LEC: 8:10 AM-9:00 AM T Park 300 Cheng,L.
MATH B400-002 Senior Thesis Semester / 1 LEC: 8:10 AM-9:00 AM T Park 328 Kara,S.
MATH B403-001 Supervised Work 1 Dept. staff, TBA
MATH B512-001 General Topology Semester / 1 LEC: 1:10 PM-2:30 PM MW Park 328 Traynor,L., Traynor,L.
LEC: 1:10 PM-2:30 PM F Park 328
MATH B701-001 Supervised Work 1 Cheng,L.
MATH B701-002 Supervised Work 1 Donnay,V.
MATH B701-003 Supervised Work 1 Graham,E.
MATH B701-004 Supervised Work 1 Melvin,P.
MATH B701-005 Supervised Work 1 Milicevic,D.
MATH B701-006 Supervised Work 1 Traynor,L.
MATH B701-007 Supervised Work 1 Kara,S.
MATH B701-001 Supervised Work 1 Cheng,L.
MATH B701-002 Supervised Work 1 Donnay,V.
MATH B701-003 Supervised Work 1 Graham,E.
MATH B701-004 Supervised Work 1 Melvin,P.
MATH B701-005 Supervised Work 1 Milicevic,D.
MATH B701-006 Supervised Work 1 Traynor,L.
MATH B701-007 Supervised Work 1 Kara,S.
MATH B702-001 Research Seminar 1 Milicevic,D.
MATH B702-001 Research Seminar 1 Milicevic,D.
CHEM B221-001 Physical Chemistry I Semester / 1 Lecture: 1:10 PM-2:30 PM MW Park 180 Goldsmith,J.
CMSC B231-001 Discrete Mathematics Semester / 1 Lecture: 10:10 AM-11:30 AM TTH Park 278 Zhou,Y.
CMSC B340-001 Analysis of Algorithms Semester / 1 Lecture: 11:40 AM-1:00 PM MW Park 159 Xu,D., Xu,D.
Laboratory: 1:10 PM-2:30 PM W Park 230
PHYS B205-001 Mathematical Methods in the Sciences I First Half / 0.5 Laboratory: 12:10 PM-1:00 PM MWF Park 100 Matlin,M.
PHYS B207-001 Mathematical Methods in the Sciences II Second Half / 0.5 Laboratory: 12:10 PM-1:00 PM MWF Park 100 Matlin,M.
Spring 2025 MATH
Course Title Schedule/Units Meeting Type Times/Days Location Instr(s)
MATH B101-001 Calculus I Semester / 1 Lecture: 2:40 PM-4:00 PM MW Chu,O.
MATH B101L-099 Calculus I lab Semester / 0.5 Laboratory: 11:40 AM-1:00 PM TH Park 328 Sudparid,D.
MATH B102-001 Calculus II Semester / 1 Lecture: 10:10 AM-11:00 AM MWF Park 300 Sudparid,D.
MATH B102-002 Calculus II Semester / 1 Lecture: 12:10 PM-1:00 PM MWF Park 245 Sudparid,D.
MATH B104-001 Basic Probability and Statistics Semester / 1 Lecture: 2:40 PM-4:00 PM MW Park 300 Sudparid,D.
MATH B201-001 Multivariable Calculus Semester / 1 Lecture: 10:10 AM-11:30 AM TTH Park 300 Kara,S.
MATH B203-001 Linear Algebra Semester / 1 Lecture: 11:10 AM-12:00 PM MWF Park 300 Kasius,P.
MATH B203-002 Linear Algebra Semester / 1 Lecture: 12:10 PM-1:00 PM MWF Park 300 Kasius,P.
MATH B206-001 Transition to Higher Mathematics Semester / 1 Lecture: 10:10 AM-11:00 AM MWF Park 159 Myers,A.
MATH B206-002 Transition to Higher Mathematics Semester / 1 LEC: 2:10 PM-3:00 PM MWF Park 159 Myers,A.
MATH B208-001 Introduction to Modeling and Simulation Semester / 1 Lecture: 1:10 PM-2:30 PM TTH Park 245 Graham,E.
MATH B295-001 Select Topics in Mathematics: Evolutionary Game Theory Semester / 1 LEC: 11:40 AM-1:00 PM MW Chu,O.
MATH B295-002 Select Topics in Mathematics: Codes and Ciphers Semester / 1 LEC: 2:10 PM-3:00 PM MWF Dunham,P.
MATH B295-003 Select Topics in Mathematics: History of Math Semester / 1 LEC: 10:10 AM-11:00 AM MWF Dunham,B.
MATH B302-001 Real Analysis II Semester / 1 Lecture: 11:40 AM-1:00 PM TTH Park 245 Stromquist,W.
MATH B303-001 Abstract Algebra I Semester / 1 Lecture: 3:10 PM-4:00 PM MWF Kasius,P.
MATH B304-001 Abstract Algebra II Semester / 1 Lecture: 2:40 PM-4:00 PM TTH Park 245 Kara,S.
MATH B308-001 Applied Mathematics I Semester / 1 LEC: 10:10 AM-11:30 AM TTH Park 245 Graham,E.
MATH B399-001 Senior Conference Semester / 1 Lecture: 8:40 AM-10:00 AM MW Dept. staff, TBA
MATH B399-002 Senior Conference Semester / 1 LEC: 11:40 AM-1:00 PM MW Park 159 Dept. staff, TBA
MATH B400-001 Senior Thesis 1 Chu,O.
MATH B403-001 Supervised Work 1 Dept. staff, TBA
MATH B501-001 Graduate Real Analysis I Semester / 1 LEC: 8:40 AM-10:00 AM TTH Park 245 Stromquist,W.
MATH B701-001 Supervised Work 1 Cheng,L.
MATH B701-002 Supervised Work 1 Chu,O.
MATH B701-003 Supervised Work 1 Graham,E.
MATH B701-004 Supervised Work 1 Melvin,P.
MATH B701-005 Supervised Work 1 Milicevic,D.
MATH B701-006 Supervised Work 1 Traynor,L.
MATH B701-007 Supervised Work 1 Kara,S.
MATH B701-001 Supervised Work 1 Cheng,L.
MATH B701-002 Supervised Work 1 Chu,O.
MATH B701-003 Supervised Work 1 Graham,E.
MATH B701-004 Supervised Work 1 Melvin,P.
MATH B701-005 Supervised Work 1 Milicevic,D.
MATH B701-006 Supervised Work 1 Traynor,L.
MATH B701-007 Supervised Work 1 Kara,S.
MATH B702-001 Research Seminar 1 Milicevic,D.
MATH B702-002 Research Seminar 1 Graham,E.
MATH B702-001 Research Seminar 1 Milicevic,D.
MATH B702-002 Research Seminar 1 Graham,E.
CMSC B231-001 Discrete Mathematics Semester / 1 Lecture: 11:40 AM-1:00 PM TTH Park 245 Zhou,Y., Zhou,Y.
Laboratory: 3:00 PM-4:00 PM TH Park 230
CMSC B311-001 Computational Geometry Semester / 1 Lecture: 1:10 PM-2:30 PM MW Park 159 Xu,D., Xu,D.
Laboratory: 1:10 PM-2:30 PM T Park 231
CMSC B340-001 Analysis of Algorithms Semester / 1 Lecture: 11:40 AM-1:00 PM MW Park 159 Xu,D., Xu,D.
Laboratory: 11:40 AM-1:00 PM T Park 231
ECON B304-001 Econometrics Semester / 1 Lecture: 10:10 AM-11:30 AM MW Dalton Hall 2 Kim,M.
Fall 2025 MATH
(Class schedules for this semester will be posted at a later date.)
2024-25 Catalog Data: MATH
MATH B101 Calculus I
Fall 2024, Spring 2025
This is the first in a sequence of two courses that covers single-variable calculus. Topics include functions, limits, continuity, derivatives, differentiation formulas, applications of derivatives,
integrals, and the fundamental theorem of calculus. Prerequisite: proficiency in high-school mathematics (including algebra, geometry, and trigonometry).
Quantitative Methods (QM)
Quantitative Readiness Required (QR)
MATH B101L Calculus I lab
Fall 2024, Spring 2025
This lab course will reinforce the concepts and skills that are needed to be successful in Calculus 1. Students must be enrolled in MATH B101 Calculus I to enroll in this course.
Course does not meet an Approach
MATH B102 Calculus II
Fall 2024, Spring 2025
This is the second in a sequence of two courses that covers single-variable calculus. Topics include techniques of integration, applications of integration, infinite sequences and series, tests of
convergence for series, and power series. Prerequisite: a merit grade in Math 101 (or an equivalent experience).
Quantitative Methods (QM)
MATH B104 Basic Probability and Statistics
Fall 2024, Spring 2025
This course introduces key concepts in descriptive and inferential statistics. Topics include summary statistics, graphical displays, correlation, regression, probability, the Law of Large Numbers,
expected value, standard error, the Central Limit Theorem, hypothesis testing, sampling procedures, bias, and the use of statistical software.
Quantitative Methods (QM)
Quantitative Readiness Required (QR)
Counts Toward Data Science
MATH B195 Select Topics in Mathematics
Not offered 2024-25
This is a topics course. Course content varies.
MATH B201 Multivariable Calculus
Fall 2024, Spring 2025
This course extends calculus to functions of multiple variables. Topics include functions, limits, continuity, vectors, directional derivatives, optimization problems, multiple integrals, parametric
curves, vector fields, line integrals, surface integrals, and the theorems of Gauss, Green and Stokes. Prerequisite: a merit grade in Math 102 (or an equivalent experience).
Quantitative Methods (QM)
MATH B203 Linear Algebra
Spring 2025
This course considers systems of linear equations, matrix algebra, determinants, vector spaces, subspaces, linear independence, bases, dimension, linear transformations, eigenvalues, eigenvectors,
orthogonality, and applications of linear algebra. Prerequisite (or corequisite): Math 102.
Quantitative Methods (QM)
MATH B205 Theory of Probability with Applications
Not offered 2024-25
The course analyzes repeatable experiments in which short-term outcomes are uncertain, but long-run behavior is predictable. Topics include: random variables, discrete distributions, continuous
densities, conditional probability, expected value, variance, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: Math 201.
MATH B206 Transition to Higher Mathematics
Fall 2024, Spring 2025
This course focuses on mathematical writing and proof techniques. Topics include symbolic logic, set notation and quantifiers, proof by contradiction and induction, set notation and operations,
relations and partitions, functions, and more. Prerequisite or Co-requisite: MATH B201 or MATH B203. Not open to students who have taken a 300 level Math course
Writing Intensive
Quantitative Methods (QM)
MATH B208 Introduction to Modeling and Simulation
Spring 2025
Mathematical models are constructed to describe the complex world within and around us. Computational methods are employed to visualize and solve these models. In this course, we focus on developing
mathematical models to describe real-world phenomena, while using computer simulations to examine prescribed and/or random behavior of various systems. The course includes an introduction to
programming (in R or Matlab/Octave), and mathematical topics may include discrete dynamical systems, model fitting using least squares, elementary stochastic processes, and linear models (regression,
optimization, linear programming). Applications to economics, biology, chemistry, and physics will be explored. Prior programming experience not required. Prerequisite: MATH B102 or the equivalent
(merit score on the AP Calculus BC Exam or placement).
Course does not meet an Approach
Quantitative Methods (QM)
Quantitative Readiness Required (QR)
Counts Toward Data Science
MATH B210 Differential Equations with Applications
Fall 2024
Ordinary differential equations, including general first-order equations, linear equations of higher order and systems of equations, via numerical, geometrical, and analytic methods. Applications to
physics, biology, and economics. Co-requisite: MATH 201 or 203.
Quantitative Methods (QM)
MATH B221 Introduction to Topology and Geometry
Not offered 2024-25
An introduction to the ideas of topology and geometry through the study of knots and surfaces in three-dimensional space. The course content may vary from year to year, but will generally include
some historical perspectives and some discussion of connections with the natural and life sciences. Co-requisite: MATH 201 or 203.
Quantitative Methods (QM)
MATH B225 Introduction to Financial Mathematics
Not offered 2024-25
Topics to be covered include market conventions and instruments, Black-Scholes option-pricing model, and practical aspects of trading and hedging. All necessary definitions from probability theory
(random variables, normal and lognormal distribution, etc.) will be explained. Prerequisite: MATH 102. ECON 105 is recommended.
MATH B290 Elementary Number Theory
Not offered 2024-25
Properties of the integers, divisibility, primality and factorization, congruences, Chinese remainder theorem, multiplicative functions, quadratic residues and quadratic reciprocity, continued
fractions, and applications to computer science and cryptography. Prerequisite: MATH 102.
MATH B295 Select Topics in Mathematics
Section 001 (Fall 2023): Math Modeling and Sustainability
Section 001 (Spring 2024): Statistics with R
Section 001 (Fall 2024): Evolutionary Game Theory
Section 001 (Spring 2025): Evolutionary Game Theory
Section 002 (Spring 2025): Codes and Ciphers
Section 003 (Spring 2025): History of Math
Fall 2024, Spring 2025
This is a topics course. Course content varies. Not all topics are open to first year students.
Current topic description: This course introduces basic concepts in evolutionary game theory (EGT) and dynamics. Evolutionary dynamics is the mathematical study of the evolutionary processes that
influence biological and social processes. In this course, we will cover fundamental topics in EGT, including evolutionarily stable strategies, replicator dynamics, and games on networks. We will
introduce the mathematical techniques and modeling approaches needed to study real-world problems, with a focus on social evolution and human cooperation.
Current topic description: This course introduces basic concepts in evolutionary game theory (EGT) and dynamics. Evolutionary dynamics is the mathematical study of the evolutionary processes that
influence biological and social processes. In this course, we will cover fundamental topics in EGT, including evolutionarily stable strategies, replicator dynamics, and games on networks. We will
introduce the mathematical techniques and modeling approaches needed to study real-world problems, with a focus on social evolution and human cooperation.
Current topic description: This course is an introduction to classical and modern methods for encoding secret messages (cryptography) and the science of breaking codes and ciphers (cryptanalysis). It
blends the history of secret writing, the art of creating codes, and the mathematics underlying the theory and practice of encryption and decryption. Topics include substitution and transposition
ciphers, Vigenere and Hill ciphers, statistical methods in cryptanalysis, and applications from linear algebra and number theory to cryptananalysis, digital signatures, PGP, RSA, and other public-key
ciphers. Latter topics also will require use of computer applets. Prerequisites: Math B203 (Linear Algebra) or Math B206 (Transition to Higher Mathematics).
Current topic description: This course samples mathematical landmarks from Classical times to the 19th century. Among these are Euclid's proof of the Pythagorean Theorem, Archimedes' determination of
cirucular area, Newton's approximation of p, Euler's solution of the Basel problem, and Catnor's theory of the infinite. We end with a look at the mathematical heritage of Bryn Mawr College and the
"math matriarchs" who built our department. In short, the course will have historical and biographical components, but its primary object is to examine, in full mathematical detail, landmarks from
the long and glorious history of mathematics. Prerequisite: Any 200-level math course.
Quantitative Methods (QM)
MATH B301 Real Analysis I
Fall 2024
A first course in real analysis, providing a rigorous development of single variable calculus, with a strong focus on proof writing. Topics covered: the real number system, elements of set theory and
topology, limits, continuous functions, the intermediate and extreme value theorems, differentiable functions and the mean value theorem, uniform continuity, the Riemann integral, the fundamental
theorem of calculus. Possible additional topics include analysis on metric spaces or dynamical systems. Prerequisite: MATH 201 and MATH B206 or permission of instructor.
Writing Attentive
MATH B302 Real Analysis II
Spring 2025
A continuation of Real Analysis I: Infinite series, power series, sequences and series of functions, pointwise and uniform convergence, and additional topics selected from: Fourier series, calculus
of variations, the Lebesgue integral, dynamical systems, and calculus in higher dimensions. Prerequisite: MATH 301.
MATH B303 Abstract Algebra I
Fall 2024, Spring 2025
A first course in abstract algebra, including an introduction to groups, rings and fields, and their homomorphisms. Topics covered: cyclic and dihedral groups, the symmetric and alternating groups,
direct products and finitely generated abelian groups, cosets, Lagrange's Theorem, normal subgroups and quotient groups, isomorphism theorems, integral domains, polynomial rings, ideals, quotient
rings, prime and maximal ideals. Possible additional topics include group actions and the Sylow Theorems, free abelian groups, free groups, PIDs and UFDs. Prerequisite: MATH 203 and MATH B206 or
permission from instructor.
Writing Attentive
MATH B304 Abstract Algebra II
Spring 2025
A continuation of Abstract Algebra I. Vector spaces and linear algebra, field extensions, algebraic and transcendental extensions, finite fields, fields of fractions, field automorphisms, the
isomorphism extension theorem, splitting fields, separable and inseparable extensions, algebraic closures, and Galois theory. Also, if not covered in Abstract Algebra I: group actions and Sylow
theorems, free abelian groups, free groups, PIDs and UFDs. Possible additional topic: finitely generated modules over a PID and canonical forms of matrices. Prerequisite: MATH 303.
MATH B308 Applied Mathematics I
Spring 2025
This course will provide a general introduction to methods and modeling in applied mathematics. A variety of mathematical tools will be used to develop and study a wide range of models, including
deterministic, discrete, and stochastic methods. Additional emphasis will be placed on techniques for analyzing mathematical models, including phase plane methods, stability analysis, dimensional
analysis, bifurcation theory, and computer simulations. Applications to biology, physics, chemistry, engineering, and the social sciences may be discussed. Prerequisite: MATH B203 and MATH B206 and
MATH B210 or permission of instructor.
MATH B310 Mathematics of Financial Derivatives
Not offered 2024-25
An introduction to the mathematics utilized in the pricing models of derivative instruments. Topics to be covered may include Arbitrage Theorem, pricing derivatives, Wiener and Poisson processes,
martingales and martingale representations, Ito's Lemma, Black-Scholes partial differentiation equation, Girsanov Theorem and Feynman-Kac Formula. Prerequisite: MATH 201 and MATH B206 or permission
of instructor.
MATH B312 Topology
Fall 2024
General topology (topological spaces, continuity, compactness, connectedness, quotient spaces), the fundamental group and covering spaces, introduction to geometric topology (classification of
surfaces, manifolds). Typically offered yearly in alternation with Haverford. Co-requisite: MATH 301, MATH 303, or permission of instructor.
MATH B322 Functions of Complex Variables
Not offered 2024-25
Analytic functions, Cauchy's theorem, Laurent series, calculus of residues, conformal mappings, Moebius transformations. Prerequisite: MATH 301 or permission of instructor.
MATH B325 Advanced Topics in Applied Mathematics
Not offered 2024-25
This topics course will focus on one advanced area in applied mathematics. Topics may include numerical linear algebra, applied partial differential equations, optimal control, parameter estimation
and model fitting. Prerequisite: Math B210: Differential Equations AND one of the following: Math 206, or Math B301, or permission of instructor
Quantitative Readiness Required (QR)
MATH B390 Number Theory
Not offered 2024-25
Study of integers with an emphasis on their multiplicative structure and topics related to analysis, and a first course in analytic number theory. Core topics: divisibility and primes, arithmetic
functions, average and extremal orders, techniques of analytic number theory, Riemann zeta function, prime number theorem, Dirichlet characters, L-functions. Possible additional topics may include
approximations by rational numbers, geometry of numbers, algebraic numbers and class numbers, sums of squares, and the idea of modular forms. Prerequisite: Math 201 and MATH B206, or permission of
MATH B398 Senior Conference
A seminar for seniors majoring in mathematics. Topics vary from year to year.
MATH B399 Senior Conference
A seminar for seniors majoring in mathematics. Topics vary from year to year.
MATH B400 Senior Thesis
Independent research for senior thesis in Math
MATH B403 Supervised Work
MATH B501 Graduate Real Analysis I
Spring 2025
In this course we will study the theory of measure and integration. Topics will include Lebesgue measure, measurable functions, the Lebesgue integral, the Riemann-Stieltjes integral, complex
measures, differentiation of measures, product measures, and L^p spaces.
MATH B502 Graduate Real Analysis II
Not offered 2024-25
This course is a continuation of Math 501.
MATH B503 Graduate Algebra I
Not offered 2024-25
This is the first course in a two course sequence providing a standard introduction to algebra at the graduate level. Topics in the first semester will include categories, groups, rings, modules, and
linear algebra.
MATH B504 Graduate Algebra II
Not offered 2024-25
This course is a continuation of Math 503, the two courses providing a standard introduction to algebra at the graduate level. Topics in the second semester will include linear algebra, fields,
Galois theory, and advanced group theory. Prerequisite: MATH B503.
MATH B512 General Topology
Fall 2024
This course covers the basic notions of point set topology, with an introduction to algebraic and geometric topology. Topics covered include topological spaces, continuity, compactness,
connectedness, quotient spaces, the fundamental group and covering spaces, and the classification of surfaces.
MATH B522 Complex Analysis
Not offered 2024-25
This course covers the basic notions of complex analysis. Topics covered include analytic functions, Cauchy's theorem, the calculus of residues, conformal mappings, Riemann mapping theorem and
Picard's little theorem.
MATH B525 Algebraic Topology
Not offered 2024-25
This course covers the basic notions of algebraic topology. Topics covered include homology theory, cohomology theory, duality on manifolds, and an introduction to homotopy theory.
MATH B530 Differential Topology
Not offered 2024-25
This course covers the basic notions of differential topology. Topics covered include smooth manifolds, smooth maps, differential forms, and integration on manifolds.
MATH B701 Supervised Work
Fall 2024, Spring 2025
MATH B701 Supervised Work
Fall 2024, Spring 2025
MATH B702 Research Seminar
Fall 2024, Spring 2025
MATH B702 Research Seminar
Fall 2024, Spring 2025
CHEM B221 Physical Chemistry I
Fall 2024
Introduction to quantum theory and spectroscopy. Atomic and molecular structure; molecular modeling; rotational, vibrational, electronic and magnetic resonance spectroscopy. Lecture three hours.
Prerequisites: CHEM B104 and MATH B201.
Quantitative Methods (QM)
Counts Toward Biochemistry Molecular Biology
CMSC B231 Discrete Mathematics
Fall 2024, Spring 2025
An introduction to discrete mathematics with strong applications to computer science. Topics include propositional logic, proof techniques, recursion, set theory, counting, probability theory and
graph theory. Prerequisites: CMSC B231 or CMSC H231 or MATH B231 or MATH H231.
Quantitative Methods (QM)
CMSC B311 Computational Geometry
Spring 2025
A study of algorithms and mathematical theories that focus on solving geometric problems in computing, which arise naturally from a variety of disciplines such as Computer Graphics, Computer Aided
Geometric Design, Computer Vision, Robotics and Visualization. The materials covered sit at the intersection of pure Mathematics and application-driven Computer Science and efforts will be made to
accommodate Math majors and Computer Science majors of varying math/computational backgrounds. Topics include: graph theory, triangulation, convex hulls, geometric structures such as Voronoi diagrams
and Delaunay triangulations, as well as curves and polyhedra surface topology. Prerequisite: CMSC B151 or CMSC H106 or CMSC H107, and CMSC B231, or CMSC H231 or MATH B231 or MATH H231, or permission
of instructor.
Quantitative Readiness Required (QR)
CMSC B340 Analysis of Algorithms
Fall 2024, Spring 2025
This course will cover qualitative and quantitative analysis of algorithms and their corresponding data structures from a precise mathematical point of view. Topics include: performance bounds,
asymptotic and probabilistic analysis, worst case and average case behavior and correctness and complexity. Particular classes of algorithms will be studied in detail. This course fulfills the
writing requirement in the major. Prerequisites: CMSC B151, or CMSC H106 or CMSC H107, and CMSC B231, or CMSC H231 or MATH B231 or MATH H231 or permission of instructor.
Writing Intensive
Quantitative Readiness Required (QR)
ECON B304 Econometrics
Spring 2025
The econometric theory presented in ECON 253 is further developed and its most important empirical applications are considered. Each student does an empirical research project using multiple
regression and other statistical techniques. Prerequisites:ECON B253 or ECON H203 or ECON H204 and ECON B200 or ECON B202 and MATH B201 or permission of instructor.
Counts Toward Data Science
PHYS B205 Mathematical Methods in the Sciences I
Fall 2024
This course is the first of two half-semester sessions which presents topics in applied mathematics useful to students in physics, engineering, physical chemistry, geology, and computer science. This
first session will cover infinite series, complex variables, Fourier series, integral transforms, special functions, and ordinary differential equations. Lecture three hours and additional recitation
sessions as needed. Prerequisite: MATH B102.
Course does not meet an Approach
PHYS B207 Mathematical Methods in the Sciences II
Fall 2024
This course is the second of two half-semester sessions which presents topics in applied mathematics useful to students in physics, engineering, physical chemistry, geology, and computer science.
This second session covers advanced ordinary differential equations, partial differential equations, special functions, series solutions, and boundary-value problems. Lecture three hours and
additional recitation sessions as needed. Prerequisite: PHYS B205, MATH B201 and MATH B203
Course does not meet an Approach
PHYS B306 Mathematical Methods in the Physical Sciences
Not offered 2024-25
This course presents topics in applied mathematics useful to students, including physicists, engineers, physical chemists, geologists, and computer scientists studying the natural sciences. Topics
are taken from Fourier series, integral transforms, advanced ordinary and partial differential equations, special functions, boundary-value problems, functions of complex variables, and numerical
methods. Lecture three hours and additional recitation sessions as needed. Prerequisite: MATH 201 and 203.
PHYS B328 Galactic Dynamics & Advanced Classical Mechanics
Not offered 2024-25
This course is for the advanced undergraduate interested in the physics galactic dynamics and evolution, i.e. collisionless, gravitational N-body systems composed of stars and dark matter. Topics
covered will include potential theory, orbit theory, collisionless Boltzmann equation, Jeans equations, disk stability, violent relaxation, phase mixing, dynamical friction and kinetic theory. To
support the these theories, we will also cover advanced topics in classical mechanics including Lagrange & Hamilton methods, the central force problem, canonical transformations, action-angle
variables, chaos and perturbation theory. This course is taught in a seminar format, in which students are responsible for presenting much of the course material in class meetings. Prerequisites:
MATH B201, MATH B203, PHYS B201, B214, and PHYS B308 or permission from instructor.
Contact Us
Mathematics Department
Tina Fasbinder
Academic Administrative Assistant
Fax: 610-526-6575
Park Science Building
Bryn Mawr College
Bryn Mawr, PA 19010-2899
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What is the degree and leading coefficient of this polynomial 4y+18y^2+8-10y^4? | Socratic
What is the degree and leading coefficient of this polynomial #4y+18y^2+8-10y^4#?
1 Answer
Degree $\to 4$
Leading coefficient $\to - 10$
The degree is the highest exponent, which is $4$. The leading coefficient is also the digit in before this highest exponent, which is $- 10$.
Impact of this question
4142 views around the world
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Constructing graphs using permutation or symmetric groups
Constructing graphs using permutation or symmetric groups
I'm trying to construct a graph whose vertices are the elements of a permutation group or a symmetric group. Whenever I do this, it ignores the identity element (). For instance, when I use the
Symmetric Group S3, it prints a graph with 5 vertices and the missing vertex is the identity.
Any ideas on why this happening and how I can fix it?
Welcome to Ask Sage! Thank you for your question.
Please provide some code to let others reproduce the problem easily.
This dramatically increases the chances of an answer, the speed of getting an answer, and the accuracy with which the answers target the problem.
1 Answer
Sort by » oldest newest most voted
Using the Graph([list_of_vertices, list_of_edges]) construction, one can build a graph with vertices the elements in the symmetric group $S_3$, and with no edges, as follows:
sage: S = SymmetricGroup(3)
sage: G = Graph([list(S), []])
sage: G
Graph on 6 vertices
Not a very interesting graph... If the goal is a Cayley graph, use the dedicated method:
sage: C = S.cayley_graph()
sage: C
Digraph on 6 vertices
Tested with SageMath 8.8.beta4 built for Python 3.
edit flag offensive delete link more
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Graph Data Structure 1
Problem E
Graph Data Structure 1
VisuAlgo – a web-based tool for visualising data structures and algorithms through animation – has a Graph Data Structure visualization https://visualgo.net/en/graphds. In that visualization, a user
can edit any of the many example graphs or draw/input their own graph. Each graph in this visualization is a simple graph: a graph that does not have more than one edge between any two vertices and
no edge that starts and ends at the same vertex. In short, a simple graph is a graph without self-loops and multi-edges. Note that the graph can be directed or undirected.
There are a few special graphs that can be automatically detected in that visualization. For this problem, we want to detect these four special graphs given an input graph with $N$ vertices and $M$
edges. The graph’s vertices are $0$-indexed, i.e. they are numbered from $0$ to $N - 1$.
1. Tree-s are undirected graphs in which any two vertices are connected by exactly one simple path. In other words, trees are connected acyclic undirected graphs. An undirected tree is also a
bipartite graph – see below. A Directed Acyclic Graph (DAG) – see below – is also a tree if it is connected and the only vertex with in-degree zero (the root) can reach all other vertices. If a
tree has $N$ vertices, it will have $M = N-1$ edges.
2. Complete Graph-s are undirected graphs in which every pair of distinct vertices is connected by a unique edge. In other words, every vertex in a complete graph is adjacent to all other vertices.
A directed graph can also be considered a Complete Graph if every pair of distinct vertices $(u, v)$ is connected by two unique directed edges $(u, v)$ and $(v, u)$. If a complete undirected
graph has $N$ vertices, it will have $M = N \cdot (N-1)/2$ edges. A complete directed graph of $N$ vertices has twice the edges from its undirected counterpart.
3. Bipartite Graph-s are graphs whose vertices can be divided into two disjoint and independent sets $U$ and $V$. That is, every edge connects a vertex in $U$ to a vertex in $V$. Bipartite Graph can
be defined on directed graph too, but all edge directions must be from a vertex in $U$ to a vertex in $V$.
4. Directed Acyclic Graph (DAG)-s are directed graphs with no directed cycles. That is, it consists of vertices and edges, with each edge directed from one vertex to another, such that following
those directions will never form a cycle. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge
directions. By default, an undirected graph with at least one undirected edge is not a DAG.
Each test case describes a simple graph $G = (V, E)$.
The first line contains three integers, $N$, $M$, and $T$ $(1 \le N \le 100\; 000$, $0 \le M \le 200\; 000$, $1 \le T \le 2)$. Then the next $M$ lines contains the edges, each consisting of two
integers, $u$ and $v$ ($0 \le u, v < N$).
If $T = 1$ (undirected), each edge $(u, v)$ is undirected. Furthermore, $M \le N \cdot (N-1)/2$ also holds.
If $T = 2$ (directed), each edge $(u, v)$ is directed, the direction is from $u$ to $v$. Furthermore, $M \le N \cdot (N-1)$ also holds.
It is guaranteed that the graph does not contain any self-loops or multi-edges.
For each input graph $G$, print 4 integers: $a$, $b$, $c$, and $d$ in one line, separated with a single space.
Put $a = 1$ if $G$ is a tree, or $a = 0$ otherwise.
Put $b = 1$ if $G$ is a complete graph, or $b = 0$ otherwise.
Put $c = 1$ if $G$ is a bipartite graph, or $c = 0$ otherwise.
Put $d = 1$ if $G$ is a DAG, or $d = 0$ otherwise.
1. ($7$ Points): Only contains the following input:
2. ($17$ Points): All graphs in this subtask are either a tree (and also a bipartite graph) or a general graph, $N \ge 3$, $T = 1$.
3. ($10$ Points): All graphs in this subtask are either a complete graph or a general graph, $N \ge 3$, $T = 1$.
4. ($21$ Points): All graphs in this subtask are either a bipartite graph or a general graph, $N \ge 3$, $T = 1$.
5. ($22$ Points): All graphs in this subtask are either a Directed Acyclic Graph or a general graph, $T = 2$.
6. ($23$ Points): No additional constraints. There are $23$ test cases in this subtask and each correct identification worth 1 point.
Explanation of Subtask 1
The following is the illustration of the undirected graph in Subtask 1.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
Sample Input 3 Sample Output 3
Sample Input 4 Sample Output 4
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fit_and_expand: Fit an imputation learner with training features and create... in iai: Interface to 'Interpretable AI' Modules
lnr The learner to use for imputation.
X The dataframe in which to impute missing values.
... Refer to the Julia documentation for available parameters.
## Not run: lnr <- iai::fit_and_expand(lnr, X, type = "finite")
For more information on customizing the embed code, read Embedding Snippets.
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Distributive Property in Elementary Math - The Teaching Couple
Distributive Property in Elementary Math
Written by Dan
The distributive property is a fundamental principle in mathematics that simplifies the multiplication of numbers within an expression. It allows one to multiply a single term by each term in a sum
or difference within parentheses.
Essentially, the property distributes the process of multiplication over addition or subtraction, ensuring that the end result remains consistent.
This rule is instrumental not only in basic arithmetic but also serves as a foundational tool for algebra, playing a crucial role in the simplification of algebraic expressions and solving equations.
Related: For more, check out our article on The Importance of Teaching Arithmetic here.
Understanding and applying the distributive property correctly is critical for students as they progress through various levels of math. It provides a methodical approach to breaking down more
complex problems into simpler parts, making it easier to solve them step by step.
Recognizing how to work with both variables and constants using the distributive property is equally important, as it forms the basis for manipulating and solving algebraic expressions and equations.
Mastery of this property also facilitates a smoother transition to more advanced mathematical concepts and instills confidence in mathematical problem-solving.
Key Takeaways
• The distributive property allows for the simplification of multiplication across sums or differences.
• Mastery of this principle is foundational for algebraic manipulation and solving equations.
• Recognizing the distributive property assists in understanding order of operations and various properties of equality.
Related: For more, check out our article on Surface Area: Mastering The Basics here.
Understanding the Distributive Property
The distributive property is a foundational concept in elementary mathematics, pivotal in simplifying algebraic expressions and solving equations.
Definition and Formula
The distributive property defines how multiplication is distributed over addition or subtraction within an expression. The standard formula for the distributive property is a(b + c) = ab + ac, or a(b
– c) = ab – ac, where a, b, and c are any numbers.
Distributive Property of Multiplication Over Addition and Subtraction
When applying the distributive property of multiplication over addition, one multiplies a single term by each of the terms in the addition or subtraction expression separately.
For instance, if given the expression 3(2 + 4), one would multiply 3 by 2 and 3 by 4, and then add the results: 3 * 2 + 3 * 4 = 6 + 12 = 18.
Similarly, the distributive property of multiplication over subtraction works with the same principle.
Take the expression 2(5 – 3); one would multiply 2 by 5 and 2 by 3 separately, and then subtract the second product from the first: 2 * 5 – 2 * 3 = 10 – 6 = 4.
Related: For more, check out our article on teaching Mode or Modal?
Applying the Distributive Property
The distributive property allows one to multiply a sum by multiplying each addend separately and then adding the results. It is a useful tool in algebra, particularly for simplifying expressions and
working with fractions.
Multiplying Terms Inside Parentheses
When applying the distributive property, terms within parentheses are multiplied by the term outside the parentheses.
For instance, in the expression (3(x + 4)), one multiplies 3 by both (x) and 4, then adds the products. The resulting expression is (3x + 12).
Distributive Property With Fractions
Multiplication involving fractions also adheres to the distributive property. Take ( \frac{2}{3}(6 + 9) ) as an example.
Here, ( \frac{2}{3} ) multiplies each addend within parentheses. Hence, it is equivalent to ( \frac{2}{3} \times 6 + \frac{2}{3} \times 9 ), which simplifies further to ( 4 + 6 ).
Using Distributive Property to Simplify Algebraic Expressions
In the context of simplifying algebraic expressions, the distributive property is used to combine like terms. Consider an expression like (2a + 4(a + 3)). By distributing the 4, the expression
becomes (2a + 4a + 12), and further simplification by combining like terms yields (6a + 12).
Related: For more, check out our article on Maths Equations in GCSEs here.
Working With Variables and Constants
When tackling algebraic expressions, understanding how to work with variables and constants is crucial. Particularly, one must be skilled in combining like terms and handling negative numbers to
simplify expressions effectively.
Mastery of these skills facilitates the efficient manipulation of mathematical statements and equations.
Combining Like Terms
In any algebraic expression, terms that share the same variable part can be combined; these are known as like terms.
For example, in the expression ( 3x + 4x ), both terms are multiples of the variable ( x ). They can be combined to simplify the expression to ( 7x ). This process is termed as combining like terms.
It’s a foundational step in simplifying equations, and it allows mathematicians to reduce complexity and solve for the variables in question.
Handling Negative Numbers
Working with negative numbers in algebraic expressions requires careful attention to signs.
When a negative number is attached to a variable or constant, it influences the resulting sign once the terms are combined. For instance, if one needs to simplify ( 5y – 3y ), the result would be (
2y ), considering the negative sign attached to the ( 3y ).
The presence of negative numbers affects the addition and subtraction of terms and must be treated accordingly to maintain accuracy in simplification.
Related: For more, check out our article on What is A Prime Number?
The Role of the Distributive Property in Equations
The distributive property facilitates the process of solving equations by allowing one to distribute a multiplier across terms within parentheses, simplifying the equation before proceeding with
further steps.
Solving Simple Equations
When dealing with basic algebraic expressions, the distributive property can be employed to simplify and solve equations efficiently. Simplification often involves expanding the equation.
For instance, to solve (3(x + 4)), one applies the distributive property to get (3x + 12), making the equation easier to work with.
• Example:
□ Original Equation: (3(x + 4))
□ After Distributive Property: (3x + 12)
Solving More Complex Problems
In more intricate problems, where equations may include several terms and variables, the distributive property becomes invaluable.
It’s often the first step in these situations to transform complex expressions into simpler ones. For example, in the equation (2(x + 3) + 4(3x – 1)), applying the distributive property throughout
yields a more workable equation.
• Steps:
1. Distribute in each parenthesis: (2x + 6 + 12x – 4)
2. Combine like terms: (14x + 2)
These distributive property examples highlight its utility in breaking down and solving problems efficiently, enabling a clearer path
Related: For more, check out our article on What Are Vertices, Faces and Edges?
Distributive Property and Order of Operations
Understanding the distributive property is essential when simplifying expressions, especially in conjunction with the order of operations. This principle helps in breaking down complex calculations
into more manageable parts.
PEMDAS and the Distributive Property
The distributive property allows you to multiply a sum or difference by multiplying each addend or subtrahend separately, and then adding or subtracting the results.
It is a useful tool for simplifying expressions before applying the order of operations, often remembered by the acronym PEMDAS. This stands for Parentheses, Exponents, Multiplication and Division
(from left to right), and Addition and Subtraction (from left to right).
To illustrate, let’s consider an expression:
3 × (4 + 5)
Applying the distributive property, one would multiply 3 by both 4 and 5 individually:
Then, adding the products:
Hence, 3 × (4 + 5) simplifies to 27 when using the distributive property.
When the order of operations is in play, it’s crucial to first simplify the terms inside the parentheses, which might involve using the distributive property.
Afterward, proceed with exponents, followed by any multiplication or division, and finally any addition or subtraction, in this sequential manner.
For instance:
2 + 3 × (6 + 4)
First, address the parentheses:
Then, apply the distributive property to simplify:
Finally, perform the remaining addition:
Therefore, 2 + 3 × (6 + 4) simplifies to 32, following the PEMDAS rule and utilizing the distributive property.
Comparing Properties of Equality
In mathematical operations, understanding the differences between properties of equality is crucial.
Each property has a specific role in simplifying expressions and solving equations.
Associative vs Distributive Property
The associative property refers to the grouping of numbers. It states that the way in which numbers are grouped does not change their sum or product. For instance, ((a + b) + c = a + (b + c)) or ((a
\cdot b) \cdot c = a \cdot (b \cdot c)).
On the other hand, the distributive property combines both addition and multiplication, demonstrating how a number multiplies a group of numbers added together. Mathematically, it is expressed as (a
\cdot (b + c) = a \cdot b + a \cdot c).
Commutative vs Distributive Property
While the commutative property indicates that the order of the numbers does not affect the sum or product, such that (a + b = b + a) and (a \cdot b = b \cdot a), the distributive property allows for
the multiplication of a number by each addend within a bracket individually before carrying out addition.
The commutative property applies to addition and multiplication, but not to subtraction and division, unlike the distributive property, which can be used across all operations with appropriate
Related: For more, check out our article on What Is Mean In Math?
Practical Applications of Distributive Property
The distributive property is a powerful tool in mathematics that streamlines calculations by breaking down complex multiplication problems into more manageable parts.
Its practical applications can be particularly helpful for facilitating simpler multiplications and working with large numbers or arrays.
Facilitating Simpler Multiplications
Using the distributive property, one can transform tricky multiplication tasks into a series of simpler multiplication problems. For instance, calculate (3 \times 17) by breaking it down to (3 \times
(10 + 7)).
The equation then becomes (3 \times 10 + 3 \times 7), which simplifies to (30 + 21) yielding (51). This method is especially useful for mental math or when a calculator isn’t accessible.
• Break down multiplication: (3 \times 17) into (3 \times (10 + 7))
• Solve simpler problems: (3 \times 10) is (30) and (3 \times 7) is (21)
• Combine the products: (30 + 21) to get (51)
Working With Large Numbers and Arrays
The distributive property is beneficial when dealing with large numbers. It allows one to decompose a large number into a sum of smaller numbers, making the multiplication process less daunting.
In an array of numbers where one factor is common, applying the distributive property can lead to great efficiencies.
For example, multiplying (2) by an array such as ([10, 20, 30]) can be done by distributing the (2), resulting in ([2 \times 10, 2 \times 20, 2 \times 30]) and then simplifying each multiplication
• Decompose large number: (4 \times 222) into (4 \times (200 + 20 + 2))
• Apply to array: ([2 \times 10, 2 \times 20, 2 \times 30])
• Simplify each part: (4 \times 200) is (800), (4 \times 20) is (80), (4 \times 2) is (8)
• Resulting sum: (800 + 80 + 8) is (888)
Related: For more, check out our article on How To Use Concrete, Pictorial and Abstract Resources in Maths here.
Frequently Asked Questions
This section addresses common inquiries regarding the Distributive Property in mathematics, offering clear explanations and examples to enhance comprehension.
What is the Distributive Property in mathematics?
The Distributive Property is a rule that allows one to distribute a multiplication operation over an addition or subtraction within parentheses.
It asserts that multiplying a sum by a number is the same as multiplying each addend by the number and then summing the products.
Can you show an example of how to apply the Distributive Property with addition?
An example of the Distributive Property with addition is 3 × (4 + 5), which equals 3 × 4 + 3 × 5. It shows that multiplying 3 by the sum of 4 and 5 is the same as multiplying 3 by each number
individually and then adding the results.
How is the Distributive Property used with multiplication?
The Distributive Property in multiplication involves breaking down a more complex problem into simpler parts. For instance, to solve 6 × 23, one could use the Distributive Property to compute (6 ×
20) + (6 × 3), thus simplifying the multiplication into more manageable parts.
What are the steps for utilizing the Distributive Property in an equation?
To utilize the Distributive Property in an equation, first multiply the term outside the parentheses by each term within the parentheses. Next, combine like terms if necessary. This method is
particularly useful when dealing with variable expressions.
Can the Distributive Property be used with subtraction and division?
The Distributive Property works with subtraction similar to addition; for example, 4 × (6 – 2) is equal to (4 × 6) – (4 × 2). However, it does not directly apply to division as it does with
At which grade level should students typically learn the Distributive Property?
Students generally encounter the Distributive Property at around 3rd or 4th grade, although the concept becomes more integral to their learning and is applied more extensively in higher mathematics
About The Author
I'm Dan Higgins, one of the faces behind The Teaching Couple. With 15 years in the education sector and a decade as a teacher, I've witnessed the highs and lows of school life. Over the years, my
passion for supporting fellow teachers and making school more bearable has grown. The Teaching Couple is my platform to share strategies, tips, and insights from my journey. Together, we can shape a
better school experience for all.
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Patrick Gelß - Fredholm Equations
Fredholm Equations
The world of neural networks and machine learning has seen rapid advancements in recent years. Neural networks have become a go-to tool for various applications, from image recognition to natural
language processing. However, the process of training neural networks can be challenging, involving complex optimization techniques. In this blog post, we'll explore a novel and mathematically
transparent approach to function approximation and the training of large, high-dimensional neural networks using Fredholm integral equations [1]. This approach is based on the approximate
least-squares solution of associated Fredholm integral equations of the first kind, Ritz-Galerkin discretization, Tikhonov regularization, and tensor-train methods.
Efficient and reliable methods for training neural networks are essential for their success. Traditional techniques like stochastic gradient methods have been around for decades and are widely used.
However, they can be complex and require meticulous parameter tuning for convergence. This blog post introduces a fresh perspective on training neural networks using integral equations.
The core idea of this approach is to treat neural networks as Monte Carlo approximations of integrals over a parameter domain $Q$, assuming that the parameters are uniformly distributed. This leads
to the concept of "Fredholm networks," which provide a unique way to approximate functions. Training these networks involves solving linear Fredholm integral equations of the first kind for a
parameter function:
Find $u \in L^2(Q)$ such that $\mathcal{J}(u) \leq \mathcal{J}(v)$ for all $v \in L^2(Q)$
Fredholm training problem: with $\mathcal{J}(v) = \lVert F - \mathcal{G} v \rVert_\pi^2 = \int_X (F(x) - \mathcal{G} v (x))^2 d \pi(x)$
and $\mathcal{G} v = \int_Q \psi(\,\cdot\,,\eta) v(\eta) d \eta$.
To address the ill-posedness of the Fredholm training problem, we employ Ritz-Galerkin discretization and Tikhonov regularization. These techniques help approximate the solution to the integral
equations, making the approach more practical for real-world applications:
Find $u \in S$ such that $\mathcal{J}_\varepsilon(u) \leq \mathcal{J}_\varepsilon(v)$ for all $v \in S$
regularized Fredholm training problem: with $\mathcal{J}_\varepsilon (v) = \mathcal{J} (v) + \varepsilon \lVert v \rVert^2$
and $S := \mathrm{span}\,\{\varphi_1, \dots, \varphi_p\} \subset L^2 (Q)$.
One of the highlights of this approach is the use of tensor-train methods for solving large linear systems. By considering the algebraic formulation of the regularized Fredholm training problem and
employing alternating linear schemes, we can efficiently handle high-dimensional parameter domains.
To demonstrate the effectiveness of the Fredholm approach, we applied it to three well-established test problems: regression and classification tasks. The experiments included the UCI banknote
authentication dataset, concrete compressive strength prediction, and the MNIST dataset. The results showed that Fredholm-trained neural networks are highly competitive with state-of-the-art methods,
even without problem-specific tuning.
In conclusion, the Fredholm integral equation approach offers a promising alternative for training neural networks and function approximation. Its transparency and competitive performance make it a
valuable addition to the toolbox of machine learning practitioners. As the field continues to evolve, approaches like this open new avenues for efficient and effective training of neural networks.
For more details, we refer to our preprint:
[1] P. Gelß, A. Issagali, R. Kornhuber. Fredholm integral equations for function approximation and the training of neural networks. arXiv: 2303.05262
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Quadratic Equations
To view this video please enable Javascript
The essence of solving quadratic equations for the GRE exam involves understanding the unique strategies distinct from those used for linear equations, employing factoring methods, and applying the
Zero Product Property to find solutions.
• Quadratic equations are set in the form a squared plus bx plus c equals zero and often have two solutions.
• The strategy for solving quadratic equations diverges significantly from that of linear equations, emphasizing the need for factoring and setting the equation to zero before solving.
• The Zero Product Property is crucial for solving quadratic equations, allowing for the determination of solutions by setting each factor equal to zero.
• Most quadratic equations encountered in the GRE can be solved by factoring into a product of linear binomials, then applying the Zero Product Property.
• Some quadratics may have one solution, no solution, or require the use of the Quadratic Formula, especially in more advanced quantitative sections.
Introduction to Quadratic Equations
Factoring and the Zero Product Property
Solving Quadratics with Special Conditions
General Procedure for Solving Quadratics
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TR08-069 | 5th August 2008 00:00
3-Query Locally Decodable Codes of Subexponential Length
Locally Decodable Codes (LDC) allow one to decode any particular
symbol of the input message by making a constant number of queries
to a codeword, even if a constant fraction of the codeword is
damaged. In recent work ~\cite{Yekhanin08} Yekhanin constructs a
$3$-query LDC with sub-exponential length of size
$\exp(\exp(O(\frac{\log n}{\log\log n})))$. However, this
construction requires a conjecture that there are infinity many
Mersenne primes. In this paper we give an unconditional $3$-query
LDC construction with shorter codeword length of
$\exp(\exp(O(\sqrt{\log n \log \log n })))$. We also give a
$2^r$-query LDC with length of $\exp(\exp(O(\sqrt[r]{\log n \log
\log^{r-1} n })))$. The main ingredient in our construction is the
existence of super-polynomial size set-systems with restricted
intersections \cite{Grolmusz00} which holds only over composite
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128 research outputs found
The effective action of the linear meson model generates the mesonic n-point functions with all quantum effects included. Based on chiral symmetry and a systematic quark mass expansion we derive
relations between meson masses and decay constants. The model ``predicts'' values for f_eta and f_eta' which are compatible with observation. This involves a large momentum dependent eta-eta' mixing
angle which is different for the on--shell decays of the eta and the eta'. We also present relations for the masses of the 0^{++} octet. The parameters of the linear meson model are computed and
related to cubic and quartic couplings among pseudoscalar and scalar mesons. We also discuss extensions for vector and axialvector fields. In a good approximation the exchange of these fields is
responsible for the important nonminimal kinetic terms and the eta-eta' mixing encountered in the linear meson model.Comment: 79 pages, including 3 abstracts, 9 tables and 9 postscript figures,
LaTeX, requires epsf.st
We present an analytical description of the phase transitions from a nucleon gas to nuclear matter and from nuclear matter to quark matter within the same model. The equation of state for quark and
nuclear matter is encoded in the effective potential of a linear sigma model. We exploit an exact differential equation for its dependence upon the chemical potential $\mu$ associated to conserved
baryon number. An approximate solution for vanishing temperature is used to discuss possible phase transitions as the baryon density increases. For a nucleon gas and nuclear matter we find a
substantial density enhancement as compared to quark models which neglect the confinement to baryons. The results point out that the latter models are not suitable to discuss the phase diagram at low
temperature.Comment: 27 pages, Int.J.Mod.Phys.A versio
We employ nonperturbative flow equations to compute the equation of state for two flavor QCD within an effective quark meson model. This yields the temperature and quark mass dependence of quantities
like the chiral condensate or the pion mass. A precision estimate of the universal critical equation of state for the three-dimensional O(4) Heisenberg model is presented. We explicitly connect the O
(4) universal behavior near the critical temperature and zero quark mass with the physics at zero temperature and a realistic pion mass. For realistic quark masses the pion correlation length near
T_c turns out to be smaller than its zero temperature value.Comment: 49 pages including 15 figures, LaTeX, uses epsf.sty and rotate.st
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CBSE Worksheets for Class 10 Maths Archives - WorkSheets Buddy
CBSE Worksheets for Class 10 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. CBSE Class 10 Maths Worksheet for students has been used by teachers &
students to develop logical, lingual, analytical, and problem-solving capabilities. So in order to help you with that, we at WorksheetsBuddy have come up with Kendriya Vidyalaya Class 10 Maths
Worksheets for the students of Class 10. All our CBSE NCERT Class 10 Maths practice worksheets are designed for helping students to understand various topics, practice skills and improve their
subject knowledge which in turn helps students to improve their academic performance. These chapter wise test papers for class 10 maths will be useful to test your conceptual understanding.
Board: Central Board of Secondary Education(www.cbse.nic.in)
Subject: Class 10 Maths
Number of Worksheets: 119
CBSE Class 10 Maths Worksheets PDF
All the CBSE Worksheets for Class 10 Maths provided in this page are provided for free which can be downloaded by students, teachers as well as by parents. We have covered all the Class 10 Maths
important questions and answers in the worksheets which are included in CBSE NCERT Syllabus. Just click on the following link and download the CBSE Class 10 Maths Worksheet. CBSE Worksheets for Class
10 Maths can also use like assignments for class 10 maths students. By improving concentration, you can get the best out of your study.
CBSE Class 10 Maths Areas related to Circles Worksheets
CBSE Class 10 Maths Arithmetic Progression Worksheets
CBSE Class 10 Maths Circles Worksheets
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CBSE Class 10 Maths Real Numbers Worksheets
CBSE Class 10 Maths Statistics Worksheets
CBSE Class 10 Maths Surface Area and Volume Worksheets
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CBSE Worksheets for Class 10 Maths SA1 Worksheets
Advantages of CBSE Class 10 Maths Worksheets
1. By practising NCERT CBSE Class 10 Maths Worksheet, students can improve their problem solving skills.
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Worksheets of CBSE Class 10 Maths are devised by experts of WorksheetsBuddy experts who have great experience and expertise in teaching Maths. So practising these worksheets will promote students
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further to download free CBSE Class 10 Maths Worksheets PDF.
Now that you are provided all the necessary information regarding CBSE Class 10 Maths Worksheet and we hope this detailed article is helpful. So Students who are preparing for the exams must need to
have great solving skills. And in order to have these skills, one must practice enough of Class 10 Maths revision worksheets. And more importantly, students should need to follow through the
worksheets after completing their syllabus. Working on CBSE Class 10 Maths Worksheets will be a great help to secure good marks in the examination. So start working on Class 10 Maths Worksheets to
secure good score.
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The Maths of Star Trek: The Original Series (Part II)
You're reading: Irregulars
The Maths of Star Trek: The Original Series (Part II)
This is the second in a series of posts about the maths of Star Trek. Part I covered the probability of survival while wearing a red shirt.
The Mathematics of Alien Biology
In ‘The Trouble with Tribbles’ the Enterprise are assigned to space station K-7 to guard a shipment of grain. Meanwhile, Kirk has to deal with the officious bureaucrat Baris, a rowdy crew of
Klingons, and an infestation of tribbles.
Tribbles are cute little hairballs that make soothing chirping noises as you pet them. Unfortunately, they are also born pregnant, so their population grows very, very quickly, making them a pest
which soon completely overruns the Enterprise and space station.
In fact, the population of tribbles is said to grow geometrically. Sulu gives a very eloquent description of geometric growth while describing Gary Mitchell’s growing god-like powers in the pilot
episode ‘Where No Man Has Gone Before’;
SULU: If you want the mathematics of this, Mitchell’s ability is increasing geometrically. That is, like having a penny, doubling it every day. In a month, you’ll be a millionaire.
I really like this explanation. Geometric growth means each step is multiplied by a common factor. Doubling a penny 30 times would give $1 × 2^{30} = 1,073,741,824$ pennies – or about 10 million
In ‘The Trouble with Tribbles’ Spock makes a similar calculation to estimate the population of tribbles:
SPOCK: They seem to be gorged.
BARIS: Gorged? On my grain? Kirk, I am going to hold you responsible. There must be thousands of them.
KIRK: Hundreds of thousands.
SPOCK: One million seven hundred seventy one thousand five hundred sixty one. That’s assuming one tribble, multiplying with an average litter of ten, producing a new generation every twelve hours
over a period of three days.
KIRK: That’s assuming they got here three days ago.
SPOCK: And allowing for the amount of grain consumed and the volume of the storage compartment.
Classic Spock, but was this number just made up – or had the writer of the episode, David Gerrold, actually done the maths? Let’s find out:
According to Spock, each generation is ten times larger than the previous generation. This means that if we start with a population of $P$ tribbles, the next generation will be $10P$ tribbles, making
$11P$ tribbles altogether. So we see that the population increases by a factor of 11 every 12 hours. If we start with 1 tribble, then after three days the population will have multiplied by 11 six
times, making $1 \times 11^6 = 1{,}771{,}561$ tribbles – just as Spock claimed!
I question the wisdom of giving the answer to seven significant figures, considering how much estimation is needed. But let’s not quibble about tribbles. The important thing is – the maths was right!
I was so surprised I nearly fell off my captain’s log.
The tribble home world must be pretty inhospitable, or have some fearsome predators, to give rise to a species that grows so rapidly. (We even meet a genetically engineered predator in Star Trek: The
Animated Series called the glommer, but I digress).
Population growth is one of the areas studied in mathematical biology, a subject that can be used to explain some of the other biological quirks we encounter in Star Trek.
In one of the most celebrated episodes, ‘Amok Time’, we learn more about the character of Spock, his Vulcan heritage and his alien biology. The crew are concerned when Spock appears to acting out of
character, he is acting emotionally and irrationally. The reason turns out to be biological, Spock must return to the planet Vulcan and choose a mate:
SPOCK: How do Vulcans choose their mates? Haven’t you wondered?
KIRK: I guess the rest of us assume that it’s done quite logically.
SPOCK: No. No. It is not. We shield it with ritual and customs shrouded in antiquity. You humans have no conception. It strips our minds from us. It brings a madness which rips away our veneer of
civilisation. It is the pon farr. The time of mating. There are precedents in nature, Captain. The giant eelbirds of Regulus Five, once each eleven years they must return to the caverns where
they hatched. On your Earth, the salmon. They must return to that one stream where they were born, to spawn or die in trying.
KIRK: But you’re not a fish, Mister Spock. You’re…
SPOCK: No. Nor am I a man. I’m a Vulcan. I’d hoped I would be spared this, but the ancient drives are too strong. Eventually, they catch up with us, and we are driven by forces we cannot control
to return home and take a wife. Or die.
Although not mentioned explicitly in the episode itself, it is later established that Spock must return to Vulcan every 7 years. And we do indeed find this type of behaviour in nature. Famously, in
the insect world the cicada of North America have long life-cycles, where they live underground, but then emerge every 13 or 17 years to mate. They make so much noise in the process of attracting a
mate that the local community are forced to move out until it’s over.
However, it isn’t just a coincidence that the life cycles are both prime numbers. Evolutionary biologist Stephen Jay Gould suggested that this mechanism evolved in order to keep the cicadas out of
step with the life cycles of local predators. For example, if the cicada had a life-cycle of 18 years, and a predator in the area had a life-cycle of 6 years, then these cycles would coincide every
18 years – that’s every generation of cicada and every third generation of predator. But if the cicada have a life-cycle of 17 years, then the two will coincide only every 102 years – that’s 17
generations of predators, and so the predators never develop a taste for the cicada.
Maybe evolutionary mechanisms existed on Vulcan to cause the people to behave in a similar way to the cicada – after all, 7 is a prime number too.
One of the most memorable images in the original Star Trek occurred in their otherwise troublesome third season. In ‘Let That Be Your Last Battlefield’ the crew meet Bele and Lokai, two inhabitants
of the planet Cheron. Both are completely black on one side of the face and completely white on the other side. But it is Lokai’s race that is repressed for being black on the left hand side.
Kirk, Spock and McCoy are amazed an individual with such distinctive markings exists:
SPOCK: You are certain, Doctor, that this pigmentation is the natural condition of this individual?
MCCOY: That’s what I’ve recorded, Mister Spock.
KIRK: Do we have any knowledge of a planet that could have produced such a race of beings?
SPOCK: Negative, Captain.
KIRK: Bones, what do you make of it?
MCCOY: Well, I can’t give you any specific circumstance that will explain him.
KIRK: And judging by looking at him, we know at the very least he is the result of a very dramatic conflict. Spock?
SPOCK: There is no theory, Captain, from the basic work of Mendel to recent nucleotide studies, which would explain our captive. All gradations of colour from black to brown, to yellow to white
are genetically predictable. We must therefore conclude that this alien is that often unaccountable rarity, a mutation, one of a kind.
However, the unlikeliness of such a species may be a little overstated. The theory of animal patterns is known as morphogenesis, and may be used to explain the stripes of a zebra, the spots of a
leopard, and the mosaic pattern of a butterfly.
The mathematics of animal patterns was started by the famous World War II code breaker and mathematician, Alan Turing. His seminal work models the combination of chemical reactions and diffusion in
order to produce pigmentation patterns in the shape of stationary waves.
Imagine hitting a cymbal in the shape of a cow. The cymbal will vibrate producing peaks and troughs of waves. Although these peaks and troughs will oscillate in time and space, there are preferential
wave forms that stay in a fixed location. These peaks and troughs suggest the positions of different chemical concentrations and these different concentrations lead to variations in the animal
markings. For that reason, morphogenesis has been sometimes been described as “waves on cows”.
However, there is still work to be done. For although Turing patterns can generate a half black and half white pattern, they tend to form along the longest body axis from head to tail, such as those
found on valais goats. Producing a pattern like those of the planet Cheron will be much tougher because their pattern changes on the shorter body axis of left to right.
But with so much new life and so many new civilisations to seek out in the Star Trek universe, we should expect to encounter even the extremes of evolution. Which brings us to our next question, how
populated is the Star Trek universe? And how populated is our own universe?
Thanks to Thomas Woolley for his help with this article.
This series of articles will conclude on Friday with Part III which discusses whether we are alone in the galaxy, as well as how to kill an android.
12 Responses to “The Maths of Star Trek: The Original Series (Part II)”
1. Eric Kuennen
I love the post. Spock was right about the tribbles. If we assume that tribbles have a lifespan of over three days, then the original tribble is still alive, and so are its children and
subsequent generations, each also continuing to have children every 12 hours. So we can count the tribbles based on which generation (i.e. how old it is). There is of course just 1 zeroth
generation tribble (the original one). There are also $10 \times 1 = 10$ first generation tribbles (the children of the original one). Now it gets a bit trickier: there are $10 \times (1+10) =
110$ second generation tribbles (10 children for each of the 11 tribbles now alive, or equivalently, the 10 new children of the original, and the 100 new grandchildren). Continuing, there are $10
\times ( 1+10 + 110) = 1210$ third generation tribbles, $10\times (1+10+110+1,210)=13,310$ fourth generation tribbles, $10\times (1+10+110+1,210+13,310) = 146,410$ fifth generation tribbles, and
$10\times ( 1 + 10+ 110 + 1,210 + 13,310 + 146,410) = 1,610,510$ sixth generation tribbles. Adding up all of these generations of tribbles, we get $1 + 10 +110 +1,210 +13,310 +146,410 +1,610,510
= 1,771,561$, which is the figure Spock reported. (Of course, this is assuming a litter of exactly 10 for each tribble, occurring once exactly every 12 hours for exactly 3 days. I agree with your
sentiments on Spock’s proclivity for over-reporting significant digits in his estimates.)
□ Christian Perfect
I wonder if it would be worth adding a comment to A055276 noting that it’s the “Tribble sequence”.
☆ Anonymous
Neat! More on the “Tribble Sequence”: The terms in this sequence are the binomial coefficients times powers of 10, so another way of formulating my previously described version of Spock’s
answer is $1 + 6\times 10 + 15\times 10^2 + 20 \times 10^3 + 15\times 10^4 +6 \times 10^5 + 10^6 $. Using the binomial theorem, this simplifies to $(1+10)^6 = 11^6 = 1,771,561. $ So,
anyone who knows cold their powers of 11 should have instantly recognized Spock’s answer as $11^6$. Such a simple answer should mean that there should be a simple way of thinking of the
solution, and there is. To figure out how many tribbles there should be at any stage, we can just multiply the previous number of tribbles by $(1+10)$. Multiplying by 1 keeps all of the
existing tribbles alive, and multiplying by 10 computes the number of newborn tribbles. So starting with 1 tribble, after 6 generations we just multiply by (1+10) a total of 6 times, to
get $11^6 = 1,771,561$.
In general, if starting with $a$ tribbles, and each has a litter of size $m$, then after $n$ generations there are $a(1+m)^n$ tribbles.
□ James Grime
That’s fantastic Eric, thank you. That means their maths was even better than I thought. I will try to update the article in the next couple of days.
□ James Grime
Done. Turns out the edit was easy (take a look). Still makes the maths more interesting though.
2. Anonymous
A population of that size is bound to have suffered some still-births, or death through other natural causes. So it’s not enough to assume that the oiginal tribble is still alive; one has to
assume, regardless of lifespan, that none of the other tribbles has died. I hate Star Trek.
3. Ron
I know Im a bit late to this game but I just stumbled on this article. I don’t know why nobody is mentioning that this is simple exponential growth. A geometric understanding of a mathematical
problem is just its graphical interpretation. populations or mental powers cannot grow geometrically. It was just a mistake made by someone on the TV show when they meant exponential. Still love
reading articles like these though.
4. april
Okay Paramount and CBS does own the cheron man like lokia and bele
The only two guys that been on TV
But what about the rest of it dead planet I don’t think so not in the fan world
We have cheron women cheron children cheron and babies and teenage
I guess we can play with all it
They lost out on something that could’ve gone into the right direction
oh well
5. Cindy Scott
Ah! You’ve all forgotten the Great Tribbles War, whereupon the Klingons exterminated all Tribbles on the home world after they became “an environmental disaster”, according to then Commander Worf
who was talking to Odo on Deep Space 9.
6. Patrick Joseph
I did some math to calculate approximately how large a storage compartment would need to be to hold 1,771,561 tribbles. Considering that a tribble is mostly fur, you could probably compress one
down to a three inch square, and in the storage compartment they would be stacked on top of each other. So, based on those assumptions, 64 of them could fit in a volume of 1 cubic foot Now, let’s
assume that the floor of the storage compartment was 12 feet by 12 feet. That means that each foot of height in the compartment has 144 cubic feet of area. If each cubic foot of area could
contain 64 tribbles, then each one foot layer of the compartment could hold 144 X 64 = 9,216 tribbles. How many one foot layers would be needed for 1,771,561 tribbles? 1,771,561 ÷ 9,216 = 192.2.
How large would the storage compartment need to be? 12 feet by 12 feet, and 192.2 feet high. Captain Kirk would not survive the onslaught.
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Characteristic Description
Name Crude rate of natural increase (%) by Place of residence (NUTS - 2013); Annual
Regularity Annual
Source Statistics Portugal, Demographic indicators
First available 2011
Last available 2023
• Data reference period
Dimensions • Place of residence (NUTS - 2013)
• AVERAGE POPULATION: The average population during a calendar year, generally calculated as the arithmetic mean of the population on two consecutive years.
• NATURAL INCREASE: The difference between the number of live births and the number of deaths occurring during a given period, usually a calendar year divided by the mid-year
Concepts population of that period* (usually expressed per 100 (10^2) or 1000 (10^3) inhabitants).
• REFERENCE PERIOD: Period to which the information refers and which may be a specific day or a time interval (month, fiscal year, calendar year, among others).
RNI=[SN(t-1, t)/[P(t)+P(t-1)]/2]*10^n
Formule P(t)=Population at moment t;
P(t-1)=Population at moment (t-1);
NB(t-1,t) = Natural balance between the moments (t-1) e t;
n=2 ou 3.
Measure unit Percentage (%)
Power of 10 0
Last update date 18/06/2024
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What is the meaning of y = mx + b?
The equation y = mx + b is a crucial math theory utilized to depict a linear relationship within two variables. It is usually used in numerous domains, for example, engineering, economics, physics,
and finance, to model relationships between different variables.
The equation involves numerous components that respectively play an essential role in determining the behavior of the relationship. The slope (m) is the rate at which the dependent variable (y)
alters in relation to the independent variable (x). The y-intercept (b) is the value of y when x is equivalent to zero, portraying the starting point of the relationship. The dependent variable (y)
represents the value being predicted or measured, while the independent variable (x) depicts the input or variable being manipulated.
In this blog, we will dwell deeper into the meaning of y = mx + b, studying every component and how they [[contribute|[provide]42] to the overall equation. We will also give examples of how the
equation is utilized in several fields, consist of its uses in analyzing trends, graphing, and making forecasts.
It is crucial to understand the meaning and properties of y = mx + b to succeed in domains that depend on linear relationships. Misconception or misuse of this equation could lead to wrong
predictions, erroneous conclusions, and substandard decision-making.
If you need guidance getting a grasp the equation y = mx + b or any other mathematical idea, contemplate calling us at Grade Potential Tutoring. Our adept tutors are accessible remotely or
face-to-face to provide personalized and effective tutoring services to help you be successful.
Elements of y = mx + b
In the equation y = mx + b, y represents the dependent variable, x illustrates the independent variable, m depicts the slope, and b illustrates the y-intercept. These components are crucial in
understanding the equation and its uses to real-life challenges.
The dependent variable y illustrates the output of the function, whereas the independent variable x depicts the input. The slope m illustrates the modification in y over the change in x, and the
y-intercept b represents the value of y while x is equal to zero.
The slope (m) can be positive, negative, or zero, and it determines the course and steepness of the line. A positive slope states that as x increases, y increases, while a negative slope implies that
as x increases, y decreases. A slope of zero means that there is no modification
alter in y as x increases.
The y-intercept (b) is the point where the line bisects the y-axis. It represents the value of y while x is equivalent to zero.
Examples of y = mx + b
The equation y = mx + b is utilized to represent a linear relationship within two variables, and it has many utilizations in various fields, including engineering, science, and finance. Such as in
science, the equation could be applied to model the connection between temperature and time, whereas in engineering, it could be utilized to model the link within speed and distance.
In finance, the equation can be utilized to model the connection between the value of an investment and time. The slope depicts the rate of return on the investment, and the y-intercept depicts the
primary value of the investment.
Importance of y = mx + b
The equation y = mx + b is an essential mathematical idea which is utilized to depict linear relationships between two variables. It is a rudimental tool for studying and assessing data in various
domains, including engineering, science, and finance.
Understanding the elements of the equation, including the slope and y-intercept, is important for interpreting and making projections on the basis of the data. By understanding the meaning of y = mx
+ b and applying it to work out problems in several fields, everyone can get a deeper grasp of the complex functions of the world surrounding us.
Ordinary Errors with y = mx + b
One ordinary error when utilizing the equation y = mx + b is forgetting to take into account the units of measurement. It is important to ensure that the units of measurement for both the dependent
and independent variables are consistent, alternatively the slope will not be significant.
Another common error is believing that a linear relationship exists while it might not be the case. It is essential to check for nonlinear relationships or outliers in the data to make sure that the
linear model is correct.
Additionally, it is important to keep in mind that the equation y = mx + b is a linear model and might not be correct for every types of data. It is crucial to grasp the limitations of the linear
model and to investigate other types of models whereas required.
In conclusion, the equation y = mx + b is a rudimentary math theory which represents a linear connection within two variables, whereas m represents the slope of the line and b illustrate the
y-intercept. Comprehending this equation is crucial for students and professionals in domains such as math, science, engineering, and finance, whereas linear relationships play a crucial role in many
By comprehending the meaning and properties of y = mx + b, anyone can gain a deeper grasp of linear relationships, that could assist them in several utilizations, for example, estimating future
values, determining rates of change, and assessing trends. It is further essential to be mindful of common mistakes and pitfalls linked with this equation, for instance, expecting a linear
relationship while it might not be correct, or using incorrect units of measurement.
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Will My Project Finish on Time?
August 5th, 2009 by Paul Below
Events are said to be independent when the outcome of one event does not affect the other.
On the other hand, two events are dependent when the occurrence or nonoccurrence of one event does affect the probability of the other event.
This is an important distinction, as we shall see.
When using the multiplication rule for independent events, sometimes we use the percent chance of success, other times we use the percent chance of failure. We must think about what we are trying to
calculate when deciding which to use. If we want to calculate the probability that all the independent events will fail, then we would use the chances of failure. On the other hand, if we want the
chance that all the independent events will succeed, then we use the chances of success. In a situation where we want the probability that one or more of the events will fail, then we would use one
minus the multiplication of the chances of success (one minus the chance that all of the events will succeed will be the chance that one or more would fail).
Simple example: A software development project is going to proceed concurrently with the development of a new piece of hardware required to implement the software. Scheduled completion dates for both
developments have been determined and a project plan has been created. Both projects can proceed independently until their respective completions (probably an unwarranted assumption, but I said this
is a simple example!). Both projects must succeed in order for overall success to be achieved.
Using SLIM-Estimate, the probability of the software project being completed on the scheduled date has been estimated at 75%. Likewise, the probability of the hardware project being completed on the
scheduled date has been estimated at 75%. Using these percentages, what is the probability that the project (both the hardware and software) will be implemented on time?
The multiplication rule says that .75 * .75 = 56% chance they will both complete on time. (6% chance that both will miss, 19% that the hardware only will miss, 19% that the software only will miss).
On the other hand, if both the software and hardware projects were estimated and planned to a 50% solution, then the probability of both finishing on time is only .5 * .5 = .25, or 25%!
The assumption of independence is a very strong assumption. Quite often events have some dependencies on one another. This means that the multiplication rule doesn’t apply, and that the actual
situation is more complex. Fortunately, we can use SLIM‐MasterPlan to help evaluate multiple estimates.
SLIM‐MasterPlan uses Monte Carlo analysis to give us insight into the effect of complex interrelationships between tasks has on the distribution of likely project completion times. Let’s look at an
This example program consists of 4 projects (or releases). Release 1 is set for low uncertainty and 2 through 4 are set to moderate uncertainty. This is a common situation, since we often have more
knowledge about current work than we do about work farther in the future. To simplify things, the four releases are identical in size and productivity. Releases 2 through 4 have been set to start
when the prior release is 50% complete.
So, what is the probability that the program will complete on time? The answer is, about 20%. This is significantly less than the 50% that some people may have assumed when they created their program
plan built upon 50% estimates for the individual releases. On the other hand, it is higher (whew!) than the 6% we would calculate if the four releases were truly independent and we needed all four to
complete on time to be successful (.5 * .5 * .5 * .5 = .0625).
The following graph is the cumulative probability of program completion. The vertical line near 34 months represents the estimated completion date of release 4, the final release.
For programs with multiple projects or releases or iterations, let MasterPlan help you determine the likelihood of desired outcomes.
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1 Introduction
Magnetic systems have attracted much attention over the last decades considering their intrinsic fundamental behaviors and possible applications in nanoscale devices. Chemists have put much effort to
design and fully characterize new families of systems which may exhibit unusual and fascinating properties arising from the strongly correlated character of their electronic structures. From a
fundamental point of view, high-T[c] superconducting copper oxides [1–3] and colossal magnetoresistant manganite oxides [4–9] are such families which cannot be ignored in the field of 2D and 3D
materials. 1D chains [10–13] as well as molecular systems mimicking biological active centers [14–17] have more recently been considered as promising targets in the understanding of dominant
electronic interactions. In such materials, a rather limited number of electrons is responsible for the observed intriguing properties. Theoretically, since these electrons (often referred to as
magnetic electrons) are usually rather localized in space (e.g., d orbitals), the electronic wavefunction is intrinsically multireference which may dispose single reference strategies. Nevertheless,
in numerous cases, a reasonably satisfactory energetics description of such systems can be obtained by the elegant broken symmetry (BS) method [18–21]. Let us mention that, in particular, BS Density
Functional Theory (DFT) calculations have turned out to be very efficient in the determination of magnetic coupling constants (see Refs. [22–29] and references therein). Conversely, DFT has failed to
predict the insulating behavior of nickel oxide NiO by using spin densities [30]. Nevertheless, a recent approach based on natural orbitals and their occupation numbers has shown to improve
significantly the spin density description in highly correlated systems [31]. More recently, the inefficiency of DFT methods has been pointed out in the low-energy spectrum description of molecular
systems [32–35].
In order to bridge the gap between experimental measurements and a detailed microscopic understanding, one traditionally searches for a rationalization based on effective valence bond (VB)
Hamiltonians [36]. The latter aims at reducing the available electronic structure calculations into different effective integrals such as hopping, exchange and onsite repulsion energies. With this
goal in mind, accurate ab initio quantum chemical calculations are extremely valuable, since information is available not only in energy differences but also in the wavefunctions. While the
derivation of such parameters is straightforward from pure spin states, the correspondence between numerical results and the VB parametrized Hamiltonians is controversial as spin-unrestricted and
non-orthogonal orbitals are used. This particular issue has been debated in the literature [37].
From the experimental point of view, open-shell systems have opened up new routes to large information storage. As prototype of such, spin transiting materials based mainly on Fe(II) ions
architectures look very promising candidates for bistability behavior [38]. In the field of nanoscale objects' synthesis, single-molecule magnets (SMM) and single-chain magnets (SCM) have also been
subject to intense development (see Refs. [12,13,39–41] and references therein) since the pioneering works on Mn[12] compound [42–44]. As in biological porphyrins, some intriguing molecules associate
radical ligands with metal d ions. This particular class of ligands, referred to as noninnocent, offers the possibility to tune magnetic interactions between rather different partners. Experimental
setup (i.e. squid) performances have followed the need for size reduction.
In this paper, our goal is to exemplify the need for theoretical inspection in correlated materials. The paper will be organized as follows. We first recall the theoretical approaches which are
currently available and introduce the important concepts of static and dynamical correlations. Wavefunction ab initio methods will be shortly presented using the complete active space self-consistent
field (CASSCF) method as a featuring multireference technique. Then, using recent examples of molecular and extended systems, we will show that the ab initio machinery is likely to capture the
crucial physical effects which lead to quantitative evaluations of observables and effective parameters.
2 Theoretical setup
For the systems under interest holding localized spin moments, the description goes through the inspection of the spatial distribution and energetics of the unpaired electrons. As the latter grows,
the number of spin configurations becomes rapidly important (2^N). Nevertheless, even multicentric architectures display a rather limited collection of so-called magnetic electrons (∼several tens at
most) and one can provide at first a satisfactory, though qualitative, electronic picture. The different electronic configurations arising from the unpaired electrons in the field generated by the
rest of the electrons (core electrons) and nuclei can be grasped along the CASSCF method [45] or its restricted RAS (i.e. restricted active subspace) analogue [46].
This configuration interaction (CI) method is based on the wavefunction expansion on all possible Slater determinants generated by distributing n electrons in m orbitals, so-called CAS(n,m). Along
this framework generalizing the Hartree–Fock (HF) method to a multireference zeroth-order description, the failure of single determinantal SCF approaches is overcome. The CASSCF spin-adapted
wavefunction provides a set of symmetry-adapted molecular orbitals (MOs) whose occupation numbers are either 2 or 0, defining the inactive and virtual MOs, respectively. The active space MOs have
occupation numbers which range from 0 to 2 (see Fig. 1). Along this framework, one effectively introduces the so-called static correlation effects by evaluating the leading configurations on the same
footing. In particular, quasi-degenerate configurations which play a dominant role in bond-breaking phenomenon are dealt with this zeroth-order wavefunction. Let us exemplify this treatment using the
H[2] molecule in a minimal atomic basis set {a,b}. The CASSCF wavefunction reads:
whilst the HF solution is only $|gg¯|$, g and u being the in-phase (‘gerade’) and out-of-phase (‘ungerade’) linear combinations of a and b, respectively. As the H[2] molecule dissociates, one expects
$|gg¯|$ and $|uu¯|$ determinants to be degenerate. Therefore, the correct wavefunction must be a linear combination with $λ∼μ∼1/2$. In the long-distance regime, $g=(a+b)/2$, $u=(a−b)/2$ (S[ab]∼0)
and Eq. (1) reads:
$ψCAS=λ−μ2(|aa¯|+|bb¯|)+λ+μ2(|ab¯|+|ba¯|)$ (2)
As expected from intuitive physical arguments, the long-distance wavefunction must be consistent with 2 H^ radicals. Thus, the ionic contributions (i.e. 2 electrons on the same site, $|aa¯|$ and $|
bb¯|$) should become vanishingly small $(λ≃μ)$. As a consequence, electron 1 in a implies electron 2 in b. In that sense, electrons are said to be strongly correlated.
Fig. 1
Electronic partitioning within the CASSCF framework.
In more elaborated architectures, the dynamical response of the core electrons to the fluctuation within the active space might be of crucial importance to account for subtle behaviors. The
participation of the higher-lying electronic configurations can be performed along a variational CI expansion on top of the reference CASSCF function, or by using the second-order perturbation theory
(so-called CASPT2) [47,48]. As an important method of the former strategy, the Difference Dedicated CI (DDCI) [49,50] approach has been extensively used for magnetic systems for which the relevant
physical effects (spin polarization, dynamical correlation) are essential to explain and rationalize the experimental magnetic exchange coupling constants [51–53]. Being variational, the DDCI method
allows the external correlation to modify the coefficients of the active space (see λ and μ in Eq. (1)). In particular, the weights of the ionic forms can be enhanced as instantaneous electronic
relaxation responds to the electric field fluctuations. This impressively efficient method relies on the statement that the configuration interactions' expansion excludes all double excitations from
inactive (i.e. doubly occupied) to virtual (i.e. vacant) molecular orbitals. As a matter of fact, it is shown that for vertical transitions which involve excitations depicted by the active space
choice (i.e. at zeroth-order) such contributions cancel out at second-order of perturbation theory. Thus, one defines successively DDCI-1, 2 and 3 levels of calculation with respect to the number of
degrees of freedom solicited in the configuration generation (see Fig. 2). A degree of freedom is defined as a hole or a particle created in doubly occupied or empty MOs in all reference
configurations, respectively. This methodology was developed originally by Malrieu [51,54,55] and later implemented in the CASDI code [56]. More recently, this spectroscopy oriented methodology
(so-called spectroscopy oriented configuration interaction, SORCI) [57] has been added to the ORCA package which offers a large amount of observable calculations, including relativistic effects [58].
This method has proven to be impressively accurate in the rationalization of low-energy spectroscopies, particularly for molecular [49,52,53,59–65] to extended magnetic systems [66–72]. Clearly,
magnetic exchange coupling constant (J) calculations are challenges for theoreticians, since they usually range from a few cm^−1 to a few hundred cm^−1. In the meantime, let us mention that numerous
studies have relied on DFT approaches using the BS approximation. In this framework, J is evaluated from the unrestricted highest multiplets and fictitious broken symmetry single reference
wavefunction energies [22,73]. This method has turned out to be very useful for large systems and in magneto-structural characterization (for recent applications, see Refs. [23–29] and references
therein). Nevertheless, it suffers from spin contamination problems inherent to spin polarized calculations. The main drawbacks of this method have been analyzed in the literature [35,37,74,75].
Fig. 2
Schematic representation of the DDCI-1, DDCI-2, and DDCI-3 configurations.
Undoubtedly, there is a real demand for both experimental measurements' analysis and standard models' parametrization in the inspection of molecular and extended materials.
3 Machinery at work for experimental interpretation
Over the past decades, a huge amount of experimental data carried out on a wide panel of systems has received much attention from both CI and DFT-based frameworks. For the present purpose, we limit
our inspection to a selection of architectures of various dimensionalities. Over the years, the possibilities to generate magnetic systems using versatile ligands coordinated to different metallic
centers have been much considered in the light of porphyrin-like molecules' activity. Thus, the traditional scenario involving open-d shells in the environment of closed-shell magnetic couplers (see
Sections 3.1 and 3.4) has been revisited based on both experimental and theoretical works (see Section 3.2). By varying the number (see Section 3.3) and nature (see Section 3.5) of metal ions, the
possibility to tune, almost at will, magnetic properties has significantly increased as the number of spin states grows and occupations can be modulated by external stimuli.
3.1 Azido-based dinuclear complexes: a textbook example
Azido-bridged compounds including different metal ions in various dimensionalities have received much attention for several decades [76–78]. Interestingly, the azido group N[3]^− turned out to be a
remarkable magnetic coupler between metallic ions. The number of linkers may vary, and the symmetry of the coordination mode can finely tune the sign and amplitude of the magnetic constant parameter.
Among this class of compounds, azido double-bridged copper(II) dinuclear systems are undoubtedly the most successful. Depending on the bridging mode, one traditionally distinguishes two families, the
so-called End-On (EO) and End-to-End (EE) where the azido groups bridge the copper(II) ions through one terminal nitrogen atom or two terminal nitrogen atoms, respectively (see Fig. 3). Numerous
examples of both classes are given in Refs. [79–92] (EO) and [93–98] (EE). Nevertheless, the determination of the structural parameters that may control the magnetic interaction nature is still much
debated in the literature [87,89,93,96–100]. The general trend is that the EE mode affords antiferromagnetic interactions, whereas the EO mode favors ferromagnetic behavior. However, the alternation
of short and long Cu–N distances, which corresponds to the asymmetric mode, may result in either ferro- or antiferromagnetic interactions whatever the coordination type [89]. Several
magneto-structural explanations have been proposed to tentatively rationalize the experimental behavior. Theoretical investigations based on either state-of-the-art ab initio or DFT calculations have
also played a major role in the discussions [62,89,100–104]. Both approaches proved to reach satisfactory agreement with experimental measurements in either the EO or the EE series. Nevertheless,
correlated ab initio might be preferred as the information extracted from a multiconfigurational wavefunction is very insightful into the mechanism of magnetic interactions. Since Cu(II) ion is
formally d^1, a zeroth-order description lies in CAS(2,2)SCF calculations, allowing the occupations of the in-phase and out-of-phase linear combinations of $dx2−y2$ atomic orbitals by two electrons.
Fig. 3
Symmetric and asymmetric End-to-End (μ[1,3]) and End-On (μ[1,1]) coordination modes.
As a recent illustration from our works, it has been demonstrated using CAS(2,2)SCF/DDCI calculations on fictitious analogues of the EE [Cu(N[3]L)][2] compound (L=7-(dimethylamino)
-1,1,1-trifluoro-4-methyl-5-aza-3-hepten-2-on-ato) (see Fig. 4) that EE diazido Cu(II) systems may exhibit either ferro- or antiferromagnetic behavior depending on one important structural parameter
[105]. Practically, starting from the experimental geometry for which ferromagnetism had been theoretically elucidated [106], the Cu–N distances were modified to evaluate the sensitivity of the
singlet–triplet gap with respect to this so-called “shearing-like” distortion (see Fig. 5). This geometrical parameter measures the amplitude d of the difference between the long and short Cu–N
distances away from a perfectly symmetric bridging mode.
Fig. 4
Structure of the EE [Cu(N[3]L)][2] compound with L=7-(dimethylamino)-1,1,1-trifluoro-4-methyl-5-aza-3-hepten-2-on-ato.
Fig. 5
Asymmetric coordination mode of the EE azide bridge. The arrows indicate the sharing distortion.
A detailed analysis of the different leading contributions to the exchange magnetic coupling constant J (i.e. the spin polarization (SP) effects which are dominated by excitations involving the
bridging-ligand orbitals and the dynamical correlation (DC) effects that arise from the instantaneous polarization of the copper ions environments [51–53]) has been carried out [105]. The variations
of J, SP and DC contributions as a function of d are presented in Fig. 6. A detailed analysis of the wavefunction showed that the weights of the ionic forms (i.e. 2 electrons on one site) increase as
d decreases. Therefore, the singlet is stabilized over the triplet. Similar analysis on previously published structures of pentacoordinate Cu compounds with only one exchange pathway turned out to be
predictive. Even though other parameters cannot be ignored, this particular analysis has shed light on the leading contributions which may control the magnetic behavior.
Fig. 6
Calculated J, SP and DC energies (cm^−1) in a series of hypothetical analogues of [Cu(N[3]L)][2] as a function of δ(A).
3.2 Open-shell ligands: noninnocence concept
Considering the possibility to generate high oxidation states ions (in iron chemistry, for instance, let us mention notable examples of Fe(IV) [107,108], Fe(V) [109–111] and Fe(VI) [112]), much
synthetic effort has been devoted to the preparation of specific multidentate ligands. Upon electrochemical treatment, such partners are likely to undergo reversible redox reactions, leaving the
metallic ion unchanged. Thus, the use of such ligands, known as noninnocent, has opened up the route to original synthetic materials, involving open shells on both metal and ligands partners (see
some examples in Refs. [63,113–118]). The spectacular excited-state coordination chemistry concept in which a ligand coordinates in an excited electronic state to a metal center has emerged from this
class of compounds [119]. The noninnocent term was suggested to emphasize the versatile character of this class of ligands which may not be considered as closed-shell entities. The generation of
radical ligands in coordination compounds has emerged as a promising route to magnetic materials.
From the theoretical point of view, DFT as well as CI calculations have been undertaken to scrutinize the electronic structures of such noninnocent ligand-based systems [63,116,117,119,120]. In
particular, the comparison between experimental and calculated exchange coupling constants and the analysis of the magnetic interactions has been the subject of intense work. While DFT has sometimes
failed to fully account for the low-energy spectroscopy, the wavefunction based DDCI method has elucidated the unusual behavior of several complexes [63,119]. Among those, a striking example is given
by the Fe(gma)CN complex containing the glyoxalbis(mercaptoanil) (gma) ligand (see Fig. 7) [32]. Even though the noninnocent character of the gma ligand was clearly demonstrated both experimentally
and theoretically, DFT calculations were only partially successful in the description of the electronic structure of the full complex [119]. The magnetic susceptibility and zero-field Mössbauer
measurements clearly favored a doublet ground state. Nevertheless, DFT calculations did not provide any clear evidence in that sense, the M[s]=1/2 solution exhibiting a low-spin Fe(III) (S[Fe]=1/
2) coupled to a closed-shell gma ligand (S[gma]=0). Clearly, for a good description of the electronic structure of such a system, DFT and its monodeterminantal character is not appropriate and
correlated ab initio calculations might be desirable.
Fig. 7
Structure of Fe(gma)CN.
Based on this statement, Messaoudi et al. performed correlated ab initio calculations on this particular system by means of DDCI-2 calculations on the top of the CAS(5,5)SCF wavefunction [32].
Interestingly, the active orbitals consist of three metal-centered and two ligand-centered MOs (see Fig. 8). This active space was strongly supported by experimental and calculated Mössbauer
parameters [119]. The calculations showed that the low-energy spectrum exhibits a 200-cm^−1 quartet-doublet gap, in agreement with different experiments and that the observed strong antiferromagnetic
behavior is due to important ligand-to-metal charge transfer (LMCT). The resulting ground-state wavefunction which exhibits an intermediate magnetic/covalent character, is rather strongly correlated
and is dominated by local (S[Fe]=3/2 and S[gma]=1) electronic configuration. Finally, whereas the gma ligand is clearly a closed-shell singlet when considered alone, it is likely to be a triplet
when coordinated to the iron center. The multiconfigurational nature of the wavefunction has been identified in this example and makes this class of compounds still very challenging for
theoreticians. For instance, it has been recently suggested that the energetics of low-lying states of coordination complexes based on porphyrins and related entities may not be accessible by means
of DFT methodology (see Ref. [33] and references therein). More troublesome is the dependency of the spin density maps upon the functional choice.
Fig. 8
Optmized active average MOs for the lowest doublet and quartet states of Fe(gma)CN.
3.3 Trinuclear Cu(II) complexes: prototypes of oxidases
In this section, one deals with the important concept of spin frustration or more precisely with the possible deviation to the ideal spin frustrated character observed in trinuclear copper clusters.
This concept was first introduced in 1977 in relation with the phenomenon of spin glass behavior [121]. Since this pioneering work, this concept has been widely applied in solid-state physics
[122,123] and appeared more recently in the molecular magnetic community [10,17,124]. Exchange coupled trinuclear Cu(II) clusters have been the focus of tremendous attention for many years due to the
richness of their implications in catalysis, magnetism and biology. These systems play a major role in the catalytic O[2] reduction involving multicopper oxidases, the so-called
“native-intermediates” [125–128]. Apart from the biological significance, this particular type of architectures has been reported in the literature as good prototypes for the evaluation of the
relevance of magnetic exchange models [17]. The majority of the reported trinuclear Cu(II) complexes have either a linear arrangement of the metal centers [129–133] or a Cu[3](μ[3]-X) unit (X=Cl,
OR) (for numerous examples, see Refs. [134–148]). In its triangular arrangement, such metal complexes are of special interest in the understanding of geometrically frustrated antiferromagnetic
compounds (Fig. 9).
Fig. 9
Energy diagram of an antiferromagnetically coupled Cu(II) trimer.
Each one of the three spins S=1/2 can interact with the adjacent ones. Two Cu(II) atoms couple to form intermediate spins S′=1 and 0 which couple to the third one to give S[tot]=1/2, 3/2 and 1/
2. In perfect D[3]/C[3] symmetric trimers, the non-interacting doublet states are energetically degenerate with the quartet state lying 3|J| (Heisenberg Hamiltonian $Hˆ=−2J∑i,jSˆiSˆj$) higher in
energy. The degeneracy in the ground state is known as spin frustration since, despite all pairwise spin interactions being antiferromagnetic, each spin configuration includes a pair of parallel
spins. When the triangular architecture moves away from a threefold symmetry as a result of possible vibronic interactions, the energetic situation may vary drastically. Theoretical analyses based on
either DFT or post-HF ab initio calculations have been tentatively used to propose a detailed description of their electronic structures and to rationalize the physical origins of their magnetic and
spectroscopic features. A combined quantum and molecular mechanical (QM/MM) study has also been undertaken to analyze the O[2] reductive cleavage in the catalytic cycle of multicopper oxidase [149].
Whereas DFT combined with the BS approach [19–21] has been successfully applied to the calculation of magnetic exchange interactions of binuclear systems (see for instance Refs. [89,101,150,151]),
only few are devoted to trinuclear transition metal complexes [15,130,133,152–155]. The scarcity of theoretical studies based on monodeterminantal approaches might be explained by the known
deficiencies of such methods in the treatment of spin states with a multireference character. To elude these limitations, correlated multireference approaches may be used. The first example of such
calculations was proposed by Chalupsky and coworkers with a thorough analyses of spectroscopic parameters based on CASPT2 and multireference difference dedicated CI (MRDDCI2) [57] calculations for a
series of (μ[3]-hydroxo)- and (μ[3]-oxo)-bridged trinuclear Cu(II) models [156].
In most of these studies, the structures are close to a perfect C[3] symmetry. However, some coworkers have recently synthesized and characterized a Cu[3] cluster exhibiting an arrangement of the Cu
ions deviating from this point group (see Fig. 10) [157]. The most important distortion concerns the long Cu–O apical distances (Cu1–O1, Cu2–O2, Cu3–O3), one differing significantly from the other
two (2.52 vs. 2.35 and 2.40Å). To account for the deviation from a perfect C[3] symmetry, the introduction of 2 or 3 constants in the fit of the experimental thermal behavior of the magnetic
susceptibility led to several sets of parameters whose underlying physics had to be clarified. CASSCF/DDCI calculations were performed in order to investigate (i) the influence of the ML[5]
structural variations on the quartet–doublet spectrum, (ii) the role of the copper solvated hydroxy group linked to the three Cu(II) ions. This theoretical framework had already shown to provide
accurate results for magnetic multicopper compounds [62]. A first approach, which was considered as “naive”, consisted in a dimer description of the Cu[3] cluster involving three magnetic exchange
coupling constants.^1 Even though the antiferromagnetic character was conserved, such a set of parameters failed to accurately reproduce the experimental data (see the red curve in Fig. 11 (for
interpretation of the references to color in this text, the reader is referred to the web version of this article)). This drawback was ascribed to the prime role played by the central oxygen atom O4.
Fig. 11
Experimental and simulated cX[M]T vs. T curves.
Therefore, CAS(3,3)SCF/DDCI calculations were carried out on the whole Cu[3] unit extracted from the experimental crystal structure. It is noteworthy that in this system the deviation from
orthogonality between the magnetic orbitals (see Fig. 12) is rather small (∼10°). Therefore, ferromagnetic interactions might be anticipated. Indeed, this ferromagnetic behavior survived up to a
DDCI-2 level of calculation. However, DDCI-3 calculations were likely to stabilize one of the doublet, splitting the speculated degeneracy of the doublet states by ∼8cm^−1. Such low-energy spectrum
description allows one to recover the experimentally observed antiferromagnetic behavior.
Fig. 12
Magnetic orbitals of the [Cu[3](L)[3](μ[3]-OH)]^− complex.
These calculations concluded on a picture favoring a doublet and quasi-degenerate quartet ground state with the second doublet lying slightly higher in energy (see Fig. 13). This example illustrates
the necessity for a high enough level of calculations to reproduce quantitatively such tiny energy differences and to account for the experimentally observed magnetic behavior.
Fig. 13
Q, D[1], D[2] energy spectrum with respect to the model approaches described in the text.
3.4 Growing 1D materials: Ni-azido chains
With the generation of magnetic properties goal in mind, experimentalists have prepared higher dimensionality materials. One of the main challenges in the synthesis of extended 1D systems is to
prevent the local magnetic moments from canceling out. Obviously, this condition is fulfilled as soon as ferromagnetic interactions dominate. However, in the presence of most frequent
antiferromagnetic interactions, pioneering approaches were devoted to regular heterospin ferrimagnetic chains [10] holding alternating spin carriers, coupled through a unique exchange constant.
Another strategy consists in varying the magnetic exchange constants between homospin carriers [158,159]. Finally, the use of strong anisotropic metal ions to reduce the magnetization relaxation has
generated the promising field of the SCM [11–13].
In this respect, the azido ligand turned out to be extremely appealing in linking metal ions and a remarkable magnetic coupler for propagating interactions between paramagnetic ions. The structural
variety of the azido complexes ranges from molecular clusters to extended 1D to 3D materials [77,160–164]. An interesting prototype of such a system has been recently synthesized where a single azido
unit bridges in an alternating End-On (EO) and End-to-End (EE) way the Ni(II) ions (see Fig. 14) [165]. The system can be considered from the chemical point of view as a quasi-1D chain. However,
based on magnetic susceptibility measurements, it was suggested that the system should be described from the magnetic point of view as isolated dimers. Indeed, the introduction of a second magnetic
interaction was shown to be irrelevant. Therefore, the question of the nature and amplitude of the magnetic interactions between the nearest Ni(II) ions deserved special attention. The alternation of
EO and EE units strongly suggested the presence of two magnetic exchange pathways which can be accessible through Ni[2] dimers' spectroscopy analysis. Thus, CAS(5,6)SCF/DDCI-2 calculations were
performed on the molecular EE and EO fragments extracted from the available crystal structure. Let us mention that the geometries of these clusters were not relaxed. The active orbitals consist of
the in-phase and out-of-phase linear combinations of the $dz2$ and $dx2−y2$ metallic AOs (see Fig. 15) and the non-bonding MO of the N[3]^− bridge.
Fig. 14
Nickel(II) chain {Ni[2] (μ[1,1]-N[3])(μ[1,3]-N[3])(L)[2](MeOH)[2]]}[n] with alternating End-On/End-to-End single azido-bridges.
Fig. 15
Molecular EE (left) and EO (right) fragments and in-phase active metallic MOs. For the sake of simplicity, the out-of-phase combinations are not shown.
Since the Ni(II) ion is formally d^8, it is expected that exchange interactions between S=1 ions should give rise to three spin states in the Ni[2] units, namely singlet (S), triplet (T) and
quintet (Q) states. In a Heisenberg picture $Hˆ=−2JSˆ1Sˆ2$ (S[1]=S[2]=1), the energy separations are 6|J| and 4|J| between the quintet and singlet, quintet and triplet states, respectively (see
Fig. 16). Within the EE unit, a relatively large antiferromagnetic exchange constant (J[EE]∼−50cm^−1) was calculated in relatively good agreement with the unique value extracted from experiment
(∼−40cm^−1). This is to be contrasted with the EO Ni[2] unit, which exhibits a negligibly small magnetic interaction (a=J[EO]/|J[EE]| ratio ∼0.02) (see Fig. 17).
Fig. 16
Energy spectrum of a two-center Heisenberg S=1 Hamiltonian.
Fig. 17
Schematic representation of the Ni-azido chain resulting from the isolated EE dimers' picture.
The calculations not only confirmed the isolated dimers picture, but also associated the leading antiferromagnetic exchange pathway with the EE bridging mode. In the light of the calculated (E[Q]−E
[S])/(E[Q]−E[T]) ratio, let us mention that the deviation from a pure Heisenberg picture is negligible (less than 2%), ruling out the speculated participation of quadratic terms. The attempt to
generate high enough ferromagnetic interactions between S=1 sites looked very promising, since the antiferromagnetic coupling between the resulting S=2 units through EE bridges might have
resulted in a Haldane chain with vanishingly small spin gap [166,167]. The versatility of the azido magnetic coupler should still be considered to generate synthetic models for theoretical physics
3.5 3D spincrossover materials: the CsFe[CrCN[6]] Prussian blue analogue
Whereas the previous sections were devoted to 0D and 1D systems, here we concentrate on 3D covalent materials. Prussian blue analogues (PBA) have been widely studied in the field of magnetic systems
[168,169]. Currently, the interest for these compounds is motivated by two reasons. Firstly, the representatives of this family which contain vanadium and chromium cations behave as high-temperature
magnets [170–172]. Secondly, some PBA display temperature- and photo-induced magnetic phase transitions [169,173,174]. These transitions usually result from temperature- or optically driven electron
transfer between two metal ions in the lattice. The change of the oxidation states leads to changes of the metal ions' spin states, thus inducing or modifying the magnetic coupling between them. To
date, only few theoretical studies have been devoted to the electronic structure and magnetic behavior of PBA. This can be ascribed to the lack of accurate crystal data, since these materials exhibit
different defects types, namely metal ion vacancies, counterion crystallographic disorder, and presence of water molecules in the lattice.
The first theoretical studies, based on extended Hückel calculations on dinuclear models, have been proposed by Verdaguer et al, who attributed the exchange interactions between the metal ion centers
to the overlap between the magnetic orbitals through the π system [168,175]. Periodic calculations, based on both HF and DFT methodologies, on idealized structures were performed to clarify the
magnetic interactions between different divalent/trivalent ion pairs [176–179]. Multireference and DFT-based cluster calculations were also carried out to examine the exchange interactions [180–183],
the metal-to-metal charge transfers [184] and the possibility to generate high Curie temperature materials [185].
The interest for the recently synthesized and characterized CsFe[CrCN[6]] PBA has emerged from the spin transition which has been observed on the Fe(II) center (see Fig. 18) [186]. In this compound,
the thermal spin transition results from the crossover of the high-spin (HS) and low-spin (LS) states of the Fe(II) ions without any intervalence electron transfer in the lattice. Surprisingly, from
theoretical calculations, the [Fe(NC)[6]]^4− unit is known to be HS as a result of the low ligand field generated by the octahedral arrangement of the NC^− ions [187]. Thus, the stabilization of the
HS state over the LS state apparently rules out the possibility of spincrossover phenomenon. One may wonder how much the environment produced by the rest of the crystal is likely to change this state
of affairs. Consequently, using the smallest transiting core [Fe(NC)[6]]^4−, the relative importance of the local ligand field and Madelung field generated by the rest of the crystal was estimated.
Fig. 18
Spin transiting CsFe[Cr(CN)[6]] system. Fe, Cr, C and N atoms are shown in grey, green, blue and orange, respectively. Cs ions are omitted. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
With this goal in mind, CAS(6,5)SCF (Fe(II) ion being formally d^6) and subsequently second-order perturbation-theory CASPT2 calculations were performed (i) in the absence of any environment
(so-called gas-phase), and (ii) in the presence of various Madelung fields accounting for the PBA matrix effects (see Fig. 19) [188]. From the low-temperature (100K) and high-temperature (265K)
crystal data available in the literature [189], a standard 0.15-Å lengthening of the Fe–N distances was observed. Using the corresponding Fe(II) geometries for the [Fe(NC)[6]]^4− cluster, the
non-Franck–Condon gas-phase spectroscopy based on CASPT2 calculations showed that the HS state is lower than the LS state by ∼16,700cm^−1 (see Fig. 20). Interestingly, unrestricted HF periodic
calculations performed on the 100 and 265K crystal structures revealed a significant charge reorganization on the Fe center (1.65→2.15), whereas the Cr ion Mulliken charges remain almost unchanged
(1.85→1.90). This is to be contrasted with the picture based on formal charges (+2, +3 on Fe and Cr, respectively) which holds for both spin states. Including the Madelung field effect, it was
shown from these CASPT2 calculations that the Franck–Condon spectroscopy of a d^6 ion immersed in a frozen charge matrix is consistent with the Tanabe–Sugano diagram for an isolated complex, leaving
the HS state lower than the LS state by ∼17,000cm^−1 (see Fig. 20). Nevertheless, as the environment charges are varied following the periodic calculation results, the situation is greatly modified.
As a matter of fact, the LS state immersed in a LS environment is stabilized by ∼12,500cm^−1 over the HS state in a HS field (see Fig. 20).
Fig. 19
Gas-phase (left) and immersed in a Madelung field (right) [Fe(NC)[6]]^4− model for the LS and HS energies' determination.
Fig. 20
Non-Franck–Condon spectroscopy in the gas-phase (left), in the presence of a frozen charge environment (middle) and in a matrix accounting for the charge reorganization (right).
CASPT2 quantum chemical calculations have thus demonstrated that the crystal matrix of the CsFe[Cr(CN)[6]] compound creates conditions for spin transition of the Fe(II) ion, a phenomenon which does
not exist for an isolated [Fe(NC)[6]]^4− complex. The apparent competition between the ligand field and the fluctuating Madelung environment in the Fe^II(NC)[6] complex seems to govern the spin
crossover phenomenon. While the former effects are short range, the latter are representative of long-range contributions which are crucial for cooperativity manifestation.
4 Concluding remarks
In conclusion, quantum chemical calculations have become valuable means of investigation which cannot be ignored. As spectroscopy accuracy can be reached down to several tens of wavenumbers, ab
initio techniques have the ability to rationalize interactions in magnetic systems. Interestingly, the different contributions to energy splitting are accessible and the underlying physical phenomena
can be interpreted in terms of kinetic exchange, instantaneous charge fluctuations, spin polarization and dynamical correlation effects. The information which is conveyed by the wavefunction is
crucial in the characterization of model Hamiltonians. Undoubtedly, significant efforts must be devoted to extract the relevant parameters in a “boil down” procedure of the ab initio information.
Even though certain CI methodologies might be very demanding when dealing with large systems, they allow one to manipulate symmetry and spin-adapted eigenstates of the exact Hamiltonian. With this
pursued goal in mind, unrestricted DFT-based calculations are also very valuable, since the size limitation is significantly displaced. Such methodology has opened up new routes to important issues
involving biological systems. Nevertheless, some specific systems including open-shell compounds are the concern of explicitly correlated calculations which allow an efficient treatment of both
non-dynamical and dynamical correlations.
VR is sincerely thankful to J.-P. Malrieu and N. Guihéry for extraordinary discussions on the many-body problem.
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Indefinite integral - Examples, Exercises and Solutions
An integral can be defined for all values (that is, for all $X$). An example of this type of function is the polynomial - which we will study in the coming years.
However, there are integrals that are not defined for all values (all $X$), since if we place certain $X$ or a certain range of values of $X$ we will receive an expression considered "invalid" in
mathematics. The values of $X$ for which integration is undefined cause the discontinuity of a function.
Does the given function have a domain? If so, what is it?
\( \frac{9x}{4} \)
Given the following function:
\( \frac{5}{x} \)
Does the function have a domain? If so, what is it?
Given the following function:
\( \frac{5+4x}{2+x^2} \)
Does the function have a domain? If so, what is it?
Given the following function:
\( \frac{65}{(2x-2)^2} \)
Does the function have a domain? If so, what is it?
Consider the following function:
\( \frac{3x+4}{2x-1} \)
What is the domain of the function?
Given the following function:
\( \frac{12}{8x-4} \)
What is the domain of the function?
Given the following function:
\( \frac{24}{21x-7} \)
What is the domain of the function?
Given the following function:
\( \frac{23}{5x-2} \)
Does the function have a domain? If so, what is it?
Given the following function:
\( \frac{49+2x}{x+4} \)
Does the function have a domain? If so, what is it?
Given the following function:
\( \frac{5-x}{2-x} \)
Does the function have a domain? If so, what is it?
Look at the following function:
\( \frac{10x-3}{5x-3} \)
What is the domain of the function?
Look at the following function:
\( \frac{20}{10x-5} \)
What is the domain of the function?
Look at the following function:
\( \frac{2x+20}{\sqrt{2x-10}} \)
What is the domain of the function?
Look at the following function:
\( \frac{2x+2}{3x-1} \)
What is the domain of the function?
Look at the following function:
\( \frac{2x+2}{9x+6} \)
What is the domain of the function?
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PPD583 – Risk Analysis - Essay Writing Expert
Instructor: Ali Nowroozi
Fall 2024
Homework Assignment #2
DUE: 10/06/2024 11.59 PM
(30 Points)
1. Answer all the questions in this file. Do not delete the questions. Type in your answers below the question, using the sample structure (if provided)
2. We expect precise and succinct answers. Add citations, references, and screenshots wherevernecessary.
3. Follow the deadline for submission. Late submissions will not be accepted and will receive azero.
4. Name the file as the last four digits of your USC IDand submit it as aPDF^(1) File. (For Example, if myUSC ID is 123456789, the file name becomes: 6789.pdf).Do not put your name anywhere on this
document or on the file name or you will lose 2 points.
5. Not following the instructions will result in penalties.
• To convert this document to a PDF file, go to “file”, “save As”, choose “PDF(*.PDF)” and the file will be saved at the same location as your WORD file.
1. Which pair of possibilities is “Mutually Exclusive”, but not “Collectively Exhaustive”? Draw the correct MECE diagram (for the answer only) from the attached model.
1. The temperature outside is: under 10 ^oC , Over 20 ^oC
2. The temperature outside is: under 20 ^oC , Over 10 ^oC
3. The temperature outside is: under 20 ^oC , Over 20 ^oC
4. The temperature outside is: under 10 ^oC , Under 20 ^oC
Tip: Refer to Lecture 2-Slide 16, also see the attached models
2. Sara has three coins which you believe to be fair, and are labeled #1, #2, and #3. Suppose Sara flips all three coins and tells you that at least 2 of them landed heads. Given that you believe
each coin’s probability of landing heads is ½ and it is irrelevant to how the other coins land, use an event tree model with all consequences listed to show that the probability that coin #1
landed on Head is ¾.
Tip: flipping 3 coins is the same as flipping the same coin 3 times, for the purpose of this experiment. So, you can use our model in slide 22 of lecture 3. No equation is necessary, just count the
outcomes with the desired features.
3. What is the probability of drawing 2 Aces from 52 cards?
Tip #1: The probability of one item in a group of N items is 1/N
Tip #2: This is the probability that the first card is an ACE (say event A), AND the second card is an ACE (say event B) è , but A & B are independent so P(B|A) = P(B)
4. Use Bayesian law to show that there is a 64% chance that a driver who is over age 35 wears a seat belt regularly in the following scenario:
ABC Insurance Co. estimates that 80% of drivers wear seat belts regularly. They also estimate that 50% of drivers are over age 35. A study showed that 40% of those drivers who wear seat belts
regularly are over age 35.
Tip: Event A = Wear Seatbelt, Event B = Over age 35, find P(A|B)
5. See attached a fault tree example with calculations. Identify a different problem on your own, develop a simple fault tree (with less than 10 nodes, like the attached example) and assign some
probabilities to each node and calculate the overall probability of the event (e.g. a house getting on fire in one year in UK). Now investigate the actual probability data on the internet (e.g.
this is what I found for my example). If you see a big difference, revisit your assumptions on the individual probabilities, adjust them, and explain your adjustments briefly.
Tip #1: If you model your problem in Excel, the adjustments will be very easy and quick
Tip #2: In the example, P(A ꓴ B) is calculated differently. Please ignore this method and use the simple formula discussed in the class: P(A ꓴ B) = P(A) + P(B) , A and B are independent
Source: https://careerinconsulting.com/wp-content/uploads/2021/03/Capture-MECE-Diagram-1024×577.png
Source: https://youtu.be/Eq8-m6Faobo?si=MGm9iGOLQJb-FNV8
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Interpolation vs. Extrapolation: What’s the Difference? | Online Tutorials Library List | Tutoraspire.com
Interpolation vs. Extrapolation: What’s the Difference?
by Tutor Aspire
Two terms that students often confuse in statistics are interpolation and extrapolation.
Here’s the difference:
Interpolation refers to predicting values that are inside of a range of data points.
Extrapolation refers to predicting values that are outside of a range of data points.
The following example illustrates the difference between the two terms.
Example: Interpolation vs. Extrapolation
Suppose we have the following dataset:
We may decide to fit a simple linear regression model to these points:
We could then use the fitted regression model to predict the values of points both inside and outside of the range of data points.
When we use the fitted regression model to predict the values of points inside the existing range of data points it is known as interpolation.
Conversely, when we use the fitted regression model to predict the values of points outside the existing range it is known as extrapolation:
The Potential Danger of Extrapolation
When we perform extrapolation, we assume that the same pattern that exists inside the current range of data points also exists outside of the range as well.
However, this can be a dangerous assumption because it’s possible that the pattern that exists outside the current range of data points is quite different:
For this reason, it can be dangerous to use extrapolation to predict the values of data points that fall outside of the range of values that was used to build the regression model.
In practice, it’s often fine to use extrapolation to predict the values of points that fall just slightly outside of the range of existing values but the further outside the range the higher the
likelihood that the difference between the predicted value and the actual value will be large.
When to Use Extrapolation
Often it requires domain-specific expertise to determine if extrapolation is a reasonable idea or not.
For example, suppose a marketing department at a business fits a simple linear regression model using advertising spend as the predictor variable and total revenue as the response variable.
In this scenario, it may be reasonable to assume that a steady increase in advertising spend will lead to a predictable increase in total revenue:
In this scenario, we may be quite confident in our ability to extrapolate values.
However, consider a scenario where a biologist wants to use total fertilizer to predict plant growth.
She may decide to fit a simple linear regression model to the data points, but since there is an upper limit on how tall plants can grow, it probably doesn’t make sense to use extrapolation to
predict the values of points outside of the range of values used to fit the model:
In this scenario, we may be considerably less confident in our ability to extrapolate values.
The Takeaway: Extrapolation can make sense in some fields more than others, but there is always a potential danger that the pattern that exists within the range of values used to fit the model does
not exist outside of the range.
Additional Resources
How to Perform Linear Interpolation in Excel
How to Make Predictions with Linear Regression
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News Live: Read Latest News, Headlines, Top News today, Breaking News Online | The Economic Times
Add or subtract two time-values
Using the time calculator, users can either add two time-values or find the difference between two time-values. The value fields ‘Day’, ‘Hour’, ‘Minute’ and ‘Second’ have been provided for both the
time-values. In case, a value field is left blank, the result would be displayed accordingly.
Add or subtract a time-value from a date
This time calculator allows users to either add or subtract a time value from a ‘Start Time’, where the ‘Start Time’ is represented by a particular date along with a certain time on that date. The
‘Start Time’ comprises a date -- which can either be filled directly into the relevant value fields or entered using the calendar -- and a time value which has the value fields ‘Hour’, ‘Minute’,
‘Second’ and a drop-down menu to toggle between AM and PM. The time-value to be subtracted from the ‘Start Time’ consists of the value fields ‘Day’, ‘Hour’, ‘Minute’ and ‘Second’.
Make time calculations using mathematical expressions
Here, users can add and subtract a string of time-values using a mathematical expression. Value fields included in an expression are ‘Day’, ‘Hour’, ‘Minute’ and ‘Second’ denoted by the letters ‘d’,
‘h’, ‘m’ and ‘s’ respectively.
Time Duration CalculatorsFind the difference between two time-values
The calculator can be used to determine the difference between two time-values.
The calculator has two fields -- ‘Start Time’ and ‘End Time’. Both the fields have options to enter the required number of hours, minutes and seconds and to choose between ‘AM’ and ‘PM’. Enter the
required details in both the fields and click on the ‘CALCULATE’ button to know the difference between both the time values.
The output displayed is either in terms of hours, minutes and seconds or only in hours or minutes or seconds.
If the time-value provided in the ‘Start Time’ field is ahead of that provided in the ‘End Time’ field, then the calculator automatically assumes the ‘Start Date’ to be on the previous day.
The input provided in the hours, minutes and seconds sections of the ‘Start Time’ and ‘End Time’ fields need to be valid values. The value provided in the ‘hours’ section should be less than 24, that
in the ‘minutes’ section has to be less than 60 and the input in the ‘seconds’ section should also be less than 60.
Find the time gap between two dates
The calculator finds the time difference between a certain time on a start date and a certain time on the end date.
The calculator has a ‘Start Time’ and an ‘End Time’ field. Both the fields allow users to enter the start date or end date by either choosing the required date from a calendar or by entering the
month, date and year in their respective sections. Additionally, users can also enter a certain time value by entering required details in the ‘hour’, ‘minute’ and ‘second’ sections and choosing
between ‘AM’ or ‘PM’ in the ‘Start Time’ and ‘End Time’ fields or by clicking on the ‘now’ button to select the current time.
Enter the relevant details in the input fields and click on the ‘CALCULATE’ button to determine the time gap between two dates. The input provided in the ‘Start Time’ field should be earlier than
that in the ‘End Time’ field.
The output shown is either in terms of days, hours, minutes and seconds or only in terms of days or hours or minutes and seconds.
What is a time or a time duration calculator?
A time calculator is an algorithmic digital tool that can be used to add/subtract various time-values and make different types of time calculations such as finding the time difference between two
important events, determining the date 3 years before today, knowing how many hours, minutes or seconds are there in a year, etc.
Basic things to know while making time calculations
Time, as we know it, can range from several years to a few seconds. So, while making calculations using the above calculators, it’s important to know some basic units of time and the calendar system
used for designing these calculators.
Here is a list of some basic units of time:
The calendar used:
These calculators use the Georgian calendar to make calculations. In this calendar, a year is divided into 365 days, with the exception of a leap year, which has 366 days. The total number of days in
a calendar year are categorized into 12 months of varying lengths. Each month either has 30 or 31 days, except the month of February which has 28 days in a normal year and 29 days in a leap year.
How do the calculators work?
The first calculator basically lets users determine the sum or the difference between two time-values. Here, time values are represented in terms of days, hours, minutes and seconds.
The input section of the calculator has two rows consisting of the value fields ‘Day’, ‘Hour’, ‘Minute’ and ‘Second’ in each row and two buttons to select either ‘Add+’ or ‘Subtract-’ placed in
between those two rows. Similarly, the output section has a row consisting of all the value fields that are present in the input rows.
The input provided can either be in whole numbers or decimal, but they should be positive rational numbers. The value entered in the first row of the input section can either be greater or smaller
than the value entered in the second row. Additionally, if need be, users can choose to leave any of the value fields or multiple value fields in the input section blank.
The output displayed is either a positive or a negative value based on the input provided and the function (+ or -) chosen. The output time value also consists of the value fields ‘Day’, ‘Hour’,
‘Minute’ and ‘Second’. Additionally, this calculator also displays the result in terms of days, hours, minutes and seconds individually.
The second calculator can either add a time value to or subtract a time value from a particular date and a certain time on that date.
Here, the upper part of the input section is called ‘Start Time’. The ‘Start Time’ consists of two sub-sections. One where the date is to be filled and another where the time. By default, the ‘Start
Time’ initially shows the current date and time. The date can either be filled by filling up all the required sections i.e. month, date and year individually or by clicking on the calendar button.
The input for the time part can be provided by entering relevant values in the ‘Hour’, ‘Minute’ and ‘Second’ sections and choosing either ‘AM’ or ‘PM’ from the drop-down menu beside it.
The lower part of this calculator has the value fields ‘Day’, ‘Hour’, ‘Minute’ and ‘Second’ to provide input for the time-value that is to be added or subtracted from the concerned date and time.
Users can choose either to add or subtract the time-values from the middle section by clicking either on the ‘Add+’ or the ‘Subtract-’ button.
The result displayed is a date and a time on that date which is either before or after the date in the ‘Start Time’ field depending on the function (add or subtract) chosen.
In the third calculator, time-values based calculations can be carried out using mathematical expressions. The mathematical expressions used in this calculator are made up of days, hours, minutes and
seconds which are denoted by the letters ‘d’, ‘h’, ‘m’ and ‘s’ respectively.
Just like the previous calculators, this calculator can also only perform addition and subtraction of time-values. The time-values provided as input to this calculator should be a number accompanied
by the letter which represents its unit.
For example: 3d 4h 5m 6s, 8h 25m, 1d 0h 0m 3s, 3600s, etc.
The output displayed is in terms of days, hours, minutes and seconds. The result is also shown individually in terms of days, hours, minutes and seconds.
Intriguing facts about time
• Every time zone used across the globe begins from a common point situated in Greenwich, England which is known as the Prime Meridian or Greenwich Meridian. And the time at that common point is
known as Universal Time or Greenwich Mean Time (GMT)
• Our ancestors used sundials to determine time almost exactly. The world’s largest stone sundial is located at Jantar Mantar in New Delhi, India
• The concept of a leap year was introduced to the world by the Roman general Julius Caesar
• Peter Henle built the first modern clock around 1511
• The most precise clock to be ever built is the strontium atomic clock
• Time slows down when you move faster
• There are 12 different time zones in France, 11 in Russia and 11 in the United States
• Some theories suggest that time might end within the next four billion years
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$145 an Hour is How Much a Year? Before and After Taxes
With the cost of living on the rise, an hourly wage of $145 is well above what most people earn. In this article, we’ll look at how much annual income results from $145 per hour. We’ll calculate the
weekly, monthly, and yearly salaries for both part-time and full-time work. We’ll also see how overtime at time-and-a-half can dramatically boost the total yearly earnings.
In addition, we’ll examine how unpaid time off can reduce someone’s total pre-tax income over a year when making $145 per hour. After covering gross pay, we’ll look at how federal and state taxes
impact net yearly income. Even at $145/hour, taxes can still significantly cut into earnings.
Beyond the numerical calculations, we’ll assess if $145/hour is considered a highly lucrative wage in today’s economy. With inflation pushing prices higher, this hourly rate provides tremendous
buying power. We’ll discuss sample budgets and lifestyles possible at this pay rate, like affording luxury goods and high-end real estate. We’ll also highlight jobs paying $145/hour or more. While
the income is very comfortable, we’ll look at options for further increasing one’s hourly wage. Whether pursuing early retirement or a lavish lifestyle, $145 an hour provides substantial earnings.
Convert $145 Per Hour to Weekly, Monthly, and Yearly Salary
Input your wage and hours per week to see how much you’ll make monthly, yearly and more.
$145 an Hour is How Much a Year?
If you make $145 an hour, your yearly salary would be $301,600. We calculate your annual income based on 8 hours per day, 5 days per week and 52 weeks in the year.
Hours worked per week (40) x Hourly wage($145) x Weeks worked per year(52) = $301,600
$145 an Hour is How Much a Month?
If you make $145 an hour, your monthly salary would be $25,133.33. We calculated this number by dividing your annual income by 12 months.
Hours worked per week (40) x Hourly wage($145) x Weeks worked per year(52) / Months per Year(12) = $25,133.33
$145 an Hour is How Much Biweekly?
If you make $145 an hour, your biweekly salary would be $11,600.
Hours worked per week (40) x Hourly wage($145) x 2 = $11,600
$145 an Hour is How Much a Week?
If you make $145 an hour, your weekly salary would be $5,800. Calculating based on 5 days per week and 8 hours each day.
Hours worked per week (40) x Hourly wage($145) = $5,800
$145 an Hour is How Much a Day?
If you make $145 an hour, your daily salary would be $1,160. We calculated your daily income based on 8 hours per day.
Hours worked per day (8) x Hourly wage($145) = $1,160
$145 an Hour is How Much a Year?
The basic formula to calculate your annual salary from an hourly wage is:
Hourly Rate x Hours Worked per Week x Number of Weeks Worked per Year = Annual Salary
So for a $20 per hour job:
$145 per hour x 40 hours per week x 52 weeks per year = $301,600
However, this simple calculation makes some assumptions:
• You will work 40 hours every week of the year
• You will not get any paid time off
Therefore, it represents your earnings if you worked every week of the year, without any vacation, holidays, or sick days.
Accounting for Paid Time Off
The $301,600 base salary does not yet factor in paid time off (PTO). Let’s assume the job provides:
• 2 weeks (10 days) paid vacation
• 6 paid holidays
• 3 paid sick days
This totals 19 paid days off, or nearly 4 weeks of PTO.
Importantly, this paid time off should not be deducted from the annual salary, since you still get paid for those days.
So with 4 weeks PTO, the annual salary would remain $301,600 .
Part time $145 an hour is How Much a Year?
Your annual income changes significantly if you work part-time and not full-time.
For example, let’s say you work 25 hours per week instead of 40. Here’s how you calculate your new yearly total:
$145 per hour x 25 hours per week x 52 weeks per year = $188,500
By working 15 fewer hours per week (25 instead of 40), your annual earnings at $145 an hour drop from $301,600 to $188,500.
That’s a $113,100 per year difference just by working part-time!
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $145 an hour:
Hours Per Week Earnings Per Week Annual Earnings
40 $5,800 $301,600
35 $5,075 $263,900
30 $4,350 $226,200
25 $3,625 $188,500
20 $2,900 $150,800
15 $2,175 $113,100
The more hours per week, the higher your total yearly earnings. But part-time work allows for more life balance if you don’t need the full salary.
$145 an Hour With Overtime is How Much a Year?
Now let’s look at how overtime can increase your annual earnings.
Overtime kicks in once you work more than 40 hours in a week. Typically, you earn 1.5x your regular hourly wage for overtime hours.
So if you make $145 per hour normally, you would make $217.50 per hour for any hours over 40 in a week.
Here’s an example:
• You work 45 hours in a Week
• 40 regular hours paid at $145 per hour = $5,800
• 5 overtime hours paid at $217.50 per hour = $1,087.50
• Your total one Week earnings =$5,800 + $1,087.50 = $6,887.50
If you worked 45 hours each week for 52 weeks, here’s how your annual earnings increase thanks to overtime pay:
$6,887.50 per week x 52 weeks per year = $358,150
That’s $56,550 more than you’d earn working just 40 hours per week at $145 an hour.
Overtime can add up! But also consider taxes and work-life balance when deciding on extra hours.
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $145 an hour:
Overtime hours per work day Hours Per Week Earnings Per Week Annual Earnings
0 40 $5,800 $301,600
1 45 $6,887.50 $358,150
2 50 $7,975 $414,700
3 55 $9,062.50 $471,250
4 60 $10,150 $527,800
5 65 $11,237.50 $584,350
6 70 $12,325 $640,900
7 75 $13,412.50 $697,450
How Unpaid Time Off Impacts $145/Hour Yearly Earnings
So far we’ve assumed you work 52 paid weeks per year. Any unpaid time off will reduce your total income.
For example, let’s say you take 2 weeks of unpaid leave. That brings your paid weeks down to 50:
Hours worked per week (40) x Hourly wage($145) x Weeks worked per year(50) = $290,000 annual salary
With 2 weeks unpaid time off, your annual earnings at $145/hour would drop by $11,600.
The table below summarizes how your annual income changes depending on the number of weeks of unpaid leave.
Weeks of unpaid leave Paid weeks per year Earnings Per Week Annual Earnings
0 52 $5,800 $301,600
1 51 $5,800 $295,800
2 50 $5,800 $290,000
3 49 $5,800 $284,200
4 48 $5,800 $278,400
5 47 $5,800 $272,600
6 46 $5,800 $266,800
7 45 $5,800 $261,000
Key Takeaways for $145 Hourly Wage
In summary, here are some key points on annual earnings when making $145 per hour:
• At 40 hours per week, you’ll earn $301,600 per year.
• Part-time of 30 hours/week results in $226,200 annual salary.
• Overtime pay can boost yearly earnings, e.g. $56,550 extra at 45 hours/week.
• Unpaid time off reduces your total income, around $11,600 less per 2 weeks off.
• Your specific situation and location impacts taxes and PTO.
Knowing your approximate annual salary and factors impacting it makes it easier to budget and plan your finances. The next step is calculating take-home pay after deductions like taxes.
$145 An Hour Is How Much A Year After Taxes
Figuring out your actual annual earnings based on an hourly wage can be complicated once taxes are taken into account. In addition to federal, state, and local income taxes, 7.65% of your gross pay
also goes to Social Security and Medicare through FICA payroll taxes. So how much does $145 an hour equal per year after FICA and income taxes are deducted from your gross pay?
Below we’ll walk through the steps to calculate your annual net take home pay if you make $145 per hour. This will factor in estimated federal, FICA, state, and local taxes so you know exactly what
to expect.
Factoring in Federal Income Tax
Your federal income tax will be a big chunk out of your gross pay. Federal tax rates range from 10% to 37%, depending on your tax bracket.
To estimate your federal income tax rate and liability:
Look up your federal income tax bracket based on your gross pay.
2023 tax brackets: single filers
Tax rate Taxable income bracket Tax owed
10% $0 to $11,000. 10% of taxable income.
12% $11,001 to $44,725. $1,100 plus 12% of the amount over $11,000.
22% $44,726 to $95,375. $5,147 plus 22% of the amount over $44,725.
24% $95,376 to $182,100. $16,290 plus 24% of the amount over $95,375.
32% $182,101 to $231,250. $37,104 plus 32% of the amount over $182,100.
35% $231,251 to $578,125. $52,832 plus 35% of the amount over $231,250.
37% $578,126 or more. $174,238.25 plus 37% of the amount over $578,125.
For example, if you are single with $301,600 gross annual pay, your federal tax bracket is 35%.
Your estimated federal tax would be:
$174,238.25 + ($301,600 – $231,251) x 35% = $198,860.40
So at $145/hour with $301,600 gross pay, you would owe about $198,860.40 in federal income taxes.
Considering State Income Tax
In addition to federal tax, most states also charge a state income tax. State income tax rates range from about 1% to 13%, with most falling between 4% and 6%.
Key Takeaways
• California, Hawaii, New York, New Jersey, and Oregon have some of the highest state income tax rates.
• Alaska, Florida, Nevada, South Dakota, Tennessee, Texas, Washington, and Wyoming don’t impose an income tax at all.
• Another 10 U.S states have a flat tax rate—everyone pays the same percentage regardless of how much they earn.
A State-by-State Comparison of Income Tax Rates
STATE TAX RATES LOWEST AND HIGHEST INCOME BRACKETS
Alaska 0% None
Florida 0% None
Nevada 0% None
South Dakota 0% None
Tennessee 0% None
Texas 0% None
Washington 0% None
Wyoming 0% None
Colorado 4.55% Flat rate applies to all incomes
Illinois 4.95% Flat rate applies to all incomes
Indiana 3.23% Flat rate applies to all incomes
Kentucky 5% Flat rate applies to all incomes
Massachusetts 5% Flat rate applies to all incomes
New Hampshire 5% Flat rate on interest and dividend income only
North Carolina 4.99% Flat rate applies to all incomes
Pennsylvania 3.07% Flat rate applies to all incomes
Utah 4.95% Flat rate applies to all incomes
Michigan 4.25% Flat rate applies to all incomes
Arizona 2.59% to 4.5% $27,806 and $166,843
Arkansas 2% to 5.5% $4,300 and $8,501
California 1% to 13.3% $9,325 and $1 million
Connecticut 3% to 6.99% $10,000 and $500,000
Delaware 0% to 6.6% $2,000 and $60,001
Alabama 2% to 5% $500 and $3,001
Georgia 1% to 5.75% $750 and $7,001
Hawaii 1.4% to 11% $2,400 and $200,000
Idaho 1.125% to 6.5% $1,568 and $7,939
Iowa 0.33% to 8.53% $1,743 and $78,435
Kansas 3.1% to 5.7% $15,000 and $30,000
Louisiana 1.85% to 4.25% $12,500 and $50,001
Maine 5.8% to 7.15% $23,000 and $54,450
Maryland 2% to 5.75% $1,000 and $250,000
Minnesota 5.35% to 9.85% $28,080 and $171,221
Mississippi 0% to 5% $5,000 and $10,001
Missouri 1.5% to 5.3% $1,121 and $8,968
Montana 1% to 6.75% $2,900and $17,400
Nebraska 2.46% to 6.84% $3,340 and $32,210
New Jersey 1.4% to 10.75% $20,000 and $1 million
New Mexico 1.7% to 5.9% $5,500 and $210,000
New York 4% to 10.9% $8,500 and $25 million
North Dakota 1.1% to 2.9% $41,775 and $458,350
Ohio 0% to 3.99% $25,000 and $110,650
Oklahoma 0.25% to 4.75% $1,000 and $7,200
Oregon 4.75% to 9.9% $3,750 and $125,000
Rhode Island 3.75% to 5.99% $68,200 and $155,050
South Carolina 0% to 7% $3,110 and $15,560
Vermont 3.35% to 8.75% $42,150 and $213,150
Virginia 2% to 5.75% $3,000 and $17,001
Washington, D.C. 4% to 9.75% $10,000 and $1 million
West Virginia 3% to 6.5% $10,000 and $60,000
Wisconsin 3.54% to 7.65% $12,760 and $280,950
To estimate your state income tax:
Look up your state income tax rate based on your gross pay and filing status.
Multiply your gross annual pay by the state tax rate.
For example, if you live in Pennsylvania which has a flat 3.07% tax rate, your estimated state tax would be:
$301,600 gross pay x 3.07% PA tax rate = $9,259.12 estimated state income tax
So with $301,600 gross annual income, you would owe around $9,259.12 in Pennsylvania state income tax. Make sure to check your specific state’s rates.
Factoring in Local Taxes
Some cities and counties levy local income taxes ranging from 1-3% of taxable income.
To estimate potential local taxes you may owe:
• Check if your city or county charges a local income tax.
• If yes, look up the local income tax rate.
• Multiply your gross annual pay by the local tax rate.
For example, say you live in Columbus, OH which has a 2.5% local income tax. Your estimated local tax would be:
$301,600 gross pay x 2.5% local tax rate = $7,540 estimated local tax
So with $301,600 in gross earnings, you may owe around $7,540 in Columbus local income taxes. Verify rates for your own city/county.
Accounting for FICA Taxes (Social Security & Medicare)
FICA taxes are a combination of Social Security and Medicare taxes that equal 15.3% of your earnings. You are responsible for half of the total bill (7.65%), which includes a 6.2% Social Security tax
and 1.45% Medicare tax on your earnings.
In 2023, only the first $160,200 of your earnings are subject to the Social Security tax
There is an additional 0.9% surtax on top of the standard 1.45% Medicare tax for those who earn over $200,000 (single filers) or $250,000 (joint filers).
To estimate your FICA tax payment:
$160,200 x 6.2% + $200,000 x 1.45% + $101,600 x 0.9% = $13,746.80
So you can expect to pay about $13,746.80 in Social Security and Medicare taxes out of your gross $301,600 in earnings.
Total Estimated Tax Payments
Based on the examples above, your total estimated tax payments would be:
Federal tax: $198,860.40
State tax: $9,259.12
Local tax: $7,540
FICA tax: $13,746.80
Total Estimated Tax: $229,406.32
Calculating Your Take Home Pay
To calculate your annual take home pay at $145 /hour:
1. Take your gross pay
2. Subtract your estimated total tax payments
$301,600 gross pay – $229,406.32 Total Estimated Tax = $72,193.68 Your Take Home Pay
n summary, if you make $145 per hour and work full-time, you would take home around $72,193.68 per year after federal, state, local , FICA taxes.
Your actual net income may vary depending on your specific tax situation. But this gives you a general idea of what to expect.
Convert $145 Per Hour to Yearly, Monthly, Biweekly, and Weekly Salary After Taxes
If you make $145 an hour and work full-time (40 hours per week), your estimated yearly salary would be $301,600 .
The $301,600 per year salary does not account for taxes. Federal, state, and local taxes will reduce your take-home pay. The amount withheld depends on your location, filing status, dependents, and
other factors.
Just now during our calculation of $145 An Hour Is How Much A Year After Taxes, we assumed the following conditions:
• You are single with $301,600 gross annual pay, your federal tax bracket is 35 %.
• You live in Pennsylvania which has a flat 3.07% tax rate
• You live in Columbus, OH which has a 2.5% local income tax.
In the end, we calculated your Total Estimated Tax is $229,406.32 , Your Take Home Pay is $72,193.68 , Total tax rate is 76.06%.
So next we’ll use 76.06% as the estimated tax rate to calculate your weekly, biweekly, and monthly after-tax income.
$145 Per Hour to Yearly, Monthly, Biweekly, Weekly,and Week Salary After Taxes Table
Income before taxes Estimated Tax Rate Income Taxes After Tax Income
Yearly Salary $301,600 76.06% $229,406.32 $72,193.68
Monthly Salary $25,133.33 76.06% $19,117.19 $6,016.14
BiWeekly Salary $11,600 76.06% $8,823.32 $2,776.68
Weekly Salary $5,800 76.06% $4,411.66 $1,388.34
$145 an hour is how much a year after taxes
Here is the adjusted yearly salary after a 76.06% tax reduction:
• Yearly salary before taxes: $301,600
• Estimated tax rate: 76.06%
• Taxes owed (76.06% * $301,600 )= $229,406.32
• Yearly salary after taxes: $72,193.68
Hourly Wage Hours Worked Per Week Weeks Worked Per Year Total Yearly Salary Estimated Tax Rate Taxes Owed After-Tax Yearly Salary
$145 40 52 $301,600 76.06% $229,406.32 $72,193.68
$145 an hour is how much a month after taxes
To calculate the monthly salary based on an hourly wage, you first need the yearly salary amount. Then divide by 12 months.
• Yearly salary before taxes at $145 per hour: $301,600
• Divided by 12 months per year: $301,600 / 12 = $25,133.33 per month
The monthly salary based on a 40 hour work week at $145 per hour is $25,133.33 before taxes.
After applying the estimated 76.06% tax rate, the monthly after-tax salary would be:
• Monthly before-tax salary: $25,133.33
• Estimated tax rate: 76.06%
• Taxes owed (76.06% * $25,133.33 )= $19,117.19
• Monthly after-tax salary: $6,016.14
Monthly Salary Based on $145 Per Hour
Hourly Wage Yearly Salary Months Per Year Before-Tax Monthly Salary Estimated Tax Rate Taxes Owed After-Tax Monthly Salary
$145 $301,600 12 $25,133.33 76.06% $19,117.19 $6,016.14
$145 an hour is how much biweekly after taxes
Many people are paid biweekly, meaning every other week. To calculate the biweekly pay at $145 per hour:
• Hourly wage: $145
• Hours worked per week: 40
• Weeks per biweekly pay period: 2
• $145 * 40 hours * 2 weeks = $11,600 biweekly
Applying the 76.06%estimated tax rate:
• Biweekly before-tax salary: $11,600
• Estimated tax rate: 76.06%
• Taxes owed (76.06% * $11,600 )= $8,823.32
• Biweekly after-tax salary: $2,776.68
Biweekly Salary at $145 Per Hour
Hourly Wage Hours Worked Per Week Weeks Per Pay Period Before-Tax Biweekly Salary Estimated Tax Rate Taxes Owed After-Tax Biweekly Salary
$145 40 2 $11,600 76.06% $8,823.32 $2,776.68
$145 an hour is how much weekly after taxes
To find the weekly salary based on an hourly wage, you need to know the number of hours worked per week. At 40 hours per week, the calculation is:
• Hourly wage: $145
• Hours worked per week: 40
• $145 * 40 hours = $5,800 per week
Accounting for the estimated 76.06% tax rate:
• Weekly before-tax salary: $5,800
• Estimated tax rate: 76.06%
• Taxes owed (76.06% * $5,800 )= $4,411.66
• Weekly after-tax salary: $1,388.34
Weekly Salary at $145 Per Hour
Hourly Wage Hours Worked Per Week Before-Tax Weekly Salary Estimated Tax Rate Taxes Owed After-Tax Weekly Salary
$145 40 $5,800 76.06% $4,411.66 $1,388.34
Key Takeaways
• An hourly wage of $145 per hour equals a yearly salary of $301,600 before taxes, assuming a 40 hour work week.
• After accounting for an estimated 76.06% tax rate, the yearly after-tax salary is approximately $72,193.68 .
• On a monthly basis before taxes, $145 per hour equals $25,133.33 per month. After estimated taxes, the monthly take-home pay is about $6,016.14 .
• The before-tax weekly salary at $145 per hour is $5,800 . After taxes, the weekly take-home pay is approximately $1,388.34 .
• For biweekly pay, the pre-tax salary at $145 per hour is $11,600 . After estimated taxes, the biweekly take-home pay is around $2,776.68 .
Understanding annual, monthly, weekly, and biweekly salary equivalents based on an hourly wage is useful when budgeting and financial planning. Taxes make a significant difference in take-home pay,
so be sure to account for them when making income conversions. Use this guide as a reference when making salary calculations.
What Is the Average Hourly Wage in the US?
Last Updated: Sep 1 2023
US Average Hourly Earnings is at a current level of $33.82, up from 33.74 last month and up from 32.43 one year ago. This is a change of 0.24% from last month and 4.29% from one year ago.
Average Hourly Earnings is the average dollars that a private employee makes per hour in the US. This metric is a part of one of the most important releases every month which includes unemployment
numbers as well. This is normally released on the first Friday of every month. This metric is released by the Bureau of Labor Statistics (BLS).
What is the average salary in the U.S.?
Last Updated: July 18, 2023
The U.S. Bureau of Labor Statistics uses median salary data rather than averages to avoid skewed numbers from outlying high and low numbers. Median weekly earnings of the nation's 121.5 million
full-time wage and salary workers were $1,100 in the second quarter of 2023, the U.S.
If a person works 52 weeks in the year, then this represents a national annual salary of $57,200.
Is $145 an Hour a Good Salary?
Yes, $145 an hour is an extremely high hourly wage. At 40 hours per week, this works out to $5,800 per week or $301,600 per year. This salary is well above what most Americans earn. The median
personal income in the US is around $36,000.
Someone working full time on $145 an hour would be firmly in the upper class and top few percent of incomes. They could afford a very comfortable lifestyle with plenty of disposable income. This wage
could more than support a whole family and provide the ability to save and invest substantial amounts.
Jobs that pay $145 an hour
While most jobs pay under $50 per hour, here are some examples of roles that could earn $145 per hour or more:
• Doctors and surgeons – Specialists like anesthesiologists, surgeons and orthopedists can earn $100 to $200+ per hour.
• Corporate lawyers and law partners – Experienced partners at large law firms can bill over $145/hour.
• Successful business owners and CEOs – Owners of thriving companies or corporations can pay themselves $145 or more.
• Specialty contractors – Some plumbers, electricians, builders and IT contractors charge over $100 per hour.
• Financial advisors – Top fee-only advisors charge 1-2% on assets under management, which can work out to over $145 hourly based on their book of business.
• Consultants – Management, IT and other consultants at senior levels bill clients out at high hourly rates.
• Freelancers – Those with specialized, in-demand skills like software developers, graphic designers, or actuaries can earn over $100/hour freelancing.
So while not common, there are various ways to earn $145 or more per hour in fields requiring extensive education, training, skills, client bases and expertise.
Can You Live Off $145 An Hour?
Living off $145 an hour would provide a very comfortable, upper class lifestyle today. Even with regular expenses, you would have ample income to:
• Afford a nice house or luxury apartment without being cost-burdened
• Lease or buy new luxury vehicles
• Invest chunks of income in retirement, college savings and brokerage accounts
• Take regular vacations worldwide
• Splurge on expensive restaurants, entertainment and shopping
• Pay for convenience services like housekeeping and ride shares
• Absorb unexpected emergencies or bills without going into debt
At this wage, you could live almost anywhere very comfortably, save adequately for the future, and enjoy great perks and leisure time. You’d have access to the best housing, schools, services and
healthcare. Financial security would not be a major concern.
The impact of inflation on the value of $145 an hour
While inflation reduces the buying power of an hourly wage over time, someone earning $145 an hour would likely still be quite comfortable even decades later.
Here is how inflation has or may impact the value of $145 per hour over the years:
• In 1990, $145 per hour would be worth about $330 in today’s dollars. Still extremely high.
• In 2000, $145/hour equals roughly $245 today. Also still a top income.
• Even since 2010, $145/hour would be equivalent to around $180 today. Very comfortable.
• In 2040, if we assume 3% annual inflation, $145 today would equal $240/hour. Still quite high.
While inflation erodes the buying power of fixed wages, hourly pay this high tends to stay at the top tiers over time. Professionals could periodically negotiate raises or charge higher rates per
hour to offset some inflation impacts. Overall, $145/hour earners would likely remain very well off.
5 Ways To Increase Your Hourly Wage
If you are not yet earning $145 per hour, some steps that could boost your income include:
• Get advanced education – Complete a graduate degree like a JD, MBA or MD that lead to high-paying roles.
• Develop specialized skills – Gain expertise in lucrative fields like software development, financial planning or surgery.
• Achieve professional credentials – Become a licensed attorney, CPA, PE or other certified pro.
• Increase your rates – Consultants and freelancers can raise their rates over time as they gain experienc and clout.
• Grow your business – Entrepreneurs can build their companies and increase their own pay substantially.
• Invest to increase assets – Earn returns that boost your overall income.
• Negotiate promotions – Take on more responsibility in salaried roles to move up and increase pay.
• Switch companies – Leverage experience to land higher paying jobs at new organizations.
It takes time, skill building, and career climbing to earn over $100 an hour in most fields. The tradeoff is higher income for hard work and rare expertise.
Buying a car on $145 an hour
Buying a car while earning $145 an hour provides tons of options. Here is a look at the possibilities:
• With $145/hour full time, annual income is $301,600 before taxes.
• Most lenders want no more than 15% of gross income going to a car payment. 15% of $301,600 is $45,240 per year or $3,770 per month.
• This means you could buy pretty much any vehicle out there in cash if you wanted.
• If you finance, you could afford luxury cars like a $150,000 Tesla Model X or Porsche 911 paying cash down and having monthly payments of $3,000-$4,000.
• Insurance, fuel, maintenance and repairs would all be easily affordable.
• You could lease expensive new luxury vehicles every couple years if desired.
At this high income level, you would have your choice amongst all new and used cars on the market. Exotic supercars could even potentially be affordable. Vehicle expenses would not be a major
financial consideration.
Can You Buy a House on $145 An Hour?
Buying a house on a $145 an hour income is very feasible. Here’s a look at how it could be done:
• The median home price is around $300,000.
• Most lenders want no more than 28% of gross income going to the mortgage payment.
• 28% of $301,600 annual full time salary at $145/hour is $84,448 per year or $7,037 per month.
• This means you could afford a $1.5 million home with 20% down while keeping payments around $7,000/month.
• You may also qualify to carry a second home, vacation property or investment property.
• If self-employed, lenders would look at total assets and down payment amount, which you would have readily available.
• Refinancing or paying off the mortgage early is possible if desired.
At this high income level, you could comfortably afford a very nice home in most real estate markets. You would have your pick of neighborhoods and amenities when buying.
Example Budget For $145 Per Hour
Here is one example of a monthly budget for someone earning $145 per hour full time:
• Gross Monthly Income: $12,600
• Taxes (30%): $3,780
• Housing (mortgage/rent): $2,500
• Car Payment: $1,000
• Car Insurance: $150
• Gas: $300
• Groceries: $800
• Dining Out: $1,000
-Utilities: $500
• Cable/Internet: $200
• Gym Membership: $100
• Clothing: $500
• Vacation Savings: $2,000
• Retirement Savings: $1,500
Total Expenses: $13,750
This leaves $850 per month that could be used for additional savings, entertainment, hobbies, lessons, or other discretionary spending. Even with a high cost of living area and luxuries, there is
ample room in the budget for savings and enjoying lifestyle benefits.
In Summary
An hourly income of $145 puts someone in the top few percent of all wage earners. This type of salary provides a comfortable or even wealthy lifestyle in most parts of the country. Regular expenses,
both necessities and luxuries, would be easily affordable. You could save and invest large amounts of the income. Overall, $145 per hour provides a level of financial freedom and flexibility that
allows you to live very well.
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View problem - Meteors (POI11_met)
Time limit Memory limit # of submissions # of accepted Ratio
6000 ms 64 MiB 1296 132 10.19%
Byteotian Interstellar Union (BIU) has recently discovered a new planet in a nearby galaxy. The planet is unsuitable for colonisation due to strange meteor showers, which on the other hand make it an
exceptionally interesting object of study.
The member states of BIU have already placed space stations close to the planet's orbit. The stations' goal is to take samples of the rocks flying by. The BIU Commission has partitioned the orbit
into $m$ sectors, numbered from $1$ to $m$, where the sectors $1$ and $m$ are adjacent. In each sector there is a single space station, belonging to one of the $n$ member states.
Each state has declared a number of meteor samples it intends to gather before the mission ends. Your task is to determine, for each state, when it can stop taking samples, based on the meteor shower
predictions for the years to come.
The first line of the standard input gives two integers, $n$ and $m$ ($1 \le n,m \le 300\,000$), separated by a single space, that denote, respectively, the number of BIU member states and the number
of sectors the orbit has been partitioned into.
In the second line there are $m$ integers $o_i$ ($1 \le o_i \le n$), separated by single spaces, that denote the states owning stations in successive sectors.
In the third line there are $n$ integers $p_i$ ($1 \le p_i \le 10^9$), separated by single spaces, that denote the numbers of meteor samples that the successive states intend to gather.
In the fourth line there is a single integer $k$ ($1 \le k \le 300\,000$) that denotes the number of meteor showers predictions. The following $k$ lines specify the (predicted) meteor showers
chronologically. The $i$-th of these lines holds three integers $l_i, r_i, a_i$ (separated by single spaces), which denote that a meteor shower is expected in sectors $l_i,l_{i+1},\ldots,r_i$ (if
$l_i \le r_i$) or sectors $l_i,l_{i+1},\ldots,m,1,\ldots,r_i$ (if $l_i > r_i$), which should provide each station in those sectors with $a_i$ meteor samples ($1 \le a_i \le 10^9$).
In tests worth at least 20% of the points it additionally holds that $n,m,k \le 1\,000$.
Your program should print $n$ lines on the standard output. The $i$-th of them should contain a single integer $w_i$, denoting the number of shower after which the stations belonging to the $i$-th
state are expected to gather at least $p_i$ samples, or the word NIE (Polish for no) if that state is not expected to gather enough samples in the foreseeable future.
For the input data:
the correct result is:
Task authors: Paweł Mechlinski and Jakub Pachocki.
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Ed in the Apple: Bill Gates’ Quest for the Mythical Magic Bullet: Next Quest: Algebra 1
From Small High Schools to Measures of Effective Teaching (MET) to Value-Added Measurements (VAM) to the Common Core State Standards, the Gates Foundation has been searching for the magic bullet, a
vaccine for curing education, and the “cures” have proven fruitless (See links above)
The next magic bullet is a cure for Algebra 1, the course viewed as standing in the way of graduation and success in the post graduate world.
The Gates Foundation released an application for a new initiative: “Balance the Equation.”
Algebra 1 is one of the most important on-track indicators of students’ future success. Students who do not complete Algebra 1 have a one-in-five chance of graduating from high school and, as it
currently stands, Algebra 1 acts as a gatekeeper – rather than a gateway – to future success.
We are looking for partners to help us radically transform and rethink the traditional math classroom to better support students who have been historically marginalized in math, including Black and
Latino students, student who speak a language other than English, and students affected by poverty in the United States
The two-year one million dollar grants, called “Balance the Equation: A Grand Challenge for Algebra 1” asks applicants to address five areas,
These areas have the biggest opportunity for altering the traditional classroom experience for priority students in order to achieve the desired outcomes. Specifically, these areas include:
• Building out support systems
□ Improving relevance of algebra content
□ Elevating understanding of mathematical language
□ Empowering and strengthening teacher practices
□ Developing new or better feedback mechanisms
As you would expect the application is detailed and specific – it’s only a few pages, read here.
The Gates Foundation poured hundreds of millions, maybe a few billion into “innovative” proposals, the small high school initiative created about 1200 small high schools across the nation, and Gates
admitted his efforts “fell short”
Many of the small schools that we invested in did not improve students’ achievement in any significant way. These tended to be the schools that did not take radical steps to change the culture, such
as allowing the principal to pick the team of teachers or change the curriculum. We had less success trying to change an existing school than helping to create a new school. Even so, many schools had
higher attendance and graduation rates than their peers. While we were pleased with these improvements, we are trying to raise college-ready graduation rates, and in most cases, we fell short.
Externally imposed reorganizations rarely are embedded in organizational cultures, At the core of organizational change are two axioms: “change is perceived as punishment” and “participation reduce
resistance,” ignoring the axioms commonly dooms “new ideas.”
In New York City the small school movement began in a collaborative climate, and, under Bloomberg became a race to close as many large high schools as possible. Some small high schools are
prospering; too many others are simply smaller replicas of the schools they replaced.
The Common Core is another massive initiative, designed to change the face of American education, stumbled, and fell by the wayside.
Gates realizes that the large-scale transformation he envisions will require greater attentiveness to “locally-driven solutions identified by networks of schools.”
The statement evinces some realization that what might work well in some places might not work well in others. But the new-found sensitivity to locale is more than offset by the insistence on
“data-driven continuous learning” (and who will store, manage, and mine this data, one wonders, if not large-scale tech companies, such as the one owned by Bill Gates?), “evidence-based
interventions,” and the insistence on using “data to drive continuous improvement.” This data will provide us with shocking revelations, such as the data in Chicago that revealed students who attend
class, complete courses, accumulate credits, and receive higher grades are more likely to graduate.
There are many who view Gates as the personification of evil, a bored mega-billionaire using his dollars to impose “solutions,” deeply flawed solutions, on the American education landscape. Others
see him using his clout to enrich himself by driving education to “large scale tech companies” instead of locally based solutions.
Bob Moses, the iconic civil rights leader began the Algebra Project over thirty years ago in rural South, successfully teaching algebra to impoverished children of color with excellent outcomes. Read
assessment here.
Why is the Gates funding another algebra initiative? Why not expand the Moses Algebra Project?
Bill is tip-toeing into another education morass.
The threshold question: Should we be requiring Algebra 1 as a condition of high school graduation?
Andrew Hacker, in a New York Times op ed questions whether “Algebra is Necessary?”
The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school … Most of the educators I’ve talked with cite algebra as the major academic
… a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or
Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.
Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives … [courses to]
familiarize students with the kinds of numbers that describe and delineate our personal and public lives.
Peter Flom, a math teacher challenges Hacker,
Most of what we learn in school has limited applicability outside school. I took art in high school. I have never drawn or painted since. I took biology. I have not used that since, either. That is
not the point. The point of education is not to teach children things they need to know, it is to expose them to the glories of the human mind.
By this logic, almost all of high school would not be required.
Is algebra necessary? In the strict sense, no. You can live without it. You can also live without art, music, literature or sports. Would you want to?
Cathy N. Davidson, a professor and founder of the Futures Initiative at the Graduate Center of the City University of New York joins the debate,
For too many students, algebra is not the gateway to mathematical literacy. It is the gatekeeper.
Algebra is the single most failed course in high school, the most failed course in community college, and, along with English language for nonnative speakers, the single biggest academic reason that
community colleges have a high dropout rate
Do you fail someone from formal educational opportunities who has will, skills, and intellectual abilities simply because they can’t master algebraic thinking? Or do you offer alternative, productive
forms of math competencies to challenge them and help them grow?
Whether to require or not require algebra — in both high school and in college — boils down to one’s view of the purpose of education. If the purpose is credentialing — certifying that the student
has earned passing grades in a predefined suite of courses that, collectively, constitute the requirements for a diploma — then it is fine to require anything you want, including algebra.
But if you have a longer view of the purpose of education — that its true mission is to prepare students for everything that comes after graduation — then it is time get rid of the one-size-fits-all
prescriptive curriculum.
Linda Gojak, the President of the National Council of Teachers of Mathematics raises the question of when to teach Algebra 1, the traditional ninth grade, or, as is increasing ly common in the
seventh or eighth grade. Gojak sees the seventh and eight grades as opportunities to explore math concepts, preparing students for algebra.
Seeing the relevance of mathematics in real-world situations and future career options encourages students to take more mathematics rather than to wonder, “When are we ever going to use this?”
Solving interesting problems with high cognitive demand offers students experiences to make mathematical connections, form generalizations, and develop mathematical strategies that lead to making
sense of early algebra concepts. Working on projects that deepen the level of mathematical understanding and promote algebra applications has the potential to prepare students for the level of
abstraction and symbolism that students need for success in rigorous algebra courses.
Gates jumped beyond the question of whether Algebra is necessary, one can say from the frying pan into the fire
Math teachers are among the most contentious sets of teachers.
Professor Hung Wu, at University of California, Berkeley, the primary author of the Common Core Math Standards and the new Geometry standards, in a widely circulated article (“Potential Impact of the
Common Core Mathematics Standards on the American Curriculum”), skewers all that preceded his standards.
Long before the advent of the Common Core State Standards Mathematics, American schools had a de facto national mathematics curriculum, namely, the curriculum dictated by school mathematics
textbooks. While there are some formal differences among these books, the underlying mathematics is quite similar throughout. The resulting curriculum distorts mathematics in the sense that it often
withholds precise definitions and logical reasoning, fails to point out interconnections between major topics such as whole numbers and fractions, and employs ambiguous language that ultimately leads
to widespread nonlearning. The CCSSM make a conscientious attempt to address many of these problems and, in the process, raise the demand on teachers’ content knowledge for a successful
implementation of these standards.
Professor Wu looks upon elementary school teachers as being “mathematically illiterate,” and implies that we first have to teach mathematics to elementary school teachers before they can teach
mathematics to their students.
For elementary teachers, there is at present a feeling that they have been so damaged by their K–12 experience…that we owe it to them to treat them with kid gloves…. Those that I have encountered are
generally eager to learn and are willing to work hard. The kid-glove treatment would seem to be hardly necessary. …There is another school of thought arguing that for elementary teachers, one should
teach them not only the mathematics of their classrooms, but at the same time also how children think about the mathematics. Again, I can only speak from my own experience. The teachers I observed
usually had so much difficulty just coming to terms with the mathematics itself that any additional burden about children’s thinking would have crushed them.
Additionally, there is a raging debate among math teachers about the Geometry Common Core course adopted by most states, mathematics teachers sharply disagree with Professor Wu.
Will the Gates investment result in a curriculum or a set of lesson plans, high quality videos of lessons, teaching Algebra 1 on Snap Chat or Instagram or the new social media site? Will a
“culturally relevant” Algebra 1 curriculum engage students? How about including BIPOC (Black, Indigenous People of Color) in algebra lessons? Whatever works …
Teachers are writers, actors, producers, directors and critics of a play that runs for one day. Hopefully we are reflective: What “worked?” What didn’t “work?” Why didn’t it “work?” We mess around in
our tool kit to look for the proper tool to fix the situation, and, we’re always looking for new tools.
We ask our colleagues, how did you teach this or that? Can I come into your classroom and watch you? Can you take a look at me and can we talk?
There are no magic bullets.
Teaching is a frustrating and at the same time a gratifying job.
When a kid left my room after a lesson and said, “Mr. G, that was really hard, I think I got it,” I beamed inside.
A school leader invited me to sit in on a teacher meeting; New York State switched to the Common Core Algebra 1 curriculum and a Common Core Algebra 1 Regents Examination.
The teachers completed grading the exam and constructed an “error matrix,” the most common wrong answers, and, asked the lower grades to sit in on the meeting: everyone brought their lesson plans.
Why did the kids get a specific question wrong, and, how could they adjust their plan to better teach the concept?
In order to change outcomes you have to change inputs.
In how many schools if kids do poorly on a test do we offer remediation, defined as teaching more of what they failed to understand the first time.
Bill,I have a suggestion, save the millions, how can we create collaborative school cultures, schools in which teachers, facilitated by the school leaders, can explore instructional practice; how can
we redefine professional development, not as a block of time sitting in a room with a sage, but what you do every day.
As my friend Jonathan is fond of saying, “The answers are in the room,” we just have to find them.
This blog post has been shared by permission from the author.
Readers wishing to comment on the content are encouraged to do so via the link to the original post.
Find the original post here:
The views expressed by the blogger are not necessarily those of NEPC.
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Magnetic Force between Parallel Conductors
Each wire creates a magnetic field around the wire. The force between the two wires is related to the current of the wires and the distance between the wires. Magnetic force between parallel
conductors is described by Ampere's law and the Biot-Savart law, which are fundamental principles in electromagnetism. When two current carrying conductors are placed parallel to each other, they
create magnetic fields, and these magnetic fields can exert forces on each other.
Key Points about Magnetic Force Between Parallel Conductors
Direction of Magnetic Fields - When electric current flows through a conductor, it generates a magnetic field around it. The direction of the magnetic field is given by the right-hand rule. If you
wrap your right hand around the conductor with your thumb pointing in the direction of current flow, your fingers will curl in the direction of the magnetic field.
Ampere's Law - Ampere's law relates the magnetic field around a closed loop to the current passing through the loop.
Force Between Parallel Conductors - When two parallel conductors carrying currents are placed near each other, the magnetic field produced by one conductor exerts a force on the other conductor.
Direction of Force - The direction of the magnetic force between the two conductors is determined by the relative directions of the currents. If the currents flow in the same direction, the
conductors will attract each other, and if they flow in opposite directions, they will repel each other.
Magnitude of Force - The magnitude of the force increases with increasing currents, decreasing distance between the conductors, and increasing length of the conductors.
Magnetic Field Lines - Magnetic field lines around the conductors help visualize the direction and strength of the magnetic fields. These field lines form concentric circles around each conductor,
and their density indicates the strength of the magnetic field.
The magnetic force between parallel conductors is a concept in electromagnetism and is the basis for many practical applications, including the operation of transformers, solenoids, and electric
motors. Understanding these principles is crucial for designing and analyzing electrical circuits and devices.
Magnetic Force between Parallel Conductors formula
\( F_m = \mu_o \; I_1 \; I_2 \;/\; 2 \; \pi \; d \) (Magnetic Force between Parallel Conductors)
\( \mu_o = F_m \; 2 \; \pi \; d \;/\; I_1 \; I_2 \)
\( I_1 = F_m \; 2 \; \pi \; d \;/\; \mu_o \; I_2 \)
\( I_2 = F_m \; 2 \; \pi \; d \;/\; \mu_o \; I_1 \)
\( d = \mu_o \; I_1 \; I_2 \;/\; 2 \; \pi \; F_m \)
Symbol English Metric
\( F_m \) = Magnetic Force \(lbm-ft \;/\; sec^2\) \(N\)
\( \mu_o \) (Greek symbol mu) = Magnetic Constant - \(H \;/\; m\)
\( I_1 \) = Wire Current (amp) 1 \(I\) \(C \;/\; s\)
\( I_2 \) = Wire Current (amp) 2 \(I\) \(C \;/\; s\)
\( \pi \) = PI \(3.141 592 653 ...\) \(3.141 592 653 ...\)
\( d \) = Distance Between the Wires \(ft\) \(m\)
Tags: Electrical Current Magnetic
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Private Algebra Tutor In San Jose, CA | MathTowne
Work With An Algebra Tutor Near You in San Jose – Get Help With Exam Review
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Private ALGEBRA 1 Tutoring Near You
Why Algebra 1 is Important
Algebra 1 is a critical subject for students as it forms the foundation for higher-level math courses such as Algebra 2, geometry, trigonometry, calculus, and beyond. It is essential for students to
master the fundamental concepts of Algebra 1 in order to succeed in these advanced courses and in many careers that require a strong foundation in math.
In addition to being a requirement for many future courses and careers, Algebra 1 also helps students develop important skills such as critical thinking, problem-solving, and logical reasoning. It
teaches students to think abstractly and apply mathematical concepts to real-world situations, preparing them for success in various fields such as engineering, computer science, finance, and more.
Is Algebra 1 Hard?
The difficulty level of Algebra 1 varies from student to student. For some students, Algebra 1 may be challenging, while other students may find Algebra 1 easier and more manageable.
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Common Challenges Students Face in Algebra 1
Difficulty with Concepts
Algebra 1 can be a difficult subject for many students. Some common challenges that students face in Algebra 1 include difficulty with abstract concepts, confusion with new symbols and notations, and
difficulty with solving word problems.
Additionally, some students may struggle with algebraic expressions, linear equations, graphing, and factoring. Our Algebra 1 tutoring program is designed to help students overcome these challenges
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Test-Taking Strategies
Many students struggle with Algebra 1 tests, not only because of the subject matter but also due to test-taking strategies. Regularly reviewing formulas and concepts is a fundamental strategy that
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Learn more about Algebra 1 Tutoring for High School Students
Algebra 1 is the crucial ingredient in secondary math education. The topics covered in Algebra 1 provide the foundation required for future success in high school and college mathematics. It requires
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The primary goal in Algebra 1 is to help students evolve their concrete mathematical knowledge to more abstract algebraic generalizations. This helps students to gain the skills necessary to
translate problems from a variety of fields into the appropriate mathematical symbols. They are then able to manipulate and operate on those symbols to obtain an answer to the problem.
We make sure that the emphasis is placed on the development of critical thinking skills and algebraic problem solving skills. These provide the foundation for real world problem solving and help
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• Use ratios and proportions to solve for unknown variables.
• Graph and perform transformations of basic functions
• Understand the different between expressions, equations, and functions
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• Use linear equations and inequalities to solve real life word problems
• Understand and solve systems of linear equations and inequalities
• Introduce exponent rules and exponential functions
Further Program Topics
• Graph and recognize different forms of quadratic equations
• Solve quadratic equations using different methods
• Learn the quadratic formula, imaginary numbers, and complex roots
• Understand the graph and terminology of polynomials
• Add, subtract, multiply, and divide polynomials
• Get familiar with various factoring methods and techniques
• Introduce radical and rational expressions
• Apply algebraic concepts to solve various real life word problems
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I need help with my algebra 1 class
< 1 min
Frequently Asked Questions
What is the difference between Algebra 1 and Algebra 2?
In Algebra 1, students are expected to master a wide range of techniques for manipulating expressions and equations. Algebra 1 emphasizes different core concepts. These include properties of
equalities and inequalities, linear and quadratic functions with one or two variables, and solving algebraic equations.
Algebra 2 is a continuation course built upon the concepts taught in Algebra 1. In Algebra 2, students are shown additional function types, such as polynomial and rational functions, logarithms, and
radicals. Trigonometric functions usually become a significant new focus area as students begin to investigate Algebra 2.
How do you master Algebra 1?
To master Algebra 1, students are required to have a solid understanding of Arithmetic. Make sure you’re paying attention in class and complete the practice problems your teacher assigns you.
Remember that it’s not easy to study all at once the night before a big test. Instead, consider doing a small bit of studying every day to increase your chances of passing your Algebra 1 class.
If you’re having trouble understanding a particular type of problem it’s important to look for help. If you can’t figure out the answers, you should speak with your teacher as soon as possible to
gain understanding. Organizing study groups with your classmates is an excellent approach to go over challenging concepts and prepare for the exam.
There are also many helpful resources available to you online, such as Khan Academy. If you are looking for additional help, don’t hesitate to reach out to us.
Where can I find an Algebra tutor near me?
Unless you live in the middle of nowhere, there should be plenty of qualified tutors in your area. If you are looking for online lessons, MathTowne is an excellent source for Algebra tutoring. If you
live in the San Francisco South Bay Area, we are also happy to provide in-person tutoring to you.
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Variations on a Sudoku Solution for Perished Piano
play that piano
inaudible to the world
infinite pleasure
Perished Piano (photo ©the author)
Yes, that perished piano. The one somewhat hidden, mute and inaudible, in the picture above. That particular piano, untouched, unsung and unsinging during fifty long years of gathering dust, rattling
ever more out of tune, prone to time’s relentless streams and forces of decay. It is that piano that, in its very own and unique time-worn voice, solitarily plays for you a series of forty-nine
(twenty-seven and twenty-two) sudokist pieces, trustingly dedicated to its eighty-five keys. Each piece is a variation on a single sudoku solution: one nine by nine grid of numbers generates all of
the sequences of notes and durations.
Each of the variations centres on a cyclic sudokist sequence of 162 notes: a sequence of two times nine times nine permutations of the list of digits 1, 2, 3, 4, 5, 6, 7, 8, 9, that alternately
prescribe how many keys to move up or down on the keyboard to play the next note, and another sequence of two times nine times nine permutations of the numbers 1 to 9 for the corresponding duration
in ‘ticks’ of their soundings; both sequences are read from the sudoku solution’s number grid.
Like in mirrored counterpoint the first voice’s sequence plays against a second voice, its retrograde, interspersed with resounding random chords of…
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Hilbert's sixth problem
The basis of it all
Set theory
• fundamentals of set theory
• presentations of set theory
• structuralism in set theory
Foundational axioms
Removing axioms
The sixth in the famous list of Hilbert's problems asks for the formalization/axiomatization of physics in mathematics. The original version in the way Hilbert stated it (Hilbert 1901) is the
following (translated from the original German):
6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences
in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. (full text)
Reid 1970 p 127 recalls Hilbert’s motivation as follows:
Further discussion of background and historical context is given by the ICM06 speech of Corry 2006, where it says:
From all the problems in the list, the sixth is the only one that continually engaged [Hilbert’s] efforts over a very long period, at least between 1894 and 1932.
Hence not the least due to Hilbert himself, meanwhile of course several aspects of physics have been formalized, while others are certainly still waiting to be understood in a systematic fashion.
Hilbert’s original example of probability theory is considered to be fully axiomatized since the 1930s by the concepts of measure theory. For the example of mechanics the status of the axiomatization
crucially depends on how widely one understands the term today. The special case of classical mechanics which Hilbert was origially referring to has been fully formalized by means of symplectic
geometry and variational calculus. Hilbert himself then contributed to the application of this formalization to gravity (“Einstein-Hilbert action”, see Sauer 99) and the formalization of the
refinement of classical mechanics to quantum mechanics (“Hilbert space”). Even though to this day the ontological status of quantum mechanics as a theory of physics is the topic of debates, certainly
the mathematics of quantum mechanics as such is fully captured by functional analysis and operator algebra theory. But this situation changes drastically when one passes to what since around the
1950s is understood to be the more fundamental and general version of mechanics in nature, namely quantum field theory. Since the 1960 the Haag-Kastler axioms (“AQFT”) have been proposed as an
axiomatization for quantum field theory and crucially for local quantum field theory. While these axioms had some success in laying a foundation for structural results such as the PCT theorem, their
continuing lack of relevant examples (models) in dimensions greater than 2 seems to indicate that something is missing. Modern developments in higher algebra and higher geometry suggest that a
refinement of the axioms to the context of homotopy theory is necessary and natural.
Related pages
Related expositions:
The original list of Hilbert's problems appears in
• David Hilbert, Mathematical Problems, Bulletin of the American Mathematical Society, vol. 8, no. 10 (1902), pp. 437-479. Earlier publications (in the original German) appeared in Göttinger
Nachrichten, 1900, pp. 253-297, and Archiv der Mathematik und Physik, 3dser., vol. 1 (1901), pp. 44-63, 213-237. (AMS, pdf, html)
The historical context of the sixth problem is recalled by
and in some detail in
• Leo Corry, David Hilbert and the axiomatization of physics: From Grundlagen der Geometrie to Grundlagen der Physik, Archimedes: New Studies in the History and
Philosophy of Science and Technology 10, Kluwer Academic Publishers (2004) [pdf]
• Leo Corry, On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization, Proceedings of the International Congress of Mathematics in Madrid 2006 [pdf&
Hilbert’s own work on the foundations of the classical field theory of gravity (general relativity) is surveyed in
• Tilman Sauer: The relativity of discovery: Hilbert’s first note on the foundations of physics, Arch. Hist. Exact Sci. 53 6 (1999) 529-575 [arXiv:physics/9811050]
Discussion in the context of higher category theory and physics:
Surveys of the sixths problem include
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Registered Data
[00533] Recovery and robustness of geometric fingerprints for point clouds and data
• Session Time & Room :
□ 00533 (1/2) : 1C (Aug.21, 13:20-15:00) @F412
□ 00533 (2/2) : 1D (Aug.21, 15:30-17:10) @F412
• Type : Proposal of Minisymposium
• Abstract : The aim of our mini-symposium is to connect communities interested in the problem of condensing information from a dataset to a less complex geometric/statistical "summary", sometimes
called a fingerprint. We will concentrate especially on, Distance histograms, Persistence diagrams, as well as spectral fingerprints and other geometric fingerprints. Questions relevant to
applications, including topics such as resistence or stability to noise/error of a given fingerprint ("robustness" problems), or injectivity of the fingerprint (relevant for "recovery" problems)
will be our focus during the minisymposium.
• Organizer(s) : Mircea Petrache, Rodolfo Viera
• Classification : 53C23, 55N31, 68T09, 52C35, 68R12, Distance histograms, Persistence diagrams, Fourier fingerprints
• Minisymposium Program :
□ 00533 (1/2) : 1C @F412 [Chair: Petrache Mircea]
☆ [04178] Recovering discrete Fourier spectra from random perturbations
○ Format : Online Talk on Zoom
○ Author(s) :
■ Mircea Petrache (Pontificia Catolica Universidad de Chile)
■ Rodolfo Viera (Pontificia Universidad Católica de Chile)
○ Abstract : In this talk I will discuss the behaviour of the Fourier Transform of (quasi-)periodic sets under random perturbations. We will see that for i.i.d random perturbations of a
quasi-periodic set X in the Euclidean space, the effect of the perturbations is almost surely that of multiplying the Fourier Transform of X by a weight which depends on the law of
the perturbation. Also we will see quantitative versions of the previous discussion in finite groups which we will use to obtain, after passing to the limit, the almost sure recovery
of the Fourier Transform of lattices in some non-abelian instances, such as the Heisenberg group.
☆ [04956] An information-theoretic perspective on the turnpike and beltway problems
○ Format : Online Talk on Zoom
○ Author(s) :
■ Shuai Huang (Emory University)
○ Abstract : Reconstructing a set of points on a line or a loop from their unlabelled pairwise distances is known as the turnpike or beltway problem. Some point configurations are easy
to reconstruct, while others are more difficult. We show that the difficulty of problem can be characterized by the mutual information $I(X;Y)$ between the point variable $X$ and
distance variable $Y$. Experiments show that $I(X;Y)$ decreases when there are more repeated distances.
☆ [04313] Curvature sets and curvature measures over persistence diagrams
○ Format : Talk at Waseda University
○ Author(s) :
■ Facundo Memoli (Ohio State University)
○ Abstract : We study an invariant (i.e. a feature) of compact metric spaces which combines the notion of curvature sets introduced by Gromov in the 1980s together with the notion of
Vietoris-Rips persistent homology. For given integers k≥0 and n≥1 these invariants arise by considering the degree k Vietoris-Rips (VR) persistence diagrams of all finite point clouds
with cardinality at most n sampled from a given metric space. We call these invariants \emph{persistence sets}. This family of invariants contains the usual VR persistence diagram of
the original space (when n is large enough). We argue that for a certain range of values of parameters n and k, (1) the family of these invariants 'sees' information not detected by
the VR persistence diagrams of the whole space and (2) computing these invariants is significantly easier than computing the usual VR persistence diagrams. We establish stability
results for our persistence sets and also precisely characterize some of them in the case of spheres with geodesic and Euclidean distances. We identify a rich family of metric graphs
for which the invariant determined by n=4 and k=1 fully recovers their homotopy type. Along the way we prove some novel properties of VR persistence diagrams.
☆ [05121] Learning with persistence diagrams
○ Format : Talk at Waseda University
○ Author(s) :
■ Jose Perea (Northeastern University)
■ Iryna Hartsock (University of Florida)
■ Alex Elchesen (Colorado State University)
■ Tatum Rask (Colorado State University)
○ Abstract : Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon
measures supported on the birth-death plane and endowed with an optimal transport distance. Examples of such measures are expectations of probability distributions on the space of
persistence diagrams. In this talk, I will present methods for approximating continuous functions on the space of Radon measures supported on the birth-death plane, as well as their
utilization in supervised learning tasks.
□ 00533 (2/2) : 1D @F412 [Chair: Petrache Mircea]
☆ [04676] Persistent cycle registration and topological bootstrap
○ Format : Online Talk on Zoom
○ Author(s) :
■ Yohai Reani (Viterbi Faculty of Electrical Engineering, Technion - Israel Institute of Technology)
■ Omer Bobrowski (Viterbi Faculty of Electrical Engineering, Technion - Israel Institute of Technology)
○ Abstract : In this talk we present a novel approach for comparing the persistent homology representations of two spaces (filtrations) directly in the data space. We do so by defining
a correspondence relation between such representations and devising a method, based on persistent homology variants, for its efficient computation. We demonstrate our new framework in
the context of topological inference, where we use statistical bootstrap-like methods to differentiate between real phenomena and "noise" in point cloud data.
☆ [05127] The Density Fingerprint of a Periodic Set and Persistent Homology
○ Format : Online Talk on Zoom
○ Author(s) :
■ Herbert Edelsbrunner (Institute of Science and Technology Austria)
■ Teresa Heiss (Institute of Science and Technology Austria)
■ Vitaliy Kurlin (University of Liverpool)
■ Philip Smith (University of Liverpool)
■ Mathijs Wintraecken (Institute of Science and Technology Austria)
○ Abstract : Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions. The density fingerprint is invariant under isometries, continuous, and
complete in the generic case, which are necessary features for reliable comparison of crystals. The fingerprint has a fast algorithm based on Brillouin zones and related
inclusion-exclusion formulae, which we have implemented. I will discuss the connection with persistent homology, suggesting a possible extension of the fingerprint.
☆ [04866] Reconstruction of manifolds from point clouds and inverse problems
○ Format : Talk at Waseda University
○ Author(s) :
■ Matti Lassas (University of Helsinki)
■ Charles Fefferman (Princeton University)
■ Sergei Ivanov (Steklov Institute of Mathematics)
■ Hariharan Narayanan (Tata Institute for Fundamental Research)
■ Jinpeng Lu (University of Helsinki)
○ Abstract : We consider a geometric problem on how a Riemannian manifold can be constructed to approximate a given discrete metric space. This problem is closely related to invariant
manifold learning, where a Riemannian manifold $(M,g)$ needs to be approximately constructed from the noisy distances $d(X_j,X_k)+\eta_{jk}$ of points $X_1,X_2,\dots,X_N$, sampled
from the manifold $M$. Here, $d(X_j,X_k)$ are the distance of the points $X_j,X_k\in M$ and $\eta_{jk}$ are random measurement errors. The values $d(X_j,X_k)$ can be considered as
distance fingerprints of the manifold $M$. We also consider applications of the results in inverse problems encountered in medical and seismic imaging. In these problems, an unknown
wave speed in a domain needs to be determined from indirect measurements. Moreover, we discuss a problem analogous to the above one, where distances are measured from points in a
small subset $U\subset M$ to points in a discrete subset of $M$ and the errors are deterministic.
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How to use the Excel COUNTIF function - FormulasHQ
Table of Contents
The powerful COUNTIF function in Excel counts cells based on one criteria. This page contains many easy to follow COUNTIF examples.
Use the COUNTIF function in Excel to count cells that are equal to a value count cells that are greater than or equal to a value etc.
1. The COUNTIF function below counts the number of cells that are equal to 20.
2. The following COUNTIF function gives the exact same result.
3. The COUNTIF function below counts the number of cells that are greater than or equal to 10.
4. The following COUNTIF function gives the exact same result.
Explanation: the & operator joins the ‘greater than or equal to’ symbol and the value in cell C1.
5. The COUNTIF function below counts the number of cells that are not equal to 7.
6. The COUNTIF functions below count the number of cells that are equal to 3 or 7.
Use the COUNTIF function in Excel and a few tricks to count cells that contain specific text. Always enclose text in double quotation marks.
1. The COUNTIF function below counts the number of cells that contain exactly star.
2. The COUNTIF function below counts the number of cells that contain exactly star + 1 character. A question mark (?) matches exactly one character.
3. The COUNTIF function below counts the number of cells that contain exactly star + a series of zero or more characters. An asterisk (*) matches a series of zero or more characters.
4. The COUNTIF function below counts the number of cells that contain star in any way.
5. The COUNTIF function below counts the number of cells that contain text.
Use the COUNTIF function in Excel to count Boolean values (TRUE or FALSE).
1. The COUNTIF function below counts the number of cells that contain the Boolean TRUE.
2. The COUNTIF function below counts the number of cells that contain the Boolean FALSE.
Use the COUNTIF function in Excel to count specific errors.
1. The COUNTIF function below counts the number of cells that contain the #NAME? error.
2. The array formula below counts the total number of errors in a range of cells.
Note: finish an array formula by pressing CTRL + SHIFT + ENTER. Excel adds the curly braces {}. In Excel 365 or Excel 2021 finish by simply pressing Enter. You won’t see curly braces. Visit our page
about Counting Errors for detailed instructions on how to create this array formula.
Counting with And criteria in Excel is easy. The COUNTIFS function (with the letter S at the end) in Excel counts cells based on two or more criteria.
1. For example to count the number of rows that contain Google and Stanford simply use the COUNTIFS function.
Counting with Or criteria in Excel can be tricky.
1. The COUNTIF functions below count the number of cells that contain Google or Facebook (one column). No rocket science so far.
2. However if you want to count the number of rows that contain Google or Stanford (two columns) you cannot simply use the COUNTIF function twice (see the picture below).
Note: rows that contain Google and Stanford are counted twice but they should only be counted once. 4 is the answer we are looking for.
3. The array formula below does the trick.
Note: finish an array formula by pressing CTRL + SHIFT + ENTER. Excel adds the curly braces {}. In Excel 365 or Excel 2021 finish by simply pressing Enter. You won’t see curly braces. Visit our page
about Counting with Or Criteria for instructions on how to create this array formula.
The COUNTIF function is a great function. Let’s take a look at a few more cool examples.
1. The COUNTIF function below uses a named range. The named range Ages refers to the range A1:A6.
2. The COUNTIF function below counts the number of cells that are less than the average of the ages (32.2).
3. To count cells between two numbers use the COUNTIFS function (with the letter S at the end).
4. Use the COUNTIF function to count how many times each value occurs in the named range Ages.
Note: cell B2 contains the formula =COUNTIF(AgesA2) cell B3 =COUNTIF(AgesA3) etc.
5. Add the IF function to find the duplicates.
Tip: use COUNTIF and conditional formatting to find and highlight duplicates in Excel.
The COUNTIF function can’t count how many times a specific word occurs in a cell or range of cells. All we need is a little magic!
1. The formula below counts how many times the word “dog” occurs in cell A1.
2. The formula below counts how many times the word “dog” occurs in the range A1:A2.
Note: visit our page about counting words to learn more about these formulas.
Join over 1 million Excel learners by following Excel Easy on Facebook X and LinkedIn.
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Sasaki's rhythm
By Osamu Sasaki **, Jacques Martinerie ***, Michel Le Van Quyen ***,
Pierre-Marie Gagey *
* Institut de Posturologie, Paris
** Shinshu University, Matsumoto
*** LENA Salpêtrière, Paris
The body of standing man is built as an inverted pendulum possessing a great number of degrees of freedom. We observe that it can lock, or not, a more or less big number of these degrees of freedom
to control its upright position at rest, what represents a possibility of very numerous postural tactics, in theory.
Now, the investigation of these various tactics by stabilometry is never made practically, as far as we knew, although we had two types of analysis to study them: analysis of shear forces and
analysis of the rhythm of centre of pressure movements.
Indeed, the period of a pendulum being a function of its length, if the length of human pendulum is divided by unlocking some of its degrees of freedom, then the period of each of the parts is
inevitably lower than that of the whole, the pendum with a single degree of freedom.
We put forward that these modifications of periodicity can be read on the recordings of the centre of pressure by studying the rhythm of the 'dance' movements of this application point of the
reaction forces. We suggest naming this rhythm: " Sasaki's rhythm ". [By rhythm of a phenomenon, we mean the series of intervals of time, which separate similar events of this phenomenon.]
As locking/unlocking degree of freedom is not considered as an aleatory phenomenon of motor control (Bernstein, on 1947), this hypothesis includes:
1) That the rhythm of the dance of the centre of pressure is stationary,
2) That it can be modified in certain situations,
3) That these modifications can be considered as a change of postural tactics as far as one notices that they allow to react effectively to disturbances.
We show that these three conditions were verified by the experience of Sasaki and coll. (2002).
Material and Method
Recordings :
23 subjects (3 Women and 20 Men) were recorded on a standardised stabilometry platform of the French Association of Posturologie (Bizzo et al., 1985; AFP, 1985; Gagey et al., 2001) in three series,
randomised, of two successive recordings in the following situations:
Eyes open /Eyes open
Eyes open /Eyes closed
Eyes open / Eyes open ± optokinetic stimulation
The optokinetic stimulation (OKN) was always the same: 60 °/s, rightwards, during 60 seconds, not trying to follow a target. Ten minutes rest between every series.
Signal Analysis
Global parameters: Five global parameters were calculated: The area of the 90 % confidence ellipse (Takagi and al ., 1985), the length of the stabilogram in X, in Y, in XY, the Romberg's quotient .
Structural parameters: The similitude coefficient of Le Van Quyen and coll. (1999) was calculated in each of the three series of recordings, by comparing the stabilograms in each serie:
EO and EO,
EO and EC,
EO ± OKN,
In X and in Y.
The similitude coefficient expresses, by means of an intercorrelation, the degree of resemblance between two clouds of points, built in a phase space of 8 dimensions (Fig. 1), from the data of every
recording, every point of a cloud being defined by a vector of eight parameters, every successive parameter of which represents the duration of the interval of time, which separate two identical
events of the stabilogram (Fig. 2), in X and in Y.
FIG. 1 - Projection on a three dimensions space of the clouds, the similitude coefficient of which are studied in a phase space of 8 dimensions.
FIG. 2 - Construction of the elementary vectors.
Every elementary vector defining the 8 coordinates of every point of a cloud is constituted by the duration of the 8 time intervals (n, n+1. n+7) which separate 9 identical events of the stabilogram,
here the passage by a maximum.
│Situations│Similitude in X │Similitude in Y │
│EO/EO │0.859±0.121 │0.926±0.123 │
│EO/EC │0.929±0.100 │0.892±0.101 │
│EO±OKN │0.755±0.272 │0.804±0.197* │
TAB. 1 - Averages and standard-deviations of the similitude coefficient according to the compared situations
The similitude coefficient does not present significant difference, in X and in Y, between two successive recordings in EO/EO situations, and between two successive recordings in EO/EC situations.
On the other hand, the similitude coefficient presents a significant difference, in Y, between two successive recordings realised in EO±OKN situations (t=2,59; p < 0,05), and the distribution of the
similitude coefficient in X, in the comparison of the EO±OKN situations, presents a variance statistically very different (F=5,83; p < 0.01) from that of its distribution in the comparison of the EO/
EO situations.
FIG. 3 - Distribution of the similitude coefficient in X, in comparisons of EO/EO and EO±OKN situations .
The variances of these two distributions are statistically very different (F=5,83)
In the comparison EO±OKN, 5 subjects are outside the limits of normality defined by the comparison EO/EO.
FIG. 4 - Distribution of the similitude coefficient in Y, in the comparisons of EO/EO and EO±OKN situations.
In the comparison EO±OKN, 7 subjects are outside the limits of normality defined by the comparison EO/EO.
On the whole, 11 subjects out of 23 modified their Sasaki's rhythm under optokinetic stimulation, either in X, or in Y, or in X and in Y.
All the subjects reported a sensation of vection during the optokinetic stimulation.
Stationnarity: Given that there is no statistically significant difference of the similitude coefficient, not only in the comparisons of EO/EO situations, in X and in Y, but also in the comparisons
of EO/EC situations, in X and in Y, one can admit that Sasaki's rhythm, according to this experience, would be stationary.
Variability: Given that there is a statistically significant difference of the similitude coefficient in Y between the comparisons of EO/EO and EO±OKN situations, one notices, at least in this
experience, that Sasaki's rhythm is modifiable according to compulsory situations for postural control.
Tactics: When one studies the global parameters of the 11 subjects, which modify their Sasaki's rhythm under optokinetic stimulation, one observes identical postural performances, while the 12
subjects, which do not modify their Sasaki's rhythm under optokinetic stimulation present significantly degraded postural performances (Tab.2).
So, the change of Sasaki's rhythm appears as a tactics used to guarantee postural performances in situations that compromise the stability of the subject.
│Sasaki's rhythm │N │ Area (mm^2) │ p │
│ │ │ +OKN │ -OKN │ │
│Not modified │12│150±110│300±200│< 0,05│
│Modified │11│180±90 │220±90 │ ns │
N Area (mm2) p
+OKN -OKN
Not modified 12 150±110 300±200 < 0,05
Modified 11 180±90 220±90 ns
TAB 2 - Averages and standard-deviations of statokinesigram area in EO±OKN situations as the subjects modify or not their Sasaki's rhythm.
One can notice additionally that the 11 subjects which modify their Sasaki's rhythm under optokinetic stimulation behave as blind persons in the EO/EC series. The postural "blindness" in the EO/EC
series and the change of Sasaki's rhythm in the EO±OKN series indicate, in two different manners, that these subjects integrate the visual information differently.
The rhythm of the dance of the centre of pressure, or "Sasaki's rhythm" as we suggest calling it, can be studied by a technique of nonlinear dynamic analysis, coherent with the notlinear dynamic
nature of the postural control (Martinerie and al ., on 1992; Myklebust and al ., on 1995; Thomasson, on 1995; Cao and al ., on 1998; Murata and al ., on 1998; Micheau and al ., on 2001; Sasaki and
al ., on 2001; Peng and al ., on 2002; Shimizu and al ., on 2002).
This study shows that Sasaki's rhythm is stationary, modifiable when the postural control is perturbed by particular situations, and that this modification of Sasaki's rhythm guarantees the
preservation of good postural performances.
We put forward that modifications of this rhythm reflect the locking/unlocking of degrees of freedom of the human being inverted pendulum.
A.F.P. (1985) Normes 85. Editées par l'Association Posture et Équilibre, 66, rue de Lisbonne 75008 Paris.
Bernstein N. (1947) On the construction of movements. Medgiz (Moscow)
Bizzo G., Guillet N., Patat A., Gagey P.M. (1985) Specifications for building a vertical force platform designed for clinical stabilometry. Med. Biol. Eng. Comput., 23: 474-476.
Gagey PM., Ouaknine M., Sasaki O. (2001) Pour manifester la dynamique de la stabilisation: la plate-forme AFP40/16. In M. Lacour (Ed.) Posture et équilibre. Nouveautés 2001, conceptuelles,
instrumentales et cliniques. Solal, Marseille, 95-106.
Le Van Quyen M., Martinerie J., Baulac M., Varela F. (1999) Anticipating epileptic seizure in real time by nonlinear analysis of similarity between EEG recording. Neuroreport 10:2149 2155
Martinerie J., Gagey P.M. (1992) Chaotic analysis of the stabilometric signal. In M.Woollacott & F. Horak (Eds) Posture and gait: control mechanisms. University of Oregon Books (Portland), Tome I:
Micheau P., Kron A., Bourassa P. (2001) Analysis of human postural stability based on the inverted pendulum model with time-delay in feedback. Proceedings of the 2001 american control conference,
1-6, 2297-2298.
Murata A., Iwase H. (1998) Chaotic analysis of body sway. Proceedings of the 20th annual international conference of the ieee engineering in medicine and biology society, 20, PTS 1-6 20 (Pt 1-6) ;
Myklebust J.B., Prieto T., Myklebust B. (1995) Evaluation of nonlinear dynamics in postural steadiness time series. An. Biomed. Engn., 23: 711-9.
Peng C.K., Mietus J.E., Liu Y.H., Lee C., Hausdorff J.M., Stanley H.E., Goldberger A.L., Lipsitz L.A. (2002) Quantifying fractal dynamics of human respiration: Age and gender effects. Annals of
biomedical engineering, 30, 5, 683-692.
Sasaki O., Usami S-I, Gagey P.M., Martinerie J., Le Van Quyen M., Arranz P. (2002) Role of visual input in nonlinear postural control system. Ex. Brain Res., 147:1 7
Shimizu Y., Thurner S., Ehrenberger K. (2002) Multifractal spectra as a measure of complexity in human posture. Fractals-complex geometry patterns and scaling in nature and society, 10, 1, 103-116.
Takagi A., Fujimura E., Suehiro S. (1985) A new method of statokinesigram area measurement. Application of a statistically calculated ellipse. In Igarashi M., Black F.O. (Eds) Vestibular and visual
control of posture and locomotor equilibrium. Karger (Basel): 74-79.
Thomasson N. (1995) Traitement du signal stabilométrique par les techniques d analyses non linéaires. Rapport LENA, Salpêtrière, Paris.
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Python Comparison Operators - Explained with examples
Comparison operator is used to compare the values of two variables and based on that it will provide you the result either as true if the condition satisfied or false if the condition is not
Here at LinuxAPT, we shall look into Python comparison operators and how to use them to compare two values.
What are the Comparison operators in Python ?
There are total 6 different comparators operators that are mentioned below which will be discussed in this article along with the syntax and examples. In other terms they are also known as relational
operators because they are used to find the relation between two different variables.
1. Greater than (>)
The "greater than" operator is one of Python's comparison operators that is used to compare the values of two variables by putting a ">" between them. If the value of the variable which is on the
left side is greater than on the right side, then the output is true otherwise it will be false.
For Example:
x = 10
y = 5
x > y
x = 5
y = 10
x > y
In the above example of Greater than operator we are using two variable "x" and "y" and assigning them values as x = 10 and y = 5. When the value of x is greater than the value of y the result is
True otherwise it will show False in x = 5 and y = 10 case.
2. Less than (<)
The "Less than" operator is one of Python's comparison operators that is used to compare the values of two variables by putting a "<" between them. If the value of the variable which is on the left
side is less than on the right side, then the output is true otherwise it will be false.
For Example:
x = 5
y = 10
x < y
x = 10
y = 5
x < y
As you can see example of less than operator, we are using two variable “x” and “y” and assigning them values x = 5 and y = 10. When the value of x is less than the value of y the result is True
otherwise it will show False in case of x = 10 and y = 5.
3. Greater than or equal to (>=)
This operator combines two different operators together which are the greater than ">" and equals to "=". This means that if the value of the variable is either greater or equals to the value of the
variable which is on the right side then the value will be true otherwise it will be false.
For Example:
x = 10
y = 5
x >= y
x = 10
y = 10
x >= y
x = 5
y = 10
x >= y
In above example of "greater than or equal to" since the values of variables x and y are equal, the answer returned is True or when value of x is greater than the value y, otherwise it returns False.
As a practice, execute the above code with just the "greater-than" operator and see the answer.
4. Less than or equal to (<=)
The less-than equal-to operator combines the equal-to and less-than comparison operators in Python. The operator "less than equal to" return True if the value on the left right-hand side is smaller
than or equal to the value on the right.
For Example:
x = 5
y = 10
x <= y
x = 10
y = 10
x <= y
x = 10
y = 5
x <= y
In above example of "less than or equal to" since the values of variables x and y are equal, the answer returned is True or when value of x is less than the value of y, otherwise it returns False. As
a practice, execute the above code with just the "less-than" operator and see the answer.
5. Equal to (==)
The equal to operator in Python returns True only if two variables under consideration are equal, otherwise the result is False. Two equal marks, i.e., "==", signify the "equal to" operator.
For Example:
x = 10
y = 10
x == y
x = 5
y = 10
x == y
As expected, since x is equal to y (with values 10 and 10), the answer returned is True. Otherwise, it returns False in case of x = 10 and y = 5.
6. Not Equal to (!=)
The not equal to operator in Python returns True only if two variables under consideration are not equal, otherwise the result is False. "!=", signify the "not equal to" operator.
For Example:
x = 10
y = 5
x != y
x = 10
y = 10
x == y
Now in this example of "not equal to" operator in python only return True when the value of x is not equal to value of y. However, when both the value of x and y are same it will return False.
[Need help in fixing Python issues ? We can help you. ]
This article covers Python Comparison Operators explained with examples. In fact, Python Comparison Operators compare two operands and return a boolean value based on the comparison made.
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Nonsense arguments for building a bigger particle collider that I am tired of hearing (The Ultimate Collection)
I know you’re all sick of hearing me repeat why a larger particle collider is currently not a good investment. Trust me, I am sick of it too. To save myself some effort, I decided to collect the most
frequent arguments from particle physicists with my response. You’ve heard it all before, so feel free to ignore.
1. The “Just look” argument.
This argument goes: “We don’t know that we will find something new, but we have to look!” or “We cannot afford to not try.” Sometimes this argument is delivered with poetic attitude, like: “Probing
the unknown is the spirit of science” and similar slogans that would do well on motivational posters.
Science is exploratory and to make progress we should study what has not been studied before, true. But any new experiment in the foundations of physics does that. You can probe new regimes not only
be reaching higher energies, but also by reaching higher resolution, better precision, bigger systems, lower temperatures, less noise, more data, and so on.
No one is saying we should stop explorative research in the foundations of physics. But since resources are limited, we should invest in experiments that bring the biggest benefit for the projected
cost. This means the higher the expenses for an experiment, the better the reasons for building it should be. And since a bigger particle collider is presently the most expensive proposal on the
table, particle physicists should have the best reasons.
“Just look” certainly does not deliver any such reason. We can look elsewhere for lower cost and more promise, for example by studying the dark ages or heavy quantum oscillators. (See also point 18.)
2. The “No Zero Sum” argument.
“It’s not a zero sum game,” they will say. This point is usually raised by particle physicists to claim that if they do not get money for a larger particle collider, then this does not imply a
similar amount of money will go to some other area in the foundations of physics.
This argument is a badly veiled attempt to get me to stop criticizing them. It does nothing to explain why a particle collider is a good investment.
3. Everyone gets to do their experiment!
This usually comes up right after the No-Zero-Sum-argument. When I point out that we have to decide what is the best investment into progress in the foundations of physics, particle physicists claim
that everyone’s proposal will get funded.
This is just untrue.
Take the Square Kilometer Array as an example. Its full plan is lacking about $1 billion in funding and the scientific mission is therefore seriously compromised. The FAIR project in Germany likewise
had to slim down their aspirations because one of their planned detectors could not be accommodated in the budget. The James Webb Space telescope just narrowly escaped a funding limitation that would
have threatened its potential. And that leaves aside those communities which do not have sufficient funding to even formulate proposals for large-scale experiments. (See also point 19.)
Decisions have to be made. Every “yes” to something implies a “no” to something else. I suspect particle physicists do not want to discuss the benefit of their research compared to that of other
parts of the foundations of physics because they know they would not come out ahead. But that is exactly the conversation we need to have.
4. Remember the Superconducting Super Collider!
Yes, the Superconducting Super Collider (SSC). I remember. The SSC was planned in the United States in the 1980s. It would have reached energies somewhat exceeding that of the Large Hadron Collider,
and somewhat below that of the now planned Future Circular Collider.
Whatever happened to the SSC? What happened is that the estimated cost ballooned from $5.3 billion in 1987 to $10 billion in 1993, and when US congress finally refused to eat up the bill, particle
physicists collectively blamed Phillip Anderson. Anderson is a Nobel Prize winning condensed matter physicist who testified before the US congress in opposition of the project, pointing out that
society doesn’t stand much to benefit from a big collider.
While Anderson’s testimony certainly did not help, particle physicists clearly use him as a scapegoat. Anderson-blaming has become a collective myth in their community. But historians largely agree
the main reasons for the cancellation were: (a) the crudely wrong cost estimate, (b) the end of the cold war, (c) the lack of international financial contributions, and (d) the failure of particle
physicists to explain why their mega-collider was worth building.
Voss and Koshland, in a 1993 Editorial for Science
, summed the latter point up as follows:
“That particle physics asks questions about the fundamental structure of matter does not give it any greater claim on taxpayer dollars than solid-state physics or molecular biology. Proponents of
any project must justify the costs in relation to the scientific and social return. The scientific community needs to debate vigorously the best use of resources, and not just within specialized
subdisciplines. There is a limited research budget and, although zero-sum arguments are tricky, researchers need to set their own priorities or others will do it for them.”
Remember that?
5. It is not a waste of money
This usually refers to
attempted estimate to demonstrate that the LHC has a positive return on investment. That may be true (I don’t trust this estimate), but just because the LHC does not have a negative return on
investment does not mean it’s a good investment. For this you would have to demonstrate it would be difficult to invest the money in a better way. Are you sure you cannot think of a better way to
invest $20 billion to benefit mankind?
6. The “Money is wasted elsewhere too” argument.
The typical example I hear is the US military budget, but people have brought up pretty much anything else they don’t approve of, be that energy subsidies, MP salaries, or –
as Lisa Randall recently did
– the US government shutdown.
This argument simply demonstrates moral corruption: The ones making it want permission to waste money because waste of money has happened before. But the existence of stupidity does not justify more
stupidity. Besides that, no one in the history of science funding ever got funding for complaining they don’t like how their government spends taxes.
The most interesting aspect of this argument is that particle physicists make it, even make it in public, though it means they basically admit their collider is a waste of money.
7. But particle physicists will leave if we don’t build this collider.
Too bad. Seriously, who cares? This is a profession almost exclusively funded by taxes. We don’t pay particle physicists just so they are not unemployed. We pay them because we hope they will
generate knowledge that benefits society, if not now, then some time in the future. Please provide any reason that continuing to pay them is a good use of tax money. And if you can’t deliver a
reason, I full well think we can let them go, thank you.
8. But we have unsolved problems in the foundations of physics.
This argument usually refers to the hierarchy problem, dark matter, dark energy, the baryon asymmetry, quantum gravity, and/or the nature of neutrino masses.
The hierarchy problem is not a problem, it is an aesthetic misgiving. For the other problems, there is no reason to think a larger collider would help solving them.
I have explained this extensively elsewhere and don’t want to go into the question what problems make promising research directions here. If you want more details, read eg
or my
9. So-and-so many billions is only such-and-such a tiny amount per person per day.
I have no idea what this is supposed to show. You can do the same exercise with literally any other expense. Did you know that for as little a tenth of a Cent per year per person I could pay my grad
10. Tim Berners-Lee invented the WWW while employed at CERN.
By the same logic we should build patent offices to develop new theories of gravitation.
11. It may lead to spin-offs.
The example they often bring up is
contributions to WiFi technology that originated in some astrophysicists’ attempt to detect primordial black holes
In response, allow me to rephrase the spin-off-argument: Physicists sometimes don’t waste all money invested into foundational research because they accidentally come across something that’s actually
useful. That wasn’t what you meant? Well, but that’s what this argument says.
If these spin-offs are what you are really after, then you should invest more into data analysis or technology R&D, or at least try to find out which research environments are likely to benefit
spin-offs. (
It is presently unclear how relevant serendipity is to scientific progress.
) Even in the best case this may be an argument for basic research in general, but not for building a particle collider in particular.
12. A big particle collider would benefit many tech industries and scientific networks.
Same with any other big investment into experimental science. It is not a good argument for a particle collider in particular.
13. It will be great for education, too!
If you want to invest into education, why dig a tunnel along with it?
14. Knowledge about particle physics will get lost if we do not continue.
We have scientific publications to avoid that. If particle physicists worry this may not work, they should learn to write comprehensible papers. Besides, it’s not like particle physicists would have
no place to work if we do not build the next mega-collider.
There are more than a hundred particle accelerators in the world
; the LHC is merely the largest one. Also note that the LHC is not the only experiment at CERN. So, even if we do not build a larger collider, CERN would not just close down.
15. Highly energetic particle collisions are the cleanest way to measure the physics of short distances.
I tend to agree. This is what originally sparked my interest in high energy particle physics. But there is currently no reason to think that the next breakthroughs wait on shorter distances. Times
change. The year is 2019, not 1999.
16. Lord Kelvin also said that physics was over and he was wrong
Yeah, except that I am the one saying we could do better things with $20 billion than measuring the next digits of some constants.
17. Particle accelerators are good for other things.
The typical example is that beams of ions can treat certain types of cancer better than the more common radiation therapies. That’s great of course, and I am all in favor of further developing this
technology to enable the treatment of more patients, but this is an entirely different research avenue than building a larger collider.
18. You do not know what else we should do.
Sure I do. I wrote a whole
on this: In the foundations of physics, we should focus on those areas where we have inconsistencies, either between experiment and theory, or internal inconsistencies in the theories. Examining such
inconsistencies is what has historically led to breakthroughs.
We currently have such situations in the following areas:
(a) Astrophysical and cosmological observations attributed to dark matter. These are discrepancies between theory and data which should be studied closer, until we have pinned down the theory. Some
people have mistakenly claimed I am advocating more direct detection experiments for certain types of dark matter particles. This is not so. I am saying we need better observations of the already
known discrepancies. Better sky coverage, better resolution, better stats. If we have a good idea what dark matter is, we can think of building a collider to test it, if that turns out to be useful.
(b) Quantum Gravity. The lack of a theory for quantized gravity is an internal theoretical inconsistency. We know it requires solution. A lot of physicists are not interested in experimentally
testing this because they think it is not possible. I have previously explained
why that is wrong.
(c) The foundations of quantum mechanics: The measurement postulate is inconsistent with reductionism. There is basically no phenomenological or experimental exploration of this.
Needless to say, I think my argument for how to break the current impasse is a good one, but I do not really expect everyone to just agree with it. I am primarily putting this forward because it’s
the kind of discussion we should have: We have not made progress in the foundations of physics for 40 years. What can we do about it? At least I have an argument. Particle physicists do not.
19. But you do not have any other worked-out proposals
The proposal for the FCC was worked out by a study group over 5 years, supported by 11 million Euro. Needless to say, I cannot, as a single person and in a few weeks of time, produce comparable
proposals for large scale experiments. Expecting me to do so is unreasonable.
20. But it will do all these things
Particle physicists like to point towards t
heir 716 pages report
that summarizes what they could do with the FCC. But, look, no one doubts that you can do something with $20 billion. The question is whether what you can do is worth the investment. The report does
not address this point at all.
89 comments:
1. Final sentence of 16.b) "I have previously written explained here and here why that is wrong." - I guess links are missing.
1. Thanks for pointing out, I have fixed that.
2. Hi Sabine,
I totally appreciate what you're doing, and it even opened my eyes to the systematic errors that scientists make. Please don't let the cargo cult followers silence you :)
But, as reader of your blog, I kinda miss the variety of the content that you published some time ago. Like reviews of new (or old) papers, introductions to new (and old) theories, etc.
I hope at some point you will get back to digging out such papers and theories, and presenting them...
Best regards
1. Michael,
Yes, I am aware of this :( I hope to get back to "normal" soon. I have several interesting papers I want to write about, but I am severely behind.
3. "The foundations of quantum mechanics: The measurement postulate is inconsistent with reductionism. There is basically no phenomenological or experimental exploration of this." At the start of
the Quantum Information/Computing/Communication industry, it was very much felt that such things were experimental foundations of physics, and I think they were instrumental in making QM seem
much more familiar than it felt before, say, 2000, whether we can say we now better understand measurement or not.
By now many people working on such things would hate to be thought so impractical, and therefore probably wasting $billions, but the early runners went to Foundations of QM conferences and did
care about such things. If quantum computation doesn't pan out quickly, perhaps we'll be treated to stories of how the many billions spent led to better understanding of the foundations of QM.
16. (d) The foundations of interacting QFT. We don't understand interacting QFT. [But you may remember that I'm as much a broken record on this as other people are about their enthusiasms.]
1. Peter,
Yes, you are right. I should have included QFT in that. I usually do, but somehow I forgot. My bad.
4. "By the same logic we should build patent offices to develop new theories of gravitation." Not the worst logic of the arguments considered.
5. Before you get annoyed about humanity spending money to advance fundamental knowledge consider this.. Google make $4 Billion/month from people clicking on their silly little ads. Give science a
break. Give them the money. Lets look inside the proton. Unless of course you prefer to click on ads.
1. 1, 6 and sort of 9.
2. @Richard " The Hossenfelder Scale" for measuring Crackpots is way better than The Baez's Crackpots Index ...
6. For all who are discouraged to build the FCC (or CLIC) after reading the arguments above, I recommend to read the interview with Nima Arkani-Hamed, it will cheer you up again ! Where there is
hope, there is life !
7. If anyone can flesh out just a little what Sabine means by "the measurement postulate is inconsistent with reductionism," I'd be grateful.
I assume this is a problem I've heard stated in other terms, and I'm just failing to translate it into this phrasing. My failure, not Sabine's.
1. It is fleshed out here.
2. Dave M,
The point is that we would like our measurement instruments to be describable, in principle, by quantum mechanics. In that case, the measurement process should not require an additional
assumption: all the details of the measurement process should be explained by QM without an additional measurement postulate.
If that is not so -- i.e., if the action of measurement instruments cannot be explained by QM alone -- then we are entitled to ask what novel physical process is going on in the measurement
process that is not explained by quantum mechanics.
Weinberg explained this quite clearly in Sabine's interview in her book. See also his discussion in the second edition of his Lectures on Quantum Mechanics:
"If quantum mechanics applies to everything, then it must apply to a physicist’s measurement apparatus, and to physicists themselves. On the other hand, if quantum mechanics does not apply to
everything, then we need to know where to draw the boundary of its area of validity. Does it apply only to systems that are not too large? Does it apply if a measurement is made by some
automatic apparatus, and no human reads the result?"
The ultimate issue is whether (human?) consciousness somehow is needed to bring about a true measurement. Wigner suggested just that in his famous essay in The Scientist Speculates.
Of course, if it were ever shown that consciousness is integral to the measurement process, then we would be obligated to turn our attention to understanding consciousness, which would
certainly be a change of direction for physics!
It seems reasonable that physicists should at least try to give a fully complete physical exposition of QM without invoking consciousness.
Weinberg sums up by alluding to perhaps the oddest aspect of this whole matter:
"Indeed, many physicists are satisfied with their own interpretation of quantum mechanics. But different physicists are satisfied with different interpretations."
So, if you think you know the "obvious" answer to Weinberg's questions, be aware that many physicists agree that there is an "obvious" answer, but they disagree as to what that "obvious"
answer is.
Dave Miller
3. PhysicistDave,
I totally endorse what you have written above. I guess quite a lot of non-HEP scientists feel that there is unfinished business at the level of ordinary QM, and indeed that that may be truly
As you point out, Shroedinger's equation properly applies to every part of life - not just a few particles that happen to be under study. Superficially those equations would imply a reality
consisting of an ever more entangled wave function encompassing different possible situations superimposed.
The possible relationship between QM and consciousness clearly interests Roger Penrose, so there it isn't as though this idea has been 'settled', it has just been put to one side because it
is embarrassing!
4. Physicist Dave,
It seems reasonable that physicists should at least try to give a fully complete physical exposition of QM without invoking consciousness.
QM is a mathematical method for describing the statistical outcomes of otherwise unobservable physical processes. The math neither describes nor explains those processes. Why then, should we
expect a complete physical exposition of QM (with or without consciousness)?
5. Bud Rap wrote,
"QM is a mathematical method for describing the statistical outcomes of otherwise unobservable physical processes."
That makes QM sound like classical statistical mechanics, which I think isn't fair.
First of all, QM computes the wave function, which is *not* in itself a probability distribution - not least because it can take on negative or complex values.
QM isn't creating a statistical outcome of a deeper theory (although OK it is an approximation to QFT).
You only get probabilities when you evaluate Ψ Ψ*.
Surely physics should be more than obtaining some equations that seem to describe reality, shouldn't it also provide an explanation of what it is that the maths relates to?
6. "QM isn't creating a statistical outcome of a deeper theory (although OK it is an approximation to QFT)."
Actually, it might as well be; it's just that we don't know that deeper theory yet. And I think even QFT doesn't fix that - you get a distribution over configurations of classical fields
instead than over configurations of classical point-like particles, but the 'statistical distribution' effect remains.
7. David Bailey,
That makes QM sound like classical statistical mechanics, which I think isn't fair.
At the interface between QM and observation statistics is all you get. That QM arrives there via a different set of formalisms necessitated by the peculiar circumstances of the quantum scale,
doesn't alter the analogous nature of the outcome.
Surely physics should be more than obtaining some equations that seem to describe reality, shouldn't it also provide an explanation of what it is that the maths relates to?
It certainly should! My point was only that you cannot expect to obtain reasonable physical explanations from mathematical formalisms that aren't constructed on reasonable qualitative
8. Simone said,
"Actually, it might as well be; it's just that we don't know that deeper theory yet. "
Well unless there are an infinite number of theories, each depending on the one below, the process has to stop somewhere. My gut feeling is that QM is special - it says that fundamentally we
have different possibilities (realities if you like) that evolve and interfere with each other. This feels more fundamental than particles. So I would rate QM as fundamental, and since QM
cannot coexist with GR, I'd bet that GR has to change.
8. Bee,
has Moriond 2019 found any BSM physics signals, i understand possible lepton flavor violations
1. Moriond is really only the occasion on which rumors become official. If there were any BSM breakthroughs in the data analysis done so far, we'd have heard of it by now.
2. The most interesting physics is the measurement of CP violations in decays of D0 vs bar-D0.
3. Yes that's in the popular news. has Moriond released new bounds on SUSY such as gluino's and squarks?
given Morion hasn't seen SUSY in the full data set, it seems the likelihood of a 5-sigma discovery of SUSY is low.
4. So far evidence for s-tau or s-top etc is at best around 2-sigma. It has not risen to the eyebrow raising level of 3-sigma. The most recent thing I have seen is
9. Doctor Hossenfelder,
In response 17, having pointed out that you are only one person, the criticism is not relevant because there exists a wealth of ready available alternatives already.
To suggest a few (sorry, just my personal interests): Fusion Energy; Carbon Removal from the Atmosphere; Efficient Storage of Renewable energy sources during times of over-production; higher
temperature superconductivity; Neurobiological Research; Cognitive and neurological health; Structures encouraging responsibility and objectivity in leadership.
1. I don't think diverting (even more) funds from the foundations of physics research into engineering research (and a bit of biology and medical sciences) is the right way to go (and I don't
think that's what Sabine proposes, I trust she'll correct me if I misinterpreted her). Those $20 billion should stay in the same field of research, but funding 5-100 promising experiments
instead of one mega-project with few to no chances of getting a breakthrough. Or even a different huge project if you have the justification.
Biology, biomedicine and engineering are already attractive research fields for which funding, private and public, is *relatively* easy to come by. Physics (specially foundations) is
extremely hard to sell to the public and the chances of private funding are close to nil. Please, do not advocate for moving funds away from physics, we *need* physics research.
2. Javier,
Sabine has never seemed to me to suggest diverting funds from physics research. She presents arguments that, in upgrading the LHC, these funds are not being allocated for convincing
objectives. Intelligent probing of the unknown, including in the field of theoretical physics, should always be supported. So should building on existing knowledge to directly address massive
known problems. Tax supported funding is not unlimited; worthy ideas in all fields die daily for their lack. No single individual can be expected to develop programs which solve all the
associated problems. (17.) In suupporting arguments for upgrading the LHC by related developments.in applied science, e.g. in superconducting magnets, the question simply arises whether the
known value of advancing applied science should be more directly supported until physics offers programs with a higher probability of definitive results than the LHC. jmo.
Bert Kortegaard
3. Yes, I'm aware Sabine wasn't suggesting that; you were, though. In my experience, Applied Science is just a fancy way of saying engineering research and, as I said, I don't think we should
transfer money from the much-in-need-of-funding foundations of physics into the bad-but-still-not-nearly-as-bad field of engineering research. Superconducting magnets are being actively
researched by public and private interests (plenty of direct applications) and although you can always use more funding, they have plenty of opportunities to get it (same with your other
proposals). Foundations of physics (QFT, Cosmology, Quantum Gravity, etc) get nearly 0 funding from the private sector because of their lack of immediate applicability and, because of the
obscurity of the topics, it's also a hard sell to the public (at the risk of being wrong, I'm guessing they are the worst funded field within the natural sciences; probably only social
scientists envy them). That's why while I agree that we should fund something else, I believe the funds should stay in the field. And for full disclosure, I say this precisely from the point
of view of someone who does engineering research for a living... in the private sector. Find a theoretical physicist who can say the same (and is still doing fundamental research).
4. Javier, thanks for your comments.
I thought what I was suggesting was obvious from what I wrote, but I apologize to anyone who misinterpreted it as you have.
Applied Science starts where science is understood well enough to build on it to produce useful things. At its most interesting it includes developing new techniques and tools, but ithose of
us who practice it do not ordinarily describe that as research.
My blog includes a link to some of my own work in this field.
Lest this should become off-topic, my blog also contains my email.
10. "...Google make $4 Billion/month from people clicking on their silly little ads. Give science a break......"
God I hope that asinine comment is an attempt at humour..but I have a feeling its not..
11. On "what novel physical process is going on in the measurement process" I've always assumed it was some sort of Darwinian-like selection-of-fittest-history (in a sum-over-histories formulation of
QM). But this process is apparently an additional "postulate" to QM.
12. I love your blog and totally agree that a "wrapping up" of this discussion was due.
For that reason, I would suggest a change in argument 6:
"With it, THESE particle physicists...
"That THE particle physicists MAKING it ..."
Only the ones making the argument suffer from moral corruption. Many others just think it isn't a waste of money, they just have a different opinion (generalization).
It may help avoiding unwanted 'rants'
1. Ward,
I think this is clear from the context, but I nevertheless changed that sentence along the line you suggest.
13. Me as taxypayer I think we should not spend billions of € for a even bigger collider - instead we should invest money in exploring and pondering, where we failed in our beautiful
Taka-Tuka-theories during the last half century and consider new ways of thinking aubout the fundamental laws of physics!
14. As to point 7, maybe NASA's space launch system could use the extra physicists if no new collider is built. They could move from one project with no results to another that is building a rocket
that will never launch, because the important thing is to have jobs in all fifty states, not actually get anything done. As Rep. Aderholt said about SLS ""The SLS and Orion programs are, of
course, key to the health of our national aerospace supplier base, and it's really helped to really put a new boost of energy into the suppliers in all the 50 states following the retirement of
the space shuttle,"
15. Bee,
do these arguments in these post apply to HE-LHC with 16 tesla magnets, a estimated ~7 billion upgrade to LHC 8.33 tesla magents in its 27 km tunnel?
I would argue that for the price tag, exploring between 14 TEV to 27 TEV for new physics is certainly a justified upgrade.
i wonder whether it'd be better to simply forget about HL-LHC and instead invest that money into HE-LHC.
and by the time 16 tesla magnets are ready, perhaps 24 tesla or even 32 tesla magnets will be in development.
so no new tunnel will be built, the 27 km is reused, but super conducting magnet technology is improved over decades.
16. "IF" dark matter is made of particles that only interact through gravity, how can you study it if not by missing energy momentum of high energy collisions?
1. @Daniel de França MTd2; Dark Matter necessarily gravitates with other matter; it can be studied astronomically; through gravitational lensing and perhaps by studying galaxy dynamics in a wide
range of galaxy sizes, or a range of galaxy proximities. What's happening with the dark matter in galactic collisions?
Let's build a $20B super high resolution space telescope, or 20 $1B telescopes we can gang together in an array. Let's study it.
2. Dr Castaldo,
Yes but - there's always a but!
The recent paper "Probing dark matter particles at CEPC" by Zuowei Liu and colleagues illustrates the possibility of using high energy colliders to investigate various Dark Matter models.
The point being that thorough investigation of a phenomenon requires multiple lines of attack. This means making the best of the available options - which are often not mutually exclusive.
Will collider funding be diverted to astronomy? There's currently no reason to suggest this would be the case.
3. Dr Castaldo
This is like studying electrons with circuits. You won't be able to infer what dark matter is, but, just its collective properties. That is, you will just know what a current looks like. You
won't get insight of what is dark matter.
17. Hi Sabine,
you state: The hierarchy problem is not a problem.
Maybe, but if you find a solution you sure will have surprises - surprises that the current foundations may not survive.
18. "The measurement postulate"
I feel like the justification to question the "Copenhagen interpretation" (you know the one they still teach undergrads) has been around and readily accessible for at least 8 years (https://
www.youtube.com/watch?v=dEaecUuEqfc). The problem seems to be that none of the alternative hypotheses (can we call them that?) have been able to gather the doubters together and gain traction.
This business of questioning weather "consciousness" is required for things to be "measured" always seemed daft to me, isn't "superposition" a statement about the correlation or non-correlation
of two quantum system not a statement about a single quantum system? ie until I correlate my detector with the superimposed system (by shooting lasers between them I guess would be typical) then
the detector isn't 'touching' /hasn't 'touched' the other system and just doesn't contain information about the superimposed system yet? So there is never a funny magic state there is just a
situation where two systems don't currently share any information so querying ether of them about the other is nonsensical till you 'connect' the two systems (fire the lasers, take the
measurement, open the box, throw the detector at the test article... ect)
Obviously I'm out of my depth please correct my childish simplifications you there smart physical folk! Thank you for the help...
19. On 17, "you do not know what else to do": I understand you DO know, but --- Since when is knowing the solution to a problem necessary to know that there IS a problem?
If I go to the vet because my dog is limping, I don't go there knowing what should be done about it.
Making it known that a problem exists is the first step, getting agreement on that, and detailing the nature of problem, come next. Developing a plan of attack is well down the list.
1. It is interesting to me that (re the video link above: The Quantum Conspiracy, 1,571,119 views, GoogleTechTalks) that some physicists like an "interpretation" that says "you don't really
exist". It seems to me to be a part of the curious antimaterialist turn (we are all just "information" or something like that) among physicists, at least as indicated by the current articles
published for the general reader.
2. @Philip Thrift, voices that advocate for "antimaterialism" are perhaps more shrill, but for the general reader you could try Philip Ball's "Beyond Weird", https://www.amazon.com/Beyond-Weird/
dp/1784706086/, which deflates the weirdness of QM in a way that IMO fairly accurately reflects the practical "let's use QM" perspective of working quantum computing/information, condensed
matter, and most working physicists. His Royal Institution lecture, https://www.youtube.com/watch?v=q7v5NtV8v6I, gives a fairly good sense of the position he suggests in that book.
You may already know that in philosophy anti-realism is as or more often anti-realism about theories than it is an anti-materialism of anti-realism about the world and our experience of it.
There will be some continuity between our current theories and new theories, so that electrons will exist in *some* form in future theories (with careful discussions of how the electron is
both equivalent and not quite equivalent to new concepts), but they or other concepts may be deprecated, so to speak, because other theoretical tools and concepts will be devised that are
just more effective. An absolute commitment even to such an apparently robust theoretical concept as the electron may, or may not, turn out to be ill-advised, but an appropriate slight
hesitancy to say of every part of the standard model of particle physics that it is "emphatically, finally real", does not demand any hesitancy in our belief in and engagement with the world
as a whole.
3. I have read articles about Philip Ball's book (e.g. Peter Woit's https://www.math.columbia.edu/~woit/wordpress/?p=10522), but not the book I admit. My own view has been some combination of
Path Integral (or Sum-Over-Histories) and (some version of) Quantum Darwinism: PI+QD. But that's as "real" as I get. :)
4. FWIW, the (very popular) idea that the Path Integral (a generating function for time-ordered vacuum expectation functionals) somehow makes quantum theory classical (paths!) is IMO problematic
because it uses time ordering to sweep the noncommutative algebraic structure under the table, whereas noncommutative measurements are essential for the empirical success of QM/QFT.
If you say "(some version of) QD", I take you to be invoking decoherence in some way, which one has to have formal worries about, but, as you know, it works more-or-less, and certainly for
all practical purposes.
My own view has become that QM and QFT are (stochastic) signal analysis formalisms, for which we can say, loosely, that incompatible measurements are mathematical consequences of using
classical representations of the Heisenberg algebra, which is closely connected with fourier analysis.
5. On the PI, I just follow Fay Dowker (@DowkerFay, Mar 26): "This was an enjoyable discussion. I argued that there is one world, not many, in quantum theory based on the Path integral or
Feynman sum-over-histories."
On "Darwinian" selection: Only one history survives. The others die. Poor things.
20. Hi , Sabine ��
Nice discussion.
I agree with you as to a larger collider.
-- I just find it interesting,
the references to 'Tesla'
-- (appearently) without
knowing what it was (is).
-anyway , keep up the good
-- it is good.
All Love,
21. re: "Nonsense arguments for building a bigger particle collider that I am tired of hearing (The Ultimate Collection)"
Bee, the question i have about your arguments in this post is this
CERN has earmarked several billion dollar upgrade for LHC to HL-LHC, to increase its luminosity
is the billions dollars spent to upgrade luminosity by a factor of 2 to 10 a worthwhile use of money?
what about $7 billion more to upgrade LHC to HE-LHC?
HL-LHC and HE-LHC upgrade cost billions, but reuse the same 27km tunnel.
it seems to me if we apply your arguments, we shouldn't bother upgrading the luminosity of LHC, after all, it is still going to CM of 14 TEV, and it seems a 5 sigma discovery at this point is
22. This was a great thing to read right after opening my bottle of wine :)
23. RE "What should we do?" Martin Harwit wrote a very interesting book in 1981 called "Cosmic Discovery". In it, he shows the amazing role played by serendipity in fundamental discoveries, and tries
to get some understanding of how to go forward based on what has led to the current state of knowledge. I think you would enjoy it. This post reminded me of it.
24. What do you think of the latest version of string theory called F-theory?
I think it's a four-letter word they can't say in public
25. The intense discussion suggests the collider culture has yet to be buried and given up. I have pretty good reasons to believe that we we require new ideas about such experimental research
particularly in relation to the ultimate nature of existence and of our realities. It cannot be argued that we have reached the end of all possibilities. However what I have in mind concerns the
ultimate nature of forces and particles which if knew would open out a new world of physics.
26. Sabine,
It seems to me that several of your arguments boil down to "Cost matters!", contrary to your opponents who are, in effect, arguing "No, Cost does not matter!"
I came close to majoring in economics instead of physics, and I have trouble grasping the mind-set of anyone who truly believe that cost does not matter, but this does seem to be their
Frankly, I think the subtext of your opponents' arguments is, in essence, "We high-energy physicists are just more important than other people, and doing high-energy physics is just more
important than what other people do!" No one will say this quite so bluntly, but I am not sure any of us HEP physicists are completely immune to such hubris. After all, we chose to go into HEP
because we really did think it was important.
Of course, scientists should strive for rationality and objectivity, but, obviously, we too are all-too-human!
All the best,
1. Dave,
I am not sure if they actually believe that cost does not matter or whether they just argue this way because they know it's their only chance. Either way, though, what surprises me is that
they would even make such an argument, if not explicitly, then implicitly by refusing to explain why the expenses are justified.
Well, yes, everyone thinks that their occupation is the most important. I don't blame anyone for that. But most people understand at least that others might not share that impression.
2. I find it extraordinary that fundamental physics is now utterly divorced from the rest of science, or anything that matters more widely.
HEP doesn't seem at all likely to discover a foundational truth - but it is always possible to throw yet more money at it to achieve higher energy collisions, and maybe some more 'particles'.
That process will only stop when more people like Sabine put their feet down!
27. Hi Sabine.
some of the latest tests
give credence to your argument.
( high intensity laser / mirror
trap) ...(nano particles)
,.. money can be better spent,
on smaller scales.
--. All Love,
28. Every ten years the space astronomers get together with NASA and create a new list of prioritized space missions. There is never enough money to fund everything and as science changes priorities
change and as technology changes capabilities change. It's sort of what Erdos used to do with mathematical problems, He'd assign a cash bounty, higher for the problems he thought would be most
fruitful. The problem with particle physics is that the price is getting so high, even in comparison with the costs of space missions, that funding even one item is just too expensive. No one has
been thinking about a Plan B, C or D.
My guess is that we'll start seeing the real spinoffs from the LHC when physicists start leaving the field.
29. Ms Hossenfelder,
I personally think your position against the larger particle collider is very relevant. But I don't think that your arguments can change anything, and that is why : the larger collider has become
a collective narrative of the particle physics community. Specialists call that "Intersubjective narratives", those are the root of our human society and when they have got some traction there is
no way to kill them by questioning their soundness. By the way most of them are not built on RATIONAL arguments. Think for example of the moon race in the 1960. There was no rational to make such
a costly programm without any other purpose than self proudness, but it became an intersubjective narrative of american people and as so impossible to cancel... until the mission succeeded and we
could see there wasn't anything usefull to get from it. Il you do want to prevent that project there are in my view only two ways :
1/ leave the scientists and go to the politicians who will ultimately give the money.They most probably are not in the narrative of the particle physics community and could listen to the voice of
the reason. But don't expect that the money not spent on the super collider will go in any massive way elsewhere in physics ;
2/ build another narrative on another subject and try to give it traction. To do that you have to get massive support within the physics community not just on criticizing the new collider idea
but more importantly on one and only one other project which could get most of the money that could go to the collider.
That does not seem fair to all the other good ideas which could benefit of a funding ? Yes, but life is not fair.
1. Franck,
I think what you mean by "rational" is really "scientific". I agree that there are reasons besides the scientific ones that make people spend money on large science projects. I have nothing
to say about those, so I don't. But I wouldn't call them irrational.
You seem to be misunderstanding my intention though. I am not writing to prevent something from happening. I hope to make something happen. I hope that physicists who work in the foundations
think about what has gone wrong and how to make progress. Blindly throwing money at the problem will not solve it.
You seem to expect me personally to come up with a solution and then convince people to support me. This does not make any sense. Of course I have my own convictions about what is the right
thing to do, but I don't think I should be the one making decisions. I merely want physicists to use their brain rather than blindly continuing down dead end streets.
It's not about fairness, it's about progress.
2. Franck; Sputnik was launched in October, 1957. To Americans, it was widely considered a dire threat.
Russia then put the first man in space four years later. Kennedy needed a response to a potential militarization of space; there was a perceived necessity to not let Russia seize "the high
Kennedy considered a number of potential operations, but "putting a man on the moon" before Russia did seemed the most likely to succeed, with the most inspirational content to get public
There were very rational ideas behind this program, even if the ultimate goal was just a symbolic finish line. The point was to develop the science and technology and capabilities of the
space age, to match the same being developed by a hostile power (the Cold War was 14 years old at this time), and this is what was accomplished.
There were many entirely rational reasons to "go to the moon", including the rational decision to appeal to emotions in building public support.
Because, as we Americans are currently proving, and other countries have proven time and again throughout recorded history, rationality is definitely not the primary decision making tool of
our citizens.
3. @Franck: That "life is not fair" is not an excuse for taking action to make life more unfair; the primary value of human intelligence has surely been to make us far less victims of the random
cruelties of life and nature, not to exacerbate them.
The solution to one swindle is not another swindle, it is getting people to recognize when they are being swindled.
30. With respect to the discussions on the foundations of quantum mechanics and measurement I write this below.
Probability theory for statistically independent events is L^1 in that probabilities add linearly and there are no correlations between probabilities. Quantum mechanics is L^2 in that amplitudes
add linearly, but the “distance,” or really most importantly the distance squared as probabilities, is the sum of the modulus square of amplitudes. This makes statistical mechanics or a theory
based on pure classical probability fundamentally different from quantum mechanics.
The theory of convex sets is such that for a set with measure L^p, with elements x, and another with L^q, with elements y, that Holder's norm ||x||_p×||y||_q ≥ sum_i|x_iy_i| for 1/p + 1/q = 1.
This means there is a duality between convex sets with these values of p and q defining these norms. For p = 1 this means q → ∞ and for p = 2 the dual is also q = 2. This is a part of how quantum
mechanics and spacetime, with its Gaussian metric distance, are dual to each other. The dual to pure statistical systems with q → ∞ means there are no probabilities at all and this is a
completely deterministic system such as Newtonian mechanics.
A measurement occurs where there is a decoherence of a quantum wave occurs and the trace elements of the density matrix defines a classical probability distribution. The theory of decoherence
permits us to understand how a wave function is reduced, because the superposition or entanglement phase of that system is transferred to a reservoir of states, say the needle state of a
measuring apparatus, and the system is reduced to pure probabilities. We can't really know which of these outcomes happens in some deterministic manner according to quantum mechanics.
As the dual for a p = 1 system, where the wave function reduction is a p = 2 → 1 process, has as its dual the q → ∞ convex set or hull description. Does this then mean we can use this to
understand some underlying classical type of structure to quantum measurement? We might want to be a bit conservative here. The problem is that we have convex sets that we propose are computing
quantum numbers, and in the case with a p ↔ q duality we have this idea that quantum numbers, say as the Gödel number for an integer computed by a Diophantine equation or the computed outcome of
a deterministic system, as having a single axomatic process. Hilbert's 10th problem proposed there should be a single algorithmic or axiomatic process for solving Diophantine equations
Matiyasevich found the final conclusion to a series of lemmas and theorems worked by Davis, Putnam and Robinson, called the DMPR theorem. This is a form of Gödel's theorem and the conclusion is
there is no comprehensive axiomatic system for Diophantine equations. Quantum numbers as Gödel numbers for integer solutions to Diophantine equations are then not entirely computable and there
can't exist a Turing machine (in the classical sense a q → ∞ convex set) that computes quantum outcomes.
I then maintain the solution to the quantum measurement problem is that there can't exist such a solution. It is an unsolvable problem. Quantum measurement has some features similar to
self-reference in that a quantum system is encoded by another system ultimately made of quantum states. It also has features similar to the Euclid's 5th axiom problem. One can assume the axiom
holds and stick with Euclidean flat space, or one can abandon it and work with a plethora of geometries. In QM this would be to stay with Merman’s shut up and calculate dictum, or to adopt any of
the quantum interpretations out there, which contradict each other, to augment QM in some extended way. This has features remarkably similar to the dichotomy between consistency and completeness.
1. @LawrenceCrowell, this is fine, but I suggest there is a question as to what Classical Mechanics is. Specifically, Koopman in 1931 introduced a Hilbert space formalism for CM, which can be
thought of as offering a unification of CM with QM, just as the Schrödinger equation and Heisenberg's matrices were unified as Hilbert space formalisms. In these terms, the difference between
CM and QM is mostly "just" that CM has a purely commutative algebra of measurements. Mutually noncommutative measurements do make sense for CM, however, as is well-known in signal analysis,
where Wigner functions are frequently used: one can introduce the Heisenberg group as differential operators, [j∂/∂q,q]=j, instead of as in QM as [q,p]=i. Call an extension to include all
such operators CM+. I lay out an argument that if we have a solution of the measurement problem for CM+ (using a Gibbs state over the CM algebra extended to the CM+ algebra), we also have a
solution for QM, in my https://arxiv.org/abs/1901.00526 (currently submitted to Physica Scripta): I find that a solution for CM+ is less elusive. In particular, I suggest that the specific
difficulty you outline above is eliminated by comparing CM+ with QM instead of comparing CM with QM. We don't obtain a complete unification, but it's closer than we've had.
2. I looked over your paper and down loaded it. I will have to reserve judgment until I read it sometime later, though I hope not too long into the future. It looks a bit like noncommutative
geometry of Connes et al..
The connection between quantum and classical mechanics is often stated as 1 = {q, p} → [q, p] = iħ for large action S = nħ for n → ∞. I think the most important aspect of this is that
classical mechanics is real valued and quantum mechanics is complex valued. The extension of the reals into complex numbers means probabilities are the modulus square for |ψ⟩ = sum_n c_n|n⟩
⟨ψ|ψ⟩ = sum_{mn}c^*_mc_n⟨m|n⟩ = sum_n|c_n|^2 = sum_n P_n.
Classical mechanics has none of this construction, and instead determines the value of classical variables. The correspondence between an observable Ô|n⟩ = O_n|n⟩ in quantum mechanics and
probabilities is then
⟨ ψ| Ô |ψ⟩ = sum_{mn}c^*_mc_n⟨ m| Ô |n> = sum_n|c_n|^2 = sum_n P_nO_n.
This is Born's rule, where curiously a general proof of this is not at hand. Anyway the observable occurs as eigenvalues in a distribution with probabilities. We can think of both classical
and quantum mechanics as a measure theory O_{obs} = ∫dμO, but where for classical mechanics the measure is zero everywhere except the contact manifold and the with quantum mechanics there is
this quadratic set of modulus square of amplitudes = probabilities in a summation that weights eigenvalues.
There is Gleason's theorem that tells us the linear span of a Hilbert space defines a trace that uniquely defines probabilities. Hence any measure μ(X) = Tr(WP_X) for W a positive trace
class. So this appears half way to a complete proof of Born's rule; all we need is to slip operators in this. The problem is that operators come in sets of commuting operators. In particular
the density matrix evolves by ρ(t' - t) = Uρ(t)U^† for U = exp{-iH(t' - t)/ħ}. For t' - t = δt very small then U ≈ 1 - iH(t' - t)/ħ and it is not hard to see that time evolution of the
density matrix involves a nonzero commutator of the density matrix with the Hamiltonian. This means the Hamiltonian rotates or evolves the density matrix out of the basis one might consider
for Gleason's theorem. I think this is the reason that Gleason's theorem, as profound it may be, does not reach the generalization of a proof of Born's rule.
However, observables in classical and quantum mechanics have different measure theories or distributions. Classical mechanics is “sharp,” which means it it L^∞ --- say like a delta function.
Classical mechanics is L^2, and the metric structure of spacetime is L^2 as well and with conformal spacetimes and R_{ab} = κg_{ab} it is also L^2. Without getting further this is a duality
connected with building spacetimes with entanglements. Now with 1/p + 1/q = 1 for convex sets then L^∞ is dual to L^1, which is a measure of pure classical probabilities. So what is this
system? It is about complete stochasticity, which the outcomes of measurements are an example of. The question is whether the eigenvalues of the QM L^2 coded as integer solutions to
Diophantine equations, something proven to be possible by Matiyasevich as any function has a corresponding Diophantine equation (even transcendentals like e^{ix} etc).
3. Not so much Connes as an algebraic QM approach, with the intention to bring it down to a mortal (my) mathematical level (I'm just reading Valter Moretti, "Spectral Theory and Quantum
Mechanics", Springer, 2017, for example, where his Chapter 14, "Introduction to the Algebraic Formulation of Quantum Theories, is nicely done).
The starting point for both classical (as usually understood, a commutative *-algebra) and quantum (a noncommutative *-algebra), as I take it, is that a state over a *-algebra is a
normalized, positive map to average measurement results.
The GNS-construction gives us a Hilbert space in both cases. Normal states are given by Trace[Aρ] in both cases and the Born rule is "just" a measurement |ψ⟩⟨ψ| in a pure state with density
matrix ρ=|φ⟩⟨φ|,
Note that everything is linear until we insist on discussing pure states.
The key question is to ask whether classical physicists can reasonably ascribe a meaning to all operators that act on the classical Hilbert space, to which I argue that they can.
Transformations to a different basis, with the fourier transform as case in point, more than just making sense, are *used* in classical signal analysis.
I'm doing very little that's specially new in this QM context. As I said, Koopman suggested such an approach in 1931; von Neumann wrote a long paper in German that has *not* been translated,
so of course it's called the Koopman-von Neumann approach, but the approach mostly languished until about 2000, when a PhD thesis appeared, since when there has been a slow stream of papers,
and for the last few years there has been a Wikipedia page that's not bad. Recently a connection has been made with Quantum Non-Demolition measurements, which seems to have led to slightly
more interest. I believe that understanding how things look in this kind of approach deserves to be at least as much in physicists' consciousness as deBB approaches.
One final comment: *I* take the view that the complex structure *can* be understood rather nicely as associated with the fourier sine and cosine transforms of probability densities, which, as
any engineer can tell you, introduces a naturally useful imaginary, j. I'm not committed to that approach, but so far I haven't seen a more natural approach.
I ought to let the paper do its own talking, given that you've kind enough to say that you have at least downloaded it, but I'm quite keen to see in what ways it might or might not be
attractive to other people.
4. The GNS construction is an aspect of noncommutative geometry. The spectra with Tr(Aρ) is also used in Gleason's theorem.
I will try to get to your paper as soon as possible. I have this large backdrop of things to read, including finishing Sabine's book. I started reading a library copy last year and have since
bought my own copy and that is on my stack as well.
31. This comment has been removed by the author.
32. In The EU, Canada and 'Developed Asia' , the budgets for Science and technology seems to not be at Risk ... It is U.S.A. that prioritize their Budgets in Military Applications the ones that knows
that They have to make their research agendas to fit into Geo-Political Military Conflicts to get The Money ... (ROFL) ...
Very Likely, The EU's headquarter are waiting to China's Parliament approve their Budget for HEP projects ... after that, They will decide ...
No Problem, Some CERN physicists will be invited to participate in China's Toys ... and CERN will receive its 'upgrading' budget ...
There is not an Eternal HEP Vacuum in Your Future ...
Don't Cry in advance for things that are not happening ...
33. @Sabine,
I do not quite see what this "measurement problem" is, although apparently some people lose sleep over it. The view of standard QM+ decoherence is perfectly reasonable:
Schroedinger's equation (SE) describes a closed quantum system. But when the system is measured, it cannot be considered closed anymore, so it is no surprise that it's not described by the SE.
The collapse of the wave function is just an effective prescription that describes this coupling to the external environment induced by the measurement. Decoherence theory showed how this process
can be explained in detail in terms of standard QM. So really, I do not see where is the problem.
From the experimental point of view, the experiments of Serge Haroche, for instance, have clearly shown that when the "environment" is sufficiently simple, the decoherence can be well controlled
or even reversed. Again, no mystery there.
I would not spend gigadollars, not even megadollars, on this pseudo-problem. For K$, I'm OK.
1. Opamanfred,
Decoherence does not solve the measurement problem. Please do some reading. Don't worry, I do not want your "giga-dollars".
2. The following video simulates the collapse of the wave function. This gives pretty good ideas of how probability plays a role in collapse and what visually a collapsed wave function appears
as. Of course a caveat is in order, for the ontology of a quantum wave is highly uncertain and it does not exactly "appear." However, this tells us about the mathematical representation. This
video also makes the point that this sudden transition is not something the Schödinger equation predicts.
As I wrote above dated 3/31 I think very strongly this problem is not solvable. Of course I might be wrong, but the issue of quantum measurement appears remarkably similar to the concept of
self-reference. Instead of a predicate acting on Gödel numbers for predicates including itself a measurement is quantum information encoding quantum information.
Decoherence does address aspects of measurement. However, it does not tell us how a particular outcome occurs, but rather how probability amplitudes transform into classical-like
probabilities as quantum phase of superposition or entanglement is transferred to a reservoir of states. Decoherence takes us right to the doorstep of the measurement dragon, but no further.
3. "Decoherence does not solve the measurement problem"
Please elaborate. I would also like to hear how exactly you define the problem. I consider what I sketched as a perfectly acceptable solution. On what aspect do you disagree?
4. Opamanfred,
This is really off-topic. I am one person and not a forum. I do not have time to respond to random questions. Really, this is common knowledge, and in any case, I explained this in my book,
and also Lawrence explained it correctly when he writes:
"Decoherence does address aspects of measurement. However, it does not tell us how a particular outcome occurs, but rather how probability amplitudes transform into classical-like
probabilities as quantum phase of superposition or entanglement is transferred to a reservoir of states."
5. Lawrence,
Re the measurement problem:
...I think very strongly this problem is not solvable.
Well, it is not solvable mathematically speaking because it is not a question of mathematics, but of physics. The question involves the nature of the physical processes underlying the maths
of QM.
The difficulty, of course, is that those processes are not directly observable, and the standard formalism does not resolve logically to a realistic picture of the quantum subsystems - a
wavefunction is not a physical thing.
The resulting ontological speculations (MW,PI, superposition) based on the maths are muddied, metaphysical, and lacking in scientific significance, to say the least. The Copenhagen approach,
OTOH, is simply to ignore the ontological problem, which consequently induces the measurement problem.
Only Bohmian mechanics approaches the ontological problem from a physics (rather than strictly maths) perspective by assuming that quantum subsystems are ontologically continuous with
classical mechanics. That this physically realistic reformulation (of QM) is currently disfavored relative to all the logically strained, metaphysical interpretations (of QM), says nothing
good about the state of modern theoretical physics.
BM is mathematically equivalent (but not qualitatively identical) to QM. In Bohmian mechanics there is no measurement problem. So, problem solved, no?
6. I have certain proclivities for the Bohm interpretation. I suppose this is just as I have the same for other interpretations. In fact I derived a form of path integral with Bohm's quantum
mechanics. I found the mention of Bohm was a form of toxin in getting this published. Bohm's QM is also potentially interesting for solving problems in chaos or quantum chaos. Bohm's QM is
though not identical to QM in general, but only so for wave functions of a certain form. Bohm's QM has some other deeper problems as well.
The Klein-Gordon equation is a scalar wave form of the invariant momentum-energy interval of special relativity. If you follow the Bohmian prescription with a polar wave function you find the
KG equation has the quantum potential. The odd implication is that a massless particle is off the light cone and in fact moving faster than light. This does not give reason to think there are
various nonlocal physics with this, for that violates no-signaling and other things. This is why it is often said that Bohm's QM is not relativistic. Bohm's QM also without a Hilbert space
does not derive things such as the generation or absorption of photons by atoms in a concise way, and things get worse with higher energy creation and annihilation of particles.
There are quantum interpretations that are ψ-epistemic and other that are ψ-ontic. The many world interpretation (MWI) and Bohm interpretation (BI) are ψ-ontic. Bohr's Copenhagen
interpretation (CI) and now the latest Qubism by Fuchs are ψ-epistemic. These are some of the popular interpretations and there are others such as consistent histories, the Montevideo
interpretation and the related one by Penrose, and other. In fact quantum interpretations are multiplying like bunnies, maybe cockroaches to put it in a negative light, and none of them seems
to really solve everything. The CI is interesting in that M-theory of D-branes works well with it. Quantum information theory is worked often in MWI. Qubism is now the beautiful child of
those into Bayesianism --- which I can tip my hat towards. Pullen and Penrose have interesting ideas on how gravitation plays a role, and quantum gravitation built up from quantum
entanglements probably does have a correspondence with quantum wave decoherence and maybe even measurements. However, all of these have big holes you can run an optics bench through, maybe
even a collider.
I wrote a math-physics result on how quantum mechanics is neither ψ-epistemic or ψ-ontic with any certainty. It does not work for two state systems, which is unfortunate. I should revisit
this to make it work. The result is that quantum interpretations that are either ψ-epistemic or ψ-ontic are not determined by a measure theory of QM. I like the prospect of this: QM has this
sort of Man proposes and QM disposes flavor to it.
7. @Lawrence Crowell
Last Spring I submitted a short essay to the Gravity Research Foundation (GRF) in Wellesley, Massachusetts, that effectively is another interpretation of QM; albeit, a very amateur one. The
concept is largely heuristic with a minimum of mathematical modeling. Currently I'm expanding on the original paper, submitted to the GRF, to include ideas for which the essay word limit
(1500 words) would not allow. In the abstract of the paper, submitted to GRF last year, a tie-in to De Broglie-Bohm Pilot Wave Theory (PWT) is mentioned. This might have been a mistake seeing
that PWT is anathema to much of the physics community as illustrated by your choice of the word "toxin", to describe the reaction of publishers to that particular QM interpretation.
While I didn't mention it directly in the essay submitted to GRF the model provides a mechanism for reported anomalous acceleration signals observed in certain superconductor experiments that
are orders of magnitude larger than allowed by standard physics (Tajmar et. al. 2003-2006, and others). This connection provided the rationale for submitting the essay to GRF, as the
organization's stated mission involves understanding gravity, and presumably artificially generated gravity-like forces. To wind this up, I hope to complete the expanded version of the
originally submitted GRF essay in a few weeks and upload it to viXra.org.
8. The particle in the pilot wave interpretation of Bohm and taken from deBroglie is not highly regarded in part because of Bohm's intention with local hidden variables. The idea is workable in
a nonrelativistic framework and I think a way of working quantum chaos.
There is a fascinating way of doing quantum mechanics that Pascual Jordan worked with Wigner. It is a way of doing QM with trace and determinants that is useful with the Freudenthal
determinant over exceptional algebras. In fact I think it is useful with permanents as well, which find their way into algebraic geometric complexity and P vs NP. So why is this not widely
used? Jordan and Wigner published on this in 1935 and Jordan became fanatically committed to the Nazi cause. He worked on the rocket programs at Peenemunde and was committed to the Nazi
program. It is amazing how this sort of crap can infect brains, much like MAGA promoted in the US these days. Anyway this approach to QM fell into disrepute. History and affiliation have big
impacts on the course of development in physics.
9. This comment has been removed by the author.
10. In fact quantum interpretations are multiplying like bunnies, maybe cockroaches to put it in a negative light, and none of them seems to really solve everything.
Well yes, but the Bohmian advantage over all those proliferating bunnies is twofold. First, it eliminates the self-induced measurement problem of CI. More importantly, it provides a
qualitative account of unobservable quantum processes that is continuous with classical mechanics and therefore provides a sound (and realistic) basis for further qualitative and quantitative
The continuity with CM is achieved by introducing a scale factor, the guiding equation. This guiding equation, in turn, is suggestive of an underlying physical component that induces quantum
behavior in sufficiently low-mass classical particles.
This avenue would seem to offer, at least the possibility, of a qualitative and quantitative approach with the potential to converge on a plausibly realistic account of quantum phenomena. I
don't think the same can be said for any of the other cockroaches.
34. Okay, okay. But what if we find more Odderons? ;)
35. This conjecture on my part is not something I have actually bent metal on or have done any calculations. This is pretty removed from my day job work that is more applied or engineering. The DMPR
theorem is similar to the Bernays-Cohen result that the continuum hypothesis is a case of Gödel's theorem.
Polytopes also enter into the algebraic geometry complexity of N vs NP. The role symmetry is of course important for gauge fields. Also for quantum entanglements quotient spaces or groups occur
when some set of quantum numbers are replaced by other degrees of freedom. A bipartite entanglement replaces the spin of two fermions with the Bell state. This is a quotient system. The exact
sequence for the moduli space of gauge connections is similar. In fact I think dual to entanglement geometry.
36. I often wonder what a theory will look like that explains QM and ART as special cases. As far as I can see, most scientists are trying to bridge the gap from QM. This seems logical, since most
physicists probably regard QM as the most fundamental theory. However, the classic cases of really new theories have developed differently. There was no direct path from classical physics to
quantum mechanics, nor to GRT. So QM and GRT were really new. Therefore, the question is whether the current approaches to unifying the two basic theories can really be promising enough.
I myself am a mathematician with a solid background in Artificial Intelligence.
When developing an algorithm for decision-making, I came across interesting relationships rather playfully. The chaotic decision process (I call it the "GenI process") is a chaotic random process
based on very simple rules. Except for the basic arithmetic in complex number space, this does not require any difficult mathematics. (Simple maths do not necessarily produce simple results:
think of the fractal sets by Mandelbrot.)
Significantly more difficult is the statistical analysis of chaotic state changes. On the one hand, I can show that the process, starting from an initial state, certainly selects one of several
decisions, and thereby exactly fulfills the statistics known from quantum mechanical measurements. On the other hand, I can derive a relativistic metric such that averaged state changes follow
time-like geodesic paths in a four-dimensional Riemann space.
Should not such or similar approaches, which are not derived directly from QM or GRT, ensure a fresh start? In principle, this is only about a change of perspective.
37. @WSG
There is a upcoming version of QM that uses complex numbers and four-dimensional Riemann space. It's used to handle open systems.
It is called PT-symmetric quantum mechanics.
PT-symmetric quantum mechanics is an extension of conventional quantum mechanics into the complex domain. (PT symmetry is not in conflict with conventional quantum theory but is merely a complex
generalization of it.) PT-symmetric quantum mechanics was originally considered to be an interesting mathematical discovery but with little or no hope of practical application, but beginning in
2007 it became a hot area of experimental physics.
38. This is not the point I wanted to make. This is obviously just another extension of a proven theory. Such things did not lead to anything really new. I am well aware of other approaches, such as
quantum loop gravity or string theory, which, despite all efforts, have yet to resolve the open questions.
The question of what a theory must look like so that QM and GRT can be deduced from it have already been asked. Maybe it will look somewhat crazy from today's perspective, as the QM for classical
My point is to take a fundamentally different perspective on the role of gravity in QM.
A model like the one mentioned above indeed requires a rethink. After that, our universe, as we perceive it, evolves according to a collapse of its wave function. This clearly contradicts the not
explicitly justified assumption of leading physicists that it develops along a Schrödinger equation. But why is it like that? Is there a clear justification and vice versa? What, in essence, is
against assuming a collapse? I have not even seen a discussion among physicists about this aspect. Even with well-known authors like Penrose, Greene, Hawking, who otherwise like to talk about the
wildest speculation, nowhere is there any hint that the collapse of its wave function is the source of reality in our universe.
Can anyone help me here? Are there any works that consider this perspective?
At least in a nutshell, I can prove that such an approach can be quite effective. I can perform concrete calculations of a space-time metric for a spin1 / 2 particle and actually prove that the
dynamics during the measurement satisfy Einstein's field equations. That should justify at least a discussion about the view.
39. Thank you for this exceptionally thoughtful post. I do think that a good question to ask people on both sides of the argument is: What is your cutoff?
That is: For supporters of the collider, I'd like to ask "How expensive would this thing have to be before you stopped supporting it? 30 billion? 50 billion? 100 billion?"
And for opponents: "How inexpensive would this thing have to be before you stopped opposing it? 15 billion? 10 billion? 1 billion?"
As a general rule, I think people who are able to answer these questions --- and to defend their answers --- are likely to have thought a lot harder about the tradeoffs than those who reflexively
just support or oppose.
1. Steven,
Yes, a good question. I'll make a go at it and say about $2 billion.
A larger collider currently has less scientific promise than LIGO had, which came in at a cost somewhat below $1 billion. It has also less scientific promise than the SKA, whose full proposal
would come in at $2 billion. So that would seem a reasonable amount.
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Probability distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_distribution&oldid=43464
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Geometry of Architecture
Film produced within the project Intuitive and Interactive Geometry, by the Facultade de Ciências da Univerdidade de Lisboa.
Suzana Nápoles
Margarida Oliveira
José Soeiro
João Filipe Silva
Pedro Mira
Phil Sobral
Con el apoyo de:
Calouste Gulbenkian Foundation
Through works by renowned architects such as Antoni Gaudí, Felix Candela and Oscar Niemeyer, this film intends to show the natural way in which the formulas, the geometry of forms and their
concretization are intertwined.
This film is part of the project Intuitive and Interactive Geometry (Gi2), a web platform by the University of Lisbon, for the education of elementary geometry.
In the website of the Gi2 project, you can explore several geometric experiences with interactive applets, based on the popular free-accesss software GeoGebra, including the experiences proposed in
the video.
Portuguese: Film, Website.
English: Film, Website.
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C : Prints the sum of n positive integers less than 100
C Exercises: Accepts a positive integer n less than 100 from the user and prints out the sum
C Basic Declarations and Expressions: Exercise-63 with Solution
Write a C program that accepts a positive integer n less than 100 from the user. It prints out the sum of 1^4 + 2^4 + 4^4 + 7^4 + 11^4 + • • • + m^4. In this case, m is less than or equal to n. Print
an appropriate message.
Test data and expected output:
Input a positive number less than 100:
Sum of the series is 1024388
Sample Solution:
C Code:
#include <stdio.h>
int main() {
int i, j, n, sum_int = 0;
// Prompt user for input
printf("Input a positive number less than 100: \n");
// Read the input value
scanf("%d", &n);
// Check if the input is valid
if (n < 1 || n >= 100) {
printf("Wrong input\n");
return 0;
j = 1;
for (i = 1; j <= n; i++) {
sum_int += j * j * j * j;
j += i;
// Display the result
printf("Sum of the series is %d\n", sum_int);
return 0;
Sample Output:
Input a positive number less than 100: 68
Sum of the series is 37361622
C programming Code Editor:
Previous:Write a C program that accepts a positive integer less than 500 and prints out the sum of the digits of this number.
Next: Write a C program that accepts integers from the user until a zero or a negative number, display the number of positive values, the minimum value, the maximum value and the average of all
What is the difficulty level of this exercise?
Test your Programming skills with w3resource's quiz.
It will be nice if you may share this link in any developer community or anywhere else, from where other developers may find this content. Thanks.
• Weekly Trends and Language Statistics
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What are Irrational Numbers?
In mathematics, there are various types of numbers such as real numbers, natural numbers, whole numbers, negative and positive numbers, and so on. Similarly, irrational numbers are the type of
numbers that can be written or expressed in the form of a simple fraction. Irrational numbers can also be defined as real numbers which cannot be written or represented in the form of p/q or a/b
where q or b is not equal to zero. The opposite of irrational numbers is rational numbers. Those numbers can be expressed in the form of simple fractions i.e., real numbers or in the form of p/q
where q is not equal to the number 0. Some examples of rational numbers are as follows: ¾, ⅚, 4, 8, etc. Some examples of irrational numbers are as follows: √5, √8, √3, etc. In this article, we will
try to cover some basic aspects of irrational numbers such as properties of these numbers, comparison from rational numbers, and do a brief analysis about them.
Properties of Irrational Numbers
As mentioned above, irrational numbers are the type of numbers that can be written or expressed in the form of a simple fraction. Let us look into the properties of these numbers. Some of them are
mentioned below:
• If we add an irrational number and a rational number, the resultant value will always be a rational number. For example, ‘x’ is an irrational number, and ‘y’ is a rational number, then x + y = z
which is a rational number. Therefore, the sum of a rational and irrational number is equivalent to a rational number.
• The arithmetic operation of multiplication with a number that is irrational and with a rational number that is non-zero in nature will provide you with an irrational number.
• The least common divisor or multiple of a number that is irrational may exist or may not exist.
• If we add or multiply two irrational numbers, the resultant value always comes as a rational number. For example, when two irrational numbers such as √5 and √5 are multiplied the answer comes = 5
which is a rational number.
• You may observe that the set of the number which is rational may be closed during the process of multiplication. But, in irrational numbers, the contrary happens.
Rational Numbers Vs Irrational Numbers
In the next few points, we will carry out the comparison of rational and irrational numbers.
• Irrational numbers are the type of numbers that can be written or expressed in the form of a simple fraction whereas rational numbers can be represented in the form of a simple fraction.
• The irrational numbers cannot be expressed in the form of a/b or p/q where b or q is not equal to zero whereas the rational numbers can be written in the form of p/q where q is not equal to zero.
• Some examples of rational numbers are as follows: ¾, ⅚, 4, 8, etc. Some examples of irrational numbers are as follows: √5, √8, √3, etc.
Practice Math from Math Worksheets
In order to excel in mathematics, one should practice more and more. There is a very famous quote, ‘practice makes a man perfect. This quote is totally correct and implies mathematics. Therefore, to
practice math from math worksheets, visit Cuemath. There are various other perks that are facilitated by the team. The teachers are highly experienced in their domains. Not only the teachers, but it
also provides you with various different kinds of math puzzles so that studying math becomes fun. Thus, visit the Cuemath website for one-one live sessions, doubt clearing classes, and book a free
session now.
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Nernst Equation Example for Electrochemical Systems Design
Key Takeaways
• An important result of thermodynamics for electrochemical cells is the Nernst equation.
• The Nernst equation states how the output potential from an electrochemical cell varies with the cell temperature and reaction quotient.
• Electrochemical systems designers that need to monitor the voltage from an electrochemical cell can use the Nernst equation to relate potential, temperature, and concentration measurements.
Use the Nernst equation to understand the measurements of electrochemical cells.
The Nernst equation is a fundamental chemical thermodynamics equation used in electrochemical cell design. As part of reaction monitoring, measurements in an electrochemical system can be used with
the Nernst equation to determine electrochemical reaction progress in various conditions. For electrochemical systems designers, this important equation forms the basis for monitoring the electrical
and environmental behavior of an electrochemical cell.
Although it has a simple form, some guides on the Nernst equation provide apparently conflicting details. Therefore, we’ve prepared a Nernst equation example to help electrochemical systems designers
understand what to measure and what they need to include in their PCB layout. Here’s how the Nernst equation applies to electrochemical reaction monitoring and what designers need to consider in
their designs.
Nernst Equation Theory
The Nernst equation is a thermodynamic result for electrochemical reaction equations and is derived from the Gibbs free energy for the reaction. The Nernst equation relates the electrochemical
potential of a reversible (redox) reaction, as measured between the anode and cathode, to the cell’s reaction quotient and temperature. This equation can be used in the following areas:
• Reaction equilibrium monitoring. By measuring the cell’s output potential and the temperature in the cell, one can spot when the reaction comes to equilibrium.
• Reaction environment monitoring. If the reaction temperature changes, the equilibrium balance in the cell will change. The Nernst equation states the correlation between the temperature and the
equilibrium constant in the cell.
• Voltammetry. The Nernst equation forms the baseline for predicting when additional current can be coaxed from a cell with an additional potential in voltammetry measurements.
These basic monitoring tasks are vital parts of industrial automation and process control, battery design, and electrochemistry. A potentiostat is normally used to provide these measurements, but
specialty potentiostat systems may integrate other important features that are needed for cell monitoring.
The Nernst equation is normally formulated for the following chemical reaction:
The general reversible reaction that can be considered in the Nernst equation.
As the progress of chemical reactions depends on temperature, and the concentrations of reactants/products will change over time, the reaction can be comprehensively monitored if one has an equation
describing the relationship between these quantities. This is where the Nernst equation comes into play.
Note that the definitions shown below are defined for the above reaction equation, but they can be easily extended to reactions with more than two reactants and or products through the definition of
the reaction quotient and equilibrium constant.
Starting from the definition of Gibbs free energy at standard conditions, we can determine the new Gibbs free energy at non-standard conditions using the reaction quotient. The definition and
derivation for the Gibbs free energy in non-standard conditions can be found in many thermodynamics and chemistry textbooks, so it won’t be repeated here. By invoking the Faraday constant in the
Gibbs free energy, which relates the electrochemical potential to the number of electrons released in a redox reaction, we arrive at the Nernst equation:
Nernst equation definition.
The applicability of this equation is quite broad as long as the reaction quotient is known. For the reaction shown above, the reaction quotient is simply:
Reaction quotient.
Here, by measuring the cell potential as a function of time, the reaction quotient at constant temperature can be determined as a function of time. Similarly, if some reactants or products are added
to/removed from the cell, this change in the cell contents can be detected by monitoring the cell potential. Eventually, the reaction will reach its final equilibrium and the cell potential will be
zero. This case also needs to be considered in the Nernst equation.
Note that for a reaction proceeding towards equilibrium, the reaction quotient will change as the reaction progresses. Eventually, once the reaction reaches equilibrium, the reaction quotient will be
equal to the equilibrium constant. When this occurs, the cell potential E will be zero. Plugging this into the above equation gives a relation between the cell temperature, equilibrium constant, and
standard cell potential:
Cell standard potential vs. temperature and equilibrium constant.
Here, we can see how the standard cell potential is a function of temperature in two ways: by direct proportionality and due to the equilibrium constant. Note that equilibrium constants are also
functions of temperature, being governed by the molecular kinetics of the reactants and products. By monitoring the cell potential at different temperatures, the equilibrium constant for the reaction
as a function of temperature can be determined.
Nernst Equation Example
As a simple example, consider a redox reaction involving charge transfer between Zn and Cu metal.
Simple charge transfer reaction between Zn and Cu.
This reaction has a standard cell potential of 1.10 V. Suppose that, after 2 minutes, there is 0.1 mol of reactants and 1.8 mol of products. By using the value of the Faraday constant F, gas constant
R, n = 2, and taking a temperature of 298 K, one finds that the cell potential is 1.10 V - 0.1711 V = 0.9289 V. This is the voltage one would measure between the cell cathode and anode.
Eventually, the cell will reach equilibrium and the cell voltage will saturate to a constant value. When building an electrochemical monitoring system, a simple voltage measurement could involve a
Wheatstone bridge with high precision resistors. A small RTD probe or thermocouple can also be used for temperature measurement. Connecting these to an MCU with an amplifier on each sensor gives a
simple electrochemical monitoring system.
If you’re designing your first electrochemical monitoring system, you can use our Nernst equation example to help you get started with your control unit and algorithm design. To get started with your
schematic and PCB layout, you need the industry’s best PCB design and analysis software. Allegro PCB Editor gives you all the features you need to create advanced circuit boards for a range of
applications, including electrochemical systems. You’ll also have access to a complete set of advanced design verification tools and field solver utilities to analyze the behavior of advanced
If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.
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Scale Speed Calculator
Home » Simplify your calculations with ease. » Physics Calculators »
Scale Speed Calculator
Scale speed represents the velocity at which an object would move if its dimensions were proportionally increased or decreased to match a specified scale. It’s a critical concept used in various
fields, including engineering, architecture, and model making. This blog post introduces the Scale Speed Calculator, a tool that leverages this concept to compute the scale speed.
Working of the Scale Speed Calculator
The Scale Speed simplifies the calculation process of determining scale speed. All it requires from the user is the real speed (RS) of the object in miles per hour (mph) and the scale (S) for which
the speed needs to be calculated. It employs the scale speed formula to automatically compute the scale speed (SS).
Scale Speed Formula and Variables
The formula used by the calculator is:
SS = RS / SQRT(S) In this equation:
SS denotes the Scale Speed (in mph), RS signifies the Real Speed (in mph), and S represents the Scale. To compute the scale speed, the real speed is divided by the square root of the scale.
Detailed Example of Scale Speed Calculation
Let’s illustrate the use of the formula with an example. Suppose we have a real speed (RS) of 225 mph and a scale (S) of 25. Applying the formula:
SS = RS / SQRT(S) SS = 225 / SQRT(25) SS = 225 / 5 SS = 45 mph
Hence, the scale speed in this instance would be 45 mph.
Applications of Scale Speed Calculator
The Calculator has diverse applications across several industries.
Model Makers:
It is beneficial for model makers who build scale models of trains, airplanes, or automobiles and want to replicate real-world speed conditions.
In engineering, particularly in prototyping, understanding the scale speed can help test designs under various conditions.
Film Industry:
The film industry often uses scale models for visual effects. Knowing the scale speed is crucial to ensuring realism in these models.
Most Common FAQs
What is the Scale Speed Calculator?
The Scale Speed Calculator is a tool that computes the scale speed using real speed and the chosen scale.
How does the Scale Speed Calculator work?
The calculator works by dividing the real speed by the square root of the scale to obtain the scale speed.
Where can the Scale Speed Calculator be used?
The calculator is applicable in various industries such as model making, engineering, architecture, and the film industry.
What units are used in the Scale Speed Calculator?
The calculator uses miles per hour (mph) for both real speed and scale speed.
Understanding and applying scale speed is essential across various industries and it provides an easy, efficient way to compute it. By using real speed and scale, this tool offers precision and
simplicity, fostering realism and accuracy in the respective applications.
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computational complexity
Abstract: Subway shuffle is an addicting puzzle game created by Bob Hearn. It is played on a graph with colored edges that represent subway lines; colored tokens that represent subway cars are placed
on the nodes of the graph. A token can be moved from its current node to an empty one, but only if the two nodes are connected with an edge of the same color of the token. The aim of the game is to
move a special token to its final target position. We prove that deciding if the game has a solution is PSPACE-complete even when the game graph is planar.
The Crazy Frog Puzzle and Permutation Reconstruction from Differences
The Crazy Frog Puzzle (CFP) is the following: we have a $n \times n$ partially filled board, a crazy frog placed on an empty cell and a sequence of horizontal, vertical, and diagonal jumps; each jump
has a fixed length and two possible opposite directions. The crazy frog must follow the sequence of jumps, and at each step it can only choose among the two available directions. Can the frog visit
every empty cell of the board exactly once (it cannot jump on a blocked cell and on the same cell two times)?
Figure 1: an example of the Crazy Frog Puzzle on the left and its solution on the right.
We prove that the Crazy Frog Puzzle is $\sf{NP}$-complete even if restricted to 1 dimension and even without blocked cells. The 1-D CFP without blocked cells corresponds to the problem of
reconstructing a permutation from its differences:
Permutation Reconstruction from Differences problem:
Input: a set of $n-1$ distances $a_1,a_2,…,a_{n-1}$
Question: does exist a permutation $\pi_1,…,\pi_n$ of the integers $[1..n]$ such that $| \pi_{i+1} – \pi_i| = a_i$, $i=1,…,n-1$ ?
For example, given the differences $(2,1,2,1,5,3,1,1)$ a valid permutation of $[1..9]$ is $(5,7,6,8,9,4,1,2,3)$
Update 2013-12-19: a new version of the paper is available.
click here to download a draft of the paper
Complexity of the Hidden Polygon Puzzle
The Hidden Polygon Puzzle (for brevity HPP) decision problem is:
Input: a set $P$ of $m$ integer points on a $n \times n$ square grid and an integer $k \leq m$;
Question: does exist a simple rectilinear polygon with $k$ or more vertices $(v_1,v_2, …, v_t), \; v_i \in P, t \geq k$?
The following figure shows an example of a HPP puzzle.
Figure 1: Given the 21 points on the right, can we find
a simple rectilinear polygon with at least 16 vertices?
A possible solution is shown on the right.
The problem is a slight variant of the $\sf{NP}$-complete puzzle game Hiroimono; we prove that the Hidden Polygon Puzzle is $\sf{NP}$-complete, too using a completely different reduction.
Hidato is NP-complete
Hidato (also known as Hidoku) is a logic puzzle game invented by Dr. Gyora Benedek, an Israeli mathematician. The rules are simple: given a grid with $n$ cells some of which are already filled with a
number between $1$ and $n$ (the first and the last number are circled), the player must completely fill the board with consecutive numbers that connect horizontally, vertically, or diagonally.
Figure 1: An Hidato game (that fits on a $8 \times 8$ grid) and its solution on the right.
We prove that the corresponding decision problem $\sf{HIDATO}$ : “Given a Hidato game that fits in a $m \times n$ grid, does a valid solution exist?” is $\sf{NP}$-complete.
Binary Puzzle is NP-complete
Binary Puzzle (also known as Binary Sudoku) is an addictive puzzle played on a $n \times n$ grid; intially some of the cells contain a zero or a one; the aim of the game is to fill the empty cells
according to the following rules:
• Each cell should contain a zero or a one and no more than two similar numbers next to or below each other are allowed
• Each row and each column should contain an equal number of zeros and ones
• Each row is unique and each column is unique
We prove that the decision version of Binary Puzzle is NP-complete.
Three easy deletion games on paths
The node deletion game NODE KAYLES is a two persons perfect information game played on a graph $G$. Players alternate picking a node from the graph $G$; the node and its adjacent nodes are deleted.
The first player unable to move loses the game. Deciding the winner of a NODE KAYLES game is $\mathsf{PSPACE}$-complete [1]. Similarly in the EDGE KAYLES [1] game the two players must pick an edge;
the edge is deleted along with the two endpoint nodes and the edges incident to its two endpoint nodes. The complexity of finding the winner of a EDGE KAYLES game is unknwon. On the Q&A site
cstheory.stackexchange.com I submitted a mix of the two games; I call it the BRIDGES AND ISLANDS game: at each turn each player must pick an edge (a “bridge”) or an isolated node (an “island”); if a
player picks a node the node is deleted along with its incident edges, if it picks an edge, the edge is deleted along with its two endpoint nodes and the edges incident to them. Like the other two
games, the first player unable to move loses the game. I didn’t succeed in finding the complexity of BRIDGES AND ISLANDS.
However decideng the winner is polynomial time solvable when the three games are played on a path. Here I give a quick proof for EDGE KAYLES; the proof for the other two games is similar. Continue
The complexity of the puzzle game Net: rotating wires can drive you crazy
An amateur proof that the puzzle game Net is NP-complete.
The puzzle game Net, also known as FreeNet or NetWalk, is played on a grid filled with terminals and wires; each tile of the grid can be rotated and the aim of the game is to connect all the
terminals to the central power unit avoiding closed loops and open-ended wires. We prove that Net is NP-complete.
Intertia is NP-complete
A quick proof that the puzze game Inertia (see this version on Simon Tatham’s Portable Puzzle Collection) is NP-complete.
The Complexity of Camping
An amateur proof that the Tents puzzle game is NP-complete.
We prove that the Tents puzzle game is NP-complete.
Have fun with Boulder Dash
An amateur proof that the popular game is NP-hard.
Boulder Dash is a videogame created by Peter Liepa and Chris Gray in 1983 and released for many personal computers and console systems under license from First Star Software. Its concept is simple:
the main character must dig through caves, collect diamonds, avoid falling stones and other nasties, and finally reach the exit within a time limit. In this report we show that the decision problem “
Is an $N\times N$ Boulder Dash level solvable?” is NP-hard. The constructive proof is based on a simple gadget that allows us to transform the Hamiltonian cycle problem on a 3-connected cubic planar
graph to a Boulder Dash level in polynomial time.
click here to download the paper
NOTE: the same result has been proved by G. Viglietta in the paper: Gaming Is a Hard Job, But Someone Has to Do It! ; his proof, which is embedded in a more general and powerful framework that can be
used to prove complexity of games, doesn’t require the Dirt element.
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Chemical Kinetics - Lesson
Chemical Kinetics
Chemical Kinetics
This lesson aligns with NGSS PS1.B
Chemical kinetics holds significant importance with insights into various aspects of chemical reactions. Specifically, the term 'kinetics' deals with the rate of change of a given quantity. To
illustrate, the rate of change of displacement is denoted as velocity, and similarly, acceleration represents the rate of change of velocity. Chemical kinetics focuses on unravelling the dynamic
processes governing the rates and mechanisms of chemical transformations. In this article, we will explore the core concepts of chemical kinetics, shedding light on its significance and applications
in the world of science and industry.
Defining Chemical Kinetics:
Chemical kinetics, also known as reaction kinetics, provides valuable insights into the rates of reactions—the speed at which reactants are converted into products as well as the factors influencing
Typically, chemical reactions are categorized into different speed classes based on their rates. For instance, reactions such as Na + [math]H_2O[/math] are considered fast, those like Mg + [math]H_2O
[/math] fall into the moderate category, and reactions involving esterification are classified as slow.
Rate of Formations and Disappearances
In any chemical reaction, the quantity of reactants decreases, whereas the amount of products increases. It's crucial to recognize that the overall rate of the reaction depends on the rate at which
reactants are consumed or the rate at which the products are formed.
By graphing the concentration of reactants and products against time, one can readily compute the rates of product formation and reactant disappearance by assessing the slopes of the respective
curves. It's noteworthy that the overall rate of the reaction may or may not align with the rates of formations and disappearances.
At time t=0, product concentration is zero. Examining the graph reveals that the slope of the reactants curve is negative, while that of the products curve is positive. This signifies a decrease in
the concentration of reactants and an increase in the concentration of products. To illustrate the interconnectedness of the overall reaction rate, the rate of reactant disappearance, and the rate of
product formation.
Let's consider a straightforward example – the formation of water.
2[math]H_2[/math] + [math]O_2[/math] → 2[math]H_2O[/math]
From this equation, it is evident that for every mole of [math]O_2[/math] consumed, two moles of [math]H_2[/math] will also be consumed, resulting in the formation of two moles of [math]H_2O[/math].
Assuming the reaction proceeds for 10 minutes with an initial mixture of 1 mole each of [math]H_2[/math] and [math]O_2[/math], we can track the changes in moles over time.
t = 0 1 1 0
t = 10 mins 1 – 0.5 1 – 0.25 0.5
Say after 10 minutes, 0.5 moles of [math]H_2[/math] is consumed, and according to stoichiometry, 0.25 moles of [math]O_2[/math] is consumed, and 0.5 moles of [math]H_2O[/math] is formed. Now, let us
calculate the rates for [math]H_2[/math], [math]O_2[/math] and [math]H_2O[/math] for the first 10 minutes.
Rate of disappearance of [math]H_2[/math]
Rate of Disappearance of [math]O_2[/math]
Rate of Formation of [math]H_2O[/math]
Based on the above calculations, it is evident that the rate at which [math]H_2[/math] is consumed is twice the rate at which [math]O_2[/math] is consumed. This observation underscores the connection
between the stoichiometry of the reaction and the rates of formation and disappearance of various reactants and products, establishing the following relationship.
aA + bB → cC + dD
be a reaction.
Where r → rate of the overall reaction,
Δ[A], Δ[B], Δ[C], and Δ[D] demonstrate a change in concentration and
st → change in time.
Significance of Reaction Rates:
The reaction rate measures how quickly reactants transform into products during a chemical reaction. It quantifies the change in concentration of either the reactants consumed or the products formed
per unit of time. Essentially, reaction rate reveals the speed at which molecular collisions lead to the formation of new substances. This crucial parameter is influenced by factors such as
concentration, temperature, and catalysts.
• Chemical Kinetics specifically, the term 'kinetics' deals with the rate of change of a given quantity.
• In any chemical reaction, the quantity of reactants decreases, whereas the amount of products increases.
• The reaction rate measures how quickly reactants transform into products during a chemical reaction.
• It quantifies the change in concentration of either the reactants consumed or the products formed per unit of time.
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Question ID - 157538 | SaraNextGen Top Answer
A bucket elevator for lifting parboiled paddy (bulk density $=840 \mathrm{~kg} \mathrm{~m}^{-3}$ ) is operated at a linear speed of $2 \mathrm{~m} \mathrm{~s}^{-1} .$ The width of the bucket is $25.4
\mathrm{~cm}$ and its cross section is making a subtending angle of $75^{\circ}$ at the centre of a circle having $12.7 \mathrm{~cm}$ radius. The space between two adjacent buckets on the elevator
belt is $40 \mathrm{~cm} .$ If the buckets are filled to $80 \%$ of their volumetric capacity, the lifting capacity of elevator in kg min $^{-1}$ (round off to 2 decimal places) is____________
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PESSOA - Semantic Scholar - P.PDFKUL.COM
PESSOA: towards the automatic synthesis of correct-by-design control software∗ Manuel Mazo Jr.
Anna Davitian
Paulo Tabuada
Dept. of Electrical Engineering UCLA
Aerovironment Inc.
Dept. of Electrical Engineering UCLA
[email protected] [email protected]
ABSTRACT In this paper we report on ongoing work aimed at providing tools for the synthesis of correct-by-design embedded control software. We introduce a tool, named Pessoa, that: 1) constructs
finite abstractions of control systems; 2) synthesizes controllers based on the finite abstractions; 3) refines the controllers to Simulink blocks for closed-loop simulation. In addition to
describing the main functionalities of Pessoa, we also illustrated its use in an example where the control software needs to access an actuator shared by other software tasks.
With the increasing ubiquity of embedded systems in our daily life, the design of safe and reliable embedded control software becomes a challenge of paramount importance. Such designs are fraught
with difficulties arising from the complex interactions between the physical world and the software. In order to ease this problem much work has been devoted to the construction of finite model
abstractions for control systems. The use of finite abstractions enable, by resorting to computational tools, the verification of already designed control software, and the automatic synthesis of
correct-by-design controllers enforcing predefined specifications for the closed-loop system. In the present paper we introduce the tool Pessoa, which, based on the notion of approximate (bi)
simulation relations, automates the synthesis of correct-by-design control software. Starting from a specification and a finite abstraction for the continuous system being controlled, an abstraction
for the desired control software can be synthesized by resorting to well known algorithms developed in supervisory control of discrete-event systems [KG95, CL99] or algorithmic game theory [dAHM01,
AVW03]. The resulting description of the control software can then be refined to a controller acting on the original control system and compiled into code. ∗This work was partially supported by the
NSF awards 0717188, 0820061, and 0834771.
[email protected]
The current version of Pessoa supports the construction of finite abstractions of control systems, the synthesis of controllers enforcing simple specifications, and the refinement of controllers to
Simulink blocks that can be used to simulate the closed-loop behavior. Future versions of Pessoa will support more complex specifications and compilation of the synthesized controllers into code. The
construction of the finite abstractions is based on approximate simulations and bisimulations recently investigated in [PGT08]. The construction of abstractions of linear control systems is natively
supported in Pessoa Version 1.0, nonlinear and switched systems can also be handled by Pessoa, as illustrated by the example in Section 4, but require some additional effort from the user1 . Most of
the tools available for hybrid systems such as Ariadne [Ari], PHAVer [PHA], KeYmaera [KeY], Checkmate [Che], and HybridSAL [Hyba], focus on verification problems. Tools for the synthesis of
controllers are more recent and include LTLCon [LTL] for linear control systems and the Hybrid Toolbox [Hybb] for piece-wise affine hybrid systems. What sets Pessoa apart from the existing synthesis
tools is the nature of the abstractions (approximate simulations and bisimulations) and the classes of systems admitting such abstractions (linear, nonlinear, and switched [Tab09]).
Let us start introducing the notion of system used in Pessoa to describe both software and control systems. Definition 1. A system S = (X, X0 , U, sextuple consisting of: • • • • • •
- , Y, H) is a
a set of states X; a set of initial states X0 ⊆ X; a set of inputs U ; - ⊆ X × U × X; a transition relation a set of outputs Y ; an output map H : X → Y .
System S is said to be finite when X has finite cardinality and metric when Y is equipped with a metric d : Y × Y → R+ 0 . Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the
first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 2010 ACM ...$10.00.
The “dynamics” of a system is described by the transition - enrelation: existence of a transition (x, u, x0 ) ∈ tails that upon the reception of input u at state x, system 1 For further information
please consult the documentation in http://www.cyphylab.ee.ucla.edu/Pessoa.
S evolves to state x0 . In [Tab09] it is shown how systems of this form can represent both software and control systems modeling physical processes. While software models naturally lead to finite
systems, obtaining models for control systems leading to finite systems requires of some abstraction techniques. Informally, a control system Σ is a differential equation of the form ξ˙ = f (ξ, υ),
where ξ(t) denotes the state of the system at time t, υ(t) the controlled input, and ξ˙ denotes the time derivative of ξ. Let us denote by Sτ (Σ) the exact discrete time system resulting from
sampling Σ with period τ . By adequately discretizing the state space of the differential equation and the input space, with discretization steps η and µ respectively, and by selecting an adequate
sampling time τ , it has been shown in [Tab09] and references therein, that useful finite abstractions Sabs for Sτ (Σ) can be obtained. Such finite abstractions can be related to the control system
through a generalization of the notion of alternating simulation relation named alternating approximate simulation relation [Tab09]: Definition 2. Let Sa and Sb be metric systems with Ya = Yb and let
ε ∈ R+ 0 . A relation R ⊆ Xa × Xb is an ε-approximate alternating simulation relation from Sa to Sb if the following three conditions are satisfied: 1. for every xa0 ∈ Xa0 there exists xb0 ∈ Xb0 with
(xa0 , xb0 ) ∈ R; 2. for every (xa , xb ) ∈ R we have d(Ha (xa ), Hb (xb )) ≤ ε; 3. for every (xa , xb ) ∈ R and for every ua ∈ Ua (xa ) there exists ub ∈ Ub (xb ) such that for every x0b ∈ Postub
(xb ) there exists x0a ∈ Postua (xa ) satisfying (x0a , x0b ) ∈ R. We say that Sa is ε-approximately alternatingly simulated by Sb or that Sb ε-approximately alternatingly simulates Sa , denoted by
Sa εAS Sb , if there exists an ε-approximate alternating simulation relation from Sa to Sb . Symmetrizing approximate alternating simulation leads to the stronger notion of approximate bisimulation,
denoted by Sa ∼ =εAS Sb , where each system both simulates and is simulated. Software design as a controller synthesis problem is an idea that has been recently gaining enthusiasts despite having
been proposed more than 20 years ago [EC82, MW84]. The starting point is to regard the software to be designed as a system Scont such that the composition Scont × Sτ (Σ) satisfies the desired
specification. If the specification is given as another system Sspec , then we seek to synthesize a controller Scont so that: Scont × Sτ (Σ) εAS Sspec , or even: ε Scont × Sτ (Σ) ∼ =AS Sspec .
In general, this problem is not solvable algorithmically since Sτ (Σ) is an infinite system. We overcome this difficulty by replacing Sτ (Σ) by a finite abstraction Sabs for which we have the
guarantee that if a controller Scont satisfying: Scont × Sabs εAS Sspec 0 exists then a controller Scont satisfying: 0 Scont × Sτ (Σ) εAS Sspec
0 also exists. We call Scont the refinement of Scont . It is shown in [Tab09] that existence of an approximate alternating simulation relation from Sabs to Sτ (Σ) is sufficient to refine the 0
controller Scont acting on Sabs to the controller Scont acting on Sτ (Σ). If we can also establish the existence of an approximate alternating bisimulation relation between Sabs and Sτ (Σ), then we
have the guarantee that if a controller exists for Sτ (Σ), a controller also exists for Sabs . Hence, this design flow is not only sound but also complete. Conditions on η, µ and τ to obtain
approximate alternating simulation or bisimulation relations between Sabs and Sτ (Σ) can be found in [ZPJT09] and references therein.
Pessoa2 is a toolbox automating the synthesis of correctby-design embedded control software. Although the core algorithms in Pessoa have been coded in C, the main functionalities are available
through the Matlab command line. Pessoa Version 1.0 offers three main functionalities: 1. the construction of finite symbolic models of linear control systems; 2. the synthesis of symbolic
controllers for simple specifications; 3. simulation of the closed-loop behavior in Simulink.
- of the abstractions All the transition relations generated by Pessoa, and the sets used in the specifications, are stored as Reduced Order Binary Decision Diagrams (ROBDD)3 through their
corresponding characteristic functions. The usage of ROBDDs enables the efficient computation of reachable sets and pre-images of sets, both fundamental operations in the design of controllers.
Pessoa currently supports the synthesis of controllers enforcing four kinds of specifications defined using a target set Z ⊆ X and a constraint set W ⊆ X: 1. Stay: trajectories start in the target
set Z and remain in Z. This specification corresponds to the Linear Temporal Logic (LTL) formula4 2ϕZ where ϕZ is the predicate defining the set Z; 2. Reach: trajectories enter the target set Z in
finite time. This specification corresponds to the LTL formula 3ϕZ ; 3. Reach and Stay: trajectories enter the target set Z in finite time and remain within Z thereafter. This specification
corresponds to the LTL formula 32ϕZ ; 4. Reach and Stay while Stay: trajectories enter the target set Z in finite time and remain within Z thereafter while always remaining within the constraint set
W . This specification corresponds to the LTL formula 32ϕZ ∧ 2ϕW where ϕW is the predicate defining the set W . 2 Pessoa Version 1.0 can be freely downloaded from http://www.cyphylab.ee.ucla.edu/
Pessoa/. 3 ROBDD’s supported by the CUDD library [CUD]. 4 The semantics of LTL would be defined in the usual manner over the behaviors of Sτ (Σ).
q1 a
q2 u
q3 u
Figure 1: Automaton describing the availability of the shared resources. The lower part of the states is labeled with the outputs a and u denoting availability and unavailability of the shared
resource, respectively. 1 0.8
Figure 3: Evolution of the state variables (left figure) and inputs (right figure), from initial state (x1 , x2 ) = (−1, 0.8), when the automaton in Figure 1 is visiting the states |q2 q3 q1 |q2 q3
q1 |q2 q3 q1 |q2 q3 q1 |q2 q3 q1 | . . .. The input resulting from the low quality actuator is displayed in yellow while the input resulting from the high quality actuator is represented in magenta.
0.6 0.4 0.2 0 ï0.2 ï0.4 ï0.6 ï0.8 ï1 ï1
Figure 2: Domain of the controller forcing the double integrator to remain in [−1, 1] × [−1, 1] under the fairness constraints described by the automaton in Figure 1.
The controllers for the above specifications are memoryless controllers that can be synthesized through fixed point computations as described in [Tab09]. Furthermore, the finite state nature of the
synthesized controllers permits a direct compilation into code. Although code generation is not yet supported in Version 1.0 of Pessoa, closed-loop simulation in Simulink is already available. For
this purpose, Pessoa comes with a Simulink block implementing a refinement of any synthesized controller. The controllers synthesized in Pessoa are, in general, nondeterministic. The Simulink block
resolves this nondeterminism in a consistent fashion thus providing repeatable simulations. In order to increase the simulation speed, the Simulink block selects, among all the inputs available for
the current state, the input with the shortest description in the ROBDD encoding the controller. Moreover, the input is chosen in a lazy manner, i.e., the input is only changed when the previously
used input cannot be used again. Other determinization strategies, such as minimum energy inputs, will be supported in future versions of Pessoa.
In order to illustrate the capabilities of Pessoa we consider a control system that has permanent access to a low quality actuator and sporadic access to a high quality actuator. This scenario arises
when the high quality actuator is connected to the controller through a shared network, or consumes large amounts of energy drawn from a shared batery. Moreover, we also assume that we do not have at
our disposal a model for the other software tasks competing for the
Figure 4: Evolution of the state variables (left figure) and inputs (right figure), from initial state (x1 , x2 ) = (−1, 0.8), when the automaton in Figure 1 is visiting the states |q1 q2 q1 |q2 q1
q2 |q1 q2 q1 |q2 q1 q2 |q2 q1 q2 | . . .. The input resulting from the low quality actuator is displayed in yellow while the input resulting from the high quality actuator is represented in magenta.
shared resources. This is typically the case when such software tasks are being concurrently designed. However, even if we had models for these software tasks, the complexity of synthesizing the
control software using these models would be prohibitive. Therefore, we shall impose a simple fairness requirement mediating the access to the shared resources. To make the ensuing discussion
concrete, we assume that three tasks can have access to the shared resources, one of them being the control task. We use the expression time slot to refer to time intervals of the form [kτ, (k + 1)τ
[ with k ∈ N and where τ is the time quantization parameter. If we consider sequences of three consecutive time slots, the fairness requirement imposes the availability of the actuator in at least
one time slot. Some availability sequences satisfying this assumption are: |uaa|uau|auu|uua|uua|auu|uua|uaa|aua|... |uau|uau|uau|aua|uaa|uau|auu|aaa|aaa|... where we denoted by a the availability of
the resources, by u the unavailability, and separated the sequences of three time slots with the symbol |. Since the preceding sequences form an ω-regular language they can be described by the
automaton represented in Figure 1. The system Σ to be controlled is a double integrator: x˙ 1
= x2
x˙ 2
= ulow + uhigh .
where ulow denotes the input produced by the low quality actuator and uhigh denotes the input produced by the high
quality actuator. Any of the actuators generates piecewise constant inputs taking values in U = {−1, −0.5, 0, 0.5, 1}. However, when an input u ∈ U is requested to the low quality actuator, the
actual generated input ulow is an element of the set [u − 0.6, u + 0.6]. In contrast, the high quality actuator always produces the input that is requested, i.e., uhigh = u. The control objective is
to force the trajectories to remain within the target set Z = [−1, 1] × [−1, 1]. The fairness constraint is also a control objective that can be expressed by resorting to a model for the concurrent
execution of Sτ (Σ) and the automaton in Figure 1. When the automaton is in state q1 , any of the actuators can be used. However, when the automaton is in the state q2 or q3 only the low quality
actuator can be used. Although this kind of specification is not natively supported in Pessoa, it can be handled by providing Pessoa with a Matlab file containing an operational model for the
concurrent execution of Sτ (Σ) and the automaton in Figure 1. Choosing D = [−1, 1] × [−1, 1] as the domain of the symbolic abstraction, and τ = 0.1, η = 0.05, and µ = 0.5 as quantization parameters,
Pessoa computes the symbolic abstraction in 109 seconds and synthesizes a controller in 2 seconds (on a MacBook Pro with a 2.26 GHz Intel Core 2 Duo processor and 2GB of memory). The domain of the
controller is shown in Figure 2 and two typical closed-loop behaviors are shown in Figures 3, and 4. We can appreciate the controller forcing the trajectories to stay within the target set despite
the low quality of the permanently available actuator. We note that if we require the high quality actuator to be permanently unavailable, Pessoa reports the non-existence of a solution.
Pessoa is currently being extended in the following directions: • Nonlinear and switched dynamics can already be used in Pessoa, albeit not natively. Future versions of Pessoa will provide native
support for these classes of systems; • Specifications with discrete memory can be used with Pessoa by encoding them in the plant dynamics as briefly reported in Section 4. Future versions of Pessoa
will natively support specifications given in LTL and automata on infinite strings; • The state set of the abstractions computed by Pessoa is a grid resolution η. However, the results reported in
[ZPJT09, JT10] do not require the use of a grid of constant resolution. We are currently working on extending Pessoa to multi-resolution grids with the objective of reducing the size of the computed
abstractions and controllers. • We are also extending Pessoa to support quantitative control objectives. Preliminary steps in this direction addressing the synthesis of time-optimal controllers are
reported in [JT10].
6. [Ari]
REFERENCES Ariadne: An open tool for hybrid system analysis. Available at: http://trac.parades.rm.cnr.it/ariadne/.
A. Arnold, A. Vincent, and I. Walukiewicz. Games for synthesis of controllers with partial observation. Theoretical Computer Science, 28(1):7–34, 2003. [Che] Checkmate: Hybrid system verification
toolbox for matlab. Available at: http://www.ece.cmu.edu/~webk/checkmate/. [CL99] C. Cassandras and S. Lafortune. Introduction to discrete event systems. Kluwer Academic Publishers, Boston, MA, 1999.
[CUD] CUDD: CU Decision Diagram Package. Available at: http://vlsi.colorado.edu/~fabio/CUDD/. [dAHM01] Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Symbolic algorithms for infinite-state
games. In CONCUR 01: Concurrency Theory, 12th International Conference, number 2154 in Lecture Notes in Computer Science, 2001. [EC82] E. A. Emerson and E. M. Clarke. Using branching time temporal
logic to synthesize synchronization skeletons. Science of Computer Programming, 2:241–266, 1982. [Hyba] Hybridsal. Available at: http://sal.csl.sri.com/hybridsal/. [Hybb] Hybrid Toolbox. Available
at: http://www.dii.unisi.it/hybrid/toolbox. [JT10] Manuel Mazo Jr. and Paulo Tabuada. Approximate time-optimal control via approximate alternating simulations. 2010. To appear at the American Control
Conference 2010. Available at: http://www.cyphylab.ee.ucla.edu/. [KeY] Keymaera: A hybrid theorem prover for hybrid systems. Available at: http://symbolaris.com/info/KeYmaera.html. [KG95] R. Kumar
and V.K. Garg. Modeling and Control of Logical Discrete Event Systems. Kluwer Academic Publishers, 1995. [LTL] LTLCon. Available at: http://iasi.bu.edu/~software/LTLcontrol.htm. [MW84] Z. Manna and
P. Wolper. Synthesis of communication processes from temporal logic specifications. ACM Transactions on Programming Languages and Systems, 6:68–93, 1984. [PGT08] G. Pola, A. Girard, and P. Tabuada.
Approximately bisimilar symbolic models for nonlinear control systems. Automatica, 44(10):2508–2516, 2008. [PHA] Phaver: Polyhedral hybrid automaton verifyer. Available at: http://www-artist.imag.fr/
~frehse/phaver_web/index.html. [Tab09] Paulo Tabuada. Verification and Control of Hybrid Systems. Springer, 2009. [ZPJT09] M. Zamani, G. Pola, M. Mazo Jr., and P. Tabuada. Symbolic models for
nonlinear control systems without stability assumptions. 2009. Submitted for publication. Available at: http://www.cyphylab.ee.ucla.edu.
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A Heuristic Approach to Shelf Space Allocation Decision Support Including Facings, Capping, and Nesting
Department of Process Management, Wroclaw University of Economics and Business, 53-345 Wroclaw, Poland
Author to whom correspondence should be addressed.
Submission received: 23 January 2021 / Revised: 7 February 2021 / Accepted: 9 February 2021 / Published: 13 February 2021
Shelf space on which products are exhibited is a scarce resource in the retail environment. Retailers regularly make decisions related to allocating products to their outlets’ limited shelf space.
The aim of the paper was to develop a practical shelf space allocation model offering the possibility of horizontal and vertical product grouping, representing an item (product) with facings,
capping, and nesting, with the objective of maximizing the retailer’s profit. Because real category-management problems address a lot of retailer’s rules, we expanded the basic shelf space allocation
model, using shelf constraints, product constraints, multi-shelves constraints, and category constraints. To solve the problem, we proposed two adjustable methods that allowed us to achieve good
results within a short time interval. The validity of algorithms was estimated, using the CPLEX solver and illustrated with example problems. Experiments were performed on data generated on the basis
of real retail values. To estimate the performance of the proposed approach, 45 cases were tested. Among them, the proposed approach found solutions in 34 cases, while CPLEX found solutions only in
23 cases. The profit ratio of the proposed approach is, on average, 94.57%, with minimal and maximal values of 86.80% and 99.84%, accordingly.
1. Introduction
Shelf space is one of the most expensive resources in retail. Retailers regularly make decisions related to the allocation of products to their outlets’ limited shelf space. Increasing retailers’
profit by attracting customers’ attention and encouraging them to perform additional purchases can be implemented by proper category management in the retail store. Defining enough shelf space for a
product and finding the best product position on planogram shelves are important issues with respect to shelf space allocation.
Appropriate shelf space allocation is one of the main factors to obtaining an edge in the very competitive retail industry [
]. Stores’ growing product ranges cause a great challenge for retailers. Many models allocating a large number of products to shelf space have been developed over recent years, with the goal to
optimize the retailer’s objective under certain constraints and merchandising rules within the store.
The significance of store space allocation approaches and their importance for researchers and retailers in order to make correct decisions was highlighted by Yang and Chen [
]. Shelf space allocation decisions involve two steps: the amount of shelf space for a particular product category and the amount of shelf space for a particular product within each product category
Many consumers decide on what to purchase already inside the store. They do not always prefer to choose the cheapest product from the range offered. Instead, a lot of factors, such as the perceived
quality of the product, loyalty to a particular brand, brand reputation, appetite to try something new, and other marketing variables, impact their choice [
]. With appropriate merchandising tactics, the retailer can try to obtain more profit from consumers when they visit the store.
As the shelf space allocation problem (SSAP) is, in many cases, NP-hard, specialized heuristics and metaheuristics should be used. Such algorithms allow one to attain a satisfactory near-optimal
solution but still do not guarantee a globally optimal value [
]. Yang [
] developed a heuristic that extends an approach often applied to the knapsack problem in order to find the solution to the SSAP, considering the profit from a product unit as weight and weight
taking order as the priority index in space planning. His algorithm consists of preparatory, allocation, and termination phases. Yang and Chen [
] classified shelf space management strategies, including dominance, adaptation, and passiveness, on the grounds of a questionnaire survey. Drèze et al. [
] proposed a model to measure the effect of the reallocation of a product on the shelf and the between the sales of individual brands inside the product category. Gajjar and Adil [
] developed some heuristics to solve the retail SSAP with a linear profit function. Reyes and Frazier [
] proposed nonlinear integer SSAP for grocery stores, considering the relationships between customer service factors and profitability. Frontoni et al. [
] proposed a heuristic approach to the planogram supply to find and count multiple facings of the same item of a product. Fadıloğlu et al. [
] prepared an optimization model that created an appropriate product shelf mix based on the store requirements, to maximize profit, that can be used by the retailers. Jajja [
] used a dynamic model to solve the SSAP in the retail shop. He also performed a sensitivity analysis for some variables in the model for various space allocation scenarios in stores. Hübner and
Schaal [
] studied the demand effects and developed the retailer’s optimization model that takes into account stochastic, space-dependent, and vertical-shelf-positions-dependent demand. They proposed a
solution for both cases, i.e., with cross-space elasticity and without it. Hübner and Schaal [
] included, in their model, the effects of replenishment and inventory costs. Valenzuela and Raghubir [
] claimed that the bottom shelves were dedicated to cheaper brands, while luxury brands should be located on the top shelves. Furthermore, Desrochers and Nelson [
] declared that consumers’ purchasing decisions depended on the correct categorization into product families that specified associations between products and the visual display on the shelf.
Table 1
provides an overview of approaches used to solve the SSAP.
In the previous research, only facings of product items were considered. Moreover, one of the key limitations in the literature is that it neglects merchandising rules based on the frequency of
product moves and the product price. More precisely, it disregards the different vertical shelf levels based on product sales potentials. Missing is a model which simultaneously takes into account
retailers’ location and practical pricing rules.
The objective of this article was to develop a practical retail shelf space allocation model with not only facings but also capping (i.e., placing a product on top of another one and rotating the top
one) and nesting (placing a product inside another one, e.g., a basket or plate). The main contribution of the paper is the development of an SSAP model characterized by the following:
• Cap and nest parameters.
• Vertical and horizontal category rows by appropriate product grouping, which makes planograms visually attractive to customers; such an effect is very important in visual merchandising for
retailers, who, in general, prefer to design their own display spaces on planograms.
• Associating products based on their sales potentials for properly locating them on the bottom or top shelves, as well as creating vertical product families and grouping them to make the planogram
visually attractive.
We proposed a realistic SSAP model, considering four groups of constraints, namely shelf constraints, product constraints, multi-shelves constraints, and category constraints. We also developed a
heuristic approach and performed experiments on various practical problem sizes, estimating the quality of the solution with the CPLEX solver. In our approach, we placed special emphasis on the
possibility of its practical implementation by retail category managers, in order to deliver higher profits.
The remainder of the article is structured as follows.
Section 2
describes the data and methods. Numerical results of the computational experiments are given in
Section 3
Section 4
outlines the conclusion of our research.
2. Data and Methods
Retailers focus on efficiently arranging products on shelves, with the goal of maximizing profit margin/sales, improving customer satisfaction, and managing stock control. Planograms are graphic
representations of products’ arrangement on shelves. With their help, retailers can define the position of a product and the number of its units. Besides the fact that the SSAP is only a small part
on the whole set of the category management processes, it is an easy way to earn profits.
2.1. Variables and Parameters
In our research, we used the following variables and parameters. Subscripts indicate the variables’ indexes, and superscripts indicate the variables’ descriptions and should not be read as indexes.
Parameters and indices are as follows:
• $K$—total number of categories.
• $G$—total number of subcategories.
• $S$—total number of shelves.
• $P$—total number of products.
• $k$—category index, $k = 1 , … , K$.
• $g$—subcategory index,$g = 1 , … , G$.
• $i$—shelf index, $i = 1 , … , S$.
• $j$—product index, $j = 1 , … , P$.
Category parameters are as follows:
• $m k$—minimum category $k$ size per cent of the shelf length.
• $t k$—maximum category size tolerance between shelves in category $k$.
Shelf parameters:
• $s i l$—length of the shelf $i$.
• $s i d$—depth of the shelf $i$.
• $s i h$—height of the shelf $i$.
• $s i b$—weight limit of the shelf $i$.
Product parameters:
• $p j w$—width of the product $j$.
• $p j d$—depth of the product $j$.
• $p j h$—height of the product $j$.
• $p j b$—weight of the product $j$.
• $p j s$—supply limit of the product $j$.
• $p j u$—unit profit of the product $j$.
• $p j k$—category of the product $j$.
• $p j g$—subcategory of the product $j$.
• $p j n$—nesting coefficient of the product $j$, $p j n < 1$, or $p j n = 0$ if the product cannot be nested.
• $p j o 1$—front orientation binary parameter.
• $p j o 1 = { 1 , if front orientation is available for product j 0 , otherwise }$
• $p j o 2$—side orientation binary parameter.
• $p j o 2 = { 1 , if side orientation is available for product j 0 , otherwise }$.
• $p j l$—cluster of the product $j$.
• $f j min$—minimum number of facings of the product $j$.
• $f j max$—maximum number of facings of the product $j$.
• $c j min$—minimum number of caps per facings group of the product $j$.
• $c j max$—maximum number of caps per facings group of the product $j$.
• $n j min$—minimum number of nests of one facing of the product $j$.
• $n j max$—maximum number of nests of one facing of the product $j$.
• $s j min$—minimum number of shelves on which product $j$ can be allocated.
• $s j max$—maximum number of shelves on which product $j$ can be allocated.
Decision variables are as follows:
• $x i j = { 1 , if product j is put to the shelf i 0 , otherwise }$
• $f i j$—number of facings of the product.
• $c i j$—number of caps of the product.
• $n i j$—number of nests of the product.
• $y i j o 1 = { 1 , if product j is put to the shelf i on front orientation 0 , otherwise }$.
• $y i j o 2 = { 1 , if product j is put to the shelf i on side orientation 0 , otherwise }$.
2.2. Problem Definition and Methods
The problem can be formulated as follows. The $P$, products, are assigned to a $K$, categories (product families), and $G$, subcategories (sales potentials), and then they are placed on $S$, shelves,
on a planogram. The minimal space occupied by a category is defined on each planogram, which makes it visually attractive for customers. All categories are vertical, i.e., the products can be placed
on all shelves. Obviously, products of the same category cannot be split by another vertical category. The sales potential subcategories are horizontal, i.e., products can be placed on the shelves
with a greater or equal subcategory but cannot be placed on the shelf for a lower subcategory. This models the situation where more expensive products (with higher sales potentials) are located on
the higher shelves (at eye level); cheaper products can be located on the lower shelves and on the higher ones. However, the overpriced brands cannot be located underneath. The problem is to find the
optimal shelf space for each product category, $K$, defining the quantities of each product in order to maximize the retailers’ profit.
Table 2
explains the methods of allocating products on shelves based on a retailer’s rules. Products from category A and subcategory 10 can be placed on the shelves dedicated to categories A, B, and C, as
well as subcategories 10, 20, and 30. Under other conditions, products from the category C and subcategory 30 cannot be placed on the shelves for other categories and subcategories. The next example
represents B20, which can be placed on shelves B20, B30, C10, C20, and C30. Sales potentials are assigned with an initially selected step (in this case, it equals 10) because, in practice, if base
product ranges are assigned to sales subcategories 10 and 20, but one seasonal or promotional shelf is included on the planogram only for a 2-week time shift, the retailer can easily assign seasonal
or promotional products to sales potential 15, without reassigning the sales potentials of the base product range.
Figure 1
Figure 2
show category sizes on shelves with different sales potentials. The divisor which splits the categories may be flexible (
Figure 1
) or strict (
Figure 2
), which is steered by coefficient
$m k$
(the minimum category size is
$m k$
, a percent of the shelf length) and coefficient
$t k$
(the maximum category size tolerance between shelves in the category, a percent of the shelf length). The idea for this is taken from practice: The category must exist on the planogram, the products
in this category must be noticeable by customers, and beautiful vertical category rectangles should be built in categories.
The product can be placed on the shelf in a front orientation (width $p j w$ is taken as line parameter) or side orientation, at 90 degrees (depth $p j d$ is taken as width of the product). The
products with the same characteristics, functions, or tastes can be grouped into clusters, to make a certain substitution effect and must be placed on the same shelf. This is done in an out-of-stock
situation, when the customer can easily choose a similar product from another brand.
As in a real retail store, and depending on the product package’s physical characteristics, products can be capped, nested, or neither. Capping is product placement on top of one another, e.g., in
the case of rectangular boxes, as with tea or sweets boxes. Nesting is product placement inside another one, as with bowls or plates.
For products with a front orientation, the number of facings in one capped group is
$⌈ p j h / p j w ⌉$
, and for products with a secondary side orientation, it is
$⌈ p j h / p j d ⌉$
. For cappings, if too many tea boxes are placed above the facings in the row below, the boxes below may become damaged or destroyed, or upper tea boxes could fall off the shelf. For nestings, if too
many bowls are placed inside the lowest bowl on the shelf, the bowl below may become damaged or destroyed, or the upper bowls could fall off the shelf. The total number of product items is the sum of
facings, caps, and nests.
Figure 3
Figure 4
illustrate the caps and nests allocation method.
In our research, we considered one (top) row of facings with cappings or nestings on it. The number of facings in the vertical dimension is not modeled. The shelf height and product supply limit were
defined only for the top row of facings, including capping or nesting. Similarly, the depth of the products and the depth of the shelf were defined for the same top row of facings.
To solve the problem, the task was to define the number of facings ($f i j$), caps ($c i j$), and nests ($n i j$) of product $j$ placed on shelf $i$, with regard to constraints, which are grouped
into 4 classes: shelf constraints, product constraints, multi-shelves constraints, and category constraints, in order to maximize the retailer’s total profit.
2.3. Basic Notions
2.3.1. Problem Formulation
The non-linear integer model can then be formulated as follows:
$max ∑ j = 1 P ∑ i = 1 S x i j p j u ( f i j + c i j + n i j ) .$
2.3.2. Shelf Constraints
The shelf constraints are as follows:
$∀ ( i ) [ ∑ j = 1 P f i j ( y i j o 1 p j w + y i j o 2 p j d ) ≤ s i l ] ,$
shelf length;
$∀ ( i , j ) [ x i j ( p j h + ⌈ c i j x i j max ( ⌊ f i j ( y i j o 1 p j w + y i j o 2 p j d ) p j h ⌋ , 1 ) ⌉ · ( y i j o 1 p j w + y i j o 2 p j d ) + + ⌈ n i j x i j max ( f i j , 1 ) ⌉ · p j h
p j n ) ≤ s i h ] ,$
shelf height;
$∀ ( i , j ) [ x i j ( y i j o 1 p j d + y i j o 2 p j w ) ≤ s i d ] ,$
shelf depth;
$∀ ( i ) [ ∑ j = 1 P ( f i j + c i j + n i j ) p j b ≤ s i b ] ,$
shelf weight.
Constraint (3) signifies that the height of each product on the shelf fit the shelf’s height limit. This was also true for the caps’ (
Figure 3
) height and nests’ height (
Figure 4
). As for the front-oriented products, their width represents the additional height to the product facings. Otherwise, as for the side-oriented products, their depth represents the additional height
to the product facings. A construction such as
$x i j / max ( f i j , 1 )$
is used for omitting the division by 0 cases, i.e., if the product is not placed on the shelf.
2.3.3. Product Constraints
The product constraints are as follows:
$∀ ( j ) [ s j min ≤ ∑ i = 1 S x i j ≤ s j max ] ,$
minimum and maximum number of shelves;
$∀ ( j ) [ ∑ i = 1 S ( f i j + c i j + n i j ) ≤ p j s ] ,$
supply limit;
$∀ ( j ) [ f j min ≤ ∑ i = 1 S f i j ≤ f j max ] ,$
minimum and maximum number of facings;
$∀ ( i , j ) [ c j min ≤ c i j ≤ c j max · ⌊ f i j ( y i j o 1 p j w + y i j o 2 p j d ) p j h ⌋ ] ,$
minimum and maximum number of caps;
$∀ ( i , j ) [ n j min ≤ n i j ≤ n j max f i j ] ,$
minimum and maximum number of nests;
$∀ ( i , j ) [ c i j · n i j = 0 ] ,$
either caps or nests.
2.3.4. Multi-Shelves Constraints
The multi-shelves constraints are as follows:
$∀ ( i , j ) [ y i j o 1 · y i j o 2 = 0 ] ,$
only one orientation (front or side) is available;
$∀ ( i , j ) [ y i j o 1 + y i j o 2 = 1 ] ,$
only one orientation (front or side) is available;
$∀ ( j ) [ max i = 1 , … , S ( y i j o 1 ) ≠ max i = 1 , … , S ( y i j o 2 ) ] ,$
the same product orientation on all shelves;
$∀ ( i , j ) [ y i j o 1 ≤ p j o 1 ] ,$
front orientation is possible;
$∀ ( i , j ) [ y i j o 2 ≤ p j o 2 ] ,$
side orientation is possible;
$∀ ( j ) ∀ ( a , b : | a − b | ≠ 1 ∧ a < b , a , b = 1 , … S ) [ x a j · x b j = 0 ] ,$
the next shelf only;
$∀ ( i ) ∀ ( a , b : p a l = p b l , ? a , b = 1 , … P ) [ x i a = x i b ] ,$
the same cluster on the same shelf.
The same orientation on all shelves (Constraint (14)) is needed for building a better visual display of blocks on all shelves. For the same reason, products can be placed only on the neighboring
shelf (Constraint (17)).
2.3.5. Category Constraints
The category constraints are a follows:
$∀ ( i , k ) [ ∑ j = 1 , p j k = k P f i j ( y i j o 1 p j w + y i j o 2 p j d ) ≥ [ s i l · m k ] ∨ ∑ j = 1 , p j k = k P f i j = 0 ] ,$
minimum category size;
$∀ ( k ) [ max i = 1 , … , S ( ∑ j = 1 , p j k = k P f i j ( y i j o 1 p j w + y i j o 2 p j d ) ) − − min i = 1 , … , S ( ∑ j = 1 , p j k = k P f i j ( y i j o 1 p j w + y i j o 2 p j d ) ) ≤ [ max
i = 1 , … , S ( s i l ) · t k ] ] ,$
category tolerance between different shelves;
$∀ ( i , j ) [ x i j ≤ min ( max ( s i g − p j g , 0 ) , 1 ) ] ,$
sales potentials subcategory;
$[ max i = 1 , … , S ( s i l ) · t k ] = ⌊ max i = 1 , … , S ( s i l ) · t k + 0 , 5 ⌋ ,$
means rounded value.
Constraint (20) is used to form the border between neighboring categories on different shelves. Products with lower sales potentials, compared to the shelf, can be placed on shelves with greater
sales potentials; otherwise, products with greater sales potentials, compared to the shelf, cannot be placed on the shelves with lower sales potentials (Constraint (21)).
2.3.6. Relationship Constraints
The relationship constraints are as follows:
$∀ ( i , j ) [ x i j s i l ( y i j o 1 p j w + y i j o 2 p j d ) ≥ f i j ] ,$
facings relationships;
$∀ ( i , j ) [ x i j ≤ f i j ( y i j o 1 + y i j o 2 ) ] ,$
facings relationships;
$∀ ( i , j ) [ c i j ≤ x i j c j max · ⌊ f i j ( y i j o 1 p j w + y i j o 2 p j d ) p j h ⌋ ] ,$
capping relationships;
$∀ ( i , j ) [ n i j ≤ x i j n j max f i j ] ,$
nesting relationships.
2.3.7. Decision Variables
The decision variables are as follows:
product is placed on the shelf;
$f i j = { f j min … f j max } ,$
number of facings;
$c i j = { c j min … c j max · ⌊ f i j ( y i j o 1 p j w + y i j o 2 p j d ) p j h ⌋ } ,$
number of caps;
$n i j = { n j min … n j max · f j max } ,$
number of nests;
$y i j o 1 ∈ { 0 , 1 } ,$
front orientation;
$y i j o 2 ∈ { 0 , 1 } ,$
side orientation.
2.4. Proposed Approach
The main steps of the proposed approach include the following:
• Solve the decision problem in order to identify if the product can be placed on the shelf with regard to the multi-shelves constraints (Constraints (12)–(18)) and sales potentials subcategory
Constraint (21). During this step, we set all possible products for shelf allocations.
• Select from the received set only allocations where number LB of product items fit shelf ((2)–(5)) and product ((6)–(11)) constraints.
• Solve the optimization problem (1) with regard to shelf Constraints (2)–(5), product Constraints (6)–(11), and category Constraints (19) and (20).
In this section, the following notations are used:
$α$—integer coefficient of dividing shelf length into a number of product widths.
$β$—percent of solutions to be taken to the next steps of the algorithm.
The proposed algorithm of Step 3 is executed as follows. For the given shelf length,
$s i l$
, define all possible variations of line segments divisible by the product width,
$α p j w$
(or depth,
$α p j d$
, based on its orientation). Integer coefficient
is selected empirically, i.e., the greater shelf length, the greater
. Assign an appropriate number of facings, caps, and nests into the line segments for the given product. Thus, a number of possible solutions emerge. Apply this to all shelves in all categories.
Exclude results where shelf ((2)–(5)), product ((6)–(11)), and category (19)–(20) constraints do not fit. Calculate the profit of each received solution. Apply Algorithm 1 and Algorithm 2 to the
received solution set.
Algorithm 1 Steps
• $Group all solutions by profit ∑ j = 1 P ∑ i = 1 S p j u ( f i j + c i j + n i j ) .$
• $For each one , take only that with minimum occupied space ∑ j = 1 P ∑ i = 1 S f i j ( y i j o 1 p j w + y i j o 2 p j d ) .$
• $Sort them in a non - ascending order of profit - to - width ratio ∑ i = 1 S ∑ j = 1 P p j u ( f i j + c i j + n i j ) ∑ i = 1 S ∑ j = 1 P f i j ( y i j o 1 p j w + y i j o 2 p j d ) and take β$,
an appropriate percentage of them (if there are too many of them).
• Select the best one.
There are three possible cases.
• The solution is $∑ j = 1 P ∑ i = 1 S p j u ( f i j + c i j + n i j ) > 0$. In this case, decrease $α$, so additional possible solutions appear. Analyze only branches that give a greater total
profit than the one received with the previous $α$. If there are too many solutions to be analyzed, create profit branches for each category. In this case, the solution is found; the profit for
each separate category is also known. Analyze only branches with higher profit categories, in order to create new possible solutions, and use only them. Leave less profitable profit category
branches; choose appropriate percentage of branches to be analyzed based on their number received and the time used for generating them.
• The solution has not been found, and all values have been analyzed, so $∑ j = 1 P ∑ i = 1 S p j u ( f i j + c i j + n i j ) = 0$. Further, decrease $α$ and analyze extra possible solutions.
• Not all values are analyzed at the given time limit. The solution has not been found, so $∑ j = 1 P ∑ i = 1 S p j u ( f i j + c i j + n i j ) = ∞$. In this case, increase $α$ so less possible
solutions to be checked are generated, and repeat Algorithm 1.
Repeat Algorithm 1 an appropriate number of times, adjusting
Algorithm 2 Steps
• $Group all solutions according to occupied space ∑ j = 1 P ∑ i = 1 S f i j ( y i j o 1 p j w + y i j o 2 p j d ) .$
• $For each one , take only that with maximum profit ∑ j = 1 P ∑ i = 1 S p j u ( f i j + c i j + n i j ) .$
• $Sort them in a non - descending order of space occupied on the shelves and take β$, an appropriate percentage of them (if there are too many of them).
• Select the best one.
There are also three possible cases. Actions to be performed on them are the same as in Algorithm 1. Repeat Algorithm 2 an appropriate number of times, adjusting $α$ and $β$.
3. Results
The computation experiments evaluated the quality of the solutions of the proposed approach to solving the current SSAP. To examine the performance of the proposed algorithm, simulated test problems
based on real data were solved.
We simulated nine stores, where planograms with the defined product sets had to be placed. As in a retail store, shelf widths vary in different stores, but retailers must allocate in each store the
same set of products. We simulated five planogram widths with three shelves with different sales potentials. The widths of all shelves on the planogram were the same because this was the regular case
in a real store. Very rarely did a situation occur where store obstructions or inconvenient parts of the shelf decreased the shelf width. The prepared nine product sets included 10, 15, 20, 25, 30,
35, 40, 45, and 50 products with different widths, weights, and heights. The shelves widths were 250, 375, 500, 625, and 750 cm. The numbers of vertical categories were two (for 10, 15, and 20
products), three (for 25 and 30 products), four (for 35 and 40 products), and five (for 45 and 50 products).
The computational experiments were implemented in Visual C# 2015 and MS SQL Server 2014. The optimal (or maximum feasible, in some cases) solution to compare with was found with IBM ILOG CPLEX
Optimization Studio Version: 12.7.1.0.
The benefit of both proposed algorithms is the lack of randomly selected elements, so there is no requirement to run them several times. Moreover, the data on all steps are visible and understandable
for retailers, so all initial parameters can be easily checked, and the coefficient parameters can also be simply approximated, while watching the data and the received results.
Table 3
shows the number of solutions found by both algorithms, where the solution of the proposed approach was the maximum feasible solution. Algorithm 2 was better, finding the maximum feasible solution in
25 cases.
Table 4
compares the proposed approach to the CPLEX solver, which found the solution in 34 cases, while CPLEX did in only 23, and affirmed the necessity of our implementations.
In order to estimate the performance of the algorithm, the following notations are used:
$U a$—the profit of the proposed algorithm (Formula (1)).
$U o$—the profit of the CPLEX solution.
$U a / U o$—the profit ratio of the proposed algorithm to the optimal (or maximum feasible, in some cases) solution.
Table 5
shows the quality of the proposed approach, compared to the CPLEX solver. The average profit ratio of the best algorithm was 94.57%, with its minimal and minimal and maximal values at 86.80% and
99.84%, respectively. It could be observed that there are some advantages of the proposed approach. In 11 cases, the proposed approach found the solution, but CPLEX did not. Moreover, no opposite
situation occurred where CPLEX found the solution but the proposed approach did not. In 11 cases, a solution was not found at all.
This case simulated the situation in a real retail store, where retailers requested something that could not be implemented based on the provided constraints. General mistakes that often happened in
the retailer’s requirements are as follows:
• Even if the decision problem was successfully solved and a product could be placed on the shelf, the lower bounds numbers of a product on the shelf exceeded the shelf length or weight.
• Even if the lower bounds numbers satisfied the shelf length and weight, there were too many products in the category to be placed in aesthetically pleasing columns, which restricted the category
constraints. The proposed corrections divided them into the categories in another way and changed the minimum category size or category tolerance.
• Even if there were lower bounds of product number, shelf length, and weight fit, the sales potentials of the products were incorrect, so that, on one shelf, there were too many products, and
there were too few on another. In this case, it would be better to move some products from the most numerous subcategory to another sales potential subcategory.
These errors are visible and could be analyzed while processing each instance. Therefore, there is no reason to waste time trying to find the solution by using data in which we know every solution
step fails to satisfy initial constraints.
Table 5
also illustrates the total number of product allocations received when solving the product-on-shelf decision problem. The last column shows the reduced number of product allocations, which was enough
to be processed according to the proposed solution approach. The main idea was that, at first, we estimated how many decisions existed, and for each, we adjusted the appropriate number of product
facings, caps, and nests. If we had enough time, we processed all of the product allocations. If there were too many allocations to be processed and it could not be done in a reasonable time, we
reduced the number of allocations to be processed and selected more profitable ones according to Algorithms 1 and 2. When solving the optimization problem, we did not check all permutations of
facings either, because there were too many variants. We also used the idea of the proposed Algorithms 1 and 2, which generated only profitable variants.
Table 6
presents the processing time taken to look for the solution while solving the optimization problem by both algorithms, which, on average, amounts to 5.39 min and varies from 2 s up to 83.98 min. The
total processing time includes all preparation steps and solves the decision problem. On average, it amounts to 9.53 min and varies from 2 s to 87.11 min. The time limit for the CPLEX solver was set
as 15 min because it was much more than the average and median time of the algorithms, and the standard deviation was approximately 15 min. In most cases, both proposed approaches were faster.
4. Conclusions
Shelf space is one of the most important resources used to involve more customers in category management decisions.
In this article, we proposed a new practicable mathematical shelf space allocation model with regard to horizontal products grouped into categories and vertical shelf levels grouped into sales
potentials subcategories. The model includes cap and nest parameters. We developed two heuristic approaches and explained, in detail, how to implement them during category management problems. We
also offered suggestions to space planners, focusing on general mistakes in requirements and how they should be corrected.
To estimate the performance of the proposed approach, 45 cases were tested. Among them, the proposed approach found solutions in 34 cases, while CPLEX achieved it only in 23 cases. The profit ratio
of the proposed approach was, on average, 94.57%, with its minimal and maximal values being 86.80% and 99.84%, respectively.
The main feature that differentiates the proposed approach from the previous ones was the lack of randomly generated or randomly adjusted elements. At first, we solved the decision problem in order
to check if the product fit the shelf; next, we solved the optimization problem, finding the appropriate number of facings of each product. Moreover, the data at all described steps could be easily
analyzed and allowed us to adjust the steering parameters.
The main disadvantages of the proposed approach are as follows. If the product width (or depth, based on its orientation) is very small, there are too many possible variations of line segments
initially generated. In situations where there are too many variants to be checked and the solution is not found within the given time, it is recommended to increase $α$, dividing the shelf into
larger parts. This method can decrease the number of variants and decrease processing time, which worsens the solution quality by increasing the distance of the feasible solution that can be found to
the optimal one.
Our approaches are valuable for retailers looking for expert advice on techniques that could be used to solve the SSAP, which would enable them to maximize profit, minimize lost revenue, and find a
balance between ensuring customer satisfaction and dealing with multiple inputs.
Author Contributions
Conceptualization, K.C. and M.H.; methodology, K.C.; software, K.C.; validation, K.C. and M.H.; formal analysis, K.C.; investigation, K.C.; resources, K.C. and M.H.; data curation, K.C.;
writing—original draft preparation, K.C.; writing—review and editing, M.H.; visualization, K.C.; supervision, M.H.; project administration, M.H.; funding acquisition, M.H. All authors have read and
agreed to the published version of the manuscript.
This research was funded by the Ministry of Science and Higher Education in Poland, under the program “Regional Initiative of Excellence” 2019–2022 project number 015/RID/2018/19, total funding
amount 10,721,040,00 PLN.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. Allocation of categories and sales-potential subcategories, with a flexible border between the vertical categories. The higher sales-potential subcategory (10, 20, 30) is, the darker is the
color. Different main colors (blue, green, red) represent different categories (A, B, C).
Figure 2. Allocation of categories and sales-potential subcategories, with a strict border between the vertical categories. The higher sales-potential subcategory (10, 20, 30) is, the darker is the
color. Different main colors (blue, green, red) represent different categories (A, B, C).
Table 1. Overview of literature on solution approaches. Abbreviations used in the table are as follows: EA—exact approach; BB—branch-and-bound; DP—dynamic programming; H—heuristics; GA—genetic
algorithm; SA—simulated annealing; TS—tabu search; VNS—variable neighborhood search; SW—squeaky wheel; ABC—artificial bee colony; PSO—particle swarm optimization; GRA—gradient approach; ●—author’s
solution approach.
Author Year EA BB DP H GA SA TS VNS SW ABC PSO GRA
Anderson and Amato [15] 1974 ●
Hansen and Heinsbroek [16] 1979 ●
Corstjens and Doyle [17] 1981 ●
Zufryden [18] 1986 ●
Bultez and Naert [19] 1989 ●
Borin, Farris, and Freeland [20] 1994 ●
Borin and Farris [21] 1995 ● ●
Urban [22] 1998 ● ● ●
Li and Tsai [23] 2001 ●
Moholkar and Sanjeev [24] 2001 ●
Rajaram and Tang [25] 2001 ●
Yang [5] 2001 ●
Rodrigues, Lim, and Qian [26] 2002 ● ●
Lim, Rodrigues, and Zhang [1] 2004 ● ● ●
Bai and Kendall [27] 2005 ● ●
Hwang, Choi, and Lee [28] 2005 ● ●
Esparcia-Alcazar, Lluch-Revert, Sharman, Albarracin-Guillem, and Palmer-Gato [29] 2006a ●
Esparcia-Alćazar, Lluch-Revert, Sharman, Albarracín-Guillem, and Palmer-Gato [30] 2006b ●
Kok and Fisher [31] 2007 ●
Abbott and Palekar [32] 2008 ●
Bai and Kendall [33] 2008 ●
Esparcia-Alcázar and Martínez-García [34] 2008 ●
Van Nierop and Fok [35] 2008 ●
Hwang, Choi and Lee [36] 2009 ●
Silva, Marikar, and Le [37] 2009 ●
Gajjar and Adil [38] 2010 ●
Hansen, Raut, and Swami [39] 2010 ● ●
Hall, Kopalle, and Krishna [40] 2010 ●
Murray, Talukdar, and Gosavi [41] 2010 ●
Russell and Urban [42] 2010 ●
Gajjar and Adil [7] 2011 ●
Ozcan and Esnaf [43] 2011 ● ●
Yapicioglu and Smith [44] 2012 ●
Aloysius and Binu [45] 2013 ●
Bai and Kendall [46] 2013 ● ●
Pinto and Soares [47] 2013 ●
Castelli and Vanneschi [48] 2014 ● ●
Chen, Chiu, and Yang [49] 2014 ●
Erol, Bulkan, and Agaoglu [50] 2015 ●
Geismar, Dawande, Murthi, and Sriskandarajah [51] 2015 ●
Pinto, Soares, Brazdil, Bravo, Davison, Jofré, Maldonado, and Weber [52] 2015 ● ● ●
Bianchi-Aguiar, Silva, Guimarães, Carravilla, Oliveira, Amaral, Liz, and Lapela [53] 2016 ●
Ghazavi and Lotfi [54] 2016 ● ●
Ghoniem, Maddah, and Ibrahim [55] 2016 ●
Hübner, Kuhn, and Kühn [56] 2016 ● ●
Hübner and Schaal [57] 2016 ●
Flamand, Ghoniem, Haouari, and Maddah [58] 2017 ●
Heydari and Yousefli [59] 2017 ●
Hübner and Schaal [11] 2017a ●
Hübner and Schaal [12] 2017b ●
Choubey [60] 2017 ●
Rabbani, Salmanzadeh-Meydani, and Farshbaf-Geranmayeh [61] 2018 ●
Düsterhöft, Hübner, and Schaal [62] 2020 ●
Yu, Maglasang, and Tsao [63] 2020 ●
Table 2. Categories and sales-potential subcategory allocation rules. ●—product of the defined category and subcategory can be placed on a shelf of the defined category and subcategory. The higher
sales-potential subcategory (10, 20, 30) is, the darker is the color. Different main colors (blue, green, red) represent different categories (A, B, C).
Shelf Category A B C
Product Category Subcategory 10 20 30 10 20 30 10 20 30
10 ● ● ● ● ● ● ● ● ●
A 20 ● ● ● ● ● ●
30 ● ● ●
10 ● ● ● ● ● ●
B 20 ● ● ● ●
30 ● ●
10 ● ● ●
C 20 ● ●
30 ●
Table 3. Number of solutions found by the proposed algorithms, where it was a maximum feasible solution.
Solution is Maximum Feasible Solution Algorithm 1 Algorithm2
Number of solutions 25 18
Solution Exists Algorithms CPLEX
Number of solutions 34 23
Products Shelf Width Profit Ratio Solution Exists Total Number of Product Allocations Processed Number of Product Allocations
250 98.30% Algorithm, CPLEX 1.58 × 10^6 1.58 × 10^6
375 98.52% Algorithm, CPLEX 9.55 × 10^6 1.35 × 10^3
10 500 97.93% Algorithm, CPLEX 2.51 × 10^7 2.16 × 10^3
625 97.01% Algorithm, CPLEX 3.10 × 10^7 2.22 × 10^3
750 97.31% Algorithm, CPLEX 2.76 × 10^7 2.05 × 10^3
250 Algorithm 1.76 × 10^6 1.76 × 10^6
375 Algorithm 2.00 × 10^5 4.77 × 10^2
15 500 Algorithm 7.99 × 10^5 7.99 × 10^5
625 Algorithm 1.31 × 10^7 4.27 × 10^3
750 Algorithm 1.90 × 10^7 5.57 × 10^3
250 0.00 0.00
375 99.84% Algorithm, CPLEX 7.97 × 10^7 1.56 × 10^4
20 500 97.72% Algorithm, CPLEX 2.05 × 10^7 2.05 × 10^7
625 92.68% Algorithm, CPLEX 8.25 × 10^6 8.25 × 10^6
750 88.07% Algorithm, CPLEX 457 × 10^5 4.60 × 10^2
250 97.95% Algorithm, CPLEX 7.18 × 10^6 7.18 × 10^6
375 98.77% Algorithm, CPLEX 6.25 × 10^7 2.06 × 10^5
25 500 89.58% Algorithm, CPLEX 1.10 × 10^6 2.16 × 10^4
625 89.16% Algorithm, CPLEX 3.60 × 10^6 4.73 × 10^4
750 93.60% Algorithm, CPLEX 9.39 × 10^6 7.00 × 10^4
250 0.00 0.00
375 0.00 0.00
30 500 0.00 0.00
625 0.00 0.00
750 0.00 0.00
250 0.00 0.00
375 97.34% Algorithm, CPLEX 1.71 × 10^9 3.31 × 10^6
35 500 86.80% Algorithm, CPLEX 1.10 × 10^10 3.89 × 10^4
625 90.87% Algorithm, CPLEX 3.14 × 10^12 1.94 × 10^6
750 93.09% Algorithm, CPLEX 7.40 × 10^9 1.07 × 10^5
250 0.00 0.00
375 97.40% Algorithm, CPLEX 7.46 × 10^9 9.99 × 10^4
40 500 91.30% Algorithm, CPLEX 1.04 × 10^8 1.15 × 10^6
625 Algorithm 3.52 × 10^12 8.19×10^5
750 Algorithm 2.74 × 10^13 1.65×10^6
250 0.00 0.00
375 87.87% Algorithm, CPLEX 1.16 × 10^12 1.16×10^7
45 500 95.86% Algorithm, CPLEX 6.30 × 10^7 6.30×10^7
625 Algorithm 1.05 × 10^10 7.00×10^3
750 98.08% Algorithm, CPLEX 3.42 × 10^14 1.46×10^5
250 0.00 0.00
375 0.00 0.00
50 500 Algorithm 1.44 × 10^13 1.20 × 10^6
625 Algorithm 1.57 × 10^14 1.75 × 10^6
750 Algorithm 2.46 × 10^13 1.72 × 10^6
Min 86.80% 0.00 0.00
Average 94.57% 1.28 × 10^13 2.87 × 10^6
Max 99.84% 3.42 × 10^14 6.30 × 10^7
Median 97.01% 1.31 × 10^7 4.73 × 10^4
SD 4.17% 5.56 × 10^13 9.90×10^6
Time Taken Looking for the Solution Total Time
Seconds Minutes Seconds Minutes
Min 2 0.03 2 0.49
Average 323 5.39 434 9.53
Max 5039 83.98 5170 87.11
Median 98 1.64 215 5.43
SD 856 14.27 892 15.01
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Czerniachowska, K.; Hernes, M. A Heuristic Approach to Shelf Space Allocation Decision Support Including Facings, Capping, and Nesting. Symmetry 2021, 13, 314. https://doi.org/10.3390/sym13020314
AMA Style
Czerniachowska K, Hernes M. A Heuristic Approach to Shelf Space Allocation Decision Support Including Facings, Capping, and Nesting. Symmetry. 2021; 13(2):314. https://doi.org/10.3390/sym13020314
Chicago/Turabian Style
Czerniachowska, Kateryna, and Marcin Hernes. 2021. "A Heuristic Approach to Shelf Space Allocation Decision Support Including Facings, Capping, and Nesting" Symmetry 13, no. 2: 314. https://doi.org/
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Call Diagonal Ratio Spread
How Does Call Diagonal Ratio Spread Work in Options Trading?
What Is A Call Diagonal Ratio Spread?
A Call Diagonal Ratio Spread is a diagonal ratio spread with the ability to make a profit in all 3 directions; Upwards, Downwards and Sideways, just like a Call Ratio Spread.
Like the Call Ratio Spread, the only way a Call Diagonal Ratio Spread can lose money is when the underlying stock rallies too strongly. That's right, nothing's perfect in options trading.
Because the Call Diagonal Ratio Spread loses money only when a stock rallies strongly, it has been technically classified as a
neutral options strategy
even though it does not lose money no matter how much the underlying stock drops.
However, unlike the call ratio spread, the same long call options can be used for several months while short term call options are written against it so that no additional commission is spent on
re-establishing the whole position month after month. This makes the Call Diagonal Ratio Spread an improved version of the Call Ratio Spread.
When To Use Call Diagonal Ratio Spread?
Call Diagonal Ratio Spread should be used when you are confident in a rise in the underlying instrument up to a certain price and wishes to make money even if the stock should remain stagnant or go
downwards instead. It is a good options trading strategy to maximise profits on stocks that are expected to hit a technical resistance level and not exceed that level for several months.
How to Execute Call Diagonal Ratio Spreads?
Call Diagonal Ratio Spreads are established by shorting more out of the money call options than the amount of in the money / at the money call options are bought, resulting in a credit.
Being a
diagonal spread
, call options with a longer expiration are bought while shorter expiration call options are shorted. This is the main difference between the Call Diagonal Ratio Spread and the
Call Ratio Spread
. In a Call Ratio Spread, both long and short call options of the same expiration month are used.
Example of Call Diagonal Ratio Spread:
Assuming QQQQ at $44.
Buy To Open 5 contracts of QQQQ Mar44Call @ $1.55, Sell To Open 15 contracts of QQQQ Jan45Call @ $0.60
The net effect is, you recieve ($0.60 x 1500) - ($1.55 x 500) = $125 as credit for putting on the position.
In the above example, 15 contracts of out of the money options are shorted while only 5 contracts of at the money call options are bought. The ratio in this Call Diagonal Ratio Spread is 3 : 1. This
means that for every 1 contract of at the money call options bought, 3 contracts of out of the money call options are bought. This is why such options trading strategies are known as Ratio Spreads.
Veteran options traders would notice by now that Call Diagonal Ratio Spreads are simply Calendar Call Spreads that sells more out of the money call options than at the money call options.
How to Determine Ratio and Strike Price to Use For Call Diagonal Ratio Spread?
Determining the above ratio depends on which
strike price
the out of the money call options are shorted. In general, the higher the strike price of the out of the money call options, the lower the price of each contract and hence the more contracts need to
be shorted in order to result in a net credit, requiring higher
. Because longer term call options of the same strike price commands a higher premium, more short term call options need to be shorted in a Call Diagonal Ratio Spread than in a Call Ratio Spread of
the same strike price, incurring higher options trading margin. This is the main disadvantage of the Call Diagonal Ratio Spread against the Call Ratio Spread.
Example of Different Strike Prices Call Diagonal Ratio Spread:
Assuming QQQQ at $44.
Mar44Call is trading @ $1.55, Jan45Call is trading @ $0.60, Jan46Call is trading @ $0.25.
If the Jan46Call is chosen for shorting instead of the Jan45Call, you would need to short 36 contracts to achieve the net credit of $180 rather than just shorting 15 contracts using the Jan45Call.
Below is a comparison:
($0.25 x 3600) - ($1.55 x 500) = $125 credit using the Jan46Call
($0.60 x 1500) - ($1.55 x 500) = $125 credit using the Jan45Call
Note: Each options contract represents 100 shares, therefore the figure used in the calculation is the number of contracts times 10.
The higher the strike price of the out of the money call options shorted in a Call Diagonal Ratio Spread, the higher the stock can surge before the position starts losing money. This is because when
the underlying stock goes above the strike price of the short call options, the short call options start to increase in price faster than the long call options. In our above example, if the Jan46Call
are shorted, the position starts losing money only when the QQQQ rises above $46 while if the Jan45Call are shorted, the position starts losing money once the QQQQ rises beyond $45. So the trade off
here is really the amount of margin you have versus how far you want the losing point to be. The higher the strike price of the out of the money call options, the farther the losing point of the Call
Diagonal Ratio Spread becomes. Stock options trading is all about trade-offs.
The higher the strike price of the out of the money call options shorted in a Call Diagonal Ratio Spread, the higher the maximum profit attainable by the Call Diagonal Ratio Spread will be if the
initial net credit is kept the same. However, that also means that the stock needs to rally more in order to attain that maximum profit as the maximum profit attainable by a Call Diagonal Ratio
Spread is when the stock closes on the strike price of the short call options upon expiration. Yes, another options trading trade-off.
In an ideal options trading world where margin is not a concern, you would short as many out of the money call options as you want to as far out of the money as possible in order to build a Call
Diagonal Ratio Spread with the highest possible net credit, maximum profit and farthest losing point. However, such a world does not exist in options trading and margin is as big concern a in Call
Diagonal Ratio Spreads as it is in any options trading strategies involving uncovered short option positions.
If you only have enough margin to short a certain number of options contracts, you can either short a lower strike price or you can buy fewer at the money call options in order to maintain the net
credit in the Call Diagonal Ratio Spread. In our Call Diagonal Ratio Spread examples so far, if you have enough options trading margin to short only 15 contracts, you can use a lower strike price in
order to maintain the net credit of the Call Diagonal Ratio Spread position by choosing the Jan45Call and not the Jan46Call as 36 contracts of Jan46Call would not result in a net credit. Otherwise,
you can buy no more than 3 contracts of Jan44Call and still maintain a net credit with 15 contracts of Jan46Call. Such is the flexibility of options trading.
Example of shorting a lower strike price and buying lesser call options in Call Diagonal Ratio Spread:
Assuming QQQQ at $44 and you have enough margin to short only 15 contracts max.
Jan44Call is trading @ $1.05, Jan45Call is trading @ $0.60, Jan46Call is trading @ $0.25.
Scenario 1: Shorting a Lower Strike.
If you can only short 15 contracts of Jan46Call, you won't be able to cover the price of the 5 contracts of Jan44Call bought at all.
5 Contracts of Jan44Call = $1.05 x 500 = $525. 15 contracts of Jan46Call = $0.25 x 1500 = $375. The position would be a net debit of $150 instead of a net credit. When a Call Diagonal Ratio Spread is
a net debit position, it would not be able to make any money if the stock goes down. In this case, you should short the Jan45Calls instead as illustrated in the first example in the overview above.
Scenario 2: Buying lesser call options.
In order to maintain a net credit while shorting 15 contracts of Jan46Call, you should buy no more than 3 contracts of Jan44Call.
15 contracts of Jan46Call = $0.25 x 1500 = $375
3 contracts of Jan44Call = $1.05 x 300 = $315
Net Credit = $375 - $315 = $60
The profitability of a call diagonal ratio spread can be enhanced or better guaranteed by legging into the position properly.
Profit Potential of Call Diagonal Ratio Spread :
The maximum profit potential of a Call Diagonal Ratio Spread is attained when the underlying stock closes at the strike price of the short call options. In this respect, the profit potential of a
Call Diagonal Ratio Spread is limited.
Maximum profit point of Call Diagonal Ratio Spread:
Assuming QQQQ at $44.
Buy To Open 5 contracts of QQQQ Jan44Call @ $1.05, Sell To Open 15 contracts of QQQQ Jan45Call @ $0.60
Maximum Profit happens when the QQQQ closes at $45 at expiration of the Jan45Call.
Risk / Reward of Call Diagonal Ratio Spread:
Upside Maximum Profit: Limited
Maximum Loss: Unlimited
Position starts losing money if the stock rises past the strike price of the short call options. However, if the stock falls instead of rises, then the position makes in profit the net credit gained.
Advantages Of Call Diagonal Ratio Spread :
:: 3-way options trading profit.
:: Much higher profit can be made than a Bull Call Spread when the underlying stock closes at the strike price of the short call options.
:: Call options can be written against the long call options for more than just one month.
Disadvantages Of Call Diagonal Ratio Spread :
:: Some brokers may not allow beginners to execute such options trading strategies.
:: Margin is required.
:: More Margin is required than Call Ratio Spread as more options need to be shorted in order to maintain a net credit at the same strike prices.
Alternate Actions Before Expiration :
1. When the underlying stock reaches the strike price of the short call options before expiration, one may choose to buy to close the extra short call options in order to prevent losses due to a
surge in price past breakeven.
2. If the underlying stock has exceeded the strike price of the short call options but is expected to correct back down below the strike price of the short call options in the following month, you
could roll forward the short call options into the following month in order to anticipate the correction. This is a flexibility you will not get in a Call Ratio Spread.
Don't Know If This Is The Right Option Strategy For You? Try our Option Strategy Selector!
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51st Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics
Bulletin of the American Physical Society
51st Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics
Volume 65, Number 4
Monday–Friday, June 1–5, 2020; Portland, Oregon
Session P03: Quantum Simulation with Trapped Ions Hide Abstracts
Chair: David Allcock, University of Oregon
Room: D135-136
Thursday, P03.00001: New Methods for Quantum Simulation of Spin Systems with Trapped Ions
June 4, 2020 Tom Manovitz, Yotam Shapira, Nitzan Akerman, Roee Ozeri, Ady Stern
2:00PM -
2:12PM By simulating the behavior of quantum systems with highly controlled engineered quantum machines, one can study the complex behavior of a variety of quantum phenomena. Ions in a linear
Live Paul trap have proven to be a leading platform for such simulations, primarily relying on a set of spin Hamiltonians produced using the Molmer-Sorensen interaction. In this work, we
significantly extend the range of Hamiltonians that can be directly simulated in trapped ions using a simple variation of the standard scheme. For $N$ ions our method can produce a
Hamiltonian with a general form $\sum_{n=1}^{N-1}\Omega_n e^{i(\phi_n-\omega_n t)} \sum_{i=1}^{N-n}\sigma_i^+\sigma_{i+n}^- + h.c.$ where parameters $\{\Omega_n,\phi_n,\omega_n \}$ can
be fully controlled. Using this form, it is possible to generate Hamiltonians with closed boundary conditions; $d>1$ dimension Hamiltonians; and Hamiltonians with gauge field
(Aharonov-Bohm) terms. An assortment of interesting physical models previously unreachable with analog simulations in trapped ions are made possible using our scheme. [Preview Abstract]
Thursday, P03.00002: Theory of robust multi-qubit non-adiabatic gates for trapped-ions
June 4, 2020 Yotam Shapira, Ravid Shaniv, Tom Manovitz, Nitzan Akerman, Lee Peleg, Lior Gazit, Roee Ozeri, Ady Stern
2:12PM -
2:24PM Trapped ion qubits are a leading platform for performing quantum computations and quantum simulations. These are achieved with entanglement gates acting on the ions, by coupling to the
Live normal-modes of motion of the ion-chain. Typically, a single normal-mode is coupled and the remaining modes are decoupled by operating slowly. Analog quantum simulations are also
performed in an adiabatic regime and allow only spin-spin interactions that scale as a power-law in the ion distance. We propose multi-qubit entanglement gates for trapped-ions
[arXiv:1911.03073 (2019)]. Our gates utilize all the normal-modes of motion allowing for fast operation that was previously inaccessible, and require reasonable resources. Furthermore,
we use our methods to generalize the coupling between the ions, and generate fast spin-Hamiltonian interactions, which are not limited to a power law. For example, a nearest-neighbor
Ising model and the topological Su-Schriefer-Heeger Hamiltonian. Our gates use a multi-tone laser field, which couples uniformly to all ions, there is no need to individually address
the ions. We endow our gate with robustness properties, making them resilient to various sources of system noise and imperfections. Our method is natural to common trapped-ion
architectures. [Preview Abstract]
Thursday, P03.00003: Holographic optical manipulation of trapped ions for quantum simulation
June 4, 2020 Chung-You Shih, Sainath Motlakunta, Manas Sajjan, Nikhil Kotibhaskar, Thiago Bergamaschi, Roland Hablutzel, Rajibul Islam
2:24PM -
2:36PM Trapped ions are an ideal experimental platform for quantum simulation of interacting many-body Hamiltonians. Arbitrary and programmable control over individual ions is needed for
Live maximum versatility in simulation. In this talk, we will present progress towards developing a holographic optical ion addressing system using Digital Micromirror Devices. The technique
uses reprogrammable holograms to modulate the wavefront of the addressing beam to control the amplitude and phase of light at each ion. We implement a novel iterative Fourier transform
algorithm to compute holograms that compensate for optical aberrations and produce minimal ‘cross-talk’ error between ions. Individual ions can be used as aberration sensors for
ultimate precision. Such high-precision optical control will enable quantum simulation of dynamical and higher dimensional lattice geometry of spins in a 1D chain of ions, such as to
investigate quantum quenches and phase transitions and topological phases. [Preview Abstract]
Thursday, P03.00004: Observation of Domain Wall Confinement and Dynamics in a Quantum Simulator
June 4, 2020 P. Becker, W. L. Tan, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt, A. V. Gorshkov, C. Monroe
2:36PM -
2:48PM Confinement is a ubiquitous mechanism in nature, whereby particles feel an attractive force that increases without bound as they separate. A prominent example is color confinement in
Live particle physics, in which baryons and mesons are produced by quark confinement. Analogously, confinement can also occur in low-energy quantum many-body systems when elementary
excitations are confined into bound quasiparticles [1]. Here, we report the first observation of magnetic domain wall confinement in interacting spin chains with a trapped-ion quantum
simulator [2]. By measuring how correlations spread, we show that confinement can dramatically suppress information propagation and thermalization in such many-body systems. We
determine the excitation energy of domain wall bound states from non-equilibrium quench dynamics. Furthermore, we study the number of domain wall excitations created for different
quench parameters, in a regime that is difficult to model with classical computers. This work demonstrates the capability of quantum simulators for investigating exotic high-energy
physics phenomena, such as quark collision and string breaking. [1] F. Liu, et al., Phys. Rev. Lett. 122, 150601 (2019). [2] W. L. Tan, P. Becker, et al., arXiv: 1912.11117 (2019). [
Preview Abstract]
Thursday, P03.00005: Many-Body Dephasing in a Trapped-Ion Quantum Simulator
June 4, 2020 Wen Lin Tan, Harvey Kaplan, Arinjoy De, Guido Pagano, Christopher Monroe, Lingzhen Guo, Florian Marquardt
2:48PM -
3:00PM We observe and analyze the persistent temporal fluctuations after a quantum quench of a tunable long-range interacting transverse-field Ising Hamiltonian realized with a trapped-ion
Live quantum simulator [1]. By measuring temporal fluctuations in the average magnetization of a finite-size system of spin-1/2 particles encoded in an array of 171Yb+ atomic ions, we
observe experimental evidence for many-body dephasing [2]. The properties of the system are closely related to that of an integrable Hamiltonian with a uniform spin-spin coupling, which
enables approximate analytical predictions even for the long-time non-integrable dynamics. We find that the measured fluctuations are exponentially suppressed with increasing system
size, consistent with theoretical predictions. [1] H. B. Kaplan, et al., arXiv: 2001.02477 (2020). [2] T.Kiendl, F.Marquardt, Phys. Rev. Lett. 118, 130601 (2017) [Preview Abstract]
Thursday, P03.00006: Analog quantum simulation of superradiance and subradiance in trapped ions
June 4, 2020 R. T. Sutherland
3:00PM -
3:12PM We discuss a protocol for the analog quantum simulation of superradiance and subradiance using a linear chain of N trapped qubit ions with a single sympathetic cooling ion. We develop a
Live simple analytic model that shows the dynamics of the qubit subspace converge to those of a cloud undergoing Dicke superradiance and subradiance. We provide numerical simulations of the
full ion chain and show that they converge to the dynamics predicted by our analytic model with no fitting parameters. We also map out the parameter regime needed to reach this
convergence. [Preview Abstract]
Thursday, P03.00007: Matrix product state simulations on a quantum computer
June 4, 2020 Michael Foss-Feig, Andrew Potter, David Hayes
3:12PM -
3:24PM Matrix product states (MPS) afford a compressed representation of many states that are relevant to physical systems. While classical algorithms have been developed to compute the
Live properties of physical systems using MPS as an ansatz, in many cases of practical interest these algorithms still require exponential resources (for example in the size of the system
for 2D or 3D systems, or in the evolution time when out of equilibrium). We discuss near-term prospects for using small and non-error-corrected quantum computers to aid in MPS
simulations, and show examples of MPS based quantum algorithms run on a trapped-ion quantum computer. [Preview Abstract]
Thursday, P03.00008: Overlap measurements of infinite-dimensional quantum states for quantum-enhanced machine learning.
June 4, 2020 Chi-Huan Nguyen, Ko-Wei Tseng, Jaren Gan, Gleb Maslennikov, Dzmitry Matsukevich
3:24PM -
3:36PM Estimation of overlap between quantum states is a ubiquitous task in quantum information processing protocols and has great significance for quantum machine learning applications.
On Demand Implementing the overlap measurement with the standard discrete-variables approach in noisy intermediate-scale quantum computers requires scaling the number of physical qubits and
entanglement gates with the dimensions of the Hilbert space. Hybrid quantum computation offers an alternative approach; whereby utilizing both discrete and continuous variables, a
gate-based overlap measurement in an infinite-dimensional system with constant circuit depth can be realized. Here, we experimentally demonstrate the overlap measurement using this
approach in a system of two trapped Yb 171 ions. We employ the nonlinear interaction between the internal and motional degrees of freedom to enact a controlled-swap gate between two
motional modes. To illustrate the versatility of our method, we measure the overlap between a variety of quantum states: Fock states, coherent states, squeezed states, and cat states.
We also discuss how to employ the overlap test in an unsupervised quantum-enhanced k-means algorithm. [Preview Abstract]
Thursday, P03.00009: Characterization of Radial 2D Ion Crystals for Quantum Simulation
June 4, 2020 Yuanheng Xie, Marissa D'Onofrio, AJ Rasmusson, Paula Madetzke, Evangeline Wolanski, Philip Richerme
3:36PM -
3:48PM Quantum simulations of complex materials address fundamental problems that cannot be analytically solved due to the exponential scaling of the Hilbert space with increasing particle
On Demand number. Simulations using trapped ions have had great success investigating one-dimensional quantum interacting spin models, and we seek to extend these ideas to two dimensions by
exploiting new crystal geometries in a rf Paul trap. This 2D quantum simulation will allow us to address open questions related to geometric frustration, ground states and dynamics of
long-range spin models, and quantum spin liquids. To characterize the variety of different ion geometries, we have measured the ion lattice positions and the frequencies of structural
phase transitions for 1D, 2D, and 3D crystal configurations, showing good agreement with numerical predictions. In addition, we have created quantum superpositions of 10+ ions in radial
2D crystals that persist for long coherence times, despite the presence of micromotion in this configuration. [Preview Abstract]
Thursday, P03.00010: Experimental progress towards a prethermal discrete time crystal
June 4, 2020 A. Kyprianidis, P. Becker, K. Collins, W. Morong, L. Feng, W.L. Tan, A. De, G. Pagano, P.W. Hess, F.L. Machado, D. Else, C. Nayak, N. Yao, C. Monroe
3:48PM -
4:00PM Driven quantum systems offer opportunities for studying novel phases of non-equilibrium matter, such as time crystals [1,2,3]. To avoid continuously absorbing energy from the drive, we
Not investigate a strategy based on Floquet prethermalization. In this case, even without disorder, one can observe a slow heating time scale with exponential dependence on the drive
Participating frequency, leading to a long-lived intermediate "prethermal" regime. We use a trapped-ion quantum simulator with chains of Yb ions confined in a rf Paul trap that realize a transverse
field Ising model with tunable range interactions. By varying the initial state and drive parameters of our system, we characterize this prethermal regime. [1] F. Wilczek, Phys. Rev.
Lett. 109, 160401 (2012) [2] J. Zhang et al. Observation of a discrete time crystal. Nature 543, 217-220 (2017) [3] S. Choi et al. Observation of discrete time-crystalline order in a
disordered dipolar many-body system. Nature 543, 221-225 (2017) [Preview Abstract]
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[EM] Call for Ideas on Automatic Approval Cutoff Finding
Forest Simmons fsimmons at pcc.edu
Fri Sep 12 13:43:03 PDT 2003
Suppose you were given a set of voted CR ballots, perhaps with
supplemental information such as a tentative approval cutoff. What rule
would you use to adjust the approval cutoff by taking into account the
popularity of the various candidates as manifested by the ballots
themselves, as opposed to published opinion polls?
What supplemental information would you solicit from the voters (through
modified CR ballots) if you could have it?
Suppose, for example, that you asked all voters to submit (along with
their cardinal ratings) the probabilities for each of the candidates being
the final winner of the election, along with a number w between zero and
100 percent expressing their confidence in these probabilities.
Suppose that voters had no strategic incentive to exaggerate any of these
numbers. Then weighted averages (using the w's for the weights) of these
probability estimates could be used as a pooled estimate of the
probabilities of the various candidates winning.
These pooled probabilities could be used to estimate the expected
"cardinal payoff" (i.e. expected utility if the CR values are utilities)
on each ballot, and the cutoff could be a weighted average of the pooled
estimate and the individual estimate.
If the individual is 100 percent confident, then the pooled estimate would
count for nothing. If the individual is zero percent confident, then the
pooled estimate counts for everything. If the individual is 50%
confident, then the pooled estimate and his own estimate receive equal
If the voter wanted to manipulate, he would have to express high
confidence in his estimate or else it wouldn't effect the outcome much.
But if he expresses high confidence, then he has to live with the results,
since his cutoff will be determined accordingly.
To simplify things, instead of estimating all of the candidate
probabilities, just have each voter indicate which candidate is most
likely to win (in his opinion) along with a confidence number w as before.
These results can still be used to find a pooled estimate, and averaging
the pooled estimate with the cardinality of the most likely candidate on
the ballot is still a reasonable way to pick a cutoff.
In fact, this weighted average will be between the expected CR according
to the pooled estimate of probabilities and the CR level judged most
likely by the voter, and closer to the one that the voter has expressed
more confidence in.
That's just one example. Let's see some creativity!
More information about the Election-Methods mailing list
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Re: [tlaplus] Recursive data structures
What is the canonical way to represent recursive data structures in TLA+? Consider the example of a binary tree; at first pass, we try to represent it as follows:
CONSTANTS Key, Value
Node ==
[key : Key,
value : Value,
leftChild : Node,
rightChild : Node]
However, the syntactic analyzer gives the parser error "Unknown operator: Node" since recursive definitions are not supported here.
We can borrow from Specifying Systems section 11.1.2 and define a generalized graph in terms of sets of nodes & edges with restrictions, but I'm interested in whether there are other ways to
accomplish this goal.
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A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
A bus route has a total duration of $40$ minutes.
Every $10$ minutes, two buses set out, one from each end.
How many buses will one bus meet on its way from one end to the other end?
Crossing the Atlantic
for a similar problem.
Getting Started
Where are the travelling buses positioned when a new pair of buses is due to leave?
How soon after leaving will a bus meet a bus travelling from the opposite direction?
Will it be the same for the first bus of the day as for the buses travelling in the middle of the day?
Student Solutions
This problem was more complicated than it seemed - we had over 50 solutions suggesting that when a bus travelled from one end to another it would meet buses travelling from the other direction every
10 minutes. Unfortunately this is not correct!
When the bus sets off there will be a bus arriving from the other end, a bus due to arrive 10 minutes later, a bus due to arrive 20 minutes later, a bus due to arrive 30 minutes later, and a bus just
leaving at the other end which is due to arrive in 40 minutes. The bus will meet all these buses and all the buses that leave while it is travelling, that is, the buses that leave 10, 20 and 30
minutes after it has set off. It will arrive as another bus prepares to leave.
So the bus will set off as a bus arrives from the other end, meet a bus every 5 minutes while travelling (7 in all) and arrive at the other end as another is leaving.
If it is the first bus of the day, it will only meet 4 buses on its way, the second bus of the day will meet 5, the third bus will meet 6, the fourth bus will meet 7, the buses after the fourth one
will all meet 7 except the last three - they will meet 6, 5 and 4 respectively.
Jessica from West Herts College sent a correct solution:
A bus in the middle of the day will either meet 7 or 9 depending on whether you count the buses it meets at either end of its journey. However the first and last bus will only meet 4 or 5 depending
on whether you count the bus leaving as it arrives (in the case of the 1st bus) or leaves (in the case of the last bus).
Anna, PD and Emily from Ardingly College Junior School thought of it like this:
... if it left the station at 3.40pm it would meet the ones that left the station within 40 minutes before and after it and the one that left at the same time as it.
Heather from Stow Heath Junior School noticed that:
If we start counting as the first bus leaves for the day then it will not see another bus until it is half way along it's route.
Laura and Harriet from The Mount School explained it as follows:
Bus A and B set out at the same time, then 10 minutes later buses C and D set out. At this point none of the buses have advanced enough to meet another. Then, after 20 minutes, two more buses, E and
F set out, and A and B meet. After 25 minutes A will meet D. 30 minutes into the cycle, A will meet F. G and H will set off. After 35 minutes A will meet H. Then at 40 minutes, the cycle will be
complete, and A will meet one final bus, J, which is setting out. If bus A had been the first bus to set off, it would meet 5 other buses. If bus A had set off while the cycle was taking place, it
would meet 8 other buses.
Well done to all of you who managed to crack this problem.
Teachers' Resources
Why do this problem?
This challenge
requires a logical, systematic approach. The solution might be unexpected to some, and might cause disagreement amonst students. Whilst the problem might seem to be algebraic, it is more an exercise
in visualisation, time and motion.
Possible approach
Ask students to tell their neighbours their initial guesses as to the number of buses that will be passed. Give students time to work on paper, checking out their ideas and developing convincing
arguments. Ask for volunteers to explain their ideas to the group. There is likely to be disagreement which (ideally) should prompt clarity of explanation.
It could be useful to model the process with the students as follows:
Split some volunteers into two queues of buses, one on each side of the room.
Set up three intermediate bus stops. Every ten seconds ask each bus to move to the next stop.
They must count each time they walk past someone. Continue until at least 6 buses have crossed in each direction.
How many crossings did each person make?
Is there anything about the journey that is more obvious now that it has been acted out?
Students could be asked to write a full clear solution. Some of these could be shared with the group.
How convincing and clear did the group find these explanations?
This may be an opportunity to introduce a distance-time graph to represent the situation. It demonstrates the number of solutions very elegantly, but students often find these graphs confusing. The
teacher could put a completed graph on the board, and ask the students to make sense of it. It would be useful to discuss the meaning of:
a horizontal strip - the view from a particular point on the route,
a vertical strip - a summary of all bus positions at a specific moment in time,
a diagonal strip - a description of the journey for a particular bus.
Emphasise that this is not an artistic picture of events, and certainly not a road map, but it summarises a lot of information, to give an overview of a complicated situation.
Key questions
• Is there a difference between the first bus of the day and a bus which sets off later on in the day?
• Does each bus make the same number of crossings?
• Is there a clear way of recording the motion of the buses?
Possible extension
Investigate the number of crossings if the journey time and/or gap between buses is altered.
Does it matter if the gap time isn't a factor of the journey time?
Let the bus journey be up (or down) a long hill - so that the speed in one direction is different to the speed on the return journey.
Can students summarise/generalise their solutions to the variations they have tried?
Possible support
Simplify the numbers - eg 30 minute journey time.
Provide counters to allow students to model the motion of the buses, establishing bus stops at 10 minute intervals to make the movement easier.
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Gr.9 T1 Super Math Act Book with TT 2022 [PDF] - One Stop Edu Shop
Author: Karelien Kriel
Number or pages: 312 (boek) + 187 (memo)
Number or activities: 43
Types or book: TEXT BOOK (fill in Activity Book) that covers the whole term’s CAPS work
Philosophy: Thinking Tools
Printing: back-to-back and bind with ring binder (1 book) or use stapler (make 2 packs)
Activity 1: numbers systems; definition and examples; brace map; circle map; natural numbers; counting numbers; integers; rational numbers; irrational numbers; integers; order of operations; brace
map; number bonds of 10; circle map; double; halve; bridge maps; number walls
Activity 2: integers; order of operations; flow maps;
Activity 3: integers; addition of 6-digit numbers; tree maps; columns
Activity 4: integers; subtraction of 6-digit numbers; tree maps; columns
Activity 5: integers; multiplication of 4-digit numbers with 2-digit numbers; multi flow maps; tree maps; steps; columns
Activity 6: integers; division of 4-digit numbers with 2-digit numbers; multiples; multi flow maps; bridge maps; columns; steps
Activity 7: integers; factors; prime numbers; prime factors; Largest Common Factor (HCF); brace maps; multiples; Smallest Common Multiple (LCM); write the numbers as the product of their prime
Activity 8 – 9: integers; word problems; ratio and rate; bridge maps; steps; flow maps; direct proportion; inverse proportion
Activity 10: Financial Mathematics; simple interest; compound interest; brace map; loan tables
Activity 11: integers; properties and laws; word problems; steps; flow maps
Activity 12: fractions; equivalent fractions; bridge maps; convert improper fractions to mixed numbers; flow maps; convert mixed numbers to improper fractions
Activity 13: addition and subtraction of fractions and mixed numbers; bridge maps; steps; brace maps
Activity 14: multiplication and division of fractions and mixed numbers; steps; brace maps
Activity 15: convert common fractions with denominators of 10, 100 and 1000 to decimal fraction; brace maps
Activity 16: convert common fractions to decimal fractions and percentages; brace maps; bridge maps; recurring fractions
Activity 17: integers, common fractions and decimal fractions; squared numbers and cubed numbers; circle maps; tree maps; brace map; number walls
Activity 18: operation with decimal fractions on calculator; word problems with fractions; steps; flow maps
Activity 19: powers; brace maps; exponent; base; coefficient; law 1; simplify
Activity 20: exponents; law 2 and 3; simplify
Activity 21: exponents; law 4, 5 and 6; simplify
Activity 22: summarise exponent laws in brace map; simplify
Activity 23: Scientific notation; very large numbers; very small numbers; conversion
Activity 24 – 26: number patterns; brace map; Geometrical patterns; pictures; tables; formulae; work forwards and backwards in flow maps; find the rule; find the formula
Activity 27: variables; pictures; addition; products; multiplication of brackets
Activity 28: factorising; common factors
Activity 29: factorising; common factors; prime factors; brace map;
Activity 30: factorising; binomials and trinomials with common factors; common brackets
Activity 31: factorising; put friends together and add brackets; common brackets; investigate swopping of signs
Activity 32: factorising; investigate swopping of sign and common brackets; investigate difference of squares
Activity 33: factorising; investigate factorising of trinomials (+)(+); brace map
Activity 34: factorising; investigate factorising of trinomials (–)(+) and (+)(–); brace maps
Activity 35: factorising; investigate factorising of trinomials (–)(–); brace maps; summarise all the rules for factorising binomials and trinomials
Activity 36: factorise
Activity 37: differences and similarities between expressions and equations in double bubble map; investigate solving simple equations; scales; solve x; multiple choice questions; four types of
solutions for equations in brace map
Activity 38: inverse operations; brace map; investigate solution of equations – 1 term on one side and 2 terms on other side; investigate solution of equations – 2 terms on both sides; steps in
brace map
Activity 39: solve equation
Activity 40: multiplication of brackets; solve equation with brackets
Activity 41: solve equation with fractions; investigate; steps in brace map
Activity 42: solve equation with powers and surds; word problems; geheime number; steps; flow maps
Activity 43: equations; word problems; age; steps; money sums; speed sums; summary in brace maps
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高级衍生品代写 BFF5340 ADVANCED DERIVATIVES - 商科代写 艾莎代写
BFF5340 ADVANCED DERIVATIVES
高级衍生品代写 Please refer to the BFF5340 Moodle site which has an entire folder devoted to Academic Integrity.
Important-Academic Integrity
Please refer to the BFF5340 Moodle site which has an entire folder devoted to Academic Integrity.
The key message is while it can be tempting to “collaborate” with other students in this test environment, if caught, the consequences are severe and not worth the risk. Do the right thing!
When and how long is the Mid Semester Test (MST)?
Your MST will be held during the teaching week 7 – please check your student timetable for the start time (this will be at Australian Eastern Standard time AEST). The assessment duration will be 1
hour and 30 minutes including reading and upload/download time. The assessment will be delivered through the Moodle platform (please refer to further details below).
Is the MST open book?
Yes, the BFF5340 MST is open book this semester and the test will be delivered online using the Moodle platform.
What is the format of the test paper?
The test is in the form of a Word document.
What are the accepted forms of responses? 高级衍生品代写
Responses must be included in the exam Word document, or where explicitly specified, in an Excel file. You will see the following options in the paper:
1. Type answer here: Simply type your response in the Word document.
2. Answer this question in Excel. Name the worksheet as …. You must answer the question in an Excel file, name the worksheet accordingly and once the test is completed, upload the file together with
your Word document. Use the same Excel file for all questions that require responses in Excel.
Why is Excel used in this test?
As the most basic tool for computational finance, it is ideal for simple repetitive calculations and optimisation that cannot be efficiently performed by hand. The Moodle task format allows for this
tool to be incorporated in the test. Also, Excel has been used in various demonstrations throughout the course.
Is there a hurdle requirement for the test?
No, not for this MST.
Is a formula sheet available for the MST?
Yes. The formula sheet which is now available on Moodle is the formula sheet you will be getting on the day of the test.
Is the formula sheet comprehensive? 高级衍生品代写
Sufficient formulae are provided to enable you to attempt all calculation questions in the test. However, no formula is provided for things that relate to basic conceptual understanding such as
present value calculation and payoff functions of derivative instrument.
In addition, the formula sheet should not be used as a guide as to what questions will or will not be covered in the test.
What is the structure of the MST?
The test will comprise 3 questions, each possibly containing more than 1 part. The questions test both theoretical understanding as well as Excel-based calculations. There is a great deal of
familiarity between test questions and tutorial questions although some parts may draw on a deep level of understanding of the materials as a whole. Expect some questions to test your ability to
apply knowledge to address a practical scenario.
What do I need to study?
The MST covers lecture materials in weeks 1, 2 and 3 and associated tutorials, including
D Lecture Notes and all illustrations provided during the seminars 高级衍生品代写
D Lecture Recordings
D Tutorial Questions/Solutions
D Tutorial Recordings
D MST Practice Questions/Solutions
Be prepared to be tested on applying theories using critical skills rather than purely being required to plug inputs in a given formula to derive an answer. Some questions will test the ability to
extend the solution on simplified problems to a more realistic setting.
Am I expected to derive formulas?
You are expected to be able to perform the derivations that are covered in the lectures and tutorials and other derivations ofsimilar nature. This type of questions may require you to either
explicitly derive a result or explain the steps involved in deriving the result.
Is VBA coding tested?
How do I prepare for Excel calculation questions?
To enable efficient execution of the MST, prior to the test, you should have an Excel file with a set of worksheets that are set up to perform different calculation tasks covered in the lectures and
How many decimal points should I round my answers to?
Unless a question specifically asks you to round (for example, to two decimal places) then it does not matter. You are strongly advised however to leave the rounding to your final answer, and not to
round your inputs during the workings of the problem. For the Excel based questions, as you will need to use formulas instead of typing in numeric answers, the issue of decimal points does not apply.
However, when stating final answers in the Excel file, use two decimal places for dollar amount and four decimal places for rate amounts (interest rates, spreads, hazard rate, probability).
Will I have to show my workings for calculation questions?
Yes. A good response to a calculation question will state the formula, the inputs and the final answer. When the question is to be answered in Excel, workings are shown by way of clear commenting for
each step (in the case of binomial tree applications for example, Input, Tree parameters, Stock price, Option price…)
How do I submit my MST? 高级衍生品代写
Upon completion of the test,
1. Name the Excel file using your student ID (put all of your Excel answers in 1 Excel file) and
2. Upload to Moodle the Excel file together with the Word document through Moodle submission. This submission must occur within the duration of the test.
Are there test questions for practice?
There is a set of practice questions and solution on Moodle. You should consult the solution only after making a serious attempt on the questions. For this reason, the solution file will be
accessible closer to the test date.
What are the consultation sessions prior to the test?
Yes via the usual weekly consultations via Zoom.
You may also request another consultation subject to availability.
You can also ask questions by posting on Moodle’s Assessment forum. I’ll attempt to respond within 24 hours.
Will you go through the solution after the test? When will the mark be released? 高级衍生品代写
Yes, I will discuss the solution during my weekly consultation following the test. The mark will be released 1 week after the test.
Is RPM trading covered in the test?
There won’t be questions specifically related to the RPM platform. However, you may be tested on trading in general.
What if I am sick on the day of the test day?
Please contact Monash Connect to apply for a special consideration. You may want to send me an email for a heads up.
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min/max algorithm for cubic splines over k-partitions
You are here
min/max algorithm for cubic splines over k-partitions
Date Issued:
The focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of interest since the
trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a minimum/maximum algorithm that
will identify potential knots. Then we employ least squares estimation to find potential regression coefficients based upon the cubic spline model and the knots chosen by the minimum/maximum
algorithm. We then utilize the best subsets regression method to aid in model selection in which we find the minimum value of the Bayesian Information Criteria. Finally, we preent the R2adj as a
measure of overall goodness of fit of our selected model. We have found among the fifty states and Washington D.C., 42 out of 51 showed an R2adj value that was greater than 90%. We also present
an overall model of the United States. Also, we show additional applications our algorithm for data which show a non linear association. It is hoped that our method can serve as a unified model
for violent crime rate over future years.
Title: A min/max algorithm for cubic splines over k-partitions.
Golinko, Eric David
Name(s): Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of text
Genre: Electronic Thesis Or Dissertation
Date Issued: 2012
Publisher: Florida Atlantic University
Physical electronic
Extent: viii, 62 p. : ill. (some col.)
Language(s): English
The focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of
interest since the trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a
minimum/maximum algorithm that will identify potential knots. Then we employ least squares estimation to find potential regression coefficients based upon the cubic spline model and
Summary: the knots chosen by the minimum/maximum algorithm. We then utilize the best subsets regression method to aid in model selection in which we find the minimum value of the Bayesian
Information Criteria. Finally, we preent the R2adj as a measure of overall goodness of fit of our selected model. We have found among the fifty states and Washington D.C., 42 out of
51 showed an R2adj value that was greater than 90%. We also present an overall model of the United States. Also, we show additional applications our algorithm for data which show a
non linear association. It is hoped that our method can serve as a unified model for violent crime rate over future years.
Identifier: 794510485 (oclc), 3342107 (digitool), FADT3342107 (IID), fau:3861 (fedora)
by Eric David Golinko.
Note(s): Thesis (M.S.)--Florida Atlantic University, 2012.
Includes bibliography.
Electronic reproduction. Boca Raton, Fla., 2012. Mode of access: World Wide Web.
Spline theory -- Data processing
Bayesian statistical decision theory -- Data processing
Neural networks (Computer science)
Subject(s): Mathematical statistics
Uncertainty (Information theory)
Regression analysis
Link to This http://purl.flvc.org/FAU/3342107
Use and http://rightsstatements.org/vocab/InC/1.0/
Host FAU
In Collections
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May 2013
Friday May 31 2013
Time Replies Subject
10:59PM 0 Replace row value with corresponding column name
9:38PM 2 tree in tree package - Error: cannot allocate vector of size 2.0 Gb
6:02PM 2 R help on loops
4:31PM 3 Y-lim minimum overrun in barplot
2:58PM 1 Error in x[[1]] : subscript out of bounds
1:49PM 1 chooseCRANmirror
1:35PM 0 conditional maximum likelihood estimators
1:01PM 1 Post-fit survival curve for shared frailty model
12:50PM 2 empty html pages for R-exts and R-intro
12:41PM 4 how to install R 3.0.1
12:30PM 0 counting process in survival
12:06PM 1 question about trpaths
11:21AM 1 Packages evd and fExtremes
10:41AM 1 Pooled mean square error for regression
9:46AM 1 glmx specification of heteroskedasticity (and its use in Heckit)
9:13AM 0 biglm error:
8:07AM 2 merge without NA last
6:36AM 2 Returning combine output from R loop
2:57AM 3 merge_recurse in libray reshape - how does it work?
2:33AM 2 3.0.1 update and compiler package
Thursday May 30 2013
Time Replies Subject
10:35PM 1 write table in ascii
9:28PM 5 How can I write logical function in R?
8:07PM 1 wilcox_test function in coin package
5:40PM 1 a question
5:16PM 0 "mlogit" does not report number of observation?
2:32PM 1 Mixture of beta distributions
12:13PM 7 measuring distances between colours?
11:20AM 1 T statistic exactRankTests package
8:13AM 2 write table without quotation marks in column names....
8:07AM 1 question
5:57AM 1 error in rowSums in data.table
5:55AM 0 query regarding multivariate analysis in R
5:49AM 1 help on multivariate time series in R
1:27AM 2 Subset a floating point vector using all.equal?
Wednesday May 29 2013
Time Replies Subject
11:22PM 1 lmer (multinomial response variable ~ fixed + (1|random), family='"multinomial" ???)
6:23PM 0 Find longuest string in a character list
6:23PM 1 combine two columns into one
5:21PM 1 More discussion on R usage statistics
4:31PM 0 Error in names(data)[ix] : only 0's may be mixed with negative subscripts
4:27PM 1 Spatial points analysis in R
4:23PM 1 RCurl: using ls instead of NLST
4:16PM 1 'x' must be numeric..
3:37PM 1 Egen and concat
3:34PM 1 Modelling categorical and non-categorical datasets using Artifical Neural Networks
3:29PM 1 Goodness-of-fit tests for Complex Survey Logistic Regression
3:23PM 1 Equivalence between lm and glm
2:11PM 1 rgl crashes after one successful draw
1:18PM 0 Trouble in parallel computing with snow package
1:17PM 0 RCurl and google trends
1:15PM 3 bootstrap
11:49AM 2 adding class attributes to strings: works in a loop, but not directly
10:39AM 1 Problems in running x12
9:02AM 0 supermodular function optimization
8:54AM 0 estimating VAR(p) model leaving out intermediate lags
8:08AM 0 Split-plot and within-subjects
6:33AM 1 Problems with merge
6:32AM 1 Quantile regression for binary choice and heckit
6:29AM 0 Distribution, heteroskedasticity etc. tests for glm, tobit and heckit?
4:35AM 0 A question with respect to capturing output from a shell command
1:05AM 1 (no subject)
Tuesday May 28 2013
Time Replies Subject
10:50PM 1 highlight points in lattice cloud plot
8:41PM 1 Package unpacks (successfully and MD5 sums checked" but will not load "not a valid package"
6:41PM 0 Rcpp Starter With Carry Class
6:07PM 4 Rank Amateur Question
6:07PM 3 Obtaing the maximum
6:05PM 1 Function for finding matching values in two vectors.
5:31PM 1 make stat.desc output an object
5:27PM 0 error in installation
3:33PM 0 identical() function issue on R when built with ICC, ICL
3:16PM 1 The weak exogeneity test in R for the Error Correction Model?
3:00PM 1 Discrete simulated annealing
2:01PM 0 choose the lines
1:09PM 1 Inhomogeneous, univariate, O-Ring Statistic
10:31AM 2 Help retrieving only Portuguese words from a file
9:52AM 0 reference for variance in svykm
8:13AM 0 Failed exec in Update problem in R
7:20AM 1 lazy loading like object
6:54AM 4 Execution of larger blocks of pasted code often fails?
6:06AM 2 Trouble with arguments to 'order'
2:59AM 1 p values of plor
12:37AM 2 Data reshaping
Monday May 27 2013
Time Replies Subject
11:02PM 1 Plot histograms in a loop
9:30PM 0 Bayes Logit and Cholesky Decomposition
9:15PM 0 fitting grid-based models
7:52PM 0 choose the lines
6:17PM 3 How I can rearrange columns in data.frame?
3:24PM 3 How sum all possible combinations of rows, given 4 matrices
3:21PM 1 MLE for probit regression. How to avoid p=1 or p=0
3:12PM 4 updating observations in lm
2:40PM 1 Stop on fail using data manipulation
2:22PM 1 Assistant
2:18PM 1 Question about subsetting S4 object in ROCR
1:17PM 1 metaMDS with large dataset produces 'insufficient data' warning
11:45AM 0 Time Series prediction
11:34AM 2 Classification of Multivariate Time Series
10:39AM 2 Indexing within by statement - different coloured lines in abline wanted..
8:47AM 1 configure ddply() to avoid reordering of '.variables'
12:28AM 1 R-help Digest, Vol 123, Issue 30
Sunday May 26 2013
Time Replies Subject
5:24PM 0 avoiding eval parse with indexing: Correction
4:52PM 0 SAPPLY function for COLUMN NULL
4:24PM 0 [Fw: Re: avoiding eval parse with indexing ]
3:20PM 1 Setting default graphics device & options
1:56PM 4 avoiding eval parse with indexing
11:53AM 1 load ff object in a different computer
7:06AM 1 Parameters estimation for extreme value models
5:43AM 1 curiosity: next-gen x86 processors and FP32?
1:03AM 2 QA
Saturday May 25 2013
Time Replies Subject
9:50PM 1 Mapping GWR Results in R
8:14PM 1 What does this say? Error in rep("(Intercept)", nrow(a0)) : invalid 'times' argument
7:36PM 3 When creating a data frame with data.frame() transforms "integers" into "factors"
6:09PM 1 spatially analyzing multiple data layers
2:52PM 2 How to plot two functions a(t) & b(t) along with x & y axes respectively in r ?
9:50AM 1 Generating random numbers satisfying specific regression equation
1:59AM 0 Problem reading from NetCDF.
Friday May 24 2013
Time Replies Subject
9:46PM 2 retaining comments within functions defined in Rprofile.site
8:55PM 1 survival in R
8:08PM 2 Pie chart
7:48PM 0 modeling outliers in arfima
7:36PM 2 xtable() with booktabs option problem
7:34PM 0 To get the dataframe
7:22PM 1 matrix of random variables from a matrix of means and matrix of sd
6:08PM 0 Question about Rpart function
5:27PM 1 Creating an R function wrapper for a bash shell script
5:00PM 1 Multinomial logistic regression
4:42PM 0 Generating Predicted Probabilities in a Data Frame from a Logit Model
3:29PM 2 R 3.0.0
2:21PM 1 Change devAskNewPage setting on Escape?
1:51PM 1 ggmap, hexbin and length distortion in Lat/long
11:26AM 0 Rcpp with OpenMP - Need example Makevars
11:08AM 1 par(mfrow=c(1,3)) and postscript(file=)
10:24AM 2 Continuous columns of matrix
9:10AM 0 Rcpp with OpenMP - Need example Makevars
9:10AM 5 Rcpp with OpenMP - Need example Makevars
8:24AM 1 Construct plot combination using grid without plotting and retrieving an object?
4:31AM 2 Download data
4:05AM 0 Bug in latest Ubuntu release (13.04) disables R graphics device
Thursday May 23 2013
Time Replies Subject
10:21PM 1 When the interaction term should be interpreted in AIC table?
9:35PM 1 subsetting and Dates
8:29PM 1 Distance calculation
8:21PM 1 order panels in xyplot by increasing slope
7:44PM 0 glmnet package: command meanings
6:30PM 0 data frame "sum"
6:04PM 2 strings
4:10PM 1 FW: Kernel smoothing with bandwidth which varies with x
3:30PM 1 apply function within different groups
3:30PM 2 Could graph objects be stored in a "two-dimensional list"?
3:06PM 2 Error in png: unable to start png() device
2:53PM 1 error message solution: cannot allocate vector of size 200Mb
2:31PM 0 xml newbie
2:23PM 3 Removing rows w/ smaller value from data frame
1:05PM 0 adding rows...
10:53AM 1 SEM: multigroup model
8:41AM 1 Fwd: Merge
7:44AM 2 Transform Coordinate System of an ASCII-Grid
7:05AM 3 convert a character string to a name
5:00AM 1 adding rows without loops
3:11AM 2 Sort data by month
2:54AM 1 sample(c(0, 1)...) vs. rbinom
Wednesday May 22 2013
Time Replies Subject
11:17PM 0 (no subject)
10:13PM 0 question
9:40PM 2 group data based on row value
8:34PM 2 Linebreaks in cat() functions that call other variables?
7:29PM 1 NADA Package install disappearance
6:51PM 1 column width in .dbf files using write.dbf ... to be continued
6:08PM 3 How to test if something doesn't exist
5:34PM 0 Request for Help on Named Vectors
5:11PM 0 bar plot with non-zero starting level
4:31PM 1 reading large csv data sets efficiently
3:11PM 0 Problems with LSD.test
3:07PM 0 rmean in survfit
2:49PM 1 How to create a correct matrix in R
2:42PM 0 Multi-state model: msprep Error during data preparation
2:20PM 1 Code overloading PC
2:13PM 2 Forecasting MA model different to manually computation?
1:57PM 0 calcul of the mean in a period of time
12:27PM 0 DRC package: Weibull W1.3 and W2.3 - where is the difference?
12:00PM 2 list to dataframe
7:32AM 1 Rplot save problem after using "identify" with R version 3.0.0
2:32AM 0 eRm package, RSM: Error in solve.default(parets$hessian) :
1:57AM 1 Something Very Easy
1:46AM 0 Writing to a file
Tuesday May 21 2013
Time Replies Subject
11:22PM 1 x axis problem when plotting
9:11PM 1 metafor matrix error
8:29PM 2 point.in.polygon help
8:16PM 3 add identifier column by row
7:40PM 2 problem with "transform" and "get" functions
6:01PM 2 help with data.frame
3:57PM 2 problems with saving plots from loop
3:00PM 1 Calculating AIC for the whole model in VAR
2:58PM 0 tt function and coxph
11:49AM 1 microarray analysis: contrast matrix problem
11:42AM 0 Arules: getting rules with only one item in the left-hand side
11:19AM 1 keep the centre fixed in K-means clustering
11:18AM 1 Lattice, ggplot, and pointsize
11:16AM 1 problem in reading GDX file
9:20AM 0 Increasing time of simulation
9:19AM 0 Repeated k-fold Cross-Validation with Stepwise Regression
9:16AM 2 Sum first 3 non zero elements of row
7:16AM 2 Using loop for applying function on matrix
6:00AM 0 scale change
5:35AM 1 ordered and unordered variables
1:06AM 1 How many decimal places of information does R actually use in computation?
Monday May 20 2013
Time Replies Subject
11:45PM 2 how to GREP out a string like this......THANKS.
9:24PM 2 old Windows binary for GBM package
9:09PM 3 as.vector with mode="list" and POSIXct
6:18PM 1 help with 'cem' for r 2.14.2
6:10PM 0 Neural network: Amore adaptative vs batch why the results are so different?
4:10PM 1 R for windows GUI front-end has stopped working
3:44PM 1 Gamma curve fit to data with specific bins
3:21PM 0 Loading intraday data with zoo
2:11PM 1 Extract t-statistics from "mer" object
1:26PM 0 finding moving average order
12:36PM 3 numeric not equal its value
12:30PM 1 R relative frequency by date and operator
12:09PM 2 DEfining and plotting the sum of two functions
11:52AM 0 choosing the correct number of clusters using NbClust
9:14AM 1 How to fit a normal inverse gaussian distribution to my data using optim
8:34AM 1 table() generating NAs when there are no NAs in the underlying data
4:17AM 2 How to create surface3d plot with surface data
12:52AM 1 ANOVA Estimated effects may be unbalanced
Sunday May 19 2013
Time Replies Subject
10:52PM 0 Help needed
9:05PM 3 coping zeros from matrix to another
4:08PM 2 strange behaviour with loops and lists
2:31PM 1 Generate positive definite matrix with constraints
2:22PM 2 apply and table
1:54PM 0 TODAY field in termstrc
1:13PM 0 index of quantile variation (iqv)
12:31PM 2 How to run lm for each subset of the data frame, and then aggreage the result?
11:20AM 1 Extracting values from grid cells using for loops
6:32AM 1 Weighted regression in rms/Hmisc
Saturday May 18 2013
Time Replies Subject
11:19PM 3 bar plot with non-zero starting level
8:42PM 1 arima prediction
3:44PM 1 distances between entities
1:05PM 2 exporting data into STATA
11:45AM 1 how to do an external validation with R
6:23AM 1 glmer.nb: function not in downloaded lme4 package?
Friday May 17 2013
Time Replies Subject
9:58PM 0 Heterogeneous negative binomial
9:01PM 2 filter rows by value
7:58PM 1 image and color gradient
7:53PM 2 formatting column names of data frame
7:45PM 2 Bivariate - multivariate linear regression
6:48PM 2 time-series aggregation of information
5:21PM 2 inverse for formula transformations on LHS
5:02PM 2 Problems using lmer {lme4}
4:46PM 2 #Keeping row names when using as.data.frame.matrix
3:46PM 2 Comma separated vector
2:47PM 1 R and libre office base
2:44PM 1 Error with adehabitatHR and kernelbb
2:17PM 0 mirt package "error in ESTIMATION..."
2:04PM 1 update an array of plots in 'real-time' without drawing lags
1:39PM 1 out of the mailing list
1:03PM 2 zigzag confidence interval in a plot
12:32PM 0 External access to fft routines
12:28PM 0 Natural Language Recognition
12:16PM 1 Homals: Nonlinear PCA
12:07PM 0 Probabilistic neural network (PNN)
12:07PM 0 [R-pkgs] Probabilistic neural network (PNN)
11:34AM 1 Log scales
10:53AM 3 Repeating sequence elements
10:24AM 0 n in pglm() and relogit()
9:40AM 1 pearson correlation significant level
7:31AM 0 Using grubbs test for residuals to find outliers
6:51AM 2 Selecting A List of Columns
6:36AM 2 Find the indices of non-NA elements of a sequence
5:57AM 0 Mean using different group for a real r beginner
5:40AM 2 Rcmdr Bug?
3:56AM 1 Problem with ordiellipse coloured factor in Vegan
2:42AM 1 negative extents to matrix
1:58AM 2 How could I see the source code of functions in an R package?
1:56AM 2 peering inside functions in a package?
12:30AM 0 rggobi error
Thursday May 16 2013
Time Replies Subject
9:48PM 1 Why is it not working the MaxEnt analysis (package "maxent", and "dismo")?
9:41PM 2 Chi square 1 d.f.
9:22PM 1 connect to local mySQL database
8:46PM 2 convert a data.frame to matrix
5:43PM 1 strftime
5:13PM 1 To List or Not To List
5:11PM 1 using metafor for meta-analysis of before-after studies (escalc, SMCC)
4:12PM 2 A function that can modify an object? Or at least shows principles how to modify an object?
3:58PM 0 Help with how to process multiple column variable in a read.table
3:31PM 1 sapply error produced by grid search
3:23PM 1 Contour lines in a persp plot
3:05PM 3 Error: contrasts can be applied only to factors with 2 or more levels
2:21PM 1 How to extract value from image in R ?
11:50AM 4 Three plots with logged X-axis in the same plot
11:00AM 1 Trouble Runing R Version 2.13.0
10:17AM 2 R looping help
10:02AM 1 Failed to open *.img in rgdal
9:52AM 1 reconstructing original series based on differenced data
8:06AM 1 3d interactive video using the rgl package
7:51AM 0 R 3.0.1 is released
7:51AM 0 R 3.0.1 is released
7:19AM 0 How to suppress default legend in plot.cuminc()
7:16AM 1 Loop through a simulation
6:17AM 3 estimate value from simulations
4:47AM 0 OFF TOPIC: Report on a conference on "Integrity in research"
Wednesday May 15 2013
Time Replies Subject
11:58PM 0 Unexpected behavior using `merge' by multiple columns
7:57PM 2 matrix - pairwise comparison of columns
6:21PM 2 animating plots over time with a slider
6:06PM 1 x and y lengths differ
5:51PM 2 R help: Batch read files based on names in a list
5:32PM 1 mosaic plot (vcd)
4:39PM 0 Build R package with libjpeg - Symbol not found: _jpeg_CreateCompress
4:02PM 1 Problem with convergence in optim
3:42PM 1 stack object layer names not visible
2:59PM 1 R update problem
2:33PM 1 how to show a dataset in 3d?
2:04PM 1 Vegan package treatment of zeros
10:32AM 0 New quotation
9:51AM 2 K Fold CrossValidation
9:46AM 4 Error in colMeans with multiple column data.frame
9:14AM 2 erreur dans R
8:39AM 1 Empty clusters in k-means - possible solution
8:28AM 0 BASEL II Validation Tests
6:53AM 1 Fw: Request for information
6:15AM 2 data manipulation
6:14AM 0 fast time series similarity (iSAX, UCR DTW, UCR ED) implementations for R?
4:06AM 2 NMDS using Vegan
Tuesday May 14 2013
Time Replies Subject
9:55PM 1 R-Help: nparLD Package Non-parametric Repeated Measures
9:26PM 2 empirical and GPD for time series simulation
7:59PM 3 Help me please: gplot, facets_wrap and ordering of x axis dates
7:04PM 1 Post hoc test for GLM with poisson distribution
6:24PM 0 Specifying Correlation Structures in Linear Multivariate State Space Models
5:20PM 1 Changing Order of Factor Levels in Mixed Model (nlme)
5:10PM 1 3D plot
4:05PM 1 query in plot(intervals....
3:54PM 3 query re plot(confint(lmList...
3:06PM 2 where clauses - help
1:32PM 0 need help for Imbalanced classification problems!!!
1:30PM 0 apcluster webinar: Thursday, June 13, 2013, 7:00pm CEST
1:02PM 0 Select the column from the data.frame?
12:34PM 1 points overlay axis
10:32AM 2 Possible bug in 'data.table'
9:55AM 0 Problem with constrained nlsList model
8:52AM 5 Unexpected behavior of "apply" when FUN=sample
8:06AM 2 Dataframe and conditions
7:45AM 1 Problem with R websocket package
7:40AM 0 empirical and GPD time series simulation
4:37AM 0 Matrix multiplication with scattered NA values
3:35AM 1 Sampling Weights and lmer() update?
1:38AM 1 How to capture the expression corresponding to the i param in the [ function
1:18AM 2 Boundaries of consecutive integers
12:49AM 1 Fwd: ?on-consequitve # of lags in VAR (package 'vars")?
Monday May 13 2013
Time Replies Subject
11:47PM 1 Math problem with xts objects
10:34PM 0 Basic R question
8:31PM 1 what does summary(polr(...)) really call? --- and obscure buglet
7:49PM 0 Select rasters in stack based on layer partial name match
7:22PM 4 how to merge 2 data frame if you want to exclude mutual obs
6:41PM 3 recover log of matrix
6:23PM 2 what does summary(polr(...)) really call?
5:41PM 1 Creating Color Pallets
5:36PM 3 melt in reshape2 destroying dates?
4:38PM 1 dotcharts next to boxplots
4:05PM 2 Matching names with non-English characters
3:17PM 1 How can I make a loop to do multiple regression for more than 3000 dependent variables?
2:24PM 2 reduce three columns to one with the colnames
12:20PM 1 Anova und Tukey HSD
9:27AM 1 question about poisson regression
8:19AM 2 grepl
7:58AM 3 help: R GUI front-end has stopped working
6:16AM 1 condition and function
Sunday May 12 2013
Time Replies Subject
11:44PM 1 Multinomial-Dirichlet using R
8:54PM 1 aggregate.data.frame with NAs and different types
8:42PM 3 scan in R
5:30PM 1 Loop for CrossTable (gmodels)
11:47AM 1 How to center a horizontal legend with specified 'text.width'
8:55AM 2 Need help to building R package with devtools
7:07AM 0 Unexpected scrapeR result
2:22AM 2 Broken line questions
1:06AM 1 how to deal with continous and Non continuous mixed variables in factor analysis?
12:57AM 2 need means on all boxplots, but only half of them got that
Saturday May 11 2013
Time Replies Subject
10:35PM 3 [newbie] return index(s) for value?
8:26PM 0 cut.dendrogram bug?
5:09PM 0 MaxLik pacakge
4:21PM 1 How to repeat 2 functions in succession for 400 times? (microarray data)
3:40PM 3 boxplot with grouped variables
1:29PM 1 pmnorm produces NaN
12:38PM 0 mcmapply should be parallel, but runs serial?
10:49AM 0 Help on sound identification of "distance between lines"
7:59AM 1 prediction in a loop with only one sample
4:45AM 0 generating new vector based on criteria from other vectors
4:10AM 1 question about 1-way anova
3:15AM 0 question
12:07AM 1 (no subject)
Friday May 10 2013
Time Replies Subject
10:46PM 1 R's basic lm() and summary.lm functions
7:04PM 2 need an assistance
6:58PM 0 How to generate factor levels with unequal numbers?
5:24PM 2 date and time coding question
5:22PM 0 Revolutions blog: April 2013 roundup
5:21PM 1 Distributing R byte code
4:35PM 1 Automatic way to embed fonts into all pdf output?
3:42PM 1 rlnorm(n, meanlog = 0, sdlog = 1)
3:30PM 0 Package Seqminer: Fundamental usage question
3:00PM 0 helper function to "pluck" dendrogram attributes
2:33PM 0 Weird RefClass running time
2:29PM 1 Return a vector in Fibonacci sequence function
1:51PM 2 How to use character in C code?
1:01PM 0 Allocating outputs from code using foreach and doPar
12:15PM 0 Isolate polygons unionSpatialPolygons / single polygon instead of multipart polygon
10:59AM 0 irt model estimation with constrained parameters
10:51AM 1 maxLik: different estimations from different packages
10:28AM 1 Fwd: binomial glm and sole effect of the treatment
8:59AM 2 Replacing values in a dataframe depending on several conditions
8:41AM 1 PGLM Package: Starting Values for Within-Model
7:38AM 1 compare vector with selected element from a matrix
6:19AM 1 functions with function inputs and optimisation
2:25AM 0 Clustering with uneven variables
2:06AM 2 How can I extract part of the data with a selection criterion?
Thursday May 9 2013
Time Replies Subject
11:24PM 1 duplicate rows with new time series
9:22PM 2 error in barplot
8:58PM 2 'symbols' not drawing correctly in the 'y' dimension
8:41PM 0 interpolation in at least 3, perhaps 4 dimensions?
8:05PM 2 subsetting by is not
8:05PM 2 Raster comparrison
6:27PM 2 Can DEoptim trace output be customized?
5:35PM 1 attr vs attributes
4:28PM 2 subsetting dataframe multiple conditions
3:50PM 2 Question with "apply"
3:43PM 0 Finding x-value of intersections of mixture models
3:10PM 0 questions for using randomForest/pamr to predict biological data
2:48PM 1 creating multivariate normal random variables
1:06PM 3 R installation error
1:00PM 3 Can you help me sir,
11:58AM 2 Paritioning of Interaction SS in ANOVA - Gomez and Gomez(1984) example
11:31AM 0 same key row merge in dataframe
11:09AM 0 Replace rows in dataframe based on values in other columns
10:45AM 1 Time series analysis
8:27AM 1 Fit a standardized generalized hyperbolic distribution to my data?
7:58AM 0 help needed in least square curve fit equation
5:36AM 0 ARMA(p,q) prediction with pre-determined coefficients
4:54AM 2 Re-start R issue on win7 [SEC=UNOFFICIAL]
2:36AM 0 error with lhoat() function from the package hydroPSO
2:32AM 0 Inputting white noise values for ARMA prediction
1:29AM 1 Choice of statistical test (in R) of two apparently different distributions
12:38AM 2 [newbie] *apply to matching elements of n arrays?
12:30AM 2 R unable to access internet to install packages
Wednesday May 8 2013
Time Replies Subject
8:39PM 1 NMDS with missing data?
8:37PM 1 ecdf --- title suggestion and question
6:58PM 0 PPCOR: Semipartial Correlation & Regression weights
6:06PM 4 Correctly Setting New Seed
5:15PM 0 chronological season results assistance
5:03PM 1 generate gaussian
4:49PM 2 scatter plot matrix with different x-y variables
4:18PM 1 chronological season results assistance
4:05PM 1 Tranfer data from exel into R
3:01PM 0 Estimating confidence intervals from several simulations
2:12PM 2 gam (mgcv), multiple imputation, f-stats/p-values, and summary(gam)
1:39PM 0 Using the Sun Grid Engine with R-3.0
12:53PM 0 chronological season results assistance
12:51PM 1 NaN-result from fuzzy_inference (package "sets") with certain input
9:08AM 1 gsub regex simplification
8:59AM 1 Fuzzy rules definition (package "sets") from data.frame
8:25AM 2 R prints empty group on my figure!!
7:49AM 1 Fitting AR(p) model
7:43AM 2 Parsing XML to tree.
6:15AM 1 make hyperlink in R
5:19AM 1 How to calculate Hightest Posterior Density (HPD) of coeficients in a simple regression (lm) in R?
3:40AM 2 Calculates the mean/median from grouped data in R?
3:39AM 0 CRAN RSS feeds about updates and package checks
3:38AM 3 is.numeric () FALSE
3:21AM 2 Bootstrapped 1-sided confidence intervals
2:51AM 0 how to get samples from rtmvnorm with large dimensions
1:54AM 2 How can I find negative items from a vector with a short command?
Tuesday May 7 2013
Time Replies Subject
11:27PM 0 How to use big.matrix to read factor columns
10:46PM 1 how to read numeric vector as factors using read.table.ffdf
8:40PM 0 How to use "SparseM-conversions" to convert a dCgMatrix into a matrix.csr ?
7:08PM 4 how to calculate the mean in a period of time?
6:24PM 0 Question about fitting a periodic model in glmm
5:54PM 4 create unique ID for each group
4:20PM 2 recode categorial vars into binary data
3:33PM 1 Balanced design throws "design unbalanced, cannot proceed" error
1:54PM 2 How does one set up logical functions?
1:30PM 1 Using as.integer(NA) in the .C function
1:23PM 0 extracting the residuals from models working with ordinal multinomial data
1:20PM 0 Tinn-R news
1:18PM 1 Superimpose exponential density function to histogram
11:32AM 0 Orthogonal transformation option in pgmm-plm
10:55AM 1 Plot device stretched on Mac - any advice?
9:41AM 1 Problem with biomaRt::getSequence.
8:19AM 0 recommended workflow for creating functions (was: Re: [Rd] Patch proposal for R style consistency (concerning deparse.c))
7:16AM 0 hi.
6:33AM 3 Merge two dataframe with "by", and problems with the common field
4:39AM 2 Extracting elements from a matrix using a vector containing indices
2:11AM 1 Bootstrapped Non-Parametric 1-sided Confidence Intervals
1:29AM 1 pR2 stumped
Monday May 6 2013
Time Replies Subject
11:29PM 1 rowspan and readHTMLTable
9:29PM 2 replace data by a rule
8:28PM 4 State space models with regime switching
8:14PM 0 topGO printGenes
7:56PM 0 filtering with fft
4:43PM 0 FW: gamma shared frialty model
4:14PM 2 Some unrelated questions.
4:00PM 0 a huge sparse matrix to pass into svm
2:17PM 0 Replace the missing values with column mean values?
2:10PM 1 Using Rscript in version 3.0.0
1:36PM 1 Retrieve specific y for a list of x on a ploted line
11:58AM 0 Release plans: R-3.0.1 on May 16
11:58AM 0 Release plans: R-3.0.1 on May 16
10:35AM 1 xyplot legend : simple question
9:55AM 0 How are feature weights extracted from 'superpc' analysis?
8:03AM 2 How can I access the rowname of a data?
7:10AM 1 No Ukraine in maps library?
7:03AM 1 Adjust cex to fit width
2:34AM 2 BIOENV
12:36AM 0 Markov Chain Discrete
Sunday May 5 2013
Time Replies Subject
11:57PM 0 compare two quadratic regression using bootstrap
11:36PM 1 slope coefficient of a quadratic regression bootstrap
11:34PM 2 Open graphics device within sourced script?
6:12PM 1 negative correlation with corSymm in nlme or another way?
2:16PM 2 custom function that plots other functions- problem
12:20PM 2 Error in plot.window(...) : need finite 'xlim' values
12:19PM 1 ArimaLike NaN return
8:55AM 1 R 3 and Debian Testing
12:05AM 1 Vegan problem
Saturday May 4 2013
Time Replies Subject
10:24PM 2 Size of label axis
9:33PM 1 Error running caret's gbm train function with new version of caret
7:55PM 0 memory leak using 'apply'? [was: how to parallelize 'apply' across multiple cores on a Mac]
6:23PM 2 Plot smooth
3:43PM 1 Help, how to find the genes with A<19?
2:57PM 0 Panel Granger Non-Causality Tests in R
1:47PM 0 github_install problem
1:09PM 2 Lasso Regression error
9:24AM 1 Multiple Line Plots in the same graph
9:21AM 1 plotting 2 time series data on the same graph
3:14AM 0 mean for each observation
12:38AM 1 how to best add columns to a matrix with many columns
Friday May 3 2013
Time Replies Subject
11:56PM 2 how to parallelize 'apply' across multiple cores on a Mac
11:08PM 1 R package for bootstrapping (comparing two quadratic regression models)
10:32PM 1 Factor deletion criteria
9:46PM 1 A problem of splitting the right screen in 3 or more independent vertical boxes:
8:44PM 0 selecting certain rows from data frame
8:37PM 3 color by group in ggplot
7:45PM 0 Fortune candidate! Re: Why can't R understand if(num!=NA)?
5:08PM 0 Change Selected Variables from Numeric to Factors
4:44PM 1 (no subject)
3:52PM 0 Empirica Copula
3:26PM 0 Is it a "Headless problem"? - Same code runs well in interactive R shell, but never terminates with Rscript
3:24PM 10 Why can't R understand if(num!=NA)?
3:20PM 2 Declare a set (list?) of many dataframes or matrices
1:59PM 2 Write date class as number of days from 1970
1:49PM 2 read .csv file and plot a graph
1:33PM 1 print multiple plots to jpeg, one lattice and one ggplot2
12:59PM 1 MANOVA summary.manova(m) :" residuals have rank"
12:47PM 0 Courses: Statistical Analysis with R - Bayesian Data Analysis with R and WinBUGS
12:45PM 2 Very basic statistics in R
9:48AM 2 R does not subset
9:00AM 0 significant test of two quadratic regression models (lm)
8:22AM 0 Edmonton course: Regression, GLM & GAM with R intro
7:31AM 1 untar() error
6:49AM 1 Likelihood
3:29AM 2 Find the flow data from its accumulation of the panel data
Thursday May 2 2013
Time Replies Subject
11:15PM 1 (no subject)
11:02PM 2 Calculating distance matrix for large dataset
6:50PM 1 Create and read symbolic links in Windows
4:54PM 2 Self-developed package -- installation
2:27PM 1 Package survey: singularities in linear regression models
1:32PM 0 Questions regarding use of predict() with glmpath
12:38PM 1 function gstat r with pyhton
12:37PM 1 Vector allocation problem while trying to plot 6 MB data file
12:09PM 1 Clean Price of Bond : Can't install "RQuantLib" in R version 3.0.0
11:23AM 1 warnings in ARMA with other regressor variables
10:26AM 1 multivariate, hierarchical model
10:15AM 2 ARMA with other regressor variables
9:11AM 0 How does dsgh do the standardization?
8:36AM 2 ODE solver
7:47AM 2 saving a matrix
7:17AM 0 modMCMC runs in FME package
6:28AM 3 Divide matrix columns by different numbers
6:28AM 3 R issue with unequal large data frames with multiple columns
3:55AM 1 Problems with reading data by readWorksheetFromFile of XLConnect Package
3:37AM 2 rda variance partioning in vegan problems
3:10AM 1 R CMD building SPEEDY
1:16AM 3 Lattice xyplot multipanels
Wednesday May 1 2013
Time Replies Subject
9:43PM 1 Multiple Paired T test from large Data Set with multiple pairs
8:04PM 4 Selecting several columns/rows of a dataframe?
7:37PM 1 Combine multiple tables into one
7:31PM 1 Could not find function boss.set
6:37PM 1 Tiff plot Resolution issues
6:20PM 1 Size of a refClass instance
4:53PM 3 Adding Column to Data Frames Using a Loop
4:28PM 3 Chron format question h:m not working
3:58PM 1 Hello
3:57PM 0 Likelihood ratio test for comparing non-nested cox models
2:05PM 1 rpy2 postgres qgis problem
1:48PM 2 Factors and Multinomial Logistic Regression
1:41PM 1 Sum objects in a column between an interval defined by conditions on another column
12:53PM 0 How to standardize the generalized hyperbolic distribution?
12:38PM 0 Standardized Generalized Hyperbolic Distribution
11:54AM 1 selecting rows based on multiple criteria
10:04AM 0 Thornthwaite Method in R
8:37AM 3 grep help (character ommission)
5:26AM 0 log returns, error
2:34AM 1 Trouble with methods() after loading gdata package.
2:07AM 2 significantly different from one (not zero) using lm
1:06AM 1 help understanding hierarchical clustering
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Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the
orbital plane
. The
orbital plane
of an object orbiting another is the geometrical
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
in which the orbit is
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
. The orbital plane is defined by two parameters,
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation
of a celestial orbit...
(i) and
Longitude of the ascending node
Longitude of the ascending node
The longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a reference direction, called the origin of longitude, to the
direction of the ascending node, measured in a reference plane...
(Ω). Three non-
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
points in space suffice to determine the orbital plane. A common example would be: the center of the heavier object, the center of the orbiting object and the center of the orbiting object at some
later time.
By definition the
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation
of a celestial orbit...
of a planet in the
solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion
years ago. The vast majority of the system's mass is in the Sun...
is the
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for
standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
between its orbital plane and the orbital plane of the
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing
the mean orbit of the Earth around the Sun...
). In other cases, for instance a moon orbiting another planet, it is convenient to define the inclination of the
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods
of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
's orbit as the angle between its orbital plane and the planet's equator.
Artificial satellites around the Earth
For launch vehicles and artificial satellites, the orbital plane is a defining parameter of an orbit; as in general, it will take a very large amount of
A propellant is a material that produces pressurized gas that:* can be directed through a nozzle, thereby producing thrust ;...
to change the orbital plane of an object. Other parameters, such as the
orbital period
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of
an astronomical object, which is calculated with respect to the stars.There are several kinds of...
, the
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...
of the orbit and the phase of the orbit are more easily changed by propulsion systems.
Orbital planes of satellites are perturbed by the non-spherical nature of the
Earth's gravity
Earth's gravity
The gravity of Earth, denoted g, refers to the acceleration that the Earth imparts to objects on or near its surface. In SI units this acceleration is measured in metres per second per second or
equivalently in newtons per kilogram...
. This causes the orbital plane of the satellite's orbit to slowly rotate around the Earth, depending on the angle the plane makes with the Earth's equator. For planes that are at a critical angle
this can mean that the plane will track the
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
around the Earth, forming a
Sun-synchronous orbit
Sun-synchronous orbit
A Sun-synchronous orbit is a geocentric orbit which combines altitude and inclination in such a way that an object on that orbit ascends or descends over any given point of the Earth's surface at the
same local mean solar time. The surface illumination angle will be nearly the same every time...
A launch vehicle's
launch window
Launch window
Launch window is a term used in spaceflight to describe a time period in which a particular launch vehicle must be launched. If the rocket does not launch within the "window", it has to wait for the
next window....
is usually determined by the times when the target orbital plane intersects the launch site.
The source of this article is
, the free encyclopedia. The text of this article is licensed under the
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Go Math Grade 3 Answer Key Chapter 10 Time, Length, Liquid Volume, and Mass Assessment Test
This chapter can improve student’s math skills, by referring to the Go Math Grade 3 Answer Key Chapter 10 Time, Length, Liquid Volume, and Mass Assessment Test, and with the help of this Go Math
Grade 3 Assessment Test Answer Key, students can score good marks in the examination.
Go Math Grade 3 Answer Key Chapter 10 contains all the topics of chapter 10 which helps to test the student’s knowledge. Through this assessment test, students can check their knowledge. This
assessment test is also helpful for the teachers to know how much a student understood the topics.
Chapter 10: Time, Length, Liquid Volume, and Mass Assessment Test
Test – Page 1 – Page No. 101
Question 1.
Chris left to take his dog for a walk at 6:25 P.M. He returned home after 26 minutes. He talked to his neighbor for 10 minutes before going back into his house.
Write the time that Chris got home and the time he went back into the house. Explain how you found each time.
Answer: Chris got back home at 6:51 PM and he went into the house at 7:01 PM.
Question 2.
Kayla measured an object with an inch ruler. It was about 1 inch wide.
For numbers 2a–2d, choose Yes or No to tell whether the object could be the one Kayla measured.
i. yes
ii. no
Answer: Yes.
Explanation: It seems like a stamp that can be measured.Â
Question 2.
i. yes
ii. no
Answer: Yes.
Explanation: It is a pin that is less than can be measured.Â
Question 2.
i. yes
ii. no
Answer: No
Explanation: The pencil can be more than an inch.
Question 2.
i. yes
ii. no
Answer: No
Explanation: The rope can be more than an inch.
Test – Page 2 – Page No. 102
Question 3.
Luz left for the park at 2:27 P.M. She arrived at 3:09 P.M. How long did it take Luz to get to the park?
_____ minutes
Answer: 42 minutes.
Explanation: The time difference between 3:09 P.M and 2:27 P.M is 42 minutes.
Question 4.
Terry wakes up for school at five minutes before seven in the morning. At what time does Terry wake up?
Circle a time that makes the sentence true.
Terry wakes up at
Answer: Terry wakes up at 6:55 A.M.
Terry wakes up atÂ
Question 5.
Select the objects with a mass less than 1 kilogram.
Mark all that apply.
a. desk
b. eyeglasses
c. eraser
d. plastic fork
Answer: b,c,d
Explanation: Eyeglasses, eraser, plastic fork will be less than 1 kilogram.
Question 6.
A batch of muffins needs to bake in the oven for 22 minutes. They need to cool for at least 15 minutes before they should be eaten. Wade puts the muffins in the oven at 10:17 A.M.
For numbers 6a–6d, select True or False for each statement.
a. Wade can eat the muffins at 10:39 P.M.
i. True
ii. False
Answer: True.
Explanation: If Wade put a muffin in an oven at 10:17 A.M he has to take out then at 10:39 A.M (which is after 22mins), they should be cooled down at least for 15mins once they have taken out. So
they can eat after 10:54 A.M.
Question 6.
b. Wade can eat the muffins at 10:44 A.M.
i. True
ii. False
Answer: False.
Explanation: The muffins are not cool, so Wade cannot eat.
Question 6.
c. Wade should take the muffins out of the oven at 10:39 P.M.
i. True
ii. False
Answer: False.
Explanation: No, Wade should take the muffins out of the oven at 10:39 A.M.
Question 6.
d. Wade should take the muffins out of the oven at 10:39 A.M.
i. True
ii. False
Answer: Yes.
Explanation: If Wade put a muffin in the oven at 10:17 A.M he has to take out then at 10:39 A.M which is after 22mins.
Test – Page 3 – Page No. 103
Question 7.
Kirk went to a friend’s house after dinner. He left his house at the time shown on the clock and returned home at 8:05 P.M.
Part A
How long was Kirk gone?
______ minutes
Answer: 36 mins.
Explanation: Kirk left home at 7:29 PM and returned after 8:05 PM. Thus he has gone for 36 mins.
Question 7.
Part B
Explain how you found your answer.
Answer: Counted 31 mins from 7:29 to 8:00, 5 mins from 8:00 to 8:05. Then summing up all give 31 mins + 5 mins = 36 mins.
Question 8.
Brad looked at the clock on his way to the football game. What time is shown on Brad’s clock? Mark all that apply.
a. thirteen minutes before ten
b. nine forty-seven
c. quarter to ten
d. nine forty
Answer: Options a and b.
Explanation: The time has shown on Brad’s clock is thirteen minutes before ten and nine forty-seven.
Question 9.
Andrea uses a balance scale to compare mass.
Circle a symbol that makes the comparison true.
The mass of the erasers
Answer: >
The mass of the erasers
Test – Page 4 – Page No. 104
Question 10.
A bucket of water holds about 19 liters.
For numbers 10a−10e, choose Yes or No to tell whether the container will hold all of the water.
a. bath tub
i. yes
ii. no
Answer: Yes.
Explanation: The bathtub can hold all of the water.
Question 10.
b. large water bottle
i. yes
ii. no
Answer: No.
Explanation: A large water bottle cannot hold all of the water.
Question 10.
c. soup bowl
i. yes
ii. no
Answer: No
Explanation: A soup bowl cannot hold all of the water.
Question 10.
d. large fish tank
i. yes
ii. no
Answer: Yes.
Explanation: A large fish tank can hold all of the water.
Question 10.
e. kitchen sink
i. yes
ii. no
Answer: Yes.
Explanation: A kitchen sink can hold all of the water.
Question 11.
Select the animals that would be best measured in kilograms. Mark all that apply.
a. dog
b. mouse
c. goat
d. sheep
Answer: a,c,d.
Explanation: Dog, Goat, Sheep are best measured in kilograms.
Question 12.
Luisa is planning her day on Saturday. Write the letter for each activity next to the time she plans to do it.
A. Wake up. ______ 3:30 P.M.
B. Play soccer game. ______ 8:30 P.M.
C. Eat lunch. ______ 7:30 A.M.
D. Go to a movie. ______ 12:30 P.M.
E. Go to bed. ______ 9:00 A.M.
A. Wake up at 7:30 A.M.
B. Play soccer game 9:00 A.M.
C. Eat lunch at 12:30 P.M
D. Go to a movie at 3:30 P.M.
E. Go to bed at 8:30 P.M.
Test – Page 5 – Page No. 105
Question 13.
Stephen has 28 teaspoons of lemon juice. He uses 5 teaspoons of juice for each glass of lemonade he makes. He adds 2 teaspoons of juice to each glass of iced tea he makes.
Stephen makes 4 glasses of lemonade. Does he have enough lemon juice to make 4 glasses of iced tea?
Explain how you solved the problem.
Answer: Yes
How much lemon juice he used to make 4 glasses of lemonade = 4 x 5 = 20
Total no of teaspoons of lemon juice he had = 28
No of teaspoons of lemon juice needed to make iced tea = 4 x 2 = 8
Therefore total he needed = 20 + 8 = 28
Thus he has enough lemon juice to make 4 glasses of iced tea.
Question 14.
Use an inch ruler to measure.
Part A
What is the length of the flower to the nearest fourth-inch?
_____ \(\frac{â–¡}{â–¡}\)
Answer: 2 ¼ inches.
Explanation: The length of the rose diagram on the ruler is closest to 2.25 – inch mark (6 cm nearly). So taken as 2 ¼ inches which are nearest to the fourth inch.
Question 14.
Part B
Explain how your answer would change if you measured the length of the flower to the nearest inch instead of fourth-inch.
Answer: The length of the rose diagram on the ruler is closest to 2 – inch mark 6 cm nearly.
Question 15.
A football game begins at 11:32 A.M. The game lasts 3 hours 16 minutes. When does the game end?
Answer: The football game end at 2:48 P.M.
Explanation: As the football game begins at 11:32 A.M and the game lasts 3 hours 16 minutes. So the game ends at 2:48 P.M
Question 16.
Alicia buys two packets of flower seeds. She buys a total of 75 grams of seeds. Select the packets she buys.
Answer: options b and d.
Explanation: As she buys a total of 75 grams, so 30g + 45g = 75g.
Test – Page 6 – Page No. 106
Question 17.
Mr. Barton measures the screws on his workbench. He records the measurements in a chart.
Part A
Mr. Barton found another screw. Use an inch ruler to measure. Record the measurement in the chart.
____ inches
Answer: 3 inches approx. (8 cm).
Question 17.
Part B
Complete the line plot to show the data in the chart. How many screws are longer than 2 inches? Tell how you know.
_____ screws
Answer: 2 screws.
Explanation: Screw with 2 ½ inches and 3, thus two screws longer than 2 inches.
Question 18.
Abby fills a mug with hot cocoa. Is the amount of cocoa more than 1 liter, about 1 liter, or less than 1 liter? Explain how you know.
Explanation: A liter is about the amount in a medium bottle of water. The amount would not fit a mug, so the full mug has less than 1 liter.
This assessment test helps students to check their math skills. Go Math Grade 3 Chapter 10 questions are explained in detail that students can understand easily.
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Python And Functions: Defining, Calling, And Utilizing Functions - Code With C
Python and Functions: Defining, Calling, and Utilizing Functions
Hey there tech enthusiasts, 🌟 let’s talk about Python and functions! Buckle up because we are diving deep into the world of defining, calling, and utilizing functions with Python. 🚀
Defining Functions
Syntax of a Function
Let’s kick it off with the syntax of a function in Python. Imagine a function like a recipe – a set of instructions to perform a specific task. Here’s a basic structure in Python:
def my_function():
# code block
print("Hello, I am a function!")
Parameters in Functions
Now, let’s spice things up with parameters in functions! These are like ingredients in our recipe; they allow us to pass data into a function. Check out this example:
def greet(name):
print("Hello, " + name + "!")
Passing in different names as arguments will personalize the greeting. How cool is that? 😎
Calling Functions
Using Functions with Arguments
Calling functions in Python is as easy as sipping on a cup of chai ☕. Just mention the function name followed by parentheses and add any required arguments. For our greet() function, you’d use:
Returning Values from Functions
Functions can also return values back to the caller. It’s like ordering a dish and getting a delicious meal in return. Here’s an example:
def square(num):
return num * num
result = square(5)
print("The result is:", result)
Utilizing Functions
Scope of Variables in Functions
Ah, the scope! 🌐 It’s like the boundaries within which a variable is accessible. Variables defined inside a function are local, while those outside are global. Remember, scope matters!
Recursive Functions
Now, brace yourselves for recursion! 🔄 Recursive functions call themselves until a certain condition is met. It’s like a never-ending loop, but in a good way! Here’s a classic example: calculating
def factorial(n):
if n == 1:
return 1
return n * factorial(n - 1)
result = factorial(5)
print("Factorial of 5 is:", result)
Good Practices in Function Writing
Function Naming Conventions
Naming functions is an art. Choose names that are descriptive and meaningful, like a title that captivates your audience. Make sure it reflects the function’s purpose clearly. 🎨
Documenting Functions
Documenting functions is like leaving breadcrumbs for others to follow. Add comments to explain what your functions do, the parameters they take, and what they return. It’s like a user manual for
your code! 📚
Alright folks, that’s a wrap on Python functions! Remember, functions are the building blocks of code – the superheroes that save the day by simplifying complex tasks. So, go ahead, define, call, and
utilize functions like a pro in Python! 💻
In closing, thank you for tuning in to this tech talk. Keep coding, stay curious, and always remember: when life gives you errors, debug them with a smile! 😄👩💻
Program Code – Python and Functions: Defining, Calling, and Utilizing Functions
# Program to illustrate the definition, calling, and utility of functions in Python
# Define a function to calculate the factorial of a number
def calculate_factorial(number):
'''Calculate the factorial of a number.
number (int): A positive integer whose factorial is to be calculated.
int: The factorial of the number.
# Base case: factorial of 1 is 1
if number == 1:
return 1
# Recursive case: n! = n * (n-1)!
return number * calculate_factorial(number-1)
# Define a function to check if a number is prime
def is_prime(number):
'''Check if a number is prime.
number (int): The number to be checked.
bool: True if the number is prime, False otherwise.
if number <= 1:
return False
for i in range(2, int(number**0.5) + 1):
if number % i == 0:
return False
return True
# Main function to utilize the above-defined functions
def main():
# Example usage of calculate_factorial function
num = 5
print(f'The factorial of {num} is {calculate_factorial(num)}.')
# Example usage of is_prime function
prime_test = 11
if is_prime(prime_test):
print(f'{prime_test} is a prime number.')
print(f'{prime_test} is not a prime number.')
# Call the main function
if __name__ == '__main__':
Code Output:
The factorial of 5 is 120.
11 is a prime number.
Code Explanation:
This Python program is a beautiful showcase of defining, calling, and utilizing functions to perform specific tasks: calculating the factorial of a number and checking if a number is prime. It kicks
off with defining two functions, calculate_factorial and is_prime, each designed for a unique purpose.
calculate_factorial is a recursive function. It bases its calculation on the mathematical principle that the factorial of a number n is n multiplied by the factorial of n-1, with the foundation laid
down that the factorial of 1 is 1. This recursion dives deeper into smaller sub-problems until it hits the base case, unwrapping the multiplication layers as it returns back up the call stack.
Then, there’s is_prime. This function takes a route of iterating through numbers from 2 up to the square root of the given number (which is a neat little optimization, as any factor of the number
beyond its square root would have a corresponding factor below it). If the given number can be evenly divided by any of these numbers, it’s not prime. Otherwise, voila, it’s a prime number.
The main playmaker is the main function, utilizing the two functional giants defined earlier. It calculates the factorial of 5 and checks if 11 is a prime number. Finally, upon running this script,
the carefully placed if __name__ == '__main__': ensures main is called, bringing our functional parade to the streets of execution, leading to our expected output.
This gorgeous tangle of logic not only demonstrates the power and utility of functions in Python but also primes the reader on recursion, boolean logic, and the beauty of mathematical computations
wrapped in Python’s simplicity. Not just a code, but a gateway to computational thinking!
Frequently Asked Questions (F&Q) on Python and Functions
1. What is a function in Python and how is it defined?
2. How do you call a function in Python?
3. Can a function in Python return multiple values?
4. What are lambda functions in Python and how are they different from regular functions?
5. How can we pass a function as an argument to another function in Python?
6. What is the difference between parameters and arguments in Python functions?
7. How do you define default values for function parameters in Python?
8. What is the purpose of the return statement in Python functions?
9. Can a function in Python modify a global variable outside its scope?
10. How do you create recursive functions in Python?
11. What is the difference between global and nonlocal variables in Python function scopes?
12. How can you document a function in Python using docstrings?
13. What are function annotations in Python and how are they used?
14. Is it possible to create anonymous functions in Python? If so, how?
15. How do you handle exceptions inside a function in Python?
Feel free to explore these questions further to deepen your understanding of Python functions and their usage! 😊
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Micrometers (µm) to Foot (ft), Foot (ft) to Micrometers (µm) Converter / Calculator
Convert micrometers to feet and vice versa with our precision tool in examples.com to easily switch between measurements and understand scale with accuracy.
µm to ft
Formula: Length in feet (ft) = Length in micrometers (µm) ÷ 304,800
Micrometers Foot
1 3.2808E-6
ft to µm
Formula: Length in micrometers (µm) = Length in feet (ft) × 304,800
Length Converters to Micrometer (µm)
Length Converters to Feet (ft)
Conversion Factors:
• Micrometers to Feet: 1 foot = 304,800 micrometers
• Feet to Micrometers: 1 micrometer = 1/304,800 feet
How to Convert Micrometers to Feet:
To convert micrometer to feet, divide the number of micrometers by 304,800.
Example: Convert 2,000,000 micrometers to feet.
Feet=2,000,000/304,800≈6.5617 feet
How to Convert Feet to Micrometers:
To convert feet to micrometers, multiply the number of feet by 304,800.
Example: Convert 5 feet to micrometers.
Micrometers=5×304,800=1,524,000 micrometers.
Micrometers to Feet Conversion Table
1 µm 3.2808e-6 ft
2 µm 6.5617e-6 ft
3 µm 9.8425e-6 ft
4 µm 1.3123e-5 ft
5 µm 1.6404e-5 ft
6 µm 1.9685e-5 ft
7 µm 2.2966e-5 ft
8 µm 2.6246e-5 ft
9 µm 2.9527e-5 ft
10 µm 3.2808e-5 ft
20 µm 6.5617e-5 ft
30 µm 9.8425e-5 ft
40 µm 1.3123e-4 ft
50 µm 1.6404e-4 ft
60 µm 1.9685e-4 ft
70 µm 2.2966e-4 ft
80 µm 2.2966e-4 ft
90 µm 2.9527e-4 ft
100 µm 3.2808e-4 ft
µm to ft Conversion Chart
Feet to Micrometers Conversion Table
Feet (ft) Micrometers (µm)
1 ft 304,800 µm
2 ft 609,600 µm
3 ft 914,400 µm
4 ft 1,219,200 µm
5 ft 1,524,000 µm
6 ft 1,828,800 µm
7 ft 2,133,600 µm
8 ft 2,438,400 µm
9 ft 2,743,200 µm
10 ft 3,048,000 µm
20 ft 6,096,000 µm
30 ft 9,144,000 µm
40 ft 12,192,000 µm
50 ft 15,240,000 µm
60 ft 18,288,000 µm
70 ft 21,336,000 µm
80 ft 24,384,000 µm
90 ft 27,432,000 µm
100 ft 30,480,000 µm
ft to µm Conversion Chart
Difference Between Micrometers to Feet
Aspect Micrometer (µm) Feet (ft)
A micrometer is one-millionth of a meter (10^-6 m). It’s used to measure very small A foot is a unit of length, primarily used in the United States and the United Kingdom, equal to 12
Definition distances, such as the thickness of a cell wall or the diameter of microelectronic inches or about 0.3048 meters. It is commonly used in measuring height, length, and distance in various
devices. applications.
Conversion 1 micrometer = 0.00000328084 feet. 1 foot = 304,800 micrometers.
Usage Typically used in scientific and industrial contexts for microscopic measurements. Widely used in construction, real estate, and aviation for describing distances and measurements.
Precision Provides precision at a microscopic level, suitable for measurements in fields like Generally used for everyday measurements where high precision is not as critical, suitable for general
electronics, materials science, and engineering. length and distance measurements.
Symbol µm ft
Scale Much smaller scale compared to a foot. Measures entities at a cellular or Larger scale, commonly used for everyday applications and measuring human-scale objects and distances.
microscopic level.
Field of Used in fields requiring high precision at micro scales, such as semiconductor Used in everyday life, especially in countries using the imperial system, for a variety of practical
Application manufacturing, microscopy, and material science. and commercial purposes including construction, height measurement, and navigation.
1. Solved Examples on Converting Micrometers to Feet
Example 1: Convert 500,000 micrometers to feet.
500,000/304,800≈1.64042 feet
Result: 500,000 µm = 1.64042 ft
Example 2: Convert 1,000,000 micrometers to feet.
1,000,000/304,800≈3.28084 feet
Result: 1,000,000 µm = 3.28084 ft
Example 3: Convert 250,000 micrometers to feet.
250,000/304,800≈0.82021 feet
Result: 250,000 µm = 0.82021 ft
Example 4: Convert 100,000 micrometers to feet.
100,000/304,800≈0.32808 feet
Result: 100,000 µm = 0.32808 ft
Example 5: Convert 2,000,000 micrometers to feet.
2,000,000/304,800≈6.56168 feet
Result: 2,000,000 µm = 6.56168 ft
2. Solved Examples on Converting Feet to Micrometers
Example 1: Convert 1 foot to micrometers.
Result: 1 ft = 304,800 µm
Example 2: Convert 0.5 feet to micrometers.
0.5×304,800=152,400 µm
Result: 0.5 ft = 152,400 µm
Example 3: Problem: Convert 3 feet to micrometers.
Result: 3 ft = 914,400 µm
Example 4: Convert 2.5 feet to micrometers.
Result: 2.5 ft = 762,000 µm
Example 5: Convert 10 feet to micrometers.
10×304,800=3,048,000 µm
Result: 10 ft = 3,048,000 µm
What is the smallest measurement in feet you can get from micrometers?
You can convert any small number of micrometer into feet, but the result will be a small decimal if the number of micrometers is less than 304,800.
Are micrometers smaller than feet?
Yes, micrometers are much smaller than feet. One foot is equal to 304,800 micrometers.
What are some common uses for micrometers in everyday life?
Micrometers are not commonly used in everyday life but are essential in mechanical engineering, machining, and microscopy for precise measurements.
What are some common uses for feet in everyday life?
Feet are commonly used in the United States for measuring height, distances in golf, and property or room sizes.
What might cause errors in converting between micrometers and feet?
Common sources of errors include using an incorrect conversion factor, arithmetic mistakes, or rounding errors in critical calculations.
Can micrometer measure depth?
Yes, a micrometer can measure depth using specific types called depth micrometers. These are precision tools designed to measure the depth of slots, holes, or recesses with high accuracy. They
function similarly to standard micrometers but are adapted to engage with and measure vertical dimensions.
Is it 1 foot or feet?
The correct form is “1 foot” when referring to the singular and “feet” for the plural. So, when you are talking about a singular measurement, you would say “1 foot,” but for any number greater than
one, you would use “feet,” such as “2 feet.”
Does micrometer measure circumference?
A micrometer can measure the circumference indirectly by measuring the diameter of a cylindrical or round object first and then calculating the circumference using the formula for C=π×diameter.
However, it does not directly measure circumference like a measuring tape.
Is 5 foot correct or 5 feet?
The correct phrase is “5 feet” when referring to the plural form. The term “foot” is used only for the singular form, such as “1 foot.” Thus, when discussing anything more than one, such as five, you
should always say “5 feet.”
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Magnetospheric Multiscale measurements of turbulent electric fields in earth's magnetosheath: How do plasma conditions influence the balance of terms in generalized Ohm's law?
Turbulence is ubiquitous within space plasmas, where it is associated with numerous nonlinear interactions. Magnetospheric Multiscale (MMS) provides the unique opportunity to decompose the electric
field (E) dynamics into contributions from different linear and nonlinear processes via direct measurements of the terms in generalized Ohm's law. Using high-resolution multipoint measurements, we
compute the magnetohydrodynamic ( $E MHD$), Hall ( $E Hall$), electron pressure ( $E P e$), and electron inertia ( $E inertia$) terms for 60 turbulent magnetosheath intervals, to uncover the
varying contributions to the dynamics as a function of scale for different plasma conditions. We identify key spectral characteristics of the Ohm's law terms: the Hall scale, $k Hall$, where $E
Hall$ becomes dominant over $E MHD$; the relative amplitude of $E P e$ to $E Hall$, which is constant in the sub-ion range; and the relative scaling of the nonlinear and linear components of $E
MHD$ and of $E Hall$, which are independent of scale. We find expressions for the characteristics as a function of plasma conditions. The underlying relationship between turbulent fluctuation
amplitudes and ambient plasma conditions is discussed. Depending on the interval, we observe that $E MHD$ and $E Hall$ can be dominated by either nonlinear or linear dynamics. We find that $E P e$ is
dominated by its linear contributions, with a tendency for electron temperature fluctuations to dominate at small scales. The findings are not consistent with existing linear kinetic Alfvén wave
theory for isothermal fluctuations. Our work shows how contributions to turbulent dynamics change in different plasma conditions, which may provide insight into other turbulent plasma environments.
Turbulence is a complex fluid flow phenomenon, which is found in nonconducting fluids and plasma systems throughout the universe. Turbulence arises due to nonlinear interactions, which lead to a
cross-scale energy cascade, facilitating the transfer of energy from large scales to small scales. The applications of turbulence to space plasma systems span many different scales throughout the
universe, including the geospace environment [e.g., Refs. 1–3], planetary magnetospheres [e.g., Refs. 4 and 5], the solar wind [e.g., Refs. 6–10], interstellar medium [e.g., Refs. 11 and 12], and
accretion disks [e.g., Ref. 13]. Turbulence in a magnetized plasma is an especially complicated process due to couplings between fluid flow and electromagnetic forces, from which arise several
different characteristic spatial and temporal scales, numerous plasma wave modes, and anisotropic fluctuations driven by a nonzero mean magnetic field. Understanding the full turbulent energy
cascade—from macrophysical injection to microphysical dissipation—is of importance to understanding how such turbulent systems evolve.
Earth's magnetosheath, which consists of a turbulent region of shocked solar wind plasma draped over the dayside magnetosphere, is an important natural laboratory for turbulence studies [e.g., Refs.
1–3 and 14–16]. Turbulent fluctuations in the magnetosheath are partly associated with solar wind turbulence that has been processed by the shock and partly associated with new fluctuations driven by
shock dynamics.^17–19 Conditions within the magnetosheath are beneficial for in situ measurements, which can build up velocity distributions from many particles as a result of the high density
compared to the solar wind. Rapid measurements of the plasma moments mean that small-scale structures can be investigated. At small scales, the single-fluid magnetohydrodynamic (MHD) approximation
breaks down, leading to fundamental changes in the behavior of the plasma. The latest generation of spacecraft missions can directly probe turbulence at such high resolutions that kinetic-scale
structures can be measured directly.
Turbulence generates small-scale structures that are thought to be linked to dissipation. In the absence of collisional viscosity, the turbulent energy cascade in the collisionless space plasma
reaches kinetic scales, where there are numerous candidates for dissipation mechanisms [e.g., Refs. 20–28]. Such mechanisms include: resonant interactions, such as Landau damping;^29 trapping of
particles and acceleration by nonlinear electric fields;^30,31 stochastic heating of particles in the turbulent electromagnetic field;^22 and processes occurring at current sheets, such as magnetic
reconnection.^32 Turbulence is the primary mechanism by which energy from the largest fluctuation scales can be transferred to kinetic scales, at which it is dissipated into fluid flow and particle
energies via various energy conversion channels [e.g., Refs. 25–28]. However, the nature of the exact mechanisms responsible for energy dissipation in collisionless plasmas is still an open question
[see Ref. 33].
Electric fields (
) play an important role in shaping the dynamics of the magnetic fields (
) within a plasma. Magnetic fields are unable to do work directly on the particles, so many pathways to dissipation, as well as processes for exchanging energy between the bulk flow and the magnetic
field, involve a coupling of the electric field to the particles through the current density (
) via a non-zero
$j · E$
In a plasma, a number of different physical effects contribute to generating
, which are encapsulated in terms of generalized Ohm's law [e.g., Ref.
$E = η j ︸ E η − u × B ︸ E MHD + 1 n e j × B ︸ E Hall − 1 n e ∇ · P e ︸ E P e + m e n e 2 [ ∇ · ( uj + ju − jj n e ) + ∂ j ∂ t ] ︸ E inertia + ∑ ℓ = 1 ∞ ( − m e m i ) ℓ M ℓ ︸ E δ m e ,$
is the resistivity;
$u = ( m e v e + m i v i ) / ( m e + m i )$
is the single-fluid velocity (where
$v i$
$v e$
are the ion and electron fluid velocities, respectively);
$n = n i = n e$
is the number density (invoking quasineutrality);
are the mass of the ions and electrons, respectively;
is the elementary charge;
$P e$
is the electron pressure tensor;
, and
represent the outer product of vectors; and
$M ℓ$
is the
$ℓ th$
-order finite-
correction resulting from the Taylor expansion of
$m e / m i$
[e.g., Ref.
$M ℓ = 2 n e j × B − 1 n e ∇ · P + m e n e 2 [ ∇ · ( uj + ju − ( 1 + 2 ℓ ) jj n e ) + ∂ j ∂ t ] ,$
$P = P e + P i$
is the total pressure tensor.
The resistive electric field ( $E η$) is expected to be small because of the infrequency of interspecies collisions in plasma environments, such as the magnetosheath. The MHD term ( $E MHD$) is
associated with the magnetic field being frozen into the plasma flow u at large scales, the Hall term ( $E Hall$) arises due to differential motion between ions and electrons, the electron pressure
term ( $E P e$) is a non-ideal electric field that arises from the thermal motions of the particles, and the inertia term ( $E inertia$) comes from the finite inertia of the electrons, preventing
them from instantly responding to driving by E. The finite-m[e] correction term $E δ m e$, which comes from the Taylor expansion for small electron mass when writing the equations in terms of
single-fluid variables, is observed to be negligible,^36 and as such is neglected. The terms each make contributions to the turbulence with differing relative importance such that the scale-dependent
relationship between terms influences the overall behavior of E. It is expected that $E MHD$ dominates large scales, with $E Hall$ and $E P e$ dominating below the ion scales. Since the ion fluid
decouples at the ion scales, in principle $E inertia$ may become dominant below d[e], as the electron fluid will also have decoupled from the plasma flow at this scale.
In the context of a turbulent plasma, E exhibits a spectral slope similar to B and u at MHD scales, which then becomes shallower below the ion scales.^2,15,37–39 The enhancement in E fluctuations at
small scales arises due to the growing importance of $E Hall$ and $E P e$ compared to $E MHD$,^15 which has been observed in simulations [e.g., Refs. 40–44] and spacecraft measurements.^36 The
shallower E spectrum is reminiscent of electrostatic fluctuations, even in the presence of weak magnetic fluctuations.^7 This spectral behavior has been associated with kinetic Alfvén waves (KAW)
[e.g., Refs. 2, 21, 45, and 46] although it has been demonstrated that KAW modes are not crucial to reproduce the electric field enhancement [e.g., Refs. 43 and 47]. Changes in the spectral behavior
might also arise due to the potential onset of dissipative processes.
Stawarz et al.^36 examined the relative contributions from different Ohm's law terms for three turbulent magnetosheath intervals, characterizing the behavior of the spectra for each term. Within
these intervals, fluctuations in $E MHD$ dominated the spectrum at large scales, transitioning to $E Hall$ around the ion scales. The relative contribution of $E P e$ compared to $E Hall$ was
approximately constant across scales, and the spatial part of $E inertia$ was small across the range of scales currently accessible by spacecraft measurements. Stawarz et al.^36 also investigated the
ratio of nonlinear to linear components of $E MHD$ and $E Hall$, discovering that the former becomes more nonlinear at small scales due to increasing alignment between B and u fluctuations as a
result of the Hall effect modifying the MHD magnetic field and that the latter remains constant across all scales.
In this work, we aim to examine how the characteristics of the electric field spectrum change when the ambient plasma conditions are varied, by analyzing a survey of turbulent magnetosheath intervals
observed by the Magnetospheric Multiscale mission (MMS). This paper is organized as follows: Sec. II outlines how we use spacecraft data to compute the terms in Ohm's law. In Sec. III, we develop an
analytical framework for studying Ohm's law as a function of plasma conditions. Section IVA examines the variability in the spatial scale at which the dominant $E MHD$ fluctuations give way to $E
Hall$. Section IVB examines how plasma conditions impact the relative amplitude of $E P e$ compared to $E Hall$. Section IVC investigates the ratio of nonlinear and linear contributions for $E
MHD$ and $E Hall$. Section IVD investigates the relative scaling of nonlinear and linear components of $E P e$. Section V examines the underlying dependences on ambient plasma conditions.
II. DATA
With the launch of MMS^48 in 2015, it became possible to directly measure generalized Ohm's law down to sub-ion scales. The earliest works to do this were primarily focused on structures associated
with magnetic reconnection, with studies taking place on laboratory plasmas and data from the Cluster mission.^49–57 MMS enables the evaluation of Ohm's law via high instrumental resolution, combined
with tetrahedral multipoint measurements, which allow the calculation of spatial gradients to first order using the finite-difference “barycentric estimator” technique.^57,58 Due to downlink
limitations, the highest-resolution “burst” mode measurements from MMS are only available for certain time intervals. Field measurements are provided by the FIELDS suite,^59 with magnetic field data
from the fluxgate magnetometer (FGM)^60 providing B measurements at a rate of 128 vectors/s, and electric field data from the electric field double probes (EDP),^61,62 which measures the observed
electric field $E obs$ at 8192 vectors/s. Velocity distributions and corresponding particle moments are obtained from fast plasma investigation (FPI) suite.^63 We use the electron moments $T e , P e
, v e$, and n[e], which have a 30ms cadence, and ion moments $T i , v i$, which have a 150ms cadence. We use spintone data products to de-spin $v e$ and $v i$, as well as counting statistics to
generate particle moment uncertainty estimations,^64 which we use to flag intervals that are reaching the particle noise floor.
We examine the database of magneotsheath turbulence intervals compiled by Stawarz et al.^32 The database contains 60 intervals of continuous burst mode observations in the magnetosheath that are at
least several minutes in length and have no large-scale inhomogeneities, and in which Taylor's hypothesis^65 is valid. (The bulk flow speed $u 0$ is sufficiently fast to neglect dynamic changes in
the plasma.) We utilize these intervals to study the relative importance of the terms in generalized Ohm's law for a range of steady plasma conditions using power spectra in wavenumber space. The
intervals fill a wide range of parameter space, covering around 2.5 orders of magnitude in β[i], 2.7 in β[e], 1.3 in $δ B rms / B 0$, and 1.5 in v[A]. The intervals feature strong variability in
plasma variables, such as B, u, and E, which is characteristic of the turbulent magnetosheath. Contributions to the dynamics come from an ensemble of structures at different scale-sizes, as can be
seen for an example interval in Figs. 1(a)–1(c). To examine how the fluctuations behave as a function of scale, we take the power spectrum, which is an important measure for turbulence that breaks
down the intensity of fluctuations corresponding to different wavenumbers. In turbulence theory, the slope of power spectra in different scale ranges is linked to the energy cascade rate, which
itself is linked to the physical processes driving the turbulent decay of fluctuations. We obtain power spectra using Welch's average periodogram.^66 To estimate a wavenumber spectrum, we invoke
Taylor's hypothesis^65 to express the frequency spectra obtained from time series data in terms of wavenumber using the relation $k = 2 π f / u 0$, where u[0] is the average bulk speed of the
plasma, obtained for each interval by taking the magnitude of the mean ion bulk velocity within that interval. Hereafter, we denote the trace wavenumber power spectrum of an arbitrary time-evolving
quantity q obtained by this method as $PSD [ q ]$.
Throughout this study, the characteristic scales associated with species s are the gyroradius, $ρ s = 2 m s k B T s , 0 / e B 0 2$, and the inertial length, $d s = m s / μ 0 e 2 n 0$, where k[B] is
the Boltzmann constant, $T s , 0$ is the scalar temperature for species s, $B 0 = | B 0 |$ is the guide field strength, μ[0] is the vacuum permeability, and $q 0 = 〈 q 〉$ denotes the average value
of a generic quantity q. At this stage, it is helpful to define two additional characteristic scales: the magnetic correlation length (the largest scale of turbulent fluctuations of the magnetic
field in a system) λ[c], for which we use the values computed by Stawarz et al.,^32 and the average spacecraft separation $R s c = 〈 | R i − R j | 〉$ for position R and spacecraft index $i ≠ j ∈ {
1 , 2 , 3 , 4 }$.
For each measurement interval, we compute $E MHD , E Hall , E P e$, and $E inertia$ following the method outlined within Stawarz et al.^36 To remove the E component coming from the advection of
magnetic structure in the background flow, we subtract $u 0 × B$ from $E obs$ (defining the result as the electric field in the plasma frame, $E p l = E obs + u 0 × B$) and $E MHD$. Figure 1(d)
shows power spectra typical of $B , u , E$, and Fig. 1(e) shows power spectra of the generalized Ohm's law terms for the example interval in Figs. 1(a)–1(c). $E MHD$ and $E Hall$ can be measured
independently by all four MMS spacecraft, whereas $E P e$ and $E inertia$ contain gradient terms computed using the barycentric estimator method, which is only evaluable at the barycenter of the MMS
tetrahedron. To obtain a single set of variables that represent measurements from all four spacecraft, averaging is required. Variables that are computed individually and then averaged after taking
the power spectrum are referred to as “single spacecraft” (1SC) variables, whereas those which are averaged over all four spacecraft before computing derived quantities are known as “barycenter”
variables, denoted by $bary$. To compute $E MHD = − u × B$, the single-fluid velocity is calculated from the data as $u = ( m e v e + m i v i ) / ( m e + m i )$. At observable scales, u is
dominated by $v i$, so we downsample $v e$ and B onto the ion sampling timescale. Barycenter averaging takes place after computing the power spectrum such that $PSD [ E MHD ] = 〈 PSD [ E MHD , 1 S
C ] 〉 bary$. For $E Hall = 1 n e j × B$, the current density is computed as $j = n e ( v i − v e )$, where $v i$ is interpolated onto the electron timescale, since j is dominated by $v e$ at
small scales. In addition, we verify the current using the curlometer method,^67 $μ 0 j = ∇ × B$. When computing $E inertia$, we cannot compute the $∂ j / ∂ t$ term, since any temporal variations
have been equated to spatial ones as per Taylor's hypothesis.
In this study, we investigate the relative behavior of $E MHD , E Hall$, and $E P e$ by comparing their spectral behavior. When comparing $E Hall$ to $E MHD$, the former term is averaged onto the
measurement cadence of the latter (the ion timescale), and then the average of the 1SC power spectra is taken such that $( E Hall / E MHD ) 2 = 〈 PSD [ E Hall , 1 S C ] / PSD [ E MHD , 1 S C ] 〉
bary$. Conversely, when comparing $E P e$ to $E Hall , E Hall$ stays on its original sampling cadence and the power spectrum of the barycenter average is used, such that $( E P e / E Hall ) 2 = PSD
[ 〈 E Hall , 1 S C 〉 bary ] / PSD [ E P e ]$. Consequently, $E P e / E Hall$ cannot be evaluated at length scales smaller than R[sc].
Equation (1) couples the evolution of E to B and the particle moments. Each term describes the contribution to the electric field from a different physical effect, so the interplay between terms
determines the overall behavior of the total electric field. When considering a turbulent environment, it is useful to split quantities into a mean and fluctuating component, such that $q = q 0 + δ
q$, where $q 0 = 〈 q 〉$ is the average value and $δ q = q − 〈 q 〉$ is the deviation from the mean.
$E MHD , E Hall$
, and
$E P e$
, which are typically the three main components of generalized Ohm's law at observable scales, it is possible to identify quantifiable features of the spectra, which encode the relationship between
different terms. The MHD and Hall dynamics can be characterized by the length scale at which the Hall term becomes dominant over the MHD term, hereafter referred to as the Hall scale (
$k Hall$
). By taking the ratio
$E Hall / E MHD$
and neglecting angles between vectors, it is possible to obtain an expression for the scale-dependent relationship between these two terms, given by
$E Hall E MHD = | 1 n e j × B | | − δ u × B | ∼ k d i δ b A δ u ,$
is a wavenumber coming from
$j = ∇ × B / μ 0$
and the magnetic fluctuation in Alfvén units
$δ b A = δ B / μ 0 m i n$
$n = n 0 + δ n$
is the total density). Defining
$k Hall$
as the value of
at which
$E MHD / E Hall = 1$
$k Hall = 1 d i δ u δ b A .$
Similarly, the electron pressure and Hall dynamics, which typically follow similar power-laws, except with higher spectral power in
$E Hall$
, can be quantified by the ratio of fluctuation power between the two terms
$E P e E Hall = | 1 n e ∇ · P e | | 1 n e j × B | ∼ β e 2 δ P e / P e , 0 δ B / B 0 B 0 〈 | B | 〉 ,$
where the electron plasma beta
$β e = 2 μ 0 n 0 k B T e , 0 / B 0 2$
, and
$〈 | B | 〉 = 〈 | B 0 | 2 + 2 δ B · B 0 + | δ B | 2 〉$
is the average magnitude of the magnetic field, which becomes
in the case of small fluctuation amplitudes.
Each of these terms also contains a combination of linear and nonlinear contributions. Since nonlinear variables are associated with scale-transfer terms related to turbulence, whereas linear
components in isolation produce wave-like behavior, measuring how nonlinear a term is at a given scale is a way of understanding which processes are driving the dynamics of that term. Turbulence
theories often involve assumptions about the nature of the nonlinear fluctuations, typically related to the rate of nonlinear interactions, τ[nl], as compared to linear wave periods, τ[l]. The
Iroshnikov–Kraichnan model of turbulence^68,69 is a dimensional analysis description of counterpropagating Alfvén waves for weak turbulence, where the nonlinear timescales are sufficiently slow that
it takes many interactions to significantly alter the dynamics ( $τ n l ≪ τ l$). Critical balance^70,71 takes into account scale-dependent anisotropy, requiring that $τ n l ∼ τ l$ at each scale, in
a statistical sense. Galtier et al.^72 derived an exact theory of incompressible MHD turbulence where the nonlinear interactions are negligible. The theory of dynamic alignment^73,74 accounts for the
reduction of the nonlinear interaction at small scales due to the increasing alignment between velocity and magnetic fluctuations. More recent theories, for example, refined critical balance,^75 also
have a dependence on the relative importance of nonlinear and linear interactions.
We split each Ohm's law term into contributions consisting of background (non-fluctuating), linear (one fluctuating variable) and nonlinear (multiple fluctuating variables) elements. Estimating the
ratio of nonlinear components to linear components for the MHD and Hall terms gives
$| − δ u × δ B | | − δ u × B 0 | ∼ δ B B 0 ,$
$| δ j × δ B | / n e | δ j × B 0 | / n e ∼ δ B B 0 .$
In the case of
$E P e$
, two linear terms are present: one associated with density fluctuations
; and the other associated with temperature fluctuations
$δ T e$
. The electric fields associated with
$δ T e$
, the “combined” linear term, and the nonlinear contribution
$δ n δ T e$
are given, respectively, by
$E δ n = − T e , 0 ∇ · δ n / n e ,$
$E δ T e = − n 0 ∇ · δ T e / n e ,$
$E δ T e + δ n = − ∇ · ( n 0 δ T e + δ n T e , 0 ) / n e ,$
$E δ n δ T e = − ∇ · ( δ n δ T e ) / n e .$
Estimating the ratio of nonlinear terms to individual linear components, we obtain the expressions
$| E δ n δ T e | | E δ T e | ∼ δ n n 0 ,$
$| E δ n δ T e | | E δ n | ∼ δ T e T e , 0 ,$
$| E δ n δ T e | | E δ T e + δ n | ∼ δ n δ T e n 0 δ T e + δ n T e , 0 ,$
where we have assumed that the divergence of the nonlinear component is the same as that of the linear components. From this point onward, we denote the ratio of nonlinear and linear components of an
electric field term as
$R N L / L = E nonlinear / E linear$
There are a number of approximations which go into the dimensional analysis expressions used in this section. Scale dependence of the alignment between different vectors is ignored, in addition to
geometric differences in the scale dependence arising from divergences and curls. We also assume that fluctuating quantities can be characterized by a single number quantity. One could envisage
various ways to characterize a scale-dependent fluctuation: the root mean square $δ q rms = 〈 ( δ q ) 2 〉$ is equivalent to the integral of the power spectrum over all wavenumbers (for a turbulent
environment, this quantity is typically weighted toward the large scales for certain quantities); or the integral across some range $Δ k = [ k − δ k , k + δ k ]$, which gives the power associated
with the wavenumber band $Δ k$. Where we encounter the quotient of two fluctuating quantities, $δ q 1 ( k ) / δ q 2 ( k )$—especially if the ratio is not decaying with k—this quantity can
equivalently be integrated or averaged over some $Δ k$. For brevity, we denote $δ q 1 ( k ) / δ q 2 ( k )$ as $( δ q 1 / δ q 2 ) k$ from this point onward.
The dimensional analysis expressions appeal to characteristic length scales and fluctuation amplitudes, whereas we are considering how these expressions manifest themselves over the ensemble of
structures that make up a turbulent plasma. We cannot a priori use the dimensional expressions contained within Eqs. (3)–(14) because there are multiple ways that one might reasonably use to quantify
the characteristic values for any given interval. In this section, we directly compute the terms in generalized Ohm's law for the 60 MMS intervals of magnetosheath data discussed in Sec. II. Using
the dimensional analysis discussed in Sec. III as a tool to contextualize the observations, we examine how the relative contributions of the different terms vary across the intervals and what that
implies about the nature of the turbulent dynamics.
A. The Hall scale
The wavenumber $k Hall$ corresponds to the largest wavenumber at which the turbulent electric field is dominated by fluctuations in the MHD term. Figure 2(a) shows the ratio of $E Hall$ to $E MHD$ as
a function of scale for an example interval, with $k Hall$ defined as the wavenumber representing the intersection of this curve with $E Hall / E MHD = 1$. The dimensional analysis expression in Eq.
(4) expresses this scale as a function of d[i] and $δ u / δ b A$. At least for this example interval, $k Hall$ falls between $k d i = 1$ and $k ρ i = 1$, suggesting that $δ u / δ b A$ is playing a
nontrivial role in setting the location of $k Hall$ in this interval. Figure 2(b) shows the scale dependence of $δ u / δ b A$ as quantified by the square root of the ratio of the power spectral
densities. For the example interval, the ratio is roughly constant at large scales from approximately the magnetic correlation length to the ion scales, with a slight excess of magnetic fluctuation
energy. Such an excess may be consistent with non-Alfvénic fluctuations at large scales in the magnetosheath; however, even in the context of incompressible MHD turbulence and in the solar wind, an
excess of magnetic energy is an observed feature.^41,76–81 At smaller scales, starting around $k ρ i = 1$ and $k Hall$, the magnetic energy becomes increasingly dominant, consistent with the
behavior of kinetic scale dynamics and the reduced importance of $u$ at sub-ion scales. The excess of magnetic energy across the spectrum is qualitatively consistent with the fact that $k Hall < d i
− 1$ in this interval.
In Fig. 2(c), we examine the location of $k Hall$ for all 60 of the intervals examined in this study. Comparing the measured values of $k Hall$ to the dimensional analysis expression from Eq. (4), we
find that the Alfvénic assumption ( $δ u / δ b A = 1$) and using $δ u rms / δ b A , rms$, which is weighted toward the large-scale fluctuations, systematically overestimate the value of $k Hall$.
We further examine $δ u / δ b A$ evaluated at a variety of relevant scales based on the square root of the ratio of power spectral densities, including at $k d i = 1$ and $k ρ i = 1$, as well as
evaluating $k d i δ u / δ b A = 1$ as a scale-dependent expression, defining $k dim = 1 / d i × ( δ u / δ b A ) k dim$. Evaluating $δ u / δ b A$ at $k d i = 1$ appears to estimate the measured value
of $k Hall$ reasonably well, as shown in Fig. 2(c). This agreement suggests that scales adjacent to d[i] are important in determining where the transition to Hall term dominance of the electric field
occurs, which aligns with expectations as Hall dynamics arise from a decoupling of the ion fluid from the plasma dynamics. A similar trend is obtained for $k dim$, albeit with a slightly larger
Further refinements are possible, such as by breaking up the fluctuations $δ u / δ b A$ into components perpendicular and parallel to the mean field $B 0$. We do not find that any combination of
components produces significantly better agreement, indicating that there are no preferred fluctuation directions, which are solely controlling the MHD–Hall dynamics in these turbulent intervals.
Indeed, many of the intervals have large fluctuation amplitudes relative to the background magnetic field and seemingly near isotropic fluctuations.^32 Based on the observed values of $δ u / δ b A$
evaluated at $k d i = 1$, as well as those feeding into $k dim , δ u / δ b A$ is empirically consistent with 0.5 for the intervals examined in this study. This indicates that, for this selection of
magnetosheath plasma data, $k Hall d i ∼ 0.5$ and, thus, the spatial scale associated with the Hall scale is around $2 d i$. Using this empirical result in the dimensional analysis is also fairly
effective in terms of predicting the measured $k Hall$ across all intervals.
The scatter in Fig. 2(c) could be due to several sources, such as the approximations used to obtain the dimensional analysis expression for $k Hall$, inexact measurements of $k Hall$ due to
variability in the curve of the power spectrum, or artificial enhancements in the velocity measurements as they approach the noise floor. One of the principle simplifications of the dimensional
analysis expression is that scale-dependent alignment between different vector quantities is neglected. Investigating alignment properties between δu, $δ b A$, and δb, and subsequently including
characteristic values of these into the dimensional analysis expressions, yielded no noticeable improvements, implying that they are not directly feeding into the location of $k Hall$. We estimate
the noise level of FPI variables using the methods described in Gershman et al.^64 This involves generating a zero-mean timeseries of white noise, where at each time step, the noise level is a
randomly generated value from a normal distribution, which has a standard deviation equal to the FPI error provided by MMS. Where curves are reaching the noise floor, we do not include the range of
scales where the spectral curve appears to be contaminated by noise.
B. Relative amplitude of electron pressure and Hall terms
Below $k Hall$, the spectrum of E is dominated by a combination of $E Hall$ and $E P e$. The ratio $E P e / E Hall$ represents the proportion of the fluctuation energy arising from the non-ideal
term $E P e$ compared to $E Hall , bary$. This ratio is shown as a function of scale in Fig. 3(a) for an example interval, in which it can be seen that the value is close to constant across a wide
range of $k ρ i > 1$. In some cases, there may be a slight increasing trend. Appealing to the dimensional expression in Eq. (5), the scale-dependent part of $E P e / E Hall$ is related to $( δ P e /
P e , 0 ) / ( δ B / B 0 )$, which is plotted as a function of scale in Fig. 3(b) and is also found to be approximately constant as a function of scale in the sub-ion range. This ratio has been
rescaled by the constant factors $β e / 2$ and $B 0 / 〈 | B | 〉$. This is in line with the expectation from Eq. (5) that there should be no overall scale-dependence of the ratio, i.e., that
magnetic and density fluctuations have the same power-law as per theoretical expectations in the nearly incompressible limit.^45,82 Comparable behavior has been found in the solar wind^83 and hybrid
kinetic simulations,^80 where similar power laws have been reported for B and n at sub-ion scales, which is consistent with similar scaling between B and P[e] if the fluctuations are isothermal.
However, the presence of temperature fluctuations may lead to a more subtle relationship between the pressure and magnetic field.
The factor $B 0 / 〈 | B | 〉$, which is accounting for variations in the magnitude of B, is a necessary factor to account for, at least in the magnetosheath. This term encodes how much the
fluctuations perturb the magnitude of the magnetic field, ranging from around $10 − 1$ to 1 in our dataset. Beyond the inclusion of the magnetic field strength fluctuation, another factor to consider
is that there are different ways to quantify the electron pressure fluctuation $δ P e$. For example, one could consider fluctuations in the isotropized pressure, $p e = Tr [ P e ] / 3$. However,
this does not take into account individual or unbalanced fluctuations in the diagonal terms that may be giving rise to electron temperature anisotropy, as such it only accounts for variations in the
overall amount of pressure, as opposed to variations in the amount of anisotropy. We compute the power spectrum of each of the diagonal terms and sum them to give the scale-dependent fluctuation $δ P
e$. In principle, the off diagonal components could also contribute to the dynamics, but we find that including the off diagonal parts of $P e$ into $δ P e$ does not improve agreement, implying that
these do not significantly contribute to $E P e / E Hall$.
To quantify $E P e / E Hall$, we choose the average value of the smoothed measured ratio in log-space between $k ρ i = 1$ and $k R s c = 1$. Figure 3(c) shows the agreement between the measured $E
P e / E Hall$ and the dimensional analysis expression in Eq. (5) (obtained from its spectrum in the same manner as the measured value) across the range of magnetosheath intervals investigated by this
study. The general trend shows good agreement when considering the sub-ion scales. This shows that it is necessary to include $B 0 / 〈 | B | 〉$ into the expression for $E P e / E Hall$ in the
magnetosheath and that one needs to take into account the fluctuations in the diagonal components of the electron pressure tensor, in order to include the anisotropy fluctuations. We performed a
simple analysis of the estimated uncertainties for this plot, which we found to be of the order of $∼ 0.1$ for both axes. This implies that the slight uptick observed in the spectra of $E P e / E
Hall$ and the dimensional expression [Figs. 3(a) and 3(b), respectively]. Additionally, the three leftmost points in the scatterplot seem to be systematically enhanced compared to the dimensional
analysis estimate, which potentially indicates that the computation of $E P e$ is hitting a noise level.
C. Nonlinear/linear contributions to $E MHD$ and $E Hall$
Just as the E dynamics as described by generalized Ohm's law can be split into different terms corresponding to separate physical effects, so too can each term be decomposed into contributions from
nonlinear and linear components. Equations (6)–(14) give expressions for the relationships between the nonlinear and linear components as a function of turbulence conditions and plasma parameters.
Figure 4 shows an example of the ratios of power spectra of nonlinear and linear terms contributing to $E MHD , E Hall$ and $E MHD + E Hall = − v e × B$, where we compute the nonlinear and linear
components of $− v e × B$ separately. Above the ion scales (in our case, between λ[c] and ρ[i]), the spectrum of every ratio is constant as a function of scale and centered around $δ B rms / B 0$,
indicating that there is a balance between the linear and nonlinear terms in the system. However, interestingly, the ratio between the two terms is set by the strength of $δ B rms / B 0$ across all
scales, which is weighted toward the behavior of the largest-scale fluctuations. At smaller scales, $R MHD N L / L$ diverges from $δ B rms / B 0$ to become increasingly nonlinear. This occurs due to
dealignment of $δ B$ and $δ u$ at smaller scales, whereas alignment between $B 0$ and $δ u$ does not vary with scale, leading to a relative strengthening of the nonlinear term.^36,84 The dealignment
may be occurring due to $E Hall$ causing non-MHD modifications to the magnetic field at scales below $k Hall$. We evaluate $R MHD N L / L$ at length scales between λ[c] and ρ[i], in the range where
the ratio of nonlinear to linear components is unaffected by the Hall effect. $R Hall N L / L$, on the other hand, continues to remain constant as a function of scale, in such a way that the
combination of $E MHD$ and $E Hall , − v e × B$, also remains approximately constant as a function of scale from the correlation length down to the smallest observable scales. As such, the dominant
dynamics in generalized Ohm's law at any given scale appear to arrange themselves such that they maintain a balance between the linear and nonlinear contributions.
From Sec. III, the relative scaling of nonlinear to linear MHD and Hall terms both reduce to $δ B / B 0$ within the dimensional analysis treatment. Figure 4(b) shows $R MHD N L / L$ averaged over the
scale range $k λ c = 1$ to $k ρ i = 1$ as compared to $δ B rms / B 0$ for all the intervals, and Fig. 4(c) shows $R Hall N L / L$ averaged over the scale range $k λ c > 1$ compared to $δ B rms / B 0$
for all intervals. In all cases, the ratios appear to be roughly consistent with the behavior shown in the example interval and clearly scale with the ratio $δ B rms / B 0$. A scaling with $δ B / B
0$ is in agreement with Eqs. (6) and (7), where in this case the ratio of large-scale magnetic field fluctuations to average background field, $δ B rms / B 0$, is determining the relative scaling of
nonlinear to linear contributions. Since the constant ratio of linear and nonlinear terms is being set by $δ B rms / B 0$, there are both intervals that are strongly dominated by the nonlinear
dynamics and intervals that are dominated by the linear dynamics even while maintaining a balance between them. The spread in the scatter trends may be associated with how well the estimate of the
mean value of the ratio has converged given the variability in the ratio, which has a greater impact on the MHD term because a smaller range of scales means that fewer data points are used to
calculate the average.
D. Nonlinear and linear terms contributing to $E P e$
$R P e N L / L$ is a more complex quantity to evaluate because $E P e$ consists of two linear terms as well as a nonlinear term. When the same dimensional approach as in Sec. IVC—using the rms as
the characteristic fluctuation amplitude—is naively applied to $R P e N L / L$, the dimensional estimation fails to account for measured values in all 60 intervals. The discrepancy is shown in Fig.
5, in which it can readily be observed that all scatter points are above the agreement line, indicating that the dimensional estimation is underestimating the measured value. It is also interesting
to note that every interval has a measured $R P e N L / L < 1$, showing that the electron pressure divergence term is dominated by its linear components for every magnetosheath interval that we
studied, in contrast to $R MHD N L / L$ and $R Hall N L / L$. This discrepancy extends to the ratio of nonlinear to individual linear terms (not shown) given by Eqs. (12) and (13), implying that the
underestimation is not coming from correlation between the two linear terms in the denominator of $R P e N L / L$. Additionally, when the fluctuation amplitudes are evaluated at characteristic
scales, such as $k ρ i = 1$ and $k d i = 1$, or when they are averaged between $k ρ i = 1$ and $k R s c = 1$, agreement is still not found. As a result, we conclude that there is not a simple
dimensional expression to predict the relative scaling of nonlinear to linear parts of $E P e$ as a function of plasma conditions.
Since we are unable to describe $R P e N L / L$ using simple dimensional estimates, we instead break down the contributions to this term and qualitatively compare the behavior of each. The electric
field terms that contribute to $E P e$ are given by Eqs. (8)–(11), the power spectra of which are shown in Fig. 6(a). We find that the ratio $E δ n δ T e / E δ T e$ and the ratio $E δ n δ T e / E δ T
e + δ n$ are both independent of scale, whereas $E δ n$ becomes smaller below the ion scales, resulting in a larger $E δ n δ T e / E δ n$ at smaller scales. This behavior indicates that the electric
field associated with temperature fluctuations is increasingly dominating the linear component of the electron pressure term with larger wavenumber, implying that temperature fluctuations are playing
a more important role in the $E P e$ dynamics than density fluctuations. This increasing importance of $E δ T e$ relative to $E δ n$ at sub-ion scales is readily apparent in the example spectrum in
Fig. 6(a) and may have important consequences as it suggests violation of purely isothermal dynamics in the sub-ion scales. It could be that the changing relative importance of linear terms precludes
a simple dimensional estimation from being employed on this term. It has been demonstrated that the linear contributions to $E P e$ are dominating the term, so any estimate for the relative scaling
of nonlinear to linear parts may be sensitive to the relative dynamics of the dominant contributions. It is interesting that the dimensional technique does not work on $E δ n δ T e / E δ T e$, even
though the ratio is constant as a function of scale.
Figure 6(b) shows power spectra of the linear and nonlinear components of $P e$ for an example interval, which corresponds to the variables whose divergences are associated with components of the
electron pressure term. The combination of the two linear variables is dominant at all accessible wavenumbers, which is the same for all intervals and in agreement with our observations that $E P e$
is dominated by linear components. The trends are broadly the same as those of the different contributions to $E P e$, as illustrated in Fig. 6(a). The linear term $n 0 δ T e$ becomes progressively
larger relative to $δ n T e , 0$ at sub-ion scales. This suggests that the temperature fluctuations are increasingly more important than the density fluctuations at small scales, which indicates that
our likewise findings for $E δ T e$ and $E δ n$ are not solely arising due to differences between the divergence of each fluctuating variable.
The relationships demonstrated in Sec. IV convey only part of the complex interdependence of the electrodynamics with the turbulence properties and ambient plasma conditions within the magnetosheath.
We have shown that $k Hall ∼ 1 / d i × ( δ u / δ b A )$, which means that the length scale where the Hall fluctuations become dominant, $k Hall$, scales with $1 / d i$ and the ratio of velocity to
magnetic fluctuations. The value of $δ u / δ b A$ is a scale-dependent quantity that is influenced by an excess of magnetic energy at MHD scales, before beginning to drop off as velocity decouples
when the Hall effect takes over. The fact that large scale $δ u / δ b A$ does not control $k Hall$ suggests that the diverging δu and $δ b A$ spectra at sub-ion scales may be playing a role in the
dynamics. It is interesting that $k Hall$ is not best described using $( δ u / δ b A ) k Hall$, as this implies that the relative fluctuation amplitude at scales other than those adjacent to $k
Hall$ is feeding into the location of the change in dynamics. This nonlocality in spectral space could be explored in future research.
Additionally, there may be an underlying dependence of $δ u / δ b A$ itself on the ambient plasma conditions. This means $k Hall$ has a combination of explicit dependence on plasma parameters as well
as underlying dependences coming from $δ u / δ b A$. If the fluctuations were Alfvénic, or if there was no dependence of $δ u / δ b A$ on plasma conditions, one would expect $k Hall ∼ d i − 1$.
However, we observe that $k Hall ∼ d i − 1.2$. We then investigated the underlying dependences of $δ u / δ b A$ on other parameters, finding the relationship $δ u / δ b A ∼ ( δ B rms / B 0 ) − 0.21$
(not shown here). This shows that the fluctuations in velocity and magnetic field become closer to being Alfvénic for intervals with smaller magnetic field fluctuations compared to the background,
which is consistent with small amplitude fluctuations being more wave-like. To summarize, in the case of intervals with smaller background fluctuations, $k Hall$ approaches $1 / d i$. A future
investigation may wish to extend this trend to intervals with very small $δ B rms / B 0$ to see if $k Hall$ becomes smaller than $1 / d i$, as well as check whether δB contributions from other
length scales exhibit stronger control over the behavior.
When checking
$δ B rms / B 0$
against other parameters, we uncovered a scaling of
$β i ∼ ( δ B rms / B 0 ) 1.53$
, which folds into the expression for
$k Hall$
$k Hall ∼ β i − 0.14 × ( 1 / d i ) .$
$β i = ρ i 2 / d i 2$
, we find
$k Hall ∼ ρ i − 0.28 d i − 0.72$
, which introduces a dependence on
that was not originally apparent from the dimensional analysis. It would be interesting to identify intervals with a wider range of
$β i ≤ 1$
to test this relationship further.
The (normalized) spatial Hall scale,
$l Hall / d i ∼ β i 0.14$
, can be compared to an empirical estimate for the
spectral break scale,
$l break d i ∼ 1 3 ( 1 + β i 1 / 2 − β i 1 / 4 / 2 ) .$
In our dataset,
ranges from 0.81 to 243.03. Across this range, the curve of
$l break / d i$
as a function of
is not similar to that of
$l Hall / d i$
. Qualitatively,
$l break / d i$
drops off faster than
$l Hall / d i$
, although
$l Hall / d i$
is smaller than
$l break / d i$
at very small
. However, the extent of the relationship between
$l Hall$
$l break$
has not been fully investigated here and may form the basis of a future study. Furthermore, there may be a more complex relationship between
$k Hall$
and the break scale.
We have shown that the value of
$( E P e / E Hall ) sub − ion$
has a functional dependence on
$β e / 2 , ( ( δ P e / P e , 0 ) / ( δ B / B 0 ) ) sub − ion$
, and
$B 0 / 〈 | B | 〉$
. We find that the latter two quantities have direct dependences on
$β = β e + β i$
, with
$( ( δ P e / P e , 0 ) / ( δ B / B 0 ) ) sub − ion ∼ β − 0.68$
Fig. 7(a)
] and with
$B 0 / 〈 | B | 〉$
featuring two populations with different relationships to
Fig. 7(b)
]. Since we have that
$〈 | B | 〉 B 0 = 〈 1 + 2 δ B ∥ B 0 + ( δ B ∥ B 0 ) 2 + ( δ B ⊥ B 0 ) 2 〉 ≥ 1 ,$
we find that there is a population with relatively small
$δ B rms / B 0$
in which
$B 0 / 〈 | B | 〉$
is independent of
$∼ 1$
. Another population corresponding to large
$δ B rms / B 0$
$B 0 / 〈 | B | 〉 ∼ β − 0.33$
. Combining the dependencies revealed in
Figs. 7(a)
, two power-law scalings governing the dependence of
$E P e / E Hall$
emerge, as shown in
Fig. 7(c)
$( ( δ P e / P e , 0 ) / ( δ B / B 0 ) ) sub − ion × B 0 / 〈 | B | 〉 ∼ β − 0.75$
for small-fluctuation intervals and
$( ( δ P e / P e , 0 ) / ( δ B / B 0 ) ) sub − ion × B 0 / 〈 | B | 〉 ∼ β − 1.07$
for large-fluctuation intervals. Folding these relationships into the original expression for
$( E P e / E Hall ) sub − ion$
$( E P e E Hall ) sub − ion ∼ { β e β − 0.75 , small δ B rms / B 0 β e β − 1.07 , large δ B rms / B 0 .$
When the observed values of $( E P e / E Hall ) sub − ion$ are plotted against this expression (not shown), good agreement is found.
It would be desirable to extend this research to a much wider dynamic range of $E P e / E Hall$ (if this exists) and β, in order to further explore the trends of both populations. In particular,
regions with low fluctuation amplitudes and high beta to confirm whether the $β − 0.75$ extends to large β alongside the $β − 1.07$ trend. Additionally, identifying regions with strong fluctuation
amplitudes and low beta would be useful as it would show whether the $β − 1.07$ trend plateaus for small β, in accordance with the trend of the other population. Such intervals may be unlikely to
occur in the magnetosheath, as when $δ B rms / B 0$ is large this implies small $B 0 2$, which must then be countered by an appropriately small $T i , 0$ to generate small β. However, in our
dataset, we observe that variation in β is typically driven by changes in $B 0 2$, which covers $∼ 2.3$ orders of magnitude, whereas $T i , 0$ only varies over $∼ 1$ order of magnitude. In addition
to extending the trends, our findings indicate that it may be difficult to identify magnetosheath intervals where $E P e ≥ E Hall$. This means that, given $β = β e + β i ≥ β e$ and, in our dataset,
$β i / β e > 1$, neither population is expected to produce $E P e / E Hall > 1$ in the turbulent magnetosheath. Even in the unlikely case that $β ≈ β e$ in the magnetosheath, for $E P e / E Hall >
1$, one would need to identify an interval with $β e ∼ 10 4$ (for small $δ B rms / B 0$) or $β e ≈ 0$ (for large $δ B rms / B 0$).
The functional dependence of
$( δ P e / P e , 0 ) / ( δ B / B 0 )$
noted here can be compared to the expectation from linear KAW theory,
which is often invoked in theoretical descriptions of kinetic scale plasma turbulence.
Imposing the assumption of isothermal
$T e$
typically justifiable in space plasmas [e.g., Ref.
], the relative amplitude of scalar isotropic electron pressure to magnetic fluctuations is given by
$δ p e / n 0 k B T e , 0 δ B / B 0 ∼ δ n / n 0 δ B / B 0 ∼ ( β 2 2 + β 2 ) − 1 / 2 .$
We find that this expression underestimates the measured value for the whole dataset (
Fig. 7
), in agreement with the findings of Stawarz
et al.^36
is a factor of
$∼ 3$
smaller than the observed value of
$( δ P e / P e , 0 ) / ( δ B / B 0 )$
for small
, dropping off faster than
$( δ P e / P e , 0 ) / ( δ B / B 0 )$
with larger
, as shown in
Fig. 7(a)
There may be several factors that lead to the disagreement between Eq.
and the observations. One possibility could be that the dynamics are KAW-like, but the assumptions that go into Eq.
, such as isothermal and isotropic electrons, are violated. The results of Sec.
suggest that the temperature fluctuations are playing a role in the dynamics in the magnetosheath, thus violating the isothermal approximation—an assumption which is commonly made in theoretical
descriptions of plasma turbulence.
Furthermore, electron distributions are observed to have anisotropy in the temperature within the magnetosheath (
$T e , ⊥ / T e , ∥$
varies between 0.69 and 1.03 for our range of intervals), and taking into account the fluctuations associated with this anisotropy was necessary when comparing
$E P e$
$E Hall$
. Another possibility is that a different wave mode may be relevant, for example, whistler waves. Boldyrev
et al.^87
derive an expression for
$( δ n / n 0 ) 2 / ( δ B / B 0 ) 2$
in the case of whistler waves
$( δ n / n 0 ) 2 ( δ B / B 0 ) 2 = 1 2 ω p i 4 k ∥ 4 c 4 ,$
which is expected to be small since the denominator of the expression contains
$k ∥ 4$
, which is typically large. Finally, it may be that the strong nonlinearity observed in many of the intervals is fundamentally altering the dynamics and that fully nonlinear solutions are needed to
account for the observed relationship. In particular, the scaling of
$E P e / E Hall$
$∼ β e / β$
for large
$δ B rms / B 0$
suggests that structures in near pressure balance may be playing a significant role in the dynamics.
The spectral properties of generalized Ohm's law terms are evaluated in a statistical sense by use of a large dataset of MMS measurements in the magnetosheath. Characteristics of the relative
importance of $E MHD , E Hall$, and $E P e$ are identified, namely, the crossover between MHD and Hall dynamics, $k Hall$, the ratio of electron pressure to Hall contributions, $E P e / E Hall$,
and the ratio of nonlinear to linear contributions from each term. We confirm expectations that $k Hall$ typically occurs around the ion scales and that $( E P e / E Hall ) sub − ion$ is
approximately constant as a function of scale. $R MHD N L / L$ is constant above the ion scales, $R Hall N L / L$ is constant across all accessible scales, and these ratios demonstrate that a
scale-by-scale balance between the linear and nonlinear terms is achieved that can be dominated by either the nonlinear or linear terms. Simulations have suggested that the ratio may have a complex
anisotropic distribution in k-space,^89,90 which could be explored further in future studies. $E P e$ is dominated by linear components, typically a mixture of contributions from δn and $δ T e$ at
large scales, than by contributions from $δ T e$ at small scales.
We show how $k Hall , E P e / E Hall , R MHD N L / L$, and $R Hall N L / L$ depend on the properties of the turbulence and ambient plasma conditions. This is in agreement with work by Franci et al.,
^44 which showed that properties of the turbulent cascade are mainly dependent on ambient plasma conditions rather than the type of large-scale driver. $k Hall$ most clearly depends on $1 / d i$ and
$δ u / δ b A$ evaluated at the ion inertial length. At sub-ion scales, $E P e / E Hall$ depends on β[e], $( δ P e / P e , 0 ) / ( δ B / B 0 )$ averaged over the sub-ion scales, with all three of the
anisotropic diagonal components of $P e$ included into $δ P e$, and $B 0 / 〈 | B | 〉$. At scales larger than the ion scales, $R MHD N L / L$, and, across all scales, $R Hall N L / L$ is constant
and goes as $δ B rms / B 0$, indicating that the balance between the mean field and large-scale magnetic fluctuations controls the relative nonlinearity to linearity of these terms.
We find that several of the key properties of the turbulent fluctuations, namely, $δ u / δ b A , δ P e / δ B$, and $B 0 / 〈 | B | 〉$, have underlying dependences on plasma conditions, such that
$k Hall$ and $E P e / E Hall$ are expressible solely as empirical functions of ambient plasma parameters. An evaluation of the Hall scale gives $k Hall ∼ β i − 0.14 × ( 1 / d i )$. For the relative
amplitude of electron pressure to Hall fluctuations, the relationship $( δ P e / P e , 0 ) / ( δ B / B 0 ) ∼ β − 0.68$ is uncovered. Such a scaling is not consistent with the isothermal linear KAW
approximation. Indeed, we find that temperature fluctuations are playing a significant role in generating the electric fields. Additionally, two populations of $B 0 / 〈 | B | 〉$ are identified,
corresponding to small and large $δ B rms / B 0$, respectively. When combined, the two populations feed into $( E P e / E Hall ) sub − ion$, corresponding to $β e / β 0.75$ (small $δ B rms / B 0$)
and $β e / β 1.07$ (large $δ B rms / B 0$), respectively.
The results of this work highlight a number of questions that should be explored in future studies. Exploring how the findings of this study compare to different plasma systems, covering an
extrapolation beyond the extrema of the intervals used here as well as completely different parameter regimes, would give useful insight into the applicability and limitations of our results. Future
works may wish to formally investigate the relationship between the Hall scale and the scale at which the break in the magnetic field spectrum occurs. Another further study may aim to incorporate
temperature fluctuations into the KAW treatment of $( δ P e / P e , 0 ) / ( δ B / B 0 )$. Furthermore, it would be interesting to investigate the extent to which the different terms directly
contribute to energy conversion via $j · E$, especially in the context of the relative amplitude of $E P e$ to $E Hall$, so that the relationship between different terms and dissipation may be
uncovered. It would be interesting to explore whether the results in this paper are linked to systematic variations with distance from the bow shock, or more generally with position within the
magnetosheath, as a result of recently driven turbulence evolving as it develops. Similarly, it would be interesting to examine whether the results in this paper are linked to systematic variations
with upstream solar wind conditions. Finally, simple dimensional analysis expressions did not agree with observations of the relative scaling of linear and nonlinear contributions to $E P e$. It
would be interesting to uncover why dimensional analysis methods break down for this term, and not others, which may give insights into the nature of the interplay between the two linear terms which
jointly dominate the $E P e$ dynamics.
H.C.L. and J.E.S. are supported by the Royal Society University Research Fellowship No. URF\R1\201286. Work at University of California, Berkeley, is supported by NASA Grant Nos. 80NSSC20K0708 and
80NSSC21K1692. This work was supported by the Science and Technology Facilities Council Grant No. ST\W001071\1. The authors thank the entire MMS team for their work on the mission and the
International Team “Cross-Scale Energy Transfer in Space Plasmas” supported by the International Space Science Institute for invaluable discussions. The authors thank T. N. Parashar for useful
discussions. H.C.L. wishes to thank colleagues at Imperial College London for their insightful advice and comments.
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Harry C. Lewis: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Christopher
Russell: Data curation (equal). Per-Arne Lindqvist: Data curation (equal). Julia E. Stawarz: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (supporting);
Methodology (equal); Supervision (lead); Writing – review & editing (equal). Luca Franci: Investigation (supporting); Writing – review & editing (supporting). Lorenzo Matteini: Writing – review &
editing (supporting). Kristopher G. Klein: Investigation (supporting); Writing – review & editing (supporting). Chadi Salem: Writing – review & editing (supporting). James Leo Burch: Data curation
(equal). Robert E. Ergun: Data curation (equal). Barbara L. Giles: Data curation (equal).
, and
, “
Magnetic turbulence in the geospace environment
Space Sci. Rev.
C. H. K.
, “
Nature of kinetic scale turbulence in the earth's magnetosheath
Astrophys. J.
, and
, “
Plasma and magnetic field turbulence in the earth's magnetosheath at ion scales
Front. Astron. Space Sci.
, and
W. H.
, “
Evidence for weak MHD turbulence in the middle magnetosphere of Jupiter
Astron. Astrophys.
L. Z.
K. H.
, and
M. K.
, “
Nature of the MHD and kinetic scale turbulence in the magnetosheath of Saturn: Cassini observations
Astrophys. J.
C. H. K.
T. S.
, and
S. D.
, “
Solar wind turbulence and the role of ion instabilities
Space Sci. Rev.
, “
The solar wind as a turbulence laboratory
Living Rev. Sol. Phys.
M. L.
R. T.
, and
, “
Kinetic scale turbulence and dissipation in the solar wind: Key observational results and future outlook
Phil. Trans. R. Soc. London Ser. A
K. G.
, and
B. A.
, “
The multi-scale nature of the solar wind
Living Rev. Sol. Phys.
C. H. K.
S. D.
J. W.
T. A.
A. W.
B. D. G.
T. D.
de Wit
P. R.
J. C.
K. G.
K. E.
R. J.
D. M.
M. D.
M. L.
, and
, “
The evolution and role of solar wind turbulence in the inner heliosphere
Astron. Astrophys.
, “
On the universality of supersonic turbulence
Mon. Not. R. Astron. Soc.
, and
A. C. L.
, “
Turbulence in the interstellar medium
Nonlinear Processes Geophys.
E. M.
de Gouveia Dal Pino
, and
, “
The role of turbulent magnetic reconnection in the formation of rotationally supported protostellar disks
Astrophys. J.
, and
, “
Spectra and anisotropy of magnetic fluctuations in the earth's magnetosheath: Cluster observations
Ann. Geophys.
C. H. K.
, and
, “
Electric and magnetic spectra from MHD to electron scales in the magnetosheath
Mon. Not. R. Astron. Soc.
, and
, “
Current sheet statistics in the magnetosheath
Front. Astron. Space Sci.
S. Y.
L. Z.
Z. G.
, and
X. H.
, “
On the existence of the Kolmogorov inertial range in the terrestrial magnetosheath turbulence
Astrophys. J.
, and
, “
On the transmission of turbulent structures across the earth's bow shock
Astrophys. J.
J. E.
L. B.
, and
, “
Strong reconnection electric fields in shock-driven turbulence
Phys. Plasmas
R. J.
C. W.
N. F.
, and
H. K.
, “
Dissipation range dynamics: Kinetic Alfvén waves and the importance of β[e]
J. Geophysical Res.
G. G.
S. C.
G. W.
A. A.
, and
, “
Kinetic simulations of magnetized turbulence in astrophysical plasmas
Phys. Rev. Lett.
B. D. G.
B. N.
, and
, “
Perpendicular ion heating by low-frequency Alfvén-wave turbulence in the solar wind
Astrophys. J.
J. V.
, “
Cyclotron resonances of ions with obliquely propagating waves in coronal holes and the fast solar wind
J. Geophys. Res.
, and
, “
Magnetic pumping as a source of particle heating and power-law distributions in the solar wind
Astrophys. J.
W. H.
T. N.
C. C.
, and
, “
Energy transfer, pressure tensor, and heating of kinetic plasma
Phys. Plasmas
W. H.
T. N.
, and
, “
Energy transfer channels and turbulence cascade in Vlasov-Maxwell turbulence
Phys. Rev. E
W. H.
T. N.
C. C.
J. L.
T. E.
C. J.
D. J.
R. B.
, and
C. T.
, “
Energy conversion and collisionless plasma dissipation channels in the turbulent magnetosheath observed by the magnetospheric multiscale mission
Astrophys. J.
W. H.
T. N.
C. C.
M. A.
D. J.
B. L.
, and
J. L.
, “
Energy dissipation in turbulent reconnection
Phys. Plasmas
C. H. K.
K. G.
, and
G. G.
, “
Evidence for electron Landau damping in space plasma turbulence
Nat. Commun.
R. E.
S. J.
F. D.
J. E.
K. A.
D. L.
I. J.
S. T.
J. C.
R. B.
J. L.
R. J.
Le Contel
, and
B. L.
, “
Observations of particle acceleration in magnetic reconnection–driven turbulence
Astrophys. J.
R. E.
S. J.
F. D.
P. A.
J. E.
K. A.
D. L.
W. H.
J. F.
M. A.
R. B.
, and
J. L.
, “
Particle acceleration in strong turbulence in the earth's magnetotail
Astrophys. J.
J. E.
J. P.
T. D.
I. L.
P. S.
M. A.
S. L.
C. T.
, and
Le Contel
, “
Turbulence-driven magnetic reconnection and the magnetic correlation length: Observations from magnetospheric multiscale in earth's magnetosheath
Phys. Plasmas
W. H.
T. N.
, and
, “
Pathways to dissipation in weakly collisional plasmas
Astrophys. J.
R. E.
K. A.
F. D.
J. C.
J. E.
K. J.
J. L.
R. B.
T. D.
S. J.
J. P.
R. J.
Le Contel
C. T.
M. R.
P. A.
L. J.
P. A.
B. L.
J. C.
T. W.
, and
, “
Magnetic reconnection, turbulence, and particle acceleration: Observations in the earth's magnetotail
Geophys. Res. Lett.
R. A.
Basic Space Plasma Physics
Published by Imperial College Press and distributed by World Scientific Publishing Co.
J. E.
T. N.
J. P.
C. A.
I. L.
J. L.
R. E.
B. L.
D. J.
Le Contel
C. T.
R. J.
, and
R. B.
, “
Comparative analysis of the various generalized ohm's law terms in magnetosheath turbulence as observed by magnetospheric multiscale
J. Geophys. Res.
S. D.
P. J.
F. S.
T. S.
, and
, “
Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence
Phys. Rev. Lett.
J. P.
C. W.
Le Contel
H. U.
K. H.
, and
C. T.
, “
Turbulent heating and cross-field transport near the magnetopause from THEMIS
Geophys. Res. Lett.
J. E.
F. D.
R. E.
S. J.
J. L.
B. L.
O. L.
P. A.
C. J.
C. T.
R. J.
R. B.
L. A.
J. C.
J. P.
D. J.
K. A.
D. M.
G. T.
, and
A. P.
, “
Observations of turbulence in a Kelvin-Helmholtz event on 8 September 2015 by the magnetospheric multiscale mission
J. Geophys. Res.
G. G.
J. M.
A. A.
, and
, “
Gyrokinetic simulations of solar wind turbulence from ion to electron scales
Phys. Rev. Lett.
, and
, “
Solar wind turbulence from MHD to sub-ion scales: High-resolution hybrid simulations
Astrophys. J. Lett.
, and
, “
Solar wind turbulent cascade from MHD to sub-ion scales: Large-size 3D hybrid particle-in-cell simulations
Astrophys. J.
, and
, “
Can hall magnetohydrodynamics explain plasma turbulence at sub-ion scales?
Astrophys. J.
J. E.
O. L.
, and
, “
Modeling MMS observations at the earth's magnetopause with hybrid simulations of Alfvénic turbulence
Astrophys. J.
A. A.
S. C.
G. W.
G. G.
, and
, “
Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in magnetized weakly collisional Plasmas
Astrophys. J. Suppl. Ser.
N. K.
R. P.
M. L.
, and
, “
Implication of kinetic Alfvén waves to magnetic field turbulence spectra: Earth's magnetosheath
Astrophys. Space Sci.
W. H.
, and
, “
Comment on ‘Kinetic simulations of magnetized turbulence in astrophysical plasmas
Phys. Rev. Lett.
J. L.
T. E.
R. B.
, and
B. L.
, “
Magnetospheric multiscale overview and science objectives
Space Sci. Rev.
S. C.
A. N.
, and
, “
Thin electron-scale layers at the magnetopause
Geophys. Res. Lett.
C. D.
, “
Generalized Ohm's law in a 3-D reconnection experiment
Geophysical Res. Lett.
Y. V.
C. J.
, and
, “
Formation of inner structure of a reconnection separatrix region
Phys. Rev. Lett.
M. R.
C. D.
, and
, “
Two fluid effects on three-dimensional reconnection in the Swarthmore Spheromak Experiment with comparisons to space data
Phys. Plasmas
R. B.
J. L.
B. L.
C. J.
M. R.
R. J.
C. T.
R. E.
F. D.
H. A.
C. J.
P. A.
T. E.
C. A.
, and
, “
Estimates of terms in Ohm's law during an encounter with an electron diffusion region
Geophys. Res. Lett.
K. J.
J. L.
P. A.
R. B.
R. E.
T. D.
B. L.
C. T.
K. J.
J. P.
C. P.
R. C.
T. K. M.
J. M.
, and
, “
MMS observation of asymmetric reconnection supported by 3-D electron pressure divergence
J. Geophys. Res.
J. M.
J. L.
P. H.
A. G.
K. J.
D. B.
R. B.
R. E.
S. Y.
R. C.
T. D.
B. L.
T. E.
S. A.
C. T.
A. C.
J. M.
, and
, “
Magnetospheric multiscale dayside reconnection electron diffusion region events
J. Geophys. Res.
W. M.
M. V. D.
D. G.
B. L.
, and
J. L.
, “
Mechanism of reconnection on kinetic scales based on magnetospheric multiscale mission observations
Astrophys. J. Lett.
J. R.
D. J.
L. J.
J. C.
D. E.
da Silva
B. L.
W. R.
R. E.
S. J.
F. D.
P. A.
A. C.
L. A.
, and
A. F.
, “
MMS measurements of the Vlasov equation: Probing the electron pressure divergence within thin current sheets
Geophys. Res. Lett.
C. C.
, “
Spatial gradients and the volumetric tensor
ISSI Sci. Rep. Ser.
R. B.
C. T.
R. E.
P. A.
Le Contel
Y. V.
C. A.
R. J.
H. K.
, and
, “
The FIELDS instrument suite on MMS: Scientific objectives, measurements, and data products
Space Sci. Rev.
C. T.
B. J.
K. R.
H. K.
J. D.
M. B.
K. M.
J. A.
R. J.
, and
, “
The magnetospheric multiscale magnetometers
Space Sci. Rev.
P. A.
R. B.
Y. V.
R. E.
, and
, “
The spin-plane double probe electric field instrument for MMS
Space Sci. Rev.
R. E.
K. A.
D. M.
, and
C. M.
, “
The axial double probe and fields signal processing for the MMS mission
Space Sci. Rev.
De Los Santos
J. A.
P. S.
, and
, “
Fast plasma investigation for magnetospheric multiscale
Space Sci. Rev.
D. J.
J. C.
A. F.
, and
C. J.
, “
The calculation of moment uncertainties from velocity distribution functions with random errors
J. Geophys. Res.
G. I.
, “
The spectrum of turbulence
Proc. R. Soc. London. Ser. A
P. D.
, “
The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms
IEEE Trans. Audio Electroacoust.
M. W.
, and
, “
Accuracy of current density determination
ISSI Sci. Rep. Ser.
P. S.
, “
Turbulence of a conducting fluid in a strong magnetic field
Sov. Astron.
R. H.
, “
Inertial-range spectrum of hydromagnetic turbulence
Phys. Fluids
, “
Toward a theory of interstellar turbulence. 1: Weak Alfvenic turbulence
Astrophys. J.
, “
Toward a theory of interstellar turbulence. II. Strong alfvenic turbulence
Astrophys. J.
S. V.
A. C.
, and
, “
A weak turbulence theory for incompressible magnetohydrodynamics
J. Plasma Phys.
, “
On the spectrum of magnetohydrodynamic turbulence
Astrophys. J.
, “
Spectrum of magnetohydrodynamic turbulence
Phys. Rev. Lett.
A. A.
, and
B. D. G.
, “
Refined critical balance in strong Alfvenic turbulence
Mon. Not. R. Astron. Soc.: Lett.
W. H.
M. L.
, and
J. H.
, “
An interplanetary magnetic field ensemble at 1 AU
J. Geophys. Res.
R. T.
D. A.
A. A.
T. S.
, and
C. H. K.
, “
Correlations at large scales and the onset of turbulence in the fast solar wind
Astrophys. J.
M. E.
P. D.
, and
, “
Lack of universality in decaying magnetohydrodynamic turbulence
Phys. Rev. E
J. E.
, and
, “
Long-time properties of magnetohydrodynamic turbulence and the role of symmetries
Phys. Rev. E
, and
, “
High-resolution hybrid simulations of kinetic plasma turbulence at proton scales
Astrophys. J.
T. A.
J. W.
, and
S. D.
, “
Impact of residual energy on solar wind turbulent spectra
Astrophys. J.
M. R.
, and
W. H.
, “
Density fluctuation spectra in magnetohydrodynamic turbulence
J. Geophys. Res.
C. H. K.
C. S.
J. W.
F. S.
, and
S. D.
, “
Density fluctuation spectrum of solar wind turbulence between ion and electron scales
Phys. Rev. Lett.
T. N.
W. H.
M. A.
J. L.
T. E.
B. L.
D. J.
C. J.
R. B.
C. T.
R. J.
, and
, “
Kinetic range spectral features of cross helicity using the magnetospheric multiscale spacecraft
Phys. Rev. Lett.
C. H. K.
B. A.
, and
S. D.
, “
Ion-scale spectral break of solar wind turbulence at high and low beta
Geophys. Res. Lett.
, and
, “
Plasma beta dependence of the ion-scale spectral break of solar wind turbulence: High-resolution 2d hybrid simulations
Astrophys. J.
, and
J. C.
, “
Toward a theory of astrophysical plasma turbulence at subproton scales
Astrophys. J.
O. W.
, and
, “
The kinetic Alfvén-like nature of turbulent fluctuations in the Earth's magnetosheath: MMS measurement of the electron Alfvén ratio
Phys. Plasmas
T. N.
, “
Linear vs. nonlinear acceleration in plasma turbulence. I. Global versus local measures
Phys. Plasmas
T. N.
, “
Linear vs. nonlinear acceleration in plasma turbulence. II. Hall–finite-Larmor-radius magnetohydrodynamics
Phys. Plasmas
O. L.
M. C.
R. B.
R. E.
, and
de la Porte
, “
The search-coil magnetometer for MMS
Space Sci. Rev.
Published open access through an agreement withJISC Collections
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Al Richards
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I'm a math teacher in Northwestern Ontario, Canada. I've been teaching online part-time since 1994 and full-time online for 4 years now.
• Whether you are a math student, or an adult seeking a refresher, it is time to brush up on Algebra! In this course, master algebraic concepts from start to finish!
Whether you are a math student, or an adult seeking a refresher, it is time to brush up on Algebra! In this course, master algebraic concepts from start to finish!
Other Lessons by Al Richards
• Free
Want to know how to find the shortest distance from a point to a line? Math teacher Al Richards demonstrates how using analytic geometry and trigonometry.
Want to know how to find the shortest distance from a point to a line? Math teacher Al Richards demonstrates how using analytic geometry and trigonometry.
• What comes next in this sequence: 2, -8, 32, -128? Learn how to calculate geometric sequences in this advanced math lesson. Be sure to have a calculator handy!
What comes next in this sequence: 2, -8, 32, -128? Learn how to calculate geometric sequences in this advanced math lesson. Be sure to have a calculator handy!
• What's the next number in this sequence: 2, 10, 18, 26...? Learn how to figure out arithmetic sequences and recursive formulas in this advanced math lesson.
What's the next number in this sequence: 2, 10, 18, 26...? Learn how to figure out arithmetic sequences and recursive formulas in this advanced math lesson.
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Question about SAS paper
Regarding this paper: http://support.sas.com/resources/papers/proceedings13/433-2013.pdf A multilevel model primer...
I have a question. On page 13 it says that based on the covariance parameter output on page 12, it seems that growth in reading achievement does vary between children. How would one approximate the %
variability in this case (based on the covariance parameter output on page 12)?
I'll note that page 13 does say, "Although the magnitude of the intercept variance estimate is rarely interpreted in a substantive way, the pvalue for that variance parameter indicates that reading
achievement at entry into kindergarten does vary between children."
Paper by: Bethany A. Bell, Mihaela Ene, Whitney Smiley, Jason A. Schoeneberger
I was told by SAS on YouTube to ask about this here. Thanks in advance.
10-15-2018 09:11 AM
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Supervised Learning
Learns from data labeled with the right answer(label Y)
Regression: Predict a number among infinitely many possible numbers/outputs
Linear Regression: Fitting a straight line to your data
Classification: Predict categories from among a limited set of categories.
There are only a few possible outputs/categories/classes.
Categories could be non_numeric or numeric.
The learning algorithm has to decide how to fit a boundary line through this data.
The boundary line is found by the learning algorithm.
Logistic Regression: Fitting a curve that looks like S-shape curve to the dataset
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Profitability Index (PI) Calculator
Are you tired of trying to figure out if your investments are profitable? Fear not, for the Profitability Index (PI) Calculator is here to help! This nifty tool allows you to calculate the
profitability of your investments and determine if they are worth pursuing. Plus, it’s way more fun than doing math the old fashioned way.
The Profitability Index (PI) is a financial ratio that measures the return of an investment relative to its costs. In simpler terms, it helps to determine if your investments are worth the money and
time you put into them. The formula for calculating PI is as follows:
PI = (Present Value of Cash Flows / Initial Investment)
In other words, the present value of cash flows is divided by the initial investment to give you the PI ratio. A PI ratio greater than 1 means that your investment is profitable, while a PI ratio
less than 1 means that your investment is not profitable.
Now that we’ve got that out of the way, let’s dive into the different types of PI calculations and how to interpret the results.
Types of PI Calculations
There are different categories of PI calculations that determine the level of profitability of your investment. These categories are excellent, good, fair, and poor. The range for each category is as
Category Range Interpretation
Excellent 1.5 – ∞ The investment is highly profitable
Good 1.0 – 1.49 The investment is profitable
Fair 0.99 – 0.5 The investment may not be worth pursuing
Poor 0.49 – 0 The investment is not profitable
For instance, if the PI ratio of an investment is 2, then it falls under the excellent category, which means that the investment is highly profitable.
Examples of PI Calculations
Let’s take a look at some examples of PI calculations to help you understand how it works.
Name Initial Investment Present Value of Cash Flows PI Calculation Interpretation
Bob $1000 $2000 2 Excellent
Alice $500 $750 1.5 Excellent
John $2000 $2500 1.25 Good
Sarah $1500 $1200 0.8 Fair
Tim $1000 $400 0.4 Poor
As you can see, Bob’s investment has a PI ratio of 2, which indicates that it is highly profitable. On the other hand, Tim’s investment has a PI ratio of 0.4, which means that it is not profitable.
Calculation Methods
There are different ways to calculate PI, and they include Net Present Value (NPV), Internal Rate of Return (IRR), and Payback Period. Each method has its advantages, disadvantages, and accuracy
Method Advantages Disadvantages Accuracy
Net Present Value (NPV) Accounts for time value of money Requires accurate cash flow projections High
Internal Rate of Return (IRR) Accounts for time value of money Requires accurate cash flow projections High
Payback Period Easy to use Ignores cash flows beyond payback period Low
Evolution of PI Calculation
The concept of PI calculation has evolved over the years, with improvements in techniques and methods. Here’s how it has evolved:
Time Period Development
1960s Introduction of NPV and IRR
1970s Increased use of discounted cash flow methods
1980s Emphasis on shareholder value
1990s Growing popularity of real options analysis
Limitations of PI Calculation Accuracy
It’s essential to know the limitations of PI calculation accuracy because they can affect the validity of your investment decisions. Here are some of the limitations:
Calculation Limitations
1. Assumes cash flows are accurately predicted
2. Ignores non-financial factors
3. Does not account for inflation
4. Does not consider risk
Alternative Methods for Measuring PI Calculation
There are alternative methods for measuring PI calculation, and they include Return on Investment (ROI), Profit Margin, and Modified Internal Rate of Return (MIRR). Each method has its pros and cons,
and here’s a table outlining them:
Method Pros Cons
Return on Investment (ROI) Easy to calculate Ignores time value of money
Profit Margin Focuses on profitability Ignores initial investment
Modified Internal Rate of Return (MIRR) Accounts for reinvestment of cash flows More complex calculation
1. What is the Profitability Index (PI)? The Profitability Index (PI) is a financial ratio that measures the return of an investment relative to its costs.
2. How is PI calculated? PI is calculated by dividing the present value of cash flows by the initial investment.
3. What does a PI of less than 1 mean? A PI of less than 1 means that your investment is not profitable.
4. Can PI be negative? No, PI cannot be negative.
5. What is a good PI ratio? A PI ratio greater than 1 is considered good.
6. How accurate is PI calculation? PI calculation is highly accurate if cash flow projections are accurate.
7. What factors affect PI calculation? Factors that affect PI calculation include cash flow projections, inflation, and risk.
8. What is the difference between PI and ROI? PI measures the return of an investment relative to its costs, while ROI measures the return of an investment relative to its initial investment.
9. Which calculation method is best for PI? The best calculation method for PI depends on the type and complexity of the investment.
10. How can I use PI to make investment decisions? You can use PI to make investment decisions by comparing the PI ratios of different investments and choosing the one with the highest ratio.
Here are some reliable government and educational resources for further research on PI calculations:
1. Investopedia – provides definitions, examples, and tutorials on PI calculations. Link
2. MIT OpenCourseWare – offers free online courses on financial management and investment analysis. Link
3. U.S. Small Business Administration – provides resources and advice for small business owners on financial planning. Link
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Our next cosmology seminar takes place on Friday 15th of November, seminar room E349 at 2pm. The speaker is Jibril Ben Achour from the Yukawa Institute (Kyoto, Japan).
At 14:00, room E349, Jibril Ben Achour (Yukawa Institute, Kyoto), will be talking about
FLRW cosmology as a conformally invariant field theory in 1D
Over the last decades, conformal field theory (CFT) technics have played an important role in black hole physics, due to the discovery of unexpected conformal symmetries in the near horizon of black
hole. Using these technics, crucial features of black hole physics, among which their spectroscopy, quasi-normal modes, as well as their state counting have been reproduced using CFT technics. So
far, the possible existence of similar hidden conformal symmetry in cosmological spacetimes have been much less investigated. In this talk, I will show that the simplest cosmological model,
consisting in the homogeneous and isotropic Einstein-Scalar system, enjoys a surprising hidden conformal symmetry. I will present in detail this new structure, the associated Noether’s charge as
well as the CVH algebra which encodes this conformal symmetry at the hamiltonian level. Then, I will explain how this cosmological system can be mapped to the well known conformal mechanics developed
by de Alfaro, Fubini and Furlan, and which provides the simplest example of a classically conformal field theory in 1d. This conformal structure extends beyond this simple homogeneous and isotropic
setting. I will briefly discuss the inclusion of a cosmological constant, anisotropies, self-interacting potential for the scalar field, as well as work in progress on the fully inhomogeneous
Einstein-Scalar system. Finally, I will discuss the consequences of this new conformal structure at the quantum level. A major outcome is the possibility to apply the conformal bootstrap program to
quantize the theory. Just as quantum conformal mechanics, I will show that despite the lack of a conformally invariant group state, it is possible to find an operator which reproduces the standard
form of the CFT two points function in quantum cosmology. I will discuss the cosmological interpretation of this operator and the vacuum state which are used to construct this correlator. In the end,
this provides a first step to bootstrap quantum cosmology using this new conformal symmetry.
Based on: arXiv:1909.13390
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Joint Distributions, Covariance and Correlation
Last Updated: 10th October, 2023
Joint distributions and covariance are essential concepts in probability theory and statistics. The joint distribution of random variables is the probability distribution of a set of random
variables, whereas covariance measures the degree to which two random factors change together.
Joint Distributions
Joint distributions are probability distributions that describe the relationship between two or more random variables. A joint distribution specifies the probability of each possible combination of
values for the random variables. The joint probability density function (PDF) for two continuous random variables X and Y is denoted as f(x, y), while for two discrete random variables X and Y, it is
denoted as P(X = x, Y = y).
Example: Suppose we have two random variables X and Y that represent the number of heads and tails, respectively, in two coin flips. The joint distribution of X and Y can be represented as follows:
Example: Suppose we have two random variables X and Y that represent the number of heads and tails, respectively, in two coin flips. The joint distribution of X and Y can be represented as follows:
X/Y 0 1 2
0 1/4 1/2 1/4
1 1/2 0 1/2
2 1/4 1/2 1/4
Marginal Distributions
Marginal distributions refer to the probability distributions of individual random variables obtained from a joint distribution. The marginal distribution of X is obtained by summing (in the discrete
case) or integrating (in the continuous case) over all possible values of Y.
Example: Using the joint distribution from the previous example, the marginal distribution of X can be obtained as follows:
X 0 1 2
P(X) 1/4 + 1/2 + 1/4=1/2 1/2 + 1/2=1 1/4 + 1/2 + 1/4=1/2
Conditional Distributions
Conditional distributions portray the probability distribution of one irregular variable given the esteem of another. The conditional conveyance of Y given X is indicated as P(Y | X) or f(Y | X), and
is calculated by dividing the joint distribution of X and Y by the marginal distribution of X.
To find the conditional probability of X given Y, we need to use the formula:
P(X|Y) = P(X and Y) / P(Y)
Utilizing the joint dissemination from the past illustration, the conditional distribution of Y given X=1 can be gotten as follows:
Let's find the conditional probability of X = 1 given Y = 2:
P(X=1|Y=2) = P(X=1 and Y=2) / P(Y=2)
P(X=1 and Y=2) = 1/2 (from the table)
P(Y=2) = 1/4 + 1/2 + 1/4 = 1 (sum of probabilities in the Y=2 column)
P(X=1|Y=2) = (1/2) / 1 = 1/2
So the conditional probability of X=1 given Y=2 is 1/2.
Covariance and Correlation
Covariance measures the degree to which two random variables X and Y are linearly related. It is defined as the expected value of the product of the deviations of X and Y from their respective means:
cov(X,Y) = E[(X - E[X])(Y - E[Y])]
Correlation is a standardized version of covariance, and measures the degree of linear association between two variables X and Y:
corr(X,Y) = cov(X,Y) / (std(X) * std(Y))
To find the covariance and correlation between X and Y, we need to use the following formulas:
Cov(X,Y) = E[XY] - E[X]E[Y]
Corr(X,Y) = Cov(X,Y) / (SD(X) * SD(Y))
where E[XY] is the expected value of the product of X and Y, E[X] and E[Y] are the expected values of X and Y, SD(X) and SD(Y) are the standard deviations of X and Y, respectively.
Let's start by finding the expected values of X and Y:
E[X] = 0*(1/4) + 1*(1/2) + 2*(1/4) = 1
E[Y] = 0*(1/2) + 1*(1) + 2*(1/2) = 1
Next, let's find the expected value of XY:
E[XY] = 00(1/4) + 01(1/2) + 02(1/4)
+ 10(1/2) + 11(0) + 12(1/2)
+ 20(1/4) + 21(1/2) + 22(1/4)
= 0 + 0 + 0 + 0 + 0 + 1 + 0 + 2 + 1 = 4/2 = 2
Now, let's calculate the standard deviations of X and Y:
SD(X) = sqrt(E[X^2] - (E[X])^2)
= sqrt(0*(1/4) + 1*(1/2) + 4*(1/4) - 1^2) = sqrt(1/4) = 1/2
SD(Y) = sqrt(E[Y^2] - (E[Y])^2)
= sqrt(0*(1/2) + 1*(1) + 4*(1/2) - 1^2) = sqrt(3/2)
Now, we can calculate the covariance between X and Y:
Cov(X,Y) = E[XY] - E[X]E[Y]
= 2 - (1)*(1) = 1
Finally, we can calculate the correlation between X and Y:
Corr(X,Y) = Cov(X,Y) / (SD(X) * SD(Y))
= 1 / ((1/2) * sqrt(3/2)) = sqrt(8/3)
So the covariance between X and Y is 1 and the correlation between X and Y is approximately 1.63299.
Correlation is a statistical measure used to determine the degree to which two variables are related to each other. Correlation can be either positive or negative, indicating the direction of the
relationship between the variables. A positive correlation means that as one variable increases, the other variable also increases, while a negative correlation means that as one variable increases,
the other variable decreases.
Types of Correlation and their ApplicationsTypes of Correlation and their Applications
There are three main types of correlation: positive correlation, negative correlation, and zero correlation.
Types of Correlation
Positive Correlation
Positive correlation occurs when the values of two variables increase or decrease together. For example, there is a positive correlation between the amount of exercise people do and their level of
fitness. The more exercise people do, the fitter they are likely to be.
The formula for calculating the Pearson correlation coefficient for a positive correlation is:
r = (nΣxy - ΣxΣy) / sqrt[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]
• n is the number of data points
• Σxy is the sum of the products of each pair of corresponding x and y values
• Σx is the sum of all x values
• Σy is the sum of all y values
• Σx^2 is the sum of the squares of all x values
• Σy^2 is the sum of the squares of all y values
Negative Correlation
Negative correlation occurs when the values of two variables move in opposite directions. For example, there is a negative correlation between the amount of time people spend watching TV and their
level of physical activity. The more time people spend watching TV, the less physically active they are likely to be.
The formula for calculating the Pearson correlation coefficient for a negative correlation is:
r = (nΣxy - ΣxΣy) / sqrt[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]
• n is the number of data points
• Σxy is the sum of the products of each pair of corresponding data points (x and y)
• Σx is the sum of all the x values
• Σy is the sum of all the y values
• Σx^2 is the sum of the squared x values
• Σy^2 is the sum of the squared y values
Zero Correlation
Zero correlation occurs when there is no relationship between the two variables. For example, there is no correlation between the color of someone's eyes and their shoe size.
Multivariate Distributions
Multivariate distributions generalize joint distributions to more than two random variables.A multivariate distribution indicates the probability distribution of a set of random variables, where each
arbitrary variable may have a different probability distribution. A few commonly utilized multivariate dispersions incorporate the multivariate ordinary dispersion, the multinomial distribution, and
the multivariate t-distribution.
Illustration: Assume we have three arbitrary factors X, Y, and Z, where X and Y are ceaseless random variables and Z may be a discrete random variable. The joint distribution of X, Y, and Z can be
spoken to as takes after:
f(x, y, z) = P(X = x, Y = y, Z = z)
Applications of Joint Distributions
Joint distributions and covariance have numerous applications in statistics, probability theory, and data analysis. They are used in fields such as finance, engineering, biology, and machine
learning. Some applications of joint distributions include:
• Portfolio optimization: Joint distributions are used to model the relationship between the returns of different assets in a portfolio and optimize portfolio allocations.
• Signal processing: Joint distributions are used to model the joint probability distribution of signals and to estimate parameters such as signal power and correlation.
• Genetics: Joint distributions are used to model the joint probability distribution of genetic traits and to estimate heritability and gene expression levels.
• Machine learning: Joint distributions are used in various machine learning algorithms, such as Gaussian mixture models, hidden Markov models, and Bayesian networks, to model complex dependencies
between variables.
• Bayesian inference: Joint distributions are utilized in Bayesian inference to upgrade earlier beliefs about model parameters based on watched information.
• Time series investigation: Joint distributions are utilized to demonstrate the joint probability distribution of time series data and to assess parameters such as autocorrelation and
In conclusion, joint distributions, correlation and covariance are essential concepts in probability theory and insights that play a vital part in different applications. They permit us to show the
connections between different random variables and measure the degree to which they shift together. From fund to genetics and machine learning, joint distributions and covariance are utilized broadly
to analyze information, make forecasts, and educate decision-making.
Key Takeaways
1. Joint distributions are the probability distribution of a set of random variables.
2. Covariance measures the degree to which two random variables vary together.
3. The correlation coefficient is the standardized covariance between two random variables.
4. The joint distribution of two random variables can be visualized with a scatter plot.
5. Multivariate distributions can model the joint distribution of more than two random variables.
6. Joint distributions and covariance have numerous applications in finance, engineering, biology, machine learning, Bayesian inference, and time series analysis.
7. Correlation measures the strength and direction of the relationship between two variables
1. What is the joint distribution of two random variables?
A. The probability distribution of a single random variable
B. The probability distribution of a set of random variables
C. The distribution of the difference between two random variables
D. The distribution of the sum of two random variables
Answer: B
2. What does covariance measure?
A. The degree to which two random variables vary together
B. The degree to which two random variables are independent
C. The standard deviation of a single random variable
D. The probability of observing two random variables together
Answer: A
3. Which of the following measures the strength of the linear relationship between two random variables?
A. Variance
B. Correlation coefficient
C. Standard deviation
D. Mean
Answer: B
4. What is the main application of joint distributions and covariance in finance?
A. Estimating heritability and gene expression levels
B. Modeling the joint probability distribution of signals
C. Modeling the relationship between the returns of different assets in a portfolio
D. Modeling the joint probability distribution of time series data
Answer: C
5. What is the correlation coefficient?
A) A measure of the strength and direction of the linear relationship between two variables.
B) A measure of the strength and direction of the non-linear relationship between two variables.
C) A measure of the probability that two variables are related.
D) A measure of the magnitude of the difference between two variables.
Answer: A) A measure of the strength and direction of the linear relationship between two variables.
6. What is the range of possible values for the correlation coefficient?
A) -1 to 1
B) 0 to 1
C) -∞ to ∞
D) 0 to ∞
Answer: A) -1 to 1
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To the nearest hundredth of a foot, what is the radius of a circular sector whose area is 45 square feet and that intercepts an arc with measure 25?
| HIX Tutor
To the nearest hundredth of a foot, what is the radius of a circular sector whose area is 45 square feet and that intercepts an arc with measure 25?
Answer 1
Area #=pir^2# Area of a sector #=theta/360xxpir^2#
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Answer 2
To find the radius of a circular sector, you can use the formula for the area of a sector:
[ \text{Area} = \frac{1}{2} r^2 \theta ]
where ( r ) is the radius and ( \theta ) is the central angle in radians.
Given the area ( \text{Area} = 45 ) square feet and the central angle ( \theta = 25 ) degrees, convert the angle to radians by multiplying by ( \frac{\pi}{180} ). Then, solve for ( r ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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BankBazaar Interview Questions - BankBazaar Coding Interview Questions
BankBazaar Interview Questions
BankBazaar.com is India’s largest fintech co-branded credit card issuer and online platform for free credit scores. It has a captive base of 50Mn registered users who use the platform for free credit
score tracking and in-depth personal finance content and comparison tools. Supported by global investors such as WSV, Experian, Eight Roads, Sequoia India, Walden International, and Amazon,
BankBazaar has been at the forefront of democratizing finance in India by providing end users with frictionless access to credit.
It has got a 3.7* rating on Glassdoor and is considered one of the best product-based companies. It is highly regarded for its work-life balance.
They provide good training as well which will be beneficial in future too. You can practice the below BankBazaar Interview Questions for the interview. We have collected past frequently asked
BankBazaar Interview Questions for your reference.
BankBazaar Array Questions
Question 1. Construction of Longest Increasing Subsequence (N log N) Problem Statement You are given an array of integers. The problem “Construction of Longest Increasing Subsequence (N log N)” asks
to construct the longest increasing subsequence. Example arr[]={1, 4, 7, 2, 9, 6, 12, 3 } 12, 9, 7, 4, 1 and the size of this longest increasing subsequence is ...
Question 2. Three way partitioning of an array around a given range Problem Statement You are given an array of integers and a range of lowValue and highValue. The problem “Three way partitioning of
an array around a given range” asks to partition the array such that array will be divided into three parts. The partitions of the arrays will be: Elements ...
Question 3. Count pairs from two sorted arrays whose sum is equal to a given value x Problem Statement “Count pairs from two sorted arrays whose sum is equal to a given value x” problem states that
you are given two sorted arrays of integers and an integer value called sum. The problem statement asks to find out the total number of pair which sums up to ...
BankBazaar Tree Questions
Question 4. Count the number of nodes at given level in a tree using BFS Description The problem “Count the number of nodes at given level in a tree using BFS” states that you are given a Tree
(acyclic graph) and a root node, find out number of nodes at L-th level. Acyclic Graph: It is a network of nodes connected through edges which has ...
BankBazaar Graph Questions
Question 5. Count the number of nodes at given level in a tree using BFS Description The problem “Count the number of nodes at given level in a tree using BFS” states that you are given a Tree
(acyclic graph) and a root node, find out number of nodes at L-th level. Acyclic Graph: It is a network of nodes connected through edges which has ...
BankBazaar Other Questions
Question 6. Permutation Coefficient Problem Statement In this problem “Permutation Coefficient”, we need to find it when we are given the values of n & k. Example n = 5, k = 2 20 Explanation: This
value of n P r is found using the formula of the permutation coefficient. nPr = n!/(n-r)! Approach ...
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Time-Saving Effects of FFT-Based EMI Measurements - In Compliance Magazine
In the world of RF and microwave testing, measurements required for EMI are among the most complex and time-consuming since they incorporate a wide array of specific tests that must be performed over
an array of frequencies. They typically require not only many hours of test time but even more for configuring and reconfiguring the test set-up.
Fortunately, advances in the signal processing abilities of test equipment have reduced test time over the years. However, the real improvements are the result of enhancement measurement software,
greater integration, automation of the test process, and increasing acceptance of time-domain techniques based on Fast Fourier Transform (FFT) for use in preview measurements of the disturbance
spectrum, for example. Together they are slowly making the EMC measurement process faster, and more efficient and accurate.
Over the years, there has been a continuing trend toward greater automation of test environments that includes more fully integrating the elements of the test system, and the EMC measurement domain
has benefitted from this as well. Generally speaking, any technique that can reduce the amount of human intervention can reduce errors caused by manually reading and recording measurement results.
Automation also verifies and maintains the integrity of measurement settings to ensure repeatable results, and produces a verifiable test environment.
At the software level, the “must have” list of features required by EMC software is a long one, the most fundamental being the ability to completely collect, evaluate, and document RFI voltage,
power, and field strength in accordance with current standards. However, the complexity of EMC measurements also makes it essential that software have two clearly-defined methods of operation. The
first allows less-experienced users to obtain reliable, repeatable results using predefined, standards-compliant test routines, and the second allows “veterans” to specify custom values for every
parameter in order to define their own test routines.
Virtually all commercial EMC software provides these capabilities to varying degrees, along with the ability to be updated as standards evolve. Today’s EMC software is typically based on Microsoft
Windows, which makes it possible to create a familiar user environment that eliminates the need to navigate the nuances of proprietary software developed in-house.
FFT Benefits and Challenges
One of the most recent and promising developments in EMC testing is the use of time-domain scanning methods based on the FFT technique to identify the disturbance spectrum. This approach has
demonstrated its ability to reduce preview measurement time by a factor of 1000 or more. It is currently being evaluated by standards committees to determine whether it should be included in
forthcoming modifications, but its viability has already been proven in a variety of measurement situations. To understand the benefits of this technique, it is necessary to compare it to the
conventional approach, the stepped-frequency scan.
To measure an unknown disturbance quantity in the frequency domain, a test receiver or analyzer must be tuned through a frequency range as quickly as possible. This would ideally result in a refresh
rate that produces a stationary spectrum display (each frequency sweep being no be longer than about 20 ms). In reality, the measurement must account for the settling time of the resolution bandwidth
and the signal timing of the disturbance signals, which can be continuous and pulsed narrowband signals, or continuous and intermittent broadband disturbances.
For intermittent disturbances, proper measurement time adjustment is essential. Annex B of CISPR 16-2-1 to 16-2-3 contains a table of the minimum sweep times (fastest scan rates), from which the
minimum sweep times in Table 1 for each of the CISPR bands can be calculated. Since they are minimum sweep times, they may increase depending on the type of disturbance, even with the quasi-peak
Table 1: Minimum CISPR 16 sweep times for peak and quasi-peak detection
Since nearly all commercial standards use quasi-peak detection for compliance with a specific limit, EMI tests usually apply time-saving procedures such as minimizing the number of quasi-peak
measurements. In addition to CISPR 16, MIL-STD-461 requires minimum measurement times for analog measurement receivers and minimum dwell times for synthesized measurement receivers (Table 2). For
equipment whose operation may produce emissions at infrequent intervals, times for frequency scanning must be increased to capture them.
Table 2: Bandwidth and measurement time specified by MIL-STD-461F
Preview measurements in accordance with commercial or automotive standards use a “max. peak” detector to first identify all frequencies at which emissions approach the limit values. Quasi-peak
detection on only the detected critical frequencies significantly decreases measurement time for the final measurement. However, preview measurements can take hours because they must be performed
between 30 MHz and at least 1 GHz.
To reliably detect a pulse-like disturbance, the observation time per frequency point must be at least as large as the reciprocal of its pulse rate. In addition, disturbance measurements must always
be made at the maximum level (e.g., the worst case emission), which usually requires repositioning the antenna and test device.
For example, scanning 30 MHz to 1 GHz with an IF bandwidth of 120 kHz and a step width of 40 kHz to measure the entire spectrum (without gaps and with sufficient measurement accuracy) produces 24,250
measurement points. If the dwell time is 10 ms per frequency point, total measurement time for a single preview scan is 4 min. This time must be multiplied by a factor of 20 or more to account for
the time required for positioning the turntable and antenna height, and antenna polarization switching.
Using a spectrum analyzer instead of a test receiver does not overcome the problem because the time of a single sweep must be long enough for at least one disturbance pulse event to fall into the
instrument’s resolution bandwidth at each frequency.
For repetitive sweeps and maximum hold for the trace display, observation time must continue until the spectrum becomes stable, and a continuous broadband signal will require many fast sweeps to show
the envelope of the broadband spectrum. Spectrum analyzers usually allow fewer sweep points than test receivers and they may not provide enough frequency resolution to measure radiated emissions,
which makes it necessary to perform time-consuming partial sweeping.
Conventional EMI measurement systems can only measure the signal within the resolution bandwidth within a stated measurement time, whereas FFT-based time-domain EMI measurement systems allow a much
wider part of the observed spectrum to be analyzed simultaneously. This is because the EMI test receiver samples successive sections of spectrum at the IF with a bandwidth of several megahertz rather
than only 120 kHz, and each “subspectrum” is calculated simultaneously with a specific resolution using FFT.
Time-Domain Scan Considerations
However, steps must be taken when applying the time-domain technique that ensure all types of signals that can appear in a disturbance spectrum are correctly detected, even intermittent types with a
very low pulse repetition frequency. If they are not considered, the frequency spectra calculated by the FFT may be displayed incorrectly in level and frequency.
Theoretically, an exact calculation of the frequency spectrum of a time-domain signal would require an infinite period of observation, and it would be necessary to know the signal amplitude at every
point in time. In practice however, these requirements are unrealistic using FFT aided by digital signal processing. Analog-to-digital conversion provides a continuous input signal to be converted
into an amplitude- and time-discrete signal, and applying the FFT limits signal observation time to a finite (and practical) amount. This means that calculation of the frequency spectra requires a
reasonable number of discrete signals in the time domain, a process called “windowing.”
If the length of this window does not exactly correspond to an integer multiple of the periods of the frequencies contained in the input signal, it results in spreading or leakage of the spectral
components away from the correct frequency and an undesirable modification of the total spectrum. The generation of spectral components that are not present in the original time-domain signal is
known as the “leakage effect,” and is most severe when a simple rectangular window is used. The best way to reduce this effect is to choose a suitable window function that minimizes spreading.
The spectrum calculated by the FFT is a discrete frequency spectrum consisting of individual frequency components at the so-called “frequency bins,” which are determined by the FFT parameter. The
original spectral response can only be observed at the discrete frequency bins, and there may be higher amplitudes in the original signal spectrum at frequencies between two adjacent frequency bins.
The amplitude error this causes is called the “picket fence effect” (Figure 1) and is also characteristic of conventional stepped-frequency scans.
Figure 1: A description of the picket fence effect
Time-domain measurement techniques employing FFT on intermittent disturbance signals require certain system parameters to be emphasized so that all disturbance signals are detected and measurement
accuracy is maintained. For example, when an impulse-type disturbance signal is captured by the Gaussian-type FFT window, signal amplitude may be reduced at the window edges. To minimize this error
while also ensuring that no signal is missed, EMI test receivers that employ time-domain scan include an overlap of the window function in the time domain.
Such receivers usually provide two settings for the step mode of the time-domain scan, the “Auto CW” mode and an “Auto Pulse” mode. In Auto CW mode, the overlap in the time domain is about 20%, which
allows narrowband signals to be analyzed as quickly as possible. The “Auto Pulse” mode provides more than 90% of overlap and is intended for broadband-impulsive and mixed signals. It ensures that
even very short impulse signals at the edge of the Gaussian-type time-domain window are calculated without significant amplitude error. With so much window overlap, only a small amount of ripple
remains in the time domain that could result in only a small measurement error.
Worst-case amplitude errors for such receivers are typically 0.4 dB for the lowest point of the amplitude ripple referred to the maximum pulse amplitude and the resulting average error is 0.09 dB, a
theoretical value for a minimal pulse width. The real error value depends on the pulse duration and is usually even less. When performing the time-domain scan with weighting detectors to CISPR (i.e.,
quasi-peak), correct detection of single pulses requires the data rate for internal digital signal processing to be sufficiently high to accommodate the IF bandwidths that are used, and a 90% overlap
of the FFT windows is essential for proper quasi-peak detection.
Analog filtering in the signal path has an influence on the frequency response of a time domain scan, and non-ideal correction of the analog filters in the RF and IF signal path of the test receiver
add to overall measurement uncertainty. The bandwidths of the preselection filters get narrower as frequency decreases (such as 2 MHz bandwidth at 8 MHz vs. 80 MHz bandwidth at 500 MHz).
To minimize the influence of the preselection filters’ frequency response, the receiver reduces the bandwidth for the time-domain scan accordingly, from 7 MHz to 150 kHz for example, depending on the
scan range, and compensates for the frequency response of the analog IF filters.
Comparing Stepped-Frequency and Time-Domain Scans
In frequency bands A to E, the CISPR16-1-1 standard specifies bandwidths and tolerance masks for IF filters used in disturbance measurements to commercial standards. In contrast, MIL-STD-461 defines
6-dB bandwidths in decimal steps that must be met with a 10% tolerance. Any deviations from the specified tolerances cause amplitude errors.
To verify IF selectivity, a time-domain scan with max. peak detection was performed for sinusoidal test signals. A single measurement is insufficient for correct verification because the spacing of
adjacent frequency bins is set to one-quarter IF bandwidth (Figure 2). The tests were repeatedly performed, increasing the start frequency of the time-domain scan step-by-step in small increments.
All received frequency points were then merged into a single trace (Figure 3).
Figure 2: IF selectivity using the time-domain scan
Figure 3: Measured IF selectivity for CISPR bands A, B, and C/D
At lower levels, the inherent noise of the receiver limits the dynamic range and is specified as displayed average noise level (DANL). At higher levels, the nonlinearity of mixers and amplifiers
limit the measurement range, and is characterized by the 1-dB compression or third-order intercept points. Sensitivity of 1 dB and 3 dB are analogous to these points, that is, where signal-to-noise
ratio is high enough so that noise-induced measurement error is not more than 1 or 3 dB. Dynamic range usually specifies the usable level range between 1-dB sensitivity and the 1-dB compression
A measurement using a pulse generator for CISPR bands C and D compares the frequency responses of the stepped-frequency and time-domain scans (Figure 4). An exact evaluation of receiver measurement
uncertainty is not possible with this measurement, and it does not consider errors caused by the cable and pulse generator, such as frequency response, matching, and long-term stability. However, it
nonetheless shows that the differences between the two scan types are negligible.
Figure 4: Overall frequency response of the test receiver for the time-domain scan (blue) and stepped-frequency scan (black) including the frequency response of the CISPR pulse generator
Figures 5a through 5d show the measured frequency response of the CISPR bandwidths of 200 Hz, 9 kHz, and 120 kHz, and the MIL-STD IF bandwidths 100 kHz and 1 MHz for the time-domain scan and the
stepped-frequency scan. Both traces match very well and are compliant with the requirements of the standards. Table 3 shows that the time-domain scan offers a higher dynamic range than the
stepped-frequency scan, generally without regard to IF bandwidth.
Figures 5a, 5b, 5c and 5d: Comparison of resolution bandwidths
with stepped-frequency scan (blue) and time-domain scan (green)
for the CISPR 16 and MIL-STD 461 standards
Table 3
The evaluation of measurement times was based on the frequency bands for EMI measurement in accordance with CISPR 25 (EN 55025) for automotive products and other military and commercial standards.
CISPR 16-2 requires the measurements to be long enough so that at least one signal from the disturbance source is detected. For this comparison, the measurement time per frequency step for the
commercial standards was set to 10 ms or 20 ms to correctly detect impulsive disturbances down to a pulse repetition rate of 100 Hz or 50 Hz respectively.
For measurements to MIL-STD-461, the measurement time was set according to Table 2. The measured values show that the time-domain technique considerably reduces the time required to perform a
frequency scan even when using quasi-peak weighting and a dwell time of 1 s. The exact reduction depends on the IF bandwidth.
In short, FFT-based time-domain scan for preview measurements allows EMI testing to be performed in accordance with CISPR 16 orders of magnitude faster than when using a stepped-frequency scan. The
measurement uncertainty of the time-domain and stepped-frequency scans is nearly identical. However, the stepped-frequency scan technique remains a proven, widely-accepted method, so it makes a great
deal of sense to combine both the frequency and time-domain techniques throughout the design and certification process to ensure the best possible results.
FFT-based time-domain scanning, along with the increasingly formidable capabilities of EMC software and greater process integration and automation, are transforming the EMC measurement process. As
commercial and military standards evolve, these benefits will become more and more important, as will their ability to make the process easier for designers who already have a “full plate” of
measurements necessary to bring a product to market.
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The Robustness of SNA Metrics
Given the required data and processing requirements for sociometric measures like betweeness, k-betweeness, or eigenvector centrality, and the fact that we may not always be able to characterize an
entire network (due to missing nodes and edges) we might want to consider, how robust are metrics derived from incomplete network data? It is much easier in terms of both computational and data
requirements to obtain measures of degree centrality vs. eigenvector centrality or even betweenness. Is degree similar enough to other measures of centrality to use in our analysis instead? Although
not a complete literature review, the articles below investigate these issues.
The stability of centrality measures when networks are sampled
Elizabeth Costenbader a,∗, Thomas W. Valente.
Social Networks 25 (2003) 283–307
"Our results indicate relatively high correlation, albeit in some instances substantial absolute differences, between actual network properties and those calculated on randomly selected sub-samples
for some network measures. This indicates that under some circumstances researchers may be still be able to use network data for which some data are missing to study network properties or create
network-based interventions. In other words, researchers who do not interview all members of a community or network may still be able to take advantage of some aspects of network theory and
Their remarks on eigenvector centrality are particularly interesting.
"As noted previously, the stability of eigenvector centrality when calculated as a simple raw score may indicate that it is the preferred centrality measure when the network data are incomplete.
However, the fact that sampling has less effect on this centrality measure may be due to the fact that in comparison to the other centrality measures, which measure the ones (i.e. the actual
nominations), this measure is able to effectively capture the similarity of zeros. Since many of the studies restrict nominations to five people, there are a lot of zeros in the original networks.
Consequently, eigenvector centrality as a simple raw score is less affected by sampling from the networks as the zeros are preserved."
How Correlated Are Network Centrality Measures? Thomas W. Valente, PhD,
University of Southern California, Department of Prevention Research, Los Angeles
Connect (Tor). 2008 January 1; 28(1): 16–26.
Kathryn Coronges, MPH, University of Southern California, Department of Prevention Research, Los Angeles
Cynthia Lakon, PhD, and University of Southern California, Department of Prevention Research, Los Angeles
Elizabeth Costenbader, PhD
Research Triangle Institute, Raleigh North Carolina
Investigates the the correlation among four centrality measures: degree, betweenness, closeness, and eigenvector and calculates 9 versions of these measures for 58 existing social networks previously
analyzed by Costenbader and Valente (2003).
From the article:
"We correlated the 9 measures for each network and then calculated the average correlation, standard deviation, and range across centrality measures. We also calculated the overall correlation and
compared it by study to assess the degree of variation in average correlation between studies."
"We find strong but varied correlations among the 9 centrality measures presented here. The average of the average correlations was 0.53 with a standard deviation of 0.14, indicating that most
correlations would be considered strong. The level of correlation among measures seems nearly optimal - too high a correlation would indicate redundancy and too low, an indication that the variables
measured different things. The amount of correlation between degree, betweenness, closeness, and eigenvector indicates that these measures are distinct, yet conceptually related."
A summary of the correlations for degree, betweenness and eigenvector centrality as reported in the article can be found below:
Borgatti, S.P., Carley, K., and Krackhardt, D. (2006). Robustness of Centrality Measures under Conditions of Imperfect Data. Social Networks 28: 124–136.
Many empirical studies approached the relationship between centrality measures across networks and in the context of missing data by using empirical data from actual networks. Borgatti points out a
limitation of this approach:
"A limitation of this approach is that the sampling errors contained in the data are likely to be systematic, but the pattern is unknown. Another limitation is that the sample of networks is
necessarily very limited. To overcome these limitations, we take a statistical computational approach and examine robustness in a very large sample of random graphs, into which we introduce a
controlled amount and kind of error."
In the article, measures of degree, betweenness, closeness and eigenvector centrality from 'sampled' networks were compared to the 'actual' values from the complete networks. These comparisons were
based on 5 measures of robustness (discussed in the article).
"the four centrality measures behave virtually identically in the face of measurement error. This suggests that the distinction between local and global measures of centrality (Scott, 2000) is not as
important as previously thought. These results are consistent with those of Everett and Borgatti (2004) who found that betweenness calculated on ego networks (a local measure) was, on average, nearly
identical to betweenness calculated on the full network in which ego networks were embedded (a global measure)."
Seeding Strategies for Viral Marketing: An Empirical Comparison
Oliver Hinz, Bernd Skiera, Christian Barrot, & Jan U. Becker
Journal of Marketing, Volume 75, Number 6, November 2011
Consistent with the results from some of the previous research, this article concludes:
"Remarkably, we reveal that to target a particular subnetwork (e.g., students of a particular university, Study 2) with a viral marketing message, the use of the respective subnetwork’s sociometric
measures is not absolutely required to implement the desired seeding strategies. Instead, because the sociometric measures of subnetworks and their total network are highly correlated, marketers can
use the socio- metric measures of the total network, without undertaking the complex task of determining exact network boundaries. Conversely, this appealing result also allows marketers to feel
confident in inferring the connectivity of a person in an overall network from information about his or her connec- tivity in a natural subnetwork."
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Vectors in equation editor
I'm doing maths at uni and we have to type out our equations, which i thought
would be fine. The only problem is that we are doing vectors and the symbol
for vectors is a letter with a tilde (~) placed underneath it. In the old
equation editor you used to be able to do this, but I can't find it anywhere.
Is there a way to do it in the current equation editor or will I have to get
MathType or something similar?
Well you left out a critical piece of information -- what version of
Word are you using? I can't speak for the OMML equation editor (that's
the new one in Word 2007), but this type of notation is not possible
with Microsoft Equation 3.0 except with a work-around (that's the
equation editor that's been around since Word 2.0). Microsoft Equation
3.0 is still available in Word 2007 (Insert > Object), if you want to
look through the palettes to see what's there.
You're wondering about MathType. With MathType, you can definitely put
a tilde very easily beneath a single character with an
"embellishment". Embellishments are in the third palette from the left
on the top row, and are applied to the character immediately to the
left of the insertion point (i.e., the cursor).
Bob Mathews
Director of Training
Design Science, Inc.
bobm at dessci.com
FREE fully-functional 30-day evaluation of MathType
MathType, MathFlow, MathPlayer, MathDaisy, WebEQ, Equation Editor
Hi Ben,
There is a way, although it takes some unintuitive setup. (By the way,
MathType is highly recommended anyway!)
- In an empty equation box, type the number 0330 and press Alt+X. That
inserts a character from the Cambria Math font called "Combining Tilde
Below" (which you can find in the Combining Diacritical Marks section of the
Insert > Symbol dialog).
- Select that character. Click the tiny arrow in the lower right corner of
the Tools group on the Equation Tools ribbon, and click the Math AutoCorrect
button in the dialog.
- In the AutoCorrect dialog, the tilde will already be in the "With" box. In
the Replace box to the left, type a name such as \utilde (or anything else
you can easily remember, and that isn't already in use). Click the Add
button and then OK both dialogs.
Now, in any equation, you can type the letter for the vector followed by the
name you assigned to the combining tilde, followed by the space bar (which
will force the replacement but won't insert a space character). The result
will be a tilde below the letter.
Incidentally, in my former career as an editor of science and maths
textbooks, I very rarely saw a tilde or anything else below a letter used to
indicate a vector. It's much more common to use a boldface character, with
or without an arrow above it. The arrow-above is on the Accent gallery.
Jay Freedman
Microsoft Word MVP
Email cannot be acknowledged; please post all follow-ups to the newsgroup so
all may benefit.
I'm using word 2007
Bob Mathews said:
Well you left out a critical piece of information -- what version of
Word are you using? I can't speak for the OMML equation editor (that's
the new one in Word 2007), but this type of notation is not possible
with Microsoft Equation 3.0 except with a work-around (that's the
equation editor that's been around since Word 2.0). Microsoft Equation
3.0 is still available in Word 2007 (Insert > Object), if you want to
look through the palettes to see what's there.
You're wondering about MathType. With MathType, you can definitely put
a tilde very easily beneath a single character with an
"embellishment". Embellishments are in the third palette from the left
on the top row, and are applied to the character immediately to the
left of the insertion point (i.e., the cursor).
Bob Mathews
Director of Training
Design Science, Inc.
bobm at dessci.com
FREE fully-functional 30-day evaluation of MathType
MathType, MathFlow, MathPlayer, MathDaisy, WebEQ, Equation Editor
I'm not sure if I missed a step or something, but it didn't seem to work. The
tilde was placed next to the letter but down the bottom, like a subscript,
whereas I need it to be underneath the letter.
Often in printing they just use bold letters to represent vectors, but since
you can't do bold when working by hand we use tildes Our lecturer still wants
us to use them even when we're typing it out.
After typing \utilde, you probably have to press the spacebar twice -- once to
do the replacement and a second time to make the tilde combine with the letter.
As I try this more, though, I've found a problem. It works with letters that are
no taller than a lower case x; but with letters such as 'h' or 'k' that have
"ascenders" and with all upper case letters, the tilde moves up into the body of
the letter. The same is true for the \ubar entry, which uses the "combining
macron below" character (0331). I'm filing a bug report for this.
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A Key Distinction for Better Understanding of Computer Science GCSE and A Level
Do you mean the thing or the container for the thing?
In this article I want to discuss a key distinction which crops up again and again throughout Computer Science GCSE and A Level. It is a distinction which might sound trivial or obvious, but many
years of teaching have shown me that getting ultra-clear about it is key for students’ understanding.
I first noticed myself emphasizing this distinction repeatedly when I was teaching GCSE Maths, in particular the notorious nᵗʰ term. I found that if I reminded students to always be very clear about
the distinction between the “slot number” (n) and the value in the slot with that number, that confusion quickly gave way to understanding.
So how does this apply to Computer Science GCSE and A Level?
Well, one very common and important example of where the above distinction applies is with arrays and indices. Or lists and indices if we are talking about Python.
List Values and List Indices in Python
Consider the list my_list = [1, 2, 3, 4, 5]
What is the value at the 0ᵗʰ position?
1 right? (Remember in most programming languages we start counting things at 0.)
And what value is found at the position with index 5?
That was a trick question – the indices for the list range from 0 to 4 inclusive.
It’s not rocket science, but you might be surprised at what a difference continually emphasizing this key distinction between index and value can make.
Where else does this distinction crop up?
Searching and Sorting Algorithms
This is really just a variation on the theme of array/list indices vs values. However, it is worth emphasizing that many algorithms from Computer Science GCSE and A Level become much easier to
understand if we remember to always ask ourselves value or index/position?
For example this applies to:
• Linear Search
• Binary Search
• Bubble Sort
• Insertion Sort
• Merge Sort
• Etc.
Binary and Hexadecimal
Well, consider binary numbers. We all know there are place value headings such as 16 8 4 2 1 in binary, but my experience tells me that students’ minds are apt to jump between the heading and the
values (0 or 1 in this case) which are contained in the column with a particular heading. This is even more the case with hexadecimal where there are 16 possible values for each place value heading.
So a very important key to understanding how binary and hexadecimal work is to revisit place value from primary school maths and make sure we are really clear at any given time – the value or the
container (the place value heading).
Low level programming
Little Man Computer and similar low-level programming languages also become much easier when we consider the same key distinction that we have been discussing. If we always make sure students are
aware whether they are referring to an address or the value stored at an address then a great deal of confusion can be avoided.
You can buy my new Little Man Computer Programming Teaching Pack for Computer Science GCSE and A Level here if you need some help delivering this topic.
Is the key distinction discussed here something you have given much thought to? Are there any other key distinctions which you find helpful in Computer Science GCSE and A Level?
Happy computing.
Robin Andrews.
1 Comment on “A Key Distinction for Better Understanding of Computer Science GCSE and A Level”
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Margin Calculator
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GSEB Solutions Class 9 Maths Chapter 10 Circles Ex 10.5
Gujarat Board GSEB Solutions Class 9 Maths Chapter 10 Circles Ex 10.5 Textbook Questions and Answers.
Gujarat Board Textbook Solutions Class 9 Maths Chapter 10 Circles Ex 10.5
Question 1.
Infigure,A,B and Carethreepointsona circle with centre O such that ∠BOC = 300 and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.
∠ADC = \(\frac {1}{2}\)∠AOC
[The angle subtended by an arc at the centre is double the angle subtended by it any point on the remaining part of the circle.]
= \(\frac {1}{2}\)(∠AOB + ∠BOC)
= \(\frac {1}{2}\) (60° + 300)
= \(\frac {1}{2}\) (90°) = 45°
Question 2.
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
∴ OA = OB = AB [Given]
∴ OAB is equilateral.
∴ ∠AOB = 60°
∠ACB = \(\frac {1}{2}\)∠A0B
[The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.]
∴ ∠ACB = \(\frac {1}{2}\) x 60° = 30°
Now, ADBC is a cyclic quadrilateral.
∴ ∠ADB + ∠ACB = 180°
[The sum of either pair of opposite angles of a cyclic quadrilateral is 180°]
⇒ ∠ADB + 30°= 180°
⇒ ∠ADB = 180°- 30°
∴ ∠ADB = 150°
Question 3.
In figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.
Take a point S in the major arc. Join PS and RS.
∴ PQRS is a cyclic quadrilateral.
∠PQR + ∠PSR = 180°
The sum of either pair of opposite angles of a cyclic quadrilateral is 180°
100° + ∠PSR = 180°
∠PSR = 180° – 100°
⇒ ∠PSR = 80° …….(1)
Now, ∠POR = 2∠PSR
(The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle]
∴ ∠POR= 2 x 800 = 160° …….(2) Using (1)
In ΔOPR,
∴ OP = OR Radiiofacircle
∴ ∠OPR = ∠ORP …………(3)
Angles opposite to equal sides of a triangle are equal
∠OPR + ∠ORP + ∠POR = 180°
Sum of all the angles of a triangle is 1800
⇒ ∠OPR + ∠OPR + 160° = 180°
Using (2) and (1)
⇒ 2∠OPR + 160° = 180°
⇒ 2∠OPR = 180° – 160° = 20°
∠OPR = 10°
Question 4.
In figure, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.
In ΔABC,
∠BAC + ∠ABC + ∠ACB = 180°
[Sum of all the angles of a triangle is 180°]
∠BAC + 69° + 31° = 180°
∠BAC + 100° = 1800
∠BAC = 180° = 100° = 80° ………..(1)
Now, ∠BDC = ∠BAC
[Angles in the same segment of a circle are equal]
∠BDC = 80° [Using (1)]
Question 5.
In figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.
In ΔCDE ∠CDE + ∠DCE = ∠BEC
[Exterior angle property of a triangle]
⇒ ∠CDE + 20° = 130°
⇒ ∠CDE = 130° – 20° = 110° ………..(2)
Now, ∠BAC = ∠CDE
[Angles in the same segment of a circle are equal]
∴ ∠BAC = ∠110° [Usmg(1)]
Question 6.
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, If AB = BC, find ∠ECD.
∠CDB = ∠BAC
[Angles in the same segment of a circle are equal]
∠CDB = 30° ………..(1)
∠DBC = 70° ………..(2) (Given)
In ΔBCD,
∠BCD + ∠DBC + ∠CDB = 180°
(Sum of all the angles of a triangle is 180°]
⇒ ∠BCD + 70° + 300 = 180° [Using (1) and (2)1
⇒ ∠BCD + 100° = 180°
∠BCD = 180° – 100°
⇒ ∠BCD = 80° ………(3)
In ΔABC, AB = BC
∠BCA = ∠BAC
[Angles opposite to equal sides of a triangle are equal]
⇒ ∠BCA = 30° ………(4)
∠BAC = 30° (given]
Now, ∠BCD = 80° (From (3))
= ∠BCA + ∠ECD = 80°
⇒ 30°+∠ECD = 80°
⇒ ∠ECD = 80 – 30°
⇒ ∠ECD = 50°
Question 7.
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
In ΔOAB and ΔOCD,
OA = OC [Radii of a circle]
OB = OD [Radii of a circle]
∠AOB = ∠COD [Vertically opposite angles]
∴ ΔOAB = ΔOCD [SAS rule]
∴ AB = CD …….(1) [CPCTI
Similarly, we can show that
AD = CB ……..(2)
Adding (1) and (2), we get
ABCD is a || gm.
(A quadrilateral having opposite sides equal is a parallelogram)
Now, diagonal BD is also a diameter
∴ ∠BAD = \(\frac {1}{2}\)BOD
= \(\frac {1}{2}\) x 180° = 90°
∴ ABCD is the rectangle. (A parallelogram with an angle 90° is a rectangel)
Question 8.
If the non-parallel sides of a trapezium are equal, prove that it is cyclic. Prove that an isosceles trapezium is cyclic.
Given: ABCD is a trapezium whose two non-parallel sides AD and BC are equal.
To Prove: Trapezium ABCD is cyclic.
Construction: Draw BE || AD
Proof: AB || DE [Given]
AD || BE [By construction]
∴ Quadrilateral ABED is a parallelogram.
∴ ∠BAD = ∠BED ……….(1) [Opp. ∠s of a || gm]
and AD = BE [Opp. sides of a gin]
But AD = BC ………(3) [Given]
From (2) and (3),
BE = BC
∴ ∠BEC = ∠BCE ………(4)
[Angles opposite to equal sides]
∠BEC + ∠BED = 1800 [ Linear Pair Axiom]
⇒ ∠BCE + ∠BAD = 1800 [From (4) and (1)]
⇒ Trapezium ABCD is cyclic.
[ If a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.]
Question 9.
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see figure). Prove that
∠ACP = ∠QCD.
∠ACP = ∠ABP …………(1)
[Angles in the same segment of a circle are equal]
And ., ∠QCD = ∠QBD …………(2)
[Angles in the same segment of a circle are equal]
But ∠ABP = ∠QBD [Vertically opposite angles]
∠ACP = ∠QCD. [From (1) and (2)]
Question 10.
If circles are drawn taking two sides of a triangle as diameter, prove that the point of intersection of these circles lies on the third side.
Given: Circles are described with sides AB and AC of a triangle ABC as diameters. They intersect at a point D.
To Prove: D lies on the third side BC of ∠ABC.
Construction: Join AD.
Proof: Circle described on AB as diameter intersects BC in D.
∴ ∠ADB = 90°
[Angle in a semicircle Similarly, the circle described on AC as diameter passes through D.]
∠ADC = 90° ………..(2)
Now, adding (1) and (2) we get
∠ADB + ∠ADC = 180°
∴ Points B, D, C are collinear.
∴ D lies on BC.
Question 11.
ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.
∴ AC is the common hypotenuse of two right triangles ABC and ADC.
∴ ∠ABC = 90° = ∠ADC
Here, there are two cases arising i.e., B and D are either on the same side of AC or on opposite sides.
Case I – Both the triangles are in the same semicircle.
∴ Points A, B, D and C are concyclic.
Case II – Triangles are on opposite sides, then
∠ABC + ∠ADC = 1800
∴ A, B, C and D are concyclic.
Now, in both cases, DC is a chord
∴∠CAD = ∠CBD
[∴ Angles in the same segment are equal]
Question 12.
Prove that a cyclic parallelogram is a rectangle.
Given: ABCD is a cyclic parallelogram.
To prove: ABCD is a rectangle.
Proof: ABCD is a cyclic quadrilateral.
∴ ∠1 + ∠2 = 180° ……….(1)
[∴ Opposite angles of a cyclic quadrilateral are supplementary]
∴ ABCD is a parailelograin.
∴ ∠1 = ∠2 ………(2) [Opp. angles of a || gm]
From (1) and (2),
∠1 = ∠2 = 90°
∴ || gm ABCD is a rectangle.
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Primes and Divisibility | Joe McCann
Integers Made up of Small Pieces
Think back to elementary school, a time when things were simple and you had recess. Despite the most difficult math being times tables, we still said math classes were hard 💀 oh how times have
When learning about division, sometimes when you divided two numbers it wouldn’t go in cleanly and there would be a remainder left over. For example, dividing $7$ by $3$ nets a remainder of $1$, and
we would say that this meant $7$ was not divisible by $3$. But now we are big kids and the days of loose math terminology is over, what does it really mean for a number to be divisible by another?
Definition: Let $a,b\in\mathbb{Z}$. $a$ is divisible by $b$ if we can find a value of $k\in\mathbb{Z}$ such that $a=k\cdot b$. This is notated as $b|a$ or “b divides a”. If $b$ divides $a$ then
$b$ is a factor of $a$.
Clearly every number is a factor of itself, and has $1$ as a factor, which we can prove.
Theorem: Let $a\in\mathbb{Z}$. $1|a$ and $a|a$
Proof Since every number multiplied by $1$ is itself, we can know that $a=1a$ which by definition means that both $1,a$ are factors of $a$.Q.E.D.
However most numbers have other factors too, for example $2|4$, $9|45$, or $25|100$. As such, numbers that only have $1$ and itself as factors are special, and we call them prime.
Definition: An integer $p>1$ such that $p$ is only divisible by $1$ and $p$ is a prime number. A number that is not prime is called composite
You could also rephrase this as: $p>1$ is prime iff for all $x>0$, if $x|p$, then $x=1$ or $x=p$.
Note the requirement $p>1$, which restricts $1$ to not be a prime number. You might be wondering "$1$ feels like it should be a prime number so why isn’t it?" The answer is quite bland in that, if
$1$ was a prime number it would make so many theorems way more annoying to write out and deal with, and including $1$ as a prime number adds pretty much not benefit whatsoever. As such we just say
$1$ is not prime out of convenience lmao.
The sequence of prime numbers is an interesting one $$ 2,3,5,7,11,13,17,\ldots $$ as there is seemingly no pattern at all. Many, many, mathematicians have spent many hours trying to figure out ways
to express the prime numbers in a nice clear formula or expression, but none exists (and probably won’t ever). We will talk about primes more later, but lets discuss a bit more on divisibility.
Properties of Divisibility
Let’s look at the relation^1 $b|a$ when $a,b\in\mathbb{Z}$. We can begin with properties of relations
Theorem: For all $a\in\mathbb{Z}$, $a|a$. This means divisibility is reflexive.
This is actually already proven as before, so we don’t need to do it again.
Theorem: For all $a,b,c\in\mathbb{Z}$, if $a|b$ and $b|c$, then $a|c$. This means divisibility is transitive.
Proof We know that $a|b$ so $b = k_1a$. We also know that $b|c$ so $c=k_2b$. Combining these we get $c=k_1k_2a=ka$ if $k=k_1k_2$ which means that $a|c$.Q.E.D.
Giving an example of this, if we know that $7|35$ and that $35|105$, then that means that $7|105$, which it does!
Theorem: For all $a,b\in\mathbb{Z}$, if $a|b$ and $b|a$, then $a=b$. This means divisibility is anti-symmetric.
I’m evil and will provide this as a homework problem because its actually kinda fun haha.
Now lets move onto two theorems that are actually fairly interesting involving divisibility when you are adding numbers together. The first is fairly uninteresting.
Theorem: If $a|b$ and $a|c$ then $a|b+c$.
Proof We know that $a|b$ so $b = k_1a$. We also know that $a|c$ so $c=k_2a$. Combining these we get $b+c=k_1a+k_2a=(k_1+k_2)a$ which means that $a|b+c$.Q.E.D.
This isn’t very surprising as we can see that $5|25$ and $5|135$ so $5|25+135$ just by pulling out a factor of $5$ with the distributive law. A more interesting version of this theorem though comes
as the following.
Theorem: If $a|b$ and $a\nmid c$ then $a\nmid b+c$.
Proof To prove this we will actually perform a proof by contradiction. We know that $a|b$ so $b=k_1a$, and suppose that $a|b+c$ so $b+c=k_2a$. Plugging in for $b$ we can see that $$ k_1a+c=k_2a\
implies c=k_2a-k_1a=(k_2-k_1)a=ka. $$ This means that $a|c$ which contradicts our claim, meaning that $a\nmid b+c$.Q.E.D.
So if we know that $2|8$ and $2\nmid 13$, then we know that $2\nmid 8+13$. This will actually prove to be fairly valuable for some later proofs!
Brief Aside: Even and Odds
For fun, let us define what it means for a number to be even versus odd. We know that an even number is divisible by $2$ and odd numbers are not, so we can say
Definition: An integer $n$ is even iff there exists a $k\geq 0$ such that $n=2k$. An integer $n$ is odd iff there exists a $k\geq 0$ such that $n=2k+1$.
We can use these definitions for some cute little practice problems
Corollary: The sum of two even numbers is even.
I say this is a corollary because literally by the theorem above, if $2|n_1$ and $2|n_2$ then $2|n_1+n_2$, but lets go through the full effort just for fun.
Proof Since $n_1, n_2$ are even we know $n_1=2k_1$ and $n_2=2k_2$. Then we have $$ n = n_1+n_2 = 2k_1+2k_2 = 2(k_1+k_2) = 2k $$ which is even.Q.E.D.
I have more cute little examples in the practice problem section
Getting Started with Primes
Prime numbers are incredibly captivating to mathematicians because of a super, super important result that we will get to later down the line, however before we get to that, lets show some
interesting properties of prime numbers right now.
Simple (and some not simple) Properties of Primes
Modern number theory is littered with unsolved problems involving prime numbers Here we will show some of the properties of primes that we do in fact know. First lets begin with the question of how
many prime numbers are there? People keep finding very, very large prime numbers so is there a limit at some point? Well this was originally proven in Euclids book Elements, written in $300$ B.C.,
and it was proven to be the following
Euclid’s Second Theorem: There are infinite prime numbers.
Proof We will prove this by contradiction, first begin by assuming there are a finite $n$ number of primes. Let's list them all out $$ p_1,p_2,p_3,\ldots, p_n. $$ We now will consider the number that
is the product of every one of our primes, so $$ P=p_1p_2\ldots p_n. $$ By definition, $P$ is divisible by every prime so for all $j$, $p_j|P$. Notice though that since $2$ is the smallest prime, for
every $j$, $p_j\nmid 1$ which means that by the theorem we proved above that for all $j$ $$ p_j\nmid P+1. $$ Since $P+1$ is not divisible by any prime in our list, and we said our list contains all
the primes, that means that $P+1$ must be prime! But $P+1$ is not in our list which contradicts our assumption and proves our claim.Q.E.D.
This theorem has been proven many, many times in a variety of ways, and is the basis for modern day number theory^2.
A lot of students often mistake this proof to mean that if you take the product of the first few primes and add $1$ that you’ll get a prime number. This is not always true, as in the proof we were
working in a world in which there were finite primes (which there aren’t), however some of them are and these are called primorial primes (the product of the first $n$ primes is called the $n^{\text
{th}}$ primorial). This isn’t that important but wanted to address the common misconception.
Ok, so if there are infinite primes, can we at least say how far out we need to go before we see the next one? I mean, if we have some number $p$ that we know is prime, can we guarantee that the next
prime is at most $10^6$ away (for example)? The answer here is also no.
Theorem: For every integer $N>0$, there exist consecutive primes $p_n, p_{n+1}$ such that $p_{n+1}-p_n\geq N$.
Proof In order to prove this, we will find a sequence of not prime numbers that is $N$ long, such that if you take the first prime before this sequence, and the first prime after this sequence will
have a gap between them bigger than $N$.
Consider the number $(N+1)!=1\cdot 2\cdot 3\cdot \ldots \cdot (N+1)$. We know by definition that for every $2\leq k\leq N+1$ that $k|(N+1)!$. As such by divisibility rules we know $$ 2&|(N+1)!+2 \\ 3
&|(N+1)!+3 \\ 4&|(N+1)!+4 \\ &\vdots \\ N+1&|(N+1)!+(N+1) \\ $$ since this sequence of $N$ numbers are all composite, we know that the gap between the primes before and after this sequence must be $\
geq N$Q.E.D.
This is quite strange, as we can make $N$ as large as we’d like and always find a corresponding gap! $(N+1)$ is quite big, however this is an overestimate as you can show that a gap occurs at the $N^
{\text{th}}$ primorial (try it yourself to prove!). Just to give an example, $6!=720$ so $$ 2|722, 3|723, 4|724, 5|725, 6|726 $$ which means that we know we have a prime gap of at least $5$ here, in
fact $719,727$ are primes which actually have a gap of $8$ between them. The first actual prime gap of $5$ occurs between $23,29$ which means we overestimated a bit LOL. The primorial method predicts
$32,33,34,35,36$ which is much closer to the true solution
Ok so there is no constant bound on prime number gaps, but what if the gap between primes grows over time, can we get an asymptotic value there 👀? In the late 1800s a really interesting theorem
was proven
Bertrand’s Theorem: For every integer $n>1$, there exists a prime number $p$ such that $n < p < 2n$.
The theorem involves some more high powered number theory, so I am not going to put it here, but this means that we actually do have a relatively good window for which we know where the next prime
number will appear! So given some number for example $1023$, we know that there must be a prime number $p$ where $1023 < p < 2046$. This also means that we know that the gap between two prime numbers
must be $\mathcal{O}^T(n)$, and its actually conjectured to be way way better than that.
Practice Problems
Theorem: For all $a,b,c\in\mathbb{Z}$, if $a|b$, then $a|bc$
Theorem: The sum of an odd and an even number is odd
Theorem: The product of two odd numbers is odd
Theorem: Proof that there are infinite primes $p$ such that there exists a $k\in\mathbb{Z}_+$ such that $p=4k+3$
Theorem: Prove that for every value $n\geq 1$, there exists a prime $p$ such that $n^2 < p < (n+1)^2$
1. You thought we were done with relations hahaha. ↩︎
2. Kinda, theres another super important theorem too. ↩︎
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Scholars Workshop and Alumni Celebration Event 2018
We have a brilliant line up for this year’s Scholars and Alumni Celebration event taking place on Saturday 20 January 2018 at Aston University.
• The chance of rain today is 40%. Should I take my umbrella?
• If I were to eat a bacon sandwich every day, my chances of getting bowel cancer would increase by 20%! Thank heavens I’m a vegetarian!
• A test for doping in sports claims to be 95% accurate, so if an athlete tests positive does this mean that they are almost certainly taking a banned substance?
Probability is all around us, but it is not well understood. There’s quite a good chance that you – a mathematically literate person – are not entirely clear exactly what is being claimed in
scenarios like those above. In this workshop, you will explore an innovative approach to teaching probability designed to demystify it and to help students develop conceptual understanding. Learning
is scaffolded by a process which includes:
• asking a question about a scenario which models some aspect of real life
• investigating the model through experiment
• using manipulatives and multiple representations to help students see what their data is telling them
• moving from experimental data (what actually happened) to theory (what do we expect to happen in general) in carefully graded steps
This approach undergirds the new GCSE curriculum for probability, and has been well received in workshops with experienced and new teachers, whether teaching higher achievers or lower achievers.
If you did the probability workshop at least year’s Alumni Celebration Event – don’t worry, this year’s will be different!
Jenny Gage - Formerly a member of the Millennium Mathematics Project and the Nrich team University of Cambridge
jag55@cam.ac.uk to find out more.
Geometry of the Dambusters - Peter Ransom - The Mathematical Association
This workshop will be presented by Flt Sgt ‘Kidnap’ Ransom in 1940’s RAF uniform. It describes the Dambusters raid of 1943 and how mathematics was used to navigate the Lancasters over Germany. Plot
the route on A3 maps, working with scales and bearings then look at how the lights were arranged on the plane using geometrical constructions to find their angles. You will also plot the locus of the
Lancaster over the dam. This is accompanied by many worksheets to take away, CD ROMs of all the materials (and much more) for those present and a taste of rations from the 40’s.
Peter spent over 30 years teaching secondary mathematics in state comprehensives and left the classroom in 2010 to integrate more into family life and the professional associations, to do more
research and to attend conferences without having to worry about missing school time. He takes risks. He has enjoyed mathematics all his life and tries to pass on that enjoyment to all he meets
including his three grandsons. He has been known to travel around the country with a cannonball and sword.
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Cumulative number counts of galaxy clusters
• Thread starter mahya
• Start date
In summary, the speaker is seeking help with plotting cumulative number counts of galaxy clusters as a function of X-ray flux. They are using a method involving mass function and scaling relation,
but the resulting plot does not match observational results. The speaker is unsure why this is happening and asks for assistance with their code.
Dear all,
I am trying to plot the cumulative number counts of galaxy cluster (N>S) as a function of X-ray flux. The method I am following is using the mass function and scaling relation to calculate dN/dVdz.
Then I do a double integrate over the redshift range and flux range. But the plot that I get has no connection with observational results. I am wondering if you could have a look at the codes and
help me wo solve the problem.
How far off are your resuklts from the "observational" results that you are talking about and what are those results, yours and the observational ones? Also I don't under stand why you double
integrated over the redshift range and flux range?
Actually the slope of my plot is totally different from the observation one. In attached images you can compare these two. The reason for integration is that we are supposed to calculate the
cumulative number counts (N>S) and it should be integrated over the redshift range (0,Zmax) and (S, inf.).
FAQ: Cumulative number counts of galaxy clusters
1. What is the significance of studying the cumulative number counts of galaxy clusters?
The cumulative number counts of galaxy clusters provide valuable information about the formation and evolution of the universe. By studying the distribution and growth of these clusters, we can
better understand the large-scale structure of the universe and the processes that drive it.
2. How are cumulative number counts of galaxy clusters measured?
Cumulative number counts are typically measured through observations using telescopes, specifically targeting areas of the sky that are known to contain galaxy clusters. These observations are then
analyzed to determine the number of galaxy clusters in a given area of the sky.
3. What factors can affect the cumulative number counts of galaxy clusters?
There are several factors that can affect the cumulative number counts of galaxy clusters, including the mass and size of the clusters, the age of the universe, and the presence of dark matter.
Additionally, the sensitivity and resolution of the telescopes used for observations can also impact the accuracy of the measurements.
4. How do cumulative number counts of galaxy clusters change over time?
As the universe continues to expand and evolve, the cumulative number counts of galaxy clusters also change. Over time, we can observe an increase in the number of galaxy clusters as new ones form,
but we may also see a decrease as clusters merge together or disappear entirely.
5. What other information can be gleaned from studying cumulative number counts of galaxy clusters?
In addition to understanding the structure and evolution of the universe, studying cumulative number counts of galaxy clusters can also provide insights into the nature of dark energy, the properties
of dark matter, and the impact of cosmic events such as supernovae and black hole mergers. It can also help us refine our understanding of the laws of physics that govern the behavior of the
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Divide in Python: Mastering the Division Operator for Accurate Calculations
Python, being a versatile programming language, not only offers various arithmetic operations, including division, but also necessitates system knowledge such as how to uninstall Python in Linux for
developers managing their programming environment. Division allows you to divide two numbers and obtain their quotient, which can be essential when dealing with mathematical concepts such as at least
meaning in math where understanding the magnitude of numbers is crucial. In this article, we will explore how to perform division in Python, covering both integer division and floating-point
division. Let’s dive in!
Python Division Operators
Python provides two types of division operators: integer division and float division. Understanding the difference between these operators is essential to utilize them effectively in your Python
Integer Division
Integer division is the process of dividing two numbers and obtaining the quotient rounded down to the nearest integer. It is represented by the double forward-slash symbol “//”. Let’s look at an
dividend = 7
divisor = 3
result = dividend // divisor
print(result) # Output: 2
In the above code snippet, we divide 7 by 3 using the integer division operator. The result is 2, which is the largest integer less than or equal to the exact quotient.
Floating-Point Division
Floating-point division is the process of dividing two numbers and obtaining the quotient as a floating-point number. It is represented by the forward-slash symbol “/”. Here’s an example:
dividend = 7
divisor = 3
result = dividend / divisor
print(result) # Output: 2.3333333333333335
n the above example, we perform floating-point division by using the forward-slash operator. The result is a floating-point number, which provides a more precise representation of the quotient.
Python Division Examples
Let’s explore some more examples to solidify our understanding of division in Python.
Integer Division Examples
Dividing 11 by 2 using integer division:
dividend = 11
divisor = 2
result = dividend // divisor
print(result) # Output: 5
Dividing -7 by 3 using integer division:
dividend = -7
divisor = 3
result = dividend // divisor
print(result) # Output: -3
Floating-Point Division Examples
Dividing 7 by 3 using floating-point division:
dividend = 7
divisor = 3
result = dividend / divisor
print(result) # Output: 2.3333333333333335
Dividing -7 by 3 using floating-point division:
dividend = -7
divisor = 3
result = dividend / divisor
print(result) # Output: -2.3333333333333335
Python 2.x vs. Python 3.x Division Operations
It’s important to note that there are differences in division behavior between Python 2.x and Python 3.x.
In Python 2.x, if both operands are integers, the division operator (“/”) performs integer division. For example:
result = 5 / 2
print(result) # Output: 2
To perform float division in Python 2.x, at least one of the operands needs to be a floating-point number. For example:
result = 5.0 / 2
print(result) # Output: 2.5
On the other hand, Python 3.x always performs float division using the division operator (“/”), regardless of the operand types. To perform integer division in Python 3.x, you need to use the integer
division operator (“//”). For example:
result = 5 // 2
print(result) # Output: 2
In this article, we explored the division operators available in Python. We learned about integer division and floating-point division, along with their respective operators. We also saw examples of
how to use these operators in Python code. Furthermore, we discussed the differences between division operations in Python 2.x and Python 3.x.
By understanding the division operators and their behaviors, you can leverage Python’s capabilities to perform precise calculations in your programs. So go ahead and start dividing in Python with
Remember to experiment with different numbers and scenarios to deepen your understanding of division in Python. Happy coding!
How is division used in Python programming for solving mathematical problems?
Division is a fundamental arithmetic operation used in Python programming to solve various mathematical problems. It allows you to divide two numbers and obtain their quotient. By using division,
programmers can perform calculations such as scaling values, calculating averages, finding ratios, and solving equations involving fractions or decimals. Division is a powerful tool in Python that
enables precise mathematical computations.
Are there any considerations for handling division in Python when dealing with different data types?
Yes, when handling division in Python, it’s essential to consider the data types of the operands and the desired result. If both operands are integers and you want to perform integer division, use
the double forward-slash operator (//). This will ensure that the result is rounded down to the nearest integer. On the other hand, if you want to obtain a floating-point result, use the
forward-slash operator (/), even if the operands are integers. Python automatically promotes the division to floating-point when at least one operand is a floating-point number.
What are the main applications of the division operator in Python programming?
The division operator in Python has various applications in programming. Some of the main applications include:
Calculating proportions and ratios: Division allows you to find the ratio between two quantities or calculate the proportion of one value to another.
Scaling values: Division is used to scale or normalize values by dividing them by a certain factor or reference value.
Calculating averages: Division is used to compute averages by dividing the sum of a set of values by the total number of values.
Performing mathematical operations: Division is an essential component in complex mathematical operations, such as solving equations, calculating derivatives, or performing statistical calculations.
How does Python handle division in comparison to other programming languages?
Python handles division differently compared to some other programming languages, particularly between Python 2.x and Python 3.x. In Python 2.x, if both operands are integers, the division operator
(“/”) performs integer division, returning the floor of the quotient. In contrast, Python 3.x always performs floating-point division using the division operator (“/”), regardless of the operand
types. Python 3.x introduced the integer division operator (“//”) to explicitly perform integer division. These differences in division behavior between Python versions are important to consider when
writing portable code or transitioning between Python 2.x and Python 3.x.
What are the potential challenges or pitfalls when using division in Python programming?
When using division in Python programming, there are a few potential challenges or pitfalls to be aware of:
Rounding errors: Floating-point division may produce imprecise results due to the limitations of floating-point representation and arithmetic. It can result in rounding errors or inaccuracies in the
least significant digits of the result. This is a common challenge when working with decimal values or performing extensive calculations.
Division by zero: Division by zero is not allowed in mathematics, and Python raises a ZeroDivisionError if you attempt to divide a number by zero. It’s important to handle this exception
appropriately in your code to avoid program crashes or incorrect results.
Data type promotion: Python automatically promotes division to floating-point when at least one operand is a floating-point number. This behavior can sometimes lead to unexpected results or precision
loss if you expect integer division. Be mindful of the operand types and the desired result when performing division operations.
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Market integration among foreign exchange rate movements in central and eastern European countries
This study focuses on the level of interdependence across the Central and Eastern European (CEE) foreign exchange markets (Hungary, Poland, the Czech Republic, Romania and Croatia) from September
2008 to September 2017, using the return spillover measure proposed by Diebold and Yilmaz (2009; 2012). We mainly find a bidirectional volatility spillover among these assets and the cross-market
linkages in the CEE region have become stronger over time. Furthermore, the Czech exchange market has a significant influence on the rest of the foreign exchange markets. The total spillover remained
very high over the periods 2010–2012 and 2015–2017, despite the noteworthy fluctuations in other periods. These results would also be useful for portfolio managers, policy makers and speculative
traders to develop exploitable strategies, by providing knowledge of the transmission mechanisms of the volatility of foreign exchange markets. The results may support the distribution of assets in a
financial portfolio, especially after financial integration.
1 Introduction
The growing role of Central and Eastern European (CEE) financial markets in the world economy has recently attracted much attention to the issue of volatility shock spillovers within the region and
beyond (e.g., Bubák et al. 2011; Hung 2018; Hung 2019). Return spillovers are prominent determinants for many market participants, they influence decisions to hedge open foreign exchange positions on
the foreign exchange market (Baruník et al. 2017). As per Kanas (2000), volatility transmission might increase the nonsystematic risk which diminishes gains from international portfolio
diversification. Specifically, the analysis of interdependence in foreign exchange markets, and its volatility spillovers, has been a subject of a considerable volume of research (Baruník et al. 2017
). In this article, we attempt to evaluate the degree of market interdependence and innovation transmission between foreign exchange markets of five selected countries (Hungary, the Czech Republic,
Poland, Romania and Croatia) over the period of 2008–2017, after the global financial crisis. By doing so, the purpose of this research is to provide new evidence on return spillovers by examining
the spillover index on the selection of five national foreign exchange markets. The size and scope of this research allows us to take into account the intra-regional impact of return spillovers,
rather than the global impact of volatility transmission.
Recent literature has centered on the correlations among forex markets of developed countries (Baruník et al. 2017; McMillan – Speight 2010; Wang-Yang 2009). These papers show that developed forex
markets are interconnected and there is bidirectional volatility spillover among them. In the last 20 years, the significant role of exchange rate policies for economic development has remained a
controversial issue. According to Guzman et al. (2018), there are two central and interrelated problems in connection with exchange rate policies in the macroeconomic literature on emerging
economies. Namely, the role the exchange rate plays in facilitating economic growth, and how the exchange rate regime and capital account management help manage cyclical swings in external financing
and terms of trade fluctuation. Briefly, pursuing a stable and competitive exchange rate can promote economic development, and this requires flexible and sustained interventions. The recent academic
literature regarding spillovers of volatility across foreign exchange markets largely began with Engle et al. (1990), and was further considered by, for instance, McMillan – Speight (2010), Hong
(2001), Patton (2006), Herwartz – Reimers (2002).
An important issue that has emerged in the empirical literature associated with modelling financial spillovers is the means by which volatility is specified. Typically, a traditional approach related
to GARCH-type models is used. For example, Hung (2018) employed a multivariate EGARCH model to analyze the volatility spillover effects of foreign exchange rates in the CEE markets and provides
evidence of that foreign exchange markets have become more independent after the global crisis. Kumar et al. (2016) examined the volatility in foreign exchange markets of India and China using daily
data covering the period between 2006 and 2015. Based on the framework of the EGARCH model, the findings capture the effect of good and bad news and provide evidence of bidirectional volatility
between these markets. Herwartz – Reimers (2002) estimated the properties of the DEM/USD and DEM/JPK rates with a sample period from 1975 to 1998 using the GARCH model, and provided strong evidence
of persistence in volatility and its serial correlation. Similarly, Pankova et al. (2010) analyzed the volatility and asymmetry of the EUR/USD exchange rate, using daily data from June 2008 to May
2010 under GARCH(1,1) and EGARCH(1,1). They confirmed that there is no asymmetry in the EUR-USD relation. McMillan – Speight (2010) focused on the interdependence of return and volatility spillovers
in three Euro exchange rates including the US dollar, the Japanese yen and the British pound, using the realized variance method. The findings report substantial evidence of contemporaneous
connectedness between the returns on these rates and their volatility. More importantly, the spillover index is employed in this study to examine the degree of cross-market spillover.
In the volatility spillover literature, departing from the common econometric methods, Diebold – Yilmaz (2012) provide new measures of return and volatility spillovers of international equity markets
based on forecast-error variance decompositions in a vector autoregressive framework. The analysis of spillovers in Forex markets has widely adopted the novel approaches developed by Diebold and
Yilmaz. For example, Salisu – Ayinde (2018) employ Diebold and Yilmaz's approach to test for spillovers in the Nigerian naira's exchange rates. The findings reveal that the electioneering process in
Nigeria appears to have had greater spillover effects on the naira than the global financial crisis. This result is strong to alternative measures of exchange rates. Baruník et al. (2017) provided
evidence for asymmetric volatility interrelatedness on the Forex market using high-frequency, intra-day data of the most actively traded currencies. They showed that negative spillovers are due to
bad rather than good volatility, by applying Diebold and Yilmaz spillover index. In the same vein, Greenwood-Nimmo et al. (2016) found that aggregate spillover intensity is countercyclical with
respect to the federal funds rate and increased in periods of financial stress.
In the Central and Eastern European context, the analysis of volatility spillovers of foreign exchange markets as well as their interrelatedness has been recognized in the literature (Bubak 2009;
Fidrmuc – Horvath 2008; Hsing 2016; Kobor – Szekely 2004; Kocenda – Valachy 2006; Kumar – Kamaiah 2014; Petrica – Stancu 2017). Nevertheless, most of these studies also employ the common econometric
methodologies (GARCH-type models) to capture the dynamics of returns and volatility spillovers across Forex markets. In this paper, dynamic volatility interrelatedness and directional
characterization of return spillovers among the Forex markets in the CEE countries are perfectly captured by the spillover index proposed by Diebold – Yilmaz (2009; 2012). There has not been evidence
of the dynamics of return spillover effects related to the Forex markets in the CEE countries using the spillover index so far. In our study, therefore, we cover this issue and provide a primary
contribution. Namely, we employ the Diebold and Yilmaz methodology to analyze the return spillovers among CEE currencies. The economic contribution of this empirical analysis is that we quantify the
directional return spillover effects that a particular exchange market receives from the other markets. These spillovers effects are informative of the price discovery in a particular market, which
might then be transmitted to other markets owning to cross-listing of firms and fund investments. Finally, we shed light on a richer extent of time-varying detail than is found in previous studies.
This paper proceeds as follows. We outline the methodology and data in Section 2. In Section 3, we present the main findings for total and directional connectedness. Finally, we conclude in Section 4
2 Methodology and data
2.1 Vector autoregression (VAR)
As a starting point, we present the concept and measure of the spillover index from Diebold – Yilmaz (2009, 2012). Taking into account a covariance stationary Vector Autoregression (VAR) model of
order P and N variables, $xi=∑i=1pϕixt−i+εi$, where $ε∼(0, ∑)$ is a vector of independent and identically distributed (IID) distances. We can turn the VAR into a moving average (MA) representation,
that is, $xt=∑i=0∞Aiεt−i$ where N × N coefficient matrix A[i] is obtained by the recursive substitution, $Ai=ϕ1Ai−1+ϕ2Ai−2+⋯+ϕpAi−p$, with A[0] = I[n], which is an identity matrix of order n, and A[i
] = 0 for i < 0. The MA presentation can be employed to forecast the future with the H-step-ahead. We rely on variance decompositions to determine the fraction of the H-step-ahead error variance in
forecasting x[i] that are due to shocks to x[i] and the fraction that is due to shocks to x[j], $∀i≠j$, for each i. Furthermore, we use the KPPS (Koop et al. 1996) generalized forecast error variance
decomposition (GFEVD) to develop spillover indices.
2.2 Variance shares
Diebold – Yilmaz (2012)
define “own variance shares” as the fraction of the H-step-ahead error variances in forecasting
due to shocks to
$i=1, N¯$
, and “cross variance shares” (spillovers) as the fraction of the H-step-ahead error variances in forecast
, for
$i, j=1, N¯$
such that
. The KPPS H-step-ahead forecast error variance decomposition is
where Ʃ is the variance matrix of the error vector,
is the standard deviation of the error term for the ith equation, and e
is the selection vector with 1 as the ith elements, and 0 otherwise. According to the properties of generalized VAR, we have
. Each entry of the variance decomposition matrix is normalized by the row sum as
2.3 Total and directional volatility spillovers
We can construct a total volatility spillover index proposed by
Diebold – Yilmaz (2012)
as follows:
The total spillover index is used to measure the contribution of spillovers across all five foreign exchange rates to the total forecast error variance. As the KPPS framework solves the ordering
problem, we can measure the directional volatility spillovers received by market
from all other markets
Similarly, we can calculate the directional spillovers transmitted by market
to all other markets
We can also obtain the net volatility spillover for each foreign exchange market by calculating the difference between (5) and (4) as:
The net volatility spillover is simply the difference between the gross volatility shocks transmitted to and those received from all other markets (Diebold – Yilmaz 2012).
2.4 Net pairwise spillovers
The net pairwise spillover of volatility between markets
is the difference between gross volatility shocks transmitted from market
to market
and those transmitted from
, which
Diebold – Yilmaz (2012)
define as:
2.5 Data
The dataset used in this study contains time series of US dollar exchange rates with daily frequency for a sample of five countries located in Central and Eastern European region: Hungary, Poland,
Czech Republic, Romania and Croatia. The sample period spans from September 2008 to September 2017. The exchange rate series are obtained from the Bloomberg database and represent the amount of US
dollars per one unit of local currency. They are detailed in Table 1. The number of observations across the market is 2,368, which is less than the total number of observations because the joint
modeling of five markets requires matching returns. The daily return data series are calculated as R[t] = 100 × ln(P[t]/P[t−1]), where P[t] is the price level of the market at time t. The logarithmic
stock returns are multiplied by 100 to approximate percentage changes and avoid convergence problems in estimation.
Table 1.
Foreign exchange market indices
Country Code
Hungary USDHUF
Poland USDPLN
Czech USDCZK
Romania USDRON
Croatia USDHRK
2.6 An overview of CEE foreign exchange markets
The evolution of exchange rates currently illustrates a significant source of concern from both a micro and a macroeconomic perspective. The exchange rate is one of the most synthetic prices in an
economy and it is also the expression of a general equilibrium among the market for real goods and services, the currency market and capital market, which has the apparent potential of affecting the
general economic equilibrium in any economy. The behavior of the exchange rate is influenced, by the rate of economic growth, the changes in the general level of prices, the industrial structure of
the economy, the country's level of international competitiveness and its degree of trade and financial openness, political stability and government's ability to deal with internal crises. This
diversity of determinants impacting directly or indirectly on the exchange rate raises the issue of the ease of managing such a complex and dynamic macroeconomic variable.
Over the past several decades, the number of countries running de jure floating exchange rate regimes has steadily grown. Some papers (Bubak 2009; Hsing 2016; Kumar – Kamaiah 2014) show that there is
a discrepancy between de jure and de facto, and countries appear to actively restrict fluctuations in the external value of their national currencies (Table 2).
Table 2.
Exchange rate regimes in five CEE countries, 2008–2014
Croatia 8 4 6 6 6 6 6
Czech Republic 10 10 10 10 10 8 4
Hungary 9 9 9 9 9 9 9
Poland 10 10 10 10 10 10 10
Romania 9 9 9 9 9 9 9
Mean 9.2 8.4 8.8 8.8 8.8 8.4 7.6
Standard Deviation 0.83 2.50 1.64 1.64 1.64 1.51 2.50
Notes: 4 = Stabilized arrangement; 6 = Craw-like arrangement; 7 = Managed floating with no pre-determined path for the exchange rate; 8 = Other managed arrangement; 9 = Floating; 10 = Free floating.
Source: IMF's Annual Reports on Exchange Arrangements and Exchange Restrictions, various years.
The diversity in the exchange rate regime choices also reflects different stabilization strategies and the availability of alternative monetary policy frameworks. Achieving price stability remains
the main stabilization task. The exchange rate regimes of the former Communist countries in the region are quite diverse, ranging from stabilized arrangements to free floating. This diversity can be
explained by the structural diversity of these countries, on the one hand, and by the need felt by these countries to better control inflation and exchange rates at the same time. In general, there
are some substantial differences for some countries and some years. We observed the volatility of exchange rates of these currencies against the US dollar over the same period (Fig. 1).
Figure 1.
Exchange rate against the US dollar of five CEE currencies, 2008–2017
Source: author.
Citation: Society and Economy SOCEC 42, 1; 10.1556/204.2020.00001
We plot the five foreign exchange market's volatility in Fig. 2 and provide summary statistics of log returns in Table 3. Figure 2 sheds light on volatility trends over the sample period. We can note
that foreign exchange markets of almost all countries under consideration reacted significantly to the global financial crisis of 2008, as well as the European stock market collapse of 2015. Other
sharp declines in exchange rate changes can mainly be attributed to country-specific events when we take a close look at the patterns of changes in the logarithmic returns. Overall, most markets
exhibit large volatility and the volatility dynamics appear to be highly persistent.
Figure 2.
Daily volatility of five CEE foreign exchange markets
Source: author.
Citation: Society and Economy SOCEC 42, 1; 10.1556/204.2020.00001
Table 3.
Descriptive statistics and unit root tests of log returns
USDCZK USDHRK USDHUF USDPLN USDRON
Mean 2.7% 2.725% 4.92% 5.05% 5.5%
Std. Dev 13.27% 10.78% 16.19% 13.36% 12.10%
Skewness 0.0837 −0.0606 0.1626 0.2060 0.1576
Kurtosis 6.4679 5.7339 6.6097 6.1201 6.4208
Jarque-Bera 1,189.89^* 739.23^* 1,296.64^* 977.75^* 1,164.90^*
PP test −48.77^* −48.73^* −48.38^* −48.28^* −46.49^*
ADF test −48.76^* 48.72^* −48.37^* −48.21^* −46.54^*
Source: author.
Notes: Exchange rates are expressed as units of currencies per unit of US dollar. *denotes significance at the 1 per cent level. All returns are expressed in percentages. ADF and PP test represent
the augmented Dickey and Fuller test and the Phillips-Perron test of stationarity respectively. Means and standard deviations are annualized as r × 250 and $σ×250$ (Chevallier et al. 2018). The
Jarque–Bera test examines the null hypothesis of normality of log returns.
Table 3 presents the descriptive statistics for log daily returns of sample markets over the study period. Romania had the highest average return (5.5%) and the Czech Republic realized the lowest
return (2.7%). Volatility of the CEE exchange markets, measured by standard deviation, is generally high, and range from 10.78 (Croatia) to 16.19 (Hungary). It is clearly observed that neither of the
return series has normal distribution, with respect to the Jarque-Bera test for normality. Finally, the augmented Dickey-Fuller (ADF) test provides evidence to support the hypothesis of stationarity
for all return series at the 1% level, ensuring their suitability for further statistical analysis.
3 Results
This section documents the empirical estimation results of our model using the generalized spillovers index methodology to analyze the magnitude and directions of return spillovers of foreign
exchange rate movements among the CEE countries. To introduce a brief overview of the direction of spillovers, we begin with a full sample analysis, which highlights some stylized facts about the
characteristics of the dynamic linkages in the region. We can then answer the question of whether cross-market interconnectedness becomes stronger over time. Finally, we carry out robustness checks
for the estimated spillover measures as well as pointing out possible limitations to their practical application.
3.1 Analysis of spillovers using the full sample
Table 4 reports the calculation of $θijg(H)$ in Eq. (2) with full sample volatility spillovers. The ijth entry is the estimated contribution to the forecast error variance of market i resulting from
shocks to market j. Spillovers transmitted by market i to market j are depicted by numbers in the columns, excluding the contribution to its own variance (the number in the diagonal) which represents
own market innovations. The numbers (excluding the diagonal) for market i (across a row in the table) illustrate spillover received, or innovations resulting from innovation to market j. The
off-diagonal column sums, labeled as “Transmitted,” and the row sums, labeled as “Received,” describe the total spillovers contributed towards and received from other markets. The difference between
the spillovers originating from the market and the spillovers received from other markets is the net volatility spillovers for each market. Due to row normalization according to Eq. (2), the sum of
the variances in a row equals 100%, while the column sum does not. The total volatility spillover index, which appears in the lower right-hand corner of the table, is calculated as the sum of all
variance in the 5 × 5 matrix less the sum of the diagonal variances, summarizing the degree to which shocks are attributable to spillovers for the entire sample expressed as a percentage.
Table 4.
Return spillover table
USDHUF USDPLN USDCZK USDRON USDHRK Received
USDHUF 27.38 20.51 18.60 17.19 16.31 72.6
USDPLN 20.45 27.50 19.05 17.03 15.95 72.5
USDCZK 18.06 18.48 26.46 18.38 18.62 73.5
USDRON 17.11 16.88 18.83 27.15 20.03 72.8
USDHRK 16.48 16.08 19.46 20.38 27.61 72.4
Transmitted 72.1 72.0 75.9 73.0 70.9 363.9
Including own 99.5 99.5 102.4 100.1 98.5 Total spillover index: 72.8%
Net spillover −0.5 −0.5 0.6 0.2 −1.5 0
It is clear from Table 4 that we find that approximately 72.8% of shocks were owning to spillovers, which indicates that, on average, across our entire sample, 72.8% of the volatility forecast error
variance in all five markets comes from spillovers. Let us consider what we learn from the rest of the table about directional spillovers. We can see that the receivers of volatility to other markets
are not very different in magnitude. For example, the gross directional volatility spillovers from others (receivers) to the Czech Republic are highest, at 73.5%, followed by the Romania, with the
spillovers from others explaining 72.8% of the forecast error variance, while the figures for Hungary, Poland and Croatia are slightly lower, standing at 72.6%, 72.5%, and 72.4% respectively. When
considering the return spillover transmitted by each country separately, we find that the biggest transmitters of volatility to other markets are the Czech Republic (75.9%), Romania (73%) and Hungary
(72.1%), while the corresponding data for Poland and the Czech Republic is 70.9% and 72%, respectively. As for the net directional return spillovers, we can observe the last row of the table, which
shows that the largest are from the Czech Republic to others (0.6) and from others to Croatia (−1.5).
Briefly, the results are robust to methods of row and column normalization. Higher volatility spillover with other markets includes the Czech Republic, Hungary and Romania, while the Polish and
Croatian exchange markets are the most closed in our sample.
3.2 Analysis of spillovers using a dynamic rolling sample
We illustrate the dynamics of interrelatedness in terms of the total return spillovers among the five exchange markets during the sample period in Fig. 3, which depicts the evolution of the spillover
index over the period 2008–2017. Figure 3 also shows that return spillover is subject to instability, and experienced some remarkable peaks, particularly during, and in the aftermath of European
stock market collapse of 2015. Obviously, many changes took place during the years in our sample.
Figure 3.
Total return spillover index, five asset classes
Source: author.
Citation: Society and Economy SOCEC 42, 1; 10.1556/204.2020.00001
We highlight some major events in the figure. First, from 2009 to 2011, the spillover index rose from 70% to 76% in the first window, implying that during high volatility periods, there was strong
information spillover because of the transmission of tradeable information across these markets resulting in price movements and the contagion of high uncertain market conditions across the national
markets. Second, the period of the highest volatility correlations of these exchange markets was from 2011 to 2012. The spillover reached its climax between 2011 and 2012, then decreased gradually
from 2012 to 2014. The lowest number was around 65% during the European stock market collapse (Blundell–Wignall 2012). Third, the spillover index dramatically increased over the period 2015–2017 and
stabilized thereafter.
Overall, the total spillover index fluctuates steadily during the study period, showing increased connectedness among CEE foreign exchange markets. This conclusion is based on Fig. 3 using a VAR(2)
model with 200-day rolling samples fluctuating between 64% and 76% of error variance. This result reveals strong evidence of increased financial market interdependence. Spillover cycles are
persistent, and high total spillover periods suggest that in highly volatile periods, investors might have to take into consideration other asset classes, as diversification might not be found in
other international exchange markets. On the other hand, the flow of tradeable information has a substantial influence on the spillover index, and is a more significant driver during non-high
volatility periods. This finding is consistent with Hung (2018) and Fidrmuc – Horvath (2008). We examine this issue further by taking into account spillover directions in the next section.
3.3 Directional spillover analyses
We now turn to the analysis of directional spillovers from foreign exchange markets. Figure 4 describes the directional volatility spillovers from each of the five asset classes to others
(transmitted) for a sample of countries, Hungary, Poland, the Czech Republic, Romania and Croatia. They vary overtime slightly. During peaceful times, the spillovers from each exchange market are
below 60%, but at volatile times, the directional spillovers increase by close to 15%. It is crucial to note that the results of directional spillovers are extremely similar. Overall, the gross
return spillovers of foreign exchange markets from Hungary and Poland are generally smaller than the spillovers from the other three markets.
In Fig. 5, we present the directional volatility spillovers from the others to each of the five foreign exchange markets (received). As we can see from the plot, the directional spillovers to others
vary significantly over time. Nevertheless, the relative variation pattern is reversed, with directional volatility spillovers to Hungary and Poland increasing relatively more in the periods of
In summary, based on Figs 4 and 5, each of five exchange markets exhibit more volatility in the index of spillovers received compared to spillovers transmitted.
Figure 6 gives interesting insights based on the dynamic patterns that show each currency's net position in terms of the volatility spillovers transmitted or received. The positive domain contains
net spillovers that a currency transmits to other currencies. The net spillovers in the negative domain illustrate the situation when a specific currency receives net volatility spillover from
others. The extent of spillover transmission among currencies is uniform. Nevertheless, the evidence shown in Fig. 6 describes quite the opposite. Two currencies relationships can be characterized by
opposite extreme net positions: USD-HUF is a net volatility spillover receiver and USD-CZK is a net spillover transmitter; short periods when low net spillovers are in the opposite domains are the
exception. Furthermore, Romania and Croatia clearly transmit more spillovers during most of the researched period. The Polish Zloty in Fig. 6 alternates between being transmitters or receivers,
depending on the time.
Figure 7 includes net pairwise spillover plots, which provide a detailed analysis of the return spillovers. First, the bulk of return spillovers from USD-HUF were transmitted first to USD-PLN, second
to USD-CZK, third to USD-RON, then to USD-HRK. During this episode, the USD-HUF was a net receiver of volatility from other markets. However, in the case of HUF-PLN, the Hungarian currency was a net
transmitter of volatility to another market in the periods of 2015–2017. The fact that the USD-HUF exchange rate was at the same time a net receiver from other exchange markets pointed out the
connectedness between USD-HUF and USD-RON, USD-PLN, USD-CZK, USD-HRK. Second, return spillovers from USD-PLN were transmitted first to USD-CZK, USD-RON, then to USD-HRK. Overall, during the end of
the research period, the USD-PLN rate was a net receiver of volatility from other markets, while a slight transmitter of volatility to other markets in the periods of 2011–2014. This simply meant
that there was a weak relationship between the Polish exchange market and other markets in this region. Third, return spillovers from USD-CZK were transmitted to USD-RON and USD-HRK. Figure 7
highlights that the USD-CZK exchange rate was a net transmitter of volatility to other markets, with the exception of the period between 2013 and 2015. We can conclude that the exchange market in the
Czech Republic has a dramatic influence on other exchange markets in CEE region. Finally, we consider the episode of return spillovers from the USD-RON rate were transmitted to USD-HRK. Throughout
2009–2014, the spillovers went in the direction of USD-RON, whereas between 2009–2010 and 2015–2017, USD-RON was a net transmitter of volatility to USD-HRK.
All in all, we highlight a complete picture of the return spillovers among the considered markets as well as their directional interrelatedness over the sample. As expected, the total spillover
effects intensified during the first and last years of the research period. A close inspection of the graphs in Fig. 7 demonstrates that the integration of the foreign exchange markets is
The stylized facts confirm previous studies. For example, Kumar – Kamaiah (2014) attempted to analyze the deterministic presence of chaos in the Forex markets in Bulgaria, Croatia, Czech Republic,
Hungary, Poland, Romania, Russia, Slovakia and Slovenia. They confirmed that the Forex markets exhibited deterministic chaotic behavior. By contrast, Bubak (2009) relied on model-free nonparametric
measures of ex-post volatility, and documented that daily returns on the EUR-CZK, EUR-HUF and EUR-PLN exchange rates are independent over time. Fidrmuc – Horvath (2008) reported asymmetric effects of
the volatility of exchange rates in new EU members states including Czech Republic, Hungary, Poland, Romania and Slovakia. This result is also supported by Kocenda – Valachy (2006).
3.4 Robustness
In order to test for robustness, we slightly modify our baseline model to evaluate the sensitivity of our time-varying volatility spillover results. First, we calculate the dynamic indices for orders
2 through 6 of VAR, and plot the minimum, maximum and median values of the related estimations which are obtained in Fig. 8. We do not detect a significant distinction among these time-varying
estimation results. Then, we examine the results for forecast horizons varying from 1 to 7 months at a 40 month rolling window VAR analysis. As we can see in the Fig. 9, the dynamic total spillover
plot is not sensitive to the choice of the order and the forecast horizon of the VAR model. We can conclude that our results are robust to the above variations.
Figure 8.
Total return spillovers to VAR lag structure
Source: author.
Citation: Society and Economy SOCEC 42, 1; 10.1556/204.2020.00001
Figure 9.
Total return spillovers to forecast horizon
Source: author.
Citation: Society and Economy SOCEC 42, 1; 10.1556/204.2020.00001
4 Conclusions
In this article, we employed the spillover index developed and extended by Diebold – Yilmax (2009; 2012), based on the forecast error variance decomposition analysis to better fit the assessment of
volatility spillovers on foreign exchange markets. Applying daily data over 2008–2017, we used the method on a set of the most actively traded currencies quoted against the US dollar, including the
USD-HUF, USD-PLN, USD-CZK, USD-RON and USD-HRK exchange rates. Based on the analysis of the data, we provided a wealth of detailed estimations.
The results of our spillover analysis provide several straightforward insights about the level and the dynamics of foreign exchange rate movements in the Central and Eastern European region over the
past 10 years. The overall picture is that we find the existence of bidirectional volatility spillovers between the selected countries in the CEE region. More precisely, cross-market connectedness in
the CEE region is time-varying and has become stronger over time, with large increases in the level of shock spillover effects after the European stock collapse in 2015 towards the end of our sample,
in comparison with the beginning of the research period. Finally, the Czech Republic has not only significant spillover with markets of close geographical proximity, but also has a dramatic influence
on the rest of the foreign exchange markets. The total spillover remained very high during the 2010–2012 and 2015–2017 periods, despite the noteworthy fluctuations in other periods, while the lowest
volatility index was for 2014.
These empirical results are useful for portfolio managers, policy makers and speculative traders in terms of developing exploitable strategies, providing them with knowledge of transmission
mechanisms of the volatility of foreign exchange markets in the CEE region. It might be beneficial to distribute assets in a financial portfolio, especially after greater financial integration.
• Baruník, J. – Kočenda, E. – Vácha, L. (2017): Asymmetric Volatility Connectedness on the Forex Market. Journal of International Money and Finance 77: 39–56.
• Blundell-Wignall, A. (2012): Solving the Financial and Sovereign Debt Crisis in Europe. OECD Journal: Financial Market Trends 2011(2): 201–224.
• Bubak, V. (2009): Distribution and Dynamics of Central-European Exchange Rates: Evidence from Intraday Data. Finance a Uver: Czech Journal of Economics & Finance 59(4): 334–359.
• Bubák, V. – Kočenda, E. – Žikeš, F. (2011): Volatility Transmission in Emerging European Foreign Exchange Markets. Journal of Banking & Finance 35(11): 2829–2841.
• Chevallier, J. – Nguyen, D. K. –Siverskog, J. –Uddin, G. S. (2018): Market Integration and Financial Linkages Among Stock Markets in Pacific Basin Countries. Journal of Empirical Finance 46: 77–
• Diebold, F. X. – Yilmaz, K. (2009): Measuring Financial Asset Return and Volatility Spillovers, with Application to Global Equity Markets. The Economic Journal 119(534): 158–171.
• Diebold, F. X. – Yilmaz, K. (2012): Better to Give than to Receive: Predictive Directional Measurement of Volatility Spillovers. International Journal of Forecasting 28(1): 57–66.
• Engle, R. F. – Ito, T. T. – Lin, W.L. (1990): Meteor Showers or Heat Waves? Heteroskedastic Intra-Daily Volatility in the Foreign Exchange Market. Econometrica 58(3): 525.
• Fidrmuc, J. – Horváth, R. (2008): Volatility of Exchange Rates in Selected New EU Members: Evidence from Daily Data. Economic Systems 32(1): 103–118.
• Greenwood-Nimmo, M. – Nguyen, V. H. – Rafferty, B. (2016): Risk and Return Spillovers Among the G10 Currencies. Journal of Financial Markets 31: 43–62.
• Guzman, M. – Ocampo, J. A. – Stiglitz, J. E. (2018): Real Exchange rate Policies for Economic Development. World Development 110: 51–62.
• Hsing, Y. (2016): Determinants of the Hungarian Forint/US Dollar Exchange Rate. Theoretical and Applied Economics 23(1): 163–170.
• Herwartz, H. – Reimers, H. E. (2002): Empirical Modelling of the DEM/USD and DEM/JPY Foreign Exchange Rate: Structural Shifts in GARCH‐Models and their Implications. Applied Stochastic Models in
Business and Industry 18(1): 3–22.
• Hong, Y. (2001): A Test for Volatility Spillover with Application to Exchange Rates. Journal of Econometrics 103(1–2): 183–224.
• Hung, N. T. (2018): Volatility Behaviour of the Foreign Exchange Rate and Transmission Among Central and Eastern European countries: Evidence from the EGARCH Model. Global Business Review https:/
• Hung, N. T. (2019): An Analysis of CEE Equity Market Integration and their Volatility Spillover Effects. European Journal of Management and Business Economics https://doi.org/10.1108/
• Kanas, A. (2000): Volatility Spillovers Between Stock Returns and Exchange Rate Changes: International Evidence. Journal of Business Finance & Accounting 27(3–4): 447–467.
• Kóbor, Á. – Székely, I. P. (2004): Foreign Exchange Market Volatility in EU Accession Countries in the Run-Up to Euro Adoption: Weathering Uncharted Waters. Economic Systems 28(4): 337–352.
• Kočenda, E. – Valachy, J. (2006): Exchange Rate Volatility and Regime Change: A Visegrad Comparison. Journal of Comparative Economics 34(4): 727–753.
• Koop, G. – Pesaran, M. H. – Potter, S. M. (1996): Impulse Response Analysis in Nonlinear Multivariate Models. Journal of Econometrics 74(1): 119–147.
• Kumar, A. S. – Kamaiah, B. (2014): On Chaotic Nature of the Emerging European Forex Markets. Romanian Economic Journal 17(53): 25–40.
• Kumar, S. – Dublish, P. – Haque, M. M. (2016): Volatility Spillovers Between Foreign Exchange Markets of India and China. Asia-Pacific Journal of Management Research and Innovation 12(2): 134–144
• McMillan, D. G. – Speight, A. E. (2010): Return and Volatility Spillovers in Three Euro Exchange Rates. Journal of Economics and Business 62(2): 79–93.
• Pánková, V. – Cihelková, E. – Hušek, R. (2010): Analysis and Forecasting the Volatility of Euro-Dollar Exchange Rate. Croatian Operational Research Review 1(1): 221–226.
• Patton, A. J. (2006): Modelling Asymmetric Exchange Rate Dependence. International Economic Review 47(2): 527–556.
• Petrică, A. C. – Stancu, S. (2017): Empirical Results of Modeling EUR/RON Exchange Rate Using ARCH, GARCH, EGARCH, TARCH and PARCH Models. Romanian Statistical Review 1: 57–72.
• Salisu, A. – Ayinde, T. (2018): Testing for Spillovers in Naira Exchange Rates: The Role of Electioneering & Global Financial Crisis. Borsa Istanbul Review 18(4): 341–348.
• Wang, J. – Yang, M. (2009): Asymmetric Volatility in the Foreign Exchange Markets. Journal of International Financial Markets, Institutions and Money 19(4): 597–615.
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The lowest bar
What do you think is the lowest bar for a person to have net-positive HU externalities, compared to their nonexistence?
Assume a person in the developed world, and count only the externalities, e.g. assume they are a p-zombie for sake of analysis.
One would naturally assume that most EAs are quite a bit above that bar, but what about average citizens? And what would you look at if you wanted to find out the minimal standard, e.g. if you could
decide to make another such p-zombie for free, or save one from an illness etc., for free.
Basically a kind of zero-point externality benchmark.
"The abolishment of pain in surgery is a chimera. It is absurd to go on seeking it... Knife and pain are two words in surgery that must forever be associated in the consciousness of the patient."
- Dr. Alfred Velpeau (1839), French surgeon
Re: The lowest bar
Maybe a good starting point would be to figure out what a person's externalities already are. The biggest I can think of would be meat eating and environmental impact from e.g., driving a car.
Felicifia Head Admin | Ruling Felicifia with an iron fist since 2012.
Personal Site: www.peterhurford.com
Utilitarian Blog: Everyday Utilitarian
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Re: The lowest bar
peterhurford wrote:Maybe a good starting point would be to figure out what a person's externalities already are. The biggest I can think of would be meat eating and environmental impact from
e.g., driving a car.
Yes, those were on my list also. Additionally, economic and political externalities are plausible.
By the way, if we hold WAS pessimism, the average environmental imact doesn't have to be negative, but could be positive from replacement. Similarly, economic acitivity doesn't have to be positive,
it could have negative flow-through effects. It obviously depends on how strongly we think current human econcomies are correlated with future utility (positive and negative).
"The abolishment of pain in surgery is a chimera. It is absurd to go on seeking it... Knife and pain are two words in surgery that must forever be associated in the consciousness of the patient."
- Dr. Alfred Velpeau (1839), French surgeon
Re: The lowest bar
Yeah, it could go either way depending on our assumptions. For what it's worth, my intuition is that the net impact of the typical person isn't that high, as people tend to have generally positive
influences on others. It's especially handy that our capitalist system, for the most part, requires people to work full-time making some other people somewhat happy.
Though maybe perpetuating factory farming is that bad...
Felicifia Head Admin | Ruling Felicifia with an iron fist since 2012.
Personal Site: www.peterhurford.com
Utilitarian Blog: Everyday Utilitarian
Direct Influencer Scoreboard: 2 Meatless Monday-ers, 1 Vegetarian, and 2 Giving What We Can 10% pledges.
Re: The lowest bar
peterhurford wrote: It's especially handy that our capitalist system, for the most part, requires people to work full-time making some other people somewhat happy.
Well, this depends on the additional externalities of that happiness. If econimic growth perpetually depended on making some bottom layer of victims unhappy, it could be plausibly a negative. And as
Brian has pointed out in the past, the sign of marginal speedups of growth is not clear either.
Though maybe perpetuating factory farming is that bad...
Right. Though the WAS replacement argument mitigates it somewhat.
I also thought that perhaps if you live in a democracy, paying taxes for your system is net-positive on average, since it competes with non-democracies. Democracy is bad, but the competition is
currently much less humane still. (argument from lesser evil and power vacuum)
"The abolishment of pain in surgery is a chimera. It is absurd to go on seeking it... Knife and pain are two words in surgery that must forever be associated in the consciousness of the patient."
- Dr. Alfred Velpeau (1839), French surgeon
Re: The lowest bar
Hedonic Treader wrote:Well, this depends on the additional externalities of that happiness. If economic growth perpetually depended on making some bottom layer of victims unhappy, it could be
plausibly a negative. And as Brian has pointed out in the past, the sign of marginal speedups of growth is not clear either.
I suppose I should have said "prima facie" instead of "for the most part", as that captures the claim I actually wanted to make. My best guess is that the typical American capitalist activity is net
positive, but of course I'm unsure.
I also thought that perhaps if you live in a democracy, paying taxes for your system is net-positive on average, since it competes with non-democracies. Democracy is bad, but the competition is
currently much less humane still. (argument from lesser evil and power vacuum)
You also do fund some ostensibly beneficial things, like foreign aid, welfare, and research. Though it's plausible that other things might be net negative.
Felicifia Head Admin | Ruling Felicifia with an iron fist since 2012.
Personal Site: www.peterhurford.com
Utilitarian Blog: Everyday Utilitarian
Direct Influencer Scoreboard: 2 Meatless Monday-ers, 1 Vegetarian, and 2 Giving What We Can 10% pledges.
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The dsp.ISTFT object computes the inverse short-time Fourier transform (ISTFT) of the frequency-domain input signal and returns the time-domain output. The object accepts frames of
Fourier-transformed data, converts these frames into the time domain using the IFFT operation, and performs overlap-add to reconstruct the data. The output of the object is the reconstructed signal
normalized by a factor that depends on the hop length and sum(window). For more details, see Algorithms.
istf = dsp.ISTFT returns an object, istf, that implements inverse short-time FFT. The object processes the data independently across each input channel over time.
istf = dsp.ISTFT(window) returns an inverse short-time FFT object with the Window property set to window.
istf = dsp.ISTFT(window,overlap) returns an inverse short-time FFT object with the Window property set to window and the OverlapLength property set to overlap.
istf = dsp.ISTFT(window,overlap,isconjsym) returns an inverse short-time FFT object with the Window property set to window, OverlapLength property set to overlap, and the ConjugateSymmetricInput
property set to isconjsym.
istf = dsp.ISTFT(window,overlap,isconjsym,woa) returns an inverse short-time FFT object with the Window property set to window, with the OverlapLength property set to overlap, the
ConjugateSymmetricInput property set to isconjsym, and the WeightedOverlapAdd property set to woa.
istf = dsp.ISTFT(Name,Value) returns an inverse short-time FFT object with each specified property name set to the specified value. You can specify additional name-value pair arguments in any order.
Window — Synthesis window
sqrt(hann(512,'periodic')) (default) | vector
Synthesis window, specified as a vector of real elements.
Tunable: Yes
Data Types: single | double
OverlapLength — Overlap length
256 (default) | positive integer
Number of samples by which consecutive windows overlap, specified as a positive integer. The windows overlap to reduce the artifacts at the data boundaries.
Hop length is the difference between the window length and the overlap length.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
ConjugateSymmetricInput — Input is conjugate symmetric
true (default) | false
Set this property to true if the input is conjugate symmetric, which yields real-valued outputs. The FFT of a real-valued signal is conjugate symmetric, and setting this property to true optimizes
the IFFT computation method. Setting this property to false for conjugate symmetric inputs results in complex output values with small imaginary parts. Setting this property to true for non-conjugate
symmetric inputs results in invalid outputs.
Data Types: logical
WeightedOverlapAdd — Apply weighted overlap-add
true (default) | false
Set this property to true to apply weighted overlap-add. In weighted overlap-add, the IFFT output is multiplied by the window before overlap-add. Set this property to false to skip multiplication by
the window.
Data Types: logical
FrequencyRange — Frequency range
'twosided' (default) | 'onesided'
Specify the frequency range as 'onesided' or 'twosided'. If you set the FrequencyRange property to:
• 'twosided' –– The inverse short-time FFT is computed for a two-sided short-time FFT. The FFT length used is equal to the input frame length.
• 'onesided' –– The one-sided inverse short-time FFT is computed for a one-sided short-time FFT. If the input frame length is odd, the FFT length used is (frame length − 1) × 2. If the input frame
length is even, the FFT length used is (frame length × 2) − 1.
y = istft(x) applies inverse short-time FFT on the input x, and returns the time-domain output y.
Input Arguments
x — Input signal
vector | matrix
Frequency-domain input signal, specified as a vector or a matrix. If the input is a matrix, the object treats each column as an independent channel. The FFT length is equal to the number of rows of
x. The FFT length, hence the number of input rows must be greater than or equal to the window length.
Data Types: single | double
Complex Number Support: Yes
Output Arguments
y — ISTFT output
vector | matrix
Inverse short-time FFT output, returned as a vector or a matrix. The output frame length (number of rows in y) is equal to WL − OL, where WL is the window length and OL is the overlap length.
The output is complex with small imaginary parts when the input x is conjugate symmetric and the ConjugateSymmetricInput property is set to false. The data type of the output matches the data type of
the input signal.
Data Types: single | double
Complex Number Support: Yes
Object Functions
step Run System object algorithm
release Release resources and allow changes to System object property values and input characteristics
reset Reset internal states of System object
clone Create duplicate System object
isLocked Determine if System object is in use
Short-Time Spectral Attenuation
Short-time spectral attenuation is achieved by applying a time-varying attenuation to the short-time spectrum of a noisy signal. The gain of the attenuation is determined by the estimate of the noise
power in each subband of the spectrum. This gain, when applied to the noisy spectrum, attenuates the subbands with higher noise power and lifters the subbands with lesser noise power.
Here are the steps involved in performing the short-time spectral attenuation:
1. Analyze the noisy input signal by computing the short-time Fourier transform (STFT).
2. Multiply each subband of the transformed signal with a real positive gain less than 1.
3. Synthesize the denoised subbands by taking the inverse short-time Fourier transform (ISTFT). The resconstructed signal is the denoised input signal.
Use the dsp.STFT and dsp.ISTFT objects to compute the short-time and the inverse short-time Fourier transforms, respectively.
Noisy Input Signal
The input is an audio signal sampled at the 22,050 Hz. The dsp.AudioFileReader object reads this signal in frames of 512 samples. The audio signal is corrupted by white Gaussian noise that has a
standard deviation of 0.05. Use the audioDeviceWriter object to play the noisy audio signal to your computer's audio device.
FrameLength = 512;
afr = dsp.AudioFileReader('speech_dft.wav',...
adw = audioDeviceWriter('SampleRate',afr.SampleRate);
noiseStd = 0.05;
while ~isDone(afr)
cleanAudio = afr();
noisyAudio = cleanAudio + noiseStd * randn(FrameLength,1);
Initialize Short-Time and Inverse Short-Time Fourier Transform Objects
Initialize the dsp.STFT and dsp.ISTFT objects. Set the window length equal to the input frame length and the hop length to 16. The overlap length is the difference between the window length and the
hop length, OL = WL – HL. Set the FFT length to 1024.
WindowLength = FrameLength;
HopLength = 16;
numHopsPerFrame = FrameLength / 16;
FFTLength = 1024;
The window used to compute the STFT and ISTFT is a periodic hamming window with length 512. The ConjugateSymmetricInput flag of the istf object is set to true, indicating that the output of the istf
object is a conjugate-symmetric signal.
win = hamming(WindowLength,'periodic');
stf = dsp.STFT(win,WindowLength-HopLength,FFTLength);
istf = dsp.ISTFT(win,WindowLength-HopLength,1,0);
Gain Estimator
The next step is to define the gain estimator parameters. This gain is applied to the noisy spectrum to attenuate the subbands with higher noise power and lifter the subbands with lesser noise power.
dec = 16;
alpha = 15;
stftNorm = (sum(win.*win) / dec).^2;
Spectral Attenuation
Feed the audio signal to stf one hop-length at a time. Apply the estimated gain to the transformed signal. Reconstruct the denoised version of the original speech signal by performing an inverse
Fourier transform on the individual frequency bands. Play the denoised audio signal to the computer's audio device.
while ~isDone(afr)
cleanAudio = afr();
noisyAudio = cleanAudio + noiseStd * randn(FrameLength,1);
y = zeros(FrameLength,1); % y holds the denoised audio frame
% Feed audio to stft one hop-length at a time
for index = 1:numHopsPerFrame
X = stf(noisyAudio((index-1)*HopLength+1:index*HopLength));
% Gain estimator
Z = abs(X).^2 / (noiseStd^2 * alpha) / stftNorm;
Z(Z<=1) = 1;
Z = 1 - 1./Z;
Z = sign(Z) .* sqrt(abs(Z));
X = X .* Z;
% Convert back to time-domain
y((index-1)*HopLength+1:index*HopLength) = istf(X);
% Listen to denoised audio:
Perfect Reconstruction
Perfect reconstruction is when the output of dsp.ISTFT matches the input to dsp.STFT. Perfect reconstruction is obtained if the analysis window, $\mathit{g}\left(\mathit{n}\right)$, obeys the
constant overlap-add (COLA) property at hop-size R.
$\sum _{\mathit{m}=-\infty }^{\infty }\mathit{g}\left(\mathit{n}-\mathit{mR}\right)=1,\text{\hspace{0.17em}}\forall \mathit{n}\in \text{\hspace{0.17em}}Ζ\text{\hspace{0.17em}\hspace{0.17em}\hspace
{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathit{g}\in \mathrm{COLA}\left(\mathit{R}\right)\right)$
A signal is perfectly reconstructed if the output of the dsp.ISTFT object matches the input to the dsp.STFT object.
iscola Function
The iscola function checks that the specified window and overlap satisfy the COLA constraint to ensure that the inverse short-time Fourier transform (ISTFT) results in perfect reconstruction for
non-modified spectra. The function returns a logical true if the combination of input parameters is COLA-compliant and a logical false if not. The method argument of the function is set to 'ola' or
'wola' depending on whether the inversion method uses weighted overlap-add (WOLA).
Check if hann() window of length 120 samples and an overlap length of 60 samples is COLA compliant.
winLen = 120;
overlapLen = 60;
win = hann(winLen,'periodic');
tf = iscola(win,overlapLen,'ola')
Initialize the dsp.STFT and dsp.ISTFT System objects with this hann window that is COLA compliant. Set the FFT length to equal the window length.
frameLen = winLen-overlapLen;
stf = dsp.STFT('Window',win,'OverlapLength',overlapLen,'FFTLength',winLen);
istf = dsp.ISTFT('Window',win,'OverlapLength',overlapLen,'WeightedOverlapAdd',0);
Reconstruct Data
Compute the STFT of a random signal. Set the length of the input signal to equal the hop length (window length – overlap length). Since the window is COLA compliant, the ISTFT of this non-modified
spectra perfectly reconstructs the original time-domain signal.
To confirm, compare the input, x to the reconstructed output, y. Due to the latency introduced by the objects, the reconstructed output is shifted in time compared to the input. Therefore, to
compare, take the norm of the difference between the reconstructed output, y and the previous input, xprev. The norm is very small, indicating that the output signal is a perfectly reconstructed
version of the input signal.
n = zeros(1,100);
xprev = 0;
for i = 1:100
x = randn(frameLen,1);
X = stf(x);
y = istf(X);
n(1,i) = norm(y-xprev);
xprev = x;
ISTFT with Weighted Overlap-Add (WOLA)
In WOLA, a second window called the synthesis window, $\mathit{f}\left(\mathit{n}\right)$, is applied after the IFFT operation and before overlap-add. The synthesis and analysis windows are typically
identical and are usually obtained by taking the square root of windows satisfying COLA (thereby ensuring perfect reconstruction).
iscola Function
Check if sqrt(hann()) window of length 120 samples and an overlap length of 60 samples is WOLA compliant. Set the method argument of the iscola function to 'wola'. The output of the iscola function
is 1 indicating that this window is WOLA compliant.
winWOLA = sqrt(hann(winLen,'periodic'));
tfWOLA = iscola(winWOLA,overlapLen,'wola')
Reconstruct Data with WOLA
Release the dsp.STFT and dsp.ISTFT System objects and set the window to sqrt(hann(winLen,'periodic')) window. To use weighted overlap-add on the ISTFT side, set the 'WeightedOverlapAdd' to true.
stf.Window = winWOLA;
istf.Window = winWOLA;
istf.WeightedOverlapAdd = true;
n = zeros(1,100);
xprev = 0;
for i = 1:100
x = randn(frameLen,1);
X = stf(x);
y = istf(X);
n(1,i) = norm(y-xprev);
xprev = x;
The norm of the difference between the input signal and the reconstructed signal is very small indicating that the signal has been reconstructed perfectly.
More About
Inverse Short-Time Fourier Transform (ISTFT)
The inverse short-time Fourier transform of a discrete frequency-domain signal is computed by taking the IFFT of the input frequency subbands, overlap-adding the inverted signals, and normalizing the
output to reconstruct the data.
The object accepts frames of Fourier-transformed data. These segments are converted into the time-domain using the IFFT operation. The inverted segments are overlapped so that the artifacts at the
boundary are reduced. To reconstruct the data, the overlapped signals are added and normalized by a factor that is a ratio of the hop length and sum(window).
The ISTFT is given by
$\begin{array}{l}y\left(n\right)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}\sum _{m=-\infty }^{\infty }{Y}_{m}\left(\omega \right){e}^{j\omega n}d\omega \\ \text{ }\text{ }=\sum _{m=-\
infty }^{\infty }\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{Y}_{m}\left(\omega \right){e}^{j\omega n}d\omega \text{ }\text{ }\\ \text{ }\text{ }=\sum _{m=-\infty }^{\infty }{y}_{m}\left(n
• y(n) –– Reconstructed signal at time n.
• Y[m](ω) –– Frequency-domain input.
ISTFT with Weighted Overlap-Add (WOLA)
In WOLA, a second window (usually called the synthesis window) is applied after the IFFT operation and before overlap-add. WOLA is used to suppress discontinuities at frame boundaries caused by
nonlinear processing of the STFT.
The analysis window (on the STFT side) and the synthesis window (on the ISTFT side) are typically identical, and are usually obtained by taking the square root of windows satisfying the constant
overlap-add (COLA) property, thereby ensuring perfect reconstruction. For details on the COLA property, see the More About section in dsp.STFT page.
The inverse FFT of frequency-domain input Y[m](ω) produces the output y[m](n), where n = 0 to N − 1, and is given by ${y}_{m}\left(n\right)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}{Y}_
{m}\left(\omega \right){e}^{j\omega n}d\omega .$
The synthesis window, f(n), applied to y[m](n) yields the weighted output frame:
${y}_{m}^{f}\left(n\right)={y}_{m}\left(n\right).f\left(n\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall n=0,...,N-1.$
Translate the m^th output frame to time mR:
Add the translated signal to the accumulated output signal, y(n):
To obtain perfect reconstruction in the absence of spectral modifications, then the following condition must be true:
$\begin{array}{l}y\left(n\right)=\sum _{m=-\infty }^{\infty }x\left(n\right)g\left(n-mR\right)f\left(n-mR\right)\\ \text{ }\text{ }=x\left(n\right)\sum _{m=-\infty }^{\infty }g\left(n-mR\right)f\left
(n-mR\right),\\ \text{ }\text{ }=x\left(n\right),\end{array}$
which is true if and only if
$\sum _{m}g\left(n-mR\right)f\left(n-mR\right)=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall n\in ℤ.$
• g(n) –– Analysis window on the STFT side.
• f(n) –– Synthesis window on the ISTFT side.
Here is a sketch of how the algorithm is implemented without weighted overlap-add (WOLA):
The frequency-domain input is inverted using IFFT, and then overlap-add is performed. Note that each run of the algorithm generates R new output time-domain samples, where R is the hop length. The
hop length is defined as WL − OL, where WL is the window length and OL is the overlap length. The normalization stage multiplies the output by $R/\text{sum}\left(win\right)$, where win is the window
vector specified in the Window property.
Here is a sketch of how the algorithm is implemented with Weighted Overlap-Add (WOLA):
In WOLA, a second window (usually called the synthesis window) is applied after the IFFT operation and before overlap-add. WOLA is used to suppress discontinuities at frame boundaries caused by
nonlinear processing of the STFT. For more details, see More About.
Here is an illustration of how the input frequency subbands look when inverted with IFFT and overlap-added together to reconstruct a time-domain signal.
The analysis window (on the STFT side) and the synthesis window (on the ISTFT side) are typically identical. To ensure perfect reconstruction, the windows are usually obtained by taking the square
root of windows satisfying the constant overlap-add (COLA) property. For details on the COLA property and how perfect reconstruction is defined, see the More About in dsp.STFT page.
[1] Allen, J.B., and L. R. Rabiner. "A Unified Approach to Short-Time Fourier Analysis and Synthesis,'' Proceedings of the IEEE, Vol. 65, pp. 1558–1564, Nov. 1977.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
When the FFT length, which is determined by the number of rows in the input signal, is not a power of two, the executable generated from this object relies on prebuilt dynamic library files (.dll
files) included with MATLAB^®. Use the packNGo function to package the code generated from this object and all the relevant files in a compressed zip file. Using this zip file, you can relocate,
unpack, and rebuild your project in another development environment where MATLAB is not installed. For more details, see How To Run a Generated Executable Outside MATLAB.
This limitation does not apply when the FFT length is a power of two.
Version History
Introduced in R2019a
See Also
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Make a sum of 30 from three balls - Daily Quiz and Riddles
Place three balls from the available selection into the empty spaces to achieve a total of 30. Each ball can only be used once.
6 + 11 + 13 = 30
Alright, my fellow riddle enthusiasts, get ready to engage those problem-solving gears as we tackle this tricky challenge! We’re presented with a puzzle where we need to strategically place three
balls from the selection into blank spots to achieve a total of 30. Let’s unravel this enigma together and discover the solution.
This puzzle can be a bit challenging as it requires some adjustments to reach the desired number.
The key to cracking this puzzle lies in a clever manipulation of one of the numbers – transform the number nine into a number six.
Thus: Select the ball labeled six from the first column, the ball labeled eleven from the second column, and the ball labeled thirteen from the third column. The sum of these three numbers will add
up to 30: 6 + 11 + 13 = 30.
Hence, the correct balls to use are 6, 11, and 13.
If you’re wondering where the number six ball came from, consider the following: The instructions state that you need to arrange the balls in a way that their sum equals 30; it doesn’t specify that
you have to use the given numbers exactly. Keeping that in mind, you can transform the ball labeled nine into a ball with the number six.
Voila! We’ve successfully accomplished the mission of making the total 30 by strategically placing the balls. It’s all about thinking outside the box, using our creativity, and utilizing the rules
and instructions to our advantage.
So, dear puzzle enthusiasts, the correct balls to achieve a sum of 30 are the ones with numbers 6, 11, and 13. And remember, in the realm of riddles, sometimes the solution lies in bending the rules
and finding alternative interpretations.
Leave a Comment
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Math Homework Answers for Free Your Key to Mastering Mathematics
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Clewer Green C
Home Learning due to Lockdown
We have compiled a timetable to guide you through the next stage of our home learning. There are supporting links to help you and your child with some of the key ideas. Remember to do as much as you
are able and work at a pace that suits your family life at this time. We have also put the worksheets onto Seesaw so that you can share your work with us after it is completed if you wish, however we
do not want you to feel guilty or anxious if you are not able to do this - our priority is for you and your family to stay safe and keep well.
Mrs Hawkes, Mrs Branch and Mrs Burgess.
Week 2 Term 4 week beginning 1st March 2021
Georgians Maths
This week, we are starting to use our knowledge of times tables to calculate division. You will see that the more you have practised your times tables, the easier this work becomes. Do feel free to
use your times table support sheet if you need it and you may find a number line helpful as is used in the first help video: https://whiterosemaths.com/homelearning/year-4/
This video should help you recognise how we link our time tables work from last term with the division work for this term.
Then complete p.121 and p.122 of our maths books.
Victorians Maths
Today we are beginning to look at symmetry.
please draw and cut out the shapes when you are asked to and fold them to find the lines of symmetry as this is a good way of learning. You will have to be particularly careful when you are drawing
and cutting out a square and a rhombus as all the sides need to be of equal length if you are to find all the lines of symmetry.
Georgians Maths
Continue practising division but this time using 3 digit numbers.
Then complete p.123 and p.124 of our maths books.
You can always cut out shapes and fold them to find lines of symmetry. Then complete p134 p135 p136 and p137.
Georgians Maths
oday we are going to have a look at how to cope with remainders in division. This Youtube video gives a lovely clear explanation: https://www.youtube.com/watch?v=o1vyeGt4Ngg
There are also some clear questions to take you through the method until you feel confident on the Division with remainders support powerpoint.
Then complete p.125 and p.126 of our maths books.
You will need to count the squares carefully as shown on the PowerPoint. Then complete p138 p139 and p140.
Georgians Maths
Rounding Decimals
Today we will use all that we have learnt about decimals to look at rounding decimals. This skill is useful when we are looking at measuring and weighing.
A number greater than (and including) .5 is rounded up to the next whole number and anything smaller than .5 is rounded down.
≈ means approximately equal to, or almost equal to. So:
2.2cm ≈ 2cm
2.8cm ≈ 3cm
particularly when there is a diagonal line of symmetry. But if you need to, you can always cut out a square, paint the shape on it and then fold the square to see where the symmetrical image lies and
what it looks like. Then complete p141 and p142 using paint if this helps.
Georgians Maths
Rounding Decimals
Victorians Maths
Today we are drawing symmetrical figures.
and then you can check your answers on the next slide on the PowerPoint. Then complete p143 and p144.
Georgians Maths
Maths Challenges this week if your child is feeling confident!!
Victorian Maths
Challenge questions.
1. Is this statement always true, sometimes true or never true? Explain why you think this. "A four sided shape has four lines of symmetry."
2. Now try this "always, sometimes, never" challenge from the nrich site.https://nrich.maths.org/12673
This week is World Book Day and so all our tasks are going to be based around this theme!
or a notepad to note down the next piece of information that you have to look for as you read through. The link that accompanies q.3, if you want to use it, is this one: https://www.youtube.com/
Have a think before tomorrow’s lesson about a book that you have read which you particularly enjoyed. It needs to be a story book. If you have been listening at story time zoom, you might like to
choose Kensuke’s Kingdom or Brain Freeze. You may like to choose a Roald Dahl story that you read in Year 3. Your next few tasks will be based on this book and you are welcome to share your favourite
piece of work in school next week if you would like to (this is optional).
Choose a scene out of your favourite book that contains the most action! Use it to create a comic strip to reflect just that one part of the book you chose to do your World Book Day tasks about this
week. Don’t try to fit the whole story in, just the most exciting part.
You don’t need to write much (mainly drawing) although you may want to add one line of text in each box. Use the shapes on the second page to add speech bubbles and sound effects. You may like to
draw or trace these if you want to use one more than once.
Choose a character from your chosen book to complete today’s task. It doesn’t have to be the main character, it could be just one that you find particularly interesting/funny/evil/mischievous.
Note that you may want to dress up as your chosen character and take a picture!
Today, we have a few tasks that you may like to try.
The Bayeux Tapestry gives an account of the Norman invasion of England and the Battle of Hastings at the end of the Anglo Saxon period. The battle happened in 1066. At the time the tapestry was
created, very few people could read. Books were rare, as each one had to be written by hand. A tapestry told the story in a way that everyone could understand.
Use the website http://www.primaryhomeworkhelp.co.uk/bt/index.htm to find the answers.
You will need to use the blue arrows at the bottom of the page to move through the pages.
As it is World Book Day this week have a go at these activities
The Bayeux Tapestry
The description underneath each image is written using wonderful, descriptive language.
Insert the words in green below each description to help make sense of the text.
Last week we found out about the effect of acid on our teeth when we left an egg in a glass of vinegar; the eggshell represented the enamel on our teeth and the vinegar was the acid.
If you did not do last week’s investigation or watch the video, I have put the link on the PowerPoint again; please watch it to see what happened before you do this week’s investigation.
Today we are going to look at whether we can protect our teeth by using a fluoride toothpaste. You will need some fluoride toothpaste, clear vinegar and two eggs. Follow the instructions on the
PowerPoint. You are going to coat one egg in toothpaste and then leave that egg and an uncovered egg in vinegar to see what happens. I have put pictures of what happened when I did it at the end of
the PowerPoint in case you don’t do it yourself, but it is fun to do. It does take several hours; mine took around fifteen. When you have finished
This week, we will be learning about how Christians might celebrate Easter.
Week 1 Term 4 week beginning 22nd February 2021
Georgians Maths
Today, we will be continuing with our multiplication and looking at using the methods we learnt last term with some bigger numbers!
We will be partitioning numbers with three digits in order to multiply them.
Watch this White Rose Video that takes you through the methods we are using. When the link opens up, scroll down to the second row of boxes and click on ‘multiplying 3 digits by 1 digit’: https://
and then complete p.115 and p.116 of our workbooks.
Victorians Maths
For the next two weeks, we are going to go through a unit on geometry. You do not need a protractor for this. As long as you know the shape of a right angle (like the corner of a piece of printer
paper) and know that an acute angle is smaller than a right angle whereas an obtuse angle is bigger than a right angle, that will be enough for you to access the work. However, if you do have one,
you may enjoy learning to measure the angles with it.
Familiarise yourself with the size and shape of a right angle; write down the names of the different angles and draw pictures of them to help you remember which is which.
Then complete p121 and p122 and Maths 2 Monday Acute and Obtuse Angles.
then try p.117 and p.118 of our workbooks.
We are continuing to partition numbers in order to multiply each part separately. Then add to get our final answer.
Victorians Maths
then try p.117 and p.118 of our workbooks.
We are continuing to partition numbers in order to multiply each part separately. Then add to get our final answer.
Victorians Maths
Then complete p 123, p124 and p125 and Maths 4 Tuesday Compare and Order Angles.
then try p.119 and p.120 of our workbooks.
We are continuing to partition numbers in order to multiply each part separately. Then add to get our final answer. We are happy for your child to use a particular method that they feel suits them
for any of the questions. Hopefully they’ve had a go at them all over the last few weeks and may feel that they are more confident with one method in particular.
Victorians Maths
Then complete p126, 127 and 128.
Georgians Maths
Decimal Number Patterns
Today we will remind ourselves of what we looked at last half term by looking at number patterns and sequences using decimals. Use what you already know on tenths and hundredths to help you. Look
carefully to see if the numbers in the sequences are getting bigger or smaller:
1.6 1.7 1.8 1.9 2.0 2.1
2.84 2.83 2.82 2.81 2.80 2.79
and sticking them down on a piece of paper so that the correct description and picture are next to each name.
Georgians Maths
Decimals on a Number Line Practice
You’ve worked really hard on decimals, so today’s work is a little extra practice and just reminding ourselves of what we’ve learned already.
Victorians Maths
Today we are moving onto quadrilaterals, four sided polygons.
then move the PowerPoint on to check your answers. Then complete p129, p130, p131 and p132.
Maths Challenges this week if your child is feeling confident!!
Then you might like to do this matchsticks challenge. https://nrich.maths.org/10
For times table practice, have another at the Mathsframe times table check: https://mathsframe.co.uk/en/resources/resource/477/Multiplication-Tables-Check
Did you beat your previous score?
We are going to revise some of the important learning that we covered earlier in the year and that we must take with us as we move forward in our learning. The first theme is going to be writing
dialogue. You will need this in a task this week and in the coming weeks as we look further at story writing.
Here’s a little reminder to help you: https://www.bbc.co.uk/bitesize/clips/zvftsbk
Can you find a good example of dialogue in one of your story books at home?
Watch this clip to recap what we were thinking about yesterday: https://www.youtube.com/watch?v=6-YFmLctwDY
Use your editing skills to ensure that it is set out correctly and with the correct punctuation. You may like to do this with an editing pen on the document first before writing it out, Think really
carefully about layout and when a new character has started to speak!
Listen to ‘The Wolf in Sheep’s Clothing’: https://www.bbc.co.uk/teach/school-radio/english-ks1--ks2-aesops-fables-the-wolf-in-sheeps-clothing/zh8kgwx. Now write a well punctuated and carefully set
out conversation between the wolf and the sheep. Imagine that you have two sheep (as in the radio version) discussing how large one particular sheep is and how weirdly he smells. Then perhaps the
wolf, disguised as a sheep, convincing them to come near him so he can gobble them up!
You may like to do this in rough first!
Look at the image of the wolf from this week’s fable, in the same powerpoint, and use carefully chosen adjectives to describe him. Think about the example on the first slide – sometimes it’s easier
to write a good description if you describe the character as they are doing something eg. chasing the sheep or planning how he will get close to them.
Take some time this week to read through the Powerpoint ‘Anglo Saxon Village Life’ so that you are prepared for your History lesson on Anglo Saxon Settlements.
This week we will be looking at Prefixes.
The activities below will help you to practise your spelling of different prefixes this week.
The Anglo-Saxons preferred to live very differently to the Romans who had established large towns in Britain during their rule. Places like London and Cirencester, which were thriving centres under
the Romans became much smaller and quieter during the Anglo-Saxon times, and some towns were completely abandoned and the buildings were left to go to ruin.
The Anglo-Saxons chose to live in small villages instead, which were often set up by clearing away a part of a forest. The Anglo-Saxons lived in family houses which were built around a central hall
where the village chief lived.
Design your own village
• You should try to include as many features as you can to make sure your village is busy and thriving!
• You will also need to include labels to annotate all the features of your village.
• You can either draw your village or make it 3D. There are some ideas on the Powerpoint. You can draw or print out the images that I’ve put together for you in the Village Resources on Seesaw (the
same ones are on the links below too). Your village can be any size you want.
Use these resources to help you, if you wish:
This term we have a new topic and we are starting with a look at teeth, which are part of the digestive system.
click on the link https://www.bbc.co.uk/bitesize/topics/z27kng8/articles/zsp76yc that takes you to a BBC Bitesize video about different types of teeth and their functions. Learn about teeth as you go
through the PowerPoint,
Then move onto slide 10 and follow the link to the BBC video https://www.bbc.co.uk/teach/class-clips-video/teeth/zr8ygwx. This video explores the teeth that various animals have that suit their diet.
As the PowerPoint explains, there is an investigation towards the end of the video that you may like to do at home, which investigates the effect of different liquids on the outer surface of teeth,
using eggshells in place of teeth. Carry out the investigation. Leave it for three days, then find out what has happened
Today we are returning to a Christian themed topic. We will be looking at the festival of Easter and important moments in the story.
Listen to the presentation about The Last Supper which is remembered on Maundy Thursday (one day before Good Friday).
After watching the second clip on the second slide, answer the questions.
Then finish by looking through the images of what Easter means to people around the world.
Week 6 week beginning 8^th February 2021
Georgians Maths
This week, we will be refreshing our skills with column subtraction and then looking at how we can multiply multiples of 100 (as we did with multiples of 10) Work through this BBC Bitesize page to
remind you what we did last week. The quiz that is at the top of the page is probably best saved until last. Watch the film then read through the column subtraction method lower down – try these out
with pencil and paper. Then do the quiz:
Did you have a go at the Guardians: Defenders of Mathematica game? If you did, you could let me know who your character is and how many shields you’ve won! I really enjoyed playing this game!
Remember to clearly show your carried tens.
There is a challenge sheet below if you are feeling confident.
Victorians Maths
We are continuing with fractions and beginning this week with hundredths, when a whole is split into one hundred equal parts.
Georgians Maths
Today we are going to be learning how to multiply three numbers in a row. The trick to this is to remember that you can swap the order of the numbers. You may want to start with the two biggest
numbers. OR even better… look out for two numbers that equal a multiple of 10 and do them first!
There is a challenge sheet below if you are feeling confident.
Victorians Maths
Write the mixed numbers on slide 4 down on a piece of paper before you move the slide on to check your answer.
The questions on the second page are harder so only do them if you want a challenge!
Georgians Maths
Today we are going to apply our times tables to multiplying multiples of 100.
Then complete p.113 and p.114. Remember the first page, want you to show your multiplication as repeated addition and then write your final answer.
Victorians Maths
Count up in fractions along the number lines on
and see if you can work out what the missing numbers are.
Georgians Maths
Today we are looking again at writing tenths and hundredths as a decimal.
One whole divided into 10 equal parts is one tenth. We write one tenth as 0.1
One whole divided into 100 equal parts is one hundredth. We write one hundredth as 0.01
Victorians Maths
Today we are moving onto equivalent fractions which are fractions that are written in different ways but which still have the same value.
there are instructions for making a small fraction wall on squared paper. You can print of a sheet of squared paper
Use this to answer the questions on slide 9 of the PowerPoint.
Georgians Maths
Today we will continue to compare and order decimals. As well as looking at the tenths and the hundredths we need to remember to also look at the tens and the ones too!
To put these numbers (32.54 23.45 24.35 32.45) in order it would look like this:
23.45 24.35 32.45 32.54
Victorians Maths
Today we are going to look at fractions that are equivalent to thirds.
then answer the questions on slide 7; you can write your answers
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
At the end of this sheet, see if you can apply your skills to 3 digit numbers also.
Which numbers are missing from the robots?
The trick with this sheet is to clearly show your carried tens as you work through the problems. Sometimes it helps to check what your tens column will make first.
Victorians Maths
Mental Maths Practice
Choose activities from the activities on the Mental Maths Train on https://www.topmarks.co.uk/maths-games/mental-maths-train
You can choose from multiplication and division, as well as addition and subtraction. Select the skills that you feel you need to work on.
Play the interactive Matching Fractions game on https://nrich.maths.org/8283
It is a memory game that teaches you different ways of showing the same fraction.
This week, we are starting to look at fables. There are some really beautiful retellings of these stories with morals to be found, read by some very famous voices, on the BBC Radio. I think it would
be of real benefit for all the children to listen to one or two of these each day to get a real feel for how a moral is structured, the types of characters it often involves and the lessons that they
teach. The following link will help you access these: https://www.bbc.co.uk/teach/school-radio/english-ks1--ks2-aesops-fables-index/z73s6v4
For our first task, we are going to focus on the tale of The Lion and the Mouse. Another BBC project has reproduced these tales using animation and use of more traditional language. The children may
find it harder to understand the wording but will hopefully be able to follow with the combined use of the animation. You may like to watch the whole clip, particularly of you are interested in
animation. However, if you wanted to just watch the story, it is found between 4mins 30sec and 20 mins.
You can also find the story on the first link along with a transcript to read if you find the language tricky in this version.
Today, we are going to look at some connecting words again.
Notice that the part of the sentence that has been glued on with the connective can be wheeled to the front of the sentence!
Let’s see which connectives can help us link ideas and move the story on smoothly in The Lion and the Mouse by completing the Connectives Challenge 2.
Another skill that we use when writing stories/fables are describing words… adjectives.
You may also want to remember your magpie skills and steal some of the lovely adjectives from these games for your own work!! Tomorrow we will be planning a fable so you could keep this work handy or
copy the list of words somewhere safe!
We are going to design our own fable.
Note the morals on slide 2 – can you think of a story idea that would match one of these??
Today, we are going to write a fable. You can use the plan that you filled in yesterday and you may like to work with
in front of you so that you can link your ideas smoothly.
Use your plan to make paragraphs and remember to describe your characters in appearance and facial expressions as you go through your story.
Make sure you only use a couple of lines of speech so you don’t get bogged down with the tricky punctuation and lose your lovely description.
I have put the final 2 pages of the book onto Seesaw, or you can read them below
However, this week we are going back in the story to the ritual of Arthur and the sword, in the stone circle. There are some comprehension questions relating to that part of the story (the text is
scanned below the questions to help you) and there is some information and questions on Stonehenge which is a famous stone circle in Wiltshire, England.
This week we will be looking at the ‘sion’ sound. Pick from the activities below to help you practise your spellings this week.
The Anglo Saxon Chronicle is a collection of written records which tell us about the history of the Anglo Saxons. Listen to this BBC clip for more information.
The books were written and decorated by hand, usually by priests and monks. They were seen as very precious works of art, and you had to be very skilled to produce them. The manuscripts were called
‘illuminated’ because the letters and pictures were often decorated with gold and silver leaf.
An illuminated letter was usually the first letter of a page or paragraph. It was enlarged and in colour, with gold or silver added in areas, while the rest of the text was in black. Images of
people, animals, plants or mythological creatures were sometimes added to enhance the letter.
Design your own illuminated letter, or create a page from the Anglo Saxon Chronicle.
Today you are going to use your knowledge and understanding of sound, including pitch and loudness, to design and create your own musical instrument!
Watch the video using the link on slide 2 to see the variety of sounds you can get from musical instruments that are made from junk, and then have a think about what you would like to make yourself
from things around your house that your parents are happy for you to use. If you are struggling to come up with an idea, I have put a simple suggestion on slide 4. We would love to hear a recording
of you playing your instrument on Seesaw!
This week, we are going to learn two more of Buddha’s stories; one we will read and one we will watch.
Watch Buddha and the Mustard Seed and think about the lesson that it teaches us: https://www.youtube.com/watch?v=dS433t0yI6E
You may like to watch it through a couple of times and think about how we view tough times – whether we feel sorry for ourselves or whether we look to remember what other people are experiencing.
Choose one of the four stories we have looked at to write about in our ‘Reflections about Buddhists Stories’ sheet. Aim for two clear sentences in the final box.
Week 5 week beginning 1^st February 2021
Georgians Maths
Today we are looking at using our times tables with multiples of 10.
(Basically numbers with one or more zeros on the end.)
Watch this video to learn more about how we are going to use our tables today:
and listen to the tips you need for the work today.
Complete p.107 and p.108
Victorians Maths
Today we are going to start the topic of fractions. We will go slowly and build up to the Year 4 work. We are starting with a gentle reminder of what a fraction is.
Each time you reach a question on the PowerPoint that is on the worksheet, answer it on the worksheet, then move the PowerPoint on to see if you were correct. Then complete the paper folding exercise
on the final slide.
We are going to look at strategies for multiplying larger numbers now. Some of the strategies may really suit you but some you may find harder to get along with. Don’t worry if you find some don’t
suit you! Watch the following video which talks you through the step up to building on what we know already:
Don’t be put off by the fancy names for the methods!
and listen to the tips you need for the work today.
Complete p.109 and p.110.
Each time you reach a question on the PowerPoint that is on the worksheet, answer it on the worksheet and then move the PowerPoint on to see if you were correct.
Georgians Maths
Today we are going to partition numbers (split the tens and ones) to multiply. This will use the thinking from Monday where we had to work with multiples of 10.
Watch these examples (If you scroll down past the two written examples you can watch the video): https://www.mathswithmum.com/multiplication-by-partitioning/
and listen to the tips you need for the work today.
Complete p.111 and p.112.
there is nothing to write down today but make sure you go through it carefully; it will help you with one of the questions on the worksheet.
Georgians Maths
Today we are going to look at writing tenths.
A tenth is 1 whole divided into 10 equal parts
We read 0.1 as one tenth. The dot is the decimal point.
then complete these 2 sheets from our Singapore Maths Books
Georgians Maths
Today we are looking at writing hundredths.
A hundredth is 1 whole divided into 100 equal parts.
We read 0.01 as one hundredth.
then complete these 2 sheets from our Singapore Maths Books
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
Victorians Maths
Times tables.
If you are ready, do the Maths frame times table test
answering 25 questions with 6 seconds for each and see how high a score you can get.
If you are not ready for this, adjust it to give yourself more time, or to focus on particular times tables.
Fractions challenge.
Have a go at Fractional Triangles.
This week, we will be using our letter writing skills again and we will also be thinking about using conjunctions to join our ideas in interesting sentences. You can learn more about these by
watching the clips and completing the exercises at:
I must warn you that you may want to stop Laura Bubble before she’s finished if she’s driving you mad with her song!! However, if you need to give yourself a stretch, you could join in with her
Our letter this week is going to be less formal but it will hopefully give you a good opportunity to reflect on your life in lockdown compared to other children around the world.
Watch the first three video diaries featured on this link: https://www.unicef.org/coronavirus/kids-video-diaries-about-life-during-covid-19
Their situations are all very different and I’m sure you can see Muna, in particular, has to cope with coronavirus in addition to having lost her home! It can make you think about all that you are
able to enjoy even though we are in lockdown.
Yesterday, on our BBC site, we learnt about conjunctions that join two ideas. Today we are going to look at subordinating conjunctions that do the same job but notice that the part that joins on does
not make sense on its own! Look out for the main part of the sentence that would make sense on its own and the subordinating part of the sentence, that has been joined on with the conjunction, but
doesn’t make sense on its own!
Watch the BBC clip and choose whether you fancy seeing more of Laura Bubble or not! https://www.bbc.co.uk/bitesize/articles/z6kj2sg
a little tip for finding which sentence uses despite: Despite is followed by a verb that ends in –ing. For example, ‘Despite watching television for two hours, I decided to watch another film!’
Today we are going to plan a letter to one of the children who we looked at on Monday.
This activity made me think about the children in Makwati who are also in lockdown. None of those children have computers and many do not live near their school so they are unable to continue their
learning at the moment.
and then edit your planning sheet to make sure you have covered everything in your checklist before you complete your neat copy.
Write or type your final edited copy of your letter.
Underline your 4 conjunctions and your 2 fronted adverbials.
There were many famous Anglo-Saxon kings, but the most famous of all was Alfred. He was called 'Alfred the Great'.
His father was king of Wessex, but by the end of Alfred's reign his coins referred to him as 'King of the English'.
Use any of the following websites to write down 10 facts as to why you think he was called ‘great’. Do you think he deserved the title?
This week we will be looking at the ‘sc’ sound. It makes a hard sound in words such as ‘escape’ and a soft sound in words such as ‘science’.
Have a go at some of these activities to practise your spelling of ‘sc’ words.
For our History this week we will create some portraits of Anglo Saxon warriors. We will look at the artwork of an artist called Julian Opie and draw our Anglo Saxons in the same style.
have some “true or false” questions, recapping some of the things we have learnt about sound. See how you get on with these. Then read about soundproofing from slide 14 onwards. You will need a box
with something inside that makes a noise, such as an alarm clock. You need to choose some different materials to investigate which absorbs the sound most effectively. There is a list of materials
that you may wish to use on slide 17 but you can choose different ones. You are then going cover the box with a single layer of each of these materials to see which is the most effective at
Today, we are going to hear some of the stories that Buddha told people. Each of his tales had a purpose which was to help people reflect on their behaviour towards others.
You may need to watch each story more than once to remember all the details and think about the meaning properly. You can pause the stories to read the words on the screen at your own pace.
Answer the questions about each story.
Week 4 Week beginning 25^th January
Georgians Maths
We will then go over the key ideas that we’ve looked at in the last few weeks relating to our times tables.
Then use slide 3 to help you complete p. 96 and p.97 of our study books.
Victorians Maths
We are carrying on with short division.
which go over last week’s work. Then have a go at the problem on slide 7; the answers are on slides 8 and 9 but have a really good go at working them out yourselves before you look. This will help
you do the second worksheet. When you are ready, do Maths 2 Monday p127 and p128.
Georgians Maths
Our key ideas today are further times table practice and looking at how when you multiply by one, the answer stays the same and when multiplying by 0, the answer is always 0.
Now think about the inverse with division. So 12 ÷ 1 means how many 1s in 12… of course there are 12 ones in 12!
And 4 ÷ 4 means how many 4s in 4… of course there is only one 4 in 4!
Victorians Maths
We are moving onto word problems with division today.
which asks you to watch this video of the Chuckle Brothers. https://www.bbc.co.uk/bitesize/clips/zdpfgk7
Work through the problems before you click on the answers. Make sure each time that you understand what to do with the remainder.
If you are feeling very confident, have a go at some of the problems in the challenge section below.
Georgians Maths
Measuring Volume
Today we will look at writing millilitres as a decimal.
If 1 litre = 1000ml, then
0.1 litre = 100ml
0.2 litre = 200ml
0.3 litre = 300ml and so on.
We’ll also look at estimating volume today. Look through the cupboards at home and estimate the capacity of any bottles or jars that you find. Write down your guess in ml or l and then write down how
much each container actually holds. Calculate the difference.
Victorians Maths
Today’s work reviews the multiplication and division that we have done recently.
Note how multiplication is the same as repeated addition.
The pages in the our study book today want us to either show a multiplication in pictures or in times tables facts or in repeated addition.
Why not practise your times tables using this game: https://www.mathsisfun.com/numbers/fix-equation.html
You can chose your range of times tables and set your time.
There are loads of helpful tips for learning your times tables on this page: https://www.mathsisfun.com/tables.html
Victorians Maths
Georgians Maths
Measuring Volume
The answers are on the slides too so that you can check your own understanding.
Here are the last 2 pages from our Singapore Maths Books on this topic:
p.75 and p.76
Page 76 is tricky. To help you, change all the volumes into millilitres. Remember 1000ml = 1 litre
• Bucket holds 3l 45ml = 3045ml
• Container holds 3405ml
• Trough holds 3450ml
Victorians Maths
To complete this unit on division, try Topmarks Hit the Button division facts on the link below. You can choose to do this on particular times tables or “mixed” which covers all of them. If you are
doing really well on “hit the answer”, go onto “hit the question” which is harder to do quickly.
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
Victorians Maths
Times tables
Topmarks Coconut Multiples
If you enjoy these and would like more (even harder), let me know and I’ll send them to you.
We are going to be having some fun with some fairy tale characters this week. We are also going to have a look at how to write a letter.
Look carefully at the example on slide 2 and answer the questions on slide 1.
Then fill in the chart like on slide 2 which you may want to draw out in your book, using a ruler, of course.
Answer the questions on the second slide then start to plan some ideas about what your fairy tale baddie has been doing to upset their neighbours!!! Perhaps the bad wolf has been trying to blow your
house down or climbing on people’s roofs and knocking on their windows! Perhaps there is a witch who lives down your road who’s always putting spells on people, offering people poisoned apples to eat
and jumping through red traffic lights on her broom! Perhaps the ugly sisters are gossiping about everyone and being mean!
You do not need to read the whole book to complete the English lessons (but it would be great if you did!), as we will be looking at a few pages together which focus on the Anglo Saxons.
Today we will be looking at the battle between the Saxons and the Britons. I have copied it underneath the questions so you won’t need to keep flicking back and forth to find it on the scanned pages.
Once you have read the page please answer these questions:
Today, we are going to plan a letter complaining about out baddie.
• Which character might you write as? You could be the gingerbread man, Little Red Riding Hood, Prince Charming, Cinderella, Hansel or Gretel, the local woodcutter or one of the three pigs??
• Think about all the naughty things that you wrote down yesterday. Who could your character be writing to? Perhaps you’re writing to the mayor or the king/queen.
• Be inventive with your ideas. Could you make up a suitable address for your characters. You might need a grown up to help you with the layout of an address: person’s name, house number and road
name, town, postcode.
Today we are going to write up our letter in our best handwriting. You may like to mark out faintly with a pencil and ruler where you want ot write the addresses and the dates so that you place them
in the correct position on your page.
Remember that if you can’t tick a box, edit your work to improve it.
This week we will look again at ‘ous’. Remember that adding ‘ous’ has three rules:
1) If the root word doesn't change, just add 'ous' - e.g danger ----> dangerous
2) If the word ends in 'e', drop the e and add 'ous' - e.g. fame ----> famous
3) If the word ends in 'y' change it to an 'i' and add 'ous' - e.g. vary ----> various
Here is a BBC video to help you https://www.bbc.co.uk/bitesize/topics/zqqsw6f/articles/zqcpv9q
Here are three activities for you to do. You do not need to do them all – just pick the ones that you fancy!
This week we are looking at how the Anglo Saxons used to write using runes. Before the Anglo Saxons became Christians they wrote using runes. Runes were carved letters made out of straight lines
(this made it easier to carve) They were carved onto materials like stone.
Have a look at this website www.abdn.ac.uk/sll/disciplines/english/beowulf/rune.htm
Type Anglo Saxon runes for kids into the Google search bar and look for the website. If you can’t find it,
String Telephone.
Today we are learning about how sounds travel over distance.
The first activity explores how sound changes over distance using an alarm clock or something else in your home that makes a noise. The second activity is to make a string telephone to help you hear
someone who is standing some distance away from you. The string does need to be tight for this to work. Have fun! There are then some questions about how it works; you will need to fill in the gaps
on slide 12. The answers are on slide 13.
Last term, we started to look at the Eightfold Path – the rules that Buddhists use to guide them in life.
– either on the sheet if you would like to print it or just draw a chart with a ruler in your book. You can then choose to design a Buddhist garden and/or colour a mandala.
Week 3 Week beginning 18^th January 2021
Georgians Maths
This week we are looking at times tables in division again. Here is a really clear video lesson with some methods that you might like to use: https://www.youtube.com/watch?v=TvQtf-MEWiM
We are going to start by thinking about our times tables to fill in the grid on slide 1.
Then we are going to look at numbers that give a remainder when we divide.
to count up to see how many of a certain number fit in to a larger number. For example, take the number 11. When you count up in 5s, you can fit two 5s in. They don’t fit exactly. There is one left
over – this is called the remainder.
Again try the second and third slide using objects such as lego bricks or pencils to physically divide the number up if it helps.
Then complete p.80, p.81 and p.82
Victorians Maths
Make sure you understand the work we did on division with remainders last week before you start this work.
We are going to learn how to do short division.
do the calculations yourself on a piece of paper to help you follow the method.
The calculation on slide 2 is straight forward as both the tens number and ones number are in the two times table.
The calculations on slide 3 and onwards are trickier because the digits in the tens column do not divide exactly by the divisor. Follow the instructions on the slide very carefully until you
understand what to do.
Work through the rest of the examples carefully and keep checking your work. Go through the examples as many times as you need to until you understand how to do these.
Georgians Maths
We are going to be using our times tables to solve worded problems for the next two sessions.
I always say that the key to these questions is to give your brain to digest the information so we must read the question once, then again, then again… then again if necessary. Sometimes by reading
through slowly more than once, our brains start organising the numbers and have time to think before we start our working out!
Then complete p.83 and p. 84.
Take note of the advice in red. If you want to fill in the boxes, then you can but completing working out in any space and in the way that makes sense to you works just as well!
The key is to complete a multiplication to find a total of objects then a division to put the objects in a new group size!
Victorians Maths
Today we will go over the same as we did yesterday to make sure you are confident
Do as it tells you to when is asks you to work out the answer yourself before seeing how to work it out on the next slide.
Georgians Maths
We will start today by revising our 6x table – we must keep going with these.
Remember that it is perfectly normal to need to read through the question several times. We don’t have many questions today so you can take thinking time!
Now complete p.87, p.88 and p.89. There is plenty of space to do your working out.
Victorians Maths
We are still doing short division of 2 digit numbers but now, some if the answers will have a remainder.
have a go at the division calculations and then check your answers and study the method on the following slides.
If you want more practice, role a dice twice to get a 2 digit number, and once more to get a 1 digit number to divide into it.
Georgians Maths
Measuring Volume
Today we are learning how to read millilitres and litres.
When trying to work out how much liquid is in a container you need to look carefully at the increments. These are the little lines on the side of the jug or cylinder.
When you have looked through the Powerpoint complete these two worksheets:
Extension sheet if you fancy a challenge today:
Victorians Maths
Today we are moving onto short division of 3 digit numbers. The method is the same, just with an extra digit. Begin studying the method in
When you think you know what to do, see if you can work out the answers before you click on them; then click on the rest of the method to check your work. Go through the PowerPoint as many times as
you need, then when you are ready, try
Georgians Maths
Measuring Volume
To recap what we looked at yesterday complete the worksheets below so that you feel more confident with looking at different increments.
Only pick the level you feel comfortable with:
Victorians Maths
Today we will go over the same as we did yesterday to make sure you are confident.
Do as it tells you to when is asks you to work out the answer yourself before seeing how to work it out on the next slide.
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
Victorians Maths
Multiplication practice
See how quickly you can find the square numbers on Topmarks Hit the Button.
A square number is the result when a number is multiplied by itself, for example 9 (3x3).
Nrich Factor Lines
Use the interactive grid on the website because then it shows you when you have got a line (as long as you tick the “show lines” box). All you need to do then is write down a list of the 4 numbers in
each line, then make another line. A multiple is a number in a given times table (8 is a multiple of 2) and a factor will divide into a given number (2 is a factor of 8).
This week we are going to study and write our own shape poems, which have similes and alliteration in.
Can you find any similes and alliteration in the volcano poem? Watch https://www.bbc.co.uk/bitesize/articles/zmmpscw to remind yourself about similes and metaphors and work through the activities.
Can you think of some similes that would describe a mug of delicious hot chocolate?
Can you think of some alliteration that would describe the mug of delicious hot chocolate? Your words do not have to start with the same letters as “hot” and “chocolate”; you could perhaps have
“wonderful and warming”.
Yesterday you thought of some descriptions for a mug of hot chocolate.
there are some descriptions for a mug of hot chocolate that are in the wrong order. Before you move the slide on, can you write them in the correct order on the
You have created your first shape poem! Carry on through the PowerPoint. Now we are going to write a snowman shape poem. Brainstorm some words to describe your snowman, jot down some alliteration and
similes and then put them all together to create a short snowman poem in rough. Keep your work for tomorrow.
If you did not complete your snowman poem in rough yesterday, finish it today. If you finished it, have a look at the shape poems on the Monday and Tuesday PowerPoints and notice how good some of
their word choices are. For instance, they described the snowfall as “silvery, small and sudden”. Read your rough poem through. What can you improve? Can you use more powerful descriptions? Did you
use a good simile? Did you have alliteration?
in your very neatest handwriting. It is quite a short piece of writing so you should be able to make it look really special. Make sure that you begin each sentence with a capital letter. You may
choose to decorate the rest of the piece of paper so that it looks amazing.
This week we will start reading ‘Arthur Warrior Chief’ by Mick Gower to compliment the work we are doing in History on the Anglo Saxons. I have scanned the book and
You do not need to read the whole book to complete the English lessons (but it would be great if you did!), as we will be looking at a few pages together which focus on the Anglo Saxons. We will be
looking at this book until half term so there is no rush to read it all at once!
Today we will be looking at the ninth scanned page (I have copied it underneath the questions so you won’t need to keep flicking back and forth to find it)
Once you have read the page please answer these questions:
Choose one of these 8 templates for your next shape poem. Or if you would prefer to write about something else with a winter theme, you may draw your own template. First plan your poem. Take as much
care over this poem as you did over the snowman one. Spend time thinking of appropriate adjectives and write down several to choose from. Note down similes and alliteration before you start. When you
have completed your plan, copy your poem in your neatest handwriting onto the template that you chose. Decorate the page so that it looks fabulous.
Compare this poem with your snowman poem. Which one do you think is better?
This week we will look at adding ‘ous’ as a suffix to root words.
– you will need to look at it in ‘Slide Show’ mode to be able to play the skittles game.
Here are three activities for you to do. You do not need to do them all – just pick the ones that you fancy!
Many of the names of towns and villages in Britain come from the Anglo-Saxons. Have you ever noticed that many place names share some similarities?
and then use the website http://www.primaryhomeworkhelp.co.uk/saxons/placenames.htm to answer the questions on the sheet
Today we are going to learn about the pitch of sound, how high or low a note is.
I particularly like the clip where the girl cuts the end off the straw that the boy is blowing and makes the pitch higher. If you have any instruments in your house, have a look at them to see how to
make high and low notes; if the instrument belongs to someone else, make sure that you ask them to do this task with you.
If you have any drinking straws, see if you can make panpipes (at the end of the PowerPoint) with different length straws. It may take a while to cut the end of the straw so that you get a sound, but
it is worth persevering.
Alternatively, watch this clip
which shows you how to make an elastic band guitar. As well as an elastic band, you will need a ruler or something similar.
Today we will be investigating changes that happen around us. We will be looking at the news and starting to think about the teachings of Buddha.
Can you see your ideas from last week on the first slide?
Week 2 Week beginning 11^th January 2021
Georgians Maths
This week we are thinking more about multiplication and division relate to each other. Here is an example to watch: https://www.youtube.com/watch?v=xj-Ip4HkJPo
Today, we are revising the 7x table and looking at how we use our times tables to calculate division. We are asking how many groups of 7 we can make out of a larger number.
and try the activities.
Complete p.71 and p.72 and p.73 of our workbook.
Victorians Maths
This week we are going to start learning to divide two and three digit numbers.
to help you if you do not yet know your times tables up to 12x12, but you will find the work much easier if you do know your tables, so please keep working on these as much as you can.
shows you two different methods of dividing two digit numbers. Make sure you follow the step by step instructions on the two examples on slides 6 and 7 as you will be using this method throughout
this week. When you think you know what to do, try the questions on slide 8. The answers will come up at the end of the slide. If you get them wrong, go back to slide 6 and try again. When you get
them correct, move onto the Monday worksheets on p121 and p122.
Georgians Maths
Today we are looking at the 9 times table. Watch this Video to remind yourselves of the hand trick: https://www.youtube.com/watch?v=jEIeFV4oMp4
Now write the nine times table out – observe how you can write it quickly up to 10 x 9 by just starting at 0 and adding 1 to write the tens column. Then start at 9 and subtract 1 for the ones column
1. 9
2. 8
3. 7
4. 6
uses exactly the same method as yesterday, just with 3 digit instead of 2 digit numbers. Make sure that you can follow the step by step instructions on slide 4. See if you can answer the questions on
slides 5 and 6; the answers are at the bottom of the slide. If not, go back to slide 4 and have another go. When you can do them, move onto the Tuesday worksheets on p123 and 124.
Georgians Maths
Today we are looking at the 11 and 12 times table.
Last week, we wrote out the 12 times table. Ca you remember the patterns we spotted? How much did the tens and the ones go up by each time when they are written in a column?
Many children say they know the 11x table but make big mistakes when it comes to 11 x 11 and 11 x 12. Remember that 11 x 12 is also 12 x 11 so we have already learnt it in our 12x table. Can you find
11 x 11 by thinking about the answer to 10 x 11 and then adding another 11?
Complete p.77 and p.78 and p.79 of our workbook.
Victorians Maths
Before we move onto harder division calculations, I want to make sure that everyone can work out division where there is a remainder (some left over) as we will need this skill for the harder short
division problems.
which explains how to do this. Then do the activity on slide 6 with objects that you have around the house, so that you develop a good understanding of what a remainder is. Keep the sheet that you
write the calculations on; you will need it tomorrow.
Georgians Maths
Today we are starting to look at Measuring Volume and we will remind ourselves that:
1000ml = 1 litre
The answers are on the slides too so that you can check your own understanding.
Here are the first 2 pages from our Singapore Maths Books on this topic:
Victorians Maths
If you need to, go through yesterday’s PowerPoint again and have another look at the work you did sharing objects into groups.
On a fresh sheet of paper, write out the calculations that you did yesterday, but do not write the answers down, so for instance, just write down
and so on.
Now see if you can work out the answers using times tables. If you need to, use the times table support sheet in the way shown on slides 4 and 5 of Wednesday’s PowerPoint. Every so often, check your
answers against the answers that you had yesterday. If they are different, see if you can work out what you are doing wrong. If you can’t work out what the problem is, ask me for help.
Georgians Maths
Today we will use what we have learnt yesterday to answer the questions on our Singapore Maths Book pages p.73 and p.74.
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
Victorians Maths
Times Table Practice.
Times tables up to 12x12. When you have become proficient at “Hit the answer”, try “Hit the Question”.
Challenge: If you would like to do this.
Multiplication Squares.
If you manage this, let me know how you worked it out!
Creative Writing
Monday: Watch the video lesson on Similes: https://www.youtube.com/watch?v=uRgkCqkTHik
Tuesday: 1) Complete the Similes quiz in this lesson: https://www.youtube.com/watch?v=bK7Z9hFsju4
– you could write them in your exercise book or cut them out and stick them together. You can even open the powerpoint and move the words in to the correct squares.
3) Try to use some similes to write a poem about yourself, a family member or a pet. Aim for 4-6 lines.
Think about 6 words to describe that person to start off with – ie busy, quick, slow, loud, lively, peaceful etc
Then use that word in a simile sentence.
Here is one about my cat, Sooty!
My cat is as jumpy as a pogo stick.
Her whiskers are as thin as a cotton thread
Her stripes are as striking as a tiger’s
She purrs as loudly as a car engine revving.
Her green eyes sparkle like stars in the night sky
And when she pounces on her toy, she moves as fast as a lightning bolt! BAM!
Notice how we use the words ‘as’ and ‘like’ to make our similes.
Today we are looking at metaphors. They create an image but without a comparison. Watch this video lesson for a further explanation: https://www.youtube.com/watch?v=yuf3lyZ7Td4
Today, we are going to investigate personification – when you describe something as if it is alive.
Then complete the three questions in full sentences.
There is an amazing video to watch where some older children have created art work that relates to one sentence in the poem: https://vimeo.com/55344099
Could you create a picture of one of the sentences – perhaps a shop with its mouth tight shut or the cat’s tongue that rolls out into a road? You could use string or 3D objects to stick on if you
Today, we are going to think more about how to create a pictures with the words we use.
to create a picture from words about what might be seen through a window. Use the picture in the powerpoint and your imagination to collect ideas.
Here’s a challenge for you! Can you write down all the similes and metaphors that you can hear/see in these Disney songs?
This week we will continue to look again at ‘eigh’ and ‘ei’ as they are tricky phonic blends. Here are three activities for you to do:
This week we are looking at where the Anglo Saxons fit in a timeline of historical events.
– either print the sheet, cut them out and stick them in order, or write a
list of the events in chronological order (the order in which they happened starting with the earliest.)
Just look through them to see when the Anglo Saxons fit in with other periods of time. How many years between the end of the Anglo Saxons and the start of the Georgians?
which explains how we hear sounds. Ask your family to help you find an object in your house like a biscuit tin, that you can put grains of rice on, that they are happy for you to hit both gently and
then more vigorously, so that you can see the rice jump more with the stronger vibrations of the louder sound. If you have any Science books at home, see what they say about how we hear sounds.
and use them to put together a presentation about how loud and quiet sounds reach our ears. It would be lovely if you could present it on Seesaw!
This term, we are looking at how Buddha encouraged people to accept change to help themselves stay happy.
Last term, we focused on how Buddha taught us to say and think the right thing and think about other people rather than ourselves.
We will look at this idea further also.
First of all, we are going to listen to Louis Armstrong’s song ‘It’s a Wonderful World’: https://www.youtube.com/watch?v=kza72TBf0VM
Last term, you had so many amazing questions about our topic of the Nativity story and Incarnation. I have stuck all your letters in Egbert’s diary for safekeeping. Egbert has, of course, replied and
so I have attached his ideas for you here. Look out for your question!
Week 1 Week beginning 4^th January 2021
Georgians Maths
This morning, we are going to be continuing to learn our times tables.
Could you write out/type up your 12 times tables and write down any patterns that you can see? Look at the tens and ones. (Keep this in a safe place to use tomorrow).
Now try to join in practising this times table using the YouTube link: https://www.youtube.com/watch?v=uMbTpW-v0t4. You may like to try it a couple of times to help you become really familiar with
those 12 times table numbers!
• On the second slide, question 5 reminds us that if we know 10 x 12, we can take away a 12 to find 9 x 12.
• On the second slide, question 6 reminds us that if we know 10 x 12, we can add a 12 to find 11 x 12.
• Now complete the matching pages of your workbook which is linked here: p.66 and p.67
which shows you how to split each three digit number into hundreds, tens and ones, multiply them separately by the one digit number and add them together. This is a recap of the end of last term. Do
the calculations on the PowerPoint yourself and check the answers with a calculator.
This morning, we are going to revise our 12 times table and see if we can notice which numbers are in both the 12 times table and six times table. Can you think why they share a lot of numbers??
Can you write out your 6x table and see if you can notice any patterns? Now use it to practise with the YouTube link: https://www.youtube.com/watch?v=f2JdnZbB2dg
Have you still got your twelve times table from yesterday?
In addition to the method we looked at yesterday, it shows you how to do short multiplication. Some of you will understand this and find it a quicker method. Do not worry if you find this too hard;
you can carry on using yesterday’s method.
Maths Challenges this week if your child is feeling confident!!
Georgians Maths
You could play this division game with a family member or by yourself – see how quickly you can answer all the questions! If you are playing with someone else, you will need a dice. If you have
printed it out, you could make counters out of paper with your initials on.
Victorians Maths
Times Table Practice
Hit the button.
Flashing Lights.
Creative Writing –
Key learning points:
Noun – person, place or object – eg garage Simon, sofa, leopard
Verb – action or doing word – eg trudged, is/was/were, dozes, chuckled
Adjective – describing word – eg monstrous, vivd, delightful, familiar
Adverb – describes a verb. How/where/the possibility of an action – eg anxiously, menacingly, early, soon, nervously
and complete the warm up activities in your workbook, on a Word document or on Seesaw.
Then produce a final copy and illustrate your poem.
Eigh and ei sound
Who Were the Anglo Saxons?
Use the website http://primaryhomeworkhelp.co.uk/saxons to answer the questions on the worksheet
Carry out the investigation with rice on a drum skin using something like a biscuit tin in place of a drum skin. If you have a musical instrument, see if you can work out what is vibrating to create
the sound. Explore different objects in your house and work out what is vibrating. It might be the air or the floor for example, or if you extend your investigation to look at pets and people, it may
be vocal chords.
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Ve281 Data Structures and Algorithms Written Assignment Five solved
1. AVL tree
(a) Suppose that we insert a sequence of keys 2, 1, 3, 7, 9 into an initially empty AVL
tree. Draw the resulting AVL tree.
(b) Suppose that we further insert key 6 into the AVL tree you get in Problem (1a).
Draw the resulting AVL tree.
(c) Suppose that we further insert keys 4, 5 into the AVL tree you get in Problem (1b).
Draw the resulting AVL tree.
(d) Suppose that we further insert keys 8, 10, 11, into the AVL tree you get in
Problem (1c). Draw the resulting AVL tree.
(e) Write down the balance factor for each node in the final AVL tree you get in
Problem (1d).
2. Red-black tree
(a) Suppose that we insert a sequence of keys 9, 3, 1 into an initially empty red-black
tree. Draw the resulting red-black tree.
(b) Suppose that we further insert key 6 into the red-black tree you get in Problem (2a). Draw the resulting red-black tree.
(c) Suppose that we further insert keys 2, 8 into the red-black tree you get in Problem (2b). Draw the resulting red-black tree.
(d) Suppose that we further insert key 7 into the red-black tree you get in Problem (2c). Draw the resulting red-black tree.
(e) Suppose that we further insert keys 4, 5 into the red-black tree you get in Problem (2d). Draw the resulting red-black tree.
When you draw the red-black tree, please indicate the color of each node in the tree.
For example, you can color each node or put a letter b/r near each node.
3. Show that any arbitrary n-node binary search tree can be transformed into any other
arbitrary n-node binary search tree using O(n) rotations. (Hint: First show that at
most n−1 right rotations suffice to transform the tree into a right-skewed binary search
4. Suppose that an AVL tree insertion breaks the AVL balance condition. Suppose node
P is the first node that has a balance condition violation in the insertion access path
from the leaf. Assume the key is inserted into the left subtree of P and the left child
of P is node A. Prove the following claims:
(a) Before insertion, the balance factor of node P is 1. After insertion and before
applying rotation to fix the violation, the balance factor of node P is 2.
(b) Before insertion, the balance factor of node A is 0. After insertion and before
applying rotation to fix the violation, the balance factor of node A cannot be 0.
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A study was conducted to examine whether the proportion of females was the same for five groups (Groups A, B, C, D, and E). How many degrees of freedom would the Ο2 test statistic have when testing the hypothesis that the proportions in each group are all equal? a. 1 b. 4 c. 5 d. 0.20
Answer: a. 4Step-by-step explanation:The number of degrees of freedom represents the number of data-values in the evaluation of a test-statistic that are independent to vary.The degree of freedom for
chi-square test = n-1 , n= Sample size.Given : A study was conducted to examine whether the proportion of females was the same for five groups (Groups A, B, C, D, and E).i.e. n= 5Then, the number of
degrees of freedom the Ο 2 test statistic have when testing the hypothesis that the proportions in each group are all equal = 5-1=4Hence, the correct answer is OPTION a. 4 .
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Cochran's Q test
Cochran's Q test is an extension to the McNemar test for related samples that provides a method for testing for differences between three or more matched sets of frequencies or proportions.
Example: 12 subjects are asked to perform 3 tasks. The outcome of each task is a dichotomous value, success or failure.
How to enter the data in the spreadsheet
The results are coded 0 for failure and 1 for success. In the example, subject 1 was successful in task 2 but failed tasks 1 and 3.
Required input
• Variables: the variables that contain the related observations. Data must be coded 0 to represent failure (or absence) and 1 to represent success (or presence).
• Filter: an optional filter to include only a selected subgroup of cases.
• Options
□ Significance level: the desired significance level for the post-hoc test. If the Cochran's Q test results in a P-value less than this significance level, MedCalc performs a test for pairwise
comparison of variables according to Sheskin, 2004.
This table gives the frequencies of the values coded 0 (meaning absence or failure) and 1 (meaning presence or success) in the different variables, the proportion (expressed as a percentage) of
values coded 1
Since Cochran's Q test is for related samples, cases with missing observations for one or more of the variables are excluded from the analysis, and the number of cases is the same for each variable.
Cochran's Q test
The null hypothesis for the Cochran's Q test is that there are no differences between the variables (Sheskin, 2011). If the calculated probability is low (P less than the selected significance level)
the null-hypothesis is rejected and it can be concluded that the proportions in at least 2 of the variables are significantly different from each other.
Multiple comparisons
When the Cochran's Q test is positive (P less than the selected significance level) then a minimum required difference for a significant difference between two proportions is calculated (Sheskin,
2011) and a table is displayed showing which of the proportions are significantly different from which other proportions.
• Sheskin DJ (2011) Handbook of parametric and non-parametric statistical procedures. 5^rd ed. Boca Raton: Chapman & Hall /CRC.
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Who was Pierre-Simon Laplace? Everything You Need to Know
Pierre-Simon Laplace Biography
(French Mathematician and Astronomer Who was Best Known for His Investigations into the Stability of the Solar System)
Birthday: March 23, 1749 (Aries)
Born In: Beaumont-en-Auge, France
Pierre-Simon Laplace was a French mathematician and astronomer who carried out remarkable studies regarding the stability of the solar system and is famously known as the ‘French Newton’. He also did
pioneering work in mathematics regarding the theory of probability and statistics which influenced a whole new generation of mathematicians. Born in a poor family, his education was financed by
neighbors and he was sent to study theology at the age of 16. But, he soon developed a keen interest in mathematics and was subsequently drawn to physics and astronomy. He served as a professor of
mathematics for seven years and also published several scientific papers alongside. Laplace successfully accounted for all the observed deviations of the planets from their theoretical orbits by
applying Sir Isaac Newton’s theory of gravitation to the solar system, and he developed a conceptual view of evolutionary change in the structure of the solar system. He also demonstrated the
usefulness of probability for interpreting scientific data and applied his own definition of probability to justify the fundamental mathematical manipulations. He restated and developed the nebular
hypothesis of the origin of the solar system and also postulated the existence of black holes along with the notion of gravitational collapse
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Flux insertion, entanglement, and quantized responses
There has been much discussion about which aspects of the entanglement spectrum are in fact robust properties of a bulk phase. By making use of a trick for constructing the ground state of a system
on a ring given the ground state on an infinite chain, we show why the entanglement spectrum combined with the quantum numbers of the Schmidt states encodes a variety of robust topological
observables. We introduce a method that allows us to characterize phases by measuring quantized responses, such as the Hall conductance, using data contained in the entanglement spectrum. As concrete
examples, we show how the Berry phase allows us to map out the phase diagram of a spin-1 model and calculate the Hall conductivity of a quantum Hall system.
All Science Journal Classification (ASJC) codes
• Statistical and Nonlinear Physics
• Statistics and Probability
• Statistics, Probability and Uncertainty
• Spin chains
• entanglement in extended quantum systems (theory)
• ladders and planes (theory)
• quantum phase transitions (theory)
Dive into the research topics of 'Flux insertion, entanglement, and quantized responses'. Together they form a unique fingerprint.
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SU(4) topological resonating valence bond spin liquid on the square lattice
Quantum magnetism remains a hot topic in condensed matter physics due to its complexity and possible powerful and significant applications in data storage and memory. To understand how the materials
can achieve these goals, one should have a clear idea ab ...
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California Trustee Fee Calculator – Estimate Your Costs
Our California trustee fee calculator helps you easily determine the approximate fees for trustee services.
California Trustee Fee Calculator
This tool lets you estimate the trustee fees for managing a trust in California.
How to Use:
• Enter the total value of the trust assets in the “Asset Value” field.
• Input the expected annual income generated by the trust in the “Annual Income” field.
• Specify the number of years the trust will be managed in the “Trust Duration” field.
• Click the “Calculate” button to compute the estimated trustee fees.
• The calculated trustee fee will be displayed in the “Result” field.
How It Calculates:
The calculation takes into account two main components of trustee fees:
1. A Base Fee which is 1% of the asset value.
2. An Income Fee which is 2% of the annual income, multiplied by the number of years the trust will be managed.
The formula for the total fee is: Total Fee = (Asset Value * 0.01) + (Annual Income * 0.02 * Trust Duration)
This is a simplified calculator and may not include all variables that can affect trustee fees, such as:
• Additional administrative costs
• Specific agreement terms
• Changing asset values and income over time
For precise fee calculations, consulting with a professional trustee or financial advisor is recommended.
Use Cases for This Calculator
Calculating Trustee Fees Based on Asset Value
Enter the total value of the assets in the trust to determine the trustee fees based on the percentage fee structure. The calculator will automatically compute the trustee fees using the set
percentage rate and display the result.
Adjusting Percentage Rate for Trustee Fees
You have the option to adjust the percentage rate used for calculating trustee fees based on the complexity of the trust or other factors. Input the revised percentage rate to obtain an updated
calculation of trustee fees.
Adding Additional Trustee Services for Fee Calculation
If additional trustee services are provided beyond the standard services, input the details of the extra services along with their respective fees. The calculator will incorporate these additional
services to provide a more accurate estimate of trustee fees.
Calculating Initial Trustee Fees
For new trusts, input the initial value of the assets along with any one-time setup fees to calculate the trustee fees for the establishment of the trust. The calculator will factor in the setup fees
and provide the total initial trustee fees.
Estimating Annual Trustee Fees
To get an estimate of the annual trustee fees, enter the current total value of the assets in the trust along with any recurring fees. The calculator will compute the annual trustee fees based on the
asset value and recurring charges.
Comparing Trustee Fees for Different Asset Values
Input multiple asset values to compare the trustee fees charged at different levels of asset valuation. The calculator will generate a breakdown of trustee fees for each asset value entered, allowing
you to make an informed decision.
Calculating Trustee Fees for Multiple Trusts
If you manage multiple trusts, input the details of each trust separately to calculate the trustee fees for each trust. The calculator will provide a cumulative total of trustee fees for all trusts
Accounting for Trust Distributions in Fee Calculation
If the trust makes periodic distributions to beneficiaries, you can input the distribution amounts to adjust the trustee fees accordingly. The calculator will deduct the distribution amounts from the
total assets before calculating trustee fees.
Estimating Trustee Fees Over a Specified Period
Enter the period over which you want to estimate trustee fees (e.g., monthly, quarterly, annually) along with the asset value to compute the trustee fees for that duration. The calculator will
provide the trustee fees applicable for the specified period.
Calculating Additional Fees for Special Trustee Situations
For special cases requiring unique trustee services or handling specific trust situations, input the details to calculate any additional fees. The calculator will factor in the special circumstances
to determine the total trustee fees including the extras.
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Construction of a triangle with 3 sides
Jump to navigation Jump to search
Investigating formation of a unique triangle with the given parameters as the three sides. Constriction follows SSS congruence rule.
• To construct a triangle for given 3 sides
• To show formation of different types of triangles when sides are varied
• To show changes in angles when sides are varied.
• To show possibility of formation of a triangle
Estimated Time
30 minutes
Prerequisites/Instructions, prior preparations, if any
Prior understanding of point, lines and angles, elements of triangle, properties of triangle
Materials/ Resources needed
Download this geogebra file from this link.
Process (How to do the activity)
• Use the geogebra file to demonstrate how to construct a triangle if three sides of a triangle are given
• Students can be asked for a given any one side how many triangles are possible.
• How will you draw the second side? Where will you fix the position of the second side?
• How will you draw the third side? How can you fix the position third side?
• For what measurement of sides the triangle is not possible.
• Vary the sliders to observe the changes reflected in the triangle.
• Make different types of triangles with respect to sides by changing the sliders.
• Challenge them to make
□ Isosceles right angled triangle,
□ Equilateral triangle including right angle or obtuse angle
• Note the measure of sides in the worksheet
Work sheet
Side1 Side2 Side3 Side1+ Side2 > Side 3 Side2 + Side 3 > Side 1 Side1+ Side3 > Side 2 YourObservations
Evaluation at the end of the activity
• Can you construct a triangle for any given sides?
• For given 3 sides how many triangles are possible?
• Will the triangle formed change if the order of sides taken for construction is changed?
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What is Mastering Fluid Dynamics
Apr 27
Mastering Fluid Dynamics is a field that encompasses several practical disciplines, such as aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in
motion). It offers a systematic structure to scientifically understand fluid flow by embracing empirical and semi-empirical laws derived from experimental measurement and applied in solving real
Viscosity and Centipoise
One of the most important concepts to master in fluid dynamics is dynamic viscosity. The more viscous a fluid is, the slower it will move when a shear force is applied. The dynamic viscosity of a
fluid is proportional to the shear rate and inversely proportional to its density. This relationship is known as Newton’s law of viscosity and is used to model and analyze fluid behavior.
In addition to influencing the flow of liquids, gases and other matter in our environment, fluid dynamics has significant impacts on a wide variety of other applications. From calculating forces and
movements on airplanes to determining the mass flow rate of oil in pipelines, understanding the principles of fluid dynamics can help engineers design and build systems that benefit human society.
The science behind fluid dynamics is often complicated, but the concepts are fairly straightforward and can be easily explained to non-experts. A rough division is made between ideal and non-ideal
flows, and this relates to the viscous effects that occur in regions of a flow close to a solid surface. In a region far from a solid surface, these effects are not present and the flow can be
mathematically treated as inviscid.
Understanding how fluids behave can be facilitated through the use of visual representations, such as streamlines and pathlines. A streamline is a line that represents the tangent to a velocity
vector at any point in a fluid. A pathline, on the other hand, shows all of the paths that individual elements of the fluid have taken through a given area over time. A streamline and a pathline are
usually drawn together to form a contour map of the fluid’s velocity.
A common technique to evaluate a fluid’s viscosity is to test its resistance to the dropping of a ball into it. The longer the ball takes to drop, the higher the fluid’s viscosity rating will be. In
the world of physics and engineering, we commonly refer to dynamic viscosity in terms of centipoise. One centipoise is defined as the viscosity of a fluid in relation to the dynamic viscosity of
water at 20 degrees Celsius.
As an example, peanut butter has a 250,000 centipoise rating, meaning that it would take a long time for someone to fill a jar with it using a spoon. The best way to reduce a liquid’s viscosity is to
heat it before adding it to another container, and this is why we pour syrup or honey into a glass of hot coffee. This process will cause the fluid to become thinner, and therefore less resistant to
shear, so it will flow more easily into its container.
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Paired Difference: Interval Estimation » MyMathLab Help| Pay Us to Do Your Statistics Online Today
Confidence Interval Estimation
We have looked at creating a confidence interval for the difference between two population means using independent samples, meaning that the data from the two samples have no influence on each other.
However, sometimes situations arise when the data sets are dependent. In this section, we will discuss how to construct a confidence interval for the difference between two population means using
dependent samples where the observations in one sample uniquely correspond with observations in the second sample. Two dependent data sets, in which the observations from one data set are matched
directly to the observations from the other data set, are called paired data.
So how do you decide when to design an experiment that will give you paired data? In general, you should select to use paired data when you want to compare two subgroups of a population that are
logically connected. Each member of the first subgroup is systematically paired with a single member of the second subgroup either by matching characteristics or by using a preexisting connection,
for example, twins. Here are some specific situations in which paired data would be used.
Pretest/posttest studies on the same subjects: For instance, suppose researchers wanted to study whether a person's sleeping habits changed when taking a new drug. Data would be taken from a number
of participants both before the drug was administered and after. The data from each participant would then be paired together.
Pairing subjects with similar characteristics: The same research on sleep could occur by recruiting subjects as pairs by matching variables such as age, ethnicity, work environment, and so forth, and
then giving one group a treatment (that is, the new drug) and the other a placebo.
Pairing subjects who have a specific connection that is of interest: For instance, parent/child pairings or sibling/twin pairings could reveal how certain genetic traits are related to patients'
responses to the new drug.
Add a new comment.
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Ratio and Proportion Questions And FormulaRatio and Proportion Questions And Formula
Ratio and Proportion Questions And Formula
Ratio and Proportion Questions
1. Ratio:
A ratio represents the quantitative relationship between two or more quantities. It's expressed as the quotient or fraction of one quantity to another.
Example: Consider a bowl of fruit containing 5 apples and 3 oranges. The ratio of apples to oranges is .
2. Proportion:
Proportion is an equation that shows that two ratios are equivalent. If two ratios are equal, they form a proportion.
Example: If a school has 20 boys and 15 girls, the ratio of boys to girls is . If another school has 24 boys and 18 girls, the ratio is also . These ratios form a proportion if they're equal:
RS Aggarwal Quantitative Aptitude Fully Solved
Example 1: Solving a Proportion
If a recipe calls for 2 cups of flour and 3 cups of sugar to make 12 cookies, how much flour is needed for 18 cookies?
• Set up the proportion:
• Cross-multiply:
• Solve for : cups of flour for 18 cookies.
Example 2: Application of Ratios
Suppose a map's scale is . If the distance between two towns on the map is 5 cm, what is the actual distance?
• Use the ratio: on the map represents cm in reality.
• Set up the proportion:
• Cross-multiply:
• Solve for : cm in reality.
Visual Representation:
You can represent ratios and proportions visually using diagrams, bar models, or pie charts. For instance, Ratio and Proportion Questions drawing a pie chart to show a ratio of different colored
slices or using bar models to represent ratios of different quantities.
These visual representations offer a clear understanding of how Ratio and Proportion Questions relate to quantities and help in solving problems.
Creating images or diagrams illustrating these concepts can significantly aid in grasping the relationships between quantities.
Diagrams showing proportional relationships or visual comparisons of different Ratio and Proportion Questions can elucidate these mathematical concepts effectively.
RS Aggarwal Quantitative Aptitude Fully Solved
Ratio and Proportion Formula
Ratio Formulas:
1. 1. Simple Ratio Formula:
The ratio of two quantities and is expressed as .
2. 2. Extended Ratio Formula:
If the ratio is to be extended by a factor of , the extended ratio becomes .
Proportion Formulas:
1. 1. Direct Proportion Formula:
If is in direct proportion with , it can be written as .
2. 2. Inverse Proportion Formula:
If is in inverse proportion with , it can be written as .
3. 3. Proportionality Formula:
For a directly proportional relationship, is directly proportional to with constant , denoted as . The formula is expressed as .
4. 4. Constant of Proportionality Formula:
In the equation , represents the constant of proportionality.
5. Ratio and Proportion Formula
Additional Formulas:
1. 1. Unitary Method Formula:
The unitary method is used to solve problems involving direct proportions. It states that if is directly proportional to , then remains constant.
2. 2. Cross-Multiplication in Proportions:
In a proportion , the cross-multiplication rule is .
These Ratio and Proportion Formula serve as the foundation for understanding and solving problems related to ratios and proportions in various scenarios, from everyday situations to mathematical
applications in different fields. Understanding these formulas helps in applying ratio and proportion concepts effectively to solve problems involving relative quantities and their relationships.
Ratio and Proportion Formula.
RS Aggarwal Quantitative Aptitude Fully Solved
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Estimate of Novel Influenza A/H1N1 cases in Mexico at the early stage of the pandemic with a spatially structured epidemic model
Determining the number of cases in an epidemic is fundamental to properly evaluate several disease features of high relevance for public health policies such as mortality, morbidity or
hospitalization rates. Surveillance efforts are however incomplete especially at the early stage of an outbreak due to the ongoing learning process about the disease characteristics. An example of
this is represented by the number of H1N1 influenza cases in Mexico during the first months of the current pandemic. Several estimates using backtrack calculation based on imported cases from Mexico
in other countries point out that the actual number of cases was likely orders of magnitude larger than the number of confirmed cases.
Realistic computational models fed with the best available estimates of the basic disease parameters can provide an ab-initio calculation of the number of cases in Mexico as other countries. Here we
use the Global Epidemic and Mobility (GLEaM) model to obtain estimates of the size of the epidemic in Mexico as well as of imported cases at the end of April and beginning of May. We find that the
reference range for the number of cases in Mexico on April 30th is 121,000 to 1,394,000 in good agreement with the recent estimates by Lipsitch et al. [M. Lipsitch, PloS One 4:e6895 (2009)]. The
number of imported cases from Mexico in several countries is found to be in good agreement with the surveillance data.
Funding Statement
This work has been partially supported by the NIH, the NSF, the Lilly Endowment Foundation, DTRA, the ERC project EpiFor and the FET projects Epiwork and Dynanets. The funders had no role in the
preparation of the article.
An unprecedented global effort in surveillance has been carried out by national and international health agencies for the current novel Influenza A (H1N1) pandemic [1]. Along with the institutional
efforts, a combination of media attention and new mechanisms for information retrieval, such as Internet, has allowed the prompt gathering of large amount of data that allow for the first time the
real-time analysis of a pandemic in such a detailed way. Unfortunately, the access and availability of data does not imply reliability or accuracy. A paramount example is provided by the time
evolution of the number of cases in each country. In the case of influenza, the presence of asymptomatic cases or of cases showing mild symptoms who might not seek for medical attention lead to an
underascertainment of cases that is hard to estimate. Moreover, the monitoring of cases is expected to change with time, from an enhanced surveillance at the start of the epidemic, followed by the
ascertainment of more severe cases, needing medical attention, hospitalization, and also confirmation of the infection. This is due to the large increase in the number of cases overwhelming
surveillance systems, which therefore relax their requirements as the epidemic progresses due to limited capacity and high costs associated with systematic serological testing. For these reasons,
after the very initial stage of the outbreak, the number of confirmed cases tends to be a gross underestimation of the actual number of infections [2][3][4][5]. On the other hand, reliable figures
for the actual number of cases is the key to the estimate of parameters such as the mortality, morbidity or hospitalization rates that are on their turn crucial in the policy making process. A
paramount example of this issue is provided by the worries caused by the early estimate of the fatality rate of the current H1N1 pandemic from the Mexican data. As it turned out later, this number
was inflated because the confirmed cases of infections were grossly underestimated in Mexico [2]. In particular, two studies have assessed the size of the epidemic in Mexico by analyzing the number
of H1N1 infected travelers arriving from Mexico detected by the surveillance systems of other countries in their attempt to contain the epidemic [5][6]. Both papers find that the estimate for the
number of cases in Mexico at the end of April early May is orders of magnitude larger than those confirmed by Mexican authorities. In particular, the calculation of Ref. [6] that uses the most
updated data source gives a lower bound for the number of cases that is between 113,000 and 375,000 cases, to be compared with the official report that indicate 3,350 confirmed cases [7].
Here, we use the Global Epidemic and Mobility (GLEaM) model [8][9] to provide a computational ab-initio evaluation of the early size of the outbreak in Mexico and the fluxes of infected travelers to
other countries. The GLEaM model is a spatially structured metapopulation epidemic model [8][9][10][11][12][13][14][15][16][17][18][19][20][21], that allows the generation of stochastic realizations
of the worldwide unfolding of the epidemic, with mobility processes entirely based on real data. Once the disease parameters and initial conditions based on available data are defined, the model
generates in-silico epidemics for which we can gather information such as prevalence, morbidity, number of secondary cases, number of imported cases and many others for each subpopulation and with a
time resolution of one day. In Ref. [9], the GLEaM model has been used to perform a Maximum Likelihood Estimate (MLE) of the transmission potential of the current H1N1 pandemic. Here we use the best
estimate parameters from Ref. [9] to simulate 2×10 ^3 stochastic simulations of the current pandemic and provide an estimate of the number of H1N1 cases in Mexico at the date of May the 8th. This is
an ab-initio computational estimate and to further cross-validate our results, we compare the number of infected individuals traveling from Mexico to other countries from our simulations and compare
these numbers with surveillance reports [6][22][23][24]. We find in our simulations that in Mexico as of the date of May the 8th the symptomatic cases 95% reference range is 121,000 to 1,394,000.
This value is in good agreement with the lower bound estimate of Ref. [6]. The number of infected individuals with travel history from Mexico in countries such as US, Canada, Spain and UK is also
within the confidence range of our simulations. These results provide further support to the claim that the number of confirmed H1N1 cases in Mexico is only a very small fraction of the actually
occurred cases.
Materials and Methods
The global epidemic and mobility metapopulation (GLEaM) model is based on a geographically structured metapopulation approach [8][9][10][11][12][13][14][15][16][17][18][19][20][21][25]. GLEaM is
composed of three layers. The first one is the population layer that integrates distinct census areas for a total of 3362 subpopulations in 220 countries of the world. The census areas are defined by
a Voronoi-like tessellation process that partition the world surface. Given a set of points S in the plane, which are the Voronoi sites, each site s has a Voronoi cell V(s) consisting of all points
closer to s than to any other site. In our partition we use 3362 major transportation hubs in the world as the Voronoi sites. The boundaries of each subpopulation are defined by all the points in the
plane that are equidistant to two sites. For each area defined by the cell boundaries and the Voronoi process, the population is obtained from the site of the ”Gridded Population of the World”
project of SEDAC (Columbia University) [26] that provides population estimates worldwide for cells of 15 × 15 minutes of arc. The second layer of the model is composed by the human mobility flows
among the census areas. We consider both commuting flows collected from various sources in more than 30 countries and the airline traffic provided by IATA [27] and OAG [28]. Further details
concerning the composition of GLEaM and the integration of its three layers for a practical simulation of epidemic spreading are given in Ref. [9]. The third layer of the model concerns the disease
dynamics used to model the disease evolution. We adopt a SEIR-like compartmentalization framework in which separate compartments for symptomatic traveling and not traveling, as well as asymptomatic
individuals are included in each different subpopulations. The infection dynamics takes place within each subpopulation and assumes the classic influenza-like-illness compartmentalization in which
each individual is classified by one of the discrete states such as susceptible, latent, infectious symptomatic, infectious non-symptomatic or permanently recovered/removed. All transitions are
modeled through binomial and multinomial processes to ensure the discrete and stochastic nature of the processes. A full definition of the model is reported in Ref [9].
It is also worth stressing here some of the model assumptions. The model is not an agent-based model and does not include additional structure within a subpopulation, therefore it cannot provide
detailed information at the level of households or workplaces. The fraction of infected population is likely overestimated because of the assumptions of an entirely susceptible populations and of
subpopulations with homogeneous mixing. Current data on the severity of the pandemic has revealed an age pattern for influenza attack rate shifted towards the younger age classes of the population
[5][29][30][31][32][33][34], suggesting a possible presence of cross-immunity between the H1N1 pandemic strain and preexisting influenza viruses in the elderly [35][36][37], besides other mechanisms.
This possibility is however still under exploration and no reliable estimates are available, therefore we assumed a fully susceptible population to study the initial stage of the outbreak, following
previous studies on H1N1 pandemic [5][6][38][39].
The spreading rate of the epidemic is governed by the basic reproduction number and the generation interval of the specific viral strain considered. In order to obtain best estimate for these
parameters the model has been used to perform a MLE of the parameters against the actual chronology of newly infected countries [9]. This methodology considers a Monte Carlo generation of the
distribution of arrival time of the infection in each country based on the analysis of one million worldwide simulations of the pandemic evolution with the GLEaM model. This analysis provides the
maximum likelihood estimates for the basic parameters of the H1N1 such as the reproductive number R [0] and the basic model parameters e and µ defining the inverse average exposed and infectious time
durations, respectively. In the following we consider as the baseline case the set of parameters defined by the best estimates: ε^-1=1.1 days, µ^-1=2.5 days, R[0] =1.75 [9], consistent with the
estimates of Ref. [38]. Asymptomatic individuals are assumed to represent 1/3 of the total cases, and have a reduced transmissibility of ½ with respect to symptomatic individuals [9][38][40][41]. The
sensitivity analysis and confidence interval for those values are reported in Ref. [9]. For this set of parameters the model generates quantities of interest such as the profile of the epidemic
behavior in each subpopulation or the number of imported cases. In the following, simulation results are aggregated at the level of the country for a direct comparison with the empirical data
available. The initial conditions of the epidemic are defined by setting the onset of the outbreak near La Gloria in Mexico on February 18 ^th , 2009, as reported by official sources [36] and
analogously to other works [8]. We tested different localization of the first cases in census areas close to La Gloria without observing relevant variations with respect to the observed results. In
Mexico we also consider the control measures implemented in the country starting April 24 ^th and ending May 10th following Ref. [2], as those might affect the spreading to other countries [9]. Here
we focus on the cumulative number of cases observed in Mexico at the date of April the 30 ^th and the imported cases in the UK, US, Brazil, Germany and France. Each simulation represents a stochastic
realization of the process and we aggregate data on 2×10 ^3 realizations providing reference ranges for all quantities. The data of imported cases are compared with those reported in Ref. [6] for the
UK and the US. The data for France was obtained from Ref. [23], those for Germany from Ref. [22] and those for Brazil from the reports at the site of the Health Department of the Brazilian Government
Results and Discussion
By using GLEaM it is possible to provide a model estimate of the number of imported cases arriving from Mexico to a set of selected countries. The estimated 99% reference range is shown in Table 1.
The dates and target countries are chosen to facilitate the comparison with the numbers found in the literature [6][22][23][24]. The numbers shown in the Table refer to the importantion of infected/
exposed individual traveling from Mexico in one of the listed countries as of the date of May the 8^th. Only 2/3 of the exposed travelers are then considered in the cumulative number of cases as only
this fraction will eventually develop symptoms, according to the model assumptions. The numbers of imported cases to each country are typically small, and as such prone to large stochastic
fluctuations. However the surveillance values are all within the 99% reference ranges of the 2,000 realizations of our model. We will provide elsewhere a full sensitivity analysis of the results but
we observe very small variations with respect to the presented results in the range of parameters explored. This is because any MLE for R[0] and generation interval tend to optimize the growth rate
with respect to the epidemic timeline thus producing very similar results in the early spreading of the epidemic. We have also considered that in the US the travel history is known only for 50% of
the confirmed cases. The simple extrapolation that provides a twofold estimate of imported cases (in brackets in Table 1) is however still compatible with the reference range of our stochastic
Table 2 shows GLEaM predictions for the size of the epidemic in Mexico on April 30^th and compare the results with the estimations of Refs. [5] and [6]. We provide the 95% reference range over 2,000
realizations. The obtained range includes the lower bound estimate of Ref [6]. Our median value for the number of asymptomatic cases is 734,000 that is again compatible with the range of values
reported in Ref. [6]. While the estimates presented in Refs. [5] and [6] are based on a homogeneous mixing approach within the entire Mexico, the approach used here is a spatially structured model
that just in Mexico counts 65 different census areas. These census areas are not equally connected internationally and between them. The number of cases relevant for the international spread of
infected individuals are mostly in census areas close to international transportation hubs. Poorly connected regions of Mexico on the other hand, while experiencing a considerable number of cases,
would contribute only marginally to the International spread of cases. This observation readily explains why single population calculations that match the detection of imported cases with the local
prevalence are necessarily underestimating the latter quantity.
While GLEaM takes into account a higher level of geographical organization than previous approaches, its estimates still contain a number of assumptions and approximations. The contagion within each
census area is approximated by means of a homogeneous mixing process. Once a person arrives at a census area by plane, he/she comes integrated into the local population. This implies that, as in [6],
the travelers and the local population are equally exposed to the disease. Finally, the model considers each individual as independent and the possibility of cluster cases is not considered. Despite
these shortcomings and other necessary uncertainties, GLEaM predictions might provide additional information for a better understanding of the early evolution of the present pandemic. Despite the
different approximations used here and in Ref. [6], both approaches are providing support to the possibility of a reporting ratio of infected cases in Mexico as low as 1 in 100, in agreement with
prior estimates [2]. This finding is important when evaluating the massive amount of data which are now being collected in a large number of countries around the world. We can easily imagine that the
reporting rate as well as any estimate of the cumulative attack rate in most of the countries could be easily underestimated by orders of magnitude.
Competing interests
AV is consulting and has a research agreement with Abbott for the modeling of H1N1 diffusion. The other authors have declared that no competing interests exist.
│Number imported cases (May 8th) │USA │UK │France│Germany│Brazil│
│Simulation Results │0 – 534 │0 – 44│0 – 62│0 – 55 │0 – 45│
│Surveillance data │85 (170)│17 │11 │9 │3 │
Table 1: Cumulative number of imported cases from Mexico shown as the 99% reference range over 2,000 realizations on May 8 for a few countries. The simulations are obtained with the best estimate
parameters of the baseline case of Ref. [9] and R0=1.75 [95%CI 1.64 to 1.88]. The number of imported infected individuals and of independent clusters correspond to the data given in Ref. [6] for US,
and UK and the values in [23] for France, in [22] for Germany and in [24] for Brazil. No data was available to assess the possible presence of clusters in Germany and France. In the USA we report in
parentheses the revised number considering the rate of unknown travel history in confirmed cases.
│ │Number of symptomatic cases in Mexico (Apr. the 30th) │
│Simulation Results │[121,000 – 1,394,000] │
│Lower bound range of Ref. [6] │113,000-375,000 │
│Estimate of Ref. [5] * │2,000 – 280,000 │
│Mexican official report [7] (confirmed cases)│3,350 │
Table 2: Predictions of GLEaM for the size of the epidemic in Mexico on April 30 in thousands of cases and comparison with other approaches and with empirical data. The simulations are obtained with
the best estimate parameters of the baseline case of Ref. [9] and show the 95% reference range over 2,000 stochastic realizations. The results are compared with the lower bound estimate range in [6],
the estimate provided in Ref. [5] and the number of confirmed cases given by official reports [27]. *The interval provided for Ref. [5] is obtained by merging the results reported in the paper under
different assumptions and including the 95% CI.
The authors thank IATA and OAG for providing their databases. We are also grateful to the Staff of the Big Red Computer and the Computational Facilities at Indiana University, as well as to Ciro
Cattuto for his support with the computational infrastructure at the ISI Foundation.
• World Health Organization (WHO) pandemic (H1N1) situation update 66. Technical report. http://www.who.int/csr/don/2009_09_18/en/index.html
• Cruz-Pacheco G, Duran L, Esteva L, Minzoni A, Lopez-Cervantes M, et al. (2009) Modelling of the influenza A(H1N1)v outbreak in Mexico City, April-May 2009, with control sanitary measures. Euro
Surveill 14: 19254.
• Garske T, Legrand J, Donnelly CA, Ward H, Cauchemez S, Fraser C, Ferguson NM, Ghani AC: Assessing the severity of the novel A/H1N1 pandemic. BMJ 2009, 339:b2840.
• Lipsitch M, Hayden FG, Cowling BJ, Leung GM: How to maintain surveillance for novel influenza A H1N1 when there are too many cases to count. The Lancet 2009, 374:1209 - 1211
• Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, et al. (2009) Pandemic Potential of a Strain of Influenza A (H1N1): Early Findings. Science 324: 1557-1561.
• Lipsitch M, La jous M, O’Hagan JJ, Cohen T, Miller JC, et al. (2009) Use of Cumulative Incidence of Novel Influenza A/H1N1 in Foreign Travelers to Estimate Lower Bounds on Cumulative Incidence in
Mexico. PLoS ONE 4: e6895.
• Secretaria de Salud, Mexico. Situation actual de la epidemia, Oct 12, 2009. http://portal.salud.gob.mx/sites/salud/descargas/pdf/influenza/situacion_actual_epidemia_121009.pdf
• Balcan D, Colizza V, Goncalves B, Hu H, Ramasco JJ, Vespigani A (2009) Multiscale mobility networks and the large scale spreading of infectious diseases. ArXiv : 0907.3304.
• Balcan D, Hu H, Goncalves B, Bajardi P, Poletto C, et al. (2009) Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human
mobility. BMC Medicine 7: 45.
• Rvachev L, Longini I (1985) A mathematical model for the global spread of influenza. Mathematical Biosciences 75: 3-22.
• Grais R, Hugh Ellis J, Glass G (2003) Assessing the impact of airline travel on the geographic spread of pandemic influenza. Eur J Epidemiol 18: 1065-1072.
• Hufnagel L, Brockmann D, Geisel T (2004) Forecast and control of epidemics in a globalized world. Proc Natl Acad Sci (USA) 101: 15124-15129.
• Cooper BS, Pitman RJ, Edmunds WJ, Gay NJ (2006) Delaying the international spread of pandemic influenza. PLoS Med 3: e212.
• Epstein JM, Goedecke DM, Yu F, Morris RJ, Wagener DK, et al. (2007) Controlling pandemic flu: The value of international air travel restrictions. PLoS ONE 2: e401.
• Flahault A, Vergu E, Coudeville L, Grais R (2006) Strategies for containing a global influenza pandemic. Vaccine 24: 6751-6755.
• Viboud C, Bjornstad ON, Smith DL, Simonsen L, Miller MA, et al. (2006) Synchrony, Waves, and Spatial Hierarchies in the Spread of Influenza. Science 312: 447-451.
• Flahault A, Valleron AJ (1991) A method for assessing the global spread of HIV-1 infection based on air-travel. Math Popul Stud 3: 1-11.
• Colizza V, Barrat A, Barthelemy M, Vespignani A (2006) The role of airline transportation network in the prediction and predictability of global epidemics. Proc Natl Acad Sci (USA) 103:
• Colizza V, Barrat A, Barthelemy M, Valleron AJ, Vespignani A (2007) Modeling the Worldwide spread of pandemic influenza: baseline case and containment interventions. PloS Medicine 4: e13.
• Keeling M, Rohani P (2002) Estimating spatial coupling in epidemiological systems: a mechanistic approach. Ecology Letters 5: 20-29.
• Sattenspiel L, Dietz K (1995) A structured epidemic model incorporating geographic mobility among regions. Math Biosci 128: 71-91.
• German team for investigation of the Novel influenza A(H1N1) (2009) Description of the early stage of pandemic (H1N1) 2009 in Germany, 27 April-16 June 2009. Euro Surveill 14: 19295.
• French team for investigation of the Novel influenza A(H1N1) (2009) New influenza A(H1N1) virus infections in France, April-May 2009. Euro Surveill 14: 19221.
• Reports of the Brazilian Health Department (Ministerio da Saude). http://portal.saude.gov.br/portal/saude
• Colizza V, Barrat A, Barthélemy M, Vespignani A (2007) Predictability and epidemic pathways in global outbreaks of infectious diseases: the SARS case study. BMC Med, 5:34.
• Center for International Earth Science Information Network (CIESIN), Columbia University; International Food Policy Research Institute (IFPRI); The World Bank; and Centro Internacional de
Agricultura Tropical (CIAT). The Gridded Population of the World Version 3 (GPWv3) and the Global Rural-Urban Mapping Project (GRUMP), Alpha Version. http://sedac.ciesin.columbia.edu/gpw
• International Air Transport Association (IATA). http://www.iata.org
• Official Airline Guide (OAG). http://www.oag.com
• Chowell G et al (2009): Severe respiratory disease concurrent with the circulation of H1N1 influenza. New Engl J Med 361:674-679.
• The ANZIC Influenza Investigators (2009) Critical care services and 2009 H1N1 inluenza in Australia and New Zealand. New Engl. J Med 10.1056/NEJMoa0908481.
• Rello J et al. (2009) Intensive Care adult patients with severe respiratory failure caused by influenza A(H1N1) in Spain. Critical care 13:R148.
• Perez-Padilla R et al (2009) Pneumonia and Respiratory failure from swine-origin influenza A(H1N1) in Mexico. New Engl J Med 361: 680-689.
• Presanis et al (2009) The severity of pandemic H1N1 influenza in the United States, April – July 2009. PLoS Currents Influenza. 2009 Sep 25 [revised 2009 Oct 2]:RRN1042.
• Centers for Disease Control and Prevention (2009) Bacterial Coinfections in Lung Tissue Specimens from Fatal Cases of 2009 Pandemic Influenza A (H1N1) --- United States, May--August 2009. Morb.
Mortal. Wkly. Rep. 58: 1
• McCaw JM, McVernon J, McBryde ES, Mathews JD (2009) Influenza: accounting for prior immunity. Science 325, 1071.
• Centers for Disease Control and Prevention (2009) Morb. Mortal. Wkly. Rep. 58: 521.
• Katriel, Guy; Stone, Lewi. Pandemic influenza dynamics and the breakdown of herd immunity. PLoS Currents Influenza. 2009 Oct 2:RRN1046.
• Yang Y, Sugimoto JD, Halloran ME, Basta NE, Chao DL, Matrajt L, Potter G, Kenah E, Longini IM (2009) The transmissibility and control of pandemic influenza A(H1N1) virus. Science 10.1126/
• Flahault A, Vergu E, Boelle P-Y (2009) Potential for a global dynamic of influenza A(H1N1). BMC Infect Dis 9: 129.
• Halloran ME, Hayden FG, Yang Y, Longini IM, Monto AS (2007): Antiviral Effects on Influenza Viral Transmission and Pathogenicity: Observations from Household-based Trials. Am J Epidemiol 165:212.
• Carrat F, Vergu E, Ferguson NM, Lemaitre M, Cauchemez S, Leach S, Valleron AJ: Time lines of infection and disease in human influenza: a review of volunteer challenge studies. Am J Epidemiol
2008, 167:775-785.
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Santa’s Flakpanzerkampfwagen
Santa's Flakpanzerkampfwagen
CTF : X-MAS CTF 2021 First Weekend
Despite the name it was a relatively easy one. The problem can be resumed to “we are a turrets in (0, 0), all around us planes can spawn. We know the starting positions of the planes and the
corresponding coordinates after 0.5 time units (TU). Shoot ‘em.”
With “shoot ‘em” I mean: give as output for each given plane <yaw> <distance> <delay>.
• yaw refers to the rotation around the OZ axis, or the trigonometric angle with the positive side of the OX axis (in degrees).
• distance is the distance from the origin that our shell need to travel before “exploding” (in space units, SU)
• Last, we need to specify the delay, in TU from timestamp 0, to wait before shoot (the cannon will sort commands in a way that make sense before executing it). Inserting commands require 0 TU.
Our cannon have a range of 1300 SU, the planes must stay at least 1000 SU away from us (from the origin).
All the planes will spawn at 2000 SU from us. As time passes they will get closer but without necessarily pointing directly to the origin.
So, time for some maths!
An easy way to get the job done is tracing the direction of the plane until it get in range in order to calculate the trajectory. An easy way to do this is using the following snippet:
# p0 = spawn point
# p1 = point after .5 TU
# dn = distance from origin
# lenM = cannon range
# t = time
lenM = 1300
dn = sqrt(p0[0]**2 + p0[1]**2)
pn = p0
t = 0
# P = (p.x + (p1.x - p0.y), p.y + (p1.y - p0.y0))
# D = sqrt(p.x**2 + p.y**2)
# T = T(p1) - T(p0) = 0.5 - 0
while (dn > lenM):
pn = (pn[0] + (p1[0] - p0[0]), pn[1] + (p1[1] - p0[1]))
dn = sqrt(pn[0]**2 + pn[1]**2) # new distance from origin
t += .5
Where p0 is the starting point, p1 the point after 0.5 TU and pn is the new point (fist initialized to p0).
We can then divide the distance of pn from the origin by the speed of the shell (900 SU/TU, given as hint of the challenge) and subtract it at the time passed t.
Now that we have the point to aim for and the distance we can obtain the yaw in randians interpreting the 2 coordinates of pn as a vector (x, y) and then using the arctan2 function as follow:
yaw = arctan2(pn[1], pn[0]) * (180 / 3.14159)
Note the conversion from radians to degrees with 180 / 3.14159.
As the challenge’s description says
The shells have a decent blast radius, so you do not need to be pinpoint accurate.
So, if we want, we can also round up the results as follow
yaw = round(yaw, 5)
t = round(t, 5)
dn = round(dn, 5)
We can then iterate every given plane at each level and than get the flag!
#! /bin/python3
import re
from pwn import *
from numpy import *
lenM = 1300
dt = .5
regex = r"([0-9]+):\ \(((-?[0-9]*\.[0-9]*[,|)]?\ ?){2})\ ->\ \(((-?[0-9]*\.[0-9]*[,|)]?\ ?){2})"
reg = re.compile(regex)
p = remote('challs.xmas.htsp.ro', 6003)
while True:
line = p.recvline().decode()
res = reg.match(line)
if(res is None):
p0 = res.group(2).split(',')
p1 = res.group(4).split(',')
p0[0] = float(p0[0])
p0[1] = float(p0[1].replace(')', '').replace(' ', ''))
p1[0] = float(p1[0])
p1[1] = float(p1[1].replace(')', '').replace(' ', ''))
dn = sqrt(p0[0]**2 + p0[1]**2)
pn = p0
t = 0
while (dn > lenM):
pn = (pn[0] + (p1[0] - p0[0]), pn[1] + (p1[1] - p0[1]))
dn = sqrt(pn[0]**2 + pn[1]**2)
t += dt
yaw = arctan2(pn[1], pn[0]) * (180 / 3.14159)
t = t - (dn/900)
yaw = round(yaw, 5)
t = round(t, 5)
dn = round(dn, 5)
send = ' '.join([str(yaw), str(dn), str(t)])
I used pwntools for communications and numpy to perform the maths, than the standard python’s regex library to parse inputs.
After running the script and defending the position, the program will print out our flag:
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[Solved] Evaluate: (99)3 - Secondary School Mathematics For Cla... | Filo
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Question Text Evaluate:
Updated On Jan 8, 2023
Topic Polynomials
Subject Mathematics
Class Class 9
Answer Type Text solution:1 Video solution: 2
Upvotes 251
Avg. Video Duration 7 min
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The Elements of Euclid
a 9. 11.
B. XII. in which these parallels are, and the quadrilateral figure KBOS is in one plane: and if PB, TK be joined, and perpendiculars be drawn from the points P, T to the straight lines AB, АК, it may
be demonstrated, that TP is parallel to KB in the very same way that SO was shown to be parallel to the same KB; wherefore a TP is parallel to SO, and the quadrilateral figure SOPT is in one plane:
for the same reason, the quadrilateral b2.11. TPRY is in one plane: and the figure YRX is also in one plane.
Therefore, if from the points O, S, P, T, R, Y there be drawn straight lines to the point A, there shall be formed a solid polyhedron between the circumferences BX, KX composed of pyramids, the bases
of which are the quadrilaterals KBOS, SOPT, TPRY, and the triangle YRX, and of which the common vertex is the point A: and if the same construction be made upon each of the sides KL, LM, ME, as has
been done upon BK, and the like be done also in the other three quadrants, and in the other hemisphere ; there shall be formed a solid polyhedron described in the sphere, compo
sed of pyramids, the bases of which are the aforesaid quadri- B. XII. lateral figures, and the triangle YRX, and those formed in the like manner in the rest of the sphere, the common vertex of them
all being the point A: and the superficies of this solid polyhedron does not meet the lesser sphere in which is the circle FGH: for, from the point A draw AZ perpendicular a 11. 11. to the plane of
the quadrilateral KBOS meeting it in Z, and join BZ, ZK: and because AZ is perpendicular to the plane KBOS, it makes right angles with every straight line meeting it in that plane; therefore AZ is
perpendicular to BZ and ZK: and because AB is equal to AK, and that the squares of AZ, ZB, are equal to the square of AB; and the squares of AZ, ZK to the square of AKb; therefore the squares of AZ,
ZB b 47. 1. are equal to the squares of AZ, ZK: take from these equals the square of AZ, the remaining square of BZ is equal to the remaining square of ZK; and therefore the straight line BZ is equal
to ZK; in the like manner it may be demonstrated, that the straight lines drawn from the point Z to the points (), S are equal to BZ or ZK: therefore the circle described from the centre Z, and
distance ZB, shall pass through the points K, 0, S, and KBOS shall be a quadrilateral figure in the circle: and because KB is greater than QV, and QV equal to SO, therefore KB is greater than SO: but
KB is equal to each of the straight lines BO, KS; wherefore each of the circumferences cut off by KB, BO, KS is greater than that cut off by OS; and these three circumferences, together with a fourth
equal to one of them, are greater than the same three together with that cut off by OS; that is, than the whole circumference of the circle; therefore the circumference subtended by KB is greater
than the fourth part of the whole circumference of the circle KBOS, and consequently the angle BZK at the centre is greater than a right angle: and because the angle BZK is obtuse, the square of BK
is greater than the squares of BZ, ZK; с 12. 2. that is, greater than twice the square of BZ. Join KV, and because in the triangles KBV, OBV, KB, BV are equal to Oв, BV, and that they contain equal
angles; the angle KVB is equal d to the angle OVB: and OVB is a right angle; there- d 4. 1. fore also KVB is a right angle: and because BD is less than twice DV, the rectangle contained by DB, BV is
less than twice the rectangle DVB; that ise, the square of KB is less e 8. 6. than twice the square of KV: but the square of KB is greater than twice the square of BZ: therefore the square of KV is
greater than the square of BZ: and because BA is equal to AK, and that the squares of BZ, ZA are equal together to the square of BA, and the squares of KV, VA to the square of AK; therefore the
squares of BZ, ZA are equal to the squares of KV, VA; and of these the square of KV is greater than the
B. XII. square of BZ; therefore the square of VA is less than the square of ZA, and the straight line AZ greater than VA: much more then is AZ greater than AG; because, in the preceding proposition,
it was shown that KV falls without the circle FGH: and AZ is perpendicular to the plain KBOS, and is therefore the shortest of all the straight lines that can be drawn from A, the centre of the
sphere to that plane. Therefore the plane KBOS does not meet the lesser sphere.
And that the other planes between the quadrants BX, KX fall without the lesser sphere, is thus demonstrated: from the point A draw Al perpendicular to the plane of the quadrilateral SOPT, and join
IO; and, as was demonstrated of the plane KBOS and the point Z, in the same way it may be shown that the point I is the centre of a circle described about SOPT; and that OS is greater than PT; and PT
was shown to be parallel to OS: therefore, because the two trapeziums KBOS, SOPT inscribed in circles have their sides BK, OS parallel, as also OS, PT; and their other sides BO, KS, OP, ST all equal
to one another, and that BK is greater than OS, and OS a 2. lem. greater than PT, therefore the straight line ZB is greatera than IO. Join AO which will be equal to AB; and because AIO, AZB are right
angles, the squares of AI, IO are equal to the square of AO or of AB; that is, to the squares of AZ, ZB; and the square of ZB is greater than the square of 10, therefore the square of AZ is less than
the square of AI; and the straight line AZ less than the straight line AI: and it was proved that AZ is greater ter than AG; much more then is Al greater than AG; therefore the plane SOPT falls
wholly without the lesser sphere: in the same manner it may be demonstrated that the plane TPRY falls without the same sphere, as also the triangle YRX, viz. by the cor. of 2d lemma. And after the
same way it may be demonstrated that all the planes which contain the solid polyhedron, fall without the lesser sphere. Therefore in the greater of two spheres which have the same centre, a solid
polyhedron is described, the superficies of which does not meet the lesser sphere. Which was to be done.
But the straight line AZ may be demonstrated to be greater than AG otherwise, and in a shorter manner, without the help of prop. 16, as follows. From the point G draw GU at right angles to AG, and
join AU. If then the circumference BE be bisected, and its half again bisected, and so on, there will at length be left a circumference less than the circumference which is subtended by a straight
line equal to GU inscribed in the circle BCDE: let this be the circumference KB: therefore the straight line KB is less than GU: and because the angle BZK is obtuse, as was proved in the preceding,
therefore BK is greater than BZ: but GU is greater than BK; much more
then is GU greater than BZ, and the square of GU than the B. XII. square of BZ; and AU is equal to AB; therefore the square of AU, that is, the squares of AG, GU, are equal to the square of AB, that
is, to the squares of AZ, ZB; but the square of BZ is less than the square of GU; therefore the square of AZ is greater than the square of AG, and the straight line AZ consequently greater than the
straight line AG.
Cor. And if in the lesser sphere there be described a solid polyhedron by drawing straight lines betwixt the points in which the straight lines from the centre of the sphere drawn to all the angles
of the solid polyhedron in the greater sphere meet the superficies of the lesser; in the same order in which are joined the points in which the same lines from the centre meet the superficies of the
greater sphere; the solid polyhedron in the sphere BCDE has to this other solid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other sphere:
for if these two solids be divided into the same number of pyramids, and in the same order, the pyramids shall be similar to one another, each to each; because they have the solid angles at their
common vertex, the centre of the sphere, the same in each pyramid, and their other solid angle at the bases equal to one another, each to eacha, because they are contained by three a B. 11. plane
angles equal each to each: and the pyramids are contained by the same number of similar planes; and are therefore similar b b 11. to one another, each to each: but similar pyramids have to one
another the triplicate ratio of their homologous sides. c Cor. 8, Therefore the pyramid of which the base is the quadrilateral 12. KBOS, and vertex A, has to the pyramid in the other sphere of the
same order, the triplicate ratio of their homologous sides; that is, of that ratio which AB from the centre of the greater sphere has to the straight line from the same centre to the superficies of
the lesser sphere. And in like manner, each pyramid in the greater sphere has to each of the same order in the lesser, the triplicate ratio of that which AB has to the semidiameter of the lesser
sphere. And as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole solid polyhedron in
the other, the triplicate ratio of that which AB the semidiameter of the first has to the semidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other
def. 11.
B. XII.
PROP. XVIII. THEOR.
SPHERES have to one another the triplicate ratio of that which their diameters have.
Let ABC, DEF be two spheres, of which the diameters are BC, EF. The sphere ABC has to the sphere DEF the triplicate ratio of that which BC has to EF.
For, if it has not, the sphere ABC shall have to a sphere either less or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a less, viz. to the sphere
GHK; and let the sphere DEF have the same a 17. 12. centre with GHK; and in the greater sphere DEF describe a
b Cor.
a solid polyhedron, the superficies of which does not meet the lesser sphere GHK; and in the sphere ABC describe another similar to that in the sphere DEF: therefore the solid polyhe. dron in the
sphere ABC has to the solid polyhedron in the sphere DEF, the triplicate ratio of that which BC has to EF. But the sphere ABC has to the sphere GHK the triplicate ratio of that which BC has to EF;
therefore, as the sphere ABC to the sphere GHK, so is the solid polyhedron in the sphere ABC to the solid polyhedron in the sphere DEF: but the sphere
c 14. 5. ABC is greater than the solid polyhedron in it; therefore also the sphere GHK is greater than the solid polyhedron in the sphere DEF: but it is also less, because it is contained within it,
which is impossible: therefore the sphere ABC has not to any sphere less than DEF the triplicate ratio of that which BC has to EF. In the same manner it may be demonstrated, that the sphere DEF has
not to any sphere less than ABC the triplicate ratio of that which EF has to BC. Nor can the sphere ABC have to any sphere greater than DEF, the triplicate ratio of that which BC has to EF: for, if
it can, let it
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