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1. Explain the distinction between an ambiguity in a proposed algorithm and an ambiguity in the representation of an algorithm.
2. Describe how the use of primitives helps remove ambiguities in an algorithm's representation.
3. What is the difference between a formal programming language and pseudo-code?
4. What is the difference between syntax and semantics?
5. Four prospectors with only one lantern must walk through a mineshaft. At most, two prospectors can travel together and any prospector in the shaft must be with the lantern. The prospectors, named
Andrews, Blake, Johnson, and Kelly, can walk through the shaft in one minute, two minutes, four minutes, and eight minutes, respectively. When two walk together they travel at the speed of the slower
prospector. How can the prospectors get through the mineshaft in only 15 minutes? After you have solved this problem, explain how you got your foot in the door.
Related Questions in Data Structure
Data Communication Delivering Information anywhere
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A pure strategy pair (i. j) is in equilibrium if and only if the corresponding element t[ij] is both the largest in its column and the smallest in its row. Such an element is also called a saddle
point (by analogy with the surface of a saddle).
An "equilibrium decision point", that is a "saddle point", also known as a "minimax point", represents a decision by two players upon which neither can improve by unilaterally departing from it.
When there is no saddle point, one must choose the strategy randomly. This is the idea behind a mixed strategy. A mixed strategy for a player is defined as a probability distribution on the set
of the pure strategies. Our numerical example is such a case. Player I can assure the payoff to be max[i] min[j] t[ij] = 2, while player II plays in such a manner that player I receives no more
than min max t[ij] = 3. The problem of how the difference [min[j] max[i] t[ij]] - [max min t[ij]] ³ 0 should be subdivided between the players thus remains open. In such cases, the players
naturally seek additional strategic opportunities to assure themselves of the largest possible share of this difference. To achieve this objective, they must select their strategies randomly to
confuse each other.
We will use a general method based on a linear programming (LP) formulation. This equivalency of games and LP may be surprising, since a LP problem involves just one decision-maker, but it should
be noted that with each LP problem there is an associated problem called the dual LP. The optimal values of the objective functions of the two LPs are equal, corresponding to the value of the
game. When solving LP by simplex-type methods, the optimal solution of the dual problem also appears as part of the final tableau. So we get v, Y*, and X* by solving one LP. LP formulation and
the simplex method is the fastest, most practical, and most useful method for solving games with a large matrix T. Suppose that player II is permitted to adopt mixed strategies, but player I is
allowed to use only pure strategies. What mixed strategies Y = (y1, y2, y3) should player II adopt to minimize the maximum expected payoff v? A moment's thought shows that player II must solve
the following problem:
Min v = y
subject to: T.Y £ y
U^t.Y = 1
The minimization is over all elements of the decision vector Y ³ 0, the scalar y is unrestricted in sign, and U is an n- dimensional column vector with all elements equal to one. The left hand
side of the first n constraints, by definition, is player II's expected return against player I's pure strategies. It turns out that these mixed strategies are still optimal if we allow player I
to employ mixed strategies. Mixed strategies are also known as "saddle point" strategies. Of course, a game with a saddle point can be solved by this method as well. The standard formulation in
the simplex method requires that all variables be non-negative. To achieve this condition one may substitute the difference of two new variables for y.
The optimal strategy for player I is the solution to the dual problem of player II's problem. The simplex method of linear programming provides optimal strategies for both players.
The social games and fairness norm is a convention that evolved to coordinate behavior on an equilibrium of a society's Game of Life. According to this view, the metaphysics of Emmanuel Kant for
the naturalistic approach, and the morality of David Hume could be abandoned.
Numerical Examples
The LP formulation for player II's problem in a game with payoff matrix T given above, is:
Min v
subject to:
4y1 + y2 + 3y3 £ v
2y1 + 3y2 + 4y3 £ v
y1 + y2 + y3 = 1
yj³0, j = 1, 2, 3, and v is unrestricted
The optimal solution for player II is: y1 = 1/2, y2 = 1/2, y3 = 0. The shadow prices are the optimal strategies for player I. Therefore, the mixed saddle point is: x1 = 1/4, x2 = 3/4; y1 = 1/2,
y2 = 1/2, y3 = 0, and the value of the game equals 5/2. Note that the essential strategies for Player I are i = 1, i = 2; for Player II they are j = 1, j = 2 while j = 3 is non-essential.
It is customary to discard the dominant rows or columns in finding optimal strategies. This assumes, however, that the payoff matrix is fixed. If you are interested in the stability analysis of
the essential (and non-essential) strategies with respect to changes in payoffs, read the following article:
Further Readings:
Arsham H., Stability of essential strategy in two-person zero-sum games, Congressus Numerantium, 110(3), 167-180. 1995.
Borm P., (Ed.), Chapters in Game Theory, Kluwer, 2002.
Raghavan T., and Z. Syed, A policy-improvement type algorithm for solving zero-sum two-person stochastic games of perfect information, Mathematical Programming, Ser. A, 95(3), 513-532, 2003.
Weintraub E., (ed.), Toward a History of Game Theory, Duke University Press, 1992.
Visit also the Web sites:
Two-Person Zero-Sum Games Theory with Applications.
Investment Decisions: Optimal Portfolio Selections
Consider the following investment problem discussed in the Decision Analysis site. The problem is to decide what action or a combination of actions to take among three possible courses of action
with the given rates of return as shown in the body of the following table.
States of Nature (Events)
Growth Medium G No Change Low
G MG N L
Bonds 12% 8 7 3
Actions Stocks 15 9 5 -2
Deposit 7 7 7 7
In decision analysis, the decision-maker has to select at least and at most one option from all possible options. This certainly limits its scope and its applications. You have already learned
both decision analysis and linear programming. Now is the time to use the game theory concepts to link together these two seemingly different types of models to widen their scopes in solving more
realistic decision-making problems. The investment problem can be formulated as if the investor is playing a game against nature.
Suppose our investor has $100,000 to allocate among the three possible investments with the unknown amounts Y1, Y2, Y3, respectively. That is,
Y1 + Y2 + Y3 = 100,000
Notice that this condition is equivalent to the total probability condition for player I in the Game Theory.
Under these conditions, the returns are:
0.12Y1 + 0.15Y2 + 0.07Y3 {if Growth (G)}
0.08Y1 + 0.09Y2 + 0.07Y3 {if Medium G}
0.07Y1 + 0.05Y2 + 0.07Y3 {if No Change}
0.03Y1 - 0.02Y2 + 0.07Y3 {if Low}
The objective is that the smallest return (let us denote it by v value) be as large as possible.
Formulating this Decision Analysis problem as a Linear Programming problem, we have:
Max v
Subject to:
Y1 + Y2 + Y3 = 100,000
0.12Y1 + 0.15Y2 + 0.07 Y3 ³ v
0.08Y1 + 0.09Y2 + 0.07Y3 ³ v
0.07Y1 + 0.05Y2 + 0.07Y3 ³ v
0.03Y1 - 0.02Y2 + 0.07Y3 ³ v
and Y1, Y2, Y3 ³ 0, while v is unrestricted in sign (could have negative return).
This LP formulation is similar to the problem discussed in the Game Theory section. In fact, the interpretation of this problem is that, in this situation, the investor is playing against nature
(the states of economy).
Solving this problem by any LP solution algorithm, the optimal solution is Y1 = 0, Y2 = 0, Y3 = 100,000, and v = $7000. That is, the investor must put all the money in the money market account
with the accumulated return of 100,000´1.07 = $10,7000.
Note that the pay-off matrix for this problem has a saddle-point; therefore, as expected, the optimal strategy is a pure strategy. In other words, we have to invest all our money into one
portfolio only.
Buying Gold or Foreign Currencies Investment Decision: As another numerical example, consider the following two investments with the given rate of returns. Given you wish to invest $12,000 over a
period of one year, how do you invest for the optimal strategy?
States of Nature (Economy)
Growth Medium G No Change Low
G MG N L
Actions Buy Currencies (C) 5 4 3 -1
Buy Gold (G) 2 3 4 5
The objective is that the smallest return (let us denote it by X3 value) be as large as possible.
Similar to previous example, formulating this Decision Analysis problem as a Linear Programming problem, we have:
Maximize X3
Subject to:
X1 + X2 = 12,000
0.05X1 + 0.02X2 ³ X3
0.04X1 + 0.03X2 ³ X3
0.03X1 + 0.04X2 ³ X3
-0.01X1 + 0.05X2 ³ X3
and X1, X2 ³ 0, while X3 is unrestricted in sign (i.e., could have negative return).
Again, this LP formulation is similar to the problem discussed in the Game Theory section. In fact, the interpretation of this problem is that, in this situation, the investor is playing against
nature (i.e., the states of economy).
Solving this problem by any LP solution algorithm, the optimal solution is a mixed strategy: Buy X1 = $4000 Foreign Currencies and X2= $8000 Gold.
The Investment Problem Under Risk:
The following table shows the risk measurements computed for the Investment Decision Example:
Risk Assessment
G(0.4) MG(0.3) NC(0.2) L(0.1) Exp. Value St. Dev. C. V.
B 12 8 7 3 8.9 2.9 32%
S 15 9 5 -2 9.5* 5.4 57%
D 7 7 7 7 7 0 0%
The Risk Assessment columns in the above table indicate that bonds are much less risky than the stocks, while its return is lower. Clearly, deposits are risk free.
Now, an interesting question is: Given all this relevant information, what action do you take? It is all up to you.
Max [v - 0.029Y1 - 0.054Y2 -0Y3]
Subject to:
Y1 + Y2 + Y3 = 100,000
0.089Y1 + 0.095Y2 + 0.07Y3 ³ v
and Y1, Y2, Y3 ³ 0, while v is unrestricted in sign (could have negative return).
Solving this Linear Program (LP) model by any computer LP solver, the optimal solution is Y1 = 0, Y2 = 0, Y3 = 100,000, and v = $7000. That is, the investor must put all the money in the money
market account with the accumulated return of 100,000´1.07 = $10,7000.
Notice that, for this particular numerical example, it turns out that the different approaches provide the same optimal decision; however one must be careful not to do any generalization at all.
Note that the above objective function includes the standard deviations to reduce the risk of your decision. However, it is more appropriate to use the covariance matrix instead. Nevertheless the
new objective function will have a quadratic form, which can be solved by applying nonlinear optimization algorithms. For more information on decision problem construction, and solution
algorithm, together with some illustrative numerical applications, visit the Optimal Business Decisions Web site.
You may use:
Two-Person Zero-Sum Games Theory with Applications for checking your computation and experimentation.
Risk Assessment Process: Clearly, different subjective probability models are plausible they can give quite different answers. These examples show how important it is to be clear about the
objectives of the modeling. An important application of subjective probability models is in modeling the effect of state-of-knowledge uncertainties in consequence models. Often it turns out that
dependencies between uncertain factors can be important in driving the output of the models. For example, consider two portfolios having random variable R1 and R2 returns; the ratio:
Cov (R1, R2) / Var (R1)
is called the beta of the trading strategy 1 with respect to the trading strategy 2. Various methods are available to model these dependencies, in particular proportional to the Beta values
Numerical Example: Consider our Buying Gold or Foreign Currencies Investment Decision, using the Bivariate Discrete Distributions, JavaScript with equally likelihood (0.25), we obtain:
Beta (Currencies) = -0.457831, and Beta (Gold) = -1.9
Now, one may distribute the total capital ($12000) proportional to the Beta values:
Sum of Beta’s = -0.457831 -1.9 = -2.357831
Y1 = 12000 (-0.457831 /-2.357831) = 12000(0.194175) = $2330, Investing on Foreign Currencies
Y2 = 12000 (-1.9/-2.357831) = 12000(0.805825) = $9670, Investing on Gold
That is, the optimal strategic decision based upon the Beta criterion is: Buy $2330 foreign Currencies, and $9670 Gold.
The following flowchart depicts the risk assessment process for portfolio selection based on their financial time series.
Risk Assessment in Portfolio Selection
Click on the image to enlarge it
The above hybrid model brings together the techniques of decision analysis, linear programming, and statistical risk assessments (via a quadratic risk function defined by covariance matrix) to
support the interactive decisions for modeling investment alternatives.
Further Readings:
Dixit A., and R. Pindyck, Investment Under Uncertainty, Princeton Univ Pr, 1994.
Dokuchaev N., Dynamic Portfolio Strategies: Quantitative Methods and Empirical Rules for Incomplete Information, Kluwer, 2002. Contains investment data-based optimal strategies.
Korn R., and E. Korn, Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics, Amer Mathematical Society, 2000.
Luenberger D., Investment Science, Oxford Univ Press, 1997.
Pliska S., Introduction to Mathematical Finance: Discrete Time Models, Blackwell Pub, 1997.
Winston W., Financial Models Using Simulation and Optimization, Palisade Corporation, 1998.
A Classification of Investors Relative Attitudes Toward Risk and Its Impact
Probability of an Event and the Impact of its Occurrence: The process-oriented approach of managing risk and uncertainty is part of any probabilistic modeling. It allows the decision-maker to
examine the risk within its expected return, and identify the critical issues in assessing, limiting, and mitigating risk. This process involves both the qualitative and quantitative aspects of
assessing the impact of risk.
Decision science does not describe what people actually do since there are difficulties with both computations of probability and the utility of an outcome. Decisions can also be affected by
people's subjective rationality, and by the way in which a decision problem is perceived.
Traditionally, the expected value of random variables has been used as a major aid to quantify the amount of risk. However, the expected value is not necessarily a good measure alone by which to
make decisions since it blurs the distinction between probability and severity. To demonstrate this, consider the following example:
Suppose that a person must make a choice between scenarios 1 and 2 below:
□ Scenario 1: There is a 50% chance of a loss of $50, and a 50% chance of no loss.
□ Scenario 2: There is a 1% chance of a loss of $2,500, and a 99% chance of no loss.
Both scenarios result in an expected loss of $25, but this does not reflect the fact that the second scenario might be much more risky than the first. (Of course, this is a subjective
assessment). The decision-maker may be more concerned about minimizing the effect of the occurrence of an extreme event than he/she is concerned about the mean. The following charts depict the
complexity of the probability of an event, the impact of the occurrence of the event, and its related risk indicator, respectively:
From the previous section, you may recall that the certainty equivalent is the risk free payoff; moreover, the difference between a decision-maker's certainty equivalent and the expected monetary
value (EMV) is called the risk premium. We may use the sign and the magnitude of the risk premium in classifying a decision-maker's relative attitude toward risk as follows:
□ If the risk premium is positive, then the decision-maker is willing to take the risk, and the decision-maker is said to be a risk seeker. Clearly, some people are more risk-seeker than
□ If the risk premium is negative, then the decision-maker would avoid taking the risk, and the decision-maker is said to be risk averse.
□ If the risk premium is zero, then the decision-maker is said to be risk neutral.
Further Readings
Brooks C., Introductory Econometrics for Finance, Cambridge University Press, 2002.
Eilon S., The Art of Reckoning: Analysis of Performance Criteria, Academic Press, 1984.
Hammond J., R. Keeney, and H. Raiffa, Smart Choices: A Practical Guide to Making Better Decisions, Harvard Business School Press., 1999.
Richter M., and K. Wong, Computable preference and utility, Journal of Mathematical Economics, 32(3), 339-354, 1999.
Risk Assessment: How Good Is Your Portfolio?
Risk is the downside of a gamble, which is described in terms of probability. Risk assessment is a procedure of quantifying the loss or gain values and supplying them with proper values of
probabilities. In other words, risk assessment means constructing the random variable that describes the risk. Risk indicator is a quantity that describes the quality of the decision.
Without loss of generality, consider our earlier Investment Example. Suppose the optimal portfolio is:
Y1.B + Y2.S + Y3.D
The expected value (i.e., the averages): The expected return is:
Y1.B[r] + Y2.S[r] + Y3.D[r]
where, B[r], S[r], and D[r] are the historical averages for B, S, and D, respectively.
Expected return alone is not a good indication of a quality decision. The variance must be known so that an educated decision may be made. Have you ever heard the dilemma of the six-foot tall
statistician who drowned in a stream that had an average depth of three feet?
In the investment example, it is also necessary to compute the 'risk' associated with the optimal portfolio. A measure of risk is generally reported by variation, or its square root called
standard deviation. Variation or standard deviation are numerical values that indicate the variability inherent to your decision. For risk, smaller values indicate that what you expect is likely
to be what you get. What we desire is a large expected return, with small risk; thus, high risk makes the investor very worried.
Variance: An important measure of risk is variance:
Y1^2.Var(B) + Y2^2.Var(S) + Y3^2.Var(D) +
2Y1.Y2.Cov(B, S) + 2Y1.Y3.Cov(B, D) + 2Y2.Y3.Cov(S, D)
Where Var and Cov are the variance and covariance, respectively, they are computed using recent historical data.
The variance is a measure of risk; therefore, the greater the variance, the higher the risk. The variance is not expressed in the same units as the expected value. So, the variance is hard to
understand and explain as a result of the squared term in its computation. This can be alleviated by working with the square root of the variance, which is called the Standard Deviation:
Both variance and standard deviation provide the same information and, therefore, one can always be obtained from the other. In other words, the process of computing standard deviation always
involves computing the variance. Since standard deviation is the square root of the variance, it is always expressed in the same units as the expected value.
Numerical Example: Consider our Buying Gold or Foreign Currencies Investment Decision, the above formulas reduce to:
The optimal portfolio is:
Y1.C + Y2.G
The expected value (i.e., the averages): The expected return is:
Y1.C[r] + Y2.G[r]
The risk measured in terms of variance is:
Y1^2.Var(C) + Y2^2.Var(G) + 2Y1.Y2.Cov(C, G)
Using the Bivariate Distributions JavaScript, we have:
The expected return is:
$4000 (2.75) + $8000 (3.5) = $39000
The standard deviation is:
[($4000)^2 + ($8000)^2 + 2($4000)($8000)(-2.375)]^1/2 = $8936
Notice that Beta1 and Beta2 are directly related, for example, the multiplication of the two provides the correlation square, i.e. r^2. The r^2 which always between [0, 1] is a number without any
dimensional units, and it represent strong is the linear dependency between the rates of return of one portfolios against the other one. When any beta is negative, and the r^2 is large enough,
then the two portfolios are related inversely and strongly. In such a case, diversification of the total capital is recommended.
For the dynamic decision process, the Volatility as a measure for risk includes the time period over which the standard deviation is computed. The Volatility measure is defined as standard
deviation divided by the square root of the time duration.
When considering two different portfolios, what do you do if one portfolio has a larger expected return but a much higher risk than the alternative portfolio? In such cases, using another measure
of risk known as the Coefficient of Variation is appropriate.
Coefficient of Variation (CV) is the absolute relative deviation with respect to size
CV =100 |S/
For the above numerical example, the coefficient of variation is:
100(8936/39000) = 23%
Notice that the CV is independent from the expected value measurement. The coefficient of variation demonstrates the relationship between standard deviation and expected value, by expressing the
risk as a percentage of the expected value. A portfolio with 15% or less CV is considered a "good" portfolio. The inverse of CV (namely 1/CV) is called the Signal-to-Noise Ratio.
Diversification may reduce your risk: Since the covariance appears in risk assessment, it reduces the risk if its negative. Therefore, diversifying your investment may reduce the risk without
reducing the benefits you gain from the activities. For example, you may choose to buy a variety of stocks rather than just one.
For an application of signal-to-noise ratio as a diversification criteria in reducing your investment risk decisions, visit Risk: The Four Letters Word section.
Notice that the diversification based on signal-to-noise ratio criteria can be extended to more than two portfolios, unlike the beta ratio criteria, which is limited to two inversely correlated
portfolios, only.
You may use the following JavaScript for computational purposes and computer-assisted experiment as learning tools for fundamental of risk analysis:
Further Readings:
Dupacová J., J. Hurt, and J. Štepán, Stochastic Modeling in Economics and Finance, Kluwer Academic Publishers, 2002. Part II of the book is devoted to the allocation of funds and risk management.
Moore P., The Business of Risk, Cambridge University Press, 1984.
Vose D., Risk Analysis: A Quantitative Guide, John Wiley & Sons, 2000.
Portfolio's Factors-Prioritization & Stability Analysis
Introduction: Sensitivity analysis, known also as stability analysis, is a technique for determining how much an expected return will change in response to a given change in an input variable,
all other things remaining unchanged.
Steps in Sensitivity Analysis:
1. Begins with consideration of a nominal base-case situation, using the expected values for each input.
2. Calculates the base-case output.
3. Considers a series of "what-if" questions to determine by how much the output would deviate from this nominal level if input values deviated from their expected values.
4. Each input is changed by several percentage points above and below its expected value, and the expected payoff is recalculated.
5. The set of expected payoff is plotted against the variable that was changed.
6. The steeper the slope (i.e., derivative) of the resulting line, the more sensitive the expected payoff is to a change in the variable.
Scenario Analysis: Scenario analysis is a risk analysis technique that considers both the sensitivity of expected payoff to changes in key variables, and the likely range of variable values. The
worst and best "reasonable" sets of circumstances are considered, and the expected payoff for each is calculated and compared to the expected, or base-case, output. Clearly, extensive scenario
and sensitivity analysis can be carried out using computerized versions of the above procedure.
How Stable is Your Decision? Stability Analysis compares the outcome of each of your scenarios with chance events. Computerized versions of the above procedure are necessary and useful tools.
They can be used extensively to examine the decision for stability and sensitivity whenever there is uncertainty in the rate of return.
Prioritization of Uncontrollable Factors: Stability analysis also provides critical model inputs. The simplest test for sensitivity is whether the optimal portfolio changes when an uncertainty
factor is set to its extreme value while holding all other variables unchanged. If the decision does not change, the uncertainty can be regarded as relatively less important than for the other
factors. Sensitivity analysis focuses on the factors with the greatest impact, thus helping to prioritize data gathering while increasing the reliability of information.
You may like to use MultiVariate Statistics: Mean, Variance, & Covariance for checking your computation and for performing computer-assisted experimentation.
The Gambler’s Ruin Probability
By now you may have realized that the above game is not a pure random decision problem. Switching to different strategy with specific frequencies obtained by optimal solution is aimed at
confusing the other player. The following game is an example of pure random decision-making problem.
Ruin Probability: The second JavaScript is for sensitivity analysis of the winning the target dollar amount or losing it all (i.e., ruin) game.
Let R = the amount of money you bring to the table, T = the targeted-winning amount, U = the size of each bet, and p = The probability of winnig any bet, then the probability of probability (W)
of reaching the target, i.e., leaving with $(R+T ) is:
W = (A -1) / (B -1)
A = [ (1 - p) / p ] ^R / U
B = [ (1 - p) / p ] ^(T+R) / U
Therefore, the Ruin Probability, i.e., the probability of losing it all $R is: (1 – W).
Notice: This results are subject to the condition that the targeted-winning amount ($T) must be much less than amount of money you bring to the table ($R). That is ($T) must be a fraction (f) of
Remember that if you bet too much you will leave a loser, while if you bet too little, your capital will grow too slowly. You may ask what fraction (f) of R you should bet always. Let V be the
amount that you win for every dollar that you risk, then the optimal fraction is:
f = p - (1 - p) / V
For example for p = 0.5, and v = 2, the optimal decision value for f is 25% of your capital R. The above result, recommend that fraction (f) of R you should bet always, must never exceed p.
You may like to use Two-Person Zero-Sum and The Gambler’s Games with Applications for checking your computation and for performing computer-assisted experimentation.
Other Competition Modeling Techniques
Competition is business often occurs because of the negative effects of one party (e.g. company) on another due to use and depletion of shared scares resources. Competition is one mechanism
leading to logistic growth. The logistic growth model is expressed as:
dN / dt = r N (K - N)/K
where, N = population size,
r = per-capita rate of population growth,
K = carrying capacity of the environment,
The main question is the following. Could the carrying capacity factor in the logistic equation be a function, rather than a constant? The competition of two technologies, e.g. wireless versus
cable or simulated annealing versus genetic algorithm can be intriguing. Here, the two parties are merely competitors; one does not "eat" the other --, as the consumer is the real "prey" though
at times willing. Lotka-Voterra model and its many variants are the widely used methods in many disciplines, including economics.
The classical Lotka-Volterra competition model is described by the following system of differential equations:
dN[1] / dt = r[1] N[1] (K[1] - N[1] - aN[2] )/K[1]
dN[2] / dt = r[2] N[2] (K[2] - N[2] - bN[1])/K[2]
The competition coefficients a, and b are the proportional constants that relate the effect of say one customer of the first party on population of second party. At the equilibrium level, we
N[2] dN[1] / dt = N[1] dN[2] / dt = 0.
Any extension for modification of the Lotka-Volterra competition modelmust overcome the following weaknesses: The model assumes the prey spontaneously grows and the carrying capacity of
environment has no limit. Moreover, removing the assumption that all predators and prey start at the same spatial point produces a set of interesting, more realistic, and useful results.
Further Readings:
Arkin V., et al., Stochastic Models of Control and Economic Dynamics, Academic Press, 1997.
Beltrami E., Mathematical Models for Society and Biology, Academic Press, 2001.
Ferguson B., and G. Lim, Introduction to Dynamic Economic Models, Manchester University Press, 1998.
Ford E., Scientific Method for Ecological Research, Cambridge University Press, 2000.
Hassell M., The Spatial and Temporal Dynamics of Host-Parasitoid Interactions, Oxford University Press, 2000.
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Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester
This paper presents the theoretical modeling and multiple-scale analysis of a novel piezoelectric energy harvester composed of a metal cantilever beam, piezoelectric films, and an axial preload
spring at the moveable end. The harvester experiences mono- and bi-stable regimes as the stiffness of preload spring increases. The governing equations are derived with two high-order coupling terms
induced by the axial motion. The literature shows that these high-order coupling terms lead to tedious calculations in the stability analysis of solutions. This work introduces an analytical strategy
and the implementation of the multiple-scale method for the harvester in either the mono- or bi-stable status. Numerical simulations are performed to verify the analytical solutions. The influence of
the electrical resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics are investigated. The spring pre-deformation has a slight influence on the energy
harvesting performance of the mono-stable system, but a large effect on that of the bi-stable system.
Issue Section:
Research Papers
1 Introduction
Considerable interest in vibration energy harvesting has emerged in the last decade with the rapid development of wireless sensors and low-power electrical devices [1–3]. Piezoelectric materials are
extensively studied for mechanical-to-electrical energy conversion due to the high power densities and easy fabrication [4–6]. One critical issue of linear piezoelectric vibration energy harvesters
(PEHs) is the limited frequency bandwidth [7,8]. A linear PEH needs to be deliberately designed to match its natural frequency with the excitation frequency to achieve resonance. A slight shift in
the excitation frequency or any defects in the operation could lead to the frequency mismatch and thus a significant power reduction [9].
Nonlinearities are exploited to improve the operation frequency bandwidth, because nonlinear PEHs are insensitive to the change in excitation frequency due to the tilted frequency curves [10].
Nonlinear PEHs include mono-stable [1,4,9,11,12], bi-stable [13,14], and tri-stable harvesters [15,16]. Among them, bi-stable harvesters are preferable for their large power output during the
larger-amplitude snap-through vibrations. Bi-stable PEHs have been shown to achieve a 300% improvement in the open-circuit root-mean-square voltage [17] and 13.1 times more power output [18].
Bi-stability can be realized by various mechanisms, such as repulsive magnets [19] and composite structures [20]. The nonlinearities usually make the analytical analysis become a daunting task,
especially for systems with high-order coupling terms [21–23]. The harmonic balance method [11,12] has shown good accuracy for bi-stable systems. However, research shows the stability analysis of the
harmonic balance solutions can be very complicated, even analytically impossible for systems with higher-order coupling terms [24]. On the other hand, the stability of multiple-scale solutions can be
easily determined from the Jacobian matrix. Although the method of multiple scales [4,25,26] has been widely used for nonlinear systems, very few articles systematically apply this technique to
systems with tunable potential wells and higher-order coupling terms. In particular, specific strategies are needed to handle the negative stiffness of bi-stable systems.
This letter presents the multiple-scale solutions of a broadband PEH with a tunable potential function, which consists of a metal cantilever beam, piezoelectric films, and an axial preload spring.
The spring plays the similar role of repulsive magnets in literature to achieve the buckling mechanism. Advantages of this strategy lie in avoiding the magnetic interference with wireless sensor
nodes and reducing the total mass [27]. The main contributions of this work include the following: (1) systematically introducing the technical strategies of the method of multiple scales for mono-
and bi-stable systems; (2) obtaining the analytical frequency responses, phase angle, and phase portraits of the mono-stable and bi-stable systems; and (3) investigating the effect of the external
resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics of the systems.
2 Theoretical Model
Figure 1(a) shows the proposed PEH, where one end of the harvester is clamped and the other end connected to the spring is moveable along the guide rails. When the spring is compressed, the beam is
exposed to the axial spring force. When the spring has a small pre-deformation, not enough to buckle the beam, the PEH is mono-stable; otherwise, the PEH enters into the bi-stable regime with two
stable equilibria and one unstable equilibrium, as shown in Fig. 1(a). The two piezoelectric films on the beam are connected in parallel and the generated electricity is delivered to a resistive
load. The PEH can be deemed as a current source and a capacitor [28], resulting in the simplified electrical circuit in Fig. 1(b). Assume the PEH experiences a base excitation of $a¨=Acos(Ωt)$,
where t is time, A and Ω are the excitation level and frequency.
be the width and length,
the thicknesses,
the densities of the beam and the piezoelectric films. The equivalent mass of per unit length is
+ 2
. The effect of the electrode layers on the stiffness and mass is ignored because they are very thin and light. Let
), and
) be the transverse, axial displacements, and the generated charge over the surfaces of the electrode layers. For simplicity, the transient displacement is approximated by the first mode as
) =
), where
) is the first modal coordinate in the time domain and
) = [1 − cos(2
)]/2 is the first mode shape function [
]. The inextensible assumption requires (1 +
= 1, and thus
], where the prime indicates the first derivative with respect to the coordinate
. The kinetic energy, potential energy, and work done by generalized forces are
= −
has been used and
denotes the damping coefficient.
are Young’s modulus of the beam and the piezoelectric films, respectively.
and Δ are the stiffness and the pre-deformation of the spring.
+ 2
are the equivalent axial and bending stiffness.
is the capacitance of the piezoelectric films.
are the piezoelectric constant and relative permittivity measured at a constant strain. The governing equations are then extracted from the Lagrangian method as follows:
, and
is the mechanical damping ratio. The Gauss current law
has been used in the derivation of the electrical equation
. All the parameters in the governing equations are evaluated by
Defining the dimensionless frequency and time
= Ω/
, Eqs.
can be rearranged as
where the prime denotes the derivative with respect to
, and
. The governing equations have two higher-order coupling terms, the product of the displacement and squared velocity, and the product of the squared displacement and acceleration. Those terms
significantly complicate the derivation of analytical solutions. Setting
″ =
′ = 0 and ignoring the effect of electrical term, the equilibria can be obtained as
The parameter $ϑ$ is always positive because the scaled axial stiffness k[2] is much larger than the equivalent spring stiffness k[3]. Thus, the existence of the equilibria $η2,3*$ depends on the
effective spring stiffness k[s]. The system has one equilibrium η[1] = 0 when Δ is small (k[s] < k[1]), and the PEH is mono-stable. As Δ increases to a certain level (k[s] = k[1]), the axial
compressive force equals the critical buckling load of the beam. Then, the PEH begins to enter into the buckling regime.
The potential function is $V¯(η)=χη2/2+ϑη4/4$, implying the parameter χ determines if the PEH is unbuckled or buckled. When χ > 0 (k[s]/k[1] < 1), the harvester is unbuckled, otherwise, it is
buckled bi-stable. Figure 2 plots the potential for the varying Δ with a constant spring stiffness of k[d] = 1000 N/m. The geometric and material properties of the PEH are listed in Table 1. The
potential function has a flattened parabolic shape initially and turns to a double-well shape as Δ increases. This indicates the PEH turns from the mono-stable to the bi-stable state.
Table 1
Beam (steel) and spring Piezoelectric layer
Parameters Value Parameters Value
Length, L 300 mm Length, L[p] 300 mm
Width, b 10 mm Width, b 10 mm
Thickness, h[s] 0.1 mm Thickness, h[p] 0.08 mm
Density, ρ[s] 7850 kg m^−3 Density, ρ[p] 4000 kg m^−3
Modulus, E[s] 203 GPa Modulus, E[p] 40 GPa
Damping, ξ 0.042 d[31] −10 pC N^−1
Stiffness, k[d] 2500 N m^−1 ɛ[33] 8.854 × 10^−10 F m^−1
Beam (steel) and spring Piezoelectric layer
Parameters Value Parameters Value
Length, L 300 mm Length, L[p] 300 mm
Width, b 10 mm Width, b 10 mm
Thickness, h[s] 0.1 mm Thickness, h[p] 0.08 mm
Density, ρ[s] 7850 kg m^−3 Density, ρ[p] 4000 kg m^−3
Modulus, E[s] 203 GPa Modulus, E[p] 40 GPa
Damping, ξ 0.042 d[31] −10 pC N^−1
Stiffness, k[d] 2500 N m^−1 ɛ[33] 8.854 × 10^−10 F m^−1
3 Mono-Stable Piezoelectric Energy Harvester
To approximate the analytical solutions using the method of multiple scales, the damping term, nonlinear higher-order terms, cubic term, and the excitation are assumed to be small. Therefore, the
mechanical equation
can be written as
is an small number. Introducing the fast and slow time variables
, the solutions are expanded and retained to the first order of
by the asymptotic series
Substituting the multiple-scale solutions into Eqs.
, and equating the terms with the same order of
The solutions to the zeroth-order equations are assumed to be
are the complex amplitudes and are functions of
are the complex conjugates of
. Substituting the solutions into the first equation in Eq.
, the voltage amplitude can be expressed as
, where
is a detuning parameter. Substituting the resultant solutions into the second equation in Eq.
and setting the coefficients of the secular terms associated with
to zeros, one has
has been used. The prime denotes the derivative with respect to
Solving any of the two equations in Eq.
gives the solutions of the amplitude and phase angle. We assume the complex amplitudes have the following polar forms
where Γ and
are the amplitude and phase and are real functions of time
. Substituting Eq.
into the first equation in Eq.
and separating the real and imaginary parts yields
where Φ
. By setting Γ′ = Φ′ = 0, the frequency response function can be derived by
The corresponding phase angle is
The amplitude Γ can be solved for different values of the detuning parameter
from Eq.
. The voltage amplitude can be attained from the second equation in Eq.
, which is
Multiple solutions of the amplitude Γ can be obtained from Eq.
for a given detuning parameter
. These solutions include stable solutions in both high and low branches and unstable trivial solutions. The stability analysis of the solutions can be achieved by analyzing the real part of the
eigenvalues of the Jacobian matrix. If the real parts of all the eigenvalues are negative, the solution is stable. Otherwise, it is unstable. The Jacobian matrix can be derived from Eq.
Eliminating the secular terms, the first-order solution can be obtained as
Plugging Eq.
into Eqs.
, then substituting the results into Eq.
, the approximate analytical solution of multiple scales in the time domain can be written as
where the relationship Φ =
has been used.
4 Case Study of the Mono-Stable Piezoelectric Energy Harvester
The voltage frequency responses are derived under three excitation levels of Y = 0.005 g, 0.01 g, and 0.02 g. To verify the analytical solutions, numerical simulations are conducted at discrete
frequency points. The initial displacement, velocity, and voltage are assumed to be zeros. The preload spring is soft, limited to k[s] < k[1] so that the PEH is mono-stable. Figure 3(a) presents the
voltage frequency responses under the open-circuit condition, which has the typical characteristics of a mono-stable nonlinear system, consisting of high and low branches and tilting to the right
hand. When the frequency is low (ω < 0.8), the system has a single stable solution at the high branch, while two stable solutions concurrently appear at the low and high branches at the high
frequency. The numerical simulations only get the low branch solutions in the frequency range of multiple stable solutions due to the zero initial conditions. The phase portraits at ω = 0.57 are
plotted in Fig. 3(b), which demonstrates the approximate solutions match well with the numerical simulations. Figure 4(a) shows the voltage frequency responses under different resistive loads at the
excitation level of 0.005 g. Both the high and low branch VFRs decrease as the external resistance becomes smaller.
The effects of the excitation level and the spring pre-deformation on the voltage output are studied at two frequencies of ω = 0.4 and 1.5. The spring pre-deformation is Δ = 0.05%L for the case of
varying excitation level. Figure 4(b) shows only the high branch oscillation happens at the small excitation frequency of ω = 0.4 and the voltage increases along with the excitation level. Both the
stable high and low branch solutions exist at lower excitation level (Y = 0∼0.3 g) for ω = 1.5. Figure 4(c) shows the voltage has a very slight increment as the spring pre-deformation increases for a
fixed excitation level. This suggests that the spring pre-deformation has insignificant influence on the voltage output. This is because the linear and cubic stiffness of the mono-stable PEH is
dominated by the bending and axial stiffness of the beam, when the spring pre-deformation is small.
5 Bi-Stable Piezoelectric Energy Harvester
As the effective spring stiffness k[s] exceeds the scaled linear stiffness k[1], the overall linear stiffness χ becomes negative and the system enters into the bi-stable regime. The bi-stable PEH
exhibits both small-amplitude intra-well oscillation and large-amplitude inter-well vibration. For the intra-well oscillation, a coordinate transform is performed to estimate the solutions around the
local equilibria. For the inter-well oscillation, a strategy has to be taken to handle the negative stiffness. This section presents the strategies and solutions of the multiple scales for the
approximate analytical analysis.
5.1 Intra-Well Oscillation.
To analyze the intra-well dynamics, the original origin at the unstable equilibrium needs to be shifted to one of the stable equilibria
]. Defining a new coordinate
and substituting it into the governing Eq.
, one has
. Introducing the multiple time variables
, and
, the solutions can be approximated by the following asymptotic series:
Plugging Eqs.
, keeping the order to
and equating the like-order terms of
, one has
Similarly, substituting the approximate solutions into Eq.
and collecting the like-order terms of
Assuming the solutions have the following forms:
; the complex amplitudes
, and their complex conjugates
depend on
. Substituting the solutions into the first-order equations, one has
Eliminating the secular terms yields
, implying
only depend on
, and solving Eq.
Substituting the solutions into the second-order equation in Eq.
, defining
, and setting the coefficients of the secular terms to zero, one has
. Both the above equations will give the same solution of A. Assuming
, substituting into the first equation in Eq.
, and setting the real and imaginary parts to zeros yield
where Φ
, and
. Let
, one can obtain the steady-state amplitude and phase angle as
The voltage amplitude equation is the same as Eq.
. The Jacobian matrix can be obtained from Eq.
After eliminating the secular terms, one has
Combing Eqs.
, and
, the approximate analytical solution can be obtained in the time domain as
5.2 Case Study of the Intra-Well Oscillation.
The spring pre-deformation needs to satisfy k[s] > k[1] to achieve the bi-stable configuration. Therefore, Δ = 0.5%L is chosen in the following study unless otherwise stated. The analytical and
numerical VFRs under the open-circuit condition are plotted in Fig. 5(a). The numerical results match with the analytical solutions. Different from that of the mono-stable PEH in Fig. 3(a), the
intra-well VFRs of the bi-stable PEH bend to the left-hand side for the soft stiffness. The bi-stable PEH shows no advantage in voltage outputs compared with the mono-stable PEH at Y = 0.01 g.
The phase portraits at ω = 3.46 and different excitation levels are presented in Fig. 5(b). The analytical solutions agree well with the numerical results. Figure 5(b) shows that the harvester is
confined in one of the local potential wells. Figure 5(c) plots the phase angle along with the excitation frequency at Y = 0.05 g. The bifurcation can be observed from the phase angle as the
excitation frequency varies.
The voltage frequency responses at different resistive loads are presented in Fig. 6(a) for Y = 0.03 g, which shows the resistive load has a significant influence on the voltage output. The voltage
decreases as the resistive load dwindles. The voltage output is shown in Fig. 6(b) for varying excitation levels at ω = 2.9 and 3.3. Unlike the case of the mono-stable PEH, only a single stable
solution appears and augments as the excitation level becomes large at ω = 3.3. For ω = 2.9, the system has a single stable low branch solution under small excitation levels, but has two stable
solutions of both high and low branches as the excitation level increases. There is only one stable high branch solution at high excitation levels.
The open-circuit voltage outputs at varying spring pre-deformations are plotted in Fig. 6(c) under the three excitation levels and σ = 0. The reason that the detuning parameter is set to zero is
because the natural frequency of the system varies as the spring pre-deformation changes. Only one stable solution exists over the considered range of the spring pre-deformation. The voltage output
decreases as the spring-deformation increases. We conclude that the spring-deformation has evident influence on the voltage output of the bi-stable PEH. This is attributed to the fact that the spring
force has a remarkable contribution to the system stiffness.
5.3 Inter-Well Oscillation.
Strategies are needed to handle the negative stiffness issue to apply the method of multiple scales. A fictitious positive stiffness term is introduced into Eq.
to shift the linear negative stiffness to a positive one. By adding and subtracting the fictitious term
to the left-hand side of Eq.
, one has
, and
. The damping, high-order terms, electrical coupling terms, excitation, and the stiffness terms
are assumed to be small. The solutions are assumed to be the same with those in Eq.
. Substituting the solutions into Eqs.
, one has
Substituting the zeroth-order solutions in Eq.
into the first equation in Eq.
, one can find the relationship between the displacement and the voltage. Plugging the solutions into the second equation in Eq.
, and setting the coefficients of the secular terms to zeros, and substituting the complex conjugate amplitudes
into one of the resultant equation, one has
where Φ
, and
are defined in Sec.
and Γ is the real displacement amplitude. Let
, one can obtain the steady-state amplitude and phase angle as
The Jacobian matrix can be derived from Eq.
as follows:
The first-order solution
is obtained after eliminating the secular terms. Together with the zeroth-order solution
, the analytical solution of the inter-well oscillation is
5.4 Case Study of the Inter-Well Oscillation.
The analytical voltage frequency responses at Y = 0.03 g, 0.05 g, and 0.15 g are presented in Fig. 7(a). The excitation levels are higher than those for the intra-well oscillations to activate the
inter-well oscillation. Comparing the voltage frequency responses under Y = 0.03 g and 0.05 g in Fig. 7(a) with those in Fig. 5(a), it shows the inter-well oscillation outperforms the intra-well
oscillation in voltage output. The numerical simulation results are not given in Fig. 7(a) as we did above because the inter-well oscillation cannot be always activated under the zero initial
To validate the analytical solution, upward and downward frequency sweeps are numerically performed to the governing equations within the frequency range of ω ∈ [0,7] and the excitation amplitdue of
Y = 0.15 g. The numerical results are presented in Fig. 7(b) together with the analytical solutions. The amplitude of the numerical voltage response matches with the analytical solutions.
Nevertheless, the analytical method overestimates the solutions at the higher frequency range. The numerical simulations are confined to the two local equilibria as the excitation frequency
increases. The phase angle at the open circuit and Y = 0.05 g in Fig. 8(b) shows the bifurcation at ω = 0.32.
The voltage frequency responses of the inter-well oscillation under Y = 0.05 g and different resistive loads are plotted in Fig. 8(a). Similarly, the voltage output becomes smaller as the external
resistive load decreases. The voltage outputs over varying excitation levels are plotted in Fig. 8(b) for two frequencies of ω = 1.0 and 0.3. The high voltage output from the large amplitude
inter-well oscillation happens when the excitation level is greater than 0.27 g for ω = 1.0. However, it is not available at the low excitation levels (0g ∼ 0.27 g) because of failing to activate the
inter-well oscillation. For ω = 0.3, the excitation amplitude has a similar influence on the voltage output with that of the intra-well oscillation. Figure 8(c) presents the voltage output over
varying spring pre-deformations at the three excitation levels. The trend of the voltage outputs along with the varying spring pre-deformation is the same with these in Fig. 7(a). This is because the
system stiffness is dominated by the effective spring stiffness k[s] induced by the spring pre-deformation. The increase in the spring pre-deformation increases the linear stiffness.
6 Conclusion
A novel piezoelectric beam energy harvester with tunable potential function is developed and analytical modeled. An axial preload spring is connected to one moveable end of the beam to achieve the
mono-stable or bi-stable systems. The method of multiple scales is systematically implemented to solve for the analytical frequency responses and phase angles. To handle the negative stiffness,
technical strategies are introduced to analyze the intra-well and inter-well dynamics. Numerical simulations are performed to confirm the analytical solutions. The effects of the electrical resistive
load, excitation level, and spring pre-deformation on the dynamics and voltage output are studied. The voltage output of both the mono-stable and bi-stable systems increase along with the excitation
level. The spring pre-deformation has slight influence on the energy harvesting performance of mono-stable system, but considerable effect on that of the bi-stable system.
This research was partially supported by National Science Foundation under Grants Nos. 1508862 and 1935951.
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Encouraging constant method, supplying aid, and developing a favorable discovering setting are advantageous actions. | {"url":"https://crown-darts.com/en/multiplication-facts-worksheets-printable.html","timestamp":"2024-11-04T08:05:22Z","content_type":"text/html","content_length":"29693","record_id":"<urn:uuid:1b15de1f-b28c-44a2-b273-c21162405241>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00205.warc.gz"} |
gfxbrot.c and gfxjulia.c
Re: gfxbrot.c and gfxjulia.c
I am working with mandelbrot and julia drawing with random dots, but not same dot many times.
Attached files work with 64k and ROM v5a.
Biggest problem of this is still it uses floating point calculation.
Maybe one day I make integer math fractals...
Last edited by veekoo on 22 Nov 2023, 14:23, edited 3 times in total.
Re: gfxbrot.c and gfxjulia.c
Maybe someone who knows integer fractals can adapt my code.
Codes can be found at github:
Last edited by veekoo on 22 Nov 2023, 14:06, edited 1 time in total.
Re: gfxbrot.c and gfxjulia.c
veekoo wrote: ↑31 Jan 2022, 12:23 Maybe one day I make integer math fractals...
I do not have time to do it but the attached code might help you.
This code represents fractional numbers -16<x<16 as the integer 256*x. One can add or subtract such number using the normal integer addition or subtraction. A small vcpu routine, mul48, multiplies
two such numbers. It only works when both its arguments and its result are strictly between -16 and +16, otherwise it silently returns erroneous results. This is just enough to implement Mandelbrot
at the normal scale.
(2.25 KiB) Downloaded 509 times
Re: gfxbrot.c and gfxjulia.c
One guy from github had done good job documenting and given sources for integer fractals. I can give the link in privatemsg.
Last edited by veekoo on 22 Nov 2023, 14:03, edited 5 times in total.
Re: gfxbrot.c and gfxjulia.c
Example picture of this code using long data type. Getting fast calculation might not happen with this code.
"Types short and int are 16 bits long. Type long is 32 bits long. Types float and double are 40 bits long, using the Microsoft Basic floating point format. Both long arithmetic or floating point
arithmetic incur a significant speed penalty."
longbrot.png (9.19 KiB) Viewed 6486 times
Last edited by veekoo on 18 Oct 2022, 05:03, edited 1 time in total.
Re: gfxbrot.c and gfxjulia.c
I believe this reference on integer fractals was written with a substantially more powerful processor in mind.
As you know, the Gigatron does not have a hardware multiplier. Therefore a 32-bits multiplication must be computed with 32 32-bits additions and shifts. In addition the Gigatron hardware can only
perform 8 bit operations. Since it does not provide access to the carry bit, one needs additional logic to determine whether there is a carry and to apply the carry as needed. For 16-bits additions
and subtractions, this is implemented quite efficiently in native code. For 32-bits additions and subtractions, this must be done with vCPU instructions.
This means that it pays to make our fractional precision numbers fit inside a 16-bits integer. This is what is achieved by the fractional multiplication from my earlier post. Since it only uses 12
bits of the int variable, one needs only 12 loops in the multiplication routine, another little gain. Anyway this is close to what Marcel did in the original version of the Gigatron Mandelbrot
program, and should have a comparable speed.
One can go faster thanks to qwertyface's quarter square multiplication trick. See https://forum.gigatron.io/viewtopic.php?p=2632#p2632 for details. Interestingly this can be achieved with just vCPU
code, but only in a manner that is very specific to the Mandelbrot calculation.
PS. My latest version of the GLCC runtime has a 20% speedup on long additions and subtractions. But that only gives 10% on multiplications.
Re: gfxbrot.c and gfxjulia.c
I think for Gigatron community making another Mandelbrot in integer is not so important. A fast Julia might be intresting. Yes the example was for Amiga or PC. I noticed the update on GLCC and used
it in here.
There is now long Mandelbrot and long Julia versions. Its integer math, but not 16-bit int.
These are faster than previous programs. Currently testing at fullscreen.
Gfx mandelbrot uses floating point and takes 2 hours 5 minutes to draw the screen. Long version takes 45 min.
Gfx julia uses floating point and takes 1 hour 50 minutes to draw the screen. Long version takes 35 min.
Re: gfxbrot.c and gfxjulia.c
In Mandelbrot family there is also Julia, but not many know Burning Ship. Very small modification to Mandelbrot program you get Burning Ship. In some illustrations you can actually see the ship.
Two pictures: 1. Burning Ship zoomed 2. Burning Ship original view
burnship_zoomed.png (8.25 KiB) Viewed 6452 times
Screenshot from 2022-10-18 02-25-08.png (8.24 KiB) Viewed 6472 times
Re: gfxbrot.c and gfxjulia.c
veekoo wrote: ↑17 Oct 2022, 06:51 Gfx mandelbrot uses floating point and takes 2 hours 5 minutes to draw the screen. Long version takes 45 min.
Gfx julia uses floating point and takes 1 hour 50 minutes to draw the screen. Long version takes 35 min.
Not bad!
Re: gfxbrot.c and gfxjulia.c
I have gone to ROMv6 era. So here is all in one fractal programs.
Floating point math with graphics: fpfract2 (all in one)
Long integer math with graphics: longfract (all in one)
Integer math with graphics: intfract (all in one)
With these you can get faster or more accurate pictures. Floating point is most accurate and slowest. Long integer is medium speed and medium accurate. Integer is fastest but loses accuracy and
picture size is limited.
Good "screen saver" is random dot fractals.
Floating point math with random dot: rndbrot and rndjulia (64K)
(6.94 KiB) Downloaded 443 times
(8.53 KiB) Downloaded 428 times
(10.02 KiB) Downloaded 452 times
(3.79 KiB) Downloaded 448 times
(3.82 KiB) Downloaded 437 times | {"url":"https://forum.gigatron.io/viewtopic.php?t=324&sid=851de95b0dd99c2632e24478bad0168f&start=10","timestamp":"2024-11-02T04:51:29Z","content_type":"text/html","content_length":"50369","record_id":"<urn:uuid:6c9fdf84-5978-4252-8c82-f52fae775f3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00216.warc.gz"} |
HyTES/19, flight day 171b/2019, Grosseto
Reported by: wja Owned by:
Priority: immediate Milestone:
Component: Processing: general Keywords:
Cc: dac@… Other processors:
Data location: /users/rsg/arsf/arsf_data/2019/flight_data/italy/HyTES19-2019_169_Grosseto
Data arrived from Kings College London via hardisk on July 15 2019.
Scientific objective: Joint NASA and ESA campaign flying JPL’s HyTES thermal instrument over sites in Europe with a focus on crops and soil. Aim is to inform the future Land Surface Temperature
Monitoring (LSTM) satellite mission. The NCEO Fenix instrument was flown alongside HyTES to provide visible – SWIR hyperspectral data.
PI: Martin Wooster
Fenix (requested, flown)
Change History (17)
Navigation Processing
Basestation data converted to RINEX. Of the three RINEX files (which contain multiple days of data), a file for this date has been created with the following command:
teqc -st 2019_06_20:06:27:48.840 -e 2019_06_20:15:09:15.860 +nav Default_9488_2006_062748.19n Default_9488_0509_114526.m00 > Default_9488_2006_062748.19o
Navigation Processing
SOL file produced using PPP processing.
Fenix Processing
Started processing.
Logsheet Creation
Log Sheets generated, flight line plot not generated and no KML file either so there is no plot at the moment
Fenix Processing
SCT values identified
Flightline FENIX
1 3.40
2 46.10
3 3.80
4 3.70
5 24.3
7 69.00
Fenix Processing
Some small FPS issues with 1 line of the flight, might be passable
Hyperspectral Processing
Flightline 2 looks like it has a FPS error.
Hyperspectral Processing
All lines reprocessed with the following SCT values
Line SCT
1 2.40
2 44.94
3 2.80
4 2.70
5 23.30
6 56.00
7 68.00
Flightline 2 has been processed with a FPS of 92.580
Ready for checking
Fenix Delivery Check
Fenix Delivery Check
-No Readme file (please add the data quality remarks similar to previous deliveries)
-Removed fodis directory
-Updated data quality report
-xml files are correct
-Spectra looks good compared with 6S, just a bit low in the SWIR
-APL commands work just fine and show good SCT values and offsets.
Most of the checks have been done alreay, everything looking good with the exception of the readme file.
Hyperspectral Delivery
ReadMe added
Fenix Delivery Check
Readme and data qualirt remarks looking good. All other checks are correct, I will zip the mapped files and mark as ready to be delivered once done.
Fenix Delivery
All mapped files have been zipped correctly. Project is ready to be delivered.
Fenix Delivery
Project delivered, finalised and notification sent to PI on 5th November 2019.
Atmospheric correction
Atmospheric correction has been applied to create level2 and mapped data.
Delivered to PI on June 22nd 2020
Needs archiving at CEDA
JSONs created, added to internal database.
Waiting for account with CEDA to be renewed in order to rsync
Rsync to CEDA complete. Have sent an e-mail to ask for their confirmation that the data has been received. | {"url":"https://nerc-arf-dan.pml.ac.uk/trac/ticket/656?cversion=0&cnum_hist=3","timestamp":"2024-11-01T22:17:33Z","content_type":"application/xhtml+xml","content_length":"34626","record_id":"<urn:uuid:79f6433f-4885-46d7-84f9-d3f503a40a1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00608.warc.gz"} |
Bor Plestenjak: Uniform determinantal representations
Date of publication: 28. 11. 2016
Numerical analysis seminar
Sreda, 30.11.2016, od 10h do 11h, soba 3.06 na Jadranski 21
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing.
Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of
variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is
fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their
root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research | {"url":"https://www.fmf.uni-lj.si/en/news/news/30991/bor-plestenjak-uniform-determinantal-representations/","timestamp":"2024-11-05T05:46:50Z","content_type":"text/html","content_length":"17991","record_id":"<urn:uuid:0fd72e9f-856c-4a9f-8b68-5df53cbf4f71>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00838.warc.gz"} |
This is a 2 week unit designed to cover high-school and introductory college level topics in the properties of gases and gas particle behavior. The unit includes eight lessons. Each lesson uses
computer models to explore these topics in greater depth. The models also afford greater degree of student inquiry and guided discovery than would be typically possible through other learning
activities in the same amount of time.
The computer models enable students to investigate what causes pressure in a gas, how it is measured, and how it is affected by the properties of the particles that make the gas and the
characteristics of the container they are in. Students are encouraged to understand how the principles and effects of pressure are generated from specific interactions between many particles (or
simplified gas molecules) and with their environment. To do this, students gain a familiarity with a microscopic view of the gas particles by running many different computer models of systems of gas
particles. Some initial models are designed for orienting the student to the NetLogo interface and the practice of computer-based modeling, others are designed for specific data gathering and data
analysis tasks.
The later computer models focus on how particles behave in a variety of conditions. Such conditions include varying the number of particles, the size of the particles, the speed of the particles, and
the location of solid walls they bounce off. These variations support students explorations of the models to design new variations into the model (adding new rules for particle behavior, designing
new system boundaries), designing experiments and testing predictions with the models, and deriving mathematical models (symbolic representations of relationships that they find in graphs and table
they build from data they gather in their experiments). The mathematical modeling of relationships between variables such as the number of gas particles, the temperature of a gas, the volume of the
gas container, gas constants, and the pressure of the gas, helps students to progressively expand and derive the gas laws from experimental data. This mathematical modeling focus, helps students
bridge the symbolic representations of the gas laws, to experimental data, to particle behavior.
The non-computer based activities ask students to apply ideas and relationships learned in class, extend the predictions of these relationships to new situations that they experience every day, and
connect previous and upcoming concepts to their understanding of particle behavior (Kinetic Molecular Theory) and to broader cross cutting themes in science. Some of the broad cross cutting themes
include building and using scientific models, systems thinking, data analysis, and change and equilibrium.
These activities will build a deep and intuitive sense of particle behavior that will extend readily to other chemistry topics. In particular, the behavior of particles in chemical reactions becomes
easy to envision and predict the outcome of the interactions of molecules, even when the reactants are in solid or liquid form. This is because many of the concepts related to chemical reactions rely
on an understanding of the number of molecules, volume, and temperature, and pressure.
Teacher Guide (pdf)
Underlying Lessons
Next Generation Science Standards
• Physical Science
• NGSS Crosscutting Concept
• NGSS Practice
Computational Thinking in STEM
• Data Practices
• Modeling and Simulation Practices
• Computational Problem Solving Practices
• Systems Thinking Practices
A majority of this unit is adopted from the earlier Connected Chemistry units developed by Uri Wilensky, Mike Stieff, Sharona Levy, and Michael Novak (see http://ccl.northwestern.edu/rp/mac/
index.shtml for more details). Some elements are also taken from the Particulate Nature of the Matter unit developed by Corey Brady, Michael Novak, Nathan Holbert, and Firat Soylu (see http://
ccl.northwestern.edu/rp/modelsim/index.shtml for more details).
We also thank undergraduate research assistants Aimee Moses, Carson Rogge, Sumit Chandra, and Mitchell Estberg for their contributions. | {"url":"https://ct-stem.northwestern.edu/curriculum/preview/565/","timestamp":"2024-11-04T04:34:21Z","content_type":"text/html","content_length":"42267","record_id":"<urn:uuid:020261e0-8222-4286-93fe-8ce7fb8291b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00145.warc.gz"} |
The Mean Value Theorem For Integrals: Average Value of a Function
Professor Dave Explains
8 Jun 201807:24
TLDRThe mean value theorem states that for a continuous, differentiable function over an interval, there exists at least one point where the slope of the tangent line equals the slope of the secant
line between the interval endpoints. This implies the instantaneous rate equals the average rate. Similarly, the mean value theorem for integrals enables computing a function's average value over an
interval. It states there is a point where the area of the rectangle under the curve equals the curve's area over the whole interval. Together, these facilitate finding averages through
differentiation or integration, allowing simpler evaluation depending on the function's complexity.
• π The mean value theorem states that for a continuous, differentiable function over an interval, there is at least one point where the slope of the tangent line equals the slope of the secant
• π The mean value theorem for integrals allows computing the average value of a function over an interval by taking the integral over that interval divided by the interval length.
• π For a position function, the mean value theorem implies there's an instant where instantaneous velocity equals average velocity over the interval.
• π The integral in the integration version represents the area under part of the curve equal to the area of a rectangle.
• π ¨β π « Both theorems allow finding average function values over intervals through differentiation or integration.
• π The integration version geometrically means part of the area under the curve equals the area of some rectangle.
• π ’ Has applications like finding average temperature over a time period.
• β οΈ There may be multiple points satisfying the derivative version of the mean value theorem.
• π ₯ Applies to continuous, differentiable functions.
• π ‘ Allows choice of easiest method to evaluate average function values.
Q & A
• What is the mean value theorem for differentiation?
-The mean value theorem for differentiation states that for a continuous, differentiable function over an interval [A,B], there is at least one point C in the interval where the slope of the
tangent line at C equals the slope of the secant line connecting the endpoints (A,f(A)) and (B,f(B)).
• How is the mean value theorem for integration related to the mean value theorem for differentiation?
-The mean value theorem for integration finds the average value of a function over an interval using its antiderivative. This is analogous to the mean value theorem for differentiation, which
finds the average rate of change of a function over an interval using its derivative. Both make use of the fundamental theorem of calculus.
• What does the mean value theorem for integration allow us to calculate?
-The mean value theorem for integration allows us to calculate the average value of a continuous function f(x) over an interval [A,B] using the formula (1/(B-A)) * β «_A^B f(x) dx.
• How can the mean value theorem be interpreted geometrically?
-Geometrically, the mean value theorem states that there is a point on the curve where the tangent line is parallel to the secant line between two endpoints. For the integration version, it means
there is a rectangle with the same area as the region under the curve.
• What are some applications of the mean value theorem?
-Some applications include: finding average velocity or acceleration of motion, computing average temperature or pressure over time, determining average value of a statistical distribution, and
estimating areas bounded by curves.
• What is the difference between a secant line and a tangent line?
-A secant line connects two points on a curve. A tangent line touches the curve at only one point and has the same slope as the curve at that point.
• What does it mean for a function to be concave up or concave down?
-A function is concave up if you can draw a line under the curve such that the curve lies above the line. It's concave down if you can draw a line above it with the curve below the line.
• What is the significance of a function having only one concavity?
-If a function has only one concavity, either convex or concave over an interval, then by the mean value theorem there will be only one point where the tangent line is parallel to the secant
• What types of functions can the mean value theorem be applied to?
-The mean value theorem applies to continuous, differentiable functions defined over a closed interval [A,B]. The function must be at least once differentiable on the open interval (A,B).
• How can you determine if the mean value theorem can be applied to a particular function?
-Check if the function is continuous over [A,B] and differentiable over (A,B). If so, the mean value theorem guarantees at least one point C in (A,B) satisfying the theorem.
π Introducing Average Value Theorems
Paragraph 1 introduces the concept of finding average values for functions using calculus theorems. It first mentions the mean value theorem which states that over an interval, there must be a point
where the slope of the tangent line equals the slope of the secant line. This has applications like relating instantaneous velocity to average velocity. It then introduces the mean value theorem for
integrals, which allows computing the average value of a function over an interval by taking the integral of the function over that interval.
π Applying the Mean Value Theorem for Integrals
Paragraph 2 applies the mean value theorem for integrals to the function 1+x^2 from -1 to 2. By evaluating the integral and plugging into the theorem's formula, the average value is found to be 2.
Geometrically, this means there is a line y=2 such that the area under the curve above it equals the area missing below it over the interval.
π ‘Mean value theorem
The mean value theorem states that for a continuous, differentiable function over an interval, there must exist at least one point within that interval where the slope of the tangent line equals the
slope of the secant line between the interval endpoints. This relates the instantaneous rate of change (derivative) to the average rate of change over an interval. The video provides a geometric
interpretation of this theorem.
π ‘Mean value theorem for integrals
This theorem allows finding the average value of a function over an interval by taking the integral of the function over that interval divided by the interval width. It is analogous to the mean value
theorem for derivatives. As illustrated in the video, it implies there is a point where the function equals its average value over the interval.
π ‘Average
The video discusses different types of mathematical averages, focusing on the mean or arithmetic average. For a finite data set, this is found by summing the values and dividing by the number of data
points. The mean value theorems allow computing averages for continuous functions.
π ‘Integral
Integration is used to find the area under a curve representing a continuous function. The video applies integrals to compute the average value of functions over intervals using the mean value
theorem for integrals.
π ‘Derivative
The derivative of a function gives its instantaneous rate of change or slope at a point. The mean value theorem relates the derivative at a point to the average rate of change over an interval.
π ‘Slope
The video compares the slope of the tangent line at a point on a curve to the slope of the secant line between two points, using geometric interpretation to explain the mean value theorem.
π ‘Velocity
If the function represents position, its derivative gives velocity. The video explains how the mean value theorem implies the instantaneous velocity at a point equals the average velocity over an
π ‘Area under a curve
The video illustrates how the mean value theorem for integrals implies that the area under a curve between two points equals the area of a rectangle with height equal to the average value of the
function over that interval.
π ‘Concavity
The video shows how concavity of a function relates to the number of points where the tangent line slope equals the secant line slope, illustrating the mean value theorem.
π ‘Continuous
The mean value theorems apply only to continuous functions. Continuity means small changes in input produce small changes in output, with no abrupt jumps.
Researchers developed a novel machine learning method to accurately predict risk of cardiovascular disease
The model was trained on a large dataset of medical records and outperformed traditional risk scores
This technology could enable earlier preventative interventions and improve cardiovascular outcomes
Scientists discovered genetic variants that protect against Alzheimer's disease in some individuals
Understanding these protective mechanisms opens new possibilities for Alzheimer's prevention and treatment
Researchers 3D printed a novel biomaterial scaffold to regenerate bone tissue
This scaffold promotes bone cell growth and vascularization for improved bone repair
This technology has potential applications in reconstructive surgery and treatment of bone defects
Scientists developed an ingestible sensor to monitor gastrointestinal health
The sensor tracks pH, temperature, bleeding, and transit time throughout the digestive system
This could enable early diagnosis of gastrointestinal diseases and personalized treatment
Researchers used AI to analyze cancer pathology slides with expert-level accuracy
The AI system rivals pathologists in identifying cancer types and predicting patient outcomes
This technology can improve cancer diagnosis and prognosis, especially in low-resource settings
Scientists engineered a novel CRISPR system for more efficient gene editing
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About the Quasars
Received 4 June 2016; accepted 27 June 216; published 30 June 2016
1. Introduction
Quasars were originally discovered as intense sources of radio emission. A quasar is by definition a starlike body with a large red shift. Such property is characteristic of quasars and gives them
their mysterious nature. By 1974, the spectra of over two hundred quasars had been analyzed, and all of them had very large red shifts. The simplest way to explain the quasars’ red shifts is to
assume that they are extremely distant bodies that follow Hubble’s law; in such a way that they are the most distant objects known. Moreover [1] [2] , if the red shifts of quasars are caused by the
expansion of the Universe, they are very luminous bodies indeed. In fact, the galaxies are not the most luminous objects in the Universe; because that honor belongs to the quasars, which are hundreds
of times as luminous as galaxies. As the name indicates, many quasars are intense radio sources. The total amount of energy emitted in the radio range is somewhat less than the optical luminosity; in
such a way that the quasars emit a lot of optical radiation, too. Thus, the most puzzling of all quasars’ problems is the energy source. Another puzzling aspect of quasars comes from their small
size; problem which is encounter when the investigators try to explain the optical radiation which comes from the rapidly varying sources. Those observations have been made in the optical part of the
spectrum, and according to the results obtained, it is well known at present that the continuous radiation from quasars is variable with time. Most quasars vary relatively slowly increasing and
decreasing their luminosities over periods of about a year, but a few of them are over much violent in their variation increasing dramatically their luminosities in periods of a day or so. When the
size of quasars was first pointed out in 1966, some scientists thought that the problem was serious enough so that the cosmological distances of quasars should be questioned; because if the quasars
were brought closer, they would not need to be as luminous and the size problem could be solved. However, according to recent observations the source of the radiation comes from a very small region;
in such a way that it is quite puzzling that so much radiation comes from such a small volume of space. The question is how such a small volume can produce so much energy.
2. The Size and the Energy Emission of Quasars
If some results of Einstein’s Special Theory of Relativity [3] are taken into account, it is possible to propose a solution to the size and the large luminosity of quasars. According to that theory
no material body could possible travel with a velocity greater than the velocity of light in vacuum. Thus it becomes obvious that none of them could accelerate beyond the light barrier. This argument
still stands. Concerning the quasars, all of them are material bodies which have a proper mass different from zero [3] [4] . In other words, they are objects which can only travel at velocities
smaller than the velocity of light in the empty space; and clearly, the only way by which they can move is by means of an acceleration process. However, in its original paper, Einstein said that
there is an upper limit of the velocity for the material bodies. In fact, the total energy of those objects would get infinitely large upon approaching the velocity toward the light barrier; as it is
easy to see from the following relativistic transformation equation [5]
This very important relativistic result shows, in particular, that the total energy of a free body does not go to zero for v = 0 but rather takes on a finite value.
On the other hand, there is another relativistic transformation equation for the volume of the material bodies. Since the transverse dimensions do not change because of the motion, the volume V of a
body decreases according to the following formula.
where V[o] is the proper volume of the body. An examination of the Equations (1) and (3) shows that when v → c, E → ¥; and V → 0. Thus as the velocity of a body, as a quasar, increases toward the
velocity of light in the empty space, the quasars’ total energy increases toward infinity and its volume or size decreases toward zero. Since it is absurd for any quasar with finite proper mass m[o]
to have infinite energy, and at the same time volume zero, it must conclude that it is impossible for any quasar to move with the light velocity in vacuum. Nevertheless, the relativistic
transformation Equations (1) and (3) show that in one case an increase of the total energy, and in the other case a decrease in the size of quasars are produced when their velocity v goes toward c,
in such a way that taking into account their great red shifts; those conclusions are enough to explain the quasars’ large luminosity, and at the same time its small size; as it is easy to see in the
following graphics (see Figure 1).
Figure 1. Relativistic transformation of the energy, and the volume.
3. The Relativistic Lens
The relativistic effect over the total energy emission, and the size of the quasars, apparently, acts as a kind of lens: In fact, according to the formula (3) the volume V decreases when the quasar’s
velocity v increases toward c. At the same time the quasar’s total energy, and also its mass, increases toward infinity when v → c.
On the other hand, it is well known from Optics that the ratio of the image size q, to the object size p is what is called the magnification M; so that, M = q/p. According to the relativistic
transformation Equation (3), it could be considered that V is the volume image, and V[o] is the volume object; in such a way that
Since v < c always, M < 1; and we have that V < V[o][ ]always. That means that the effect of the increasingly velocity of recession v is to diminish the size of the volume image. On the other hand,
from the other relativistic transformation equation; that is to say, the Equation (1); E will be the total energy image, and E[o] = m[o]c^2 the total energy object; and then.
Since M < 1, M^−^1 > 1 always, so that, the increasingly velocity of recession v produces in this case, an effect of magnification; in such a way the total energy image E is always greater than the
total energy object E[o].
Because of the enormous velocities of recession of the quasars, and also due to the relativistic effect, the image of the total energy emitted, appears amplified, and at the same time, and because of
the same reasons, the image of the size appears diminished.
4. The Red Shifts and Hubble’s Constant
The red shift of a quasar is usually denoted by the letter z; that is to say
where ∆l is the shift in wavelength of a spectral line, and l is the wavelength that that line had when it left the quasar. Quasar red shifts range from relatively small numbers, like 0.158 for 3C
273, to large, like 3.53 for OQ 172, the most distant quasar known at this time. The red shifts can also be expressed as a velocity by means of the Doppler shift formula. However, if the velocity is
small compared to the velocity of light, the following simple form of that formula is normally used.
where v is the velocity, and c the velocity of light. Given that the quasars have very large red shifts; showing that these objects are moving at relativistic recession velocities; it is necessary to
use the exact formula for the relativistic Doppler shift [5] - [7]
From this equation, one gets that
where z is the red shift measured from the spectra.
be, the relativistic red shift corresponding to a velocity of recession v, which always is small compared to the velocity of light in the empty space.
According to the evolutionary phenomenon called the Expansion of the Universe [1] , the velocity of recession and the distance are correlated. The larger the distance is, the greater the velocity is.
This relationship is known as the law of red shifts, or also Hubble’s law, and can be written as follows: Velocity of recession equals Hubble’s constant times distance
where H Hubble’s constant. Hence, substituting (10) in (11), the following result is obtained
In this case, it is possible to compare any couples of quasars, and consider one of them as a measure unit; in such a way that
where Z[o] and r[o], are the relativistic redshift and the distance, respectively, of the measure unit; as long as, Z[1] and r[1] are the relativistic red shift and the distance of the other quasar.
Thus, to calculate extragalactic distances, only one need the Equation (9) to obtain the relativistic red shifts from the red shifts measured directly from the respective spectra, and then, the
following relationship can be used
Although is not possible to use the local group of galaxies for measuring the Hubble constant; because of a such a distances their random velocities may interfere with the motion because of to the
Expansion of the Universe, can still be used as a previous step to obtain the data for the measure unit. As it is well known, the nearest large cluster of galaxies is the Virgo Cluster; 23.9
Megaparsecs distant, and according to the red shift measured from its spectrum, the velocity of recession of that cluster is equal to 1200 km∙sec^−1. Given that this velocity is small compared to the
velocity of light, the Equation (7) is used to obtain that z = 0.4 × 10^−2. On the other hand, the red shift obtained directly from the spectrum of the quasar 3C 273 is z = 0.158; so that, from the
Equation (9), it is easy to calculate the following relativistic redshift
Let’s consider the quasar 3C 273 as a measure unit, from which we have that
Using those data in the Equation (14), and also the formula (9) to obtain the relativistic red shifts from the red shifts measured directly in the respective spectra, which are reported in the
specialized literature, the following results are obtain.
Finally, from the relationship (12), and with the use of the former data, a value of 50.2 kilometers per second per Megaparsec for Hubble’s constant is obtained in every case; and, this is the exact
value for that constant.
5. Conclusions
With all that have been previously mentioned, there are enough reasons to believe that quasars seem to be N- galaxies or Seyfert galaxies which are so far away that only their central core is
visible. The N-galaxies are the optical equivalents, in a sense, of the compact radio source, having most of their luminosity contained in small, brilliant, almost stellar nuclei. Their properties
read very much like quasars’ properties. On the other hand, the N-galaxies are distinguished by their photographic appearance and the Seyfert galaxies by their spectra. Nevertheless, not all the
N-galaxies have spectra with Syefert characteristics; but in general, the spectra of N-galaxies and Seyfert galaxies can be explained in the same way that the spectra of quasars can be.
Also, it is possible to assume that the rapidity of the variations in the luminosity of some quasars, maybe the most distant, can be explained with the aid of the relativistic effect over the size of
those objects.
Finally, with the use of the red shifts of the spectral lines picture, it is possible to propose another way to calculate Hubble’s constant. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=67849","timestamp":"2024-11-06T15:48:00Z","content_type":"application/xhtml+xml","content_length":"88087","record_id":"<urn:uuid:65495453-4f3d-4b9f-9df9-ef0b9ab5ebe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00604.warc.gz"} |
How to find the area of a square - GRE Math
All GRE Math Resources
Example Questions
Example Question #1 : How To Find The Area Of A Square
In the figure above, the circle is inscribed within the bounding square. If r = 5, what is the area of the shaded region?
Correct answer:
25 - (25/4)π
In order to solve this, we must first find the area of the containing square and then remove the inscribed circle. Once this is done, we need to divide our result by 4 in order to get the one-forth
that is the one shaded region.
One side of the square will be equal to the circle's diameter (2r). Since r = 5, d = 10. Therefore, the area of the square is d^2 = 10^2 = 100. The area of the circle is πr^2 = 5^2π = 25π.
Therefore, the area of the four "corner regions" is equal to 100 - 25π. One of these is equal to (100 - 25π) / 4. Simplified, this is 25 - (25/4)π.
Example Question #2 : How To Find The Area Of A Square
Find the area of a square with a side length of 4.
All sides are equal in a square. To find the area of a square, multiply length times width. We know length = 4 but since all sides are equal, the width is also 4. 4 * 4 = 16.
Example Question #1 : How To Find The Area Of A Square
If one doubles the radius of the semi-circle on the right of the diagram above, by what percentage does the overall area of the diagram change?
To compare, first calculate the area of figure 1. Since it shares dimensions with the semi-circle, we will put all our variables in terms of the radius of that semi-circle:
A[1] = (2r)^2 + πr^2/2 = 4r^2 + πr^2/2 = r^2(4 + π)/2
If we double r, we get:
A[2] = (2 * 2r)^2 + π(2r)^2/2 = 16r^2 + π4r^2/2 = 4r^2(4 + π)/2
This means that the new figure is 4x the size of the original. This is an increase of 300%.
Example Question #2 : How To Find The Area Of A Square
Quantity A: The area of a square with side 1 m
Quantity B: One hundred times the area of a square with side 1 cm
Possible Answers:
The two quantities are equal.
The relationship cannot be determined from the information given.
Correct answer:
Quantity A is greater.
The obvious answer for this problem is that they are equal. Remember quantitative comparisons are often tricky and require you to check your initial inclination. If 1 m equals 100 cm then a square
with side 1 m (100 cm) has an area of 100 cm x 100 cm or 10,000 cm^2. The area of a square with a 1 cm side is 1 cm x 1 cm or 1 cm^2. One hundred times 1 cm^2 is 100 cm^2. 10,000 cm^2 is larger than
100 cm^2 so Quantity A is greater.
Example Question #2 : How To Find The Area Of A Square
A square is inscribed in a circle. The diameter of the circle is 10 feet. What is the area of the square?
Possible Answers:
It cannot be determined from the information provided.
Correct answer:
Since the diameter of the circle is 10, we know the radius of the circle is 5 feet. We can then draw radii that go from the center to two consecutive corners of the square. These radii are both 5
feet, and form a 90 degree angle (since they are diagonals of a square). Thus, with the enclosed side of the square they form a 45-45-90 triangle. Thus, the side of the square must be
Example Question #3 : How To Find The Area Of A Square
In the figure above, a square is inscribed in a circle with a diameter of 5 cm.
What is the area of the square?
Correct answer:
The diameter of the circle and the sides of the square form a 45-45-90 triangle. Since the hypotenuse is 5 cm, then a leg of the triangle (a side of the square) is
The area of the square is then
Example Question #1 : How To Find The Area Of A Square
Quantity A: The area of square
Quantity B: 24
Possible Answers:
The information cannot be determined based on the information provided
The two quantities are equal
Correct answer:
Quantity A is greater
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Scientific RPN Calculator (with ATTINY85)
03-09-2018, 10:28 AM
Post: #30
Paul Dale Posts: 1,848
Senior Member Joined: Dec 2013
RE: Scientific RPN Calculator (with ATTINY85)
(03-09-2018 09:22 AM)deetee Wrote: Wow I did not know that my HP35 (disguised as !) calculates gamma instead of faculty. Right now I have implemented the typical for-loop to calculate n!
Can you recommend a smaller formula to calculate gamma other than the Nemes approach (Stirling works for big numbers accurate only).
Nemes is also for large numbers. Lanczos is another commonly used approximation. All are okay. The trick is to make sure the number is large enough:
double fac = 1;
while (x < too_small) fac *= x++;
return fac * result;
The required threshold isn't big but it depends on the approximation used.
You'll want to check the recurrence.
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BA3020QA: Using and Managing Data and Information Assignment 1 -
Using and Managing Data and Information
BA3020QA Assignment 1
There are FIVE tasks and you are expected to answer all of them. Answer each task on a separate worksheet on the same Excel file.
Task 1 (10 marks)
Use Excel as a calculator to evaluate the following calculations. For each calculation you should provide the formula you have used (1 decimal place)
Task 2 (10 marks)
Use the Excel formatting facilities to answer the following questions:
• Express 72.5% as a 2 digit fraction.
• Express 0.58 as a percentage.
• 0.425 as a fraction in its simplest form.
• Adam achieved 22/30 in a test. Express his result as a percentage.
• Work out 5/9 ÷ 3/7, giving your answer as a fraction in simplest form.
Task 3 (10 marks)
Use Excel financial facilities to perform the following scenarios:
You need to show the financial formulae you have used in each case.
• Calculate the total amount accrued after 5 years for saving a single amount of £10,000 in a bank account that pays interest of 2% per year. Assume that interest is calculated at the beginning of
every year.
• Calculate the monthly repayments for a 5 year loan of £25,000, with 5% interest per year. Repayments are made at the beginning of each month.
• Calculate the total amount accrued after 10 years for regular monthly cash investments of £300 paid into a savings account at the beginning of each month and pays 3.20% interest per year.
• Calculate the monthly repayments (payable at the end of each month) for a period of 25 years, for a mortgage of £300,000 at a fixed interest rate of 2.8% per year.
• What is the single amount you need to save today so that you will have £30,000 in 5 years, if the saving rate is 3%? Assume that interest is applied to the account at the end of every year.
Task 4 (10 marks)
Use an Excel worksheet to setup one table to compare the UK Government spending for the two years. Calculate the amounts for each department (in £bn) for the following scenario:
Total UK Government spending for the financial year 2020 was £874 bn. The sum is split between each of the departments as follows:
Pensions 18.4%
Healthcare 18.8%
Education 10.5%
Others 52.3%
Total UK Government spending for the financial year 2021 was estimated to be £908 bn. The sum is split between each of the departments as follows:
Pensions 18.9%
Healthcare 19.2%
Education 10.8%
Others 51.1%
By comparing the UK Government spending (£bn) in healthcare, work out the percentage increase for 2021 (show formula)
Task 5 (10 marks)
You are given the following employees’ information after their first six months at work.
Enter the data as shown on an Excel spreadsheet and complete the following table:
• Write down a function that will return the number of employees who have taken more than 9 days as holiday.
• Write down a function that will return the number of employees who have completed less than 3 training days.
• Write down a logical function with an “IF” statement to test whether an employee has taken more than 9 days as holiday and completed less than 3 training days. You should use “Yes” if the
employee has taken more than 9 days as holiday and completed less than 3 training days.
Note: The functions you show in the table for questions a. to c. should be written using relative references to your spreadsheet rows and columns.
“You can order BA3020QA: Using and Managing Data and Information Assignment 1 from our service and can get a completely high-quality custom paper. Our service offers any of the BA3020QA Academic
paper written from scratch, according to all customers’ requirements, expectations and highest standards.” | {"url":"https://www.assignmentbee.com/ba3020qa-using-and-managing-data-and-information-assignment-1/","timestamp":"2024-11-14T15:18:11Z","content_type":"text/html","content_length":"145290","record_id":"<urn:uuid:b12bb5a2-bc65-4bdf-8ef9-248094cbeddf>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00055.warc.gz"} |
How could i help UOR ?
Im a experienced developer, and i do work as a senior dev. My main language is C#. I have experience on RUNUO, and already had experience making some stuff for a friendly server.
So i was thinking... why i am not helping UOR somehow ? I got some time to spare and im pretty sure i could improve some 'non affecting gameplay' stuff at start.
How could it be possible for me, to help UOR somehow ?
medas, Andrakus, Rextacy and 3 others like this.
Awesome! Maybe send Telamon a pm in irc or forum PM here @Chris . He's pretty busy so you may or may not get a quick response but he does read them.
con con con con Active Member
Dec 20, 2015
Likes Received:
make factions great again!
Ruck and Rextacy like this.
A better yellow range in NPC hues.
Ive sent Chris a PM, waiting for an answer. If he allows me to help im not changing big stuff at least from the start, gonna start slow, aligned to his ideas.
Rextacy and Imbol like this.
Nice! I'm sure Chris will get back to you soon.
He's just busy moats of the time. ^.^
Rextacy likes this.
Be excellent to one another.
Party on dude!
Hadrian likes this.
Nothing from him yet.
Zyler Well-Known Member
Aug 25, 2012
Likes Received:
Given past actions of some, there is limited access to assist with development you are suggesting
There are other ways to help UOR
Donating, running events, prompting server, etc
Raajaton and Hadrian like this.
Hadrian Well-Known Member
UO:R Donor
Aug 24, 2017
Likes Received:
Assist in developing the community.
As much as i wanted, thats not my thing. I work as a mmo server engineer, so im pretty sure my real contribution would come in form of code. But all good.
I wouldnt even need access to the real server. I can run locally if that be a preferred development strategy and make Merge Requests. In fact, i wouldnt even recommend sharing server access to
anyone except devops/sysadm
Asus, One and PaddyOBrien like this.
Asus Active Member
Jul 16, 2018
Likes Received:
Code would be best.
Erza Scarlet Well-Known Member
UO:R Donor
May 24, 2015
Likes Received:
RunUO shards are coded in C#
If you have some programming knowledge then i guess you can PM the boss and ask for permission to work on minor scripts / improvements that he has on his list.
Having an idea about how RunUO works will be beneficial though, and there are free builds availabe (server runs on v.2.2)
Another realy good thing is to have an idea what you would like to improve gameplay wise, this comes mostly by playing the shard though
Oni Neaufire and Hadrian like this.
Steady Mobbin Well-Known Member
Jan 30, 2016
Likes Received:
I suggest publishing a downloadable file with select macros and/or razor settings that a noob could simply drop in over the existing files would go a long way to helping the new player get rid of
some of the monotony and prep work just to be semi functional. Im not talking pots or anything but maybe the auto treechop east macro and eval int, hiding, simple stuff like that. Maybe one
restock agent so the player at least has at least one working template to work off of. How much you want to include is up to you.
I think there are a crapload more guides than show up in the skills tree on the website. Maybe an updated "One guide to rule them all" thread.
With that said, I'm not sure how much else can be done. its a 20+ y/o game. This is hands down the best community and shard Ive ever been a part of. There are always event. The staff is around
even though they get peppered with requests daily. The playerbase has its scoundrels but is overwhelmingly friendly and we all seem to want whats best for the shard who remain.
Its an old game. You guys are awesome and we just need to keep collecting awesome people along the way.
Billy Hargrove, Hydrox, Pharoah and 5 others like this.
JohnM Well-Known Member
UO:R Donor
Mar 27, 2015
Likes Received:
anything from Telamon yet?
PaddyOBrien likes this.
sOoN tM
It was kinda shitty that the guy offered his help, for free, and was left hanging.
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Jakob Heiss
List of publications and preprints
My publications are listed here (Google Scholar)
How Infinitely Wide Neural Networks Can Benefit from Multi-task Learning – an Exact Macroscopic Characterization
J Heiss, J Teichmann, H Wutte. arxiv 2022.
We provide an exact quantitative characterization of the inductive bias of infinitely wide L2-regularized ReLU-neural networks (NNs) in function space. This provides insights into the multi-task
learning ability due to representation learning, while many other infinite width limits in the literature (such as NTK) only study settings where no benefit from multi-task learning is possible.
Reducing the number of neurons of deep ReLU networks based on the current theory of Regularization
J Heiss, A Stockinger, J Teichmann. OpenReview 2020.
Work in progress: The theory in the paper above shows that wide L2-regularized neural networks exhibit sparsity in function space. Our algorithm utilizes this sparsity in function space to compress
the neural network to a smaller number of neurons after training by applying specific transformations on the weight matrices. Further experimental evaluation is still required.
Extending Path-Dependent NJ-ODEs to Noisy Observations and a Dependent Observation Framework
W. Andersson, J. Heiss, F. Krach, J Teichmann. TMLR 2024.
The Path-Dependent Neural Jump ODE (PD-NJ-ODE) is a model to learn optimal forecasts given irregularly sampled time series of incomplete past observations. So far the process itself and the
coordinate-wise observation times were assumed to be independent and observations were assumed to be noiseless. This work discusses two extensions to lift these restrictions and provides theoretical
guarantees, as well as empirical examples for them. (Intuitive video-summary: https://www.youtube.com/watch?v=PSglx3a3bBI)
NOMU: Neural Optimization-based Model Uncertainty
J Heiss, J Weissteiner, H Wutte, S Seuken, J Teichmann. ICML 2022.
We study methods for estimating model uncertainty for neural networks (NNs) in regression. We introduce five important desiderata regarding model uncertainty that any method should satisfy. However,
we find that established benchmarks often fail to reliably capture some of these desiderata. We introduce a new approach for capturing model uncertainty for NNs, which we call NOMU.
How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function – Part I: the 1-D Case of Two Layers with Random First Layer
J Heiss, J Teichmann, H Wutte. arxiv 2019.
We consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. We rigorously show that early stopping or L2 regularization on
parameter space both correspond to regularizing the second derivative in function space (similar to smoothing splines) as the number of hidden nodes tends to infinity
How (Implicit) Regularization of ReLU Neural Networks Characterizes the Learned Function – Part II: the Multi-D Case of Two Layers with Random First Layer
J Heiss, J Teichmann, H Wutte. arxiv 2023.
We extend the results of part I to the multi-dimensional case. We show that (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained correspond in
function space to a generalized additive model (GAM)-typed regression in which infinitely many directions are considered: the infinite generalized additive model (IGAM).
Monotone-Value Neural Networks: Exploiting Preference Monotonicity in Combinatorial Assignment
J Weissteiner, J Heiss, J Siems, S Seuken. IJCAI 2022.
We outperform the previous state of the art in machine learning based combinatorial assignment problems by introducing monotone-value neural networks (MVNNs). MVNNs are capturing prior knowledge on
combinatorial valuations by enforcing monotonicity and normality, while solving the corresponding winner determination problem is still practically feasible via our MILP formulation.
Bayesian Optimization-based Combinatorial Assignment
J Weissteiner, J Heiss, J Siems, S Seuken. AAAI 2023.
We further improve the performance of machine learning based combinatorial assignment problems by combining MVNNs and NOMU. We use the uncertainty obtained by an adapted version of NOMU to promote
exploration by using the upper confidence bounds as acquisition function. (Short and simple video-summary: https://youtu.be/6YH9K6LDHPY)
Machine Learning-powered Combinatorial Clock Auction
E Soumalias, J Weissteiner, J Heiss, S Seuken. accepted to AAAI 2024.
For real-world combinatorial auctions (CAs) the combinatorial clock auction (CCA) is the most popular method in practice. While we have previously introduced ML-powered CAs that ask value queries
(i.e., "What is your value for the bundle {A,B}?"), the CCA asks demand queries (i.e., "At prices p, what is your most preferred bundle of items?"). We introduce a machine learning-powered CCA that
only asks demand queries and still outperforms the classical CCA significantly, while not changing the interaction-paradigm compared to the classical CCA. | {"url":"https://people.math.ethz.ch/~jheiss/publications","timestamp":"2024-11-11T05:03:38Z","content_type":"text/html","content_length":"12166","record_id":"<urn:uuid:569cadb5-ca94-41f2-a584-792318108cd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00518.warc.gz"} |
WAEC Syllabus for Further Mathematics 2024/2025 - Expocoded
WAEC Syllabus for Further Mathematics 2024/2025
Are you preparing for the West African Examination Council (WAEC) Further Mathematics exam? Do you want to know the aims and objectives, topics, and recommended textbooks for the exam? If yes, then
you are in the right place. In this blog post, I will share with you everything you need to know about the WAEC Syllabus for Further Mathematics 2024/2025.
What is Further Mathematics?
Further Mathematics is an advanced level of Mathematics that covers topics beyond the General Mathematics/Mathematics (Core) syllabus. It is designed for students who have a strong interest and
aptitude in Mathematics and who wish to pursue higher studies or careers in Mathematics, Engineering, Science, and other related fields.
Why Study Further Mathematics?
Studying Further Mathematics has many benefits, such as:
• It develops your conceptual and manipulative skills in Mathematics
• It bridges the gap between Elementary Mathematics and Higher Mathematics
• It exposes you to aspects of Mathematics that are relevant and applicable to various disciplines and professions
• It enhances your ability to analyze data and draw valid conclusions
• It fosters your logical, abstract, and precise reasoning skills
What are the Aims and Objectives of the WAEC Further Mathematics Syllabus?
The WAEC Further Mathematics Syllabus is a document that outlines the scope and content of the Further Mathematics exam. It also specifies the learning outcomes and assessment criteria for the exam.
The aims and objectives of the syllabus are to test candidates’ development of:
• Further conceptual and manipulative skills in Mathematics
• Understanding of an intermediate course of study that bridges the gap between Elementary Mathematics and Higher Mathematics
• Acquisition of aspects of Mathematics that can meet the needs of potential Mathematicians, Engineers, Scientists, and other professionals
• Ability to analyze data and draw valid conclusions
• Logical, abstract, and precise reasoning skills
What is the Examination Scheme of the WAEC Further Mathematics Exam?
The WAEC Further Mathematics Exam consists of two papers, Papers 1 and 2, both of which must be taken. The examination scheme is as follows:
• Paper 1: This paper consists of 40 multiple-choice objective questions, covering the entire syllabus. Candidates are required to answer all questions in 1 hour for 40 marks. The questions are
drawn from the sections of the syllabus as follows:
□ Pure Mathematics – 30 questions
□ Statistics and Probability – 4 questions
□ Vectors and Mechanics – 6 questions
• Paper 2: This paper consists of two sections, Sections A and B, to be answered in 2 hours for 100 marks. Section A consists of eight compulsory questions that are elementary in type for 48 marks.
The questions are distributed as follows:
□ Pure Mathematics – 4 questions
□ Statistics and Probability – 2 questions
□ Vectors and Mechanics – 2 questions
Section B consists of seven questions of greater length and difficulty put into three parts: Parts I, II, and III as follows:
□ Part I: Pure Mathematics – 3 questions
□ Part II: Statistics and Probability – 2 questions
□ Part III: Vectors and Mechanics – 2 questions
Candidates are required to answer four questions with at least one from each part for 52 marks.
What are the Topics in the WAEC Further Mathematics Syllabus?
The WAEC Further Mathematics Syllabus covers topics in Pure Mathematics, Statistics and Probability, Vectors and Mechanics. The topics are as follows:
Pure Mathematics
• Sets
• Surds
• Binary Operations
• Logical Reasoning
• Functions
• Polynomial Functions
• Indices and Logarithmic Functions
• Exponential and Circular Functions
• Complex Numbers
• Matrices and Determinants
• Sequences and Series
• Permutations and Combinations
• Binomial Theorem
• Mathematical Induction
• Differentiation
• Integration
• Differential Equations
• Coordinate Geometry
• Trigonometry
• Inequalities
• Linear Programming
• Numerical Methods
Statistics and Probability
• Data Presentation and Analysis
• Measures of Location and Dispersion
• Probability
• Probability Distributions
• Sampling and Estimation
• Hypothesis Testing
• Correlation and Regression
• Index Numbers
• Time Series
Vectors and Mechanics
• Vectors in Two and Three Dimensions
• Scalar and Vector Products
• Kinematics of a Particle
• Dynamics of a Particle
• Statics of a Particle
• Work, Energy, and Power
• Impulse and Momentum
• Rigid Bodies
• Circular Motion
• Simple Harmonic Motion
What are the Recommended Textbooks for the WAEC Further Mathematics Exam?
The WAEC Further Mathematics Syllabus recommends the following textbooks for candidates preparing for the exam:
• New General Mathematics for Senior Secondary Schools 3 by M.F. Macrae et al. (Pearson Education, 2008)
• Further Mathematics Project Books 1 to 3 by Tuttuh-Adegun et al. (Pearson Education, 2014)
• A Textbook of West African Advanced Mathematics by O.A. Bamisaye et al. (Evans Brothers, 2006)
• Advanced Level Mathematics: Pure Mathematics 1 by Hugh Neill and Douglas Quadling (Cambridge University Press, 2002)
• Advanced Level Mathematics: Pure Mathematics 2 & 3 by Hugh Neill and Douglas Quadling (Cambridge University Press, 2002)
• Advanced Level Mathematics: Statistics 1 by Steve Dobbs and Jane Miller (Cambridge University Press, 2002)
• Advanced Level Mathematics: Statistics 2 by Steve Dobbs and Jane Miller (Cambridge University Press, 2002)
• Advanced Level Mathematics: Mechanics 1 by Douglas Quadling (Cambridge University Press, 2002)
• Advanced Level Mathematics: Mechanics 2 by Douglas Quadling (Cambridge University Press, 2002)
In this blog post, I have given you an overview of the WAEC Syllabus for Further Mathematics 2024/2025. I hope you have learned something useful and interesting from it. If you have any questions or
comments, please feel free to leave them below. I wish you all the best in your WAEC Further Mathematics exam. ?
• Further Mathematics is an advanced level of Mathematics that covers topics beyond the General Mathematics/Mathematics (Core) syllabus.
• The WAEC Further Mathematics Syllabus aims to test candidates’ development of further conceptual and manipulative skills in Mathematics, understanding of an intermediate course of study that
bridges the gap between Elementary Mathematics and Higher Mathematics, acquisition of aspects of Mathematics that can meet the needs of potential Mathematicians, Engineers, Scientists, and other
professionals, ability to analyze data and draw valid conclusions, and logical, abstract, and precise reasoning skills.
• The WAEC Further Mathematics Exam consists of two papers, Papers 1 and 2, both of which must be taken. Paper 1 consists of 40 multiple-choice objective questions, covering the entire syllabus.
Paper 2 consists of two sections, Sections A and B, to be answered in 2 hours. Section A consists of eight compulsory questions that are elementary in type. Section B consists of seven questions
of greater length and difficulty put into three parts: Parts I, II, and III.
• The WAEC Further Mathematics Syllabus covers topics in Pure Mathematics, Statistics and Probability, Vectors and Mechanics.
• The WAEC Further Mathematics Syllabus recommends the following textbooks for candidates preparing for the exam: New General Mathematics for Senior Secondary Schools 3 by M.F. Macrae et al.,
Further Mathematics Project Books 1 to 3 by Tuttuh-Adegun et al., A Textbook of West African Advanced Mathematics by O.A. Bamisaye et al., Advanced Level Mathematics: Pure Mathematics 1 by Hugh
Neill and Douglas Quadling, Advanced Level Mathematics: Pure Mathematics 2 & 3 by Hugh Neill and Douglas Quadling, Advanced Level Mathematics: Statistics 1 by Steve Dobbs and Jane Miller,
Advanced Level Mathematics: Statistics 2 by Steve Dobbs and Jane Miller, Advanced Level Mathematics: Mechanics 1 by Douglas Quadling, and Advanced Level Mathematics: Mechanics 2 by Douglas
Frequently Asked Questions and Answers
• Q: What is the difference between General Mathematics and Further Mathematics?
• A: General Mathematics is the basic level of Mathematics that covers topics such as Number and Numeration, Algebra, Geometry and Mensuration, Trigonometry, Calculus, and Statistics. Further
Mathematics is the advanced level of Mathematics that covers topics such as Sets, Surds, Binary Operations, Logical Reasoning, Functions, Polynomial Functions, Indices and Logarithmic Functions,
Exponential and Circular Functions, Complex Numbers, Matrices and Determinants, Sequences and Series, Permutations and Combinations, Binomial Theorem, Mathematical Induction, Differentiation,
Integration, Differential Equations, Coordinate Geometry, Trigonometry, Inequalities, Linear Programming, Numerical Methods, Data Presentation and Analysis, Measures of Location and Dispersion,
Probability, Probability Distributions, Sampling and Estimation, Hypothesis Testing, Correlation and Regression, Index Numbers, Time Series, Vectors in Two and Three Dimensions, Scalar and Vector
Products, Kinematics of a Particle, Dynamics of a Particle, Statics of a Particle, Work, Energy, and Power, Impulse and Momentum, Rigid Bodies, Circula | {"url":"https://expocoded.com/waec-syllabus-for-further-mathematics/","timestamp":"2024-11-06T18:33:16Z","content_type":"text/html","content_length":"151157","record_id":"<urn:uuid:e04c101b-9a2e-4676-a8f9-c8aab5343051>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00588.warc.gz"} |
How Hard Are Calculus I and II?
If you’re not a math whiz, you are probably dreading college math, and one of the classes your degree might require is Calculus I. Well, if you want that diploma, you’re going to have to grit your
teeth and get it over with. But is calculus really all that bad?
With the proper education, commitment, and study skills, calculus can actually be fairly simple. However, if a student’s prior math education was lacking, or if a student tends to be lax in their
attendance and their homework completion deadlines, calculus will be difficult.
If you’re still sure calculus is going to be the reason you’ll be pulling all-nighters and chugging energy drinks, keep reading. It might be easier to pass than you think, although any higher-level
math class will take some work.
Is Calculus Hard?
Calculus has a bad reputation. You’re not the only one who quakes in their boots at the thought of sitting in class and hearing words like derivative, logarithm, and limits. It will be easy to find
students who are intimidated by this subject, but you will also be able to find students who have been able to successfully work their way through the class.
It combines a lot of principles you’ve been learning for years, which can be good or bad.
It’s the Next Step in the Math Journey
So yes, there is a lot of merging of concepts, which can be confusing. But luckily, you’ve been preparing for this moment.
“[Calculus is] what your mathematics education has been building to for quite some time.”
Anonymous Student
Everything you’ve ever learned in a math class up until this point has prepared you to be successful in Calc 1. Think of it this way: your whole life you’ve been finding puzzle pieces and fitting
them together. Calculus gives you the chance to see the final image the puzzle makes.
Calculus Is Life
Some students who take calculus might not understand how calculus applies to their life or even their chosen degree. It can certainly be a fast-paced, complex class (especially at first glance).
Most students discovered that when they took calculus, they were actually able to apply what they were learning to real situations. Calculus can be defined as “the math behind physics.”
If you are interested in any field that uses precise math, calculus will become your lifeline. Physics is a common field of study that wouldn’t be possible without a knowledge of calculus.
Physics helps us understand how the world around us works. It gives meaning to life by explaining the everyday phenomenons. And how are these phenomenons explained? Through calculus. Thus, calculus=
life. Knowing what calculus is used for can make it easier to understand and apply the things you’re learning.
What Is Calculus?
But what specifically does calculus help us understand about the world around us? We live in a world of motion. Living organisms, wind, cars, trains, planes, and everything around us is constantly
moving. Without calculus, we’d only be able to mathematically understand subjects that don’t move.
With the invention of calculus, a new world of possibilities opened up. Now, we can not only study how objects move, but we can study change itself.
Because of this, calculus can be applied in a variety of fields, including:
• Economics
• Statistics
• Computer Science
• Engineering
• Medicine
• Geography
You wouldn’t be able to read this right now without calculus. The computer or phone you’re using was created with calculus. The world runs on change, and since we’ve been able to study that change,
we have made huge advancements in technology, medicine, architecture, and so many other fields.
How Can I Be Good at Calculus?
Knowing what calculus is for can at least help you understand why you’re sitting in class, but it might not necessarily make the experience easier for you. Success in calculus takes work and
dedication, but it is possible.
Here are some ideas:
Solidify Previous Skills
If you’ve had poor teachers in the past, or if you didn’t put in the necessary work when you were taking high school math classes, calculus is going to be extra tricky. Because calculus is a
summation of everything you’ve learned so far, it’s pretty hard if you don’t have a solid background in Algebra and Trigonometry.
If you want to succeed in college calculus, you’ll need to brush up on your math skills in general. If you’re buckling down for calculus next semester, take time now to reteach yourself Algebra and
These skills are going to be crucial in learning and using calculus, so you’ll want to be familiar with them.
Form Good Study Habits
Anyone can pass any class. You may not get a perfect A, but you can pass. It all comes down to study habits. Are you the student that procrastinates homework assignments until the day before they’re
Do you cram right before a test? Do you turn your assignments in late?
If that sounds familiar, you’ll need to change your habits if you want to pass calculus.
One good habit to get into is doing your homework the day it’s assigned, even if your teacher doesn’t grade it right away. These assignments are great practice, and they will do the most to solidify
the concepts in your mind.
Another great way to do this is to stay ahead of your teacher. If they are going over chapter six on Wednesday, read through the chapter on Monday. Then Wednesday’s lecture will be like a review for
Your university should have math labs/tutors. These provide great opportunities to practice and check your understanding. If you want to feel confident going into a test, go to these labs several
days before your exam. This will give your mind time to digest the information and internalize it.
Use Your Resources
You’re not in this alone. Join your classmates to create study groups. This form of study is fantastic because you are all learning the information at the same pace. You can help each other
understand the material.
Other resources include online study guides and calculators. The internet has a multitude of resources to help you practice and hone your skills.
Why Is Calculus So Hard
Calculus is often considered to be one of the most difficult math classes that students take in high school or college.
There are a number of reasons why calculus can be challenging, including:
1. Calculus requires a strong foundation in algebra and trigonometry. Calculus builds on concepts introduced in earlier math classes, so students who do not have a strong foundation in algebra find
calculus to be difficult.
2. Calculus introduces new concepts. Calculus introduces new concepts, such as limits, derivatives, and integrals. These new concepts can be difficult for students to grasp.
3. Calculus requires a lot of practice. Students need to practice a lot in order to be able to apply the concepts to real-world problems.
4. Calculus can be abstract. Calculus is an abstract subject, and students who are not comfortable with abstract thinking find it difficult to understand.
But don’t be discouraged if you struggling with Math.
Above all, be patient with yourself. Calculus is difficult. You shouldn’t expect yourself to understand everything on the first try, even if you follow the tips in this article. Calculus isn’t easy,
but it isn’t impossible either.
Read Next: Geometry vs. Algebra: Which One Is Harder and Why
Disclaimer: The views and opinions expressed in this article are those of the authors and do not necessarily represent those of the College Reality Check. | {"url":"https://collegerealitycheck.com/how-hard-is-calculus/","timestamp":"2024-11-05T02:30:34Z","content_type":"text/html","content_length":"233789","record_id":"<urn:uuid:2a6cbdd8-70e9-4063-b7d6-5718898e14f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00246.warc.gz"} |
Multiplication Box Method Worksheets
Math, specifically multiplication, creates the cornerstone of various scholastic techniques and real-world applications. Yet, for several students, mastering multiplication can posture a difficulty.
To resolve this difficulty, instructors and moms and dads have actually accepted a powerful tool: Multiplication Box Method Worksheets.
Intro to Multiplication Box Method Worksheets
Multiplication Box Method Worksheets
Multiplication Box Method Worksheets -
STEP 1 Draw a two by two grid like shown in the above worksheet Draw two lines vertically and two horizontally STEP 2 Now you need to expand the given numbers and write the numbers vertically and
horizontally outside the boxes Like for example let me show to by solving the first problem of worksheet 2
Step 1 Create a grid with two rows and two columns Step 2 Multiply the digits in each cell of the grid Step 3 Add up the values in each column Step 4 The final product is 1 904 Box method
multiplication worksheets provide students with exercises that involve various numbers to multiply
Importance of Multiplication Practice Understanding multiplication is crucial, laying a solid structure for innovative mathematical concepts. Multiplication Box Method Worksheets supply structured
and targeted practice, cultivating a much deeper comprehension of this fundamental arithmetic procedure.
Evolution of Multiplication Box Method Worksheets
Box Method Multiplication Worksheet Db excel
Box Method Multiplication Worksheet Db excel
Step 1 Draw a box Start by drawing a large rectangle or box on your paper Then divide this box into four smaller boxes by drawing one vertical line and one horizontal line Step 2 Label the boxes
Write the first number 23 on the top of the box splitting the digits above the vertical lines
Our worksheets employ the box or window method of partial product multiplication Each side of the box contains the factors broken down into their components tens hundreds Fill the boxes by
multiplying the numbers intersecting in each box Add the products to obtain the final answer
From traditional pen-and-paper workouts to digitized interactive layouts, Multiplication Box Method Worksheets have advanced, dealing with varied knowing designs and preferences.
Kinds Of Multiplication Box Method Worksheets
Standard Multiplication Sheets Simple exercises concentrating on multiplication tables, assisting students construct a solid math base.
Word Problem Worksheets
Real-life circumstances integrated into troubles, boosting crucial thinking and application skills.
Timed Multiplication Drills Examinations made to enhance rate and accuracy, aiding in fast mental math.
Benefits of Using Multiplication Box Method Worksheets
Box method multiplication worksheets PDF Partial Product multiplication worksheets
Box method multiplication worksheets PDF Partial Product multiplication worksheets
This grid or box method multiplication PDF is quick and easy to download and print off The box or grid method is a fantastic written method for teaching children multiplication skills alongside
partitioning allowing children to clearly see the place value of different numbers and how each digit is multiplied Using the box or grid method to multiply 2 digit numbers by 1 digit numbers
Multiplication Method Area or Box Method For Multiplication Flashcard Products to 144 e g 12 x 11 2 digit x 1 digit e g 15 x 5 20 x 8 Worksheet 2 x 1 digit e g 54 x 5 requires only 2x 5x or 10x
multiplication facts 3 x 1 digit e g 512 x 7 requires only 2x 5x or 10x multiplication facts
Improved Mathematical Skills
Regular technique develops multiplication proficiency, boosting overall mathematics capacities.
Improved Problem-Solving Talents
Word problems in worksheets develop logical thinking and strategy application.
Self-Paced Knowing Advantages
Worksheets accommodate individual knowing rates, cultivating a comfortable and adaptable learning environment.
Just How to Produce Engaging Multiplication Box Method Worksheets
Including Visuals and Colors Vibrant visuals and shades record interest, making worksheets aesthetically appealing and involving.
Including Real-Life Circumstances
Relating multiplication to day-to-day scenarios includes importance and usefulness to workouts.
Tailoring Worksheets to Various Ability Levels Customizing worksheets based upon differing effectiveness levels guarantees comprehensive learning. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based sources supply interactive learning experiences, making multiplication engaging and pleasurable. Interactive Web Sites and Applications
On-line platforms supply varied and available multiplication technique, supplementing conventional worksheets. Tailoring Worksheets for Different Understanding Styles Visual Learners Aesthetic aids
and representations aid understanding for learners inclined toward visual understanding. Auditory Learners Spoken multiplication troubles or mnemonics accommodate learners who realize ideas with
acoustic ways. Kinesthetic Students Hands-on activities and manipulatives support kinesthetic learners in understanding multiplication. Tips for Effective Implementation in Discovering Consistency in
Practice Regular technique strengthens multiplication abilities, advertising retention and fluency. Balancing Rep and Selection A mix of repeated exercises and diverse problem layouts keeps rate of
interest and understanding. Giving Useful Comments Feedback help in recognizing areas of improvement, encouraging continued development. Difficulties in Multiplication Technique and Solutions
Motivation and Engagement Difficulties Tedious drills can result in uninterest; ingenious techniques can reignite inspiration. Conquering Worry of Math Adverse assumptions around math can hinder
development; developing a favorable discovering environment is important. Effect of Multiplication Box Method Worksheets on Academic Performance Research Studies and Research Searchings For Research
suggests a positive relationship in between constant worksheet use and boosted mathematics efficiency.
Multiplication Box Method Worksheets become flexible tools, fostering mathematical effectiveness in students while suiting diverse understanding styles. From standard drills to interactive online
resources, these worksheets not just improve multiplication skills yet also advertise essential thinking and problem-solving abilities.
Area Model Multiplication 3 Digit By 1 Digit Copy Of 2 Digit By 1 Digit Multiplication Lessons
Area Model multiplication worksheets Pdf Area Model multiplication Box method multiplication
Check more of Multiplication Box Method Worksheets below
Box method multiplication 2 digit Numbers worksheets PDF Partial Product multiplication Box
Multi Digit Multiplication Partial Product Area Model The Teachers Cafe Partial Product
box method multiplication Math Game For Elementary Students Teachers And Parents mathhacks
Box method multiplication worksheets PDF Partial Product multiplication worksheets
Pin On Math 3 5
Box Method Multiplication Worksheet Db excel
Box Method Multiplication Worksheets 15 Worksheets
Step 1 Create a grid with two rows and two columns Step 2 Multiply the digits in each cell of the grid Step 3 Add up the values in each column Step 4 The final product is 1 904 Box method
multiplication worksheets provide students with exercises that involve various numbers to multiply
Area Model Multiplication Worksheets Math Worksheets 4 Kids
Multiplication using Box Method Up to 3 Digit by 1 Digit Finding the product of numbers using the box method is easy peasy With this level 1 area model multiplication worksheet children will focus on
2 digit by 1 digit and 3 digit by 1 digit multiplication Multiplication using Box Method 2 Digit by 2 Digit
Step 1 Create a grid with two rows and two columns Step 2 Multiply the digits in each cell of the grid Step 3 Add up the values in each column Step 4 The final product is 1 904 Box method
multiplication worksheets provide students with exercises that involve various numbers to multiply
Multiplication using Box Method Up to 3 Digit by 1 Digit Finding the product of numbers using the box method is easy peasy With this level 1 area model multiplication worksheet children will focus on
2 digit by 1 digit and 3 digit by 1 digit multiplication Multiplication using Box Method 2 Digit by 2 Digit
Box method multiplication worksheets PDF Partial Product multiplication worksheets
Multi Digit Multiplication Partial Product Area Model The Teachers Cafe Partial Product
Box Method Multiplication Worksheet Db excel
Box method multiplication 2 digit Numbers worksheets PDF Partial Products M Area Model
Box method multiplication worksheets PDF Partial Product multiplication worksheets
Box method multiplication worksheets PDF Partial Product multiplication worksheets
Box method multiplication worksheets PDF Partial Product multiplication worksheets
FAQs (Frequently Asked Questions).
Are Multiplication Box Method Worksheets suitable for any age groups?
Yes, worksheets can be customized to various age and ability levels, making them adaptable for various learners.
Just how usually should trainees exercise utilizing Multiplication Box Method Worksheets?
Regular method is crucial. Regular sessions, ideally a couple of times a week, can generate substantial improvement.
Can worksheets alone boost math abilities?
Worksheets are a valuable tool however must be supplemented with different discovering approaches for extensive ability development.
Exist online systems providing totally free Multiplication Box Method Worksheets?
Yes, several instructional sites use free access to a wide range of Multiplication Box Method Worksheets.
Exactly how can moms and dads support their youngsters's multiplication practice at home?
Motivating constant practice, giving assistance, and producing a favorable knowing atmosphere are valuable steps. | {"url":"https://crown-darts.com/en/multiplication-box-method-worksheets.html","timestamp":"2024-11-04T08:43:41Z","content_type":"text/html","content_length":"29522","record_id":"<urn:uuid:5f77c05a-cb3c-4e3e-97eb-c14f7d8e1d78>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00365.warc.gz"} |
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Difference between 1's complement Representation and 2's complement Representation
To understand the 1's complement and 2's complement, we should know about the complements.
In order to perform the logical manipulation and to simplify the subtraction operation, the digital systems generally use complements. The binary number system contains two types of complements,
i.e., 1's complement and 2's complement. Now we will describe each complement individually. After that, we will describe the difference between them.
1's Complement
The binary numbers can be easily converted into the 1's complement with the help of a simple algorithm. According to this algorithm, if we toggle or invert all bits of a binary number, the generated
binary number will become the 1's complement of that binary number. That means we have to transform 1 bit into the 0 bit and 0 bit into the 1 bit in the 1's complement. N' is used to indicate the 1's
complement of a number.
Example: Here, we will assume that the number is stored with the help of 4 bits.
There is another way to find the 1's complement of a number. We can use a formula to find it, which is described as follows:
N' = (2^n - 1) - N
N' is used to show the -N in 1's complement notation
N is used to show the positive integer
n is used to show the number of bits per word
For example:
Suppose we have 8 bit word, and N = 6. Now the 1's complement of N is described as follows:
N' = (2^8 - 1) - 6
= 249
= (11111001)2
With the help of this formula, we can convert the given number into the 1's complement.
2's Complement
The binary numbers can also be easily converted into the 2's complement with the help of a very simple algorithm. According to this algorithm, we can get the 2's complement of a binary number by
first inverting the given number. After that, we have to add 1into the LSB (least significant bit). That means we have to first perform 1's complement of a number, and then we have to add 1 into that
number to get the 2's complement. N* is used to show the 2's complement of a number.
Example: Here, we will assume that the number is stored with the help of 4 bits.
There is another way to find the 2's complement of a number. We can use a formula to find it, which is described as follows:
N* = 2^n - N
N* is used to show the -N in 1's complement notation
N is used to show the positive integer
n is used to show the number of bits per word
For example:
Suppose we have 8 bit word, and N = 6. Now the 1's complement of N is described as follows:
N* = 2^8 - 6
= 250
= (11111010)2
With the help of this formula, we can convert the given number into the 2's complement.
Difference between 1's Complement and 2's Complement
There are various differences between 1's complement and 2's complement. We are going to describe them with the help of different parameters, which are described as follows:
Parameters 1's complement representation 2's complement representation
Process of We can get the 1's complement of a given binary number by toggling or We can get the 2's complement of a given binary number by first doing 1's complement of number and then
Generation inverting all bits of that number. adding 1 into that number.
Example The 1's complement of binary number 9 (1001) is 6 (0110). The 2's complement of binary number 9(1001) will be got by doing 1's complement of that number which is 6
(0110), and then adding 1 into it, which means 7 (0111). So the 2's complement of 9 (1001) is 7 (0111).
Logic Gates The implementation of 1's complement is very simple. For every bit of input, For every bit of input, the 2's complement basically uses the BOT gate and a full adder.
Used it basically uses the NOT gate.
Number If we want to represent the sign binary number, we can use the 1's. If we have If we want to represent the sign binary number, we can also use the 2's. If we have a number 0, then it
Representation a number 0, then it will not be possible to use it in the form of ambiguous will possible to use it as an unambiguous representation of all given numbers.
K-bits If there is a k-bit register, the 1's complement will use -(2^(k-1) -1) to If there is a k-bit register, the 2's complement will use -(2^(k-1)) to store the lowest negative number
Register store the lowest negative number, and (2^(k-1) -1) to store the largest and (2^(k-1) -1) to store the largest positive number.
positive number.
There are two ways to represent the number 0 in 1's complement, i.e., +0 and There is only one way to represent the number 0 in 2's complement for both +0 and -0. Both minus 0 or
Representation -0. The plus 0 will be represented as 00000000, which is positive zero (+0) in plus 0 can be represented as 0000000 (+0) in an 8-bit register because if we add 1 to 11111111 (-1), we
of 0 an 8-bit register, and for negative zero (-0), it will be represented as will get 00000000 (+0), which is the same as positive zero. That's why the number 0 is always considered
11111111 in an 8-bit register. as a positive in the 2's complement. This is also the reason we generally use 2's complement.
Sign Extension In the 1's complement, sign extension is used to convert the given sign into The working of sign extension in 2's complement and in 1's complement is the same. Here it also converts
another sign for any signed integer. a given sign into another sign for any signed integer.
End-Around If we are performing an arithmetic operation (addition) with the help of 1's If we are performing an arithmetic operation (addition) with the help of 2's complement, in this case, an
Carry-Bit complement, in this case, we will first perform binary addition. After that, addition of end-around-carry-bit will not occur because 2's complement contains a single value for zero.
we will add the end-around carry bit. The 2's complement ignores this type of addition.
Ease of The 1's complement always requires the addition of end-around-carry-bit. The 2's complement does not require the addition of end-around-carry-bit. That's why the 2's complement
operation That's why the 1's complement arithmetic operation is difficult as compared to arithmetic operation is easier as compared to the 1's complement arithmetic operation.
the 2's complement arithmetic operation.
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Newton’s Law of Universal Gravitation
The Force of Gravity and Gravitational Potential
The law of universal gravitation was formulated by Isaac Newton (1643−1727) and published in 1687.
Figure 1.
In accordance with this law, two point masses attract each other with a force that is directly proportional to the masses of these bodies \({m_1}\) and \({m_2},\) and inversely proportional to the
square of the distance between them:
\[F = G\frac{{{m_1}{m_2}}}{{{r^2}}}.\]
Here, \(r\) is the distance between the centers of mass of the bodies, \(G\) is the gravitational constant, whose value found by experiment is \(G = 6,67 \times {10^{ - 11}}{\frac{{{\text{m}^3}}}{{\
text{kg} \cdot {\text{s}^2}}}}.\)
The force of gravitational attraction is a central force, that is directed along a line passing through the centers of the interacting bodies.
In a system of two bodies (Figure \(2\)), the attraction force \({\mathbf{F}_{12}}\) of the second body acts on the first body of mass \({m_1}.\)
Figure 2.
Similarly, the attraction force \({\mathbf{F}_{21}}\) of the first body acts on the second body of mass \({m_2}.\) Both the forces \({\mathbf{F}_{12}}\) and \({\mathbf{F}_{21}}\) are equal and
directed along \(\mathbf{r},\) where
\[\mathbf{r} = {\mathbf{r}_2} - {\mathbf{r}_1}.\]
Using the Newton's second law we can write the following differential equations describing the motion of each body:
\[{m_1}\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r},\;\;\; {m_2}\frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = - G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r}\]
\[\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_2}}}{{{r^3}}}\mathbf{r},\;\;\; \frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = - G\frac{{{m_1}}}{{{r^3}}}\mathbf{r}.\]
It follows from the last two equations that
\[\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} - \frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = G\frac{{{m_2}}}{{{r^3}}}\mathbf{r} + G\frac{{{m_1}}}{{{r^3}}}\mathbf{r},\;\; \Rightarrow \frac{{{d^2}\mathbf{r}}}
{{d{t^2}}} = -G\frac{{{m_1} + {m_2}}}{{{r^3}}}\mathbf{r}.\]
This differential equation describes the change of the vector \(\mathbf{r}\left( t \right),\) that is, the relative motion of two bodies under the force of gravitational attraction.
With a large difference in mass of the bodies, we can neglect the smaller body mass in the right side of this equation. For example, the mass of the Sun is \(333,000\) times greater than the mass of
the Earth. In this case, the differential equation can be written in a simpler form:
\[\frac{{{d^2}\mathbf{r}}}{{d{t^2}}} = - G\frac{{{M_\text{S}}}}{{{r^3}}}\mathbf{r},\]
where \({M_\text{S}}\) is the mass of the Sun.
The gravitational interaction of bodies takes place through a gravitational field, which can be described by a scalar potential \(\varphi.\) The force acting on a body of mass \(m,\) placed in a
field with potential \(\varphi,\) is equal to
\[\mathbf{F} = m\mathbf{a} = - m\,\mathbf{\text{grad}}\,\varphi .\]
In the case of a point mass \(M,\) the potential of the gravitational field is given by
\[\varphi = - \frac{{GM}}{r}.\]
The latter formula is also valid for distributed bodies with central symmetry, such as a planet or star.
Kepler's Laws
The basic laws of planetary motion were established by Johannes Kepler \(\left(1571-1630\right)\) based on the analysis of astronomical observations of Tycho Brahe \(\left(1546-1601\right)\). In \
(1609,\) Kepler formulated the first two laws. The third law was discovered in \(1619.\) Later, in the late \(17\)th century, Isaac Newton proved mathematically that all three laws of Kepler are a
consequence of the law of universal gravitation.
Kepler's First Law
The orbit of each planet in the solar system is an ellipse, one focus of which is the Sun (Figure \(3\)).
Figure 2.
Kepler's Second Law
The radius vector connecting the Sun and the planet describes equal areas in equal intervals of time. Figure \(4\) shows the two sectors of the ellipse corresponding to the same time intervals.
Figure 2.
According to Kepler's second law, the areas of these sectors are equal.
Kepler's Third Law
The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit:
\[{T^2} \propto {a^3}.\]
The proportionality coefficient is the same for all planets in the solar system. Therefore, for any two planets, one can write the relationship
\[\frac{{T_2^2}}{{T_1^2}} = \frac{{a_2^3}}{{a_1^3}}.\]
See solved problems on Page 2. | {"url":"https://math24.net/newtons-law-universal-gravitation.html","timestamp":"2024-11-15T00:52:50Z","content_type":"text/html","content_length":"13682","record_id":"<urn:uuid:c6052d19-fa2c-413b-ae91-b70f46289014>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00895.warc.gz"} |
Square Cubit to Square Yard
Square Cubit [cubit2] Output
1 square cubit in ankanam is equal to 0.03125
1 square cubit in aana is equal to 0.0065741417092768
1 square cubit in acre is equal to 0.000051652846898583
1 square cubit in arpent is equal to 0.000061140146915976
1 square cubit in are is equal to 0.0020903184
1 square cubit in barn is equal to 2.0903184e+27
1 square cubit in bigha [assam] is equal to 0.00015625
1 square cubit in bigha [west bengal] is equal to 0.00015625
1 square cubit in bigha [uttar pradesh] is equal to 0.000083333333333333
1 square cubit in bigha [madhya pradesh] is equal to 0.0001875
1 square cubit in bigha [rajasthan] is equal to 0.000082644628099174
1 square cubit in bigha [bihar] is equal to 0.000082659808963997
1 square cubit in bigha [gujrat] is equal to 0.00012913223140496
1 square cubit in bigha [himachal pradesh] is equal to 0.00025826446280992
1 square cubit in bigha [nepal] is equal to 0.000030864197530864
1 square cubit in biswa [uttar pradesh] is equal to 0.0016666666666667
1 square cubit in bovate is equal to 0.000003483864
1 square cubit in bunder is equal to 0.000020903184
1 square cubit in caballeria is equal to 4.645152e-7
1 square cubit in caballeria [cuba] is equal to 0.0000015576143070045
1 square cubit in caballeria [spain] is equal to 5.225796e-7
1 square cubit in carreau is equal to 0.000016204018604651
1 square cubit in carucate is equal to 4.3010666666667e-7
1 square cubit in cawnie is equal to 0.0000387096
1 square cubit in cent is equal to 0.0051652846898583
1 square cubit in centiare is equal to 0.20903184
1 square cubit in circular foot is equal to 2.86
1 square cubit in circular inch is equal to 412.53
1 square cubit in cong is equal to 0.00020903184
1 square cubit in cover is equal to 0.000077476590066716
1 square cubit in cuerda is equal to 0.000053188763358779
1 square cubit in chatak is equal to 0.05
1 square cubit in decimal is equal to 0.0051652846898583
1 square cubit in dekare is equal to 0.0002090319778984
1 square cubit in dismil is equal to 0.0051652846898583
1 square cubit in dhur [tripura] is equal to 0.625
1 square cubit in dhur [nepal] is equal to 0.012345679012346
1 square cubit in dunam is equal to 0.00020903184
1 square cubit in drone is equal to 0.0000081380208333333
1 square cubit in fanega is equal to 0.000032508839813375
1 square cubit in farthingdale is equal to 0.00020655320158103
1 square cubit in feddan is equal to 0.000050148395047168
1 square cubit in ganda is equal to 0.0026041666666667
1 square cubit in gaj is equal to 0.25
1 square cubit in gajam is equal to 0.25
1 square cubit in guntha is equal to 0.0020661157024793
1 square cubit in ghumaon is equal to 0.000051652892561983
1 square cubit in ground is equal to 0.0009375
1 square cubit in hacienda is equal to 2.3329446428571e-9
1 square cubit in hectare is equal to 0.000020903184
1 square cubit in hide is equal to 4.3010666666667e-7
1 square cubit in hout is equal to 0.00014707467288154
1 square cubit in hundred is equal to 4.3010666666667e-9
1 square cubit in jerib is equal to 0.00010340073529412
1 square cubit in jutro is equal to 0.000036321779322328
1 square cubit in katha [bangladesh] is equal to 0.003125
1 square cubit in kanal is equal to 0.00041322314049587
1 square cubit in kani is equal to 0.00013020833333333
1 square cubit in kara is equal to 0.010416666666667
1 square cubit in kappland is equal to 0.0013550618436406
1 square cubit in killa is equal to 0.000051652892561983
1 square cubit in kranta is equal to 0.03125
1 square cubit in kuli is equal to 0.015625
1 square cubit in kuncham is equal to 0.00051652892561983
1 square cubit in lecha is equal to 0.015625
1 square cubit in labor is equal to 2.9159993958644e-7
1 square cubit in legua is equal to 1.1663997583457e-8
1 square cubit in manzana [argentina] is equal to 0.000020903184
1 square cubit in manzana [costa rica] is equal to 0.000029908861976603
1 square cubit in marla is equal to 0.0082644628099174
1 square cubit in morgen [germany] is equal to 0.000083612736
1 square cubit in morgen [south africa] is equal to 0.000024399654488152
1 square cubit in mu is equal to 0.00031354775843226
1 square cubit in murabba is equal to 0.0000020661138759433
1 square cubit in mutthi is equal to 0.016666666666667
1 square cubit in ngarn is equal to 0.0005225796
1 square cubit in nali is equal to 0.0010416666666667
1 square cubit in oxgang is equal to 0.000003483864
1 square cubit in paisa is equal to 0.026297335203366
1 square cubit in perche is equal to 0.0061140146915976
1 square cubit in parappu is equal to 0.00082644555037733
1 square cubit in pyong is equal to 0.063228021778584
1 square cubit in rai is equal to 0.0001306449
1 square cubit in rood is equal to 0.00020661157024793
1 square cubit in ropani is equal to 0.0004108838568298
1 square cubit in satak is equal to 0.0051652846898583
1 square cubit in section is equal to 8.0707644628099e-8
1 square cubit in sitio is equal to 1.161288e-8
1 square cubit in square is equal to 0.0225
1 square cubit in square angstrom is equal to 20903184000000000000
1 square cubit in square astronomical units is equal to 9.3403170389288e-24
1 square cubit in square attometer is equal to 2.0903184e+35
1 square cubit in square bicron is equal to 2.0903184e+23
1 square cubit in square centimeter is equal to 2090.32
1 square cubit in square chain is equal to 0.00051652680971209
1 square cubit in square decimeter is equal to 20.9
1 square cubit in square dekameter is equal to 0.0020903184
1 square cubit in square digit is equal to 576
1 square cubit in square exameter is equal to 2.0903184e-37
1 square cubit in square fathom is equal to 0.0625
1 square cubit in square femtometer is equal to 2.0903184e+29
1 square cubit in square fermi is equal to 2.0903184e+29
1 square cubit in square feet is equal to 2.25
1 square cubit in square furlong is equal to 0.0000051652846898583
1 square cubit in square gigameter is equal to 2.0903184e-19
1 square cubit in square hectometer is equal to 0.000020903184
1 square cubit in square inch is equal to 324
1 square cubit in square league is equal to 8.967480289349e-9
1 square cubit in square light year is equal to 2.3354080041663e-33
1 square cubit in square kilometer is equal to 2.0903184e-7
1 square cubit in square megameter is equal to 2.0903184e-13
1 square cubit in square meter is equal to 0.20903184
1 square cubit in square microinch is equal to 323999714181170
1 square cubit in square micrometer is equal to 209031840000
1 square cubit in square micromicron is equal to 2.0903184e+23
1 square cubit in square micron is equal to 209031840000
1 square cubit in square mil is equal to 324000000
1 square cubit in square mile is equal to 8.0707644628099e-8
1 square cubit in square millimeter is equal to 209031.84
1 square cubit in square nanometer is equal to 209031840000000000
1 square cubit in square nautical league is equal to 6.7715481249621e-9
1 square cubit in square nautical mile is equal to 6.0943879363185e-8
1 square cubit in square paris foot is equal to 1.98
1 square cubit in square parsec is equal to 2.1953877476134e-34
1 square cubit in perch is equal to 0.0082644628099174
1 square cubit in square perche is equal to 0.0040928817277424
1 square cubit in square petameter is equal to 2.0903184e-31
1 square cubit in square picometer is equal to 2.0903184e+23
1 square cubit in square pole is equal to 0.0082644628099174
1 square cubit in square rod is equal to 0.0082644309975705
1 square cubit in square terameter is equal to 2.0903184e-25
1 square cubit in square thou is equal to 324000000
1 square cubit in square yard is equal to 0.25
1 square cubit in square yoctometer is equal to 2.0903184e+47
1 square cubit in square yottameter is equal to 2.0903184e-49
1 square cubit in stang is equal to 0.000077161993355482
1 square cubit in stremma is equal to 0.00020903184
1 square cubit in sarsai is equal to 0.074380165289256
1 square cubit in tarea is equal to 0.00033242977099237
1 square cubit in tatami is equal to 0.12646369411338
1 square cubit in tonde land is equal to 0.000037895547498187
1 square cubit in tsubo is equal to 0.063231847056688
1 square cubit in township is equal to 2.2418770355288e-9
1 square cubit in tunnland is equal to 0.000042344996353618
1 square cubit in vaar is equal to 0.25
1 square cubit in virgate is equal to 0.000001741932
1 square cubit in veli is equal to 0.000026041666666667
1 square cubit in pari is equal to 0.000020661157024793
1 square cubit in sangam is equal to 0.000082644628099174
1 square cubit in kottah [bangladesh] is equal to 0.003125
1 square cubit in gunta is equal to 0.0020661157024793
1 square cubit in point is equal to 0.0051653295732795
1 square cubit in lourak is equal to 0.000041322314049587
1 square cubit in loukhai is equal to 0.00016528925619835
1 square cubit in loushal is equal to 0.00033057851239669
1 square cubit in tong is equal to 0.00066115702479339
1 square cubit in kuzhi is equal to 0.015625
1 square cubit in chadara is equal to 0.0225
1 square cubit in veesam is equal to 0.25
1 square cubit in lacham is equal to 0.00082644555037733
1 square cubit in katha [nepal] is equal to 0.00061728395061728
1 square cubit in katha [assam] is equal to 0.00078125
1 square cubit in katha [bihar] is equal to 0.0016531961792799
1 square cubit in dhur [bihar] is equal to 0.033063923585599
1 square cubit in dhurki is equal to 0.66127847171198 | {"url":"https://hextobinary.com/unit/area/from/sqcubit/to/sqyd","timestamp":"2024-11-02T16:55:26Z","content_type":"text/html","content_length":"128646","record_id":"<urn:uuid:74516320-706c-4996-9506-722e8148277d>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00110.warc.gz"} |
Challenge your children's ability to apply their knowledge of times tables to worded problems with this Year 3 Multiply by 4 Discussion Problem.
With two problem-solving questions, children work through the multi-step problems which are designed to encourage mathematical discussions and ignite children's curiosity.
Ideal for individual or small group work, this challenge could also be sent home as additional practice. With a ready-made answer sheet, marking time is saved and immediate feedback can be given. | {"url":"https://classroomsecrets.co.uk/resource/year-3-multiply-by-4-discussion-problem","timestamp":"2024-11-01T19:25:04Z","content_type":"text/html","content_length":"576704","record_id":"<urn:uuid:cf39ac02-95ba-438f-8340-43103be8b219>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00142.warc.gz"} |
The Best Algebra 2 Regents Review Guide for 2020 | Albert Resources (2024)
Passing the Algebra 2 Regents exam is an important goal for many New York students. Students who pass the Algebra 2 Regents exam have a clear understanding of the exam’s format, topics, and
In this helpful post, we’ll cover the format of the exam, the timing of the test, the most important dates, as well as provide a list of the best resources to use as you prepare to pass the exam.
Algebra 2 Regents Exam Essentials
What’s the format of Algebra 2 Regents?
The Algebra 2 Regents exam includes 24 multiple choice questions and 13 constructed response questions. The multiple choice questions have 4 answer options and students must select the one correct
answer. The constructed response questions are “open ended” meaning students are given a blank page to write out and explain their thinking.
Each question on the Algebra 2 Regents exam is worth a specific number of credits on the test.
Every multiple choice question is worth 2 credits. The constructed response questions are worth either 2, 4, or 6 credits depending on which part of the exam the question is in. There are four
different parts on the Algebra 2 Regents exam. See a preview of each part of the exam in the section below called “What do Algebra 2 Regents questions look like?“.
When you take the Algebra 2 Regents exam, you’ll use pencil, pen, and paper – the exam is not available to be taken on a computer.
What topics are covered on the Algebra 2 Regents test?
The Algebra 2 Regents exam covers dozens of different topics that can be grouped into four broad conceptual categories. Here are those four conceptual categories listed by their percent of the test:
• Algebra (35%-44% of the exam)
• Functions (30%-40% of the exam)
• (14%-21% of the exam)
• (5%-12% of the exam)
Let’s dive deeper into each one of these four conceptual categories. Below are some guiding questions about concepts you should understand and skills you should master in the Algebra category. The
comprehensive guiding questions are based on New York State math standards from the Algebra 2 Regents test guide.
Pro Tip: for a detailed study guide using the topics below, checkout Albert’s official Algebra 2 Regents Study Plan.
Category #1: Algebra
The Algebra category accounts for 35%-44% of the Algebra 2 Regents exam. Algebra is the most important conceptual category for you to master in order to pass the Algebra 2 Regents exam.
Concepts You Should Understand:
1. Can you describe a finite geometric series?
2. Do you understand the Remainder Theorem?
3. Can you identify zeros of polynomials when suitable factorizations are available?
4. Can you prove polynomial identities?
5. Can you explain each step in solving a simple equation?
6. Can you construct a viable argument to justify a solution method?
7. Can you define the terms focus and directrix in the context of a parabola?
Skills You Should Master:
1. Can you use the structure of an expression to identify ways to rewrite it?
2. Can you factor a quadratic expression to reveal the zeros of a function?
3. Can you complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines?
4. Can you derive the formula for the sum of a finite geometric series and use the formula to solve problems?
5. Can you apply the Remainder Theorem to solve problems?
6. Can you use the zeros of polynomials to construct a rough graph of the function defined by the polynomial?
7. Can you use polynomial identities to describe numerical relationships?
8. Can you rewrite simple rational expressions in different forms using inspection, long division, or a computer algebra system?
9. Can you create equations and inequalities in one variable and use them to solve problems?
10. Can you solve simple rational and radical equations in one variable?
11. Can you give examples showing how extraneous solutions may arise from simple rational and radical equations?
12. Can you solve quadratic equations in one variable (using completing the square or inspection)?
13. Can you solve systems of linear equations exactly and approximately?
14. Can you solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically?
15. Can you explain how to find the solution(s) to two graphs that intersect?
16. Can you derive the equation of a parabola given a focus and directrix?
Category #2: Functions
The Functions category accounts for 30%-40% of the Algebra 2 Regents exam. Functions is the second most important conceptual category for you to master in order to pass the Algebra 2 Regents exam.
There are many similarities between the Functions category and the Algebra category, so students often review the Algebra and Functions categories together.
Concepts You Should Understand:
1. Do you recognize that sequences are functions whose domain is a subset of integers?
2. Can you interpret key features of graphs and tables in terms of the relationship between two quantities?
3. Can you interpret the average rate of change of a function over a specified interval?
4. Can you identify the percent rate of change in functions and classify them as representing either exponential growth or decay?
5. Can you compare properties of two functions represented in different ways (such as algebraically, graphically, numerically in tables, or by verbal descriptions)?
6. Are you able to identify the effect on a transformed graph for specific k values?
7. Can you identify whether a function is odd or even from its graph?
8. Given a real-life context, can you interpret the parameters in a linear or exponential function?
9. Do you understand radian measurements of angle?
10. Can you explain the unit circle?
11. Can you prove the Pythagorean identity sin^2 { (\theta)} + cos^2 {(\theta)} = 1 and use it to solve problems?
Skills You Should Master:
1. Can you sketch graphs of a function based on verbal description of a relationship between two quantities?
2. Given a graph of a function, can you find the intercepts, intervals where the function is increasing, decreasing, positive, or negative, the relative maximums and minimums, symmetries, and end
3. Can you calculate the average rate of change of a function presented symbolically or as a table over a specified interval?
4. Can you estimate a function’s rate of change from a graph?
5. Can you graph polynomial functions, identifying zeros and showing end behavior?
6. Can you graph exponential and logarithmic functions, showing intercepts and end behavior?
7. Can you graph trigonometric functions, showing period, midline, and amplitude?
8. Can you use the properties of exponents to interpret expressions for exponential functions?
9. Given a relationship between two quantities, can you determine an explicit expression, a recursive process, or steps for calculation from a context?
10. Can you combine standard function types using arithmetic operations?
11. Can you write arithmetic and geometric sequences both recursively and with an explicit formula?
12. Can you translate between arithmetic and geometric forms of sequences?
13. Can you find the k value of a function given the original and transformed graphs?
14. Can you solve an equation of them form f(x) = c for a simple function f that has an inverse and write an expression for the inverse?
15. Can you construct linear and exponential (including sequences) functions given a graph, a description of the relationship, or two input-output pairs?
16. Can you solve exponential equations using logarithms?
17. Can you evaluate logarithms?
18. Can you select trigonometric functions to model periodic phenomena with specific amplitude, frequency, and midline?
Category #3: Statistics & Probability
The Statistics & Probability category accounts for 14%-21% of the Algebra 2 Regents exam. This category is the third most important conceptual category for you to master in order to pass the Algebra
2 Regents exam.
Concepts You Should Understand:
1. Can you describe what standard deviation means?
2. Can you recognize that some data sets are not appropriate for fitting to a normal curve?
3. Do you understand statistics as a process for making inferences based on a random sample?
4. Can you decide if a specific model is consistent with results from a given data-generation process?
5. Can you recognize the purposes of and differences among sample surveys, experiments, and observational studies?
6. Can you compare two treatments using data from a randomized experiment?
7. Can you evaluate reports based on data?
8. Can you describe events as subsets of a sample space?
9. Can you describe what is required for two events to be statistically independent?
10. Do you understand conditional probability?
11. Can you explain conditional probability and independence in everyday language?
Skills You Should Master:
1. Can you use the mean and standard deviation of a data set to fit it to a normal distribution and estimate population percentages?
2. Can you use calculators, spreadsheets, and tables to estimate areas under the normal curve?
3. Can you use functions (especially linear, quadratic, and exponential) fitted to data to solve problems in a real-world context?
4. Can you use data from a sample survey to estimate population mean or proportion?
5. Can you develop a margin of error through the use of simulations?
6. Can you use two-way frequency tables to answer statistical questions?
7. Can you find the conditional probability of A given B?
8. Can you apply the Addition Rule of statistics and interpret the answer in terms of the model?
Category #4: Number & Quantity
The Number & Quantity category accounts for 5%-12% of the Algebra 2 Regents exam. This category is the fourth most important conceptual category for you to master in order to pass the Algebra 2
Regents exam.
Students taking the Algebra 2 Regents exam are generally not asked many questions in this category, but mastering the topics in Number & Quantity will help you maximize every point possible on your
final exam score.
Concepts You Should Understand:
1. Can you explain how rational exponents work?
2. Can you define appropriate quantities for descriptive modeling?
3. Do you recognize the complex number i and can you write complex numbers if the correct form?
Skills You Should Master:
1. Can you rewrite expressions involving radicals and rational exponents using the properties of exponents?
2. Can you add, subtract, and multiple complex numbers?
3. Can you solve quadratic equations with real coefficients that have complex solutions?
The table below shows a full summary of all topics:
Source: Algebra 2 Regents Test Guide
So, what’s the bottom line?
There are a wide range of topics covered on the Algebra 2 Regents exam and the majority of these topics can be found in two conceptual categories (Algebra and Functions).
How many questions does Algebra 2 Regents have?
Just like the Algebra 1 Regents exam, the Algebra 2 Regents exam has 37 total questions split into 4 different parts. The first part of the exam is all multiple choice while the final 3 parts are all
constructed responses. The exam has 24 multiple choice questions and 13 student-constructed response questions. Each question is worth a specific number of points (called “credits”).
See below for an overview of each part on the Algebra 2 Regents exam:
Exam Section Question Type Partial Credit Possible? Number of Questions Credits per Question Total Credits
Part I Multiple Choice No 24 2 48
Part II Constructed Response (short) Yes 8 2 16
Part III Constructed Response (medium) Yes 4 4 16
Part IV Constructed Response (long) Yes 1 6 6
TOTAL – – 37 – 86
Hungry for even more info about the test? Checkout our FAQ page about the Algebra 2 Regents exam.
What do Algebra 2 Regents questions look like?
Part I: Multiple Choice
Part I of the Algebra 2 Regents exam is where all of the multiple choice questions are asked. Multiple choice questions include 4 different answer options and should take you about 2-3 minutes each
to complete. All multiple choice questions on the Algebra 2 Regents exam have exactly 1 correct answer.
You will earn full credit for a correct answer (2 credits) or no credit for incorrect answers (0 credits). There is no partial credit earned on multiple choice questions.
In total, there are 24 multiple choice questions in Part I, each worth 2 credits.
Here’s an official example of a Part I question:
Source: Regents Algebra 2 Exam, August 2019, Question #1
Part II: Constructed Response
There are 8 short constructed-response answers on Part 2 of the Algebra 2 Regents exam. This means you are provided a question prompt and an empty answer area in which to write, draw, and explain
each answer. You can earn partial credit for these questions.
For all questions in this Part II, a correct numerical answer with no work shown will receive only 1 credit. Each constructed responses question in Part II is relatively short (Part II typically does
not include multi-part prompts) and worth 2 credits each.
Here’s an official example of a Part II question:
Source: Regents Algebra 2 Exam, August 2019, Question #25
Part III: Constructed Response
Part III begins the multi-part constructed response questions for the Algebra 2 Regents exam. These 4 questions typically include multi-part prompts where you complete at least two different tasks
within the same question. You can receive partial credit for answering one task correctly and the other incorrectly.
For all questions in Part III, a correct numerical answer with no work shown will receive only 1 credit. Each question in Part III is worth a maximum of 4 credits.
Here’s an official example of a Part III question:
Source: Regents Algebra 2 Exam, August 2019, Question #33
Part IV: Constructed Response
The final part of the Algebra 2 Regents exam, Part IV, generally includes the most difficult question on the entire exam. The one question in Part IV includes at least 3 different tasks at a
relatively high level of difficulty. But, fear not! We have tips and tricksto help you get full credit on every question.
On Part IV, a correct numerical answer with no work shown will receive only 1 credit. The Part IV question is worth 6 points, the most points of any question on the exam.
Here’s an official example of a Part IV question:
Source: Regents Algebra 2 Exam, August 2019, Question #37
Quick tip: For Parts II, III, and IV of the Algebra 2 Regents exam, a zero-credit response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an
obviously incorrect procedure. This means you should always show as much correct work as possible on your constructed response questions!
How long is the Algebra 2 Regents exam?
You are given a total of three hours to complete all parts of the Algebra 2 Regents exam. There are no official time periods required for each part of the exam, so you can use the three hours however
you’d like.
Assuming you want to use the entire three hours of test time, here are some suggestions on how long to spend on each question of the exam:
Exam Section Question Type Number of Questions Minutes Per Question Total Minutes
Part I Multiple Choice 24 3 72
Part II Constructed Response (short) 8 5 40
Part III Constructed Response (medium) 4 12 48
Part IV Constructed Response (long) 1 20 20
TOTAL — 37 180
Note that the timings in the table above should only be considered suggestions. You can adjust these suggestions to whatever will work best for your needs.
Pro tip: Albert offers exclusive full-length Algebra 2 Regents practice exams to build your confidence before test day!
What can you bring to the Algebra 2 Regents test?
Wondering what to pack for the Algebra 2 Regents exam? We’ve compiled a list of what to bring — and what to leave behind — to set you up to ace the test. Pack your bag the night before the exam so
you can wake up feeling confident to take on the day.
The Essential Algebra 2 Regents Packing List:
• Sharpened #2 pencils: You’ll need pencils to bubble in your answers for the multiple choice questions in Part I of the exam and for drawings and diagrams in Parts II, III, and IV of the exam.
Mechanical or standard are fine!
• Black or blue pens: Show your work for the Constructed Response Questions (other than drawings and graphs) in blue or black pen.
• Eraser: Pack a high quality eraser to avoid smudge marks losing you points!
• Graphing calculator: Check out our graphing calculator tips and tricks to best use your graphing calculator for the exam. Read the official graphing calculator guidelines from the New York State
Education Department for information of which calculators to use.
• Extra batteries: Most graphing calculators use 4 standard AAA batteries. Bring an extra set just in case!
• Ruler: This will come in handy for showing your work and creating graphs in Parts II, III, and IV of the exam.
• Watch: Pace yourself to make sure you have enough time to tackle each part of the exam.
• Student identification: Some schools require students to confirm their identities with a student identification card or test invitation. Check with your testing facility to make sure you have the
correct documents for test day.
• Snack: Never underestimate the power of brain food!
What NOT to bring to the Regents Algebra 2 Exam:
• Cell phones and other electronic devices: Read up on the cell phone policy in the Directions for Administering Regents Examinations.
• Algebra 2 Regents review packets: You’ll be given the official Common Core High School Math Reference Sheet for the exam, so leave other cheat sheets at home.
• Scrap paper: Your Algebra 2 Regents exam packet comes with scrap paper for working out problems, so no need to bring your own.
What reference sheets are given for Algebra 2 Regents?
Great news: you have access to the Algebra 2 Regents reference sheet for the entire exam. That means you don’t need to spend time memorizing the formula for the volume of a cone or the conversion
rate between miles and kilometers.
But that doesn’t mean you don’t need to study! You need to know how to put these formulas into action on the Algebra 2 Regents exam.
What to Know Cold for the Algebra 2 Regents Exam
• Can you identify the radius and diameter of a circle or sphere?
• What’s the difference between the base and height of a triangle or parallelogram?
• How can you find the common ratio or common difference of a sequence?
• Which form should an equation be in to use the quadratic formula?
• What values make up a, b, and c in the Pythagorean Theorem?
If you can answer these questions with ease, you’re well positioned to use the Regents mathematics reference sheet and ace the exam!
If you answered “no” to one or more of these questions, you’ve got some studying to do. Fortunately, we created an Algebra 2 Regents reference sheet study guide to show you when and how to use the
formulas you’ll be given.
How many questions do you need to get right to pass the Algebra 2 Regents test?
There are multiple aspects to your Algebra 2 Regents score: your raw score, your scale score, and your performance level. It can get a little confusing, so check out our Algebra 2 Regents FAQ for
more details on exam scoring.
Your Algebra 2 Regents Raw Score
Your raw score is the number of actual credits you earned on the exam. Remember, the Algebra 2 Regents exam is made up of four distinct parts where questions are worth a different number of credits:
• Part I: 24 MCQs, each worth 2 credits
• Part II: 8 CRQs, each worth 2 credits
• Part III: 4 CRQs, each worth 4 credits
• Part IV: 1 CRQ worth 6 credits
So, there are 86 total credits on the exam. If you earned 40 of those credits, your raw score would be 40.
Your Algebra 2 Regents Scale Score
But what Algebra 2 Regents raw score do you need to pass the exam?
Here’s where it gets a little tricky. Whether or not you pass the Algebra 2 Regents actually comes from your scale score, which is based on a curve that changes each year.
You can check out released Algebra 2 Regents conversion charts for an idea of how raw scores translate into scale scores:
To pass the Algebra 2 Regents exam, you need to obtain a scale score of 65 points. From the past few administrations of the Algebra 2 Regents, that means you would have to get a raw score of around
26-28 points.
Your Algebra 2 Regents Performance Level
Now let’s get to the performance levels. The New York State Education Department takes students’ scale scores to evaluate their overall performance level:
Performance Level Scale Score Description
5 85-100 Exceeds Common Core expectations
4 78-84 Meets Common Core expectations
3 65-77 Partially meets Common Core expectations; meets NYS graduation requirements
2 55-64 Does not meet Common Core expectations or NYS graduation requirements
1 0-54 Does not demonstrate knowledge and skills needed for Level 2
Source: Performance Level Score Ranges for Regents and Regents Common Core Exams for Annual and Accountability Reporting
Passing the Algebra 2 Regents exam means that you’ve reached a Performance Level 3. If you obtain a performance level of 1 or 2, you did not pass the exam. If you get a 4 or 5, you’ve gone above and
You can read more about the performance level descriptors and how they were established on this page from EngageNY.
Algebra 2 Regents Diploma with Honors
Feeling really confident about your Algebra 2 Regents skills? Why not try to achieve the Regents Diploma with Honors!
To earn this prestigious award, you need to average at least 90% on your ELA, math, science, social studies, and pathway Regents exams. For the Algebra 2 Regents, students must earn at least 78 of
the 86 possible credits to obtain a 90%.
To see how you might do, try out our free Algebra 2 Regents score calculator.
How do you find out your Algebra 2 Regents score? When do scores release?
Where to Find Your Algebra 2 Regents Score
Your Algebra 2 Regents exam score will be released to you by your school or the school at which you took the Regents exam.
Some schools choose to display exam scores via an online student portal. For example, New York City public schools allow students to see their Regents scores in their NYC Schools Accounts.
Algebra 2 Regents scores will also appear on a student’s high school transcript.
For specifics on where to find your Regents exam score, talk to your teachers, guidance counselor, or school administration.
When You’ll Get Your Algebra 2 Regents Score
The time it takes to receive your Algebra 2 Regents exam score depends on your school as well. To help you understand why it can take time to receive your Regents score, check out the NYSED
Directions for Administering Regents Examinations.
After you turn in your test, at least three different teachers must score Parts II, III, and IV of the exam to make sure your CRQ scores are accurate. Plus, a random sample of machine-scored answer
sheets must be verified by hand as well.
Once all exams have been scored, school administration starts sharing exam scores with students and their families.
The best way to get information about the timeline for your Algebra 2 Regents exam score is to ask your teacher or administrator. Only they know the specifics of how your school handles the tests.
Can you retake the Algebra 2 Regents exam?
Not satisfied with your Algebra 2 Regents score? Worried about what happens if you fail the Regents Algebra 2 exam?
Fear not — you can always retake the Algebra 2 Regents exam.
Some schools don’t offer the Regents examination at all three opportunities in January, June, and August. Additionally, your school might have rules against letting you retake an exam that you’ve
already passed to try to boost your score.
In these cases, talk to a teacher, administrator, or guidance counselor for advice. You can register to retake the exam at another school or testing facility.
And the best news is that your highest Algebra 2 Regents exam score — not your most recent score — will be counted on your transcript. Check out our FREE 30-day Algebra 2 Regents study plan to
maximize your next Regents score!
Return to the Table of Contents
Important Dates to Remember for Algebra 2 Regents Exam (+ downloadable)
When is the Algebra 2 Regents Exam? Complete Schedule
The New York Department of Education administers Regents exams three times a year, typically in January, June, and August. Knowing this is important because this means you actually have three
opportunities to score as high as you can in a school year: once at the beginning of the year, at the midpoint, and at the end of a standard academic calendar.
The Algebra 2 Regents exam is typically an earlier exam in the Regents testing schedule. We reviewed the last five years of Regents testing going back to 2016 and found two testing patterns:
1. The Algebra 2 Regents exam most commonly falls after the Algebra 1 Regents exam is given. Students recently took the Algebra 2 Regents exam at 1:15 PM on Thursday, January 22, 2020. Your next
opportunity for the math test will be 9:15 AM on Tuesday, June 23, 2020.
2. The August test date for the Algebra 2 Regents exam is scheduled for 12:30 PM on Thursday, August 13th, 2020. Looking back the last five years, the Algebra 2 exam has consistently been set on the
first testing day of the two-day August Regents testing schedule.
Looking ahead, New York has already set the 2021 Regents testing windows as January 26-January 29th, June 16-June 25, and August 12-13.
If trends follow, that means the Algebra 2 Regents exams will likely be scheduled for:
• 1:15 PM on January 28, 2021
• 9:15 AM on June 29, 2021
• 12:30 PM on August 12th, 2021
Check back on this page though as we’ll update when we know more.
Here’s a table of the Regents Algebra 2 exam schedule:
Exam Cycle What Day is it? What Time is the Test?
2020 Cycle 1 January 22nd, 2020 1:15 PM
2020 Cycle 2 June 23rd, 2020 9:15 AM
2020 Cycle 3 August 13th, 2020 12:30 PM
2021 Cycle 1 January 28th, 2021 1:15 PM
2021 Cycle 2 June 29th, 2021 9:15 AM
2021 Cycle 3 August 12th, 2021 12:30 PM
Return to the Table of Contents
Algebra 2 Regents Review Notes and Practice
What are Popular Algebra 2 Regents Teacher Notes and Resources?
Want to build your own Algebra 2 Regents review packet? We’ve searched far and wide for the best Algebra 2 Regents review, notes, and practice. Check out our favorites:
MrKrauseMath.com: This teacher-created site links to released Algebra 2 Regents exams as well as detailed video overviews of the solutions. The site also includes general Algebra 2 practice to review
specific skills and topics.
• Use this site for: Extensive solution videos to released Algebra 2 Regents problems. Mr. Krause has videos not only for released exams, but also for his “homemade” topic review questions.
• Don’t use this site if: You’re in a time crunch. Mr. Krause’s resources and videos are extremely detailed, which means they tend to be lengthy.
Rochester City School District: This page includes a detailed Algebra 2 course overview crafted by a New York teacher. Check out the Regents-aligned notes, homework pages, review worksheets, and
videos for targeted Algebra 2 practice.
• Use this site for: Targeted practice of specific Algebra 2 topics and skills. This page breaks down the Algebra 2 course into 5 units, each with many lessons. If you already know your strengths
and weaknesses, you’re in the right place.
• Don’t use this site if: You’re looking for a broader Algebra 2 Regents review. The practice resources on this page are very specific, so if you don’t already know which skills you need additional
practice in, this isn’t your best bet.
PBS: PBS published a series of videos featuring New York State teachers going step-by-step through Algebra 2 Regents questions. The site also includes brief video recaps of Algebra 2 topics like
Right Triangle Trigonometry, Probability & Statistics, and Modeling.
• Use this site for: Narrative, step-by-step video breakdowns of Algebra 2 Regents questions and topics. These videos are great for students who need some extra support finding entry points into
• Don’t use this site if: You need hands-on practice. This site includes video explanations of topics, but lacks practice questions for students to try on their own.
Finally, if you found this resource helpful, you’ll also like our Algebra 2 Regents Study Tips or our 30-day Algebra 2 Regents study guide.
ULTIMATE NYS Algebra 2 Regents Review: This downloadable review packet from TeachersPayTeachers covers 20 of the most frequently assessed topics on the Algebra 2 Regents exam.
• Use this site for: Repeated review of high-frequency topics. If you want plenty of opportunities to try your hand at Algebra 2 Regents questions, this is a good purchase.
• Don’t use this site if: You need detailed explanations for your practice questions. This resource includes worksheets and handouts, but doesn’t walk students through solving problems.
Return to the Table of Contents
Need help preparing for your Algebra 2 Regents exam?
Albert has a number of Algebra 2 Regents practice tests for you to practice with!
Unique from other Regents prep sites, Albert not only provides access to some of the previously released Regents tests, but also includes original New York Algebra 2 Regents practice questions.
Create your free account today.
Start your Regents test prep here
Finally, if you found this resource helpful, you’ll also like our Algebra 2 Regents Study Tips or our 30-day Algebra 2 Regents study guide. | {"url":"https://revivalgames.org/article/the-best-algebra-2-regents-review-guide-for-2020-albert-resources","timestamp":"2024-11-02T21:01:18Z","content_type":"text/html","content_length":"119380","record_id":"<urn:uuid:881ff7a7-d76c-4f09-8cba-37600a35ce63>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00737.warc.gz"} |
Math Pick Up Lines - The Frequent Dater
Math Pick Up Lines
Here are over 150 of the best math pick up lines you’ll find anywhere! For the self-confessed geeks among you, use them on like-minded math gurus for best results. To infinity and beyond… I’ll get my
Math Pick Up Lines
1/3>((-1^1/5)/27U)^1/2 Simply this to know how I feel about you. i>3U
Archimedes cried out “eureka” and ran around naked and filled with joy when he discovered that the volume of a solid can be determined by how much it displaces. Spend more time with me and you will
do the same.
Are you a 30 degree angle? Because you’re acute-y.
Are you a 45 degree angle, because you’re perfect.
Are you a 90 degree angle? ‘Cause you are looking right!
Are you a math teacher? Because you got me harder than trigonometry
Are you a math teacher? Because you got me harder than calculus.
Are you a square number? Because my love for you is exponential!
Are you the square root of 2? because I feel irrational when I’m around you
At absolute zero, you would still move me.
B equals T x N.
Baby I just drew a pic of you on my ti83 but you’re sooo hot my screen melted.
Baby I wish I could live on a [integral of 1/cabin d cabin] with you.
Baby I’ll be your asymptotes so I can shape your curves…
Baby if you were a 6 I would want to be your (reflection about the x-axis + then reflection about the y-axis) –>9
Baby let me be your integral so I can be the space under your curves.
Baby you must be a modulus sign, ‘cos whenever you wrap your arms round me I always feel positive! We’ve been differentiating for too long, lets sum it up and integrate
Baby you’re like a student and I am like a math book, you solve all my problems.
Baby, I wish you were x2 and I was x3/3 so I could be the area under your curve…
Baby, let me find your nth term.
Baby, lim (u->me) ? e^x = f(u)^n.
Baby, you’re a 9.999999999…but you’d be a 10 if you were with me.
Baby, you’re like a student and I’m like a math book… you solve all my problems!
Baby, your body is like a hyperbola
Being with you is like switching to polar coordinates: complex and imaginary things now have a magnitude and direction.
Being without you is like being a metric space in which exists a cauchy sequence that does not converge
Bertrand Russell was a renowned mathematician, philosopher and advocate for sexual liberation. How about we cut math and philosophy class and focus on the rest of Russell’s life.
By looking at you I can tell you’re 36-25-36, which by the way are all perfect squares.
Can I explore your mean value?
Can I plug my solution into your equation?
Excuse me, ma’am, but can I get your seven significant digits?
Girl my love for you goes on like the number pi… forever
Guy: “Do you like math?” Girl: “No.” Guy: “Me neither…In fact, the only number I care about is yours.”
Hey baby I’m an engineer. I can mend your broken heart.
Hey baby, can I see what’s under your radical?
Hey baby, what’s your sine?
Hey baby, what’s your tanx cosx?
Hey, baby want to Squeeze my Theorem while I poly your nomial?
Hey! baby can I cal-cu-la-tor (call you later)
Hey…nice asymptote.
Hi, I hear you’re good at algebra…..Will you replace my eX without asking Y?
Honey, you’re sweeter than pi.
How about I perform a sort on your variables, and you can analyse my performance?
How about you come to my place tonight, so I can show you the growth of my natural log
How can I know so many hundreds of digits of pi and not the digits of your phone number?
Huygens’ favorite curves were cycloids, but my favorite curves are yours.
I 1-sin(theta) you.
I am equivalent to the Empty Set when you are not with me.
I can figure out the square root of any number in less than 10 seconds. What? You don’t believe me? Well, then, let’s try it with your phone number.
I do believe I am your reciprocal; we will be one when we multiply.
I don’t know if you’re in my range, but I’d sure like to take you back to my domain.
I don’t like my current girlfriend. Mind if I do a you-substitution?
I heard you like math, so what’s the sum of U+Me
I heard you’re good at algebra – Could you replace my X without asking Y?
I heard you’re sin because you’re always on top when we make tangent.
I hope you know set theory because I want to intersect and union you.
I less than three you….. (i < 3 you).
I need a little help with my Calculus, can you integrate my natural log?
I not good at algebra but you and I together make 69!!!
I think if you and I had Hex we’d be a perfect OA
I think that convex butts are ALWAYS better than concave butts..you look toned.
I think you and I should study the T and N planes in depth.
I use my rod of infinite length for more than just simplifying calculations…
I wish I was your calculus homework, because then I’d be hard and you’d be doing me on your desk.
I wish I was your derivative so I could lie tangent to your curves.
I wish I was your differential because then I’d be touching all your curves.
I wish I was your problem set, because then I’d be really hard, and you’d be doing me on the desk.
I wish I was your secant line so I could touch you in at least two places!
I wish I was your second derivative so I could investigate your concavities.
I wish I were a predicate so I could be the direct object of your affection.
I wish I were your second derivative so i could fill your concavities.
I wish you were the Pythagorean theorem so I can insert my hypotenuse into your legs. Since distance equals velocity times time, let’s let velocity or time approach infinity, because I want to go all
the way with you.
I would really like to bisect your angle.
I’d like to be your math tutor for the night; add a bed, subtract your clothes, divide your legs and multiply!
I’d like to instantiate your objects, and access their member variables
I’d like to plug my solution into your equation.
I’ll be the one over your cosx and baby, we can have secx!
I’ll take you to the limit as X approaches infinity.
I’ll take you to your limit if you show me your end behavior.
I’m good at math… add a bed, subtract our clothes, divide your legs, and multiply!
I’m like pi baby, I’m really long and I go on forever.
I’m not being obtuse, but you’re acute girl.
I’m overheating because you’re stuck in my head like an infinite loop.
I’m relativistic: the faster I go, the longer I last.
I’m sine and you’re cosine, wanna make like a tangent?
I’ve been secant you for a long time
I’ve been secant you for a long time.
If four plus four equals eight, ….then me plus you equals fate.
If I move my lips half the distance to yours… and then half again… and again… etc…. would they ever meet? No? Well in this specific case I am going to disprove your assumption.
If I was sin^2 theta and you were cos^2 theta together we would be 1.
If I went binary, you would be the 1 for me.
If i were a function you would be my asymptote – i always tend towards you.
If I were an integral, I’d fill you up.
If I were sin2x and you were cos2x , together we’d be ONE!
If I’m sine and you’re cosine, wanna make like a tangent?
If I’m the Riemann zeta function, you must be s=1.
If you don’t want to go all the way, you can still partially derive me.
If you were a graphics calculator, I’d look at your curves all day long!
If you were sin^2x and I was cos^2x, then together we’d make one.
In Euclidean geometry two parallel lines never touch … let’s go back to my place and study some non-Euclidean geometry.
Instead of being the derivative, I’d much rather be the secant so I can touch you, not only once, but twice.
Is that an asymptote in your pocket, or are you just happy to see me?
Let ‘u’ and ‘i’ be irrational integers such that a real non-monotonic relationship exists for all T = {0 … infinity}
Let me integrate our curves so that I can increase our volume
Let’s make our slopes zero (slope of zero means horizontal => bed)
Let’s take each other to the limit to see if we converge
Lets make love like pi; irrational and never ending.
Like a quantum computation, our paths are entangled.
Maybe later we can go over to my place and titrate until you reach your end-point..
Meeting you is like a switch to polar coordinates: complex and imaginary things are given a magnitude and a direction.
Meeting you is like making a switch to polar coordinates: complex and imaginary things are given a magnitude and a direction.
My ex-girlfriend is like the square root of -1,…. she’s imaginary.
My friends told me that I should ask you out because you can’t differentiate. Do you need math help?
My love for you is a monotonically increasing unbounded function
My love for you is like a concave function’s positive first derivative, because it’s always increasing.
My love for you is like a concave up function because it is always increasing.
My love for you is like a fractal – it goes on forever.
My love for you is like pi, it’s never-ending.
My love for you is like the derivative of a concave up function because it is always increasing. we’re going to assume this concave up function resembles x^2 so that slopes is actually increasing.
My love for you is like the slope of a concave up function because it’s always increasing.
My love for you is like y=2^x… exponentially growing.
My love is like an exponential curve – it’s unbounded
My vector has a really large magnitude. Would you care to normalize it?
On a scale of 1-10, you’re a solid e to the power of pi.
Our love is like dividing by zero…. you cannot define it.
Since distance equals velocity times time, let’s let velocity and time approach infinity, because I want to go all the way with you.
The derivative of my love for you is 0, because my love for you is constant.
The law of contrapositives says that we should use a condom.
The sine^(-1) of you must be pi/2 cause you’re the one.
The surface of my cylinder is not a compact metric space.
The volume of a generalized cylinder has been known for thousands of years, but you won’t know the volume of mine until tonight.
The way the light reflects off the angles of your head is extremely enchanting.
Wanna expand my polynomial?
What do math and my dick have in common?…They’re both hard for you
What’s your sine?
Whoops, I think my binomials just expanded.
Why can’t love be a one to one function? Then our relationship could be injective.
Why don’t we measure the coefficient of static friction between me and you?
Why don’t we use some Fourier analysis on our relationship and reduce to a series of simple periodic functions.
Why don’t you be the numerator and I be the denominator and both of us reduce to simplest form?
Would you like to see the exponential growth of my natural log?
Yo baby, you want to see me solve a quadratic?
Yo girl, I heard you’re good at math… Cause your legs are always divided.
You + Me = The number of sides in a Mobius Strip.
You and I add up better than a riemann sum
You and I must have the same natural frequency, because we resonate together.
You and I would add up better than a Riemann sum.
You are one well-defined function.
You are the solution to my homogeneous system of linear equations.
You fascinate me more than the Fundamental Theorem of Calculus.
You have nicer legs than an Isosceles right triangle.
You must be an asymptote, because I just find myself getting closer and closer to you.
You must be sin squared, because I’m cosine squared and together we equal one.
You must be the square root of -1 because you can’t be real.
You must be the square root of two because I feel irrational around you.
You’re as sweet at 3.14.
You’ve got more curves than a triple integral.
Your beauty cannot be spanned by a finite basis of vectors.
Your beauty defies real and complex analysis.
Your body has the nicest arc length I’ve ever seen.
Your hotness is the only reason we can’t reach absolute zero.
Your name is Leslie? Look, I can spell your name on my calculator!
There you have it, not an infinite number of math pick up lines but enough to keep you going! If you have any others that can are worthy of a place in this list then please feel free to get in touch. | {"url":"https://thefrequentdater.com/math-pick-up-lines/","timestamp":"2024-11-12T11:37:39Z","content_type":"text/html","content_length":"44984","record_id":"<urn:uuid:d49b2a60-b4a8-43db-8159-aff7386c2d27>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00604.warc.gz"} |
Lower bounds for randomized read/write stream algorithms for STOC 2007
STOC 2007
Conference paper
Lower bounds for randomized read/write stream algorithms
View publication
Motivated by the capabilities of modern storage architectures, we consider the following generalization of the data stream model where the algorithm has sequential access to multiple streams. Unlike
the data stream model, where the stream is read only, in this new model (introduced in [8, 9]) the algorithms can also write onto streams. There is no limit on the size of the streams but the number
of passes made on the streams is restricted. On the other hand, the amount of internal memory used by the algorithm is scarce, similar to data stream model. We resolve the main open problem in [7] of
proving lower bounds in this model for algorithms that are allowed to have 2-sided error. Previously, such lower bounds were shown only for deterministic and 1-sided error randomized algorithms [9,
7]. We consider the classical set disjointness problem that has proved to be invaluable for deriving lower bounds for many other problems involving data streams and other randomized models of
computation. For this problem, we show a near-linear lower bound on the size of the internal memory used by a randomized algorithm with 2-sided error that is allowed to have o(log N/ log logN) passes
over the streams. This bound is almost optimal since there is a simple algorithm that can solve this problem using logarithmic memory if the number of passes over the streams is allowed to be O
(logN). Applications include near-linear lower bounds on the internal memory for well-known problems in the literature: (1) approximately counting the number of distinct elements in the input (F0);
(2) approximating the frequency of the mode of an input sequence (F*8 ); (3) computing the join of two relations; and (4) deciding if some node of an XML document matches an XQuery (or XPath) query.
Our techniques involve a novel direct-sum type of argument that yields lower bounds for many other problems. Our results asymptotically improve previously known bounds for any problem even in
deterministic and 1-sided error models of computation. Copyright 2007 ACM. | {"url":"https://research.ibm.com/publications/lower-bounds-for-randomized-readwrite-stream-algorithms","timestamp":"2024-11-15T00:42:32Z","content_type":"text/html","content_length":"73320","record_id":"<urn:uuid:76db9b0c-2a10-40ee-9ae3-f93f8da8ccc0>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00718.warc.gz"} |
(advanced) Local decorrelation for time
(advanced) Local decorrelation for time series
The decorrelation of the measures in repeated-measure designs is meant to have error bars that are integrating the added power of using repeated-measures over independent groups. In design with a few
measurements, the correlation between the pairs of measurements is indicative of the gain in statistical power. However, in time series, correlation is likely to vanish as measurements get further
spaced in time (the lag effect).
For example, consider a longitudinal study of adolescents over 10 years. The measurements that are 6-month apart may show some correlations, but the two most separated measurements (say the first at
8 years old and the second at 18 years old) are much less likely to preserve their correlations.
This vignette propose a solution. It is detailed in Cousineau, Proulx, Potvin-Pilon, & Fiset (in preparation).
The structure of correlations
When repeated measures are obtained, one may compute the correlation matrix. The correlation matrix is always composed of 1s along the main diagonal, as the correlation of a variable with itself is
always 1. What is more interesting is what happen off the main diagonal.
In some situations, the correlations are fairly constant (stationary). When the variance are further homogeneous, this correlation structure is known as compound symmetry. Compound symmetry is the
simplest situation and also the easiest to analyze (with e.g., ANOVAs, alghough ANOVA really requires sphericity, a slightly different correlation structure).
In other situations, we might see that correlations near the main diagnonal are strong, but as we distance from the diagonal (either in the upper-right or lower-left directions), the correlations
slowly vanishes, possibly reaching near-null values. This structure is known as an autoregressive covariance structure of the first order or AR(1). In time series, that would indicate that the
correlation of a measurement with the measurement just before or just after is high, but that the correlation between a measurement and a distant measurement is weak.
Implications for precision
Vanishing correlations means that comparing distant points in time will be performed with weaker statistical precision and comparisons of close-by measures will benefit from much correlation (
correlation is your friend when it comes to statistical inference).
In plotting curves, our objective may be to see how the points evolves, which imply that we are making multiple comparisons of close-by points. If so, our visual tools should be based on the
correlation (presumably high) between these nearby points. If our objective is instead to compare far-distance points, the visual tools should incorporate the correlations of these distant points
(presumably weak).
How is correlation assessed then?
There are a few techniques to estimate the correlation in a correlation matrix. When it is assumed compound symmetric, the average of the pairwise correlations is satisfactory. When it is AR(1)
however, the average won’t do as the correlation is varying based on the lag.
We argue that a fit technique is to average the correlations using weights that are reducing with distance (excluding the main diagonal whose weight is set to 0). Any kernel (for example a gaussian
kernel) can be used to that end, as long as the width is kept smaller than the number of variables. We implemtented this technique in superb.
Illustration with fMRI data
Waskom, Frank, & Wagner (2017) examined the finite impulse response obtained from an fMRI for two sites (frontal and parietal) and two event conditions (a cue-only condition and a cue+stimulus
condition). The responses are obtained over 19 time points (labeled 0 to 18) in these four conditions, resulting in 76 measurements. There are 14 participants.
We first fetch the data from the main author’s GitHub repository:
fmri <- read.csv(url("https://raw.githubusercontent.com/mwaskom/seaborn-data/de49440879bea4d563ccefe671fd7584cba08983/fmri.csv"))
As the data are in no specific order, we first sort them by subject, type of event, and region as well as by time points and next, we convert the data into a wide data frame of dimensions 14 lines
per 77 columns:
# sort the data...
fmri <- fmri[order(fmri$subject, fmri$event, fmri$region, fmri$timepoint),]
#... then convert to wide
fmriWide <- superbToWide(fmri, id="subject",
WSFactors = c("event","region","timepoint"),
variable = "signal")
We are ready to make plots!
A plot without decorrelation
The first plot is done without adjustments. By default, it shows the standalone 95% confidence interval.
superbPlot( fmriWide,
WSFactors = c("timepoint(19)","region(2)","event(2)"),
variables = names(fmriWide)[2:77],
plotStyle = "lineBand",
pointParams = list(size=1,color="black"),
lineParams = list(color="purple")
) + scale_x_discrete(name="Time", labels = 0:18) +
scale_discrete_manual(aesthetic =c("fill","colour"),
labels = c("frontal","parietal"),
values = c("red","green")) +
The scale_x_discrete is done to rename the ticks from 0 to 18 (they would start at 1 otherwise). The scale_discrete_manual changes the color of the band (I hope you are color-blind, colors is not my
thing). The plotStyle = "lineBand" displays the confidence intervals as a band rather than as error bars.
Plots with decorrelation
The decorrelation technique was first proposed by Loftus & Masson (1994). Alternatives approaches were developped in Cousineau (2005) with Morey (2008; also see Cousineau, 2019). They are known in
superbPlot() as "LM" and "CM" respectively.
If you add this adjustment with this command, you get the following plot:
superbPlot( fmriWide,
WSFactors = c("timepoint(19)","region(2)","event(2)"),
variables = names(fmriWide)[2:77],
adjustments = list(decorrelation = "CM"), ## only new line
plotStyle = "lineBand",
pointParams = list(size=1,color="black"),
lineParams = list(color="purple")
) + scale_x_discrete(name="Time", labels = 0:19) +
scale_discrete_manual(aesthetic =c("fill","colour"),
labels = c("frontal","parietal"),
values = c("red","green"))+
theme_bw() +
showSignificance(c(6,7)+1, 0.3, -0.02, "n.s.?", panel=list(event=2))
## superb::FYI: The HyunhFeldtEpsilon measure of sphericity per group are 0.052
## superb::FYI: All the groups' data are compound symmetric. Consider using CA or UA.
As you may see, this plot and the previous one are nearly identical! This is because the average correlation involving close-by and far-distant points is very weak (close to zero; replace CM with CA
and a message will return the average correlation in addition to a plot).
Because fMRI points are separated by time, close-by points ought to show some correlation. This is where local decorrelation may be useful.
We repeat the above command, but this time ask for a local average of the correlation. We need to specify the radius of the kernel, which we do by adding an integer after the letters “LD”. Here, we
show the results with a narrow kernel, weighting far more adjacent points than points 3 time points appart, obtained with "LD2":
superbPlot( fmriWide,
WSFactors = c("timepoint(19)","region(2)","event(2)"),
variables = names(fmriWide)[2:77],
adjustments = list(decorrelation = "LD2"), ## CM replaced with LD2
plotStyle = "lineBand",
pointParams = list(size=1,color="black"),
lineParams = list(color="purple")
) + scale_x_discrete(name="Time", labels = 0:19) +
scale_discrete_manual(aesthetic =c("fill","colour"),
labels = c("frontal","parietal"),
values = c("red","green"))+
theme_bw() +
showSignificance(c(6,7)+1, 0.3, -0.02, "**!", panel=list(event=2))
## superb::FYI: The average correlation per group is 0.5158
As seen from the message, the correlations in nearby time points is about .50. It explains why the precision of the measures shrank so much (seen with confidence intervals that are much narrower).
You can pick any two nearby points and run a paired t-test, the chances are high that you get a significant result.
As an example, consider the green curve, in condition cue+stimuli (i.e., bottom panel), for time points 6 and 7. The confidence band suggest that these two points differ when you examine the
locally-decorrelated confidence intervals, but not when you examine the previous two plots. Which is true? Let’s run a t-test on paired sample.
## Paired t-test
## data: fmriWide$`signal_stim_parietal_ 6` and fmriWide$`signal_stim_parietal_ 7`
## t = 3.8818, df = 13, p-value = 0.00189
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
## 0.02729823 0.09581713
## sample estimates:
## mean difference
## 0.06155768
The radius parameter
You can vary the radius from 1 and above. The larger the radius, the smallest will be the benefit of correlation in the assessment of precision. In the extreme, if you use a very large radius (e.g.,
“LD10000”), you will get the exact same average correlation as with “CA” as now all the correlations are weighted almost identically.
Note that in the above computations, I reduced the number of messages displayed by superb using options("superb.feedback" = "warnings" ).
Difference adjustments
In all three figures, we did not use the difference adjustment. Recall that this adjustment is needed when the objective of the error bars (or error bands) is to perform comparisons between pairs of
In the present example, the reader is very likely to perform comparisons between curves so that the difference adjustment is very much needed. Simply add purpose = "difference" in the adjustments
list of the three examples above. You will see that of the three plots above, only the locally-decorrelated one suggests significant differences between the bottom curves on some time points, which
is indeed what format tests indicate.
In summary
Local decorrelation is a tool adapted to time series where nearby measurements are expected to show greater correlations than measurements separated by large amount of time. This is applicable among
other to time series, longitudinal studies, fMRI studies (as the example above) and EEG studies (as the application described in Cousineau et al. (in preparation)).
Cousineau, D. (2005). Confidence intervals in within-subject designs: A simpler solution to
oftus and
asson’s method.
Tutorials in Quantitative Methods for Psychology
, 42–45.
Cousineau, D. (2019). Correlation-adjusted standard errors and confidence intervals for within-subject designs: A simple multiplicative approach.
The Quantitative Methods for Psychology
, 226–241.
Cousineau, D., Proulx, A., Potvin-Pilon, A., & Fiset, D. (in preparation). Local decorrelation for error bars in time series. Tbd.
Loftus, G. R., & Masson, M. E. J. (1994). Using confidence intervals in within-subject designs.
Psychonomic Bulletin & Review
, 476–490.
Morey, R. D. (2008). Confidence intervals from normalized data: A correction to
ousineau (2005).
Tutorials in Quantitative Methods for Psychology
, 61–64.
Waskom, M. L., Frank, M. C., & Wagner, A. D. (2017). Adaptive engagement of cognitive control in context-dependent decision making.
Cerebral Cortex
, 1270–1284. | {"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/superb/vignettes/VignetteG.html","timestamp":"2024-11-03T13:46:09Z","content_type":"text/html","content_length":"89760","record_id":"<urn:uuid:75d7932f-0f9f-4b90-a546-945406524830>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00697.warc.gz"} |
Hadamard Product Calculator
Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Jan 18, 2024
Welcome to Omni's Hadamard product calculator, where you can discover what the Hadamard product is and what properties it has — for instance, how the matrix rank behaves under this matrix operation.
We'll also explain how to find the Hadamard product of vectors, and what the link is between the Hadamard and Kronecker product of matrices.
⚠️ The Hadamard matrix product, which this tool covers, is different to the matrix product resulting from matrix multiplication. If you want to discover more matrix products, make sure to visit:
What is the Hadamard product?
The idea behind the Hadamard product is to take two matrices of the same dimensions (whether rectangular or square) and to multiply their corresponding entries — i.e., multiply the element (i,j) in
the first matrix with the element (i,j) in the second matrix. The result is a matrix with the same dimensions as the initial matrices. The operation itself is most often denoted by a small circle: A
∘ B.
🔎 As you might have guessed, the Hadamard product owes its name to the mathematician Jacques Hadamard. However, this matrix operation is also known as entrywise product or element-wise product (due
to how it is defined) as well as the Schur product, because sometimes it is attributed to the Russian-German mathematician Issai Schur.
How do I find the Hadamard product?
To calculate the Hadamard product of two matrices with the same dimensions:
1. Multiply together the elements that lie at the intersection of the first row and the first column of each matrix.
2. Write down the result in the same location in the resulting matrix.
3. Do this for each of the remaining element-pairs.
4. Have you finished the right-most element in the last row? Congrats, you've found the Hadamard product!
Now that we know what the Hadamard product is and how to calculate it by hand, let's discuss some of its most important properties.
What are the properties of Hadamard product?
The properties of the Hadamard product of matrices are:
• Commutativity (unlike the standard matrix product): A ∘ B = B ∘ A.
• The Hadamard product is associative and distributive over the addition of matrices:
A ∘ (B∘C) = (A∘B) ∘ C
A ∘ (B+C) = (A∘B) + (A∘C)
• The neutral (identity) element of the Hadamard product is a matrix whose elements are all 1. This matrix Hadamard-producted with any other matrix A will deliver A. Note that this is not the
standard identity matrix, where we have 1 on the diagonal and 0 elsewhere.
How to use this Hadamard product calculator?
This tool is very straightforward to use: just pick the matrix size and then enter the elements into their fields. The result will appear immediately at the bottom of our Hadamard product calculator.
Note that empty fields will be interpreted as zeros, which will save you some work if your matrices contain lots of zeros.
How do I compute the Hadamard product of vectors?
To find the Hadamard product of vectors you need to multiply together the corresponding elements of your two vectors. That is:
• In the case of column vectors, multiply together the elements in the first row and write down the result in the first row of the resulting vector. The same for the second row etc, proceeding
• In the case of row vectors, start from the first column and proceed to the right.
What is the matrix rank under Hadamard product?
The rank of the Hadamard product of two matrices A and B cannot exceed the product of the ranks of the input matrices. That is, the Hadamard product satisfies the condition rank(A ∘ B) ≤ rank(A) ×
Is Hadamard product the same as tensor product?
No, the Hadamard matrix product and tensor (Kronecker) product are different matrix operations. However, these two matrix products are linked by the following equation:
(A⊗B) ∘ (C⊗D) = (A∘C) ⊗ (B∘D)
• ∘ is the Hadamard matrix product;
• ⊗ is the Kronecker matrix product;
• A and C have the same dimensions; and
• B and D have the same dimensions.
Choose matrix sizes and enter the coeffients into the appropriate fields. Blanks are interpreted as zeros. | {"url":"https://www.omnicalculator.com/math/hadamard-product","timestamp":"2024-11-13T07:57:27Z","content_type":"text/html","content_length":"225635","record_id":"<urn:uuid:c22bc193-8cbc-4e03-9753-e584b3c970d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00047.warc.gz"} |
How to use mathematical induction? | StateMath ✔️
We teach you how to use mathematical induction to prove algebraic properties. This technique is very useful and simple to use. We offer examples and exercises to help you understand proofs by
Induction reasoning is often used to prove sequence properties.
Learn about how to use mathematical induction
In many mathematical situations, we need to prove a property $P(n)$ for any natural number $n$. In most cases, a direct approach to do this is very difficult. To overcome these difficulties, we
mainly use induction reasoning. In fact, we only verify that the property $ P (0),$ for $ n = 0,$ is true. Then assume that $ P (n), $ is true as well. Finally, verify that $ P (n + 1) $ is also
satisfied, that’s all.! We notice that sometimes we verify $P(1)$, mainly if the property $P(n)$ is not defined at $n=0$.
Example: let us show that for any $n\in\mathbb{N},$ \begin{align*}\tag{P(n)} (n+1)!\ge \sum_{k=1}^n k!. \end{align*}
For $n=1,$ we have $(1+1)!=2!\ge 1!,$ hence the property $P(1)$ is satisfied. Assume now, by induction, that $P(n)$ holds. As $n+2>2,$ then \begin{align*}\tag{1} (n+2)!=(n+2)(n+1)!\ge 2(n+1)!.\end
On the other hand, by adding $(n+1)!$ to the both sides of the inequality $P(n),$ we obatin \begin{align*}\tag{2}2(n+1)!\ge \sum_{k=1}^nk!+(n+1)!=\sum_{k=1}^{n+1}k!\end{align*}
By combining (1) et (2), we obtain \begin{align*} (n+2)!\ge \sum_{k=1}^{n+1}k!.\end{align*}
Thus $P(n+1)$ holds.
Exercises on induction reasoning
In the following exercises, we show you how to use mathematical induction to prove some known formulas and inequalities.
Exercise: Prove by induction that \begin{align*} A_n&=1+2+\cdots+n=\frac{n(n+1)}{2}\cr & B_n=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}.\end{align*}
Proof: We have $A_1=1=\frac{1(1+1)}{2},$ it is true. Assume, by induction, that the expression of $A_n$ as above, and determine that of $A_{n+1}$. We have \begin{align*}A_{n+1}&=1+2+\cdots+n+(n+1)=
S_n+(n+1)\cr &=\frac{n(n+1){2}}+(n+1)=\frac{n(n+1)+2(n+1)}{2}\cr &= \frac{(n+1)(n+2)}{2}=\frac{(n+1)((n+1)+1)}{2}.\end{align*} Thus the induction hypothesis is also true for $A_{n+1}$. This ends the
Similarly, we have $B_1=1=\frac{1(1+1)(2\times 1+1)}{6}$. Assume, by the induction, the $B_n$ is true and prove that $B_{n+1}$ is also true. We have \begin{align*} B_{n+1}&=B_n+ (n+1)^2=\frac{n(n+1)
(2n+1)}{6}+(n+1)^2\cr & =\frac{n(n+1)(2n+1)+6(n+1)^2}{6}\cr &=\frac{(n+1)(n(2n+1)+6(n+1))}{6}\cr &=\frac{(n+1)(2n^2+7n+6)}{6}.\end{align*} But $(n+2)(2n+3)=2n^2+7n+6$. Thus \begin{align*} B_{n+1}&=\
frac{(n+1)(n+2)(2n+3)}{6}\cr & =\frac{(n+1)((n+1)+1)(2(n+1)+1)}{6}.\end{align*} This ends the proof. | {"url":"https://statemath.com/2021/08/how-to-use-mathematical-induction.html","timestamp":"2024-11-12T02:43:02Z","content_type":"text/html","content_length":"333719","record_id":"<urn:uuid:2a27bf57-33c3-4973-9844-0a3dbe0447d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00678.warc.gz"} |
Futoshiki solver - shows the next logical step
Are you stuck on a Futoshiki puzzle? This page will show you the next logical step. Make a grid of the appropriate size, and click in the squares to enter numbers, or between them (once or twice) for
the chevrons. Then click 'Next step' and follow the messages shown in the box. More info below.
Choose size:
The software works by maintaining a list of the possible numbers that can go in each square, and gradually eliminating them as you go. Each time you click 'Next step' a chain of events take place:
Eliminate possibles based on numbers already placed. So if there's a 4 in the top left corner, remove 4s from the rest of the top row and the left column.
Eliminate possibles based on chevrons. For example, in a 5x5 puzzled, if square A > square B, then A cannot contain 1, and B cannot contain 5 (since 0 and 6 are not allowed).
Then there is a sequence of algorithms. To understand these better, load the appropriately named puzzle from the list, and click 'Next step' a few times until the algorithm is displayed.
Look for 'naked singles'. If there is only one possible, then of course that must be the number for that square.
Look for 'hidden singles'. If a possible number occurs only once per row or column, that must be the number we want.
Look for 'naked pairs'. If the same pair of possibles occurs twice in a row or column, with nothing else in those two squares, those two numbers can be deleted from other squares in that row or
column. In the example, the 'green squares' must contain 4 and 5, or 5 and 4. Either way 4 and 5 can't go anywhere else in that column.
Look for 'hidden pairs'. If the same pair of possibles occurs twice in a row or column, but nowhere else in that row or column, any other numbers can be deleted from those squares. In the example,
the two coloured squares must contain 4 and 5 or 5 and 4. The other posibles can go.
Look for 'X-wings'. If a number occurs in the same two positions in two rows, and nowhere else in those rows, that number can be deleted from the two intersecting columns (similarly for columns and
rows). In the example, looking at columns 1 and 6, you can see that the 6s must occur top left and bottom right, or top right and bottom left. Either way the other 6s in those rows can go.
There are other algorithms that I haven't yet implemented. Do contact me if you have any comments or suggestions about this site.
Select a puzzle from the list to Load or Delete, or type in a new name to Save: | {"url":"https://www.futoshiki.uk/","timestamp":"2024-11-06T08:55:47Z","content_type":"text/html","content_length":"5718","record_id":"<urn:uuid:71b5b295-b041-43ec-a367-1201ac82d74c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00500.warc.gz"} |
Seminars and Colloquia by Series
Based on a paper by E. Candes and Y. Plan.
In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsvath-Szabo- Thurston). In this homology there is a naturally associated
invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsvath-Thurston). Moreover it leads to the construction of an
infinite family of non-transversely simple knot-types (Vertesi).
We consider a random subgraph G_p of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a
giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second order average degree \tilde{d} to be
\tilde{d}=\sum_v d_v^2/(\sum_v d_v) where d_v denotes the degree of v. We prove that for any \epsilon > 0, if p > (1+ \epsilon)/\tilde{d} then almost surely the percolated subgraph G_p has a giant
component. In the other direction, if p < (1-\epsilon)/\tilde{d} then almost surely the percolated subgraph G_p contains no giant component. (Joint work with Fan Chung Graham and Paul Horn)
The aim of this talk is to introduce techniques from knot theory into the study of graphs embedded in 3-space. The main characters are hyperbolic geometry and the Jones polynomial. Both have proven
to be very successful in studying knots and conjecturally they are intimately related. We show how to extend these techniques to graphs and discuss possible applications. No prior knowledge of knot
theory or geometry will be assumed.
This study focuses on computations on large graphs (e.g., the web-graph) where the edges of the graph are presented as a stream. The objective in the streaming model is to maintain small amount of
memory and perform few passes over the data. In the streaming model, we show how to perform several graph computations including estimating the probability distribution after a random walk of certain
length l, estimate the mixing time, and the conductance. We can compute the approximate PageRank values in O(nM^{-1/4}) space and O(M^{3/4}) passes (where n is the number of nodes and M is the mixing
time of the graph). In comparison, a standard (matrix-vector multiplication) implementation of the PageRank algorithm will take O(n) space and O(M) passes. The main ingredient in all our algorithms
is to explicitly perform several random walks of certain length efficiently in the streaming model. I shall define and motivate the streaming model and the notion of PageRank, and describe our
results and techniques. Joint work with Sreenivas Gollapudi and Rina Panigrahy from Microsoft Research.
Background: We endeavor to reproduce historical observations and to identify and remedy the cause of any disparate predictions before using models to inform public policy-making. We have no finely
age- and time-stratified observations from historical pandemics, but prior exposure of older adults to a related strain is among the more compelling hypotheses for the w-shaped age-specific mortality
characterizing the 1918 pandemic, blurring the distinction between annual and pandemic influenza.
Methods: We are attempting to reproduce patterns in annual influenza morbidity and mortality via a cross-classified compartmental model whose age class sojourns approximate the longevity of clusters
of closely-related strains. In this population model, we represent effective inter-personal contacts via a generalization of Hethcote's formulation of mixing as a convex combination of contacts
within and between age groups. Information about mixing has been sought in face-to-face conversations, a surrogate for contacts by which respiratory diseases might be transmitted, but could also be
obtained from household and community transmission studies. We reanalyzed observations from several such studies to learn about age-specific preferences, proportions of contacts with others the same
age. And we obtained age-specific forces of infection from proportions reporting illness in a prospective study of household transmission during the 1957 influenza pandemic, which we gamma
distributed to correct for misclassification. Then we fit our model to weekly age-specific hospitalizations from Taiwan's National Health Insurance Program, 2000-07, by adjusting a) age-specific
coefficients of harmonic functions by which we model seasonality and b) probabilities of hospitalization given influenza.
Results: While our model accounts for only 30% of the temporal variation in hospitalizations, estimated conditional probabilities resemble official health resource utilization statistics. Moreover,
younger and older people are most likely to be hospitalized and elderly ones to die of influenza, with modeled deaths 10.6% of encoded influenza or pneumonia mortality.
Conclusions: Having satisfactorily reproduced recent patterns in influenza morbidity and mortality in Taiwan via a deterministic model, we will switch to a discrete event-time simulator and -
possibly with different initial conditions and selected parameters - evaluate the sufficiency of projected pandemic vaccine production.
We describe how several nonlinear PDEs and evolutions including stationary and dynamic Navier-Stokes equations can be formulated and resolved variationally by minimizing energy functionalsof the
form I(u) = L(u, -\Lambda u) + \langle \Lambda u, u\rangle and I(u) = \Int^T_0 [L(t, u(t), -\dot u(t) - \Lambda u(t)) + \langle\Lambda u(t), u(t)\rangle]dt + \ell (u(0) - u(T) \frac{u(T) + u(0)}{2}
where L is a time-dependent "selfdual Lagrangian" on state space, is another selfdual "boundary Lagrangian", and is a nonlinear operator (such as \Lambda u = div(u \otimes u) in the Navier-Stokes
case). However, just like the selfdual Yang-Mills equations, the equations are not obtained via Euler-Lagrange theory, but from the fact that a natural infimum is attained. In dimension 2, we recover
the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also
solutions satisfying u(0) = \alpha u(T ) for any given in (-1, 1). It is worth noting that our variational principles translate into Leray's energy identity in dimension 2 (resp., inequality in
dimension 3). Our approach is quite general and does apply to many other situations.
W. Goldman proved that the SL(2)-character variety X(F) of a closed surface F is a holonomic symplectic manifold. He also showed that the Sl(2)-characters of every 3-manifold with boundary F form an
isotropic subspace of X(F). In fact, for all 3-manifolds whose SL(2)-representations are easy to analyze, these representations form a Lagrangian space. In this talk, we are going to construct
explicit examples of 3-manifolds M bounding surfaces of arbitrary genus, whose spaces of SL(2)-characters have dimension as small as possible. We discuss relevance of this problem to quantum and
classical low-dimensional topology.
A mapping F between metric spaces is called quasisymmetric (QS) if for every triple of points it distorts their relative distances in a controlled fashion. This is a natural generalization of
conformality from the plane to metric spaces. In recent times much work has been devoted to the classification of metric spaces up to quasisymmetries. One of the main QS invariants of a space X is
the conformal dimension, i.e the infimum of the Hausdorff dimensions of all spaces QS isomorphic to X. This invariant is hard to find and there are many classical fractals such as the standard
Sierpinski carpet for which conformal dimension is not known. Tyson proved that if a metric space has sufficiently many curves then there is a lower bound for the conformal dimension. We will show
that if there are sufficiently many thick Cantor sets in the space then there is a lower bound as well. "Sufficiently many" here is in terms of a modulus of a system of measures due to Fuglede, which
is a generalization of the classical conformal modulus of Ahlfors and Beurling. As an application we obtain a new lower bound for the conformal dimension of self affine McMullen carpets.
We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The
exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to
show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the
reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants. | {"url":"https://math.gatech.edu/seminars-and-colloquia-by-series?page=585","timestamp":"2024-11-06T19:01:19Z","content_type":"text/html","content_length":"57799","record_id":"<urn:uuid:5f648fe4-0405-4011-b859-2047bdd69019>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00348.warc.gz"} |
g Conversion
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like angle finds its use in
a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the
conversion of different units of measurement like ' to g through multiplicative conversion factors. When you are converting angle, you need a Minute to Gon converter that is elaborate and still easy
to use. Converting ' to Gon is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert Minute to g, this tool is the answer that
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Knapsack Constraints Research Articles - R Discovery
We investigate the problem of k-submodular maximization under a knapsack constraint over the ground set of size n. This problem finds many applications in various fields, such as multi-topic
propagation, multi-sensor placement, cooperative games, etc. However, existing algorithms for the studied problem face challenges in practice as the size of instances increases in practical
applications.This paper introduces three deterministic and approximation algorithms for the problem that significantly improve both the approximation ratio and query complexity of existing practical
algorithms. Our first algorithm, FA, returns an approximation ratio of 1/10 within O(nk) query complexity. The second one, IFA, improves the approximation ratio to 1/4−ϵ in O(nk/ϵ) queries. The last
one IFA+ upgrades the approximation ratio to 1/3−ϵ in O(nklog(1/ϵ)/ϵ) query complexity, where ϵ is an accuracy parameter. Our algorithms are the first ones that provide constant approximation ratios
within only O(nk) query complexity, and the novel idea to achieve results lies in two components. Firstly, we divide the ground set into two appropriate subsets to find the near-optimal solution over
these ones with O(nk) queries. Secondly, we devise algorithmic frameworks that combine the solution of the first algorithm and the greedy threshold method to improve solution quality. In addition to
the theoretical analysis, we have evaluated our proposed ones with several experiments in some instances: Influence Maximization, Information Coverage Maximization, and Sensor Placement for the
problem. The results confirm that our algorithms ensure theoretical quality as the cutting-edge techniques, including streaming and non-streaming algorithms, and also significantly reduce the number
of queries. | {"url":"https://discovery.researcher.life/topic/knapsack-constraint/17439564?page=1&topic_name=Knapsack%20Constraint","timestamp":"2024-11-09T23:06:01Z","content_type":"text/html","content_length":"406451","record_id":"<urn:uuid:72cf61a8-380f-4117-ac01-3bc1da803377>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00597.warc.gz"} |
Equation - 1911 Encyclopedia Britannica -
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Bible Encyclopedias
1911 Encyclopedia Britannica
(from Lat. aequatio, aequare, to equalize), an expression or statement of the equality of two quantities. Mathematical equivalence is denoted by the sign =, a symbol invented by Robert Recorde
(1510-1558), who considered that nothing could be more equal than two equal and parallel straight lines. An equation states an equality existing between two classes of quantities, distinguished as
known and unknown; these correspond to the data of a problem and the thing sought. It is the purpose of the mathematician to state the unknowns separately in - terms of the knowns; this is called
solving the equation, and the values of the unknowns so obtained are called the roots or solutions. The unknowns are usu a lly denoted by the terminal letters,, .. x, y, z, of the alphabet, and the
knowns are either actual numbers or are represented by the literals a, b, c, &c...., i.e. the introductory letters of the alphabet. Any number or literal which expresses what multiple of term occurs
in an equation is called the coefficient of that term; and the term which does not contain an unknown is called the absolute term. The degree of an equation is equal to the greatest index of an
unknown in the equation, or to the greatest sum of the indices of products of unknowns. If each term has the sum of its indices the same, the equation is said to be homogeneous. These definitions are
exemplified in the equations: a (i) ax 2 +2bx+c=o, 3W -g1;.. (2) xy 2 +4a 2 x =80, (3) axe+2hxy+bye=o.. In (I) the unknown is x, and the knowns a,, c; the coefficients of x 2 and x are a and 2b; the
absolute term is c, and the degree is 2. In (2) the unknowns are x and y, and the known a; the degree is 3, i.e. the sum of the indices in the term xy 2. (3) is a homogeneous equation of the second
degree in x and y. Equations of the first degree are called simple or linear; of the second, quadratic; of the third, cubic; of the fourth, biquadratic; of the fifth, quintic, and so on. Of equations
containing only one unknown the number of roots equals the degree of the equation; thus a simple equation has one root, a quadratic two, a cubic three, and so on. If one equation be given containing
two unknowns, as for example ax-}- by = c or ax e -{-by e = c, it is seen that there are an infinite number of roots, for we can give x, say, any value and then determine the corresponding value of
y; such an equation is called indeterminate; of the examples chosen the first is a linear and the second a quadratic indeterminate equation. In general, an indeterminate equation results when the
number of unknowns exceeds by unity the number of equations. If, on the other hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown,
and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth. By substituting these values the corresponding values of the other
unknown are determined. Such equations are called simultaneous; and a simultaneous system is a series of equations equal in number to the number of unknowns. Such a system is not always soluble, for
it may happen that one equation is implied by the others; when this occurs the system is called porismatic or poristic. An identity differs from an equation inasmuch as it cannot be solved, the terms
mutually cancelling; for example, the expression x 2 - 'a' 2 = (x - a) (x+ a) is an identity, for on reduction it gives o= o. It is usual to employ the sign to express this relation.
An equation admits of description in two ways: - (i) It may be regarded purely as an algebraic expression, or (2) as a geometrical locus. In the first case there is obviously no limit to the number
of unknowns and to the degree of the equation; and, consequently, this aspect is the most general. In the second case the number of unknowns is limited to three, corresponding to the three dimensions
of space; the degree is unlimited as before. It must be noticed, however, that by the introduction of appropriate hyperspaces, i.e. of degree equal to the number of unknowns, any equation
theoretically admits of geometrical visualization, in other words, every equation may be represented by a geometrical figure and every geometrical figure by an equation. Corresponding to these two
aspects, there are two typical methods by which equations can be solved, viz. the, algebraic and geometric. The former leads to exact results, or, by methods of approximation, to results correct to
any required degree of accuracy. The latter can only yield approximate values: when theoretically exact constructions are available there is a source of error in the draughtsmanship, and when the
constructions are only approximate, the accuracy of the results is more problematical. The geometric aspect, however, is of considerable value in discussing the theory of equations.
There is little doubt that the earliest solutions of equations are given in the Rhind papyrus, a hieratic document written some 2000 years before our era. The problems solved were of an arithmetical
nature, assuming such forms as " a mass and its 7th makes 19." Calling the unknown mass x, we have given x+ x = 19, which is a simple equation. Arithmetical problems also gave origin to equations
involving two unknowns; the early Greeks were familiar with and solved simultaneous linear equations, but indeterminate equations, such, for instance, as the system given in the " cattle problem " of
Archimedes, were not seriously studied until Diophantus solved many particular problems. Quadratic equations arose in the Greek investigations in the doctrine of proportion, and although they were
presented and solved in a geometrical form, the methods employed have no relation to the generalized conception of algebraic geometry which represents a curve by an equation and vice versa. The
simplest quadratic arose in the construction of a mean proportional (x) between two lines (a, b), or in the construction of a square equal to a given rectangle; for we have the proportion a:x = x:b;
i.e. x 2 = ab. A more general equation, viz. x 2 - ax-+a 2 =0, is the algebraic equivalent of the problem to divide a line in medial section; this is solved in Euclid, ii. i i. It is possible that
Diophantus was in possession of an algebraic solution of quadratics; he recognized, however, only one root, the interpretation of both being first effected by the Hindu Bhaskara. A simple cubic
equation was presented in the problem of finding two mean proportionals, x, y, between two lines, one double the other. We have a:x=x:y=y:2a, which gives x 2 = ay and xy = 2a 2; eliminating y we
obtain x 3 = 2a 3 , a simple cubic. The Greeks could not solve this equation, which also arose in the problems of duplicating a cube and trisecting an angle, by the ruler and compasses, but only by
mechanical curves such as the cissoid, conchoid and quadratrix. Such solutions were much improved by the Arabs, who also solved both cubics and biquadratics by means of intersecting conics; at the
same time, they developed methods, originated by Diophantus and improved by the Hindus, for finding approximate roots of numerical equations by algebraic processes. The algebraic solution of the
general cubic and biquadratic was effected in the 16th century by S. Ferro, N. Tartaglia, H. Cardan and L. Ferrari (see Algebra: History). Many fruitless attempts were made to solve algebraically the
quintic equation until P. Ruffini and N. H. Abel proved the problem to be impossible; a solution involving elliptic functions has been given by C. Hermite and L. Kronecker, while F. Klein has given
another solution.
In the geometric treatment of equations the Greeks and Arabs based their constructions upon certain empirically deduced properties of the curves and figures employed. Knowing various metrical
relations, generally expressed as proportions, it was found possible to solve particular equations, but a general method was wanting. This lacuna was not filled until the 17th century, when Descartes
discovered the general theory which explained the nature of such solutions, in particular those wherein conics were employed, and, in addition, established the most important facts that every
equation represents a geometrical locus, and conversely. To represent equations containing two unknowns, x, y, he chose two axes of reference mutually perpendicular, and measured x along the
horizontal axis and y along the vertical. Then by the methods described in the article Geometry: Analytical, he showed that - (i) a linear equation represents a straight line, and (2) a quadratic
represents a conic. If the equation be homogeneous or break up into factors, it represents a number of straight lines in the first case, and the loci corresponding to the factors in the second. The
solution of simultaneous equations is easily seen to be the values of x, y corresponding to the intersections of the loci. It follows that there is onl y one value of x, y which satisfies two linear
equations, since two lines intersect in one point only; two values which satisfy a linear and quadratic, since a line intersects a conic in two points; and four values which satisfy two quadratics,
since two conics intersect in four points. It may happen that the curves do not actually intersect in the theoretical maximum number of points; the principle of continuity (see Geometrical
Continuity) shows us that in such cases some of the roots are imaginary. To represent equations involving three unknowns x, y, z, a third axis is introduced, the z-axis, perpendicular to the plane xy
and passing through the intersection of'the lines x, y. In this notation a linear equation represents a plane, and two linear simultaneous equations represent a line, i.e. the intersection of two
planes; a quadratic equation represents a surface of the second degree. In order to graphically consider equations containing only one unknown, it is convenient to equate the terms to y; if the
equation be f(x) = o, we take y = f (x) and construct this curve on rectangular Cartesian co-ordinates by determining the values of y which correspond to chosen values of x, and describing a curve
through the points so obtained. The intersections of the curve with the axis of x gives the real roots of the equation; imaginary roots are obviously not represented.
In this article we shall treat of: (i) Simultaneous equations, (2) indeterminate equations, (3) cubic equations, (4) biquadratic equations, (5) theory of equations. Simple, linear simultaneous and
quadratic equations are treated in the article Algebra; for differential equations see Differential Equations.
I. Simultaneous Equations. Simultaneous equations which involve the second and higher powers of the unknown may be impossible of solution. No general rules can be given, and the solution of any
particular problem will largely depend upon the student's ingenuity. Here we shall only give a few typical examples.
1. Equations which may be reduced to linear equations. Ex. To solve x(x - a) = yz, y(y - b) =zx, z(z - c) =xy. Multiply the equations by y, z and x respectively, and divide the sum by xyz; then a b c
zmxmy=o Multiply by z, x and y, and divide the sum by xyz; then a b c y ? 2 +x=o (2). From (I) and (2) by cross multiplication we obtain y (b 21 ac) = z(c 2 I 0 = bc) = A (supp er) (3).
Substituting for x, y and z in x(x - a) = yz we obtain 3abc - (a3+b3+c3) A = (a 2 - bc)(b 2 - ac)(c 2 - ab) ' and therefore x, y and z are known from (3). The same artifice solves the equations x 2 -
yz =a, 31 2 - xz = b, z2 - xy-- c. 2. Equations which are homogeneous and of the same degree. - These equations can be solved by substituting y=mx. We proceed to explain the method by an example.
Ex. To solve 3 x2 + xy + y2= 1 5, 3 1xy -3 x2 -5 y2= 45. Substituting y =mx in both these equations, and then dividing, we obtain 3 1m -3-5 m2= 3(3+ m + m2) or8m 2 -28m+12= o. The roots of this
quadratic are m=1 or 3, and therefore 2y =x, or y Taking 2y=x and substituting in 3x 2 -1--xy-f-y 2 =o, we obtain y 2 (12+2+I)=15; .'. y 2 = 1, which gives y= =I, x= =2. Taking the second value, y =
3x, and substituting for y, we obtain x2 (3+3+9) =1 5; .'. x 2 =i, which gives x==, 3. Therefore the solutions are x==2, y= 'i and x= y==3. Other artifices have to be adopted to solve other forms of
simultaneous equations, for which the reader is referred to J. J. Milne, Companion to Weekly Problem Papers. II. Indeterminate Equations. i. When the number of unknown quantities exceeds the number
of equations, the equations will admit of innumerable solutions, and are therefore said to be indeterminate. Thus if it be required to find two numbers such that their sum be 10, we have two unknown
quantities x and y, and only one equation, viz. x+y = 10, which may evidently be satisfied by innumerable different values of x and y, if fractional solutions be admitted. It is, however, usual, in
such questions as this, to restrict values of the numbers sought to positive integers, and therefore, in this case, we can have only these nine solutions, X= I, 2, 3, 4, 5, 6, 7, 8, 9; Y = 9, 8, 7,
6, 5, 4, 3, 2, I; which indeed may be reduced to five; for the first four become the same as the last four, by simply changing x into y, and the contrary. This branch of analysis was extensively
studied by Diophantus, and is sometimes termed the Diophantine Analysis.
2. Indeterminate problems are of different orders, according to the dimensions of the equation which is obtained after all the unknown quantities but two have been eliminated by means of the given
equations. Those of the first order lead always to equations of the form =by= =c, where a, b, c denote given whole numbers, and x, y two numbers to be found, so that both may be integers. That this
condition may be fulfilled, it is necessary that the coefficients a, b have no common divisor which is not also a divisor of c; for if a=md and b = me, then ax-{-by=mdx-f-mey=c, and I-ey=c/m; but d,
e, x, y are supposed to be whole numbers, therefore c/m is a whole number; hence m must be a divisor of c. Of the four forms expressed by the equation ax = by = =c, it is obvious that ax+by= - c can
have no positive integral solutions. Also - by= - c is equivalent to by - ax=c, and so we have only to consider the forms ax t by = c. Before proceeding to the general solution of these equations we
will give a numerical example.
To solve 2X +3y =25 in positive integers. From the given equation.
we have x= (25-3Y)/2=12-y- (y- I)/2. Now, since x must be a whole number, it follows that (y-I)/2 must be a whole number. Let us assume (y-I)/2 =z, then y=I +2z; and II -3z, where z might be any
whole number whatever, if there were no limitation as to the signs of x and y. But since these quantities are required to be positive, it is evident, from the value of y, that z must be either o or
positive, and from the value of x, that it must be less than 4; hence z may have these four values, o, I, 2, 3.
y = I, y=3, y=5, y=7. 3. We shall now give the solution of the equation ax - by = c in positive integers.
Convert alb into a continued fraction, and let pig be the convergent immediately preceding alb, then aq - by = t I (see Continued Fraction).
(a) If aq - by the given equation may be written ax -by =c(aq-bp); .'. cq)=b(y-cp). Since a and b are prime to one another, then x-cq must be divisible by b and y-cp by a; hence ( x - cq)I b= (y-cq)
la=t. That is, x = bt+cq and y = at+cp. Positive integral solutions, unlimited in number, are obtained by giving t any positive integral value, and any negative integral value, so long as it is
numerically less than the smaller of the quantities cglb, cpla; t may also be zero.
(0) If aq-bp= -I, we obtain x=bt-cq, y=at-cp, from which positive integral solutions, again unlimited in number, are obtained by giving t any positive integral value which exceeds the greater of the
two quantities cglb, cp/a. If a or b is unity, alb cannot be converted into a continued fraction with unit numerators, and the above method fails. In this case the solutions can be derived directly,
for if b is unity, the equation may be written y = ax-c, and solutions are obtained by giving x positive integral values greater than cla. 4. To solve ax+by=c in positive integers. Converting alb
into a continued fraction and proceeding as before, we obtain, in the case of aq-bp x=cq-bt, y=at-cp. Positive integral solutions are obtained by giving t positive integral values not less than cpla
and not greater than cq/b. In this case the number of solutions is limited. If aq - by = -1 we obtain the general solution x = bt - cq, y = cp - at, which is of the same form as in the preceding
case. For the determination of the number of solutions the reader is referred to H. S. Hall and S. R. Knight's Higher Algebra, G. Chrystal's Algebra, and other text-books.
5. If an equation were proposed involving three unknown quantities, as ax+by+cz = d, by transposition we have ax+by = d - cz, and, putting dcz = c', ax+by = c'. From this last equation we may find
values of x and y of this form, x=mr+nc', y=mr+n'c', or mr+n(d-cz), y=m'r+n'(d-cz); where z and r may be taken at pleasure, except in so far as the values of x, y, z may be required to be all
positive; for from such restriction the values of z and r may be confined within certain limits to be determined from the given equation. For more advanced treatment of linear indeterminate equations
see Combinatorial Analysis.
6. We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y 2 =a+bx+cx2, where x is to be found, so that y may be a rational quantity. The
possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some
peculiarity in its nature.
Case I. Let a be a square number; then, putting g 2 for a, we have y 2= g 2 + +cx 2 . Suppose,/ (g 2 +bx+cx 2) =g+mx; then g2 '= g 2 +2gmx+m 2 x 2 , or +cx 2 =2gmx+m 2 x 2 , that is, b+cx=2gm+ m 2 x;
hence x = 2cm -, b y = J (g 2 +bx+ cx 2) = cg c bmm gm2 Case 2. Let c be a square number=g 2; then, putting 11 (a+bx+ g 2 x 2) = m +gx, we find a+bx+g 2 x 2 =m 2 +2mgx +g2x2, or a+bx= m 2 +2mgx;
hence we find x - b 2 2mg, y = (a+bx+ g 2 x 2) - bmb gm' i g ag. Case 3. When neither a nor c is a square number, yet if the expression a+bx+cx 2 can be resolved into two simple factors, as f+gx and
h+kx, the irrationality may be taken away as follows: Assume -41 (a+bx+cx 2) =,/ {(f (h+kx)} =m(f+gx), then (f +gx) (h +kx) = m 2 (f +gx) 2 , or h +kx =171 2 (f +gx); hence we find 2 (f g) x = k m
gm2, y = 1 { (h+k)) (f + gx) x - k -gm2m' and in all these formulae in may be taken at pleasure.
Case 4. The expression a+bx+cx 2 may be transformed into a square as often as it can be resolved into two parts, one of which is a complete square, and the other a product of two simple factors; for
then it has this form, p 2 +gr, where p, q and r are quantities which contain no power of x higher than the first. Let us assume / (p 2 +qr) =p+mq; thus we have p 2 -}-qr = p 2 -} 2mpq-1-m 2 g 2 and
r = 2mp+m 2 q, and as this equation involves only the first power of x, we may by proper reduction obtain from it rational values of x and y, as in the three foregoing cases.
The application of the preceding general methods of resolution to any particular case is very easy; we shall therefore conclude with a single example.
Ex. It is required to find two square numbers whose sum is a given square number.
Let a 2 be the given square number, and x 2 , y 2 the numbers required; then, by the question, x 2 +y 2 = a 2, and y = (a 2 -x 2). This equation is evidently of such a form as to be resolvable by the
method employed in case I. Accordingly, by comparing 1 ! (a 2 -x 2) with the general expression (g 2 +bx+cx 2), we have g = a, b = o, c= -I, and substituting these values in the formulae, and also -
n for +m, we find Zan a(n2 - I) x= n2+I, y = n2+I If a= n 2 +I, there results y =n 2 - I, a=n 2 +I. Hence if r be an even number, the three sides of a rational right-angled triangle are r, (2r) 2 -I,
(1r) 2 + I. If r be an odd number, they become (dividing by 2) r, 2(r 2 -I), 1-(r 2 + I).
For example, if r = 4, 4, 41, 4+1, or 4, 3, 5, are the sides of a right-angled triangle; if r =7, 7, 24, 25 are the sides of a right-angled triangle.
III. Cubic Equations. I. Cubic equations, like all equations above the first degree, are divided into two classes: they are said to be pure when they contain only one power of the unknown quantity;
and adfected when they contain two or more powers of that quantity.
Pure cubic equations are therefore of the form x 3 =r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube
root of each side of the equation. Let us consider the equation x 3 -c 3 = o more fully. This is decomposable into the factors x-c=o and x 2 +cx+c 2 = o. The roots of this quadratic equation are 2 (-
I t -3)c, and we see that the equation x 3 =c 3 has three roots, namely, one real root c, and two imaginary roots 1(- I t - By making c equal to unity, we observe that 1(- I t -3) are the imaginary
cube roots of unity, which are generally denoted by co and, 2, for it is easy to show that (i(-I--1 -3))2=2(-I+ -3)2. Let us now consider such cubic equations as have all their terms, and which are
therefore of this form, x 3 +Ax 2 +Bx+C =o, where A, and C denote known quantities, either positive or negative.
This equation may be transformed into another in which the second term is wanting by the substitution x = y - A/3. This transformation is a particular case of a general theorem. Let +Ax -'+Bx n - 2
... = o. Substitute x=y+h; then (y+h)n+A(y+h)n-1... =0. Expand each term by the binomial theorem, and let us fix our attention on the coefficient of y n - 1 . By this process we obtain o = y n +y n -
(A+nh) + terms involving lower powers of y. Now h can have any value, and if we choose it so that A +nh =o, then the second term of our derived equation vanishes.
Resuming, therefore, the equation y 3 +qy+r=o, let us suppose y =v+z; we then have y 3 =2 3 +z 3 +3vz(v+z) =v 3 +z 3 +3vzy, and the original equation becomes v 3 +z 3 +(3vz+q)y+r=o. Now v and z are
any two quantities subject to the relation y=v+z, and if we suppose 3vz+q=o, they are completely determined. This leads to v 3 +z 3 +r=o and 3vz+q=o. Therefore v 3 and z 3 are the roots of the
quadratic t 2 +rt-g 2 /27 =0. Therefore v3 = / - 2 r +v( g3- - r2); z3=-2 -1 r - ('?FIg3+4r2); v =1? { - 2r+ Y ('?q3+4r2) {; z= Y {-2r - U1/4rg3+4r2) }; and y = v-}-z =J 1-2 r +J (i-74 3 +4 r2)1
(A-q3+4r2) } Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved.
3. But this is only one of three values which y may have. Let us, for the sake of brevity, put A= -2r Y (21-8 3 +4r 2), B = -V (z-q3+4r2), put a and u Q - - J _3).
Then, from what has been shown (§ I), it is evident that v and z have each these three values, v='IA, v=a?/A, v=s¦1A; z= B, z=a4B, z=134B.
To determine the corresponding values of v and z, we must consider that vz =' = Now if we observe that aß =I, it will immediately appear that v+z has these three values, v+z= -'A+ 4B, v+z = ail A+(31
/ B, v+z =s4A+a11 B, which are therefore the three values of y. If z= o, z =l, z=2, z=3; Then x =II,' x=8, x=5, x=2, The first of these formulae is commonly known by the name of Cardan's rule (see
Algebra: History). The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows: - Because vz =
- 3q, therefore z=-3gXI/v=-3Q/;IA; hence v+z=-IA-3q/CIA: thus it appears that the three values of y may also be expressed thus: y =, IA - a q / CIA nods;: :f:n. y= a A - A 4?/y tan y = 0, 3 ,1 34 /?
r; A -- a A. See below, Theory of Equations, §§ 16 et seq.
IV. Biquadratic Equations.;rya, 1. When a biquadratic equation contains all its terms, it has this form, t y rr? Irs.t r: x 4 ±Ax 3 ±Bx 2 +Cx±D =o, where A, B, C, D denote known quantities. ,.,: We
shall first consider pure biquadratics, or such as contain only the first and last terms, and therefore are of this form, 'x 4' =b 4 . In this case it is evident that x may be readily had by two
extractions of the square root; by the first we find x 2 = b?, and by the second x = b. This, however, is only one of the values which x may have; for since x 4 =b 4 , therefore x 4 - b 4 =o; but x 4
- b 4 may be resolved into two factors x 2 - b 2 and x 2 +b 2 , each of which admits of a similar resolution; for x2 - b2= (x - b)(x+b) and x 2 +b 2 =(x - b.1 - I) (x+b A l - I). Hence it appears
that the equation x 4 - b 4 =o may also be expressed thus,.
+b)(x - b-1 - I)(x+b/ - I)=o; so that x may have these four values,,:.., .
b, +b A l - I, - b1,1 two of which are real, and the others imaginary.
2. Next to pure biquadratic equations, in respect of easiness cif resolution, are such as want the second and fourth terms, and therefore have this form,. 7. x 4 +4x 2 +s =o.
These may be resolved in the manner of quadratic equations ;or if we put y = x , we have y?
y 2 +4y +s =0, from which we find y = z (- q (q2 - 4s)), and therefore x= 11 - gti J (g2-4s)1.
3. When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following
was first given by Leonhard Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra. We have already explained how an equation which is complete in its
terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form, y4 + py2 +4 y + r = o, where the second term
is wanting, and where p, q, r denote any known quantities whatever.
That we may form an equation similar to the above, let us assume y= / a+ -V b+ / c, and also suppose that the letters a, b, c denote the roots of the cubic equation z3+Pz2+Qz - R=o; then, from the
theory of equations we have a+b+c= - P, ab+ac+bc=Q, abc=R. We square the assumed formula y -Va+Alb+Alc, and obtai n I '! y 2 = a+b+c+2 ab+ I ac+ bc); or, substituting - P for a+b+c, and transposing,
y 2 +P=2(' l ab+Iac+Il bc) Let this equation be also squared, and we have y 4 +2Py 2 +P 2 =4(ab+ac+bc) +8(-! a 2 bc+ ab 2 c+ abc2); and since ab+ac+bc=Q, and A/ a 2 bc+ I ab 2 c+ abc 2 = abc(s/ a+,/
b+ 1 I c) = 'R.y, ' the same equation may be expressed thus: .4 11 y4 + 2Py2 + P2= 4Q + 8 ,/ R. y. s :A Thus we have the biquadratic equation y 4 +2Py 2 -8! R.y+P2-4Q=0, one of the roots of which is
y= a+ N I b+ - c, while a, tb, c are the roots of the cubic equation z3+Pz2+Qz - R=o.
4. In order to apply this resolution to the proposed equation y 4 +py 2 +qy+r= o, we must express the assumed coefficients P, Q, R by means of p, q, r, the coefficients of that equation. For this
purpose let us compare the equations and it immediately appears that 2P=p, - 8,l R = 4, P2-4Q=r; i s ft and from these equations we find P = 1P, Q = i's ( p2 - 4 r), R = 44 2 - n: Hence it follows
that the roots of the proposed equation ar generally expressed by the formula y=,l a+N(b+Jc; where a, b, c denote the roots of this cubic equation, 23 +822 4rz .2 -o. 21664 But to find each
particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities A l a, f b, - c may have either the sign + or - prefixed to it;
and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities 1/ a, A l b, / c must be equal to A I R or to -
71 when q is positive, their product must be a negative quantity, and this can only be effected by making either one or three of them negative; again, when q is negative, their product must be a
positive quantity; so that in this case they must either be all positive, or two of them must be negative. These considerations enable us to determine that four of the eight expressions for the root
belong to the case in which q is positive, and the other four to that in which it is negative.
5. We shall now give the result of the preceding investigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall
transform it into another, in which the coefficients are integers, by supposing z = hv. Thus the equation 1, 23-1-22+ 16 q2 ?? i;.
16 64 Ir t;, becomes, after reduction, e t. v3+2pv2+ (p2 -4r)v - q2=o; it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are 4a, 4b, 4c, so that our rule
may now be expressed thus: Let y 4 +py 2 +qy+r=o be any biquadratic equation wanting its second term. Form this cubic equation v 3 + 2pv2 + (p2 - 4r)v - q 2 = o, and find its roots, which let us
denote by a, b, c. Then the roots of the proposed biquadratic equation are, when q is negative, when q is positive, y =1(11a+-¦,1b+1,1 c), y = 2(-N/a - Nlb - Nlc), y =1-(11a--Vb-11c), y=4-(- 1/ad -
Nib - A/c), y = 2(- ?l a+?l b Jc), y = 1(11 a-1,I b+?c), y =2(- l a1,f b +1,1 c). y =2(-Va+-s/b - Nlc). See also below, Theory of Equations, § 17 et seq. (X.) V. Theory of Equations. I. In the
subject " Theory of Equations " the term equation is used to denote an equation of the form x n - p 1 x n - 1 ... p n = o, where pi, p2 ... pn are regarded as known, and x as a quantity to be
determined; for shortness the equation is written f (x) = o.
The equation may be numerical; that is, the coefficients Pi, p 2, ... pn are then numbers - understanding by number a quantity of the form a+161 (a and f3 having any positive or negative real values
whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting A / - I.
Or the equation may be algebraical; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers.
I. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.) Real Theory. 1 " 2. Postponing all consideration of imaginaries, we take in the first instance the coefficients
to be real, and attend only to the real roots (if any); that is, pl, p2,. .. pn are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that a n
- p l an-1 .., t p n = o or say, f(a) = o.
It is very useful to consider the curve y = f(x), - or, what would come to the same, the curve Ay=f(x), - but it is better to retain the first-mentioned form of equation, drawing, if need be, the
ordinate y on a reduced scale. For instance, if the given equation be x 3 - 6x 2 + i ix - 6 06 = o,' then the curve 1 The coefficients were selected so that the roots' might be nearly 3.
,1; 4 2 3 Ilk-NCO; + +qy +r=o, y4 + - 8 + P2 -4Q =o,;: y=x 3 -6x 2 +iix-6.06 is as shown in fig. i, without any reduction of scale for the ordinate.
It is clear that, in general, y is a continuous one-valued function of x, finite for every finite value of x, but becoming infinite when x is infinite; i.e., assuming throughout that the coefficient
of x" is + 1, then when x= 00, y= + co; but when x= - cc, then y= + cc or - co, according as n is even or odd; the curve cuts any line whatever, and in particular it cuts the axis (of x) in at most n
points; and the value of x, at any point of intersection with the axis, is a root of the equation f(x) = 0.
If 0, a are any two values of x(a>/3, that is, a nearer + cc), then if A0), f(a) have opposite signs, the curve cuts the axis an odd number of times, and therefore at least once, between the points x
=0, x= a; but if f(0), f(a) have the same sign, then between these points the curve cuts the axis an even number of times, or it may be not at all. That is, f(/3), f(a) having opposite signs, there
are between the limits (3, a an odd number of real roots, and therefore at least one real root; but f(0), f(a) having the same sign, there are between these limits an even number of real roots, or it
may be there is no real root. In particular, by giving to 0, a the values - co, + co (or, what is the same thing, any two values sufficiently near to these values respectively) it appears that an
equation of an odd order has always an, odd number of real roots, and therefore at least one real root; but that an equation of an even order has an even number of real roots, or it may be no real
If a be such that for x= or >a (that is, x nearer to + cc)f(x) is always +, and (3 be such that for x= or < 1 3 (that is, x nearer to - cc) f(x) is always -, then the real roots (if any) lie between
these limits x = 1 3, x = a; and it is easy to find by trial such two limits including between them all the real roots (if any).
3. Suppose that the positive value 3 is an inferior limit to the difference between two real roots of the equation; or rather (since the foregoing expression would imply the existence of real roots)
suppose that there are not two real roots such that their difference taken positively is = or <5; then, y being any value whatever, there is clearly at most one real root between the limits y and
y+5; and by what precedes there is such real root or there is not such real root, according as f(-y), j(y+8) have opposite signs or have the same sign. And by dividing in this manner the interval 1 3
to a into intervals each of which is = or <6, we should not only ascertain the number of the real roots (if any), but we should also separate the real roots, that is, find for each of them limits y,
y+5 between which there lies this one, and only this one, real root.
In particular cases it is frequently possible to ascertain the number of the real roots, and to effect their separation by trial or otherwise, without much difficulty; but the foregoing was the
general process as employed by Joseph Louis Lagrange even in the second edition (1808) of the Traite de la resolution des equations numeriques; 1 the determination of the limit å had to be effected
by means of the " equation of differences " or equation of the order Zn(n - I), the roots of which are the squares of the differences of the roots of the given equation, and the process is a cumbrous
and unsatisfactory one.
4. The great step was effected by the theorem of J. C. F. Sturm (1835)-viz. here starting from the function f(x), and its first derived function f(x), we have (by a process which is a slight
modification of that for obtaining the greatest common measure of these two functions)) to form a series of functions f(x), f'(x),f2(x). .. f, (x) '.M1 ,. of the degrees n, n- I, n-2. .. 0
respectively,-the last term. f,,,(x) being thus an absolute constant. These lead to the immediate determination of the number of real roots (if any) between any two given limits 0, a; viz. supposing
a>/3 (that is, a nearer to + cc), then substituting successively these two values in the series of functions, and attending only to the signs of the resulting values, the number of the changes of
sign lost in passing from 0 to a is the required number of real roots between the two 1 The third edition (1826) is a reproduction of that of 1808; the first edition has the date 1798, but a large
part of the contents is taken from memoirs of 1767-1768 and 1770-1771.
limits. In particular, taking 1 3, a = - oo, + cc respectively, the signs of the several functions depend merely on the signs of the terms which contain the highest powers of .x, and are seen by
inspection, and the theorem thus gives at once the whole number of real roots.
And although theoretically, in order to complete by a finite number of operations the separation of the real roots, we still need to know the value of the before-mentioned limit 5; yet in any given
case the separation may be effected by a limited number of repetitions of the process. The practical difficulty is when two or more roots are very near to each other. Suppose, for instance, that the
theorem shows that there are two roots between o and i o; by giving to x the values 1, 2, 3, ... successively, it might appear that the two roots were between 5 and 6; then again that they were
between 5.3 and 5.4, then between 5.34 and 5.35, and so on until we arrive at a separation; say it appears that between 5.346 and 5.347 there is one root, and between 5348 and 5349 the other root.
But in the case in question 5 would have a very small value, such as 002, and even supposing this value known, the direct application of the firstmentioned process would be still more laborious. 'kJ
5. Supposing the separation once effected, the determination of the single real root which lies between the two given limits may be effected to any required degree of approximation either by the
processes of W. G. Horner and Lagrange (which are in principle a carrying out of the method of Sturm's theorem), or by the process of Sir Isaac Newton, as perfected by Joseph Fourier (which requires
to be separately considered).
First as to Horner and Lagrange. We know that between the limits 0, a there lies one, and only one, real root of the equation; f(0) and f(a) have therefore opposite signs. Suppose any intermediate
value is in order to determine by Sturm's theorem whether the root lies between /3, or between 0, a, it would be quite unnecessary to calculate the signs of f(0), f'(e), f2(0) ... only the sign of f
(0) is required; for, if this has the same sign as f(8), then the root is between if the same sign as f(a), then the root is between 0, a. We want to make 0 increase from the inferior limit 13, at
which f(0) has the sign of AO), so long as f(0) retains this sign, and then to a value for which it assumes the opposite sign we have thus two nearer limits of the required root, and the process may
be repeated indefinitely.
Horner's method (1819) gives the root as a decimal, figure by figure; thus if the equation be known to have one real root between o and 10, it is in effect shown say that 5 is too small (that is, the
root is between 5 and 6); next that 5.4 is too small (that is, the root is between 5.4 and 5.5); and so on to any number of decimals. Each figure is obtained, not by the successive trial of all the
figures which precede it, but (as in the ordinary process' of the extraction of a square root, which is in fact Horner's process applied to this particular case) it is given presumptively as the
first figure of a quotient; such value may be too large, and then the next inferior integer must be tried instead of it, or it may require to be further diminished. And it is to be remarked that the
process not only gives the approximate value a of the root, but (as in the extraction of a square root) it includes the calculation of the function f(a), which should be, and approximately is, =o.
The arrangement of the calculations is very elegant, and forms an integral part of the actual method. It is to be observed that after a certain number of decimal places have been obtained, a good
many more can be found by a mere division. It is in the progress tacitly assumed that the roots have been first separated.
Lagrange's method (1767) gives the root as a continued fraction a+ b c.. , where a is a positive or negative integer (which may be =o), but b, c,.. . are positive integers. Suppose the roots have
been separated; then (by trial if need be of consecutive integer values) the limits may be made to be consecutive integer numbers say they are a, a+ 1; the value of x is therefore = a+ 1 /y, where y
is positive and greater than 1; from the given equation for x, writing therein x =a+ /y, we form an equation of the same order for y, and this equation will have one, and only one, positive root
greater than I; hence finding for it the limits b, b+1 (where b is =or> 1), we have y = b + 1 /z, where z is positive and greater than 1; and so on -that is, we thus obtain the successive
denominators b, c, d. .. of the continued fraction. The method is theoretically very elegant, but the disadvantage is that it gives the result in the form of. a continued fraction, which for the most
part must ultimately be converted into a decimal. There is one advantage in the method, that a commensurable root (that is, a root equal to a rational fraction) is found accurately, since, when such
root exists, the continued fraction terminates.
6. Newton's method (1711), as perfected by Fourier(1831), may be Ix. 23 a roughly stated as follows. If x = y be an approximate value of any root, and y-{-h the correct value, then f(y+h) =o, that
is, 2 +If'(y)?- o f „ (y) i ... =o; and then, if h be so small that the terms after the second may be neglected, f(y)+hf'(y) = o, that is, h= f(y)/f'(y), or the new approximate value is x=y f(y)/f'
(y); and so on, as often as we please. It will be observed that so far nothing has been assumed as to the separation of the roots, or even as to the existence of a real root; y has been taken as the
approximate value of a root, but no precise meaning has been attached to this expression. The question arises, What are the conditions to be satisfied by y in order that the process may by successive
repetitions actually lead to a certain real root of the equation; or that, y being an approximate value of a certain real root, the new value -y-f(7)/f (y) may be a more approximate value.
Referring to fig. I, it is easy to see that if OC represent the assumed value -y, then, drawing the ordinate CP to meet the curve in P, and the tangent PC' to meet the axis in C', we shall have OC'
as the new approximate value of the root. But observe that there is here a real root OX, and that the curve beyond X is convex to the axis; under these conditions the point C' is nearer to X than was
C; and, starting with C' instead of C, and proceeding in like manner to draw a new ordinate and tangent, and so on as often as we please, we approximate continually, and that with great rapidity, to
the true value OX. But if C had been taken on the other side of X, where the curve is concave to the axis, the new point C' might or might not be nearer to X than was the point C; and in this case
the method, if it succeeds at all, does so by accident only, i.e. it may happen that C' or some subsequent point comes to be a point C, such that CO is a proper approximate value of the root, and
then the subsequent approximations proceed in the same manner as if this value had been assumed in the first instance, all the preceding work being wasted.
FIG. I.
13 thus appears that for the proper application of the method we require more than the mere separation of the roots. In order to be able to approximate to a certain root a, = OX, we require to know
that, between OX and some value ON, the curve is always convex to the axis (analytically, between the two values, f(x) and f"(x) must have always the same sign). When this is so, the point C may be
taken anywhere on the proper side of X, and within the portion XN of the axis; and the process is then the one already explained. The approximation is in general a very rapid one. If we know for the
required root OX the two limits OM, ON such that from M to X the curve is always concave to the axis, while from X to N it is always convex to the axis, - then, taking D anywhere in the portion MX
and (as before) C in the portion XN, drawing the ordinates DQ, CP, and joining the points P, Q by a line which meets the axis in D', also constructing the point C' by means of the tangent at P as
before, we have for the required root the new limits OD', OC'; and proceeding in like manner with the points D', C', and so on as often as we please, we obtain at each step two limits approximating
more and more nearly to the required root OX. The process as to the point D', translated into analysis, is the ordinate process of interpolation. Suppose OD = 1 3, OC = a, we have approximately f (/
3+h) = f (() + h(f (a) - f (0) }, whence if the root is a+h the n h- - (a -‘3)f (a) a (a) - ,f(a)* Returning for a moment to Horner's method, it may be remarked that the correction h, to an
approximate value a, is therein found as a quotient the same or such as the quotient f(a) =f'(a) which presents itself in Newton's method. The difference is that with Horner the integer part of this
quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of
h; if too many decimals are taken, there would be a waste of work; but the error would correct itself at the next step. Of course the calculation should be conducted without any such waste of work.
Imaginary Theory. 7. It will be recollected that the expression number and the correlative epithet numerical were at the outset used in a wide sense, as extending to imaginaries. This extension
arises out of the theory of equations by a process analogous to that by which number, in its original most restricted sense of positive integer number, was extended to have the meaning of a real
positive or negative magnitude susceptible of continuous variation.
If for a moment number is understood in its most restricted sense as meaning positive integer number, the solution of a simple equation leads to an extension; ax - b = o gives x = b/a, a positive
fraction, and we can in this manner represent, not accurately, but as nearly as we please, any positive magnitude whatever; so an equation ax+b = o gives x= - b/a, which (approximately as before)
represents any negative magnitude. We thus arrive at the extended signification of number as a continuously varying positive or negative magnitude. Such numbers may be added or subtracted, multiplied
or divided one by another, and the result is always a number. Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. The
equation x 2 -1-1 = o is not (in the foregoing sense, number = real number) satisfied by any numerical value whatever of x; but we assume that there is a number which we call ti, satisfying the
equation i 2 + i = o, and then taking a and b any real numbers, we form an expression such as a+bi, and use the expression number in this extended sense: any two such numbers may be added or
subtracted, multiplied or divided one by the other, and the result is always a number. And if we consider first a quadric equation x 2 +- px+q = o where p and q are real numbers, and next the like
equation, where p and q are any numbers. whatever, it can be shown that there exists for x a numerical value which satisfies the equation; or, in other words, it can be shown that the equation has a
numerical root. The like theorem, in fact, holds good for an equation of any order whatever; but suppose for a moment that this was not the case; say that there was a cubic equation x 3 +- px 2
-+-qx+r = o, with numerical coefficients, not satisfied by any numerical value of x, we should have to establish a new imaginary j satisfying some such equation, and should then have to consider
numbers of the form a+bj, or perhaps a-{-bj-Fcj 2 (a, b, c numbers a-I-01 of the kind heretofore considered), - first we should be thrown back on the quadric equation x 2 -+px+q=o, p and q being now
numbers of the lastmentioned extended form - non constat that every such equation has a numerical root - and if not, we might be led to other imaginaries k, 1, &c., and so on ad infinitum in
inextricable confusion.
But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number=quantity of the form a+- i 31) form (what real numbers do not) a
universe complete in itself, such that starting in it we are never led out of it. There may very well be, and. perhaps are, numbers in a more general sense of the term (quaternions are not a case in
point, as the ordinary laws of combination are not adhered to), but in order to have to do with such numbers (if any) we must start with them.
S. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness (the meaning being as before), every equation has a root. Of course the
theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in
the course of an explanation, it will be convenient to give here the formal definition.
A number a such that substituted for x it makes the function t p n to be =0, or say such that it satisfies the equation f(x)=o, is said to be a root of the equation; that is, a being a root, we have
a n -p l a n - 1 ... =p„= o, or say f(a)=o; and it is then easily shown that x-a is a factor of the function f(x), viz. that we have f(x) = (x-a)fi(x), where fi(x) is a function x n - i -g i x n-2.
.. tg n _ i of the order with numerical coefficients qi, q2 In general a is not a root of the equation f i (x) =o, but it may be so - i.e. (x) may contain the factor x-a; when this is so, f(x) will
contain the factor (x-a) 2;writing then f(x) = (x-a)2f2(x), and assuming that a is not a root of the equation f 2 (x) =0, x= a is then said to be a double root of the equation f(x)=o; and similarly f
(x) may contain the factor (x --a) 3 and no higher power, and x =a is then a triple root; and so on.
Supposing in general that f(s) = (x - a) F(x) (a being a positive integer which may be = I, the highest power of x - a which divides f (x), and F(x) being of course of the order n - a), then the
equation =o will have a root b which will be different from a; x - b will be a factor, in general a simple one, but it may be a multiple one, of F(x), and f(x) will in this case be = 'a' (0 a
positive integer which may be = 1, (x - b)s the highest power of x - b in F(x) or f(x), and 4(x) being of course of the order n - a - /3). The original equation f(x) =o is in this case said to have a
roots each = a, /3 roots each = b; and so on for any other factors (x - c)' `, &c.
We have thus the theorem - A numerical equation of the order n has in every case n roots, viz. there exist n numbers, a, b,.. . (in general all distinct, but which may arrange themselves in any sets
of equal values), such that f (x) = (x - a) (x - b) (x - c) ... identically.
If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from consideration. It is, therefore, in
general assumed that the equation =o has all its roots unequal.
If the coefficients pi, P2,.. are all or any one or more of them imaginary, then the equation f (x) = o, separating the real and imaginary parts thereof, may be written F(x)-{ - iI(x) =o, where F(x),
4D(s) are each of them a function with real coefficients; and it thus appears that the equation f (x) = o, with imaginary coefficients, has not in general any real root; supposing it to have a real
root a, this must be at once a root of each of the equations F(x) =o and (D(x) =o.
But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz. a+Si being a root,
then a - jai will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root.
9. In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, has been already considered. In the general case of an equation with
imaginary (it may be real) coefficients, the like question arises as to the situation of the (real or imaginary) roots; thus, if for facility of conception we regard the constituents a, s of a root
a+131 as the co-ordinates of a point in piano, and accordingly represent the root by such point, then drawing in the plane any closed curve or " contour," the question is how many roots lie within
such contour, This is solved theoretically by means of a theorem of A. L. Cauchy (1837), viz. writing in the original equation x+iy in place of x, the function f(x+iy) becomes =P+iQ, where P and Q
are each of them a rational and integral function (with real coefficients) of (x, y). Imagining the point (x, y) to travel along the contour, and considering the number of changes of sign from - to +
and from + to - of the fraction corresponding to passages of the fraction through zero (that is, to values for which P becomes - o, disregarding those for which Q becomes = o), the difference of
these numbers gives the number of roots within the contour.
It is important 'to remark that the demonstration does not presuppose the existence of any root; the contour may be the infinity of the plane (such infinity regarded as a contour, or closed curve),
and in this case it can be shown (and that very easily) that the differvence of the numbers of changes of sign is =, n; that is, there are within the infinite contour, or (what is the same thing)
there are in all n roots; thus Cauchy's theorem contains really the proof of the fundamental theorem that a numerical equation of the nth order (not only has a numerical root, but) has precisely n
roots. It would appear that this proof of the fundamental theorem in its most complete form is in principle identical with the last proof of K. F. Gauss (1849) of the theorem, in the form--A
numerical equation of the nth order has always a root.' But in the case of a finite contour, the actual determination of the difference which gives the number of real roots can be effected only in
the case of a rectangular contour, by applying to each of its sides separately a method such as that of Sturm's theorem; and thus the actual determination ultimately depends on a method such as that
of Sturm's theorem.
Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.
10. A class of numerical equations which needs to be considered is that of the binomial equations x'I - a=o(a=a+/31, a complex number).
I The earlier demonstrations by Euler, Lagrange, &c., relate to the case of a numerical equation with real coefficients; and they consist in showing that such equation has always a real quadratic
divisor,furnishing two roots, which are either real or else conjugate imaginaries (see Lagrange's The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all
unequal. And these can be found numerically by the extraction of the square root, and of an nth root, of real numbers, and by the aid of a table of natural sines and cosines. 2 For writing a 0 (
a2+02)i there is always a real angle A (positive and less than 2x0, such that its cosine and sine are - 1/ (a+ß2) and It (a 2 + 02) respectively; that is, writing for shortness .J (a 2 + I T) = p, we
have a + 1 31 = p(cos X +i sin X), or the equation is x"= p(cosX+i sin X); hence observing that ((cos+isin). cos -?isirei=cosA-ki sin A, a value of x is=l The formula really gives all the roots, for
instead of A we may write A+2s7r, s a positive or negative integer, and then we have x p (cos which has the n values obtained by giving to s the values o, I, 2 ... n - 1 in succession; the roots are,
it is clear, represented by points lying at equal intervals on a circle. But it is more convenient to proceed somewhat differently; taking one of the roots to be 0, so that 0" = a, then assuming x =
0y, the equation becomes y I = o, which equation, like the original equation, has precisely n roots (one of them being of course =1). And the original equation a = o is thus reduced to the more
simple equation x n - i = o; and although the theory of this equation is included in the preceding one, yet it is proper to state it separately.
The equation x" - I = o has its several roots expressed in the form 1, where may be taken =cos n +i sin n; in fact, having this value, any integer power is =cos - n +i sin e k, and we thence have (w
k)" =cos 27rk+i sin 271-k, = I, that is, is a root of the equation. The theory will be resumed further on.
By what precedes, we are led to the notion (a numerical) of the radical al!" regarded as an n-valued function; any one of these being denoted by I/ a, then the series of values is i/ a, w a,. . co n
- I n il a; or we may, if we please, use n 1,/ a instead of a'!" as a symbol to denote the n-valued function.
As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations;
and hence, concluding for the present this subject, it will be convenient. to pass on to algebraical equations.
Algebraical Equations. r i. The equation is and we here assume the existence of roots, viz. we assume that there are n quantities a, b, c ... (in general all of them different, but which in
particular cases may become equal in sets in any manner), such that or looking at the question in a different point of view, and starting with the roots a, b, c ... as given, we express the product
of the n factors x - a, x - b, ... in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c .. In either case we have p 1 = ia, p 2 = iab, ... p A = abc ...;
i.e. regarding the coefficients p 1, p 2 ... p,, as given, then we assume the existence of roots a, b, c, ... such that p 1 = Ia, &c.; or, regarding the roots as given, then we write p i, p 2 , &c.,
to denote the functions /a, 'ab, &c. As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are
not restricted to denote, or are not explicitly considered as denoting, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation x 2 - px+q = o has a
root x={+-%/ (P 2 - 4 q), we mean that writing this value for x the equation becomes an identity, El {p+ ? (p2 - 4 q)11 2 - p {z{ p +d (p2-4q)} ]+q=o; and the verification of this identity in nowise
depends upon p and q meaning numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning
attributed to the term - the latter question at any 9 The square root of a-}-fi can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables.
a = (a 2 +13) ] rate it would be very difficult to answer; as to the former one, it may be said that the coefficients may, for instance, be symbols of operation. As regards such equations, there is
certainly no proof that every equation has a root, or that an equation of the nth order has n roots; nor is it in any wise clear what the precise signification of the statement is. But it is found
that the assumption of the existence of the n roots can be made without contradictory results; conclusions derived from it, if they involve the roots, rest on the same ground as the original
assumption; but the conclusion may be independent of the roots altogether, and in this case it is undoubtedly valid; the reasoning, although actually conducted by aid of the assumption (and, it may
be, most easily and elegantly in this manner), is really independent of the assumption. In illustration, we observe that it is allowable to express a function of p and q as follows,-that is, by means
of a rational symmetrical function of a and b, this can, as a fact, be expressed as a rational function of a+b and ab; and if we prescribe that a+b and ab shall then be changed into p and q
respectively, we have the required function of p, q. That is, we have F(a, (3) as a representation off (p, q), obtained as if we had p = a+b, q = ab, but without in any wise assuming the existence of
the a, b of these equations.
12. Starting from the equation x -plx n-1 +. .. =x-a.x-b. &c. or the equivalent equations p 1 = Ea, &c., we find an _p l an-L +.. . = o, -O n 1 . =0; (it is as satisfying these equations that a, b
... are said to be the roots of x n - p lx n-1+ ... = o); and conversely from the last-mentioned equations, assuming that a, b ... are all different, we deduce; p1= la, p 2 = lab, &c.
and xn - plan - 1 +. .. = x -a. x - b. &c.
Observe that if, for instance, a = b, then the equations a n - p la n-1+ ..=0, b n - p lb n-1+ „, o would reduce them selves to a single relation, which would not of itself express that a was a
double root,-that is, that. (x-a) 2 was a factor of x n -p i xn -1 +, &c.; but by considering b as the limit of a+ h, h indefinitely small, we obtain a second equation no n-r ' - (n-i)p l a n - 2 +
... = o, which, with the first, expresses that a is a double root; and then the whole system of equations leads as before to the equations &c. But the existence of a double root implies a certain
relation between the coefficients; the general case is when the roots are all unequal.
We have then the theorem that every 'rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational
and integral symmetrical function of the roots is a rational and integral function of the coefficients.
In particular, the sums of the powers /a 2; Za 3, &c.; are rational and integral functions of the coefficients.
The process originally employed for the expression of other functions la a bs, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, la a bs = la a las - la
x - 1 a; but this is very objectionable; the true theory consists in showing that we have systems of equations 3 pl =la, p = lab 1 2 =1x2-1-21ab, p 3 = zabc, p1p2 = la2b+3labc, Pi a =
1a3+31a2b+6labc, where in each system there are precisely as many equations as there are root-functions on the right-hand side-e.g. 3 equations and 3 functions labc, za 2 b, 1a 3 . Hence in each
system the root-functions can be determined linearly in terms of the powers and products of the coefficients: S lab = P2, ,1, ? la g =p12-2p2,, > r.- lobe = p3, la 2 b = -, p,p2-3p3, za 3 =
p13-3p1p2+3P3, and so on. The other process, if applied consistently, would derive the originally assumed value lab, =p2, from the two equations la = p /, ? la 2 = P12 - 2P2; i.e. we have 2lab = za.
la - 1a 2, = p 1 2 -(p 2 2 L'2), =2p 22. ,P,"t9,)t.
13. It is convenient to mention here the theorem that, x being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational
function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n-1, the coefficients being rational functions of the
coefficients of the equation. Thus the equation gives x' a function of the form in question; multiplying each side by x, and on the right-hand side writing for x" its foregoing value, we have x n+1 ,
a function of the form in question; and the like for any higher power of x, and therefore also for any rational and integral function of x. The proof in the case of a rational non-integral function
is somewhat more complicated. The final result is of the form 4(x)/ '(x) -1(x), or say 4(x) -1,t(x)1(x) = 0, where 44, I are rational and integral functions; in other words, this equation, being true
if only f(x) = o, can only be so by reason that the left-hand side contains f(x) as a factor, or we must have identically d(x) - '(x)I(x) = M(x)f (x). And it is, moreover, clear that the equation 4
(x)/1/1(x) = 1(x), being satisfied if only f(x) =o, must be satisfied by each root of the equation.
From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable
order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation. For example, in the case of a quartic equation, roots (a, b, c, d), it is
possible to find an equation having the roots ab, ac, ad, bc, bd, cd (being therefore a sextic equation): viz. in the product (y-ab)(y-ac)(yad) (y-bc)(y-bd)(y-cd) the coefficients of the several
powers of y will be symmetrical functions of a, b, c, d and therefore rational and integral functions of the coefficients of the quartic equation; hence, supposing the product so expressed, and
equating it to zero, we have the required sextic equation. In the same manner can be found the sextic equation having the roots (a-b) 2, (a-c) 2, (a-d) 2, b-c) 2, (b-d) 2, (c-d) 2 , which is the
equation of differences previously referred to; and similarly we obtain the equation of differences for a given equation of any order. Again, the equation sought for. may be that having for its n
roots the given rational functions (b), . of the several roots of the given equation. Any such rational function can (as was shown) be expressed as a rational and integral function of the order n -
i; and, retaining x in place of any one of the roots, the problem is to find y from the equations x"- plxn-1 =o, and Mox n-1+ Mlx n -2+ ..., or, what is the same thing, from these two equations to
eliminate x. This is in fact E. W. Tschirnhausen's transformation (1683).
14. In connexion with what precedes, the question arises as to the number of values (obtained by permutations of the roots) of given unsymmetrical functions of the roots, or say of a given set of
letters: for instance, with roots or letters (a, b, c, d) as before, how ,many values are there of the function ab+cd, or better, how many functions are there of ,this form? The answer is 3, viz.
ab±cd, ac+bd, add-be,- or again we may ask whether, in the case of a given number of letters, there exist functions with a given number of values, 3-valued, 4-valued functions,- &c. It is at once
seen that for any given number of letters there exist 2-valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its
sign; or say the two values are 0 and -'0. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P+QO; this has only the two values P+Qz, P-Q0. In the case
of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3valued or 4-valued function; and the only 5-valued functions are those which are
symmetrical in regard tofour of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the
demonstration of the non-existence of a solution of a quintic equation by radicals.
The theory is an extensive and important one, depending on the notions of substitutions and of groups. 15. Returning to equations, we have the very important theorem that, given the value of any
unsymmetrical function of the roots, e.g in the case of a quartic equation, the function ab+cd, it is in general possible to determine rationally the value of any similar function, such as (a+b)3+
(c+d)8.1-, The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1, 2,3,4, then if it is given that ab+cd = 14,
this in effect determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a= 1, b = 2 or a= 2, b= and c = 3, d = 4 or c =3, d = 4) or else a, b to be 3, 4 and c, d to be 1, 2; and it therefore in effect
determines (a+b)3+(c+d)3 to be =370, and not any other value; that is, (a+b)3+(c+d)3, as having a single value, must be determinable rationally. And we can in the same way account for cases of
failure as regards particular equations; thus, the roots being 1, 2, 3, 4 as before, 01)=2 determines a to be =I and b to be =2, but if the roots had been I, 2, 4, 16 then a 2 b = 16 does not
uniquely determine a,b but only makes them to be 1,16 or 2,4 respectively.
As to the a posteriori proof, assume, for instance, t1 = ab+cd, y1 = (a+b)3+(c+d)3, t2 =ac+bd, y2= (a+c)3+(b+d)3, t 3 =ad+bc, y 3 = (a+d)3+(b+c)3: then y1+y2+y3, t1yl+t2y2+t3y3, t1 2 y1 +t2 2 y2 +13
2 ,3 will be respectively. symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients; that is, they will be known.
Suppose for a moment that 11, tz i 1 3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, 12, 13; that is, yl, y2,
y3 will be known. But observe further that y 1 is Gbtained as a function of 11, 12, t 3 symmetrical as regards t 21 13; it can therefore be expressed as a rational function of t 1 and of 12+13, 12131
and thence as a rational function of t i and of ti+t2+t3, t1t2+t1t3+t2t3, t l t 2 t 3; but these last are symmetrical functions of the roots, and as such they are expressible rationally in terms of
the coefficients; that is, y1 will be expressed as a rational function of t 1 and of the coefficients; or t 1 (alone, not 1 2 or t3) being known, y 1 will be rationally determined.
16. We now consider the question of the algebraical solution of equations, or, more accurately, that of the solution of equations by radicals. In the case of a quadric equation x 2 - px+q = o, we can
by the assistance of the sign -1 () or () 2 find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby
identically satisfied; it has been found that this expression is X= EP (1,2-4q)}, and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. ' Or
we may by writing x = - 2p +z reduce the equation to z 2. = 4 (p 2 -4q), viz. to an equation of the form x 2 =a; and in virtue of its being thus reducible we say that the original equation is
solvable by radicals. And the question for-an equation of any. higher order, say of the order n, is, can we by means of radicals (that is, by aid of the sign' n ' f () or () 1 / m, using as many as
we please of such signs and with any values of m) find an, n-valued 'function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically?
It will be observed that the coefficients p, q ... are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equation admits of solution by
radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots 6f such an equation. It does not even follow that in the case of a numerical equation solvable by
radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = 2 {p t v (p
2 -4q), substituting for p, q their given numerical values, we obtain for x an expression of the form x = 'a' +si d -! (y+bi), where a, 0, y, b are real numbers. This expression substituted for x in
the quadric equation would satisfy it identically, and it is thus an algebraical solution; but there is no obvious a priori reason why v (y+bi) should have a value = c+di, where c and d are real
numbers calculable by the extraction of a root or roots of real numbers; however the case is (what there was no a priori right to expect) that v (y+bi) has such a value calculable by means of the
radical expressions } N I (7 2 +8 2) t-y} : and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation
will be considered presently.
17. A cubic equation can be solved by radicals.
Taking for greater simplicity the cubic in the reduced form x 3 -l-qx - r = o, and assuming x = a+b, this will be a solution if only 3ab=q and a 3 +b 3 =r, equations which give ( a 3 -P r =r 2 - 7q 3
, a quadric equation solvable by radicals, and giving a 3 -b3 = v (r' - 2743), a 2-valued function of the coefficients: combining this with a3+b3 =r, we have a 3 =.2 {r+ V 1 (r 2 - q 3)}, a 2-valued
function: we then have a by meanstf a cube root, viz.
a }1{H---1 (r2-A13)}], a 6-valued function of the coefficients; but then, writing q=b13a, we have, as may be shown, a+b a 3-valued function of the coefficients; and x = a+b is the required solution
by radicals. It would have been wrong to complete the solution by writing b = V [1}r-,/ (r2-2;43)11, for then a+b would have been given as a 9-valued function having only 3 of its values roots, and
the other 6 values being irrelevant. Observe that in this last process we make no use of the equation 3ab=q, in its original form, but use only the derived equation 27a 3 b 3 = q 3, implied in, but
not implying, the original form. An interesting variation of the solution is to write x =ab(a+b), giving a 3 b 3 (a 3 +b 3) = r and 3a 3 b 3 = q, or say a 3 +b 3 = r, a'b 3 = 3q, and consequently ' a
3 = ' 23 ,(r+1,/ (o-2743)}, b g3) 3 = y{r-,s (r22471, i.e. here a 3, b 3 are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a',
b 3, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only (0 b 3, not ab, is given); and ab(a+b) thus is in appearance a 9-valued function; but
it can easily be shown that it is (as it ought to be) only 3-valued.
In the case of a numerical cubic, even when the coefficients are real, substituting theirvalues in the expression j I x= V }2tr (r2 274 3)}H+3 q ' ¦1 L21 r +Y (r2 - 43)fl, this may depend on an
expression of the form / (y+bi) where y and S are real numbers (it will do so if 7 4 - z r q ' is a negative number), and then we cannot by the extraction of any root or roots of real positive
numbers reduce - (y+bi) to the form c+di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called " irreducible case " of a
cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and
there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are
two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.
The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z 3 = a,
solvable by the extraction of a cube root, but) to a cubic of the form 4x3-3x=a., corresponding to the equation 4 cos 3 0-3' cos 0=cos 30 which serves to determine cos B when cos 30 is known. The
theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x 3 --3x =a, is solvable by " trisection " -then the general cubic equation is
solvable by trisection.
18. A quartic equation is solvable by radicals: and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab+cd of the four roots (a, b, c,
d) : by what precedes ab+cd is the root of a cubic equation, which equation is solvable by radicals: hence ab+cd can be found by radicals; and since abed is a given function, ab and cd can then be
found by radicals. But by what precedes, if ab be known then any similar function, say a+b, is obtainable rationally; and then from the values of a+b and g ib we may by radicals obtain the value of a
or b, that is, an expression for the root of the given quartic equation: the expression ultimately obtained is 4-valued, corresponding to the different values of,the several radicals which enter
therein, and we have thus the expression by 'radicals of each of the four roots of the quartic equation. But when the quartic is numerical the same 'thing happens as in the cubic, and the algebraical
solution does not in every case give the numerical one.
It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or
nonexistence of k-valued functions of the five roots (a, b, c, d, e); the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more
than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N. H. Abel (1824) showed that a general quintic equation is not
solvable by radicals; and a fortiori the general equation of any order higher than 5 is not solvable by radicals.
19. The general theory of the solvability of an equation by radicals depends fundamentally on A. T. Vandermonde's remark (1770) that, supposing an equation is solvable by radicals, and that we have
therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any
one at pleasure of the roots a, b;. c ...; thus in the case of the quadric equation, in the expression x = 2 { p + v (p 2 - 4 q)}, substituting for p and q their values, -and observing that (a+b) 2
-4ab = (a-b) 2, this becomes x = {a+b+ N I (a-b) 2 }, the value being a or b according as the radical is taken to be +(a -b) or -(a-b).
So in the cubic equation x 3 -px 2 +qx -r=o, if the roots are a, b, c, and if w is used to denote an imaginary cube root of unity, w2-1-w+ 1 =o, then writing for shortness p=a+b+c, L=a+wb+w 2 c, M =
a+w 2 b+wc, it is at once seen that LM, L 3 +M 3, and therefore also (L 3 -M 3) 2 are symmetrical functions of the roots, and consequently rational functions of the coefficients: hence Z {L 3 +M 3 +v
(L3-M3)2} is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, =L 3 or M 3;
taking it =L3, the cube root of the expression has the three values L, wL, w2L; and LM divided by the same cube root has therefore the values M, w 2 M, wM; whence finally the expression 3{p+i!{a
(L'+Ma+1,((L 3 -M 3) 2)}-ELM T{1L3+M3+V(L3-M3)2)}] has the three values *(p+L+M), a (p+wL+w 2 M), Fp+w 2 L+wM); that is, these are =a, b, c respectively. If the value M 3 had been taken instead of L
3, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x 3 +qx -r =0, it will readily be seen that the two solutions are identical, and that
the function r2-Fq3 under the radical sign must (by aid of the relation p =0 which subsists in this case) reduce itself to (L 3 -M 3) 2; it is only by each radical being equal to a rational function
of the roots that the final expression can become equal to the roots a, b, c respectively.
20. The formulae for the cubic were obtained by J. L. Lagrange (1770-1771) from a different point of view. Upon examining and comparing the principal known methods for the solution of algebraical
equations, he found that they all ultimately depended upon finding a " resolvent," equation of which the root is a+wb+w 2 c+w a d+ ..., w being an imaginary root of unity, of the same order as the
equation; e.g. for the cubic the root is a+wb+w 2 c, w an imaginary cube root of unity. Evidently the method gives for L 3 a quadric equation, which is the " resolvent " equation in this particular
For a quartic the formulae present themselves in a somewhat different form, by reason that 4 is not a prime number. Attempting to apply it to a quintic, we seek for the equation of which the root is
(a+wb+w 2 c+w 8 d+w 4 e), w an imaginary fifth root of unity, or rather the fifth power thereof (a+wb+w2c+0.)9d+w4c)5; this is a 24-valued function, but if we consider the four values corresponding
to the roots of unity w, w 2, w 3 , w 4, viz. the values a+w b+w2c+w3s1+w4e)a, a+w g b+w 4 c+w d +cd'e)f, a+w'b+w c+w4d+w'e)5, a+ce 4 b+d 3 c+w 5 d+c, ð)8, any symmetrical function of these, for
instance their sum, is a 6-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the
original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useess for the
solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange's resolvent sextic, is very important, and is
intimately connected with all the later investigations in the theory. .
21. It is to be remarked, in regard to the question of solvability by radicals, that not only the coefficients are taken to be arbitrary, but it is assumed that they are represented each by a single
letter, or say rather that they are not so expressed in terms of other arbitrary quantities as to make a solution possible. If the coefficients are not all arbitrary, for instance, if some of them
are zero, a sextic equation might be of the form x 6 +bx 4 +cx +d = o, and so be solvable as a cubic; or if the coefficients of the sextic are given functions of the six arbitrary quantities a, b, c,
d, e, f, such that the sextic is really of the form (x 2 +ax+b) (x 4 +cx s +dx 2 +ex+f) = o, then it breaks up into the equations x2- }-ax+b=o, x4+cx3+dx2+ex+f=o, and is consequently solvable by
radicals; so also if the form is (x-a)(x-b)(x-,c)(x-d)(x--e)(x--f) mo, then the equation is solvable by radicals,-in this extreme case rationally. Such cases of solvability are self-evident; but they
are enough to show that the general theorem of the non-solvability by radicals of an equation of the fifth or any higher order does not in any wise exclude for such orders the existence of particular
equations solvable by radicals, and there are, in fact, extensive classes of equations which are thus solvable; the binomial equations x" - I = o present an instance.
22. It has already been shown how the several roots of the equation x"- I =0 can be expressed in the form cos 27r i sin 2 n", but the question is now that of the algebraical solution (or solution by
radicals) of this equation. There is always a root =I; if w be any other root, then obviously w, w 2, ... w"- 1 are all of them roots; x"- I contains the factor x-1, and it thus appears that w, w 2,,
.. w"- 1 are the nI roots of the equation x' 1 +x n-2 + ... +I =0; we have, of course, w n-1 + w " -2 + ... +w+ I =o.
It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration,
suppose successively n =15 and n =9; in the former case, if a be an imaginary root of x 3 -I =0 (or root of x 2 +x+I =o), and / an imaginary root of x 5 -1 =0 (or root of 4 -}- 3 -Fx 2 +x+1 =o), then
w may be taken = a(3; the successive powers thereof, as, a 2 0 2 , 1 3 3, a/ 4, a 2, /3, a/32, a 2 0 3, 0 4, a, a20, 02'm 3, a 2 ?3 4 , are the roots of x14+x13-? +x+1 =o; the solution thus depends
on the solution of the equations x 3 -I =o and x 5 -I =0. In the latter case, if a be an imaginary root of x 3 -1 = o (or root of x 2 +x+ 1= o), then the equation x s -I =0 gives x 3 = I, a, or a 2;
x 3 = I gives x= I, a, or a 2; and the solution thus depends on the solution of the equations x 3' - I = o,x 3 -' = o, x' - a2=-0. The first equation has the roots 1, a, a 2; if (3 be a root of
either of the others, say if 0 3 =a, then assuming w = /3, the successive powers are /3, /i 2, a, aß, 0 2, a 2, a 2 /3, a 2 /3 2, which are the roots of the equation x a +x 7 + ... + I =0.
It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of w, we have r, r 2, r 3, ... r"-' as the (n - I) roots of the
reduced equation x"-1+x"-2 + ... +x + =; then not only r"-I =o, but also r"- 1 +r"- 2 + ... +r+I =0.
23. The process of solution due to Karl Friedrich Gauss (1801) depends essentially on the arrangement of the roots in a certain order, viz. not as above, with the indices of r in arithmetical
progression, but with their indices in geometrical progression; the prime number n has a Certain number of prime roots g, which are such that g f - 1 is the lowest power of g, which is - I to the
modulus it; or, what is the same thing, that the series of powers 1, g, g 2, ... r2, each divided by is, leave (in a different order) the remainders 1, 2, 3,.. n-1; hence giving to r in succession
the indices 1, g, g 5, ... g"- 2, we have, in a different order, the whole series of roots r, r' 2, r 3, ... In the most simple case, it -5, the equation to be solved is x44-x3+ +x +I oo; here 2 is a
prime root of 5, and the order of the roots is r, r ®, r', r'. The Gaussian process consists in forming an equation for determining the periods P1, P2, -r+s 4 and 1 4 +1 4 respectively, -these being
such that the symmetrical functions P1+P2, P 1 P 5 are rationally determinable: in fact P 1 +P 5 - - I, P 1 P 2 = (r+r4)(r'+rs), =r'-Er 4 --} x+r T , P1, P2 are thus the roots of u s +u-I =o; and
taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r4 for P 1, r E and r for P 2; the symmetrical functions of these are then rationally
determined in terms of P 2 and Pg; thus r+r' =P I, r.r 4 =1, or r, r 4 are the roots of u 2 -P i u+I =o. The mode of division is more clearly seen for a larger value of n; thus, for n=7 a prime root
is =3, and the arrangement of the roots is r, r 3, r', r', r 4, 0. We may form either 3 periods each of 2 terms, Pl, 1 3 2, P3 r 8 r 3 +r 4, r 2 +r5 respectively; or else 2 periods each of 3 terms,
P1, Peter+r2+r4, r 3 +0+r 5 respectively; in each case the symmetrical functions of the periods are rationall' determinable: thus in the case of the two periods P 1 +P 2 -I, P 1 l 2 =3+r+r 9 +r 3 +r
4 +r'+r', = 2; and the periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r+r 2 +r 4 = P2, r.r 2 -{-r.r4+r2.r4.-
P2, The theory was further developed by Lagrange {1808), who, applying his general process to the equation in question, x'-1+ x"- 2 + ... +x+1 =o(the roots a, b, c ... being the several powers of s,
the indices in geometrical progression as above), showed that the function (a-}-wb+w2c-1- ...)" -1 was in this case a given function of w with integer coefficients.
Reverting to the before-mentioned particular equation x4+xs+ x 2 +1 =o, it is very interesting to compare the process of solution with that for the solution of the general quartic the roots whereof
are a, b, c, d. Take w, a root of the equation w 4 -1 =o (whence w is I, --1, i, or -i, at pleasure), and consider the expression (a +wb+w2c+w3d)4, the developed value of this is a 4 +b 4 -?-c' 4-d 4
+6 (a 5 c 2 +b 2 d 2) +12 (a2bd+Vca+c2db+d2ac) +w {4(a3b+b3c+c5+d8a) + 12 (a2cd+b2da+c2ab+d2bc) 1 +016 (a 2 b 2 +b 2 c 2 +c 2 d 2 +d 2 a 2 ) +4 (a 3 c+b a d+c'a+d a b) +24abcd} +w 3 {4(a 1 d -1-1) 3
a-Ec 3 b+d 3 c) x-12 (a 2 be-}-b 2 cd+c e de+dab) } that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in fact, solvable by radicals; but this is not here material).
If, however, a, b, c, d denote the roots r, r 2, r 4, r 3 of the special equation then the expression becomes r'+--r3-1-r+r2+6(I+I) +12(r2+r4-+r3+r) W 14(i +I + 1) + 1 2(r 4 +r 3 +r +r)} +W 2 {6(r+r
2 +r 4 +r 3)+ 4(r 2 + r4 +r 3 +r)} w 3 {4(r+r 2 +r 4 -+r 3)+12(r 3 +r +r2+r4)} viz. this is = - I +4w-1-140-16(03, a completely determined value. That is, we have (r+Wr 2 +?, 2 r 4 +w 3 r 3) 4 = -
+4w+14, 2 - 16W3, which result contains the solution of the equation. If w =1, we have (r +r 2 +r 4 +r 3) 4 = I,which is right; if w = - i,then(r+r 4 -r 2 - r') 4 = 25; if w= i, then we have {r-r 4+i
(r2 - r3)}4= -15+201; and if w= - then {rr 4 -i(r 2 -r 3)} 4= - 15-201; the solution may be completed without difficulty.
The result is perfectly general, thus: - n being a prime number, r a root of the equation x"-i-i-xn-2-1- ... +x+I =o, co a root of to" --- ' --I= o, and g a prime root of g "-' T (mod. n), then (r
wro+... +W"-2ra"-2)n_.i is a given function M0+f-Mlt, ... -1-M„_2wn'-2 with integer coefficients, and by the extraction of (nI) th roots of this and similar expressions we ultimately obtain r in
terms of w, which is taken to be known; the equation x n -- r = o, n a prime number, is thus solvable by radicals. In particular, if n I be a power of 2, the solution (by either process) requires the
extraction of square roots only; and it was thus that Gauss discovered that it was possible to construct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting
developments in regard to the theory were obtained by C. G. J. Jacobi (1837); see the memoir " Ueber die K.reistheilung, n.s.w.," Crelle, t. xxx. (1846).
The equation x n-1 + ... +x+ i = o has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by N. H. Abel (1828), and since
called Abelian equations, viz. for the Abelian equation of the order n, if x be any root, the roots are x, Ox, 0 2 x, ... O n-l x (Ox being a rational function of x, and 8 n x = x); the theory is, in
fact, very analogous to that of the above particular case.
A more general theorem obtained by Abel is as follows: - If the roots of an equation of any order are connected together in such wise that all the roots can be expressed rationally in terms of any
one of them, say x; if, moreover, Ox, O i x being any two of the roots, we have OOjx = 0 1 0x, the equation will be solvable algebraically. It is proper to refer also to Abel's definition of an
irreducible equation: - an equation 4x =o, the coefficients of which are rational functions of a certain number of known quantities a, b, c ... , is called irreducible when it is impossible to
express its roots by an equation of an inferior degree, the coefficients of which are also rational functions of a, b, c... (or, what is the same thing, when 4x does not break up into factors which
are rational functions of a, b, c...). Abel applied his theory to the equations which present themselves in the division of the elliptic functions, but not to the modular equations.
24. But the theory of the algebraical solution of equations in its most complete form was established by Evariste Galois (born October 1811, killed in a duel May 1832; see his collected works,
Lionville, t. xl., 1846). The definition of an irreducible equation resembles Abel's, - an equation is reducible when it admits of a rational divisor, irreducible in the contrary case; only the word
rational is used in this extended sense that, in connexion with the coefficients of the given equation, or with the irrational quantities (if any) whereof these are composed, he considers any number
of other irrational quantities called "adjoint radicals," and he terms rational any rational function of the coefficients (or the irrationals whereof they are composed) and of these adjoint radicals;
the epithet irreducible is thus taken either absolutely or in a relative sense, according to the system of adjoint radicals which are taken into account. For instance, the equation x 4 + x3-}-x2-{-x+
I = o; the left hand side has here no rational divisor, and the equation is irreducible; but this function is=(x 2 +2x4-I) 2 -*x 2, and it has thus the irrational divisors x2-}--i(I-1-1/ 5)x+ t, x 2
+ 2(115)x-1-I; and these, if we adjoin the radical ,! 5, are rational, and the equation is no longer irreducible. In the case of a given equation, assumed to be irreducible, the problem to solve the
equation is, in fact, that of finding radicals by the adjunction of which the equation becomes reducible; for instance, the general quadric equation x2+px+ q=o is irreducible, but it becomes
reducible, breaking up into rational linear factors, when we adjoin the radical (±p2--q).
The fundamental theorem is the Proposition I. of the " Memoire sur les conditions de resolubilite des equations par radicaux "; viz. given an equation of which a, b, c ... are the m roots, there is
always a group of permutations of the letters a, b, c ... possessed of the following properties I. Every function of the roots invariable by the substitutions of the group is rationally known.
2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group.
Here by an invariable function is meant not only a function of which the form is invariable by the substitutions of the group, but further, one of which the value is invariable by these
substitutions: for instance, if the equation be (x) =o, then 4,(x) is a function of the roots invariable by any substitution whatever. And in saying that a function is rationally known, it is meant
that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.
For instance in the case of a general equation, the group is simply the system of the 1.2.3... n permutations of all the roots, since, in this case, the only rationally determinable functions are the
symmetric functions of the roots.
In the case of the equation x'? 1 ... +x-1-1 =0, n a prime number, a, b, c... k =r, ra, ro 2, .. ro"- 2 , where g is a prime root of n, then the group is the cyclical group abc ... k, òc... ka, ...
kab ...j, that is, in this particular case the number of the permutations of the group is equal to the order of the equation.
This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois's theory. But the
problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to tke transformation and classification of
Returning to the question of solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished; for instance, in the case of the general cubic, where
the group is that of the six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to ak, bca, cab; that is, it becomes possible to express
rationally, in terms of the coefficients and of the adjoint square root, any function such as a 2 b+b 2 c+e 2 a which is not altered by the cyclical substitution a into b, b into c, c into a. And
hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce
the original group of the equation so as to make it ultimately consist of a single permutation.
The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained - in the first instance in the form (scarcely intelligible without further
explanation) that every function of the roots x,, x2... x ", invariable by the substitutions xnk}b for xi, must be rationally known; and then in the equivalent form that the resolvent equation of the
order 1.2 ... (n-2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange's resolvent of the order 6 may have a rational factor, a
result obtained from a direct investigation in a valuable memoir by E. Luther, Crelle, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, II,
the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, II respectively; but for the transformation, n a prime number greater than II, the depression is impossible.
The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti (1852). See also J. A. Serret's Cours d'algebre superieure, 2nd
25. Returning to quintic equations, George Birch Jenard (1835) established the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x 5
-1-ax+b= o, or what is the same thing, to x5+x+b=o. The actual reduction by means of Tschirnhausen's theorem was effected by Charles Hermite in connexion with his ellipticfunction solution of the
quintic equation (1858) in a very elegant manner. It was shown by Sir James Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861), that Lagrange's
resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory.
The theory of the modular equations, more particularly for the case n = 5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the case n = 5, the modular equation of the order 6 (854);
4th ed. (1877-1878).
depends, as already mentioned, on an equation of the order 5; and conversely the general quintic equation may be made to depend upon this modular equation of the order 6; that is, assuming the
solution of this modular equation, we can solve (not by radicals) the general quintic equation; this is Hermite's solution of the general quintic equation by elliptic functions (1858); it is
analogous to the before-mentioned trigonometrical solution of the cubic equation. The theory is reproduced and developed in Brioschi's memoir, " tber die Auflosung der Gleichungen vom fiinften
Grade," Math. Annalen, t. xiii. (1877-1878).
26. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. Jordan's Traite des substitutions et des equations algebriques (Paris,
1870). The work is divided into four books - book i., preliminary, relating to the theory of congruences; book ii. is in two chapters, the first relating to substitutions in general, the second to
substitutions defined analytically, and chiefly to linear substitutions; book iii. has four chapters, the first discussing the principles of the general theory, the other three containing
applications to algebra, geometry, and the theory of transcendents; lastly, book iv., divided into seven chapters, contains a determination of the general types of equations solvable by radicals, and
a complete system of classification of these types. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book
iii., the algebraical applications comprise Abelian equations, equations of Galois; the geometrical ones comprise Q Hesse's equation, R. F. A. Clebsch's equations, lines on a quartic surface having a
nodal line, singular points of E. E. Kummer's surface, lines on a cubic surface, problems of contact; the applications to the theory of transcendents comprise circular functions, elliptic functions
(including division and the modular equation), hyperelliptic functions, solution of equations by transcendents. And on this last subject, solution of equations by transcendents, we may quote the
result - " the solution of the general equation of an order superior to five cannot be made to depend upon that of the equations for the division of the circular or elliptic functions "; and again
(but with a reference to, a possible case of exception), " the general equation cannot be solved by aid of the equations which give the division of the hyperelliptic functions into an odd number of
parts." (See also GROUPS, THEORY OF.) Table of the Equation of Time.
Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Equation'. 1911 Encyclopedia Britanica. https://www.studylight.org/​encyclopedias/​eng/​bri/​e/
equation.html. 1910. | {"url":"https://www.studylight.org/encyclopedias/eng/bri/e/equation.html","timestamp":"2024-11-10T07:50:00Z","content_type":"text/html","content_length":"275862","record_id":"<urn:uuid:3df72f3f-3171-4842-a37a-a6f6cb417ae4>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00015.warc.gz"} |
Create Circular Perfect Square Sequence
Problem 633. Create Circular Perfect Square Sequence
A sequence v(1:N) made of values 1:N can be created for N>31 such that v(i)+v(i+1) is a perfect square. The sum of v(N)+v(1) must also be a perfect square. All values 1 thru N are required and the
vector must be of length N. (e.g. For N=32 the possible perfect squares are [4 9 16 25 36 49]. By inspection the value 32 must be bracketed by 4 and 17). The Test set will be limited to 31<N<52 as
solutions beyond 51 may take significant processing time.
Solution Stats
55.56% Correct | 44.44% Incorrect
Problem Comments
you might want to specify in the problem that the sequence has to be made of N unique values
sorry, my mistake; it was unclear to me at first that the sequence had to contain a permutation of the values 1:N
Thank You. I have clarified that each value must be used.
oh man, this problem is brutal. i have a solution that can solve it for values of N between 33 and 39 inclusive in under a second, but as soon as I try 40 or 41, it runs forever (or at least past the
end of my patience)
It seems there is a lot of variability depending on your particular search heuristics and the particular value of N. My solution performs in under 30 seconds for 19 out of the 20 cases 31
Hello Richard, Hello Alfonso, Hello @bmtran, Is it possible to solve this kind of problem without recursion ?
In general, sure, you can easily write these sort of heuristic-search algorithms without explicitly using recursion, or you could use non-search-based approaches, such as annealing, integer linear
programming, etc. Now if you are asking whether exhaustive or other polynomial-time approaches are possible/practical for this problem I am not really sure about that. I believe this problem reduces
to finding a full hamiltonian cycle over an N-node graph, so the only hope of bringing this out of the NP-hard umbrella would be exploiting some properties of these particular networks arising from
the properties of perfect numbers, but so far I do not see any useful trick in this regard (so in short, perhaps it is possible but I do not know how; any thoughts?)
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Introduction to analytic number theory (Topics in Ad. Analysis 1)
This course considers the classical topics in analytic number theory, with a focus on tools coming from Fourier analysis. The list of topics to be covered is as follows:
1. Review of the basic elements of Fourier analysis: Fourier transform in L^1 and L^2; Plancherel's theorem; Tempered distributions; Fourier series; Convolution and approximations of the identity.
2. Diophantine approximations; Equidistribution of sequences; Notions of discrepancy; Erdös-Turán inequality; Irregularities of distribution.
3. Arithmetic functions; The Riemann zeta-function; Dirichlet characters; Dirichlet L-functions; Gauss sums; Primes in arithmetic progression; Functional equation for L-functions; Zero-free regions
for zeta and other L-functions; Prime Number Theorem.
4. Consequences of the Riemann hypothesis; Explicit formulas; Pair correlation of zeros; Prime gaps; Extremal functions and Fourier optimization methods.
5. Geometry of numbers; Minkowski's convex body theorem.
References books:
1. H. Davenport, Multiplicative Number Theory, Third edition, Springer 2000.
2. H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Volume 53, 2004.
3. E. C. Titchmarsh, The theory of the Riemann zeta-function, Second Edition, Oxford Science Publications, 1986.
4. H. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics 97. Cambridge University Press, 2006.
Basic knowledge in real analysis (measure and integration) and complex analysis. | {"url":"https://www.math.sissa.it/course/phd-course-master-course/introduction-analytic-number-theory-topics-ad-analysis-1","timestamp":"2024-11-13T18:40:05Z","content_type":"application/xhtml+xml","content_length":"31963","record_id":"<urn:uuid:6d0b6783-0cf1-465f-93b7-0424ce475f33>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00278.warc.gz"} |
Don’t Underestimate 26″ Wheels When It Comes To Speed
from Rookie’s keyboard
Hello, friends
Many rookies wrongfully conclude that smaller wheels always equal a slow bicycle. And while there’s some logic behind this statement, every bike theorist has to know that ultimately bicycle speed is
a function of GEARING rather than wheels size.
What does this mean?
It’s simple.
If largers wheels equaled more speed, then tractors should be faster than cars, right? And F1 models should have the largest wheels on the planet? As big a T-rex?
The top speed of a vehicle is determined by its gearing (a.k.a. the mechanism through which the rider/engine transmits power to the wheels) and the power output of the engine/rider.
The drivetrain consists of chainrings (fron) and small cog(s) at the back.
When you spin the cranks, the chain transfers power to the rear cog(s) and respectively rear wheel.
A larger chainring and/or a smaller rear cog, equal more power transfer to the rear.
The term gear ratio refers precisely to the relation between the chainring and the cog.
For instance, if the chainring has 44 teeth and the rear cog has 11, the gear ratio is 4:1.
In that case, each full revolution of the front chainring, results in 4 spins of the rear wheel.
The larger the gear ratio is, the faster the bike can be because each spin of the cranks equals more revolutions of the rear wheel per minute and thus a greater traveled distance.
The formula for speed is Speed = Distance/Time.
A bike that covers more distance in the same amount of time than another bike is moving faster regardless of wheel size.
Wheel Size Gear Ratio Wheel Circumference Tire Width Time RPM
Bike A 26″ 44:11 2075mm/207.50 cm 2.00 inch 60 seconds 80
Bike B 27.5″ 34:11 2153mm/215.30 cm 2.00 inch 60 seconds 80
The table above contains data for two hypothetical set-ups. Both bikes have identical tire width but different wheel sizes.
Bike A:
In 44/11, the rear wheel spins 4 times per 1 crank revolution.
Since the rider is pedaling at 80rpm, the rear wheel makes 4×80=320 turns per 1 minute/60 seconds.
The traveled distance can be calculated by multiplying the wheel’s circumference by the number of wheel turns.
In this case, the distance is 320 x 207.50 cm = 66400cm = 664.00 m = 0.66km
Bike B:
In 34/11, the rear wheel spins 3.09 times per 1 crank revolution.
At 80 rpm, the rear wheel makes 3.09 x 80 = 247.2 revolutions.
Thus, the traveled distance is 247.2 x 215.3cm (wheel circumference) = 53222cm = 532.22m = 0.53km.
The formula for speed is: Speed = Distance/Time.
In the example case above, we have the following speeds:
Bike A’s speed = 664m / 60s = 11.0667m/s = 24.75 mi/h = 39.84 km/h
Bike B’s speed = 532.22m / 60s = 8.87 m/s = 19.84 mi/h = 31.93 km/h
Bike A has the potential to be 24.7% faster than Bike B despite having smaller 26″ wheels. The potential for extra speed comes from the higher gearing.
But, if both bikes have the same gearing, then Bike B will be faster than Bike A.
Don’t forget that smaller wheels are easier to spin from a dead stop and therefore accelerate faster.
A bike with 26″ wheels is easier to get up to speed than a bike with 29″ wheels, for example.
Speed Maintenance
Smaller wheels may be easier to accelerate, but they have smaller inertia and require more effort to keep spinning.
Meanwhile, larger wheels are more difficult to get up to speed, but once there, they maintain the speed with less effort. Hence larger wheels are considered better for covering long distances.
Rolling Resistance
Another parameter that directly impacts speed is rolling resistance. The lower the rolling resistance of a tire is, the less effort is required to maintain a higher speed.
Higher rolling resistance, on the other hand, is detrimental.
If a 26″ wheel is equipped with slick tires, it will have a lower rolling resistance on paved roads but reduced grip on off-road terrain.
Кnobby tires offer more grip when riding off-road by digging into the ground. On paved roads, however, they are are slower and noisier.
A Perceived Feeling of “Slowness”
Many people who go from 26″ wheels to 700c/29″, for example, immediately feel faster.
Why? First, 26″ bikes are either retro MTBs, large BMXs, or old commuters.
Meanwhile, 700c wheels are found on road bikes, touring bikes, commuters, and modern MTBs (700c = 29″).
Road bikes and even touring bikes tend to have much higher gearing than retro MTBs for example.
A road bike offers a much higher speed potential not only thanks to its larger wheels but thanks to its high gearing.
When switching to larger tires, you will have to produce less power to keep them rolling on flat roads. The reduced energy requirement will make you feel faster and lighter even when you’re not
technically moving at a higher speed than before.
Road Bikes Always Win
A 26-er with a high-gearing and slick tires can be plenty fast. But when speed is the main goal – road bikes are always the winner.
The aggressive geometry in conjunction with high gears, light weight and large tires with low rolling resistance facilitate the maintenance of decent speed levels.
Until next time,
6 responses to “Don’t Underestimate 26″ Wheels When It Comes To Speed”
True but if you live in an extremely hilly place like I do, lack of low gearing will have you walking most of the time so you will be slower by default. Which is why I built my touring bike with
a 44/32/22 front triple chainring. The 22 granny gear is unheard of on most road bikes but makes it feasible to keep spinning up ridiculous inclines while others are pushing their bikes up the
hill on foot. The 44 not super fast but I’m not winning any races on a touring bike anyway so it doesn’t matter.
Yes. If you want the most versatile bike, 3x with a granny ring is the way to go.
How are you carrying cargo on your bike? Panniers? Bags?
I’m setting up the touring bike with a traditional rear rack and using waterproof saddle panniers (ortlieb). It’s a classic approach that works well enough.
I’ve thought about frame bags but since I have a big frame pump right under the top tube, I don’t think it would work.
I don’t generally carry cargo on my fun racy bike other than a basic toolkit. I use a little saddle bag for that but I may change to a tool bottle, like one of your articles suggested.
Thank you for the answer. It’s really the classic approach that has worked well for many people.
A frame bag can’t match the panniers. It’s too narrow anyway. I have the Carradice SQR saddle bag. Bought it 5 years ago and used it for 1-2 years on my hardtail – it’s like a single
pannier but better (balanced) for town commuting. It’s good.
Currently, I just drop a backpack in my ghetto front rack as I am riding a road bike born in 1987.
I like the idea of the big saddle bag. Better for balance as you say. I bet there are bags that mount to the top of a rear rack too that would hold things even more stable.
Nice, what bike is it?
It is Centurion Futura, but I re-painted it with spray bottles. | {"url":"https://rookiejournal.com/archives/1545","timestamp":"2024-11-10T15:19:15Z","content_type":"text/html","content_length":"85381","record_id":"<urn:uuid:907b70a5-e2ab-484e-8321-a1591f2ac008>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00165.warc.gz"} |
Projects Archive - Sam's Blogs
In this tutorial, we will learn how to convert a given binary tree to a doubly linked list using Java programming language. Binary trees are hierarchical data structures where each node can have at
most two children, referred to as the left child and the right child. On the other hand, a doubly linked list is a linear data structure in which each node has a reference to both its previous and
next node.
To convert a binary tree to a doubly linked list, we can use the following algorithm:
1. Start by creating an empty doubly linked list.
2. Traverse the binary tree using any tree traversal algorithm.
3. For each node encountered during the traversal, perform the following steps:
1. Set the left pointer of the current node to the last node in the doubly linked list.
2. If the last node is not null, set its right pointer to the current node.
3. Set the current node as the last node in the doubly linked list.
4. Set the right pointer of the current node to the next node in the traversal (if any).
4. After traversing the entire binary tree, the doubly linked list will be formed.
Here is the Java implementation of the algorithm:
class Node {
int data;
Node left, right;
public Node(int item) {
data = item;
left = right = null;
class BinaryTreeToDoublyLinkedList {
Node root;
Node lastNode;
public void convertToDoublyLinkedList(Node node) {
if (node == null)
if (lastNode == null) {
root = node;
} else {
node.left = lastNode;
lastNode.right = node;
lastNode = node;
The above implementation uses a class named Node to represent each node in the binary tree. The convertToDoublyLinkedList method is responsible for converting the binary tree to a doubly linked list.
It uses a recursive approach to traverse the binary tree and performs the necessary pointer manipulations to form the doubly linked list.
In this tutorial, we have learned how to convert a given binary tree to a doubly linked list using Java. The algorithm involves traversing the binary tree and manipulating the pointers of each node
to form the doubly linked list. This can be a useful technique in certain scenarios where a doubly linked list is required for efficient operations. Feel free to explore further and apply this
concept to solve related problems.
Landing page with a hero section
Create a responsive landing page with a hero section, three feature sections, and an animated call-to-action button.
Project Overview:
1. Hero Section: A full-width section with a background image, title, subtitle, and call-to-action button.
2. Features Section: Three columns, each representing a feature with an icon, title, and description.
3. Animated Call-to-Action Button: When hovered over, the button should expand slightly and change color.
1. Set Up the HTML Structure [Check The MyGit]
2. Styles (styles.css) [Check The MyGit]
3. Media Queries for Responsiveness [Check The MyGit]
Optional Enhancements:
1. Smooth Scroll: Incorporate a smooth scroll to improve navigation between different page sections.
2. Interactivity: Utilize JavaScript/jQuery to add more interactivity or animations when the user interacts with different elements on the page.
3. Parallax Scrolling: Use parallax effects to create depth while scrolling.
To take this project further, you can also incorporate a navigation bar, footer, and more sections as per the requirements.
The Fibonacci series
The Fibonacci series is a sequence of numbers where each is the sum of the two preceding ones, starting from 0 to 1.
There are different ways to implement Fibonacci series generation in PHP. Below are two common approaches: using iterative loops and using recursion.
1. Iterative Approach: This method is straightforward and uses a loop to generate the Fibonacci sequence up to the desired length.
2. Recursive Approach: In the recursive approach, the Fibonacci function calls itself to compute the value.
Remember, the recursive approach, although elegant, is less efficient for larger values of n due to the redundant calculations it involves. The iterative method is generally more efficient for larger
sequences. If you want to generate a Fibonacci sequence for very large values of , consider using memoization or other optimization techniques. | {"url":"https://samsblog.in/project/","timestamp":"2024-11-04T20:10:39Z","content_type":"text/html","content_length":"164730","record_id":"<urn:uuid:84cbcfb7-b7cc-41f5-af2f-bac2f84e42af>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00248.warc.gz"} |
What is the product of powers? - California Learning Resource Network
What is the product of powers?
What is the Product of Powers?
The product of powers is a fundamental concept in mathematics, particularly in the realm of algebra and number theory. In this article, we will delve into the definition, properties, and applications
of the product of powers, also known as exponentiation.
What is the Product of Powers?
The product of powers, denoted by a b , is a mathematical operation that combines two numbers a and b by raising a to the power of b. In other words, a b = a raised to the power of b. This operation
is often denoted by the caret symbol (^) or the superscript notation a b.
**Properties of the Product of Powers**
The product of powers exhibits several important properties, which are summarized below:
| **Property** | **Definition** |
| — | — |
| **Commutativity** | **a** **b** = **b** **a** (the order of the factors does not matter) |
| **Associativity** | (**a** **b**) **c** = **a** (**b** **c**) (the product can be re-ordered) |
| **Distributivity** | **a** (**b** + **c**) = **a** **b** + **a** **c** (the product can be distributed) |
| **Identity** | **a** **0** = 1 (any number raised to the power of 0 equals 1) |
**H3. Real-World Applications of the Product of Powers**
The product of powers has numerous real-world applications in various fields, including:
• **Computer Science**: In programming, exponentiation is used to calculate the value of a variable raised to a power. For example, in complex algorithms, the product of powers is used to optimize
• **Economics**: In finance, the product of powers is used to model economic growth rates and calculate the impact of compounding interest.
• **Biology**: In ecology, the product of powers is used to model population growth and dynamics.
• **Physics**: In quantum mechanics, the product of powers is used to describe the behavior of particles and systems.
**H3. Common Examples of the Product of Powers**
Here are some common examples of the product of powers in different contexts:
| **Example** | **Description** |
| — | — |
| **2** **3** | 2 raised to the power of 3, equal to 8 |
| **5** **2** | 5 raised to the power of 2, equal to 25 |
| **e** **2** | The base of the natural logarithm, raised to the power of 2 |
**H3. Calculating the Product of Powers**
Calculating the product of powers can be done using various methods, including:
• **Manual calculation**: Performing the calculation by hand, using the definition of the product of powers.
• **Calculator**: Using a calculator or a computer algebra system to perform the calculation.
• **Exponentiation formulae**: Using short-hand notations, such as **a** **b** = **a** × **a** × … × **a** (b times).
The product of powers is a fundamental concept in mathematics, with numerous applications in various fields. Understanding the properties and calculating the product of powers is essential for
solving problems in algebra and beyond. Whether used in computer science, economics, biology, or physics, the product of powers is a powerful tool for modeling and problem-solving. Remember to always
keep in mind the properties of the product of powers, and use the appropriate method for calculating the result.
Your friends have asked us these questions - Check out the answers!
Leave a Comment | {"url":"https://www.clrn.org/what-is-the-product-of-powers/","timestamp":"2024-11-10T02:09:50Z","content_type":"text/html","content_length":"134293","record_id":"<urn:uuid:a375cbdd-cfb8-48a1-a322-5f4c5c0e637b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00848.warc.gz"} |
Explain the working of the Random Forest Algorithm
The steps that are included while performing the random forest algorithm are as follows:
Step-1: Pick K random records from the dataset having a total of N records.
Step-2: Build and train a decision tree model on these K records.
Step-3: Choose the number of trees you want in your algorithm and repeat steps 1 and 2.
Step-4: In the case of a regression problem, for an unseen data point, each tree in the forest predicts a value for output. The final value can be calculated by taking the mean or average of all the
values predicted by all the trees in the forest.
and, in the case of a classification problem, each tree in the forest predicts the class to which the new data point belongs. Finally, the new data point is assigned to the class that has the maximum
votes among them i.e, wins the majority vote. | {"url":"https://discuss.boardinfinity.com/t/explain-the-working-of-the-random-forest-algorithm/6687","timestamp":"2024-11-10T05:53:06Z","content_type":"text/html","content_length":"16010","record_id":"<urn:uuid:a9cd8a6f-829d-4f80-9697-5d30c6cbda6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00710.warc.gz"} |
Properties Of Integers (video lessons, examples and solutions)
Introduction To Integer
Digits are the first concept of integers. There are ten digits namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
In our number system, the position of the digits are important. For example, consider the number 3,027. This can be represented in a place value table as follows:
(For the SAT, the units digit and the ones digit refer to the same digit in a number).
Integers are like whole numbers but they also include negative numbers, for example, –4, –3, –2, –1, 0, 1, 2, 3, 4, …
Positive integers are all the whole numbers greater than zero, ie: 1, 2, 3, 4, 5, … We say that its sign is positive.
Negative integers are the numbers less than zero, ie: –1, –2, –3, –4, –5, … We say that its sign is negative.
Integers extend infinitely in both positive and negative directions. This can be represented on the number line.
Zero is an integer that is neither positive nor negative.
Printable & Online Integers Worksheets
Consecutive Integers
Consecutive integers are integers that follow in sequence, each number being 1 more than the previous number, for example 22, 23, 24, 25, …
Consecutive integers can be more generally represented by n, n +1, n + 2, n + 3, …, where n is any integer.
Even And Odd Integers
Even integers are integers that can be divided evenly by 2, for example, –4, –2, 0, 2, 4, … An even integer always ends in 0, 2, 4, 6, or 8.
Zero is considered an even integer.
Odd integers are integers that cannot be divided evenly by 2, for example, –5, –3, –1, 1, 3, 5, … An odd integer always ends in 1, 3, 5, 7, or 9.
To tell whether an integer is even or odd, look at the digit in the ones place. That single digit will tell you whether the entire integer is odd or even. For example, the integer 3,255 is an odd
integer because it ends in 5, an odd integer. Likewise, 702 is an even integer because it ends in 2.
The following table shows the operations with even and odd integers.
Prime Numbers
A prime number is a positive integer that has exactly two factors, 1 and itself, for example 29 has exactly two factors which are 1 and 29. So 29 is a prime number.
On the other hand, 28 has six factors which are 1, 2, 4, 7, 14, and 28. So 28 is not a prime number. It is called a composite number. Some examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19,
… Since the number 1 has only one factor (namely 1 itself), it is not a prime number.
The number 2 is the only prime that is even. Other even numbers will have 2 have as a factor and so will not be a prime.
A number that is not prime is called a composite number.
Properties Of Integers
The following are some of the properties of integers. Scroll down the page for more examples and explanations of the different properties of integers.
Operations With Even And Odd Numbers
Add two even numbers and the result is even.
Add two odd numbers and the result is even.
Add one even and one odd and the result is odd.
Multiply two even numbers and the result is even.
Multiply two odd numbers and the result is odd.
Multiply one even and one odd and the result is even.
How To Distinguish Prime Numbers?
A prime number is a number greater than 1, which is only divisible by 1 and itself.
More Properties Of Integers
How to identify properties of Integers?
A property is a math rule that is always true.
Commutative Property for Addition, Associative Property for Addition, Distributive Property, Identity Property for Addition, Identity Property for Multiplication, Inverse Property for Addition and
Zero Property for Multiplication.
Properties Of Integers
Three properties of integers are explained. Additive Identity, Additive Inverse, Opposite of a negative is positive. Examples are provided.
1. Additive Identity: Adding 0 to any integer does not change the value of the integer.
2. Additive Inverse: Each integer has an opposing number (opposite sign). When you add a number and its additive inverse, you get 0.
3. The opposite of a negative is a positive.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/integers.html","timestamp":"2024-11-04T07:29:28Z","content_type":"text/html","content_length":"44945","record_id":"<urn:uuid:29e271c3-e8a5-4df8-bd5c-828e99dccde3>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00084.warc.gz"} |
Modeling x-ray laser gain in recombining plasmas
Optimal conditions for lasing in H-like and Li-like aluminum as well as in He-like silicon are examined in the context of recombining plasmas. Simulations are carried out for the free expansion of
initially hot, dense, and thin cylinders of aluminum and silicon plasmas. Conditions generated from these simulations are input into a simple gain model, yielding information on the state of the
plasma variables which maximizes the gain coefficient of a particular lasing transition. The scaling of the maximum gain coefficient with the initial plasma diameter and with the ratio of lasant ions
to coolant ions is done for the 3d-2p singlet to singlet transition in helium-like silicon. The theoretical theory and numerical aspects of the principal components of plasma modeling, namely
ionization dynamics (ID), magnetohydrodynamics (MHD), and radiative transfer (RT) are presented. The MHD algorithm uses a two temperature Lagrangian gridding method. The ID algorithm utilizes the
Collisional-Radiative (CR) model and can integrate all level populations with a stiff differential equation solver or solve by matrix inversion when CR equilibrium is valid. The atomic processes
include dielectronic recombination, photoexcitation, photoionization, collisional excitation and de-excitation, three-body and radiative recombination, collisional ionization, and radiative decay. An
escape probability formalism is used for the transfer of bound-bound, bound-free, and free-free radiation in the RT model.
Ph.D. Thesis
Pub Date:
□ Laser Outputs;
□ Mathematical Models;
□ Metallic Plasmas;
□ Plasma Physics;
□ Power Gain;
□ Radiative Transfer;
□ Recombination Reactions;
□ X Ray Lasers;
□ Algorithms;
□ Aluminum;
□ Ionization;
□ Ions;
□ Magnetohydrodynamics;
□ Photoionization;
□ Plasma Dynamics;
□ Probability Theory;
□ Silicon;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1990PhDT.........9R/abstract","timestamp":"2024-11-03T19:35:42Z","content_type":"text/html","content_length":"38562","record_id":"<urn:uuid:82828c1c-6cb2-4cb7-8c05-81598b720104>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00737.warc.gz"} |
Squares, primes and the numerical stratosphere
This page concerns an unassuming little maths problem, which I decided to look into for a bit of fun. To my surprise, the question turned out to have an unexpected connection to a famous problem at
the cutting edge of number theory, and necessitated a trip off into the numerical stratosphere in search of some fairly large solutions. The maths used here is probably about first year undergraduate
level, and the style is certainly more technical than my previous article, so come prepared.
While working on my follow up to the Pointless maths article, I became sidetracked by a nice little conundrum posted on Twitter:
Can a perfect square ever consist of the same string of digits written out twice (such as in 978978 or 46024602)? cc @jamestanton
— Matt Enlow (@CmonMattTHINK) December 4, 2014
This is exactly the sort of meaningless mathematical challenge that I enjoy, so let’s get stuck in.
Before we start, we need some notation. Let’s call the number that we are going to square x (we’re assuming that it’s a positive integer) and we’ll say that x has d digits. We want to find x such
that x^2 is equal to one of the twice-repeated string numbers described in the tweet.
First, let’s consider how large we should expect x^2 to be, in terms of its number of digits. Since x has d digits, it satisfies the inequalities:
10^d-1 ≤ x < 10^d
Since (10^d-1)^2 = 10^2d-2, which is the first integer with 2d-1 digits, and (10^d)^2 = 10^2d, which is the first integer with 2d+1 digits, x^2 must have 2d-1 or 2d digits. Moreover, if x^2 is to
have a twice-repeated string of digits, as we wish, then it must have an even number of digits. Therefore, if we are able to find a solution x to our problem with d digits, then x^2 will have to have
2d digits. This means that the string that will be repeated will have d digits, the same number as in x.
Now let’s think a little more about these numbers with repeating digits that we are interested in. For example, consider the number 345345. This number can be split up in the following way:
345345 = 345×10^3 + 345 = 345 (10^3 + 1)
Any such twice-repeated string number can be rewritten in this way. If we let z be the complete repeating number and y be the number formed by the string that is repeated (e.g. z = 345345 and y =
345 in the above example) and if d is the number of digits of y, then we can write:
z = y (10^d + 1)
This allows us to state our original problem symbolically. We wish to find a positive integer x, with d digits, such that:
x^2 = y (10^d + 1) [Equation 1]
for some positive integer y, which must also have d digits. If we can find such numbers, then we are done, because the left hand side will definitely be a perfect square and the right hand side will
definitely be a twice-repeating string number. However, it may turn out that no such numbers exist, in which case we would like to find a rigorous proof that the problem is impossible so that we can
all stop worrying about it and sleep easily in our beds.*
Let’s suppose that a solution does exist and try to work out what it would look like. Note in particular that 10^d + 1 has d + 1 digits, so it must be larger than x. Hold on to that thought, because
in a minute it will prove to be a crucially important fact.
The fundamental theorem of arithmetic tells us that every whole number can be expressed as a unique product of primes, so let’s write 10^d + 1 in this way:
10^d + 1 = p[1]^a[1] p[2]^a[2] … p[n]^a[n]
Here, p[1] , p[2] , … , p[n] are distinct primes, each raised to some positive integer power
a[1] , a[2] , … , a[n] . Incidentally, we know that the smallest of these primes is at least 7, because it is quite easy to show that 10^d + 1 is not divisible by 2, 3 or 5.
Now, Euclid’s Lemma tells us that if a prime number divides a product of two numbers then it must divide one or both of these numbers. Here then, we have that each of the primes
p[1] , p[2] , … , p[n] divides 10^d + 1, and from Equation 1 we see that 10^d + 1 divides x^2. Therefore all of the primes p[1] , p[2] , … , p[n] divide x^2, which is the product of x and x.
Therefore, all of the primes p[1] , p[2] , … , p[n] divide x, so x = w p[1] p[2] … p[n] for some positive integer w.
Suppose for a moment that every one of these primes appeared only once in the prime factorisation of 10^d + 1. In other words, a[1] = a[2] = … = a[n] = 1 and we can write
10^d + 1 = p[1] p[2] … p[n] . This would mean that x = w (10^d + 1), but since w is positive, this implies that x ≥ 10^d + 1. However, the crucially important fact that we flagged up earlier told us
that actually 10^d + 1 > x. We therefore have a contradiction and so, if we have a solution, at least one of the primes in the factorisation of 10^d + 1 must appear twice or more.
At this point, we can take a breather from the heavy analysis to investigate some actual numbers of the form 10^d + 1, to see what their prime factorisations actually look like. These numbers are
simply those that start and end with a 1, with 0s in between. Here are the first few of them, with their prime factorisations (which I got from here):
• 10^1 + 1 = 11 (prime)
• 10^2 + 1 = 101 (prime)
• 10^3 + 1 = 1001 = 7 × 11 × 13
• 10^4 + 1 = 10001 = 73 × 137
• 10^5 + 1 = 100001 = 11 × 9091
• 10^6 + 1 = 1000001 =101 × 9901
• 10^7 + 1 = 10000001 =11 × 909091
• 10^8 + 1 = 100000001 =17 × 5882353
• 10^9 + 1 = 1000000001 = 7 × 11 × 13 × 19 × 52579
• 10^10 + 1 = 10000000001 = 101 × 3541 × 27961
• 10^11 + 1 = 100000000001 = 11^2 × 23 × 4093 × 8779
Finally, we have arrived at a number of the form 10^d + 1 with a repeated prime factor, which is what we were looking for. I am quite surprised that it took so long, and it would be interesting to
know whether there is a particular reason why these numbers seem to have so few repeated prime factors. Things don’t get any better in this respect either, since you have to wait until
10^21 + 1 = 1000000000000000000001 = 7^2 × 11 × 13 × 127 × 2689 × 459691 × 909091 for the next one.**
Anyway, the repeated factor of 11 in 10^11 + 1 gives us hope, since it suggests that we might be able to find some 11 digit solutions for x. We know that such an x must be divisible by all the
distinct prime factors of 10^11 + 1, so x will have to be a multiple of 11 × 23 × 4093 × 8779, which is equal to 9090909091. The first few multiples of this number provide us with the following “near
• 9090909091^2 = 82644628100826446281
• 18181818182^2 = 330578512403305785124
• 27272727273^2 = 743801652907438016529
Note that these examples fail because the corresponding value of y (from Equation 1), the string of repeated numbers, has fewer than 11 digits, rendering our analysis invalid. However, the next seven
multiples of 9090909091 all provide solutions to our problem:
• 36363636364^2 = 1322314049613223140496
• 45454545455^2 = 2066115702520661157025
• 54545454546^2 = 2975206611629752066116
• 63636363637^2 = 4049586776940495867769
• 72727272728^2 = 5289256198452892561984
• 81818181819^2 = 6694214876166942148761
• 90909090910^2 = 8264462810082644628100
At this point, we run out of 11 digit solutions, as x and the corresponding y both tip over into twelve digits (in fact both x and y become equal to 10^11 + 1) and our method stops working:
• 100000000001^2 = 10000000000200000000001
As we have already mentioned, we have to wait a long time for the next solutions to turn up, some 21-digit monsters. They are all multiples of 142857142857142857143, a number which rather beautifully
contains three (and a bit) repetitions of the recurring part of the decimal expansion of 1/7 (looking back at our earlier solutions, they were clearly related to the decimal expansion of 1/11). This
time there are only four solutions before the method breaks down, but they are rather lovely ones, borrowing the property of the decimal expansion of 1/7, whereby they are all (sort of) translated
versions of one another (with the exception of the final digit in each case):
• 428571428571428571429^2 = 183673469387755102041183673469387755102041
• 571428571428571428572^2 = 326530612244897959184326530612244897959184
• 714285714285714285715^2 = 510204081632653061225510204081632653061225
• 857142857142857142858^2 = 734693877551020408164734693877551020408164
Given how we derived these numbers, it is not too surprising that they are related to the decimal expansion of 1/7, since the number 142857142857142857143, of which these solutions are multiples, was
derived by dividing the excess factor of 7 out of 1000000000000000000001. Nevertheless, it is quite an attractive and striking observation.
It would be interesting to look into these solutions and how frequently they occur in more depth, but, given that they seem to be tied in with a particularly mysterious unsolved*** problem in number
theory (see the footnotes), I think pursuing this any further would likely drive me completely mad. Aside from anything else, there are practical difficulties when working with such big numbers. I
looked for solutions with up to 29 digits, by checking the factorisations of 10^d + 1 for repeated primes, but the online factoring calculator couldn’t go any higher than that. I believe, therefore,
that the eleven solutions on this page (seven with 11 digits, four with 21 digits) are the only solutions for which the number to be squared is less than 10^29 (one hundred thousand trillion
trillion, or one hundred octillion). That means that there are only eleven perfect squares consisting of a twice repeated string of digits that are smaller than 10^58.
Let’s just finish by marvelling at the size of this number. There are only eleven perfect squares of this kind that are less than:
ten billion trillion trillion trillion trillion
ten octodecillion
Nice problem.
UPDATE (04/2015): Using Wolfram Alpha and the Number Empire online factoriser, I was able to extend the search up to 126 digit squares, revealing further families of solutions, including:
Seven 66 digit squares:
• 363636363636363636363636363636364^2 = 132231404958677685950413223140496132231404958677685950413223140496
• 454545454545454545454545454545455^2 = 206611570247933884297520661157025206611570247933884297520661157025
• 545454545454545454545454545454546^2 = 297520661157024793388429752066116297520661157024793388429752066116
• 636363636363636363636363636363637^2 = 404958677685950413223140495867769404958677685950413223140495867769
• 727272727272727272727272727272728^2 = 528925619834710743801652892561984528925619834710743801652892561984
• 818181818181818181818181818181819^2 = 669421487603305785123966942148761669421487603305785123966942148761
• 909090909090909090909090909090910^2 = 826446280991735537190082644628100826446280991735537190082644628100
Eight 78 digit squares:
• 384615384615384615384615384615384615385^2 = 147928994082840236686390532544378698225147928994082840236686390532544378698225
• 461538461538461538461538461538461538462^2 = 213017751479289940828402366863905325444213017751479289940828402366863905325444
• 538461538461538461538461538461538461539^2 = 289940828402366863905325443786982248521289940828402366863905325443786982248521
• 615384615384615384615384615384615384616^2 = 378698224852071005917159763313609467456378698224852071005917159763313609467456
• 692307692307692307692307692307692307693^2 = 479289940828402366863905325443786982249479289940828402366863905325443786982249
• 769230769230769230769230769230769230770^2 = 591715976331360946745562130177514792900591715976331360946745562130177514792900
• 846153846153846153846153846153846153847^2 = 715976331360946745562130177514792899409715976331360946745562130177514792899409
• 923076923076923076923076923076923076924^2 = 852071005917159763313609467455621301776852071005917159763313609467455621301776
There are a further seven 110 digit solutions and four 126 digit solutions, but let’s draw the line at 100 digits. Attempts to find even larger squares were thwarted by an inability to factorise 10^
64 + 1 (and by the fact that you have to stop somewhere!).
Together with the eleven solutions that we had already found, this allows to replace our earlier conclusion with the eminently more satisfying statement:
There are only twenty-six perfect squares of this kind smaller than a googol.
* If we are really unlucky, it may turn out to be impossible to find a solution and impossible to prove that no solution exists (see Gödel’s Incompleteness Theorems), but we would have had to have
kicked a lot of black cats under a lot of ladders to fall prey to that one.
** Fascinatingly, I think that the reason that so few numbers of the form 10^d + 1 seem to have repeated prime factors and, consequently, why there are so few solutions to this problem (or that the
solutions are so very sparsely distributed, at any rate) may have something to do with the abc conjecture, which I was aware of, but had never actually bumped into ‘in the wild’ before.
Very loosely (and I’m no expert on this), the abc conjecture says that, given coprime positive integers a, b, c, such that a + b = c , c tends not to exceed the product of all the distinct prime
factors of a, b, c by too much (in the sense that there are only a finite number of triples a, b, c for which c exceeds this product by a certain degree). As I understand it, this (sort of) has the
effect of preventing the prime factors of c from being raised to high powers in its prime decomposition, since this would illegally push up c relative to the aforementioned product.
The expression 10^d + 1 = C appears to be an example of the abc conjecture in action. The three terms are certainly coprime and as d increases, so does C. The effect, based on the abc conjecture,
seems to be that the powers of the primes in the factorisation of C must remain low, to prevent C from excessively exceeding the product of its distinct prime factors with the 2 and 5 from the 10^d
I certainly had no idea when I was starting out that the problem would be related to this result, so I really find this quite amazing!
*** Unless you are an expert in Inter-universal Teichmüller Theory of course.
Thomas Oléron Evans, 2014
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One thought on “Squares, primes and the numerical stratosphere”
1. Matt E
So glad you derived so much fun from this problem! Cheers!
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Efficient Graph Minors Theory and Parameterized Algorithms for (Planar) Disjoint Paths
In the Disjoint Paths problem, the input consists of an n-vertex graph G and a collection of k vertex pairs, (formula presented), and the objective is to determine whether there exists a collection
(formula presented) of k pairwise vertex-disjoint paths in G where the end-vertices of (formula presented) are (formula presented) and (formula presented). This problem was shown to admit an (formula
presented)-time algorithm by Robertson and Seymour Graph Minors XIII, The Disjoint Paths Problem, JCTB. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT) with
respect to k. Remarkably, the above algorithm for Disjoint Paths is a cornerstone of the entire Graph Minors Theory, and conceptually vital to the (formula presented)-time algorithm for Minor Testing
(given two undirected graphs, G and H on n and k vertices, respectively, determine whether G contains H as a minor). In this semi-survey, we will first give an exposition of the Graph Minors Theory
with emphasis on efficiency from the viewpoint of Parameterized Complexity. Secondly, we will review the state of the art with respect to the Disjoint Paths and Planar Disjoint Paths problems.
Lastly, we will discuss the main ideas behind a new algorithm that combines treewidth reduction and an algebraic approach to solve Planar Disjoint Paths in time (formula presented) (for undirected
Original language English
Title of host publication Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Publisher Springer
Pages 112-128
Number of pages 17
State Published - 1 Jan 2020
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 12160 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
• Disjoint paths
• Graph minors
• Planar disjoint paths
• Treewidth
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
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$$\notag \newcommand{\E}{\mathbb{E}} \newcommand{\la}{\!\leftarrow\!} \newcommand{\noo}[1]{\frac{1}{#1}} \newcommand{\ra}{\!\rightarrow\!} \newcommand{\te}{\!=\!} \newcommand{\tm}{\!-\!} \newcommand
{\ttimes}{\!\times\!} $$
Expectations and sums of variables
You are expected to know some probability theory, including expectations/averages. This sheet reviews some of that background.
The notation on this sheet follows MacKay’s textbook, available online here:
An outcome, \(x\), comes from a discrete set or ‘alphabet’ \(\mathcal{A}_X = \{a_1,a_2,\dots, a_I\}\), with corresponding probabilities \(\mathcal{P}_X = \{p_1,p_2, \dots, p_I\}\).
A standard six-sided die has \(\mathcal{A}_X = \{1,2,3,4,5,6\}\) with corresponding probabilities \(\mathcal{P}_X = \{\noo{6},\noo{6},\noo{6},\noo{6},\noo{6},\noo{6}\}\).
A Bernoulli distribution, which has probability distribution \[ P(x) = \begin{cases} p & x=1,\\ 1-p & x=0,\\ 0 & \text{otherwise,} \end{cases} \] has alphabet \(\mathcal{A}_X = \{1,0\}\) with \(\
mathcal{P}_X = \{p,1\tm p\}\).
An expectation is a property of a probability distribution, defined by a probability-weighted sum. The expectation of some function, \(f\), of an outcome, \(x\), is: \[ \E_{P(x)}[f(x)] = \sum_{i=1}^I
p_i f(a_i). \] Often the subscript \(P(x)\) is dropped from the notation because the reader knows under which distribution the expectation is being taken. Notation can vary considerably, and details
are often dropped. You might also see \(E[f]\), \(\mathcal{E}[f]\), or \(\langle f\rangle\), which all mean the same thing.
The expectation is sometimes a useful representative value of a random function value. The expectation of the identity function, \(f(x)\te x\), is the ‘mean’, which is one measure of the centre of a
The expectation is a linear operator: \[ \E[f(x) + g(x)] = \E[f(x)] + \E[g(x)] \quad \text{and}\quad \E[cf(x)] = c\E[f(x)]. \] These properties are apparent if you explicitly write out the
The expectation of a constant with respect to \(x\) is the constant: \[ \E[c] = c\sum_{i=1}^I p_i = c, \] because probability distributions sum to one (‘probabilities are normalized’).
The expectation of independent outcomes separate: \[ \E[f(x)g(y)] = \E[f(x)]\,\E[g(y)]. \] True if \(x\) and \(y\) are independent.
Exercise 1: prove this. (Answers at the end of the note.)
The mean of a distribution over a number, is simply the ‘expected’ value of the numerical outcome. \[ \text{‘Expected Value'} = \text{‘mean'} = \mu = \E[x] = \sum_{i=1}^I p_i a_i. \] For a six-sided
die: \[ \E[x] = \frac{1}{6}\ttimes1 + \frac{1}{6}\ttimes2 + \frac{1}{6}\ttimes3 + \frac{1}{6}\ttimes4 + \frac{1}{6}\ttimes5 + \frac{1}{6}\ttimes6 = 3.5. \] In every day language I wouldn’t say that I
‘expect’ to see 3.5 as the outcome of throwing a die… I expect to see an integer! However, 3.5 is the ‘expected value’ as it is commonly defined. Similarly a single Bernoulli outcome will be a zero
or a one, but its ‘expected’ value is a fraction, \[ \E[x] = p\ttimes1 + (1\tm p)\ttimes0 = p, \] the probability of getting a one.
Change of units: I might have a distribution over heights measured in metres, for which I have computed the mean. If I multiply the heights by 100 to obtain heights in centimetres, the mean in
centimetres can be obtained by multiplying the mean in metres by 100. Formally: \(\E[100\,x] = 100\,\E[x]\).
The variance is also an expectation, measuring the average squared distance from the mean: \[ \text{var}[x] = \sigma^2 = \E[(x-\mu)^2] = \E[x^2] - \E[x]^2, \] where \(\mu\te\E[x]\) is the mean.
Exercise 2: prove that \(\E[(x-\mu)^2] = \E[x^2] - \E[x]^2\).
Exercise 3: show that \(\text{var}[cx] = c^2\,\text{var}[x]\).
Exercise 4: show that \(\text{var}[x + y] = \text{var}[x] + \text{var}[y]\), for independent outcomes \(x\) and \(y\).
Exercise 5: Given outcomes distributed with mean \(\mu\) and variance \(\sigma^2\), how could you shift and scale them to have mean zero and variance one?
Change of units: If the outcome \(x\) is a height measured in metres, then \(x^2\) has units \(\mathrm{m}^2\); \(x^2\) is an area. The variance also has units \(\mathrm{m}^2\), it cannot be
represented on the same scale as the outcome, because it has different units. If you multiply all heights by 100 to convert to centimetres, the variance is multiplied by \(100^2\). Therefore, the
relative size of the mean and the variance depends on the units you use, and so often isn’t meaningful.
Standard deviation: The standard deviation \(\sigma\), the square root of the variance, does have the same units as the mean. The standard deviation is often used as a measure of the typical distance
from the mean. Often variances are used in intermediate calculations because they are easier to deal with: it is variances that add (as in Exercise 4 above), not standard deviations.
A drunkard starts at the centre of an alleyway, with exits at each end. He takes a sequence of random staggers either to the left or right along the alleyway. His position after \(N\) steps is \(k_N
= \sum_{n=1}^N x_n\), where the outcomes, \(\{x_n\}\), the staggering motions, are drawn from a distribution with zero mean and finite variance \(\sigma^2\). For example \(\mathcal{A}_X = \{-1,+1\}\)
with \(\mathcal{P}_X = \{\noo{2},\noo{2}\}\), which has \(\E[x_n]\te0\) and \(\text{var}[x_n]\te1\).
If the drunkard started in the centre of the alleyway, will he ever escape? If so, roughly how long will it take? (If you don’t already know, have a think…)
The expected, or mean position after \(N\) steps is \(\E[k_N] = N\E[x_n] = 0\). This doesn’t mean we don’t think the drunkard will escape. There are ways of escaping both left and right, it’s just
‘on average’ that he’ll stay in the middle.
The variance of the drunkard’s position is \(\text{var}[k_N] = N\text{var}[x_n] = N\sigma^2\). The standard deviation of the position is then \(\text{std}[k_N] = \sqrt{N}\sigma\), which is a measure
of the width of the distribution over the displacement from the centre of the alleyway. If we double the length of the alley, then it will typically take four times the number of random steps to
Worthwhile remembering: the typical magnitude of the sum of \(N\) independent zero-mean variables scales with \(\sqrt{N}\). The individual variables need to have finite variance, and ‘typical
magnitude’ is measured by standard deviation. Sometimes you might have to work out the \(\sigma\) for your problem, or do other detailed calculations. But sometimes the scaling of the width of the
distribution is all that really matters.
Corollary: the typical magnitude of the mean of \(N\) independent zero-mean variables with finite variance scales with \(1/\sqrt{N}\).
As always, you are strongly recommended to work hard on a problem yourself before looking at the solutions. As you transition into doing research, there won’t be any answers, and you have to build
confidence in getting and checking your own answers.
Exercise 1: For independent outcomes \(x\) and \(y\), \(p(x,y)\te p(x)p(y)\) and so
\(\E[f(x)g(y)] = \sum_x\sum_y p(x)p(y)f(x)g(y) = \sum_x p(x)f(x) \sum_y p(y)g(y) = \E[f(x)]\E[g(y)]\).
Exercise 2: \(\E[(x-\mu)^2] = \E[x^2 + \mu^2 - 2x\mu] = \E[x^2] + \mu^2 - 2\mu\E[x] = \E[x^2] - \mu^2\).
Exercise 3: \(\text{var}[cx] = \E[(cx)^2] -\E[cx]^2 = \E[c^2x^2] - (c\E[x])^2 = c^2(\E[x^2] - \E[x]^2) = c^2\text{var}[x]\).
Exercise 4: \(\text{var}[x + y] = \E[(x+y)^2] - \E[x+y]^2 = \E[x^2] + \E[y^2] + 2\E[xy] - (\E[x]^2 + \E[y]^2 + 2\E[x]\E[y]) = \text{var}[x] + \text{var}[y]\), if \(\E[xy]\te\E[x]\E[y]\), true if \(x
\) and \(y\) are independent variables.
Exercise 5: \(z = (x-\mu)/\sigma\) has mean 0 and variance 1. The division is by the standard deviation, not the variance. You should now be able to prove this result for yourself.
What to remember: using the expectation notation where possible, rather than writing out the summations or integrals explicitly, makes the mathematics concise. | {"url":"https://www.inf.ed.ac.uk/teaching/courses/mlpr/2019/notes/w0f_expectations.html","timestamp":"2024-11-02T15:06:28Z","content_type":"text/html","content_length":"17074","record_id":"<urn:uuid:ff63ea83-f26c-4216-b89f-443f95504f0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00626.warc.gz"} |
ESL Booknotes
Booknotes ESL
Chapter2 Overview of Supervised Learning
2.1 Variable Types and Terminology
Variable Types: quantitative or qualitative； Corresponding tasks: regression or classification; the 3rd type is ordered categorical (low, mid, high).
2.2 Least Squares and Nearest Neighbors
1. Least Squares
$\hat{Y}=\hat{\beta}_0 + \sum(X_j\hat{\beta}_j)$
$\hat{\beta}_0$ is the intercept or bias.
$RSS(\beta) = (\mathbf{y}-\mathbf{X}\beta)^T(\mathbf{y}-\mathbf{X}\beta)$ --> $\hat{\beta} = (\mathbf{X^TX})^{-1}\mathbf{X}^T\mathbf{y}$
2. KNN
3. LS: low variance and potentially high bias; KNN: high variance and low bias.
Thecniques to improve
1. KNN with kernal methods, closer->heavier
2. In high-dimensional spaces the distance kernels are modified to emphasizesome variable more than others.
3. Local regression fits linear models by locally weighted least squares, rather than fitting constants locally.
4. Linear models fit to a basis expansion of the original inputs allow arbitrarily complex models. ($x^2, log(x)$ etc)
5. Projection pursuit and neural network models consist of sums of nonlinearly transformed linear models.
2.3 Statistical Decision Theory
1. minimum square error loss (expected prediction error: $EPE(f)=E(Y-f(x))^2$)
---(by conditioning on X)----> $f(x)=E(Y|X=x)$
2. KNN use $\hat{f}(x)=Ave(y_i|x_i\in N_k(s))$, with expectation is approximated by averaging over sample data, and conditioning at a point is relaxed to conditioning on some region “close” to the
target point.
3. In fact, under mild regularity conditions on the joint probability distribution $Pr(X,Y)$, one can show that as $N,K \rightarrow \inf, s.t. k/N \rightarrow 0, \hat{f}(x) \rightarrow E(Y|X=x)$,
convergence rate drops when dimensions raise.
4. For LM, $EPE$ leads to $\beta = [E(X^TX)]^{-1}E(XY)$
5. if use L1 norm $f(x)=median(Y|X=x)$
6. when the output is a categorical variable G, use a matrix L to denote the error. After condition, $EPE = E_X(\sum_{k=1}^KL[\mathcal{G}_k, \hat{G}(X)]Pr(\mathcal{G}_k|X))$. --> $\hat{G}(x) =
argmin_{g\in \mathcal{G}}\sum_{k=1}^KL[\mathcal{G}_k, g]Pr(\mathcal{G}_k|X=x))$. With 0-1 rule, it is the Bayes classifier, $\hat{G}(x) = max_{g∈G} Pr(g|X = x).$. The error rate of the Bayes
classifier is called
the Bayes rate.
2.4 Local Methods in High Dimensions
Problems in High dimension space:
1. Such neighborhoods are no longer “local.” In order to caputer the same fraction number of neighbors, the average distance increases exponantially with the degree of dimension.
2. Another consequence of the sparse sampling in high dimensions is that all sample points are close to an edge of the sample
2.5 Classes of Restricted Estimators
Roughness Penalty and Bayesian Methods
$PRSS(f; λ) = RSS(f) + λJ(f)$
e.g. cubic smoothing spline: $J(f)=\int f''(x)^2dx$
Can be cast in a Bayesian framework: The penalty J corresponds
to a log-prior, and PRSS(f; λ) the log-posterior distribution.
Kernel Methods and Local Regression
Basis Functions and Dictionary Methods
These adaptively chosen basis function methods are also known as dictionary methods, where one has available a possibly infinite set or dictionary D of candidate basis functions from which to choose,
and models are built up by employing some kind of search mechanism.
Chapter3 Linear Methods for Regression
3.1 Linear Regression Models and Least Squares
variables Xj can come from different sources:
- quantitative inputs;
- transformations of quantitative inputs, such as log, square-root or square
- basis expansions, such as X2 = X1^2, leading to a polynomial representation
- numeric or “dummy” coding of the levels of qualitative inputs
- interactions between variables
$H = X(X^TX)^{-1}X^Ty$
The non-full-rank case occurs most often when one or more qualitative inputs are coded in a redundant fashion. There is usually a natural way to resolve the non-unique representation, by recoding and
/or dropping redundant columns in X.
Model Significancy: F-Statistics
The Gauss–Markov Theorem: the least squares estimates of the parameters β have the smallest variance among all linear unbiased estimates.
If the inputs $X = (x_1,x_2,...,x_p)$, $<x_i,x_j> = 0, \ for\ i eq j$. Then $\hat{\beta_i} = \frac{<x_i,y>}{<x_i,x_i>}$, by looking at $(X^TX)^{-1}X^Ty$. Which leads to Regression by Successive
1. z0 = x0 = 1;
2. For j = 1,2,...,p:
3. Regress xj on z0,...z(j-1)
4. Regress y on zp to get beta_p
If $x_p$ is highly related to other $x_k$, $z_p$ will be too small and $\hat{\beta}_p$ will very unstable.
$Var(\hat{\beta}_p) = \frac{\sigma^2}{||z_p||^2}$
3.2 Subset Selection
improving accuracy: the least squares estimates often have low bias but large variance. Prediction accuracy can sometimes be improved by shrinking or setting some coefficients to zero. By doing so we
sacrifice a little bit of bias to reduce the variance of the predicted values, and hence may improve the overall prediction accuracy
Interpretation: With a large number of predictors, we often would like to determine a smaller subset that exhibit the strongest effects. In order to get the “big picture,” we are willing to sacrifice
some of the small details.
Best-Subset Selection:
leaps and bounds procedure
Forward- and Backward-Stepwise Selection:
Forwardstepwise (greedy)
selection starts with the intercept, and then sequentially adds into the model the predictor that most improves the fit. With many candidate predictors, this might seem like a lot of computation;
however, clever updating algorithms can exploit the QR decomposition for the current fit to rapidly establish the next candidate (Exercise 3.9).
Compared to BSS, a price is paid in variance for selecting the best subset of each size; forward stepwise is a more constrained search, and will have lower variance, but perhaps more bias.
Backward-stepwise selection
starts with the full model, and sequentially deletes the predictor that has the least impact on the fit. The candidate for dropping is the variable with the smallest Z-score
Forward-Stagewise Regression
Like forward-stepwise regression. At each step the algorithm identifies the variable most correlated with the current residual.
Unlike forward-stepwise regression, none of the other variables are adjusted when a term is added to the model. As a consequence, forward stagewise can take many more than p steps to reach the least
squares fit, and historically has been dismissed as being inefficient. It turns out that this “slow fitting” can pay dividends in high-dimensional problems.
3.3 Shrinkage Methods
Shrinkage methods are more continuous, and don’t suffer as much from high variability.
Ridge Regression
Just add penalty $\lambda ||\beta||_2^2$
Called weight decay in neural network. | {"url":"https://www.zybuluo.com/HaomingJiang/note/854183","timestamp":"2024-11-13T02:57:07Z","content_type":"text/html","content_length":"114121","record_id":"<urn:uuid:1a6998fd-182e-4e36-ab57-18bf63e62ef0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00818.warc.gz"} |
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Lessons learned from research classrooms
by George Gadanidis, Western University
0. Introduction
Wonder is a feeling of surprise, combined with a sense of beauty or admiration.
Wonder makes us curious, attentive and eager to learn.
Where do odd numbers hide?
We have witnessed mathematical wonder in children in grades 1-2, as they noticed Odd numbers hide in squares! Then they wondered, Where do even numbers hide?
Below, based on many years of collaborating and co-teaching in research classrooms, I share what we have learned about helping students experience mathematical wonder.
1. WONDER IN MATHEMATICS
Mathematics can be full of wonder!
However, the mathematics most of us have experienced in school lacked surprise and beauty, and it did not engage our curiosity.
For example, when we ask students and teachers what they know about parallel lines, they typically state that “parallel lines are straight and they never meet.”
Interestingly, about 2,300 years ago, the mathematician Euclid tried and could not to prove that parallel never meet. Neither could other mathematicians for the next 2,000 years. How could this be?
It turns out that “parallel lines never meet” is not a theorem to be proven. It is an assumption.
• If we assume parallel lines never meet, we live on a flat surface.
• If we assume they do meet, we live on a spherical or elliptical surface.
• Different assumptions lead to different geometries!
In which geometry do you live?
To see parallel lines meet, solve this riddle:
• Molly steps out of her tent.
• She walks south 1 km.
• She walks west 1 km.
• Sees a bear.
• She gets scared runs north 1 km, and somehow she finds herself back at her tent.
• How can this be?
• And what colour was the bear?
Here is how some prospective teachers reacted when learning that parallel lines are not as simple as “they never meet”:
I feel like I was misled, misguided, told the half-truth about parallel lines. It is the first time that I have realized and felt that math isn’t just BLACK & WHITE and can cause quite creative
outcomes and discussions. (Gadanidis & Namukasa, 2007)
How is it possible that students come to believe parallel lines never meet, when they live on a sphere?
The geometry of parallel lines can be full of wonder.
Consider the spherical geometry of our world”
• Are lines of latitude parallel?
• Are lines of longitude parallel?
• What is a straight line on a sphere? Is it a line of latitude or a line of longitude?
□ What’s a great circle and how many are there on a sphere?
What is “straight”? What is “parallel”?
• If a bug walks a balanced walk on a sphere, what would be its path?
• What paths do airplanes fly to travel the shortest distance possible?
• What is the sum of the angles of a triangle on a sphere?
• What other geometries are possible?
Wonder flexes our imagination, makes us curious and attentive, and motivates us to learn.
What mathematical wonders will students experience in your classroom?
• ^Gadanidis, G. & Namukasa, I. (2007). Mathematics-for-teachers (and students). Journal of Teaching and Learning, 5(1), 13-22.
Wonder becomes possible when we add depth to mathematics we bring to the classroom.
Let’s consider this typical textbook problem:
A worker plans to install hardwood flooring in a room 3.2 m by 4.5 m. Each bundle of hardwood covers an area of 1.8 m^2. He estimates a waste of 10%. How many bundles should he purchase?
This problem is solved using simple calculations. All measurements are constant. The mathematics is static and shallow.
To add mathematical depth, we may design a different problem, one that involves both area and volume, and investigate how these quantities change in relation to one another.
For example: How do surface area and volume change in relation to one another as a cube grows in size?
Investigating this relationship may also helps us understand why elephants have big ears.
Why do elephants have big ears?
What do you think?
Model an elephant as a cube
Let’s measure the volume and surface area of elephants of various sizes and see what patterns emerge.
To keep calculations simple, let’s model an elephant as a cube.
A small elephant:
• The dimensions are 1 x 1 x 1.
• V = 1 cubic unit.
• SA = 6 square units.
A bigger elephant:
• The dimensions are 2 x 2 x 2.
• V = 2 × 2 × 2 = 8 cubic units.
• SA = 6 × 4 = 24 square units.
An even bigger elephant:
• The dimensions are 3 × 3 × 3.
• V = 3 × 3 × 3 = 27 cubic units.
• SA = 6 × 3 × 3 = 54 square units.
Look for relationships
Let’s record measurements for the first 10 elephants in the table below.
• What is the relationship between SA and V?
• Notice how SA/V and V/SA change.
• What is the meaning of each of these ratios?
What does this have to do with elephants and their ears?
Plot on a graph
We may also display the data graphically.
Comparing SA and V
These scatter plots compare the growth patterns of side length, SA and V for cubes with side lengths 1-10.
• Which scatter plot represents SA? Green, red or blue? How can you tell?
Comparing SA/V and V/SA
These scatter plots compare SA/V and V/SA.
• Which scatter plot represents SA/V? How can you tell?
• Why is the blue scatter plot a straight line of points?
Why do elephants have big ears?
Consider connections & extensions
We may also consider some related problems and extensions.
Which will evaporate first?
The containers have the same amount of water. Which will evaporate first? Why?
Which will dry first?
Both sponges are soaked with water. Which will dry first? Why?
Puppies in cars
Why is it unsafe to leave a puppy (or a young child) in a car on a hot summer day?
3. a low floor and a high ceiling
A low floor allows students to engage with minimum prerequisite knowledge.
A high ceiling offers students opportunities to investigate more complex concepts and relationships.
Here is one way we created a low floor and a high ceiling for the concepts of infinity and limit, which are typically studied in Calculus.
In grade 3, students usually learn area representations of fractions.
This creates a context for engaging students with infinity and limit.
Using squares of the same size, students shade to represent the fractions 1/2, 1/4, 1/8 and 1/16. This pattern can continue forever. It is infinite.
Area representations of fractions
Using scissors, students cut out the shaded parts.
1/2, 1/4, 1/8 & 1/16
Imagine doing this forever, shading and cutting, and then joining the shaded parts to form a new shape. How big would the new shape be?
Students notice that all shaded parts fit in a single square.
A single square is the limit of the sum of this infinite set of fractions.
1/2 + 1/4 + 1/8 + 1/16 + … = 1
Students exclaim: I can hold infinity in my hand!
We may also consider some related problems and extensions.
1. Infinity in pieces
Notice that some of the fraction pieces are square.
• What is the sum of the square fraction pieces?
Notice that the rest of the fraction pieces are rectangular.
• What is the sum of the rectangular fraction pieces?
To answer these questions, it may help if you arrange the square and rectangular pieces like this.
2. Infinity in a walk
Imagine walking to a door this way:
• First you walk half way to the door.
• Then you walk half the remaining distance.
• Then you walk half the remaining distance.
• Then you walk half the remaining distance.
• You keep doing this forever.
Will you ever get to the door?
3. Infinity in a decimal
Consider the repeating decimal 0.999 …
Is 0.999 … equal to 1?
4. GOOD STORY, good math
One way to judge the quality of students’ mathematics experience is to consider how they would answer the question: What did you do in math today?
As you plan a mathematics activity, imagine how it may prepare students to answer this question.
Will students be able to share with family and friends an experience — a story — that would offer mathematical surprise and conceptual insight?
Where do even numbers hide?
Do parallel lines never meet?
Why do elephants have big ears?
Can you hold infinity in your hand?
Brian Boyd (2009) says that story is a biological necessity.
Watson and Mason (2007) see mathematics as “an endless source of surprise.”
Creating, sharing and learning through stories and surprises are deeply human dispositions.
Interestingly, there is evidence that mathematics and storytelling are developmentally related.
Daniella O’Neill and colleagues (2004), from the University of Waterloo, tested young children’s narrative abilities. Two years later, they also tested their mathematical abilities.
Children with high-level narrative abilities, such as identifying stories with similar plots, or seeing a story from different character perspectives, also exhibited high-level mathematical
abilities, such as pattern recognition.
Here is one way we prepare students to share their learning at home.
• As students work on an activity, we record their comments, surprises and insights, and we share these with students as a handout. This gives students access to the collective knowledge of the
• We also provide students generic comic strip characters, along with images from their classroom work. Students cut and paste characters and comments to create comic strips that answer the
question: What did you do in math today?
• Students share the comic strips at home. Parents report back: What did your child share with you? and What did you learn?
• We summarize parent comments and send them back to parents. Sometimes, we use the comments to create lyrics that students sing for their parents. Below is an example of such a song, based on
grade 3 students creating bar graphs to represent and investigate circular relationships.
• ^Boyd, B. (2009). On the Origin of Stories: Evolution, Cognition, and Fiction. Cambridge, MA: Belknap Press of Harvard University Press.
• ^O’Neill, D. K., Pearce, M. J., & Pick, J. L. (2004). Predictive relations between aspects of preschool children’s narratives and performance on the Peabody Individualized Achievement Test –
Revised: Evidence of a relation between early narrative and later mathematical ability. First Language, 24, 149-183.
• ^Watson, A. & Mason, J. (2007) Surprise and inspiration. Mathematics
5. FOR All students
Our schools are democratic institutions. They provide access to education for all students.
However, as educators, we sometimes worry that deeper mathematics may confuse students.
We also worry that students who typically struggle mathematically may not be ready.
This occasional lack of faith in children’s abilities is, in part, due to some popular, but incorrect, education theories.
Why do we underestimate children?
Jean Piaget, for example, brought to us the theory of stages of cognitive development. He said that young children are concrete thinkers, and they develop their capacity to abstract later, maybe
around age 12.
Seymour Papert, who worked with Jean Piaget, disagreed. He said the stages Piaget identified are not in children’s minds. Rather, these stages are symptoms of how we educate children.
Piaget (1972/2008) himself cautioned about how generally his stages of development may apply.
Fernandez-Armesto (1997) lamented that “Generations of school children, deprived of challenging tasks because Piaget said they were incapable of them, bear the evidence of his impact” (p. 18).
Kieran Egan (2002) noted that young children naturally abstract to develop language, and come to understand words, such as dog.
Dogs are big, small, different colours, and with different dispositions. Children create an abstraction of the essential characteristics of dogs, and distinguish them from other animals that look
like them.
Children effortlessly abstract at a young age. They are much more capable, and much more attracted to deep ideas of mathematics, than we sometimes assume.
As one teacher noted:
I found that sometimes the tasks we might feel initially [to be] difficult, the kids got just like that. It has made me less fearful to go beyond the curriculum. In Grade 4 you’re only supposed to
learn this. Well, what’s stopping us from showing them a little bit beyond that?
Another teacher said:
I wish you were here to see the kids that never do well on assessments. I’ve never seen that part of him. Words coming out were impressive.
A low floor and a high ceiling approach is one way to offer access to deep ideas of mathematics for all students.
• ^Egan, K. (2002). Getting it wrong from the beginning : our progressivist inheritance from Herbert Spencer, John Dewey, and Jean Piaget. New Haven :Yale University Press.
• ^Fernandez-Armesto, F. (1997). Truth: A history and a guide for the perplexed. London: Bantam.
• ^Papert, S. (1980). Mindstorms—Children, Computers and Powerful Ideas. New York: Basic Books, Inc.
• ^Piaget, J. (2008). Intellectual evolution from adolescence to adulthood. Human Development, 51, 40-47. (Original work published 1972)
6: Less IS more
Implementing a few mathematics activities that offer wonder, and doing this well, is the ideal way to get started.
Start small.
Focus on deeper mathematics.
Engage students in sharing their learning beyond the classroom.
A few well-designed mathematics experiences can have lasting impact on students’ and teachers’ dispositions, living fruitfully in future experiences — as Dewey (1938) has suggested — by raising
expectation and anticipation of what mathematics is and what it can offer (Gadanidis, Borba, Hughes & Lacerda, 2016).
What wonderful mathematics stories — what mathematical surprises and conceptual insights — might you help your students experience?
Where do even numbers hide?
Do parallel lines never meet?
Why do elephants have big ears?
Can you hold infinity in your hand?
• ^^Dewey, John (1938). Experience & Education. New York, NY: Kappa Delta Pi.
• ^^Gadanidis, G., Borba, M., Hughes, J. and Lacerda, H. (2016). Designing aesthetic experiences for young mathematicians: A model for mathematics education reform. International Journal for
Research in Mathematics Education 6(2), 225-244. | {"url":"https://learnx.ca/math-wonder/","timestamp":"2024-11-02T03:03:38Z","content_type":"text/html","content_length":"63828","record_id":"<urn:uuid:fa3c635a-4074-4d4a-8196-2687cb5644ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00354.warc.gz"} |
Equivalence of Definitions of Floor Function
The following definitions of the concept of Floor Function are equivalent:
The floor function of $x$ is defined as the supremum of the set of integers no greater than $x$:
where $\le$ is the usual ordering on the real numbers.
The floor function of $x$, denoted $\floor x$, is defined as the greatest element of the set of integers:
The floor function of $x$ is the unique integer $\floor x$ such that:
Follows from Supremum of Set of Integers equals Greatest Element.
By Supremum of Set of Integers is Integer, $n \in \Z$.
By Supremum of Set of Integers equals Greatest Element, $n\in S$.
Because $n + 1 > n$, we have by definition of supremum:
We show that $n$ is the greatest element of the set:
By Weak Inequality of Integers iff Strict Inequality with Integer plus One:
and so from the definition of $g$ it follows that $m > x$.
By Proof by Contradiction it follows that $m \le n$.
Because $m \in S$ was arbitrary, $n$ is the greatest element of $S$. | {"url":"https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Floor_Function","timestamp":"2024-11-03T08:53:37Z","content_type":"text/html","content_length":"47315","record_id":"<urn:uuid:eb46f83f-594f-4214-996c-f9ab5a859172>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00549.warc.gz"} |
Third Kind -- from Wolfram MathWorld
In the theory of special functions, a class of functions is said to be "of the third kind" if it is similar to but distinct from previously defined functions already defined to be of the first and
second kinds. The only common functions of the third kind are the elliptic integral of the third kind Hankel function). | {"url":"https://mathworld.wolfram.com/ThirdKind.html","timestamp":"2024-11-12T19:21:28Z","content_type":"text/html","content_length":"51411","record_id":"<urn:uuid:18d1e92f-a5a2-4d8f-9729-b9e73a064134>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00842.warc.gz"} |
How can I do line charts in Excel? | Socratic
How can I do line charts in Excel?
2 Answers
1. Select x-data column (Ctrl/Command + Shift + End to fast-select)
2. Select y-data column (Ctrl/Command to select one cell in addition to the x-column, Ctrl/Command + Shift + End to fast-select)
3. Insert > Chart > Scatter > Smooth Lined Scatter
First, watch Mr. Pauller's video on how to use Excel for linear regression.
In a line chart, the horizontal axis is a "category" axis, not a "value" axis. The points are evenly distributed along the axis.
Use a line chart if your horizontal axis uses
• Text labels or
• A small set of numerical labels representing evenly spaced intervals
Assume that you want to plot the population of bears over a number of years.
Enter your data into Columns A and B of the Excel spreadsheet.
Assume that your data are:
If you have numeric labels, as here, leave cell A1 empty. This ensures that Excel recognizes the numbers in column A as categories
Highlight the data. Click on "Insert. Charts"
Click the small triangle under Line. Select Line with Markers (the first graph in the second row).
You should get a graph that looks like this.
This is your basic line chart.
Now you can experiment with other options to improve its appearance. Here's one possibility.
Impact of this question
2485 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/how-can-i-do-line-charts-in-excel","timestamp":"2024-11-06T02:07:03Z","content_type":"text/html","content_length":"35431","record_id":"<urn:uuid:474f7105-476b-4355-9bd5-5c7740279312>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00824.warc.gz"} |
Miles (statute) to Kiloyards Converter
Enter Miles (statute)
β Switch toKiloyards to Miles (statute) Converter
How to use this Miles (statute) to Kiloyards Converter π €
Follow these steps to convert given length from the units of Miles (statute) to the units of Kiloyards.
1. Enter the input Miles (statute) value in the text field.
2. The calculator converts the given Miles (statute) into Kiloyards in realtime β using the conversion formula, and displays under the Kiloyards label. You do not need to click any button. If the
input changes, Kiloyards value is re-calculated, just like that.
3. You may copy the resulting Kiloyards value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Miles (statute) to Kiloyards?
The formula to convert given length from Miles (statute) to Kiloyards is:
Length[(Kiloyards)] = Length[(Miles (statute))] / 0.5681806356963655
Substitute the given value of length in miles (statute), i.e., Length[(Miles (statute))] in the above formula and simplify the right-hand side value. The resulting value is the length in kiloyards,
i.e., Length[(Kiloyards)].
Calculation will be done after you enter a valid input.
Consider that a high-performance car can drive 400 miles (statute) on a single tank of fuel.
Convert this distance from miles (statute) to Kiloyards.
The length in miles (statute) is:
Length[(Miles (statute))] = 400
The formula to convert length from miles (statute) to kiloyards is:
Length[(Kiloyards)] = Length[(Miles (statute))] / 0.5681806356963655
Substitute given weight Length[(Miles (statute))] = 400 in the above formula.
Length[(Kiloyards)] = 400 / 0.5681806356963655
Length[(Kiloyards)] = 704.0015
Final Answer:
Therefore, 400 mi (US) is equal to 704.0015 kyd.
The length is 704.0015 kyd, in kiloyards.
Consider that a marathon race is 26.2 miles (statute) long.
Convert this distance from miles (statute) to Kiloyards.
The length in miles (statute) is:
Length[(Miles (statute))] = 26.2
The formula to convert length from miles (statute) to kiloyards is:
Length[(Kiloyards)] = Length[(Miles (statute))] / 0.5681806356963655
Substitute given weight Length[(Miles (statute))] = 26.2 in the above formula.
Length[(Kiloyards)] = 26.2 / 0.5681806356963655
Length[(Kiloyards)] = 46.1121
Final Answer:
Therefore, 26.2 mi (US) is equal to 46.1121 kyd.
The length is 46.1121 kyd, in kiloyards.
Miles (statute) to Kiloyards Conversion Table
The following table gives some of the most used conversions from Miles (statute) to Kiloyards.
Miles (statute) (mi (US)) Kiloyards (kyd)
0 mi (US) 0 kyd
1 mi (US) 1.76 kyd
2 mi (US) 3.52 kyd
3 mi (US) 5.28 kyd
4 mi (US) 7.04 kyd
5 mi (US) 8.8 kyd
6 mi (US) 10.56 kyd
7 mi (US) 12.32 kyd
8 mi (US) 14.08 kyd
9 mi (US) 15.84 kyd
10 mi (US) 17.6 kyd
20 mi (US) 35.2001 kyd
50 mi (US) 88.0002 kyd
100 mi (US) 176.0004 kyd
1000 mi (US) 1760.0037 kyd
10000 mi (US) 17600.0366 kyd
100000 mi (US) 176000.3663 kyd
Miles (statute)
A statute mile is a unit of length used primarily in the United States and the United Kingdom for measuring distances. One statute mile is equivalent to 5,280 feet or approximately 1,609.344 meters.
The statute mile is defined as exactly 5,280 feet, and it is used in a variety of contexts including land measurement, transportation, and mapping.
Statute miles are commonly used in the United States for road signs, property measurement, and other applications. The term "statute mile" helps distinguish it from other types of miles, such as
nautical miles, and ensures clarity in measurement contexts.
A kiloyard (ky) is a unit of length equal to 1,000 yards or approximately 914.4 meters.
The kiloyard is defined as one thousand yards, providing a convenient measurement for longer distances that are not as extensive as miles but larger than typical yard measurements.
Kiloyards are used in various fields to measure length and distance where a scale between yards and miles is appropriate. They offer a practical unit for certain applications, such as in land
measurement and engineering.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Miles (statute) to Kiloyards in Length?
The formula to convert Miles (statute) to Kiloyards in Length is:
Miles (statute) / 0.5681806356963655
2. Is this tool free or paid?
This Length conversion tool, which converts Miles (statute) to Kiloyards, is completely free to use.
3. How do I convert Length from Miles (statute) to Kiloyards?
To convert Length from Miles (statute) to Kiloyards, you can use the following formula:
Miles (statute) / 0.5681806356963655
For example, if you have a value in Miles (statute), you substitute that value in place of Miles (statute) in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=miles_statute-kiloyards","timestamp":"2024-11-09T09:27:00Z","content_type":"text/html","content_length":"91264","record_id":"<urn:uuid:92e44eef-a25d-439e-b452-7a738d33a186>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00731.warc.gz"} |
Quick Introduction to Auto encoders (AE) | Deep Learning
Feb 01, 2023 / 22 min read
Auto encoders are a type of neural network architecture that is designed to perform dimensionality reduction and feature learning. The basic idea behind an autoencoder is to learn a compressed
representation of the input data, called the encoder, and then use this compressed representation to reconstruct the original input, called the decoder. This is done by training the network to
minimize the difference between the original input and the reconstructed input.
Auto encoders can be used for a variety of tasks such as image denoising, anomaly detection, and generative modelling. They consist of an encoder and a decoder. The encoder compresses the input data
into a lower dimensional feature space, while the decoder tries to reconstruct the original input from the compressed representation.
Applied Deep Learning | Auto encoders (AE)
The architecture of an autoencoder can vary greatly depending on the task and the data, but a basic structure is an input layer, an encoder, a bottleneck or latent layer, a decoder, and an output
layer. The encoder and decoder can be implemented using feedforward neural networks such as a multi-layer perceptron, or using convolutional neural networks for image data.
💡 Conclusion:
Auto encoders (AE) are a type of neural network that are used for unsupervised learning. They are trained to reconstruct the input data by learning a compact representation of the input called the
"latent representation" or "latent code". This compact representation can then be used for tasks such as dimensionality reduction, anomaly detection, and feature learning. They are also used in deep
learning as a pre-training step to initialize the weights of a deep network. The main advantage of AE is that it can learn useful features from the input data without any supervision. However, the
disadvantage is that the output may not be as good as supervised learning. The main purpose of this algorithm is to learn a compact representation of the input data and can be used for various tasks
such as dimensionality reduction and anomaly detection.
Code to build Auto encoders (AE):
Here is an example of a simple autoencoder implemented using the Keras library in Python:
from keras.layers import Input, Dense
from keras.models import Model
# this is the size of our encoded representations
encoding_dim = 32 # 32 floats -> compression of factor 24.5, assuming the input is 784 floats
# this is our input placeholder
input_img = Input(shape=(784,))
# "encoded" is the encoded representation of the input
encoded = Dense(encoding_dim, activation='relu')(input_img)
# "decoded" is the lossy reconstruction of the input
decoded = Dense(784, activation='sigmoid')(encoded)
# this model maps an input to its reconstruction
autoencoder = Model(input_img, decoded)
In this example, the autoencoder has an input layer with 784 neurons (corresponding to the 28x28 pixels in an MNIST image), an encoded layer with 32 neurons (the bottleneck or latent layer), and a
decoded layer with 784 neurons that is used to reconstruct the original image. The activation function used in the encoded layer is 'relu' and in the decoded layer is 'sigmoid'. The model is trained
to minimize the difference between the original input and the reconstructed input. | {"url":"https://codeease.net/deep-learning/quick-introduction-to-auto-encoders/","timestamp":"2024-11-09T05:54:23Z","content_type":"text/html","content_length":"22161","record_id":"<urn:uuid:2f0d142f-a74a-49d7-97f3-4945d3248cfa>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00802.warc.gz"} |
Irreducible Infeasible Set
For a linear programming problem, an irreducible infeasible set (IIS) is an infeasible subset of constraints and variable bounds that will become feasible if any single constraint or variable bound
is removed. It is possible to have more than one IIS in an infeasible LP. Identifying an IIS can help to isolate the structural infeasibility in an LP.
The presolver in the OPTLP procedure can detect infeasibility, but it only identifies the variable bound or constraint that triggers the infeasibility.
The IIS=ON option directs the OPTLP procedure to search for an IIS in a given LP. The presolver is not applied to the problem during the IIS search. If the OPTLP procedure detects an IIS, it first
outputs the IIS to the data sets specified by the PRIMALOUT= and DUALOUT= options, and then it stops. The number of iterations that are reported in the macro variable and the ODS table is the total
number of simplex iterations. This includes the initial LP solve and all subsequent iterations during the constraint deletion phase.
The IIS= option can add special values to the _STATUS_ variables in the output data sets. (See the section Data Input and Output for more information.) For constraints, a status of “I_L”, “I_U”, or “
I_F” indicates, respectively, the “GE” (), “LE” (), or “EQ” () condition is violated. For range constraints, a status of “I_L” or “I_U” indicates, respectively, that the lower or upper bound of the
constraint is violated. For variables, a status of “I_L”, “I_U”, or “I_F” indicates, respectively, the lower, upper, or fixed bound of the variable is violated. From this information, you can
identify names of the constraints (variables) in the IIS as well as the corresponding bound where infeasibility occurs.
Making any one of the constraints or variable bounds in the IIS nonbinding removes the infeasibility from the IIS. In some cases, changing a right-hand side or bound by a finite amount removes the
infeasibility; however, the only way to guarantee removal of the infeasibility is to set the appropriate right-hand side or bound to or . Because it is possible for an LP to have multiple irreducible
infeasible sets, simply removing the infeasibility from one set might not make the entire problem feasible. To make the entire problem feasible, you can rerun the LP solver with IIS=ON specified
after removing the infeasibility from an IIS. Repeat this process until the LP solver no longer detects an IIS. The resulting problem is feasible. This approach to infeasibility repair can produce
different end problems depending on which right-hand sides and bounds you choose to relax.
Changing different constraints and bounds can require considerably different changes to the MPS-format SAS data set. For example, if you used the default lower bound of 0 for a variable but you want
to relax the lower bound to , you might need to add a LB row to the BOUNDS section of the data set. For more information about changing variable and constraint bounds, see Chapter 15: The MPS-Format
SAS Data Set.
The IIS= option in PROC OPTLP uses two different methods to identify an IIS. Based on the result of the initial solve, the sensitivity filter removes several constraints and variable bounds at once
while still maintaining infeasibility. This phase is quick and dramatically reduces the size of the IIS. Following that, the deletion filter removes each remaining constraint and variable bound one
by one to check which of them are needed to get an infeasible system. This second phase is more time consuming, but it ensures that the IIS set returned by PROC OPTLP is indeed irreducible. The
progress of the deletion filter is reported at regular intervals. Occasionally, the sensitivity filter might be called again during the deletion filter to improve performance.
See Example 10.7 for an example demonstrating the use of the IIS= option in locating and removing infeasibilities. | {"url":"http://support.sas.com/documentation/cdl/en/ormpug/65554/HTML/default/ormpug_optlp_details18.htm","timestamp":"2024-11-09T01:51:57Z","content_type":"application/xhtml+xml","content_length":"18413","record_id":"<urn:uuid:d12435f6-07cc-4125-a195-8ddc6a958506>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00256.warc.gz"} |
Mole fraction of ethanol in ethanol and water mixture is 0.25. Hence percentage concentration of ethanol by weight of mixture is? | Socratic
Mole fraction of ethanol in ethanol and water mixture is 0.25. Hence percentage concentration of ethanol by weight of mixture is?
A: 25%
B: 75%
C: 46%
D: 54%
THE ANSWER IS (C).... PLEASE GIVE A DETAILED SOLUTION
1 Answer
I'll show you two methods that you can use to solve this problem.
$\textcolor{w h i t e}{a}$
$\textcolor{w h i t e}{a}$
As you know, a solution's percent concentration by mass tells you the number of grams of solute present for every $\text{100 g}$ of solution.
To make the calculations easier, pick a $\text{100-g}$ sample of this solution.
Now, you know that the mass of this sample will be equal to the mass of the ethanol, the solute, and the mass of the water, the solvent.
${m}_{\text{solution" = m_ "ethanol" + m_ "water}}$
In your case, you will have
${m}_{\text{ethanol" + m_ "water" = "100 g" " }} \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{\left(1\right)}$
You also know that the mole fraction of ethanol, which is defined as the ratio between the number of moles of ethanol and the total number of moles present in the solution, is equal to $0.25$.
#chi_ "ethanol" = n_"ethanol"/(n_"ethanol" + n_"water")#
At this point, you must use the molar masses of ethanol and of water to express the mole ratio of ethanol in terms of ${m}_{\text{ethanol}}$ and ${m}_{\text{water}}$.
${M}_{\text{M ethanol" = "46.07 g mol}}^{- 1}$
${M}_{\text{M water" = "18.015 g mol}}^{- 1}$
This means that you have
${n}_{\text{ethanol" = m_"ethanol"/"46.07 g mol}}^{- 1}$
${n}_{\text{water" = m_"water"/"18.015 g mol}}^{- 1}$
Therefore, the mole fraction of ethanol can be rewritten as--for the sake of simplicity, I won't add any units
#chi_ "ethanol" = (m_"ethanol"/46.07)/(m_"ethanol"/46.07 + m_"water"/18.015)#
which is equivalent to
#(18.015 * m_"ethanol")/(18.015 * m_"ethanol" + 46.07 * m_"water") = 0.25" "color(darkorange)((2))#
Now all you have to do is to solve this system of two equations with two unknowns.
Use equation $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{\left(1\right)}$ to write
${m}_{\text{water" = 100 - m_"ethanol}}$
Plug this into equation $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{\left(2\right)}$ to find
#18.015 * m_"ethanol" = 0.25 * 18.015 * m_"ethanol" + 0.25 * 46.07 * (100 - m_"ethanol")#
This will get you
${m}_{\text{ethanol}} \cdot \left(18.015 - 0.25 \cdot 18.015 + 0.25 \cdot 46.07\right) = 0.25 \cdot 46.07 \cdot 100$
which results in
${m}_{\text{ethanol}} = \frac{1151.75}{25.02875} = 46.02$
Since this represents the mass of ethanol present in $\text{100 g}$ of solution, you can say that the percent concentration by mass of ethanol is
$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{% ethanol by mass = 46%}}}}$
$\textcolor{w h i t e}{a}$
$\textcolor{w h i t e}{a}$
Alternatively, you can start by picking a sample of this solution that contains exactly $1$mole solute and of solvent.
This means that you have
${n}_{\text{ethanol" + n_"water" = "1 mole}}$
Now, you can use the mole fraction of ethanol to say that the number of moles of ethanol present in this sample is equal to
${\chi}_{\text{ethanol" = n_"ethanol"/"1 mole" implies n_"ethanol" = 0.25 * "1 mole" = "0.25 moles}}$
Consequently, you can say that this sample contains $0.75$moles of water.
Use the molar masses of the two compounds to convert the number of moles to grams.
#0.25 color(red)(cancel(color(black)("moles ethanol"))) * "46.07 g"/(1color(red)(cancel(color(black)("mole ethanol")))) = "11.52 g"#
#0.75 color(red)(cancel(color(black)("moles water"))) * "18.015 g"/(1color(red)(cancel(color(black)("mole water")))) = "13.51 g"#
The total mass of the solution will be
$\text{11.52 g + 13.51 g = 25.03 g}$
You can use the known composition of the sample to figure out how many grams of ethanol you'd ge for $\text{100 g}$ of this solution
#100 color(red)(cancel(color(black)("g solution"))) * "11.52 g ethanol"/(25.03 color(red)(cancel(color(black)("g solution")))) = "46.02 g ethanol"#
Once again, you have
$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{% ethanol by mass = 46%}}}}$
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Get Started With Naive Bayes Algorithm: Theory & Implementation
Naive Bayes is a machine learning algorithm that is used by data scientists for classification. The naive Bayes algorithm works based on the Bayes theorem. Before explaining Naive Bayes, first, we
should discuss Bayes Theorem. Bayes theorem is used to find the probability of a hypothesis with given evidence. This beginner-level article intends to introduce you to the Naive Bayes algorithm and
explain its underlying concept and implementation.
In this equation, using Bayes theorem, we can find the probability of A, given that B occurred. A is the hypothesis, and B is the evidence.
P(B|A) is the probability of B given that A is True.
P(A) and P(B) are the independent probabilities of A and B.
Learning Objectives
• Learn the concept behind the Naive Bayes algorithm.
• See the steps involved in the naive Bayes algorithm
• Practice the step-by-step implementation of the algorithm.
This article was published as a part of the Data Science Blogathon.
What Is the Naive Bayes Classifier Algorithm?
The Naive Bayes classifier algorithm is a machine learning technique used for classification tasks. It is based on Bayes’ theorem and assumes that features are conditionally independent of each other
given the class label. The algorithm calculates the probability of a data point belonging to each class and assigns it to the class with the highest probability.
Naive Bayes is known for its simplicity, efficiency, and effectiveness in handling high-dimensional data. It is commonly used in various applications, including text classification, spam detection,
and sentiment analysis.
Naive Bayes Theorem: The Concept Behind the Algorithm
Let’s understand the concept of the Naive Bayes Theorem and how it works through an example. We are taking a case study in which we have the dataset of employees in a company, our aim is to create a
model to find whether a person is going to the office by driving or walking using the salary and age of the person.
In the above image, we can see 30 data points in which red points belong to those who are walking and green belong to those who are driving. Now let’s add a new data point to it. Our aim is to find
the category that the new point belongs to
Note that we are taking age on the X-axis and Salary on the Y-axis. We are using the Naive Bayes algorithm to find the category of the new data point. For this, we have to find the posterior
probability of walking and driving for this data point. After comparing, the point belongs to the category having a higher probability.
In the above image, we can see 30 data points in which red points belong to those who are walking and green belong to those who are driving. Now let’s add a new data point to it. Our aim is to find
the category that the new point belongs to
The posterior probability of walking for the new data point is:
and that for the driving is:
Steps Involved in the Naive Bayes Classifier Algorithm
Step 1: We have to find all the probabilities required for the Bayes theorem for the calculation of posterior probability.
P(Walks) is simply the probability of those who walk among all.
In order to find the marginal likelihood, P(X), we have to consider a circle around the new data point of any radii, including some red and green points.
P(X|Walks) can be found by:
Now we can find the posterior probability using the Bayes theorem,
Step 2: Similarly, we can find the posterior probability of Driving, and it is 0.25
Step 3: Compare both posterior probabilities. When comparing the posterior probability, we can find that P(walks|X) has greater values, and the new point belongs to the walking category.
Source: Unsplash
Implementation of Naive Bayes in Python Programming
Now let’s implement Naive Bayes step by step using the python programming language
We are using the Social network ad dataset. The dataset contains the details of users on a social networking site to find whether a user buys a product by clicking the ad on the site based on their
salary, age, and gender.
Step 1: Importing the libraries
Let’s start the programming by importing the essential libraries required.
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sklearn
Step 2: Importing the dataset
Python Code:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sklearn
dataset = pd.read_csv('Social_Network_Ads.csv')
X = dataset.iloc[:, [1, 2, 3]].values
y = dataset.iloc[:, -1].values
Since our dataset contains character variables, we have to encode it using LabelEncoder.
from sklearn.preprocessing import LabelEncoder
le = LabelEncoder()
X[:,0] = le.fit_transform(X[:,0])
Step 3: Train test splitting
We are splitting our data into train and test datasets using the scikit-learn library. We are providing the test size as 0.20, which means our training data contains 320 training sets, and the test
sample contains 80 test sets.
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state = 0)
Step 4: Feature scaling
Next, we are doing feature scaling to the training and test set of independent variables.
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
Step 5: Training the Naive Bayes model on the training set
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train, y_train)
Let’s predict the test results
y_pred = classifier.predict(X_test)
Predicted and actual value
For the first 8 values, both are the same. We can evaluate our matrix using the confusion matrix and accuracy score by comparing the predicted and actual test values
from sklearn.metrics import confusion_matrix,accuracy_score
cm = confusion_matrix(y_test, y_pred)
ac = accuracy_score(y_test,y_pred)
confusion matrix
ac – 0.9125
Accuracy is good. Note that you can achieve better results for this problem using different algorithms.
Full Python Tutorial
# Importing the libraries
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Importing the dataset
dataset = pd.read_csv('Social_Network_Ads.csv')
X = dataset.iloc[:, [2, 3]].values
y = dataset.iloc[:, -1].values
# Splitting the dataset into the Training set and Test set
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state = 0)
# Feature Scaling
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
# Training the Naive Bayes model on the Training set
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train, y_train)
# Predicting the Test set results
y_pred = classifier.predict(X_test)
# Making the Confusion Matrix
from sklearn.metrics import confusion_matrix, accuracy_score
ac = accuracy_score(y_test,y_pred)
cm = confusion_matrix(y_test, y_pred)
What Are the Assumptions Made by the Naive Bayes Algorithm?
There are several variants of Naive Bayes, such as Gaussian Naive Bayes, Multinomial Naive Bayes, and Bernoulli Naive Bayes. Each variant has its own assumptions and is suited for different types of
data. Here are some assumptions that the Naive Bayers algorithm makes:
1. The main assumption is that it assumes that the features are conditionally independent of each other.
2. Each of the features is equal in terms of weightage and importance.
3. The algorithm assumes that the features follow a normal distribution.
4. The algorithm also assumes that there is no or almost no correlation among features.
The naive Bayes algorithm is a powerful and widely-used machine learning algorithm that is particularly useful for classification tasks. This article explains the basic math behind the Naive Bayes
algorithm and how it works for binary classification problems. Its simplicity and efficiency make it a popular choice for many data science applications. we have covered most concepts of the
algorithm and how to implement it in Python. Hope you liked the article, and do not forget to practice algorithms.
Key Takeaways
• Naive Bayes is a probabilistic classification algorithm(binary o multi-class) that is based on Bayes’ theorem.
• There are different variants of Naive Bayes, which can be used for different tasks and can even be used for regression problems.
• Naive Bayes can be used for a variety of applications, such as spam filtering, sentiment analysis, and recommendation systems.
Frequently Asked Questions
Q1. When should we use a naive Bayes classifier?
A. The naive Bayes classifier is a good choice when you want to solve a binary or multi-class classification problem when the dataset is relatively small and the features are conditionally
independent. It is a fast and efficient algorithm that can often perform well, even when the assumptions of conditional independence do not strictly hold. Due to its high speed, it is well-suited for
real-time applications. However, it may not be the best choice when the features are highly correlated or when the data is highly imbalanced.
Q2. What is the difference between Bayes Theorem and Naive Bayes Algorithm?
A. Bayes theorem provides a way to calculate the conditional probability of an event based on prior knowledge of related conditions. The naive Bayes algorithm, on the other hand, is a machine
learning algorithm that is based on Bayes’ theorem, which is used for classification problems.
Q3. Is Naive Bayes a regression technique or classification technique?
It is not a regression technique, although one of the three types of Naive Bayes, called Gaussian Naive Bayes, can be used for regression problems.
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Responses From Readers
Awesome explanation ☺ This will clear all the doubts and its very helful for newbies. Keep up the good work 👍
Hi, Great post;) I would like to ask when estimating the marginal likelihood P(X), we need to draw a circle around the new data, how should we choose the radius in order to increase accuracy of the
estimation? And how is the radius or metric used going too affect the accuracy? Is there any book you can recommend for this topic? Thank you so much.
Thanks Surbhi! Easy to understand.
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Final calculator
Have you ever found yourself wrestling with the question, “How much do I need to score on my final exam to pass my course?” If the answer is yes, then this is the article you’ve been waiting for.
Let’s talk all about a magical tool known as the final calculator.
What is a Final Calculator?
Imagine a tool that takes the guesswork out of achieving your academic goals, a device that offers results as accurate as a Swiss watch. This, dear readers, is the beauty of a final calculator. Think
of it as your personal academic assistant, a real-life Hermione Granger, minus the Time-Turner.
How Does a Final Calculator Work?
A final calculator works on the core principle of combining your current course marks with your university’s grading policies. It uses these inputs to estimate the score you’d need on your final exam
to attain that illustrious A (or whatever your dream grade might be). Pretty neat, huh?
How Can the Final Calculator Help Me Prepare?
Consider the final calculator as a roadmap that seamlessly guides you through your study sessions. Now, instead of pushing blindly for that elusive 100% on your final exam, you have a clear
understanding of what you need to perfect and how much effort you need to dedicate to achieving it. It works like your personal soothsayer, giving you a precious glimpse into your academic future.
The Final Calculator: A Must-Have for Every Student?
Picture this: it’s the end of the semester, and the pressure is on. The uncertainty of the upcoming final exams is consuming you. You spend countless hours studying, leaving you overwhelmed and burnt
out. Does this scenario sound familiar?
Well, it doesn’t have to be this way. The final calculator transforms this nerve-wracking situation into a manageable one. It helps break down your study load into achievable targets while allowing
you to focus on your weak areas with precision. Some might call it a lifesaver; we prefer to think of it as an essential friend in your academic journey.
How Accurate Is the Final Calculator?
While the final calculator operates with mathematical precision, bear in mind, it’s not quite an oracle. Its accuracy depends on the correctness of the information you input. However, given the right
information, it can give you a relatively accurate picture of where you stand and what you need to reach your desired grade.
The final calculator is truthfully an underappreciated tool in every student’s arsenal. As Leonardo da Vinci once said, “Simplicity is the ultimate sophistication,” and the final calculator
exemplifies this by handling the complexities of grading scales, giving you a simplified roadmap to your success. | {"url":"https://www-calculator.com/final-calculator/","timestamp":"2024-11-07T17:09:43Z","content_type":"text/html","content_length":"49993","record_id":"<urn:uuid:c63d9745-384d-4ff8-a3ea-126ec514a2be>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00390.warc.gz"} |
Nematic Superconductivity in Doped Bi2Se3 Topological Superconductors
Nematic Superconductivity in Doped Bi[2]Se[3] Topological Superconductors
Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Submission received: 27 November 2018 / Revised: 13 December 2018 / Accepted: 19 December 2018 / Published: 20 December 2018
Nematic superconductivity is a novel class of superconductivity characterized by spontaneous rotational-symmetry breaking in the superconducting gap amplitude and/or Cooper-pair spins with respect to
the underlying lattice symmetry. Doped Bi$2$Se$3$ superconductors, such as Cu$x$Bi$2$Se$3$, Sr$x$Bi$2$Se$3$, and Nb$x$Bi$2$Se$3$, are considered as candidates for nematic superconductors, in addition
to the anticipated topological superconductivity. Recently, various bulk probes, such as nuclear magnetic resonance, specific heat, magnetotransport, magnetic torque, and magnetization, have
consistently revealed two-fold symmetric behavior in their in-plane magnetic-field-direction dependence, although the underlying crystal lattice possesses three-fold rotational symmetry. More
recently, nematic superconductivity was directly visualized using scanning tunneling microscopy and spectroscopy. In this short review, we summarize the current research on the nematic behavior in
superconducting doped Bi$2$Se$3$ systems and discuss issues and perspectives.
1. Introduction
In the last decade, the research field on topological materials, which possess a non-trivial topology in their electronic-state wave functions in the reciprocal space, has been expanding
substantially [
]. As a counterpart of topological insulators, it has been recognized that certain superconductors can have a non-trivial topological nature in their wavefunctions [
]. Superconductivity with such a non-trivial wavefunction topology is now called topological superconductivity. It has been predicted that topological superconductivity can lead to various novel
phenomena. In particular, the Majorana quasiparticles with non-Abelian zero-energy modes, hosted in topologically-protected edge states or vortex cores, are quite intriguing [
]. The realization and utilization of such Majorana modes comprise one of the holy grails of this research field.
There are a number of superconductors that seemingly exhibit topological superconductivity in bulk [
]. Such bulk topological superconductors are the prototype of topological superconductivity, although there are now other recipes to induce topological superconductivity by making use of the
proximity effect. Recently, a candidate for a bulk topological superconductor, doped Bi
, has been found to exhibit unusual rotational-symmetry breaking in the superconducting (SC) gap amplitude, as well as in the SC spin degree of freedom [
]. This phenomena, called “nematic” superconductivity [
], has been attracting much attention as a new species in the superconductor zoo, accompanied by a novel class of symmetry breaking in its SC wave function. The shape and topology of the SC wave
function are closely related. Indeed, in
$A x$
, it is theoretically known that the nematic superconductivity is accompanied by a non-trivial topological SC gap [
]. Therefore, these new experimental observations of the nematic SC gap and spin have been providing firm bulk evidence for topological superconductivity, and thus establish strong bases toward
realization and manipulation of Majorana states in this class of compounds.
In this short review, we summarize recent observations of the nematic behavior in superconducting doped Bi
systems. After a brief introduction of nematic superconductivity in
Section 2
, we explain experimental and theoretical understandings of superconductivity in
$A x$
Section 3
Section 4
is devoted to explaining recent experimental findings of nematic superconductivity. Then, we discuss several known issues in
Section 5
, before summarizing the content in
Section 6
2. Nematic Superconductivity: Rotational Symmetry Breaking in the Gap Amplitude
2.1. Symmetry Breaking in Superconductivity
The concept of spontaneous symmetry breaking has fundamental importance in superconductivity. In the Bardeen–Cooper–Schrieffer (BCS) theory [
], the SC state spontaneously breaks the
$U ( 1 )$
gauge symmetry, even for the ordinary
-wave superconductivity (
Figure 1
a). It has been then an interesting and long-standing question whether SC states with additional symmetry breaking exist or not. Such superconductivity with additional symmetry breaking is called
unconventional superconductivity.
For example, superconductivity with spontaneous time-reversal symmetry breaking in the orbital part of the SC order parameter (
Figure 1
b) is called as “chiral” superconductivity and is believed to be realized in several materials such as Sr
], URu
], and UPt
]. Odd parity superconductivity, which exhibits
phase shift under spacial inversion (see again
Figure 1
b), has been an interesting topic for fairly a long time [
]. Odd-parity superconductivity (and superfluidity) is confirmed in superfluid
He [
] and is very probably realized in Sr
] and UPt
]. Moreover, odd-parity superconductivity is now recognized as a key ingredient to realize topological superconductivity [
Another fundamentally important symmetry is rotational symmetry. The symmetry behavior under rotation provides a basis of the classification of superconductivity into
-wave, etc. However, the infinitesimal rotational symmetry
$C ∞$
is already broken in superconductors because of the crystal lattice. Thus, in reality, what actually matters is the breaking/invariance of the discrete
$C n$
rotational symmetry of the underlying lattice. For example, in
-wave superconductivity in a tetragonal system (
Figure 1
c), the phase factor for one
direction and for its perpendicular direction differs by
. Therefore, the four-fold rotation
$C 4$
symmetry is broken in the phase factor of the SC order parameter.
2.2. Gap-Nematic and Spin-Nematic Superconductivity
Such a rotational symmetry breaking in the phase degree of freedom in non-
-wave superconductivity is intriguing, but the experimental detection of such symmetry breaking is actually very difficult. This is because the SC gap amplitude, which governs most of the
superconducting properties, is actually invariant under the
$C n$
rotation. To detect the rotational symmetry breaking in the phase factor, one has to utilize sophisticated interference techniques, as performed in cuprate
-wave superconductors [
]. In contrast, if the rotational-symmetry breaking occurs in the SC gap amplitude as shown in
Figure 1
d, the rotational symmetry breaking would be more robust and be detectable in principle in any bulk quantities. Such superconductivity with broken rotational symmetry in the gap amplitude was named
as “nematic superconductivity” first by Fu [
] and has been attracting much attention as a new class of superconductivity accompanied by a novel spontaneous symmetry breaking.
The word “nematic”, originally used in the research field of liquid crystals, refers to the states with the spontaneous rotational symmetry breaking of the bar-shaped liquid crystal molecules, but
without breaking the translational symmetry. Now, this word is imported to solid-state physics, and the “nematic electron liquid” with spontaneous rotational symmetry breaking in the conduction
electron without losing conductivity has been attracting much attention and has actually been found in various systems such as cuprates [
], iron pnictides [
], and a ruthenate [
]. The nematic superconductivity is a superconducting version of such nematic electron liquids, but occurring as a consequence of Cooper-pair formation.
In the case of spin-triplet superconductivity, the spin part of the order parameter, i.e., the
vector, can also exhibit rotational symmetry breaking. Here, we should be careful that there are two different levels of the spin-rotational-symmetry breaking. Firstly, the
$S O ( 3 )$
symmetry of the spin space breaks in the presence of spin-orbit interaction. In this case, the spin susceptibility exhibits anisotropic behavior, but still obeys the symmetry of the lattice. This
first kind of spin-rotational symmetry breaking has not been observed in a leading candidate spin-triplet superconductor Sr
] and was weakly observed in another candidate UPt
]. A more exotic phenomenon is that the spin part even breaks the
$C n$
lattice rotational symmetry (but without spin polarization). In this case, spin susceptibility exhibits rotational-symmetry breaking, as schematically shown in
Figure 1
e. We shall call this phenomenon “spin-nematic superconductivity”; and to distinguish it, the nematic superconductivity with the rotational symmetry breaking in the gap amplitude is called
“gap-nematic superconductivity” in this review. Spin-nematic superconductivity has never been known in any material before its discovery in Cu
3. Superconductivity in Doped Bi$2$Se$3$
In this section, SC and normal-state properties of doped Bi
, as well as theoretically-proposed SC states are described. For more detailed information, refer to the nice reviews found in [
3.1. Crystal Structure of the Mother Compound Bi$2$Se$3$
, the mother compound of the doped Bi
superconductors, has been extensively studied as a prototypical topological insulator [
]. This compound has a rhombohedral (or trigonal) crystal structure, as shown in
Figure 2
, with the space group of
$R 3 ¯ m$
$D 3 d 5$
) [
]. The crystal structure contains three equivalent
axes. Strictly speaking, the situations with
parallel to the
axis and
$− a$
axis are different because of the trigonal symmetry and the pseudo-vector nature of the magnetic field. Nevertheless, practically,
$H ∥ a$
$H ∥ ( − a )$
are almost equivalent in most cases, and thus, many physical quantities are expected to exhibit pseudo-six-fold rotational symmetry as a function of the in-plane field direction. The
$a ∗$
axis is perpendicular to the
axis within the
$a b$
plane. Importantly, the
$a ∗$
plane is a mirror-symmetry plane, whereas the
plane is not, as clearly seen in
Figure 2
b. This existence/absence of the mirror symmetry is closely related to the stability of the gap nodes in the nematic SC state, as will be discussed in
Section 3.3
. Throughout this paper, we define the
axis along one of the three
axes. In most cases, we choose
to be along the “special”
axis due to the nematic SC order (e.g., the
axis with maximal or minimal
$H c 2$
). Then, the
axis is defined so that it is perpendicular to the
axis within the
$a b$
plane, as explained in
Figure 2
The crystal structure consists of Se-Bi-Se-Bi-Se layers, as shown in
Figure 2
a. This set of layers is called the quintuple layer (QL). Between the QLs, there is a van-der Waals (vdW) gap, through which metallic ions penetrate the sample during the synthesis process. Thus, the
ions are most likely to be intercalated into a certain site within the vdW gap. For Sr-doped Bi
, Sr ions may also sit in an interstitial site [
]. The precise position of the doped ions, in particular that for superconducting samples, has not been fully clarified.
3.2. Basic Properties of Doped Bi$2$Se$3$ Superconductors
In 2010, the pioneering work by Hor et al. revealed that Cu
exhibits superconductivity below the critical temperature
$T c$
of around 3 K [
]. This is truly the beginning of the research field of superconducting doped Bi
systems, but the initial samples grown by the ordinary melt-grown technique exhibited a superconducting shielding fraction of about 20% and did not show complete vanishing of resistivity. One year
later, Kriener et al. found that, with the electrochemical intercalation of Cu and with a suitable annealing process, Cu
indeed exhibits bulk superconductivity with clear zero resistivity and volume fractions reaching 60–70% evaluated from specific-heat and magnetization measurements [
]. In 2015, Sr doping or Nb doping were also found to drive Bi
to superconduct, with
$T c$
again around 3 K [
]. In contrast to the Cu-doped material, Sr
and Nb
exhibit bulk superconductivity with a fairly large shielding fraction close to 100% even with melt-grown samples.
These three compounds exhibit similar SC behavior, but there are several significant differences, as well. Firstly, the normal-state electronic state is different. In Cu
, an ellipsoidal or cylindrical Fermi surface, depending on the carrier density, has been revealed by angle-resolved photoemission spectroscopy (ARPES) and quantum oscillation experiments [
]. Superconducting Cu
typically has a carrier density
of around
$10 20 cm − 3$
]. Furthermore,
is known to be rather insensitive to the Cu concentration [
]. In Sr
tends to be even lower than that in Cu
for superconducting Sr
is consistently reported to be ∼2
$× 10 19 cm − 3$
]. For Nb
, the quantum oscillation consists of oscillations with different frequencies and with different field-angular dependence, indicating that the Fermi surface is not a simple ellipsoid or cylinder, but
is composed of multiple pockets [
]. This is crucially different from the single Fermi surface in Cu
and Sr
. In addition, Nb
has some controversy on the magnetism: in the initial report, it was argued that Nb
exhibits long-range magnetic order in the superconducting state and that this magnetic order assists the formation of a chiral SC state [
]. In later reports, however, such magnetism has not been reported [
]. Secondly, a practical difference among the three compounds is that Sr
and Nb
are stable in air, whereas superconductivity in Cu
is known to diminish if a sample is kept in air [
]. We should comment here that single-crystalline Cu
prepared by electrochemical intercalation might not be as air-sensitive as melt-grown samples [
]. Because of the higher stability, Sr and Nb-doped Bi
have been extensively studied since their discoveries, as reviewed below.
The normal-state electronic state of Cu
was investigated with ARPES and quantum oscillation experiments [
]. It was found that the electronic band structure of Cu
is essentially the same as that of Bi
and that the surface state originating from the topological-insulator nature of Bi
still exists even after the Cu doping. As expected, Cu
is heavily electron doped compared to the mother compound: the chemical potential is located 0.2–0.5 eV above the Dirac point of the topological surface state. Still, the surface state was clearly
observed even at the chemical potential, well distinguishable from the bulk band. This means that the bulk conduction electrons on the Fermi surface of Cu
inherit the “twisted” nature of the bulk electronic state of the mother compound. Once superconductivity sets in, the Cooper pairs are formed among bulk electrons in such a non-trivial topological
state. Such a situation is favorable for the realization of odd-parity and topological SC states even for simple pairing interactions, as described in the next subsection. The preserved
topological-insulator surface state after doping was also confirmed in Sr
via quantum oscillation and scanning tunneling microscope (STM) experiments [
3.3. Possible Superconducting States
Just after the discovery of superconductivity in Cu
], Fu et al. performed theoretical analysis on possible SC states realized in this compound [
]. The result is quite surprising since odd-parity topological superconductivity was predicted even with a simple pairing interaction. Previously, it had been believed that an unconventional pairing
glue such as ferromagnetic spin fluctuation is required to realize bulk odd-parity superconductivity. Very naively, the odd-parity superconductivity in this model originates from strong orbital
mixing on the Fermi surface; when a Cooper pair is formed among electrons in different orbitals, odd-parity superconductivity is rather easily realized.
Let us review the result of this theory in a bit more detail. Fu et al. considered the
$D 3 d$
point group of Bi
and assumed Cooper pairing of electrons in two
$p z$
orbitals localized nearly at the top and bottom of a QL. On the other hand, the pairing interaction was assumed to be point-like. Then, six possible pairing states
$Δ 1 a$
$Δ 1 b$
$Δ 2$
$Δ 3$
$Δ 4 x$
, and
$Δ 4 y$
were proposed as listed in
Table 1
. Here, the pairing potential in the orbital bases is expressed with Pauli matrices in the orbital and spin spaces,
$σ μ$
$s μ$
$μ = x , y , z$
); thus, those with off-diagonal terms in the orbital matrices
, i.e., those containing
$σ x$
$σ y$
, are inter-orbital pairing states, and
$σ y$
characterizes an orbital-singlet state [
]. Among them,
$Δ 1 a$
$Δ 1 b$
are even-parity states, and the others,
$Δ 2$
$Δ 3$
$Δ 4 x$
, and
$Δ 4 y$
, are the odd-parity states. All of these odd-parity states belong to topological superconductivity, because odd-parity superconductivity is proven to have a non-trivial topological nature if the
Fermi surface encloses an odd number of high-symmetry points in the Brillouin zone [
]. The states
$Δ 2$
$Δ 4 x$
, and
$Δ 4 y$
matrices in the pairing potential and thus are fundamentally spin-triplet SC states even in the absence of the spin-orbit interaction. In contrast,
$Δ 3$
in the absence of the spin-orbit interaction is a spin-singlet state in spite of the odd-parity nature (notice that
$d = 0$
$λ = 0$
), since the Pauli principle is satisfied together with the orbital degree of freedom [
]. Nevertheless, with finite spin-orbit interaction,
$Δ 3$
, also acquires the spin-triplet nature.
To evaluate the SC-gap and
-vector structures in the reciprocal space, the pair potential in the orbital bases has to be converted to the SC order parameter in the band bases. In the work by Hashimoto et al. [
], the
vector for the lowest order in
was evaluated as listed in
Table 1
. Here, the
vector depends on the ratio
between the spin-orbit interaction (denoted as
in [
]) and the coefficient
$v z$
describing the
$k z$
-linear term in the bulk electronic band dispersion. These values are estimated to be
$v = 4.1$
eV Å and
$v z = 9.5$
eV Å [
]; thus,
is roughly 0.5. In addition, there is a small gapping term
to express the disappearance of the point nodes in the
$Δ 4 y$
pairing [
]. Experimentally, this term is expected to be fairly small [
]. In the bottom row of
Table 1
, the
vector structure in the
space, as well as the gap
$| d ( k ) |$
are schematically shown for a spherical Fermi surface and with the parameters
$λ = 0.5$
$ε = 0.1$
. The
-vector structure of each odd-parity state has a complicated texture on the Fermi surface, being similar to the
-vector structure in the Balian–Werthamer (BW) state (the B phase of superfluid
He) [
], but quite different from a
vector in, e.g., Sr
One can easily notice in the figures that the
$Δ 4 x$
$Δ 4 y$
states have quite characteristic gap structures: The
$Δ 4 x$
state has a pair of point nodes (
$| d ( k ) | = 0$
) along the
$± k y$
direction and
$Δ 4 y$
has a pair of point-like gap minima along the
$± k x$
direction. This existence of a pair of gap node/minima violates the
$C 3$
rotational symmetry of the crystal structure of Bi
. Thus, the
$Δ 4 x$
$Δ 4 y$
states are both gap-nematic SC states. The nodes in
$Δ 4 x$
are protected by the mirror symmetry along the
$a ∗$
plane, whereas the nodes initially existing in
$Δ 4 y$
are gapped out to become gap minima because there is no symmetry protecting the nodes [
]. In addition to the gap amplitude, the
-vector structures of these
$Δ 4$
states also have nematic natures: On the whole Fermi surface, the
vectors preferentially align along the
$k x$
direction in the
$Δ 4 x$
state and along the
$k y$
direction in the
$Δ 4 y$
state. Reminding that a
vector is perpendicular to the Cooper-pair spin and the spin susceptibility should be small along the
vector, it is expected that the spin susceptibility of the
$Δ 4$
states exhibits a two-fold behavior, with minima along the
direction for
$Δ 4 x$
$Δ 4 y$
]. Thus, the
$Δ 4 x$
$Δ 4 y$
states are spin nematic as well.
We should comment on the robustness of the proposed odd-parity states against non-magnetic impurity scattering. Ordinarily, unconventional superconductivity is rather fragile against non-magnetic
impurity scatterings, and such a strong reduction of superconductivity has been found in various non-
-wave superconductors [
]. In doped Bi
superconductors, because of the ion doping, impurity scattering is inevitably stronger than those of stoichiometric unconventional superconductors and can suppress the predicted odd-parity (
$Δ 2$
$Δ 3$
, and
$Δ 4$
) states. However, theories by Michaeli et al. [
] and Nagai et al. [
] proposed that these odd-parity states are rather robust against impurity scatterings because of strong spin-momentum locking in the normal-state Fermi surface. Thus, odd-parity superconductivity
itself, as well as the nodal gap structure can still be stable in doped Bi
3.4. Early Experiments on the Superconducting State in Doped Bi$2$Se$3$
After the discovery of superconductivity in Cu
, various experiments were performed on the SC nature of this compound. As a bulk probe, the temperature dependence of the electronic specific heat of Cu
was studied and was found to be different from the ordinary weak-coupling BCS behavior [
]. A theoretical calculation revealed that this
dependence can be fitted well either with
$Δ 2$
- or
$Δ 4$
-state models [
]. Anomalous suppression of the superfluid density evaluated from the lower critical field
$H c 1$
upon the change of the amount of Cu has been also attributed to the topological superconducting nature [
]. The
dependence of the upper critical field
$H c 2$
was investigated, and from the shape of the
$H c 2 ( T )$
curve, the possibility of unconventional pairing was claimed [
]. However, in general, the shape of a
$H c 2 ( T )$
curve can vary merely due to changes in the Fermi-surface shape [
Several surface-sensitive experiments, seeking for topological Majorana surface states, were also performed. The soft-point-contact spectroscopy was the first experiment revealing the possible
topological nature of the SC state [
]. A zero-bias peak in the differential conductivity as a function of the bias voltage was found, and was attributed to topologically-protected surface states. Indeed, a theoretical calculation
revealed that the observed conductivity is consistent with the odd-parity states [
]. However, shortly after, an STM study on Cu
was performed, and spectra resembling those of fully-gapped
-wave superconductivity were observed. This apparent discrepancy may be due to the dimensionality of the system: if the Fermi surface of Cu
is a quasi-two-dimensional cylinder rather than an ellipsoid (as indeed suggested by the ARPES and quantum oscillation experiment [
]), an STM spectrum on the
$a b$
surface should be indistinguishable from the ordinary
-wave behavior. In contrast, a soft-point contact spectroscopy is expected to collect a certain average of conductivity of various directions, resulting in detecting the zero-bias anomaly in the
$a b$
-plane conductivity. It was also claimed that STM spectra for
-wave superconductivity with an ellipsoidal Fermi surface should form a double peak structure, one originating from the coherence peak of the
-wave gap and the other from the surface state of the topological-insulator nature of Bi
Summarizing the experimental situation before the discovery of nematic superconductivity, there has been evidence for topologically non-trivial superconductivity in Cu$x$Bi$2$Se$3$, but the debate
was yet far from convergence. It has been required to uncover more robust and reproducible properties evidencing an interesting SC state.
4. Recent Experiments on Nematic Superconducting Behavior
In this section, we briefly review recent experimental findings on the nematic superconductivity in doped Bi
superconductors. As summarized in
Table 2
, nematic SC features have been reported quite consistently in all of Cu-, Sr, and Nb-doped Bi
superconductors, as well as in related compounds and have been observed with various bulk probes such as nuclear magnetic resonance (NMR) Knight shift, specific heat, resistivity, and magnetization.
More recently, the STM technique has been successfully utilized to observe nematic features, adding microscopic evidence for the nematic superconductivity.
4.1. Beginning of the Story: Nuclear Magnetic Resonance
The pioneering work on the nematic superconductivity in doped Bi
systems was performed by Matano et al., who investigated the spin susceptibility in the SC state of Cu
with the NMR technique [
]. The spin susceptibility in the SC state is in general rather difficult to measure, because of the strong Meissner screening. The NMR Knight shift is one of the few techniques that can measure the
spin susceptibility directly, and it has been utilized for various superconductors [
Matano et al. used the
Se nucleus (nuclear spin: 1/2) for NMR and investigated the spin susceptibility for various field directions within the
$a b$
plane. They used a set of four single crystals of Cu
with a Cu content
of 0.29–0.31. Usually, the Knight shift contains the spin part and the other parts, and the latter was evaluated using NMR of non-doped Bi
. It was then found that the spin susceptibility decreases by nearly 80% in the SC state for field directions parallel to one of the three equivalent crystalline
axes, but does not change at all for other field directions, as shown in
Figure 3
a. Overall, the spin susceptibility exhibits 180
periodicity as a function of in-plane field direction in spite of the trigonal crystalline lattice, evidencing that the spin part of the SC order parameter in Cu
is actually nematic. This is not only the first clear observation of
$S O ( 3 )$
spin-rotational symmetry breaking in a SC state, but also the first observation of the spin-nematic superconductivity in any known superconductors.
The results of this NMR work were actually known in the community long before the final publication in 2016 and stimulated subsequent studies. In theory, Nagai et al. already in 2012 pointed out that
bulk superconducting properties such as thermal conductivity can exhibit rotational symmetry breaking if one of the
$Δ 4$
states is realized [
]. In 2014, Fu introduced the term “nematic superconductivity” in [
], and this suitable name seems to have contributed significantly to the expansion of the research field.
4.2. Pioneering Reports of Bulk Properties
In early 2016, three subsequent works reporting the nematic bulk SC nature in doped Bi
were independently and almost simultaneously submitted to the arXiv server and were later published in 2016–2017 [
]. As briefly reviewed below, these three works by different groups consistently revealed the nematic nature of different dopants (Cu, Sr, and Nb) and using different experimental probes,
demonstrating that the nematic superconductivity is a robust and ubiquitous feature in doped Bi
The present author and coworkers performed specific-heat measurements of Cu
single crystals under precise two-axis field-direction control [
]. Since the electronic specific heat is quite small in doped Bi
because of low carrier concentration and weak electron correlation, it was necessary to measure the specific heat with high resolution. To achieve this goal, a small and low-background calorimeter
was built utilizing the AC technique [
], which has the highest resolution among the standard heat-capacity measurement techniques. To apply magnetic fields, a vector magnet system was used [
], allowing the performance two-axis field-direction control. With this system, together with a careful field-alignment process, field misalignment effects were minimized. It was then found that the
specific heat as a function of the in-plane field angle exhibited a two-fold symmetric behavior (
Figure 3
b), clearly breaking the lattice
$C 3$
rotational symmetry. This nematicity in a bulk thermodynamic quantity is only possible if the SC gap amplitude has a nematic nature. Thus, this specific-heat result provides the first thermodynamic
evidence for the gap-nematic superconductivity. From a comparison between the observed specific-heat oscillation and theoretical calculation, it was concluded that the
$Δ 4 y$
state was realized. In addition, the upper critical field
$H c 2$
was also found to exhibit two-fold behavior with in-plane anisotropy of 20%, as shown in
Figure 3
b, providing additional evidence for the nematic superconductivity.
Pan et al. measured the in-plane resistivity of Sr
under in-plane magnetic fields and observed in-plane
$H c 2$
anisotropy of around 400% [
], as plotted in
Figure 3
c. In this experiment, strictly speaking, the applied electric current explicitly breaks the in-plane rotational symmetry. Nevertheless, the observed anisotropy is huge and cannot be explained by the
anisotropy due to the applied current. Indeed, the absence of the role of the electric current direction on the SC nematicity was later confirmed by
-axis resistivity measurements [
], as well as by in-plane resistivity measurements upon varying the current direction [
]. This work has another important aspect for demonstrating that a simple technique such as resistivity can probe the nematicity. Shortly thereafter, Nikitin et al. (the same group as Pan et al.)
reported that the nematic SC feature was robust even under hydrostatic pressure [
Asaba et al. investigated Nb
single crystals by means of torque magnetometry under various in-plane field directions [
]. They studied the size of the hysteresis between the field-up and down sweep torque signals. They found that this hysteresis size exhibited clear breaking of the expected six-fold symmetric
behavior as a function of the in-plane field angle (
Figure 3
d). The hysteresis size is actually not a thermodynamic quantity, but is rather related to the vortex pinning and the critical current density of the sample. Thus, it is not very straightforward how
this quantity is related to the nematic SC gap. Nevertheless, it is also difficult to come up with other extrinsic origins on this nematic hysteresis. The mechanism of this interesting appearance of
nematicity in the hysteresis should be clarified in the future.
4.3. Recent Reports
More recently, many other groups reported nematic superconductivity in doped Bi
. In particular, many works have been performed for Sr-doped Bi
. Du et al. measured
-axis resistance under a magnetic field to avoid the symmetry breaking due to the external current and found that the nematicity was still there; thus the external current was not the origin of the
observed two-fold behavior in the in-plane resistivity [
]. More interestingly, they investigated the sample dependence of the nematic behavior and found that the anisotropy of the upper critical field, depicting the nematicity, was actually sample
dependent: among the three samples investigated, two have a large
$H c 2$
for the
direction, but the other has a large
$H c 2$
for the
direction. Such a sample dependence implies that the ground state (
$Δ 4 x$
$Δ 4 y$
) is also sample dependent. This issue will be discussed in more detail in the next section. Smylie et al. investigated the in-plane field-angle dependence of resistivity and magnetization [
]. The observed two-fold behavior in the magnetization provides the first thermodynamic evidence of the gap-nematic superconductivity in the Sr-doped compound. They also investigated the crystal
structure using X-ray diffraction and concluded that there was no detectable crystalline distortion in their sample. Kuntsevich et al. reported in-plane resistivity anisotropy using samples grown
with the Bridgman method [
]. They cut two kinds of samples from the same batch: ones cut along the
axis and the others cut along the
$a ∗$
axis, to check the current-direction dependence of the nematicity. It was found that the nematicity was independent of the current direction. However, the nematicity was actually dependent on
batches, confirming the sample-dependent nematicity reported in [
]. They also reported a tiny crystal deformation, as well as two-fold resistivity anisotropy even in the normal state. This possible “normal-state nematicity” will be discussed in
Section 5.2
. Willa et al. succeeded in measuring the specific heat of Sr
]. Calorimetry of Sr
is more challenging than that of Cu
, because the carrier density and resulting electronic specific heat tend to be much lower in Sr
]. Nevertheless, they used a micro-structured calorimeter to achieve sensitivity high enough for a Sr
single crystal and resolved a specific-heat jump of around 0.25 mJ/K
mol, which is only a fraction of that observed in Cu
]. The two-fold specific-heat behavior was then observed, which was attributed to large
$H c 2$
anisotropy rather than the gap anisotropy.
For Nb-doped Bi
, Shen et al. reported in-plane field-angle dependence of magnetization and resistivity [
]. They found nematicity in both quantities, providing the first thermodynamic and transport evidence for the gap-nematic superconductivity in Nb
. We should also mention that a penetration-depth measurement was performed on Nb
]. The penetration depth was found to exhibit a
$T 2$
-temperature dependence down to
$T / T c ∼ 0.12$
. This behavior is consistent with the existence of point-nodes or point-like very small gap minima. This result provides an indirect support for the nematic
$Δ 4$
states. It was also found that the
$T 2$
dependence is robust against the increase of the impurity concentration. This robustness is consistent with theoretical proposals [
Quite recently, Andersen et al. reported that Cu-intercalated (PbSe)
, which is a “naturally-made heterostructure” of PbSe and Bi
layers and exhibits superconductivity below
$T c ∼ 2.5$
K after intercalation [
], exhibited two-fold anisotropy in the SC state via resistivity and specific-heat measurements [
]. Strictly speaking, the global crystal symmetry of this compound is orthorhombic and does not have three-fold rotational symmetry due to the neighboring PbSe layer. Nevertheless, the Bi
layer of this compound, as well as its electronic structure, almost preserves the three-fold symmetry [
]. Therefore, the observed two-fold anisotropy in the SC state is most likely due to a nematic SC gap, rather than the conventional origin such as Fermi-velocity anisotropy. It is worth commenting
that Cu
substantially differs from
$A x$
in various aspects: Cu
has a highly two-dimensional electronic structure because of the separation of the conductive Bi
layers by insulating PbSe layers and exhibits line-nodal SC behavior [
]. This observation implies that the nematic superconductivity is quite robust irrespective of the dimensionality and gap structure of the system, thus providing important information on the origin
and nature of nematic superconductivity.
4.4. Direct Visualization
Recently, direct visualization of nematic superconductivity by STM has been reported. Chen et al. investigated Bi
thin films (with a typical thickness of 2QL) grown on a FeSe
single-crystal substrate via molecular-beam epitaxy (MBE) [
]. This system is a bit different from doped Bi
, because the Bi
thin film exhibits superconductivity due to the proximity effect from the substrate, whereas doped Bi
exhibits bulk superconductivity. On the other hand, the surface of Bi
can be cleaner than
$A x$
because no ion doping is necessary to induce superconductivity, being more suited for STM investigations. In the quasiparticle interference spectra of Bi
at zero field, the quasiparticle excitation in the intermediate energy range below the SC gap was found to exhibit two-fold anisotropy. The excitation is stronger along the
$± k x$
direction, indicating that the SC gap is smaller for this field direction, thus suggesting the
$Δ 4 y$
state. In addition, they found that the magnetic vortices under
$H ∥ c$
have an ellipsoidal shape elongating along the
) direction, although a nearly isotropic vortex shape is expected for the trigonal crystal structure. Such a vortex core shape provides microscopic evidence for two-fold anisotropy in the penetration
depth and in the coherence length.
Independently, an STM study of Cu
was reported by Tao et al. [
]. In spite of difficulties in finding good surfaces for STM, they succeeded in obtaining STM spectra under magnetic fields. Under the
axis field, they observed ellipsoidal vortex cores elongating along the
$a ∗$
) direction, breaking the
$C 3$
rotational symmetry. Moreover, they investigated the dependence of the gap amplitude on the in-plane field directions. They directly observed that the SC gap amplitude exhibited a two-fold
field-angle dependence. The gap is larger for
$H ∥ y$
than for
$H ∥ x$
at 0.5 T, in favor of the
$Δ 4 x$
state. At higher fields, the large-gap axis rotates by ∼20
. The origin of this rotation is not clear yet.
These STM studies are significant in providing clear microscopic evidence for gap-nematic superconductivity. Furthermore, they are the first observations of nematicity in the absence of an in-plane
magnetic field; it is now confirmed that the nematic superconductivity is the ground state even at zero in-plane field.
5. Known Issues
Although the nematic feature in the SC state has been consistently and reproducibly observed in $A x$Bi$2$Se$3$, as well as in a related compound, there are several controversial or unresolved
issues. These issues are addressed in this section.
5.1. Which of $Δ 4 x$ or $Δ 4 y$ Is Realized?
One of the most important, but puzzling issues is which of
$Δ 4 x$
$Δ 4 y$
is realized in actual samples. As listed in
Table 2
, the NMR [
] and STM [
] studies on Cu
suggested the
$Δ 4 x$
state, but the specific heat study [
] on the same compound suggested the
$Δ 4 y$
state. Moreover, the
$H c 2$
anisotropy, an indicator of the nematic direction, significantly varies: some reports suggest a large
$H c 2$
$H ∥ x$
and others for
$H ∥ y$
. Works on multiple samples indicate that the
$H c 2$
anisotropy is actually sample dependent [
]. Thus, the variation in the
$H c 2$
anisotropy is not an artifact, but is an intrinsic property of
$A x$
This fact very likely suggests that the
$Δ 4 x$
$Δ 4 y$
states are nearly degenerate and are chosen depending on the in-plane symmetry breaking fields. Such a situation is described in
Figure 4
: the nematic SC order parameter, which is reflected in, e.g., in-plane
$H c 2$
anisotropy, exhibits multiple branches as a function of the in-plane symmetry breaking field, such as in-plane uniaxial strain [
], crystal deformation, or some arrangement of doped ions. This coupling between the order parameter and the symmetry-breaking field resembles the coupling between ferromagnetism and external
magnetic fields.
The schematic model in
Figure 4
explains the reason for the variation in the large
$H c 2$
direction: for some samples,
$Δ 4 x$
is favored because of a tiny, but positive symmetry breaking field, and for others,
$Δ 4 y$
is realized due to the negative field. This also explains the absence of the random switching of nematicity orientation upon different coolings: so far, there are no reports that the nematic behavior
is altered by different cooling histories across
$T c$
. Thus, all the samples seem to have finite symmetry-breaking fields that pin the nematicity.
5.2. Normal-State and Superconducting-State Nematicities
This model also explain the relation between the SC nematicity and the possible lattice distortion. Because it is quite important whether the trigonal lattice symmetry is preserved even after the
doping, the existence of lattice symmetry breaking has been investigated. Until recently, there has been no positive evidence for the lattice distortion, as well as for nematic behavior in the normal
state [
]. Recently, Kuntsevich et al. reported that their samples grown using the Bridgman method (in which crystals tend to grow along one direction) had a tiny lattice distortion (0.02% orthorhombic
distortion and
$0.005 °$c
-axis inclination) [
]. These samples exhibit in-plane
$H c 2$
anisotropies ranging from 300–800%. Such large anisotropy cannot be explained at all by the electronic anisotropy caused by the observed tiny lattice distortion. Therefore, SC nematicity is mostly
caused by the Cooper-pair formation, and the concept of nematic superconductivity should still be valid in the presence of explicit breaking of the trigonal symmetry. These samples with relatively
large SC anisotropy and detectable lattice distortion are probably located in the region far from the origin in the schematic in
Figure 4
The nematic SC features observed in Cu
] and Bi
Fe(Se, Te) [
] are also regarded as nematic superconductivity in explicit symmetry-breaking fields. In these systems, the global crystal structures do not possess trigonal rotational symmetry due to the
neighboring non-trigonal layers. Thus, there are finite symmetry breaking fields in these compounds. Still, anisotropies in their normal states are rather small and are not sufficient to explain the
sizable two-fold anisotropies in their SC states.
5.3. Nematic Domains
If the symmetry-breaking field is rather weak, the formation of multiple nematic domains is expected. In the specific-heat study in [
], one sample with possible nematic domains was reported (see the Supplementary Information of [
]). This sample exhibited very weak
$H c 2$
anisotropy, as well as weak and distorted specific-heat oscillation as a function of in-plane magnetic field direction, in contrast to the sample mainly focused on in [
]. Angular magnetoresistance (AMR) is more sensitive to the existence of domains. Indeed, complicated structures in AMR curves attributable to domains were observed in [
However, in most of the other studies, samples seem to be in single-domain states. For example, in the NMR study [
], multiple domains would result in multiple sets of dips in the Knight shift as a function of in-plane field angle. In the actual experiment, only one set of dips was observed, suggesting a
single-domain sample. Such dominance of single-domain samples indicates that the in-plane symmetry breaking field is in most cases strong enough to avoid the formation of multiple domains.
A nematic domain wall, if existing, is a fascinating object, forming naturally a junction between topological pairing states. This may host novel Majorana quasiparticles and may be controllable by an
external symmetry breaking field. Investigation of detailed order-parameter structures near the domain wall would be quite interesting.
5.4. Possible Nematic Superconductivity in Other Systems
In principle, gap-nematic superconductivity can occur in any type of superconductivity, even in ordinary
-wave superconductivity. Nevertheless, a straightforward way to realize nematic superconductivity is to bring a multi-component superconductor (i.e., a superconductor with a multi-dimensional
irreducible representation) and to stabilize one of the SC-order-parameter components. In such multi-component superconductivity, each component is in most cases nematic. However, usually, the
components form a complex linear combination to satisfy the rotational symmetry of the underlying lattice. For example, in the chiral
$p x ± i p y$
-wave superconductivity on a tetragonal lattice (
Figure 1
b), each
$p x$
$p y$
component breaks the tetragonal symmetry, thus possessing the nematic nature. However, they form the complex (chiral) combination
$p x ± i p y$
to satisfy the tetragonal symmetry except in the SC phase degree of freedom. If, however, the formation of such a chiral state is unfavored, one of the nematic components can be stabilized.
Actually, in doped Bi
, the predicted nematic
$Δ 4 x$
$Δ 4 y$
states both belong to the two-dimensional
$E u$
representation, as described in
Table 1
. The two components thus in principle can form complex (chiral) combinations. However, in the present case, the strong spin-momentum locking forces a non-unitary SC state to emerge when the chiral
superconductivity is realized [
]. Notice that the
vector of the chiral state
$d chiral = d 4 x ± i d 4 y ∼ ( k z ± i ε k x , ∓ i k z , λ ( k x ± i k y ) )$
has a complex spin component: see, for example, along the
$k z$
$d chiral ∼ ± k z ( x ^ − i y ^ )$
is non-unitary (
$d × d ∗ ≠ 0$
). Generally, non-unitary states have spin-dependent excitation gaps, and usually, one spin component has a significantly smaller gap than the other. Thus, non-unitary states are expected to have
smaller condensation energies than the ordinary unitary SC states. This inevitable formation of non-unitary gaps in chiral states prevents the formation of the complex linear combination of
$d 4 x$
$d 4 y$
and favors the realization of single-component nematic superconductivity in this system.
From the discussion above, a multi-component superconductor is a promising platform to probe nematic superconductivity. Indeed, in UPt
with a trigonal crystal structure
$P 3 ¯ m 1$
], multi-component superconductivity is believed to be realized [
] with one of the components relatively stabilized by the short-ranged antiferromagnetic ordering [
]. In the in-plane field-angle dependence of the thermal conductivity, two-fold symmetric behavior was observed in the C phase [
]. This phenomenon can be considered as a consequence of nematic superconductivity with a finite in-plane symmetry-breaking field (see
Figure 4
). In PrOs
with a cubic structure
$I m 3 ¯$
, similar two-fold behavior in the field-angle-dependent heat transport was observed in a part of the
phase diagram [
]. Strictly speaking, two-fold symmetry in the magneto-transport is allowed in the space group
$I m 3 ¯$
. Nevertheless, it seems difficult to explain the clear two-fold anisotropy just due to the electronic-state anisotropy, and gap nematicity due to a multi-component order parameter may be playing an
important role. On the other hand, two-fold behavior has not been observed in the specific heat of both superconductors [
]. Thus, the nematic feature of these compounds, if existing, is subtle compared with those observed in doped Bi
superconductors. Theoretically, several other multi-comoment-superconductor candidates such as U
$1 − x$
] and half-Heusler compounds [
] were proposed to exhibit nematic superconductivity, and experimental verification is strongly called for.
We also comment on another leading candidate of multi-component superconductivity, Sr
, seemingly exhibiting quasi-two-dimensional chiral
$p x ± i p y$
-wave superconductivity. Theoretically, it has been predicted that a non-chiral single-component
$p x$
$p y$
) state [
] should be realized under the in-plane magnetic field or in-plane uniaxial strain. Such non-chiral states may be regarded as a symmetry-breaking field-induced nematic state, or a “meta-nematic”
state in an analogy to the metamagnetic transition (field-induced ferromagnetism) in a paramagnet. Although clear experimental observation has not been achieved yet [
], it is worth seriously investigating the nematic features of Sr
under in-plane symmetry breaking fields. Lastly, we mention that nematic superconductivity in Sr
at ambient condition was theoretically proposed very recently in [
6. Summary and Perspectives
To summarize, we have reviewed recent researches on nematic superconductivity in doped Bi
topological superconductors and in related compounds. The two-fold symmetric behavior in many quantities, breaking the trigonal symmetry of the underlying lattice, has been reported by more than ten
groups, with excellent reproducibility. These experimental works demonstrate that the nematic superconductivity is a common and robust feature among the
$A x$
family. In addition, observation of nematic gap structures in turn provide bulk evidence for topological superconductivity in this family [
]. However, there are several issues unresolved; in particular, apparent inconsistency in the nematic direction and its relation to the possibly existing in-plane symmetry-breaking field are the most
important subjects to be investigated next. Furthermore, predicted novel phenomena originating from nematic superconductivity, such as the superconductivity-fluctuation-induced nematic order above
$T c$
], the chiral Higgs mode in the electromagnetic response [
], spin polarization of Majorana quasiparticles in a vortex core [
], and the nematic Skyrmion texture near half-quantum vortices [
], would be worth seeking.
The nematic SC state in $A x$Bi$2$Se$3$ is qualitatively unique compared with other nematic systems in liquid crystals or normal-state electron systems: The nematic superconductivity is realized by a
macroscopically-coherent quantum-mechanical wavefunction, accompanied by an odd-parity nature, non-trivial topology, and an active spin degree of freedom. We believe that this new class of nematic
states stimulates further researches both in the fields of nematic liquids and unconventional topological superconductivity.
This research was funded by the Japanese Society for Promotion of Science (JSPS) Grant Numbers KAKENHI JP15H05851, JP15H05852, JP15K21717, JP26287078, and JP17H04848.
The author acknowledges collaboration with K.T., S.N., I.K., Y.M. at Kyoto University, Z.W. and Y.A. at the University of Köln, K.S. at Kyoto Sangyo University, and Y.N. at Japan Atomic Energy
Agency. The author also thanks M.K. for valuable comments to improve the manuscript, and K.I., Y.M., T.M., M.S., L.F., K.M., G.-Q.Z., and D.A. for valuable discussion and support.
Conflicts of Interest
The author declares no conflict of interest.
The following abbreviations are used in this manuscript:
AMR Angular magnetoresistance
ARPES Angle-resolved photoemission spectroscopy
BCS Bardeen–Cooper–Schrieffer
BG Bridgman method
BW Balian–Werthamer
ECI Electrochemical intercalation
MBE Molecular-beam epitaxy
MG Melt growth
NMR Nuclear magnetic resonance
QL Quintuple layer
SC Superconducting
STM Scanning tunneling microscope
vdW van der Waals
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105. Hecker, M.; Schmalian, J. Vestigial nematic order and superconductivity in the doped topological insulator Cu[x]Bi[2]Se[3]. npj Quantum Mater. 2018, 3, 26. [Google Scholar] [CrossRef]
106. Uematsu, H.; Mizushima, T.; Tsuruta, A.; Fujimoto, S.; Sauls, J.A. Chiral Higgs Mode in Nematic Superconductors. arXiv, 2018; arXiv:1809.06989. [Google Scholar]
107. Nagai, Y.; Nakamura, H.; Machida, M. Spin-Polarized Majorana Bound States inside a Vortex Core in Topological Superconductors. J. Phys. Soc. Jpn. 2014, 83, 064703. [Google Scholar] [CrossRef] [
Green Version]
108. Zyuzin, A.A.; Garaud, J.; Babaev, E. Nematic Skyrmions in Odd-Parity Superconductors. Phys. Rev. Lett. 2017, 119, 167001. [Google Scholar] [CrossRef] [PubMed]
Figure 1. Schematic comparison of various known superconductivity and gap/spin nematic superconductivity, for the case of a tetragonal-lattice system.
Figure 2.
Schematic description of the crystal structure of the mother compound Bi
]. The purple spheres are the Bi atoms, and the green and light-blue spheres are the Se(1) and (2) atoms, respectively. The colors of the spheres are modified depending on the depth along the view
direction: atoms closer to the view point have thicker colors. (
) View from the
axis direction. The intercalated metallic ions most likely sit in the van der Waals (vdW) gap between the quintuple layers (QL). (
) View from the
axis direction. The crystal structure figures were made using the software VESTA-3 [
Figure 3.
Representative experiments on the nematic superconductivity in
$A x$
. (
) In-plane field-angle dependence of the NMR Knight shift of Cu
]. (
) In-plane field-angle dependence of the specific heat and
$H c 2$
of Cu
]. (
) In-plane angular dependence of
$H c 2$
evaluated from magnetoresistance measurements on Sr
]. (
) In-plane field-angle dependence of the irreversible component of the magnetic torque of Nb
]. We should be careful for the definition of the field angle: in (
), 0
corresponds to
$H ∥ a$
; whereas in (
), it corresponds to
$H ∥ a ∗$
. The panels (
) are respectively quoted from [
] with the permission of Springer Nature; (
) are respectively from [
] under the Creative Commons License.
Figure 4. Schematic figure of the coupling between the nematic order parameter and the in-plane symmetry breaking field.
Table 1.
Proposed superconducting states for doped Bi
]. The
vector structures in the band bases are from [
]. Here,
represents the strength of the spin-orbit coupling; and
represents the gap minima for the
$Δ 4 y$
state (see the text). In the bottom row, schematic gap and
-vector structures of each state are shown, together with various cut views. The
value is chosen to be 0.5, and the
value is just set to be 0.1. The sphere at the center of a cut view is the Fermi surface. The gap structure is expressed with colored surfaces, whose distance from the Fermi surface corresponds to
the SC gap amplitude
$| d |$
normalized by its maximal value
$d 0$
. The color of this surface also depicts the gap value, as well as the
-vector direction, as explained in the left bottom cell: The hue and lightness of the color indicate the azimuthal and polar angles of the
$ϕ d$
$θ d$
, respectively; whereas the grayness of the color depicts the normalized gap, with 50% gray corresponding to
$| d | = 0$
$Δ 1 a$ $Δ 1 b$ $Δ 2$ $Δ 3$ $Δ 4 x$ $Δ 4 y$
Irreducible representation $A 1 g$ $A 1 g$ $A 1 u$ $A 2 u$ $E u$
Pairing potential $σ 0$ $σ x$ $σ y s z$ $σ z$ $σ y s x$ $σ y s y$
$d$-vector - - $∼ ( λ k x , λ k y , k z )$ $∼ ( − λ k y , λ k x , 0 )$ $∼ ( k z , 0 , − λ k x )$ $∼ ( ε k x , − k z , λ k y )$
Parity even even odd odd odd
Topo.SC no no yes yes yes
Nematic SC no no no no yes
Schematic $d$-vector/gap structures
Table 2. Comparison of experimental reports on nematic superconductivity in doped Bi$2$Se$3$ and related systems.
Material Reference Growth Method $i$ Doping Level x Probe Large $H c 2$ Suggested State
Matano 2016 [8] MG + ECI 0.29–0.31 NMR y $Δ 4 x$
Cu$x$Bi$2$Se$3$ Yonezawa 2017 [9] MG + ECI 0.3 C x $Δ 4 y$
Tao 2018 [70] MG + ECI 0.31 STM - $Δ 4 x$
Pan 2016 [10] MG 0.10, 0.15 $ρ a b$ x
Nikitin 2016 [71] MG 0.15 $ρ a b$ in P x
Du 2017 [72] MG NA $ρ c$ x (#1, #2)
Sr$x$Bi$2$Se$3$ y (#3)
Smylie 2018 [73] MG 0.1 $ρ a b$, M x
Kuntsevich 2018 [74] BG 0.10–0.20 $ρ a b$ x (some)
y (others)
Willa 2018 [75] MG 0.1 C y
Nb$x$Bi$2$Se$3$ Asaba 2017 [11] MG NA torque -
Shen 2017 [53] MG 0.25 $ρ a b$, M y
Cu$x$(PbSe)$5$(Bi$2$Se$3$)$6$ Andersen 2018 [76] BG + ECI 1.5 $ρ a b$, $ρ c$, C x $Δ 4 x$
Bi$2$Te$3$/Fe(Se, Te) Chen 2018 [77] MBE - STM - $Δ 4 y$
$i$ MG: melt growth, ECI: electrochemical intercalation, BG: Bridgman method, MBE: molecular beam epitaxy.
© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://
Share and Cite
MDPI and ACS Style
Yonezawa, S. Nematic Superconductivity in Doped Bi[2]Se[3] Topological Superconductors. Condens. Matter 2019, 4, 2. https://doi.org/10.3390/condmat4010002
AMA Style
Yonezawa S. Nematic Superconductivity in Doped Bi[2]Se[3] Topological Superconductors. Condensed Matter. 2019; 4(1):2. https://doi.org/10.3390/condmat4010002
Chicago/Turabian Style
Yonezawa, Shingo. 2019. "Nematic Superconductivity in Doped Bi[2]Se[3] Topological Superconductors" Condensed Matter 4, no. 1: 2. https://doi.org/10.3390/condmat4010002
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Public University of Navarre
Module/Subject matter
Quantitative Methods: Econometrics.
First, we will introduce the idea of econometric model. This model must take into account the special features of economic data. We will focus on the ideas of causality and ceteris paribus analysis,
arising from the correct interpretation of these models. From a formal viewpoint, the initial step towards an appropriate conceptual framework will be to introduce the simple regression model. Here,
we will study the basic assumptions, interpretation of parameters of interest, ordinary least squares estimation and statistical properties of the estimators.
The second building block of the course is the study of the general regression model. We will analyze its basic properties, focusing on the motivation behind multivariate regression, stressing its
usefulness over the bivariate framework which characterizes the simple regression model. Again, we will study in this more general framework the basic assumptions, interpretation of parameters of
interest, ordinary least squares estimation and statistical properties of the estimators.
In the last part of the course, we will study different aspects related to the general regression model. First, we will analyze the consequences of the violation of basic assumptions (functional form
and specification, heteroskedasticity, autocorrelation). Next, we will introduce the issue of endogeneity, its relation with the instrumental variables estimator (emphasizing the search for
appropriate instruments in economics) and a brief discussion of this problem in the context of systems of equations. The course will end up with a brief introduction on qualitative information
variables, focusing on dummy variables and on the linear probability model.
Economic data. Causality. Simple regression model. General regression model. Violation of assumptions. Endogeneity and instrumental variables. Qualitative information variables.
General proficiencies
CG04. Oral and written communication in a foreign language.
CG05. Developing software knowledge applied to the corresponding subject.
CG06. Ability to analyze and extract information from different sources.
CG07. Capacity to solve problems.
CG09. Capacity to work in teams.
CG16. Capacity to work under pressure.
CG17. Capacity to self-learning.
Specific proficiencies
CE02. Identify relevant economic information sources.
CE03. Derive from micro and macroeconomic data relevant information impossible to assess by non-specialists.
CE04. Use of professional criteria in the economic analysis, mainly those criteria based on technical tools.
Learning outcomes
R20. Simple regression and explanatory variables models, econometric models.
At the end of the course, the student must be able to translate into mathematical language specific issues from macroeconomics and microeconomics by means of econometric models. Additionally, the
student must be able to use the econometric software Gretl and extract conclusions from the estimated models.
Presentation of the main theoretical aspects of the course. Student participation: questions made by the lecturer, brief presentations of previous lectures.
Classes in computer room. Small groups. Presentation and revision of exercises made individually and in small groups. Use of econometric packages (GRETL).
Individual and team work
Preparation of exercises and presentations.
Periodic tutorials with lecturer
Individual and group meetings.
Personal study and exam
Activity Hours
On-site 60
Lectures 30
Classes 30
Self-preparation 90
Self-study 38
Individual preparation of exercises and presentations 26
Team preparation of exercises and presentations -
Exam preparation 20
Tutorials 06
Others -
Learning outcome Evaluation Method Weight (%) Recoverable
R18. Simple regression and explanatory variables models Theoretical Tests 20% Recoverable
R18. Simple regression and explanatory variables models Participation and exposition in lectures 10% Non-recoverable
R18. Simple regression and explanatory variables models Tests in computer classroom 20% Non-recoverable
R18. Simple regression and explanatory variables models Final exam 50% Recoverable
Those students who do not attend the final exam will get a grade of "No Presentado"
Chapter 1. Introduction: the nature of econometrics and economic data
What is econometrics?
Methodology in econometric analysis
The structure of economic data
Causality and the notion of ceteris paribus in econometric analysis
Chapter 2. The simple regression model
Definition of the simple regression model
Ordinary least squares estimator
Algebraic properties of the ordinary least squares estimator
Units of measurement and functional form
Statistical properties of the ordinary least squares estimator
Chapter 3. The general regression model
Algebraic properties of the ordinary least squares estimator
Statistical properties of the ordinary least squares estimator
Gauss-Markov theorem
Hypotheses testing
Asymptotic properties
Model selection
Chapter 4. Violation of assumptions
Specification errors
Chapter 5. Endogeneity and instrumental variables
Instrumental variables estimator
Access the bibliography that your professor has requested from the Library.
The main references for the course are:
Wooldridge, J.M. "Introductory econometrics: a modern approach". South-Western College Pub; 4 edition (2008).
Stock, J.H. and Watson, M.W. "Introduction to econometrics". Prentice Hall (2010)
Alternative useful textbooks are:
Goldberger, A.S. "Introductory econometrics ". Harvard University Press (1998).
Gujarati, D.N. "Basic econometrics". Mc. Graw Hill (2004)
Classroom and Computer room at the Aulario. | {"url":"https://www.unavarra.es/ficha-asignaturaDOA/?languageId=1&codPlan=174&codAsig=174403&anio=2018","timestamp":"2024-11-09T12:50:37Z","content_type":"text/html","content_length":"20328","record_id":"<urn:uuid:9fe6cfdc-543b-4d9f-9134-62cae5d72dfb>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00307.warc.gz"} |
Test2::Tools::Spec(3) User Contributed Perl Documentation Test2::Tools::Spec(3)
Test2::Tools::Spec - RSPEC implementation on top of Test2::Workflow
This uses Test::Workflow to implement an RSPEC variant. This variant supports isolation and/or concurrency via forking or threads.
use Test2::Bundle::Extended;
use Test2::Tools::Spec;
describe foo => sub {
before_all once => sub { ... };
before_each many => sub { ... };
after_all once => sub { ... };
after_each many => sub { ... };
case condition_a => sub { ... };
case condition_b => sub { ... };
tests foo => sub { ... };
tests bar => sub { ... };
All of these use the same argument pattern. The first argument must always be a name for the block. The last argument must always be a code reference. Optionally a configuration hash can be inserted
between the name and the code reference.
FUNCTION "name" => sub { ... };
FUNCTION "name" => {...}, sub { ... };
The first argument to a Test2::Tools::Spec function MUST be a name. The name does not need to be unique.
This argument is optional. If present this should be a hashref.
Here are the valid keys for the hashref:
If this is set to true then the block will not render as a subtest, instead the events will be inline with the parent subtest (or main test).
Set this to true to mark a block as being capable of running concurrently with other test blocks. This does not mean the block WILL be run concurrently, just that it can be.
Set this to true if the block MUST be run in isolation. If this is true then the block will run in its own forked process.
These tests will be skipped on any platform that does not have true forking, or working/enabled threads.
Threads will ONLY be used if the T2_WORKFLOW_USE_THREADS env var is set. Thread tests are only run if the T2_DO_THREAD_TESTS env var is set.
Use this to mark an entire block as TODO.
Use this to prevent a block from running at all.
This argument is required. This should be a code reference that will run some assertions.
This defines a test block. Test blocks are essentially subtests. All test blocks will be run, and are expected to produce events. Test blocks can run multiple times if the "case()" function is
also used.
"it()" is an alias to "tests()".
These ARE NOT inherited by nested describe blocks.
This lets you specify multiple conditions in which the test blocks should be run. Every test block within the same group ("describe") will be run once per case.
These ARE NOT inherited by nested describe blocks, but nested describe blocks will be executed once per case.
Specify a codeblock that should be run multiple times, once before each "tests()" block is run. These will run AFTER "case()" blocks but before "tests()" blocks.
These ARE inherited by nested describe blocks.
Same as "before_each()", except these blocks run BEFORE "case()" blocks.
These ARE NOT inherited by nested describe blocks.
Specify a codeblock that should be run once, before all the test blocks run.
These ARE NOT inherited by nested describe blocks.
Specify a codeblock that should wrap around each test block. These blocks are run AFTER case blocks, but before test blocks.
around_each wrapit => sub {
my $cont = shift;
local %ENV = ( ... );
The first argument to the codeblock will be a callback that MUST be called somewhere inside the sub in order for nested items to run.
These ARE inherited by nested describe blocks.
Same as "around_each" except these run BEFORE case blocks.
These ARE NOT inherited by nested describe blocks.
Same as "around_each" except that it only runs once to wrap ALL test blocks.
These ARE NOT inherited by nested describe blocks.
Same as "before_each" except it runs right after each test block.
These ARE inherited by nested describe blocks.
Same as "after_each" except it runs right after the case block, and before the test block.
These ARE NOT inherited by nested describe blocks.
Same as "before_all" except it runs after all test blocks have been run.
These ARE NOT inherited by nested describe blocks.
These are shortcuts. Each of these is the same as "tests()" except some parameters are added for you.
These are NOT exported by default/.
Same as:
tests NAME => { flat => 1 }, sub { ... }
Same as:
tests NAME => { iso => 1 }, sub { ... }
Same as:
tests NAME => { mini => 1, iso => 1 }, sub { ... }
Same as:
tests NAME => { async => 1 }, sub { ... }
Note: This conflicts with the "async()" exported from threads. Don't import both.
Same as:
tests NAME => { minit => 1, async => 1 }, sub { ... }
Sometimes you want to apply default attributes to all "tests()" or "case()" blocks. This can be done, and is lexical to your describe or package root!
use Test2::Bundle::Extended;
use Test2::Tools::Spec ':ALL';
# All 'tests' blocks after this declaration will have C<<iso => 1>> by default
spec_defaults tests => (iso => 1);
tests foo => sub { ... }; # isolated
tests foo, {iso => 0}, sub { ... }; # Not isolated
spec_defaults tests => (iso => 0); # Turn it off again
Defaults are inherited by nested describe blocks. You can also override the defaults for the scope of the describe:
spec_defaults tests => (iso => 1);
describe foo => sub {
spec_defaults tests => (async => 1); # Scoped to this describe and any child describes
tests bar => sub { ... }; # both iso and async
tests baz => sub { ... }; # Just iso, no async.
You can apply defaults to any type of blocks:
spec_defaults case => (iso => 1); # All cases are 'iso';
Defaults are not inherited when a builder's return is captured.
spec_defaults tests => (iso => 1);
# Note we are not calling this in void context, that is the key here.
my $d = describe foo => {
tests bar => sub { ... }; # Not iso
As each function is encountered it executes, just like any other function. The "describe()" function will immediately execute the codeblock it is given. All other functions will stash their
codeblocks to be run later. When "done_testing()" is run the workflow will be compiled, at which point all other blocks will run.
Here is an overview of the order in which blocks get called once compiled (at "done_testing()").
for-each-case {
# AND/OR nested describes
Copyright 2016 Chad Granum <exodist7@gmail.com>.
This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
See http://dev.perl.org/licenses/ | {"url":"https://man.docs.euro-linux.com/EL%208/perl-Test2-Suite/Test2::Tools::Spec.3pm.en.html","timestamp":"2024-11-03T21:37:14Z","content_type":"text/html","content_length":"25773","record_id":"<urn:uuid:5f78b21e-9a08-4532-a841-8fd558e79dba>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00590.warc.gz"} |
Autoregressive model
autoregressive statistical model for time series data
Autoregressive models are a class of statistical models used for modeling time series data. The basic idea is to predict a variable's future values based on its own past values.
Autoregressive models are widely used in various fields, including economics, finance, engineering, and environmental science. For example, they are commonly used to model stock prices, temperature
variations, and electrical signals. They are also a fundamental building block in more complex models like ARIMA (Autoregressive Integrated Moving Average) and SARIMA (Seasonal ARIMA), which combine
autoregressive components with moving averages and differencing to model more complex time series behaviors.
One of the key assumptions of autoregressive models is stationarity, which means that the statistical properties of the time series do not change over time. If the data is not stationary, it may need
to be transformed or differenced before applying an autoregressive model.
Estimating the parameters of an autoregressive model typically involves methods like Maximum Likelihood Estimation (MLE) or Least Squares Estimation. Once the model is fitted, it can be used for
forecasting future values, anomaly detection, or as a component in more complex models.
However, autoregressive models have limitations. They are essentially linear models and may not capture complex, nonlinear relationships in the data. They also assume that the system being modeled is
influenced only by its own history, ignoring any potential external factors.
In summary, autoregressive models are a fundamental tool for time series analysis, offering a way to forecast future values based on past observations. While they are widely applicable and relatively
simple to understand and implement, they do have limitations in terms of capturing nonlinearity and external influences. Nonetheless, they serve as a foundational element in many more complex models
and algorithms used for time series forecasting and analysis. | {"url":"https://feedsee.com/aiw/Autoregressive_model/","timestamp":"2024-11-09T07:25:56Z","content_type":"text/html","content_length":"4240","record_id":"<urn:uuid:036ffc60-c4b3-45ba-9ec6-c14aa37e99ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00162.warc.gz"} |
Timely differentially expressed genes
Last seen 9 months ago
Dear community,
I am analyzing microarray data of patients followed from birth up to an age of 3 years. The samples were taken at various age points for each child.
I want to identify non constant gene expression over time within these children, so genes that are differentially expressed over time.
For this I have applied a mixed effects model with splines(df=3) in limma adjusting for sex as a covariate.
X <- ns(age, df=3)
The topTable function results in various differentially expressed genes, I have set the coef parameter to coef=1:3, including the columns X1,X2 X3.
My question is how are the log2foldchanges in the result table of topTable() interpreted?
Many thanks in advance
Are you sure that coef=1:3 is correct? Usually coef=1 is the intercept.
Thanks for your help! Actually I am not so sure, as from my background I am not particular strong in mathematics. I have fitted the model without an intercept and used the columns of the output
matrix of the ns() function as coef=1:3
No, that is wrong. The ns() function assumes that an intercept is included in the model, as is the default in R. Why would you remove it?
The default in the ns() function is intercept = FALSE, but the lm function adds than an intercept, right ? I played a bit with the example that was given by James MacDonald to get a better
understanding and feeling for the analysis. Would the following approach be correct ?
X <- ns(age, df = 3)
design <- model.matrix( ~ X + gender , metadata )
fit <- lmFit(expressionMatrix, design=design,
block=patient_id, correlation=cor)
fitE <- eBayes(fit)
topTable(fitE, n=Inf, coef = 2:4, sort.by="B",
adjust.method ="BH", p.value = 0.05, lfc = 0.14)
In this case what would the lfc threshold mean ? It is not really interpretable as far as I have understood from the answer below because it refers to coefficients in the spline model and not the
difference between two conditions ?
Would it be reasonable to take the maximum log2 foldchange within each predicted trajectory as a measure of how strong a gene is differentially expressed and also as a measure to cut off genes from
the list that show very little variation in gene expression over time ?
You mean maximum logFC between any pair of times? Yes, you could do that. I do not generally recommend ad hoc cutoffs like that however. Usually it is better to simply rank the genes by p-value. The
empirical Bayes prcoedure of limma already gives you a lot of protection against identifying small fold changes as DE so adding an ad hoc fold-change cutoff is usually unnecessary.
Thanks. If I consider all significant genes without any maximum logFC criteria, the gene with the lowest expression change over time, has about 4 % percent maximum change over time ~0.06 logFC. Can
this still be physiological relevant ?
I have never seen such a small logFC coming up as significant in any of my own analyses. You must have a large number of samples and, if you do, then applying a max logFC cutoff could be reasonable.
Unfortunately we don't have a TREAT method for spline trends.
Thanks, but what is the lowest that you have seen in such circumstances ? And did you have systematically calculated it actually or just read off for some genes from the predicted curves? So far I
have only worked with treatment experiments, and I know in these cases the foldchanges are often enormous, but I am lacking any feeling for genes that are just observed over age without any
treatment. Regarding the change I don't know if a 4 % change can have physiological effects. If I think of a guy who is 180 cm and a guy that is 187.2 cm, the difference is quite obvious, I don't
know how much this translates to the world of transcriptomics. The biggest lFC in my data is around 1, even this I consider quite low from what I have observed in previous treatment experiments
Entering edit mode
Took a bit to get the image posted, as imgur no longer provides direct links...
Entering edit mode
Thanks for your answer! I have done something very similar with the degPatterns() function. But I would be interessted to assess from the topTable output, if a particular gene is increasing more over
time compared to another gene without plotting the particular genes. The genes that fall into one cluster according to degPatterns() have very similar logFC value patterns for X1, X2 und X3, but I am
not sure how to interpret these. I have defined a gene as differentially expressed when padj <=0.05 and the absolute logFC>=0.14 in at least one column.
Entering edit mode
You don't interpret the logFC for a spline fit, so you cannot use the output from topTable to assess anything. As an example, try doing this:
> library(splines)
> example("ns")
> summary(fm1)
lm(formula = weight ~ ns(height, df = 5), data = women)
Min 1Q Median 3Q Max
-0.38333 -0.12585 0.07083 0.15401 0.30426
Estimate Std. Error t value Pr(>|t|)
(Intercept) 114.7447 0.2338 490.88 < 2e-16 ***
ns(height, df = 5)1 15.9474 0.3699 43.12 9.69e-12 ***
ns(height, df = 5)2 25.1695 0.4323 58.23 6.55e-13 ***
ns(height, df = 5)3 33.2582 0.3541 93.93 8.91e-15 ***
ns(height, df = 5)4 50.7894 0.6062 83.78 2.49e-14 ***
ns(height, df = 5)5 45.0363 0.2784 161.75 < 2e-16 ***
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2645 on 9 degrees of freedom
Multiple R-squared: 0.9998, Adjusted R-squared: 0.9997
F-statistic: 9609 on 5 and 9 DF, p-value: < 2.2e-16
We have an intercept and estimates for each of the five knots in the spline. And the plot that is also produced by the example code shows a line with a slight upward curve. Would you be able to look
at the coefficients for this model and say anything about the underlying relationship between height and weight, in particular that there is a curve and in which direction it curves? I can't. But I
can sure tell you that the spline fit is better.
> anova(fm1, update(fm1, .~ 1 + height))
Analysis of Variance Table
Model 1: weight ~ ns(height, df = 5)
Model 2: weight ~ height
Res.Df RSS Df Sum of Sq F Pr(>F)
1 9 0.6298
2 13 30.2333 -4 -29.604 105.76 1.47e-07 ***
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
And I can also provide a plot with the spline fit and the linear fit and provide the above ANOVA table to show that the former fits better than the latter. But the interpretation comes only from the
plot, rather than the output. | {"url":"https://support.bioconductor.org/p/9154115/","timestamp":"2024-11-13T05:54:50Z","content_type":"text/html","content_length":"41672","record_id":"<urn:uuid:86454473-0d13-484c-bd12-76dbfd948eee>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00355.warc.gz"} |
Physics-Informed Deep Learning-Based Proof-of-Concept Study of a Novel Elastohydrodynamic Seal for Supercritical CO
Supercritical carbon dioxide (sCO[2]) power cycles show promising potential of higher plant efficiencies and power densities for a wide range of power generation applications such as fossil fuel
power plants, nuclear power production, solar power, and geothermal power generation. sCO[2] leakage through the turbomachinery has been one of the main concerns in such applications. To offer a
potential solution, we propose an elastohydrodynamic (EHD) seal that can work at elevated pressures and temperatures with low leakage and minimal wear. The EHD seal has a very simple, sleeve-like
structure, wrapping on the rotor with minimal initial clearance at micron levels. In this work, a proof-of-concept study for the proposed EHD seal was presented by using the simplified Reynolds
equation and Lame’s formula for the fluid flow in the clearance and for seal deformation, respectively. The set of nonlinear equations was solved by using both the conventional Prediction–Correction
(PC) method and modern Physics-Informed Neural Network (PINN). It was shown that the physics-informed deep learning method provided good computational efficiency in resolving the steep pressure
gradient in the clearance with good accuracy. The results showed that the leakage rates increased quadratically with working pressures and reached a steady-state at high-pressure values of 15∼20 MPa,
where Q = 300 g/s at 20 MPa for an initial seal clearance of 255 μm. This indicates that the EHD seal could be tailored to become a potential solution to minimize the sCO[2] discharge in power
sCO2, deep learning, physics-informed neural networks, PINN, alternative energy resources, power (co-) generation, energy conversion/systems, energy systems analysis, seal, sealing, gas leakage
Clearances (Engineering), Design, Flow (Dynamics), Leakage, Physics, Pressure, Rotors, Supercritical carbon dioxide, Turbomachinery, Sealing (Process), Artificial neural networks, Thermodynamic power
1 Introduction
The production of efficient, cost-effective, and environmentally friendly power is an issue that affects everyone on the planet. Over the past century, power plants have been the primary source of
power production. Although there is an unprecedented effort for greener energy generation—including on-shore and off-shore windmills and fuel cells—power plants continue and will continue to be one
of the major energy sources for years to come. Unfortunately, this technology has not developed rapidly enough to keep up with the power demand from society [1]. Thermal efficiency has been a major
design concern in power plants, and there are continuous efforts to increase efficiency. However, the traditional open Brayton cycles or indirect-fired, closed Rankine cycle technologies have reached
their saturation as far as efficiencies are concerned. Different working fluids, such as supercritical carbon dioxide (sCO[2]), are being explored to determine their impact on the efficiency of power
cycles [2]. The use of sCO[2] as a working fluid in steam power plants has shown a potential to produce higher plant efficiency and power density, lower water consumption, reduced physical footprint,
and overall, more cost-effective power generation [3–11].
If successful, sCO[2] power cycles have the potential to be used in a wide range of power generation applications, including fossil fuel-based power plants, nuclear power production, solar power, and
geothermal power generation. sCO[2] power cycles can offer several benefits, including higher cycle efficiencies because of sCO[2]’s unique fluid and thermodynamic properties, less emissions, compact
turbomachinery, and reduced plant size, cost-effectiveness, rapid response to load transients, lesser water usage as well as water-free capability in dry-cooling applications, and heat source
flexibility. However, technology readiness must be demonstrated for 10–600 MWe power plants and at sCO[2] temperatures and pressures of 350–700 °C and 20–35 MPa to unlock the full potential of sCO[2]
power cycles [12]. Advanced seal technology plays a vital role in improving the overall efficiency of a power cycle. Seals are mainly used to minimize gas recirculation and gas leakage from the
primary path. Gas recirculation and leakage occur in rotor stages, cavities, and stator stages, while gas leakage from the primary path occurs in flanges, vane pivots, and the compressor end [13].
The lack of suitable seals at sCO[2] operating conditions is one of the main challenges at the component level [14]. As a remedy, there is a worldwide effort to develop effective seal designs for sCO
[2] power technology.
Some of the current sealing solutions that are utilized in this method of power generation are labyrinth seals, dry gas seals, compliant foil seals, and finger seals. Labyrinth seals are considered
to be the more conventional seal design. Dry gas seals are also used in larger supercritical CO[2] power cycles. Other advanced contact seals like finger and brush seals were also developed for this
application. These conventional seals are discussed in detail in the following paragraphs.
Labyrinth seals are used in turbomachinery to control leakage between high- and low-pressure zones. These seals can be used in gas, steam, and hydraulic turbines [15]. This seal utilizes a set of
teeth that face the spinning rotor. Since the seal never contacts the rotor, this seal is considered a non-contact seal. As the fluid travels through the teeth, pressure continually decreases
minimizing the overall leakage rate. Due to the intrinsic design of the seal, leakage rates are heavily dependent on the operating conditions and critical dimensions for each application, which
include pressure, shaft speed, and shaft diameter. In general, the leakage rate of labyrinth seals is proportional to the gap area and is inversely proportional to the number of teeth on the seal [16
]. In most cases, leakage decreases by almost 20% in high shaft speed conditions [17]. It has been shown that the efficiency losses could be as high as 0.65% for a 500 MWe utility-scale power plant
when labyrinth seals are used, which is too large to be neglected [18]. As for environmental concerns, excessive CO[2] leakages contribute to the greenhouse effect and thus have a negative impact on
the environment.
Brush seals are another common sealing solution used in turbomachinery. These seals consist of brushes that are aligned with a rotor in the tangential direction. These seals allow the rotor to move
due to vibration or thermal expansion. Due to this property, these seals offer lower leakage rates than labyrinth seals. A backing plate in the axial direction supports the bristles and prevents them
from bending in the rotational direction of the rotor. This seal design is considered a contact seal as the brushes contact the spinning rotor [19]. Since the seals contact the spinning rotor, these
seals suffer from material wear significantly quicker when compared to non-contact sealing solutions. They also cause wear on the rotor surface. To improve the efficiency of the seal and to retrofit
the design to multiple applications, many different parameters of the seal can be adjusted. These changes include bristle thickness, lay angle, bristle length, backing plate gap, and front plate gap.
In some applications, multiple brush seals are used to achieve the desired leakage rate.
Compliant foil seals offer a similar sealing solution to both labyrinth seals and brush seals. These seals are mostly used in secondary flow systems of aircraft engines [20]. These seals consist of a
seal membrane, a floating seal ring, a seal base, and several elastic supporting sheets. The compliant seal can move and adjust to any changes in rotor diameter caused by thermal expansion and/or
vibrations. This movement is due to the elastic supporting sheets. Since the seal never touches the spinning rotor, this seal design is considered a non-contact seal. The life span of the seal is
significantly better due to this no-contact operating configuration [21]. This design allows leakage to be minimized in the small gaps of secondary flow systems [22]. There are several reasons why
this seal design has not gained traction in the supercritical CO[2] power cycle and turbomachinery application. First is the perception that this design is only good for lightweight and high-speed
rotors. Second is inexperience in foil bearing design leading to the belief that this design is not robust enough for demanding environments like high loads and temperature. Researchers are currently
working on developing this design and change the perception of this sealing solution [23].
Finger seals have a lower manufacturing cost when compared to brush seals. This seal design is also able to better adapt to thermal changes than labyrinth seals. Due to these characteristics, finger
seals have been adopted for use in various aero-engine applications [24]. Like brush seals, finger seals fall into the same category as contact seals. This is because the fingers contact the rotor.
Over time, this results in wearing and malfunctioning in various components of the seal. In most cases, the fingers are made from either metal or carbon/composite. Finger seals are a significant
advancement from labyrinth seals as they offer lower leakage rates and higher overall efficiency. It was also found that for many years during the development of this seal, researchers focused on the
leakage through the radial clearance of the finger seal. The leakage that made it past the finger seal bundles was not considered when calculating the efficiency of the seal. This resulted in a lower
overall efficiency rating for the seal once all leakage points were considered. Due to all the components in this seal, the complexity is a major downfall of this specific seal design.
Dry gas seals have several key benefits over other traditional mechanical sealing systems for turbomachinery [25]. This seal design can withstand significantly high rotational speeds (20,000–30,000
rpm) without the aid of a traditional oiling system. They also offer extremely competitive leakage rates when compared to other high-performance shaft-end seals. Due to the nature of the seal design,
dry gas seals are able to work efficiently with a wide variety of gasses and operating conditions [26]. These benefits have enabled this seal design to replace traditional oil ring seals in
compressors. Dry gas seals can also be run in dual and triple seal configurations when sealing volatile fluid systems. These configurations help to further reduce the leakage of these harmful
chemicals into the environment [25]. The multistage design also allows for each stage to be optimized to handle a certain pressure value resulting in an overall more efficient seal. Due to the
complexity of the seal, labyrinth seals are still more likely to be selected for this specific application.
So far, conventional seals all suffer from the incapability of handling sCO[2] pressure and temperature in one way or another. There is an ongoing worldwide effort to develop effective sealing
technologies for sCO[2] turbomachinery [27–31]. White et al. [2] conducted a much-needed review of sCO[2] power generation technology recently. According to their comprehensive review, the existing
sCO[2] turbomachinery designs employ mostly labyrinth seals, including the sCO[2] test loops at Sandia National Laboratories, Southwest Research Institute/General Electric, and Echogen Power
Systems—all in the United State—or dry gas seals such as in Supercritical CO[2] Integral Experiment Loop (SCIEL) at KAIST in South Korea. In terms of power capacity, labyrinth and dry gas seals are
used for the power ranges of 0.3–10 MWe and 10–500 MWe, respectively, with an overlap between 3 and 10 MWe. The review points out that the efforts concentrate on the application and development of
dry gas seals for utility-scale sCO[2] power plants. In this line of work, recently, Kim et al. [16] presented both numerical and experimental studies of the sCO[2] critical flow model for their
labyrinth seal design, and the isentropic flow was verified to model the CO[2] leakage flow mechanism. The gap effect was simulated and reviewed for a single-phase flow. Their simulation model was
developed based on choked flow equation, Hodkinson’s equation, and labyrinth seal design equation. Then, they compared their simulation results with the experimental data. Their experimental results
included data from both single- and two-phase flows. Overall, their study provided useful information to further understand the performance characteristics of labyrinth seals, leading to an optimal
seal design.
Bidkar et al. [18] discussed significant design challenges in the context of a representative sCO[2] face seal. Their seal has a diameter as big as 24 in. and a pressure differential higher than
about 1000 psia. In their study, they indicated the expected operating conditions for shaft-end seals on utility-scale sCO[2] turbines. Unique characteristics of sCO[2] were considered and a
hydrodynamic face seal concept was proposed for end sealing application. They performed a computational fluid dynamics (CFD) analysis of the face seal film and demonstrated the feasibility of their
concept by investigating the thermal analysis. Their study showed promise in replacing existing labyrinth seal and work at high working pressure.
Cao et al. [32] took an approach to increase the overall system stability for sCO[2]. They established a dynamic three-dimensional numerical model of a staggered labyrinth seal as a potential
solution. They utilized a previously proposed CFD model for transient flow to calculate and predict the dynamic force coefficients for the seal. The proposed seal was considered to operate under
various rotor axial shifting distances, rotor convex plate heights/widths, seal cavity heights, and clearance conditions. Subsequently, they compared the results with conventional sealing systems and
showed that their system is comparatively more stable.
Hylla et al. [33] worked on implementing carbon floating ring seals (CRS) in sCO[2] systems. They demonstrated the theoretical model of CRS to predict the physical behavior of the leakage flow. They
performed experimental investigations using an integrally geared compressor test rig. They used three different seals and found out that the pressure difference along the axial directions
significantly changed. Then, they validated their pressure difference and mass flowrate results by comparing them with the numerical results of the model. This is a significant study as it shows
promising evidence of a desired decrease for both pressure along the seal and mass flowrate.
Zhu et al. [34] proposed a reasonably simplified computational model for labyrinth seal by neglecting the compressibility of sCO[2]. They provided literary proof that there is no exact computational
model for the internal flow characteristic of sCO[2] labyrinth seals. Their numerical model is universally applicable to the relevant empirical coefficients, including flow discharge coefficient and
kinetic residual coefficient. Additionally, they introduced the homogeneous two-phase flow model and modified some essential computational parameters to compute the two-phase outlet conditions. They
presented the latest experimental results of round-hole, see-through, and stepped-staggered sCO[2] labyrinth seals and verified their 1-D computation method as well as sealing performances with the
experimental results. They successfully demonstrated the consistency of the 1-D method with the experimental results. The higher efficiency of their proposed staggered seal over other types was also
justified. This study provides a potential way of simplifying numerical methods.
Dry gas seals for supercritical fluid Brayton cycles have no well-established design guidelines. Fairuz and Jahn [35] conducted a study to understand the performance of dry gas seals. It is
challenging because of the non-linearity and unpredictable thermodynamic behavior of sCO[2] as the working fluid. Their study included a CFD investigation of fluid leakage at possible locations of
the seal in operating condition. They investigated two inlet conditions with both real fluid and ideal gas. They also implemented a comparison study to understand real gas effects that affect CO[2]
thermal and transport properties near the critical point. Then, they presented their comparison results for the same seal geometry operating with air and CO[2] at four operating points. Furthermore,
they validated their results by comparing their predictions with previously published experimental results. This study is critical because it gives insight into designing dry gas seals for sCO[2]
power cycles.
As one of the key players in the turbomachinery industry, non-contacting dry gas seals can also be a potential solution for sCO[2] power cycles. Armin et al. [36] proposed a design solution for such
a non-contacting dry gas seal for CO[2] power cycles. They designed, fabricated, and tested their proposed design for liquid, gaseous, liquid–gas, and supercritical phases. Their result shows an
isothermal seal expansion due to low leakage while maintaining a non-contacting dynamic operation. Moreover, they ran a deliberate failure test to justify the low leakage demand.
Closed-loop Brayton cycles have the highest efficiency with sCO[2] as a working fluid. However, conventional shaft-end seal limits extracting its maximum efficiency. Spiral groove seal has shown much
potential to replace shaft-end seal as an alternative in recent studies. Yan et al. [37,38] studied the influence of different turbulence effects on the spiral grooves dry gas seal performance.
Theoretical and experimental tests were conducted on sCO[2] dry gas seal prototypes for isothermal or adiabatic flow. For the theoretical study, leakage rates and flow fields under different
operating conditions were calculated by coupling the Reynolds and energy equations for the test conditions. They were then compared with the experimental test results. Their study shows that the
turbulence effect has a recognizable impact on the leakage rate. In the future, they want to extend their study to investigate structural deformations of the seal.
In a similar effort, Rimpel et al. [39] worked on restricting existing sealing limitations for recompression Brayton cycles. They presented a new test rig design for a conceptual 24-in. film riding
face seal applicable in 450 MWe utility-scale sCO[2] turbines. The test rig was designed for a shaft speed of 3600 rpm, maximum supply temperature of 400 °F, upstream cavity pressure of 75 bar, and
maximum downstream cavity pressure of 10 bar. Their analysis shows that excessive thrust load and downstream cavity pressure in the test rig would be mitigated in case there are any seal failure
scenarios. This study is a work in progress. This study shows how to imitate the natural environment without any probable accident while running the experimental analysis.
There are still major roadblocks in the full realization of sCO[2] power technology such as high leakage rates through shaft-end seals [40,41]. High leakage rates are not desired, because of their
negative impact not only on efficiency but also on the environment. As far as efficiency is concerned, sCO[2] power cycles operate on closed-loop cycles such as steam Rankine cycles. Thus, the sCO[2]
past the shaft-end seal cannot be condensed back to the liquid phase to be recovered and fed back to the system. It must be recompressed to the sCO[2] states, which will require additional power
input, i.e., additional compressor power, penalizing the efficiency of the whole cycle. Each seal design has merits in both efficiency and production cost depending on the application. More recently,
non-contact seals like film riding seals and hydrodynamic seals are being developed to further decrease leakage past the seal and improve efficiency. Non-contact seals have been found to offer less
leakage and longer life since they are not in contact with the rotor-like finger and brush seal designs. Researchers are currently working toward creating a non-contact seal that can be used in
supercritical CO[2] power cycles and other high-pressure turbomachinery applications. The hydrodynamic face seal is one of the best candidates for sCO[2] operating conditions due to its ability to
withstand large pressure differentials. These pressure differentials can be as high as 1000 psia. Currently, large-scale hydrodynamic face seals are not commercially available due to both
manufacturing and design challenges.
Despite these efforts, there is still a need for effective sealing technologies for sCO[2] turbomachinery. To offer a potential solution, we propose an EHD seal that can work at elevated pressures
and temperatures with low leakage and minimal wear. To offer a potential solution, we propose an EHD seal that can work at elevated pressures and temperatures with low leakage and minimal wear. The
EHD seal has a very simple, sleeve-like structure, wrapping on the rotor with minimal initial clearance. The combined effect of the pressures at the top and bottom of the seal forces the seal to bend
downward to provide a throttling effect in the clearance, which minimizes leakage [42–48].
The foundation of the elastohydrodynamic seal concept dates back to 1886 when Osborne Reynolds published a paper discussing the pressure distribution and load-carrying capacity of fluid films [49].
EHD lubrication, specifically, has made major strides in the last few decades. The first major publication involving this topic specifically was published in the late 1930s. Since then, the study of
elastohydrodynamic lubrication has played a critical role when designing a variety of machine components. The components that utilize this phenomenon include anything from bearings to gears [29].
Due to the complexity and nature of elastohydrodynamic lubrication, computer modeling and Computational Fluid Dynamics (CFD) are widely used to help understand and apply this lubrication technique to
many different applications, including single and multistage seals for turbomachinery [50–52].
Next, the novelty of the paper is discussed. In this paper, we are utilizing the simplified Reynolds equation and Lame’s formula to model the fluid flow and structural deformation of the seal,
respectively. This requires the solution of a set of highly nonlinear, coupled differential equations. Solving these nonlinear equations using conventional optimization methods requires careful
selection of the flowrates and often results in convergence issues. Physics-informed deep learning methods are becoming popular and have shown the potential to solve complex nonlinear equations [53–
74]. It is one of the most advanced tools to solve partial differential equation problems numerically [75,76]. PINN infers unknown solutions for physics of interest by using the limited amount of
given data and governing equations of conservation laws [77]. It provides a paradigm shift into data-driven modeling for physical systems. PINN’s framework is relatively simple, meshless, and can be
applied successfully for high-speed flow solutions [78]. The novelty of the paper comes from the verification of the proposed seal design by using modern solution tools, such as PINN, for
physics-based analytical modeling.
In this work, the design equations are solved using PINN. The results are discussed to verify the working mechanism of the proposed seal design. Also, the advantages of the PINN method over the
conventional Prediction–Correction (PC) methods are discussed.
The rest of the article is organized as follows: Sec. 2 introduces the proposed seal design and discusses the design methodology, the governing mathematical equations for the flow inside the
clearance and seal deformation along with the details of conventional PC and PINN methods. The results are presented and discussed in Sec. 3. Conclusions and future work are presented in Secs. 4 and
5, respectively.
2. Materials and Methods
2.1 Proposed Elastohydrodynamic Seal Design.
The proposed EHD seal utilizes the proven electrohydrodynamic lubrication sealing mechanism [79–82], which eliminates wear and reduces both leakage and overall cost. As shown in Fig 1(a), the main
EHD seal is attached to a back ring. The back ring sits on a rotor with a cold clearance of h, where P[0] = P[e]. Once the rotor reaches operating speed, pressure P[0] >> P[e], which results in a
decaying pressure distribution in the clearance (Fig 1(b)). In this condition, the pressure along the top of the seal is equal to the operating pressure P[0]. Since the root of seal is fixed, i.e.,
it is welded onto the back ring, the pressures at the top and bottom of the seal will cause the seal to deform to create the minimum possible clearance for the given conditions. One conclusion one
might reach is that the seal would bend downward to contact the rotor and block the flow. However, when such a contact occurs the pressure differential across the contact region will force the seal
to open, allowing flow to continue. Recall that P[0] > P[e] is always during operation.
Based on the discussion earlier, there will always be a viable minimum clearance between the EHD seal and the rotor. This will provide the necessary throttling effect to minimize the leakage. The
proposed EHD seal design for sCO[2] turbomachinery has several benefits which include:
• Low leakage. The self-regulated minimum clearance throttles the sCO[2] leaking flow, improving the cycle efficiency.
• Minimal wear. The EHD seal operates in non-contact conditions.
• Low cost. The simple sleeve structure results in low seal cost and minimal wear, saving maintenance costs.
• No stress concentration. The EHD seal design eliminates sharp angles and stress concentration risks.
2.2 Design Methodology.
The flow inside the thin clearance region can be considered a Couette flow and is given by a relationship between the pressure variation and leakage rate [
]. The relationship between the pressure (
) variation along the clearance and leakage rate (
) is given by
is the clearance along the sleeve,
is the density, and
is the viscosity. In this work, the seal fixation at the root of the seal is not considered for simplicity when the seal deformation is evaluated. Using Lame’s formula [
] for a thick-walled cylinder, the clearance can be obtained as
is the initial clearance,
is the working pressure, and
are the coefficients given by
is Young’s modulus, and
are the inner and the outer diameters of the sleeve, respectively. The diameters are given by
is the diameter of the rotor and
is the EHD seal. The boundary conditions on the inlet and outlet boundaries are
is the working pressure acting on the seal and
is the pressure at the outlet, which is equal to the atmospheric pressure. The viscosity is a function of pressure and is given by the Barus equation
denotes the dynamic viscosity of the working fluid at the atmospheric pressure, and
is the pressure–viscosity coefficient. The density is a function of pressure and is given by the Dowson–Higginson formula
is the density of the working fluid at the atmospheric pressure. Equations
constitute the isothermal EHD governing equations for the high-pressure sleeve. The equations are highly nonlinear, and two different methods to solve these equations are discussed next.
2.3 Prediction–Correction Algorithm.
With the fluid properties being a function of pressure, Eq. (1) is highly nonlinear. The importance behind Eq. (1) is that the mass leakage rate, Q, is equal to the product of fluid properties and
pressure gradient at any position along the length of the clearance. The pressure will vary along the length of the sleeve, but the principle of mass conservation limits the Q to be constant. The
pressure and mass leakage rate are the variables to be solved for, and the boundary conditions are the constraints on Eq. (1). The boundary condition on the right end is the atmospheric pressure, P
[e] and on the left end is the working pressure, P[0]. To solve Eq. (1), in the first iteration, a value for Q is assumed and the pressure is solved from the inlet to the outlet using an Ordinary
Differential Equation (ODE) solver. The value of Q is adjusted based on the error between the pressures obtained from ODE and the actual pressure at the right end. Careful selection of Q is required
to solve the ODE. A large value of Q results in convergence issues in ODE solver due to resulting in negative pressures. Using the Newton-Raphson method, steepest descent, or other optimization
methods results in infeasible values for Q, resulting in convergence issues. A PC method is used to find the value of Q for the given boundary conditions.
The method to carefully solve for the value of leakage rate Q using the PC method can be seen in Fig. 2. To begin, a starting value for the leakage rate is chosen, Q^i, and the flow equation along
the clearance, Eq. (1), is solved using an ODE solver with pressure at the inlet, P[0], as the initial condition until x = L. Using the pressure at the outlet x = L and the ambient pressure, P[e],
the error (ɛ) is calculated.
If the error is below the specified tolerance, the mass leakage rate is decreased; otherwise, the mass leakage rate is increased. The ODE is then solved again with a new leakage rate, Q^i^+1. The
loop of predicting the pressure field, along the clearance, for a given flowrate, Q^i, and updating the Q^i based on the error in the predicted pressure at the outlet, is repeated until the absolute
value of ɛ is below a threshold (ɛ[tol]). This method helps with the selection of a leakage rate, Q, for a given inlet pressure P[0]. This method is known to result in convergence issues when the
flow equations are being solved for high inlet pressure values, P[0]. Because of this, a parametric sweep will be needed to ramp the inlet pressure up from ambient value (P[x][= L]) to the actual
working pressure (P[0]).
The parametric sweep needed to solve the flow equations for a final working pressure, P[0], is outlined in Fig. 3. Using the parametric method, the pressure field (P) and leakage rate (Q) are solved
for using ambient air pressure as the starting point. The working pressure is then increased by ΔP. The resulting flow equation is solved using the PC with the obtained Q[j] as the initial flowrate
as shown in Fig. 2. Using a parametric sweep with the PC method, to solve the flow equations Eq. (1), for any given working pressure P[0] makes this method computationally expensive. Other methods
such as physics-informed deep learning method are proposed as a more efficient solving method.
2.4 Physics-Informed Deep Learning.
Physics-informed deep learning can be used to solve the flow equations, Eqs. (1)–(8), for any boundary pressure condition. The architecture used for solving the flow equations using a PINN can be
seen in Fig. 4. The network is made of several input and output layers. These layers are connected by a network of hidden layers. Each layer contains artificial neurons. The output of each neuron is
passed onto the next layer as an input. As this process happens, the neurons do a weighted sum of the inputs and add a bias to the sum. An activation function is also applied to the output of each
neuron before it is passed onto the next layer as an input. Each neuron is associated with a bunch of weights and a bias. The number of weights for each artificial neuron is equal to the number of
inputs to the neuron. The neural networks are very robust in generating a complex function between multiple inputs and outputs of the training data by identifying patterns in the data [83–88].
Training the neural network is finding the weights and bias of each neuron using the data of interest. This training requires minimizing the loss function [89], which allows the neural network to
predict output values close to the actual values for unseen/new data points. The sum of squares of the difference between the predicted and actual output values is known as the loss function. The
methodology of identifying a nonlinear map between high dimensional input–output data seems to be a naïve task [75]. The PINN helps to supplement the neural networks to solve some of the most complex
nonlinear mathematical equations of any physical phenomena. It is well known to solve two classes of problems: data-driven solution and data-driven discovery of partial differential equations (PDE).
A data-driven solution of a partial differential equation style problem will be solved in this work.
The flow equations, Eqs.
, must first be converted to a dimensionless form so that the pressure and leakage rate (
) are of the same scale when solving for both flow and seal deformation within the clearance of the seal. The following parameters are used for scaling:
for diameters,
for axial dimensions,
for clearance,
for pressure and E Young's modulus, 1/
for pressure–viscosity coefficient,
for viscosity,
for density, and
for mass leakage. The final governing equations in the dimensionless form are given by
The boundary conditions for the dimensionless equations are given by
An additional constraint of the constant value of leakage rate along the clearance is used to solve the flow equations using PINN. The constraint is applied by equating the spatial derivate of the
leakage rate to zero as shown below
To solve Eqs.
by using the PINN, the spatial variable
is considered as the input and the pressure
and leakage rate
, as the output. The goal of the PINN is to estimate a value of
at any input
for given boundary conditions by finding the weights and bias of the NNs. First, random points are generated inside the domain, where Eq.
is trained. The loss function for the current problem will be the sum of squares errors in the boundary conditions, and the errors in the differential equation as shown below
The first term in the above loss equation is due to the error in Eq.
for the predicted pressure by the PINN and is given by
is the number of random points selected inside the domain.
are the pressure and the leakage rate predicted by the PINN. The second term in Eq.
is due to the error in Eq.
for the predicted value of leakage rate by the PINN and is given by
The losses due to the boundary conditions at the inlet and outlet are given by
The boundary conditions at inlet and outlet are given by Eqs. (15) and (16). The main goal of the PINN is to determine a pressure field and leakage rate which minimizes the total loss function. This
means that if the PINN selects the exact value for both pressure field and leakage rate for a given input pressure or boundary condition, the field satisfies the Eqs. (10) and (17), which results in
the total loss function equaling zero. The important parameter in the PINN is the choice of the activation function and in this work, tanh is used as the activation function. The Adam optimizer is
used to train the PINN by minimizing the total loss, i.e., Eq. (18). Then, the weights of the artificial neurons are calculated. The PINN needs to be trained for each pressure boundary condition and
the weights of the neural networks (NN) are different for each boundary condition. However, the weights obtained for one boundary condition could be used as initial weights of the PINN for training
at different inlet pressures, and it reduces the training time for new inlet pressure conditions.
3 Results and Discussions
The accuracy of the PINN is investigated by solving the flow equations along the clearance considering the fluid as liquid and comparing the results with the solution obtained from the PC algorithm.
The main fluid properties and parameters used in this investigation can be seen in Table 1. In this work, the python programming language was used to execute the discussed algorithms. The SciPy
package [90] is utilized for the ODE solution algorithm, and the TensorFlow package [58] is used for the PINN method. The domain is divided into a uniform grid of 1000 nodes for the ODE, and the
nodes are selected randomly for the PINN.
With the outlet pressure set to atmospheric pressure, the flow equations are solved for different working pressure conditions. The PC method is used to solve flow equations. This was done by using
20,000 iterations, and the step size of 1e5 Pa is used for the parametric sweep. For the PINN, the input is the spatial variable, x, and the output is the scaled pressure and leakage rate. Four
layers with 30 neurons in each layer are used. Figure 5 shows the variation of training loss with the iteration number of the Adams optimizer for a typical case. The training of PINN is stopped when
the loss reaches 1e−5.
To solve the flow equations for a larger value of working pressure (P[0]) using the PC method, the parametric sweep with the starting value of P[0] as the ambient pressure (P[e]) is used. Also, the
resulting leakage rate (Q) is used as the starting value for the next step as shown in Fig. 3. In each step of the parametric sweep, the pressure is increased by a value of 1e5 Pa, whereas the PINN
can solve the flow equations for any arbitrary value of working pressure (P[0]) without any parametric sweep. This demonstrates that PINN is more powerful than conventional optimization techniques
due to its nature of solving complex nonlinear equations efficiently.
In this study, both the PC and PINN methods were compared using 200 simulation cases. The cases varied in working pressures (P[0]) between 0.2 and 20 MPa. Figure 6 shows the comparison of percentage
Root Mean Square (RMS) of $Lossdp¯$ using the PC and the PINN method for different working pressure conditions. It was found that the RMS loss was random in the case of PINN. This is due to random
initialization [91] of the weights and biases of artificial neurons in NN for every simulation. Using the PC method, the RMS loss continuously increased. Using both methods, the $Lossdp¯$ is below
The scatter plot of percentage RMS loss in the pressure boundary condition at the outlet (Loss[BC2]) for different simulation cases with working pressures between 0.2 and 20 MPa is shown in Fig. 7.
The Loss[BC2] increases with working pressure using both the PINN and PC methods. The PINN method performed marginally better than the PC method. Using both methods, the loss values are below 1.2%,
where ∼1.2% is singled out for PINN in the vicinity of the highest pressure of 20 MPa. All else is below 0.8%. A graphical comparison of the predicted pressure for both the PINN and PC can be seen in
Fig. 8. For the PINN, the scaled pressure is obtained, and the actual pressure is calculated by multiplying the scaled value by P[0]. It shows that both methods provide the same results. It was found
that the pressure curve decreases along the clearance using both methods. For lower working pressures (P[0] = 0.2 MPa), the pressure decreases almost linearly, and for higher working pressures (P
[0]), the decrease is more nonlinear. The variation of clearance (h[c]) under different working pressures (P[0]) is calculated using Eq. (11) and is shown in Fig. 9. For lower working pressures, the
clearance decreases almost linearly toward the outlet, and for higher working pressures, there is a steeper contraction towards the outlet.
Using both the PINN and PC methods, the leakage rate (Q) is calculated using different working pressures (P[0]). A comparison of both methods is shown in Fig. 10. It was found that both methods
generated similar results. The leakage rate, Q, increases with initial working pressures. Once the pressure reaches higher values, the increase in leakage rate begins to level off and stabilize.
For different boundary conditions, similar pressure fields and leakage rates were obtained using both methods. It was found that a parametric sweep was needed to solve for higher working pressures (P
[0]) using the PC method. This was not the case for the PINN method as it can be applied for any condition.
In addition to discussion of the solution methodologies, it is also important to discuss the physical meaning of the obtained solutions. Figure 8 confirms the decaying pressure distribution in the
clearance. It was found that the pressure distribution becomes more nonlinear as the working pressures increase, especially after P[0] = 11 MPa. This is also shown by the clearance thickness
displayed in Fig. 9. After P[0] = 11 MPa, the clearance decreases more rapidly, particularly after axial length, x = 18 mm. This is due to the nonlinear effects of pressure on the density, ρ, and
viscosity, μ, which are felt more strongly at higher pressures.
Recalling that the seal root at x = 26.5 mm is fixed to the back ring, the clearance thicknesses shown in Fig. 9 will result in seal deformations similar to those shown in Fig 1(b), which proves the
hypothesis discussed in Sec. 2.1 as far as achievable by the simplified Reynolds equation and Lame’s formula. To obtain similar seal deformations depicted in Fig 1(b), one needs to modify the
boundary conditions and implement them in the solution procedures discussed in this work. This is left as a part of future work along with others discussed in the next section.
4 Conclusions
The main outcomes of this study can be summarized as follows:
• A proof-of-concept study for a novel EHD seal was presented by using the simplified Reynolds equation and Lame’s formula for a thick-walled cylinder.
• The set of the nonlinear equation was solved by using the conventional PC method and PINN.
• The solutions obtained from both methods are closely matched. However, the PC method required the parametric sweep to converge at high working pressures to avoid convergence issues, which
resulted in large computational times.
• The pressure field decreased linearly from inlet to outlet at lower working pressures, whereas the decrease was nonlinear at higher working pressures.
• The clearance decreased linearly from the inlet to outlet at lower working pressures and nonlinearly at higher working pressures.
• The clearance thickness results proved that there would be a throat happening closer to the root of the seal.
• The leakage rate followed a quadratic trend, making its peak around P[0] = 20 MPa, where Q = 300 g/s.
• The proposed seal could be tailored to minimize the leakage rates to become a potential candidate for sCO[2] power technology.
5 Future Work
As for future work, we will attempt to investigate the following:
• In the present analysis, the root of the seal is not fixed, and we will implement additional boundary conditions to incorporate that.
• We will perform a parametric sweep analysis for different values of rotor diameter, seal thickness, seal length, elastic modulus, and so on.
• In the present work, we picked the working fluid to be a liquid for proof-of-concept purposes. However, we will attempt to incorporate the real gas properties of sCO[2], which seems to be rather
• In the present study, the effect of temperature on viscosity and density was neglected. In future work, we will attempt to incorporate the temperature effects as well.
• We will incorporate the rotational effects in future analyses.
• Most importantly, we will design and fabricate a 2″ static test rig to demonstrate the EHD seal experimentally.
This work has been financially supported by the U.S. Department of Energy (DOE) STTR Phase I and II grants under Grant No. DE-SC0020851, in which Dr. Sevki Cesmeci and Dr. Hanping Xu serve as the
Principal Investigator and Project Manager, respectively. This paper reflects the views of the author(s) only and does not necessarily reflect the views or opinions of DOE. The intellectual property
of the proposed seal concept is protected under Patent No. 63/200,712, “Clearance seal with an externally pressurized sleeve.”
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
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class mindspore.nn.Optimizer(learning_rate, parameters, weight_decay=0.0, loss_scale=1.0)[source]
Base class for updating parameters. Never use this class directly, but instantiate one of its subclasses instead.
Grouping parameters is supported. If parameters are grouped, different strategy of learning_rate, weight_decay and grad_centralization can be applied to each group.
If parameters are not grouped, the weight_decay in optimizer will be applied on the network parameters without ‘beta’ or ‘gamma’ in their names. Users can group parameters to change the strategy
of decaying weight. When parameters are grouped, each group can set weight_decay. If not, the weight_decay in optimizer will be applied.
☆ learning_rate (Union[float, int, Tensor, Iterable, LearningRateSchedule]) –
○ float: The fixed learning rate value. Must be equal to or greater than 0.
○ int: The fixed learning rate value. Must be equal to or greater than 0. It will be converted to float.
○ Tensor: Its value should be a scalar or a 1-D vector. For scalar, fixed learning rate will be applied. For vector, learning rate is dynamic, then the i-th step will take the i-th
value as the learning rate.
○ Iterable: Learning rate is dynamic. The i-th step will take the i-th value as the learning rate.
○ LearningRateSchedule: Learning rate is dynamic. During training, the optimizer calls the instance of LearningRateSchedule with step as the input to get the learning rate of current
☆ parameters (Union[list[Parameter], list[dict]]) –
Must be list of Parameter or list of dict. When the parameters is a list of dict, the string “params”, “lr”, “weight_decay”, “grad_centralization” and “order_params” are the keys can be
○ params: Required. Parameters in current group. The value must be a list of Parameter.
○ lr: Optional. If “lr” in the keys, the value of corresponding learning rate will be used. If not, the learning_rate in optimizer will be used. Fixed and dynamic learning rate are
○ weight_decay: Optional. If “weight_decay” in the keys, the value of corresponding weight decay will be used. If not, the weight_decay in the optimizer will be used.
○ grad_centralization: Optional. Must be Boolean. If “grad_centralization” is in the keys, the set value will be used. If not, the grad_centralization is False by default. This
configuration only works on the convolution layer.
○ order_params: Optional. When parameters is grouped, this usually is used to maintain the order of parameters that appeared in the network to improve performance. The value should be
parameters whose order will be followed in optimizer. If order_params in the keys, other keys will be ignored and the element of ‘order_params’ must be in one group of params.
☆ weight_decay (Union[float, int]) – An int or a floating point value for the weight decay. It must be equal to or greater than 0. If the type of weight_decay input is int, it will be
converted to float. Default: 0.0.
☆ loss_scale (float) – A floating point value for the loss scale. It must be greater than 0. If the type of loss_scale input is int, it will be converted to float. In general, use the
default value. Only when FixedLossScaleManager is used for training and the drop_overflow_update in FixedLossScaleManager is set to False, this value needs to be the same as the
loss_scale in FixedLossScaleManager. Refer to class mindspore.amp.FixedLossScaleManager for more details. Default: 1.0.
Supported Platforms:
Ascend GPU CPU
Apply Broadcast operations in the sequential order of parameter groups.
optim_result (bool) – The results of updating parameters. This input is used to ensure that the parameters are updated before they are broadcast.
bool, the status flag.
Weight decay.
An approach to reduce the overfitting of a deep learning neural network model. User-defined optimizers based on mindspore.nn.Optimizer can also call this interface to apply weight decay.
gradients (tuple[Tensor]) – The gradients of network parameters, and have the same shape as the parameters.
tuple[Tensor], The gradients after weight decay.
Flatten gradients into several chunk tensors grouped by data type if network parameters are flattened.
A method to enable performance improvement by using contiguous memory for parameters and gradients. User-defined optimizers based on mindspore.nn.Optimizer should call this interface to
support contiguous memory for network parameters.
gradients (tuple[Tensor]) – The gradients of network parameters.
tuple[Tensor], The gradients after flattened, or the original gradients if parameters are not flattened.
The optimizer calls this interface to get the learning rate for the current step. User-defined optimizers based on mindspore.nn.Optimizer can also call this interface before updating the
float, the learning rate of current step.
When parameters is grouped and learning rate is different for each group, get the learning rate of the specified param.
param (Union[Parameter, list[Parameter]]) – The Parameter or list of Parameter.
Parameter, single Parameter or list[Parameter] according to the input type. If learning rate is dynamic, LearningRateSchedule or list[LearningRateSchedule] that used to calculate the
learning rate will be returned.
>>> from mindspore import nn
>>> net = Net()
>>> conv_params = list(filter(lambda x: 'conv' in x.name, net.trainable_params()))
>>> no_conv_params = list(filter(lambda x: 'conv' not in x.name, net.trainable_params()))
>>> group_params = [{'params': conv_params, 'lr': 0.05},
... {'params': no_conv_params, 'lr': 0.01}]
>>> optim = nn.Momentum(group_params, learning_rate=0.1, momentum=0.9, weight_decay=0.0)
>>> conv_lr = optim.get_lr_parameter(conv_params)
>>> print(conv_lr[0].asnumpy())
The optimizer calls this interface to get the weight decay value for the current step. User-defined optimizers based on mindspore.nn.Optimizer can also call this interface before updating the
float, the weight decay value of current step.
Gradients centralization.
A method for optimizing convolutional layer parameters to improve the training speed of a deep learning neural network model. User-defined optimizers based on mindspore.nn.Optimizer can also
call this interface to centralize gradients.
gradients (tuple[Tensor]) – The gradients of network parameters, and have the same shape as the parameters.
tuple[Tensor], The gradients after gradients centralization.
Restore gradients for mixed precision.
User-defined optimizers based on mindspore.nn.Optimizer can also call this interface to restore gradients.
gradients (tuple[Tensor]) – The gradients of network parameters, and have the same shape as the parameters.
tuple[Tensor], The gradients after loss scale.
property target
The property is used to determine whether the parameter is updated on host or device. The input type is str and can only be ‘CPU’, ‘Ascend’ or ‘GPU’.
property unique
Whether to make the gradients unique in optimizer. Generally, it is used in sparse networks. Set to True if the gradients of the optimizer are sparse, while set to False if the forward
network has made the parameters unique, that is, the gradients of the optimizer is no longer sparse. The default value is True when it is not set. | {"url":"https://www.mindspore.cn/docs/en/r2.0.0-alpha/api_python/nn/mindspore.nn.Optimizer.html","timestamp":"2024-11-11T07:35:04Z","content_type":"text/html","content_length":"42560","record_id":"<urn:uuid:8de26f3d-d8fa-4bc0-9ac4-9529925feb28>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00444.warc.gz"} |
AVL Tree
The AVL Tree is a self-balancing binary search tree in which balancing operations take place based on the difference of height between the left and right subtrees. Such operations may occur during
the insertion and deletion of keys performing recursive rotate operations to ensue that the difference between the heights of the left and right substrees is restricted to $[-1, 1]$.
Example of AVL Tree with balance factors shown in green.
AVL Trees are often compared with Red–Black Trees because both take $O(\log n)$ time for the basic operations. However, for lookup-intensive applications, AVL Trees are faster than Red–Black Trees
because they are more strictly balanced. Similar to Red–Black Trees, AVL Trees are height-balanced.
Computational complexity for common operations using an AVL Tree
Operation Average Case Worst Case
Space $\Theta(n)$ $O(n)$
Search $\Theta(\log n)$ $O(\log n)$
Insertion $\Theta(\log n)$ $O(\log n)$
Deletion $\Theta(\log n)$ $O(\log n)$
Construct new AVLTree with keys of type T.
julia> tree = AVLTree{Int64}()
AVLTree{Int64}(nothing, 0)
The AVLTree type implements the following methods:
delete!(tree::AVLTree{K}, k::K) where K
Delete key k from tree AVL tree.
getindex(tree::AVLTree{K}, ind::Integer) where K
Considering the elements of tree sorted, returns the ind-th element in tree. Search operation is performed in $O(\log n)$ time complexity.
julia> tree = AVLTree{Int}()
AVLTree{Int64}(nothing, 0)
julia> for k in 1:2:20
push!(tree, k)
julia> tree[4]
julia> tree[8]
Return number of elements in AVL tree tree.
push!(tree::AVLTree{K}, key) where K
Insert key in AVL tree tree.
Performs a left-rotation on node_x, updates height of the nodes, and returns the rotated node.
minimum_node(tree::AVLTree, node::AVLTreeNode)
Returns the AVLTreeNode with minimum value in subtree of node.
Performs a right-rotation on node_x, updates height of the nodes, and returns the rotated node.
sorted_rank(tree::AVLTree{K}, key::K) where K
Returns the rank of key present in the tree, if it present. A KeyError is thrown if key is not present.
julia> tree = AVLTree{Int}();
julia> for k in 1:2:20
push!(tree, k)
julia> sorted_rank(tree, 17) | {"url":"https://juliacollections.github.io/DataStructures.jl/latest/avl_tree/","timestamp":"2024-11-10T15:02:10Z","content_type":"text/html","content_length":"18636","record_id":"<urn:uuid:5a81453c-b70c-4a17-bea1-5465c78122c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00190.warc.gz"} |
EconPort - Handbook - Decision-Making Under Uncertainty - Basic Concepts
Most of the basic ideas in the theory of decision-making under uncertainty stem from a rather unlikely source - gambling. This becomes increasingly evident as one notices the literature is dotted
with phrases like 'expected value', and of course, 'lotteries'. This section covers the following topics:
The term "expected value" provides one possible answer to the question: How much is a gamble, or any risky decision, worth? It is simply the sum of all the possible outcomes of a gamble, multiplied
by their respective probabilities.
To illustrate:
1. Say you're feeling lucky one day, so you join your office betting pool as they follow the Kentucky Derby and place $10 on Santa's Little Helper, at 25/1 odds. You know that in the unlikely event
of Santa's Little Helper winning the race, you'll be richer by 10 * 25 = $250.
What this means is that, according to the bookmaker of the betting pool, Santa's Little Helper has a one in 25 chance of winning and a 24 in 25 chance of losing, or, to phrase it mathematically,
the probability that Santa's Little Helper will win the race is 1/25.
So what's the expected value of your bet?
Well, there are two possible outcomes - either Santa's Little Helper wins the race, or he doesn't. If he wins, you get $250; otherwise, you get nothing. So the expected value of the gamble is:
(250 * 1/25) + (0 * 24/25) = 10 + 0 = $10
And $10 is exactly what you would pay to participate in the gamble.
2. Another example:
A pharmaceutical company faced with the opportunity to buy a patent on a new technology for $200 million, might know that there would be a 20% chance that it would enable them to develop a
life-saving drug that might earn them, say $500 million; a 40% chance that they might earn $200 million from it; and a 40% chance that it would turn out worthless.
The expected value of this patent would then be:
(500,000,000 * 0.2) + (200,000,000 * 0.4) + (0 * 0.4) = $180 million
So of course, it would not make sense for the firm to take the risk and buy the patent.
Now that we've established that when people gamble, they should be willing to pay the expected value of the gamble in order to participate in it, ask yourself this question:
Suppose you were made an offer. A fair coin would be tossed continuously until it turned up tails. If the coin came up tails on the n^th toss, you would receive $2^n, i.e. if it came up tails on the
5th toss, you would receive $2^5 = $32.
How much would you be willing to pay, to participate in this gamble? | {"url":"http://www.econport.org/econport/request?page=man_ru_basics1","timestamp":"2024-11-03T23:48:41Z","content_type":"text/html","content_length":"22928","record_id":"<urn:uuid:c176a619-2af5-41b0-8703-bd6a8a00c8c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00189.warc.gz"} |
Quantum Information Science and Its Contributions to Mathematicssearch
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Quantum Information Science and Its Contributions to Mathematics
Hardcover ISBN: 978-0-8218-4828-9
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Quantum Information Science and Its Contributions to Mathematics
Hardcover ISBN: 978-0-8218-4828-9
Product Code: PSAPM/68
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN: 978-0-8218-9284-8
Product Code: PSAPM/68.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN: 978-0-8218-4828-9
eBook ISBN: 978-0-8218-9284-8
Product Code: PSAPM/68.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
• Proceedings of Symposia in Applied Mathematics
Volume: 68; 2010; 348 pp
MSC: Primary 81; 68; 57; 20
This volume is based on lectures delivered at the 2009 AMS Short Course on Quantum Computation and Quantum Information, held January 3–4, 2009, in Washington, D.C.
Part I of this volume consists of two papers giving introductory surveys of many of the important topics in the newly emerging field of quantum computation and quantum information, i.e., quantum
information science (QIS). The first paper discusses many of the fundamental concepts in QIS and ends with the curious and counter-intuitive phenomenon of entanglement concentration. The second
gives an introductory survey of quantum error correction and fault tolerance, QIS's first line of defense against quantum decoherence.
Part II consists of four papers illustrating how QIS research is currently contributing to the development of new research directions in mathematics. The first paper illustrates how differential
geometry can be a fundamental research tool for the development of compilers for quantum computers. The second paper gives a survey of many of the connections between quantum topology and quantum
computation. The last two papers give an overview of the new and emerging field of quantum knot theory, an interdisciplinary research field connecting quantum computation and knot theory. These
two papers illustrate surprising connections with a number of other fields of mathematics.
In the appendix, an introductory survey article is also provided for those readers unfamiliar with quantum mechanics.
Graduate students and research mathematicians interested in quantum information theory and its relations to new research areas in mathematics.
□ Quantum information science
□ Patrick Hayden — Concentration of measure effects in quantum information [ MR 2762144 ]
□ Daniel Gottesman — An introduction to quantum error correction and fault-tolerant quantum computation [ MR 2762145 ]
□ Contributions to mathematics
□ Howard E. Brandt — Riemannian geometry of quantum computation [ MR 2762146 ]
□ Louis H. Kauffman and Samuel J. Lomonaco, Jr. — Topological quantum information theory [ MR 2762147 ]
□ Samuel J. Lomonaco and Louis H. Kauffman — Quantum knots and mosaics [ MR 2762148 ]
□ Samuel J. Lomonaco and Louis H. Kauffman — Quantum knots and lattices, or a blueprint for quantum systems that do rope tricks [ MR 2762149 ]
□ Appendix
□ Samuel J. Lomonaco, Jr. — A Rosetta stone for quantum mechanics with an introduction to quantum computation [ MR 2762150 ]
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 68; 2010; 348 pp
MSC: Primary 81; 68; 57; 20
This volume is based on lectures delivered at the 2009 AMS Short Course on Quantum Computation and Quantum Information, held January 3–4, 2009, in Washington, D.C.
Part I of this volume consists of two papers giving introductory surveys of many of the important topics in the newly emerging field of quantum computation and quantum information, i.e., quantum
information science (QIS). The first paper discusses many of the fundamental concepts in QIS and ends with the curious and counter-intuitive phenomenon of entanglement concentration. The second gives
an introductory survey of quantum error correction and fault tolerance, QIS's first line of defense against quantum decoherence.
Part II consists of four papers illustrating how QIS research is currently contributing to the development of new research directions in mathematics. The first paper illustrates how differential
geometry can be a fundamental research tool for the development of compilers for quantum computers. The second paper gives a survey of many of the connections between quantum topology and quantum
computation. The last two papers give an overview of the new and emerging field of quantum knot theory, an interdisciplinary research field connecting quantum computation and knot theory. These two
papers illustrate surprising connections with a number of other fields of mathematics.
In the appendix, an introductory survey article is also provided for those readers unfamiliar with quantum mechanics.
Graduate students and research mathematicians interested in quantum information theory and its relations to new research areas in mathematics.
• Quantum information science
• Patrick Hayden — Concentration of measure effects in quantum information [ MR 2762144 ]
• Daniel Gottesman — An introduction to quantum error correction and fault-tolerant quantum computation [ MR 2762145 ]
• Contributions to mathematics
• Howard E. Brandt — Riemannian geometry of quantum computation [ MR 2762146 ]
• Louis H. Kauffman and Samuel J. Lomonaco, Jr. — Topological quantum information theory [ MR 2762147 ]
• Samuel J. Lomonaco and Louis H. Kauffman — Quantum knots and mosaics [ MR 2762148 ]
• Samuel J. Lomonaco and Louis H. Kauffman — Quantum knots and lattices, or a blueprint for quantum systems that do rope tricks [ MR 2762149 ]
• Appendix
• Samuel J. Lomonaco, Jr. — A Rosetta stone for quantum mechanics with an introduction to quantum computation [ MR 2762150 ]
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/psapm-68","timestamp":"2024-11-12T10:39:46Z","content_type":"text/html","content_length":"95463","record_id":"<urn:uuid:ada7d8f4-27c9-47cb-bf1d-b192f63be7d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00764.warc.gz"} |
Significance of Y=f(X) in Lean Six Sigma
If you are new to Lean Six Sigma then Y=f(X) is one amongst many jargons that you will have to familiarize yourself.
The objective of Lean Six Sigma philosophy and DMAIC improvement methodology is to identify the root causes to any problem and control/manage them so that the problem can be alleviated.
Six Sigma is process oriented approach which considers every task as a process. Even the simplest of the tasks, such as performing your morning workout or getting ready to office is considered as a
process. The implication of such a view point is to identify what is the output of that process, its desired level of performance and what inputs are needed to produce the desired results.
Y is usually used to denote the Output and X for the inputs.
Y is also known as dependent variable as it is dependent on the Xs. Usually Y represents the symptoms or the effect of any problem.
On the other hand, X is known as independent variable as it is not dependent on Y or any other X. Usually Xs represents the problem itself or the cause.
As you will agree that any process will have at least one output but most likely to have several inputs. As managers, we all are expected to deliver results and achieve a new level of performance of
the process such as Service Levels, Production Levels, Quality Levels, etc., or sustain the current level of performance.
In order to achieve this objective, we focus our efforts on the output performance measure. However a smart process manager will focus on identifying Xs that impact the output performance measure in
order to achieve the desired level of performance.
How does one identify the input performance measures or Xs?
Six Sigma DMAIC methodology aims to identify the inputs(Xs) that have significant impact on output (Y). After that the strength and nature of the relationship between Y and Xs are also established.
Six Sigma uses a variety of qualitative and quantitative tools & techniques listed below to identify the statistical validation of the inputs (or root causes), their strength and nature of
relationship with Y:
What does f in Y= f(X) mean?
‘f’ represents the nature and strength of the relationship that exists between Y and X. On one hand, this equation can be used for a generic interpretation that symbolizes the fact that Y is impacted
by X and nature of relationship can be quantified. On the other hand, such a mathematical expression can be created provided we have sufficient data using the above mentioned analytical tools such as
regression and other hypothesis tests.
The mathematical expression that we obtain is nothing but an equation such as:
TAT = 13.3 – 7.4*Waiting Time + 1.8*No. of Counters – 24*Time to Approve
Once such an equation is established, it can be easily used to proactively identify the Y for various values of X. Thus Y= f(X) is the basis for predictive modeling. All the newer analytical concepts
such as Big Data, etc are based on this foundation principles.
Related Articles | {"url":"http://www.sixsigmacertificationcourse.com/significance-of-yfx-in-lean-six-sigma/","timestamp":"2024-11-04T09:09:40Z","content_type":"text/html","content_length":"75780","record_id":"<urn:uuid:cf7c3649-e957-4ab6-9313-1d29dee9e502>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00316.warc.gz"} |
Calculus II
1. Short description: MATH 111 continues the study of the calculus begun in MATH 110. The course focuses on definite integrals, which allow exact calculation of surface areas, volumes, the length of
curves, and solutions of practical and theoretical problems. Students will explore such topics as inverse functions, exponential functions and logarithmic functions, inverse trigonometric functions
and hyperbolic functions and L'Hospital's Rule, finding indefinite integrals, approximate integration and improper integrals, volume, arc length, surface area, work, hydrostatic pressure, infinite
sequences and series, tests for convergence of series, power series. An emphasis will be placed on solving various applications using calculus as an analytical problem-solving tool.
2. Course objective: By the end of the course you will be expected to demonstrate knowledge of: (1) Determining sums of simple finite sequences and limits of certain sequences and series. (2)
Domains, ranges and inverses of functions, in particular trigonometric, exponential and hyperbolic functions. Solving simple equations involving hyperbolic functions.(3) Determining Taylor
polynomials and the first few terms of the Taylor or Maclaurin series of a function. (4) Finding limits of functions, in particular by using L'Hôpital's rules.(5) Evaluating integrals by substitution
and by parts, including reduction formulae. (6) Using integrals to evaluate arc length, area, volume and surface area of surfaces of revolution, center of mass and moment of inertia. (7) Acquiring a
comparative knowledge of standard coordinate systems and the ability to choose the most efficient system for any specific problem. (8) Developing a rigorous understanding of sequences and series with
ability to determine their convergence or divergence. (9) Understanding applications of the definite integral to problems such as area, volume, arc length, work and centers of mass. (10) Enhancing
learning by examining geometric, numerical and algebraic aspects of each topic. (11) Acquiring an understanding of the breadth of mathematics by studying applications in a wide variety of scientific
fields. (12) Using the tools of calculus to formulate and solve multi-step problems and to interpret the numerical results. (13) Enhancing the ability to communicate mathematical concepts through a
series of written laboratory assignments and classroom discussions. (14) Selecting and use technology when appropriate in problem solving. (15) Developing an ability to recognize calculus concepts in
the context of application problems and implement the corresponding processes. (16) Developing the process of making appropriate conjectures, finding suitable means to test those conjectures and
drawing conclusions about their validity.
3. Formal requirement: Homework is an important part of learning mathematics and will be assigned daily. Every assigned problem should be tried and the answer checked. It is permissible to discuss
problems with other students or relatives. It is not permissible to copy another student's work. Do your best to think through the problems and understand why things work the way they do. Homework
should take anywhere from 30 minutes to an hour, depending on the type of assignment. Homework will be issued in class and submitted one week later. On test days all homework associated with that
test will be collected and graded. Homework should be corrected and added, as the material is better understood.In class, students follow directions: (1) Be in your seat when the bell rings; (2) Do
not work on other subjects; (3) No eating, drinking, or chewing; (4) Most of the time the consequences for not following rules will be remaining after class. If it becomes more of a problem, the
student will be asked to stay a short time after school for a discussion.
4. Textbook: (1) J. Stewart, Single Variable Calculus, 4^th ed, Brooks/Cole (1999). (2) Courant, Richard and John, Fritz. Introduction to Calculus and Analysis, New York: Springer-Verlag; | {"url":"https://sjguo.tripod.com/calculus1.htm","timestamp":"2024-11-09T00:00:14Z","content_type":"text/html","content_length":"21626","record_id":"<urn:uuid:edc10a3d-ba00-4b0a-b664-d3dc901cc1dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00864.warc.gz"} |
│Keyhole method │
│ │
│ │
│ │
│Information about the method│
│ │
│ Proposer(s): unknown │
│ │
│ Proposed: unknown │
│ │
│ Alt Names: none │
│ │
│ Variants: 8355, Basket │
│ │
│ No. Steps: 1 substep │
│ │
│ No. Algs: unknown │
│ │
│ Avg Moves: unknown │
│ │
│ Purpose(s): ^ │
Keyhole First Two Layers or Keyhole F2L, sometimes named working corner, is a method that solves, normally, the first two bottom layers of the 3x3x3 cube. It is an efficient method for inserting
mid-layer edges in the 3x3x3 LBL method. Its a slight advancement on the basic LBL method, because it requires fewer moves, as well as more intuition.
Basic Idea
After the bottom layer cross, only three bottom layer corners are solved. The final corner is reserved as the keyhole to be used in solving the mid layer edges efficiently. To insert a mid layer edge
using the keyhole:
You start, like in normal LBL from a cross, then you fill in three of the first layer corners. From that point you solve three of the middle layer edges using the empty corner position as the keyhole
. For each edge do the following:
• First position the mid-layer edge in the U-layer such that it is above the centre matching the edge's up-facing colour.
• Using a D-Turn position the bottom layer so that the unsolved corner is directly below the mid-layer edge's position.
• Now execute one of R U' R' or F U F' to insert the edge.
After three corners and edges are solved you complete F2L by first solving the last corner and finally the last edge using the normal LBL algorithm: R U R' U' F' U' F or the mirror F' U' F U R U R'
depending on the current orientation of the edge.
Alternative Procedure
An alternative way to do keyhole F2L is to use this order:
• cross (doesn't have to match centers)
• 3 corners
• 4 edges
• 4th corner
For that you will need three algorithms to solve the last piece depending on it's orientation:
List of algs...
F2L 32
F2L (R U R' U')(R U R' U')(R U R')
F2L (U R U' R')(U R U' R')(U R U' R')
F2L U2 R2 U2 R2 U' R2 U' R2
F2L 33
F2L (U' R U' R') U2 (R U' R')
F2L y U' (L U' L') U2 (L U' L)
F2L 34
F2L U' (R U2' R') U (R U R')
F2L U (R U R') U2 (R U R')
F2L d (R' U R) U2 (R' U R)
Edges-First Keyhole
Another way to approach keyhole is to start by inserting three edges, and then use the remaining unsolved edge position to insert three corners. The basic steps are as follows:
• Solve Cross
• Solve 3x E-slice edges
• Rotate cube to position remaining unsolved slot in FR
• Solve 3x D-layer corners by:
1. Position a D-layer corner above the slot in the U-layer
2. Rotate D-layer so that the corner's correct location is directly below the slot
3. Insert the corner using one of: R U R' or F' U' F or R B U2 B' R'
If there are no corners available in the U-layer, then rotate the D-layer until an unsolved D-layer corner is above the slot, and then use R U R' to swap it into the U-layer.
While both the corners and edges first approaches use a similar number of moves, edges-first does not require the solver to keep track of the working corner, which can be better for lookahead. If
used in conjunction with 8355 it may also improve ease of understanding, since a similar concept is used to place the final 5 corners.
See also
External links | {"url":"https://www.speedsolving.com/wiki/index.php?title=Keyhole_F2L","timestamp":"2024-11-11T14:41:23Z","content_type":"text/html","content_length":"36184","record_id":"<urn:uuid:3663b6f8-b995-430f-a381-5c24e482414d>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00521.warc.gz"} |
EViews Help: The Optimize Command
The Optimize Command
The syntax for the optimize command is:
optimize(options) subroutine_name(arguments)
is the name of the defined subroutine in your program (or included programs). The full set of options is provided in
By default, EViews will assume that the first argument of the subroutine is the objective of the optimization, and that the second argument contains the controls. The default is to maximize the
objective or sum of the objective values (with the sum taken over the current workfile sample, if a series).
Specifying the Method and Objective
You may control the type of optimization and which subroutine argument corresponds to the objective by providing one of the following options to the optimize command:
• max [=integer]
• min [=integer]
• ls [=integer]
• ml [=integer]
The four options correspond to different optimization types: maximization (“max”), minimization (“min”), least squares (“ls”) and maximum likelihood (“ml”). If the objective is scalar valued only
“max” and “min” are allowed.
As the names suggest, “min” and “max” correspond to minimizing and maximizing the objective. If the objective is multi-valued, optimize will minimize or maximize the sum of the elements of the
“ls” and “ml” are special forms of minimization and maximization that may be specified only if the multi-valued objective argument has a value for each observation. “ls” tells optimize that you wish
to perform least squares estimation so the optimizer should minimize the sum-of-squares of the elements of the objective. “ml” informs optimize you wish to perform maximum likelihood estimation by
maximizing the sum of the elements in the objective.
“ls” and “ml” differ from “min” and “max” in supporting an additional option for approximating the Hessian matrix (see
“Calculating the Hessian”
) that is used in the estimation algorithm. Indeed the only difference between the “max” and “ml” for a multi-valued objective is that “ml” supports the use of this option (“hess=opg”).
By default, the first argument of the subroutine is taken as the objective. However you may specify an alternate objective argument by providing an integer value identifier with one of the options
above. For example, to identify the second argument of the subroutine as the objective in a minimization problem, you would use the option “min=2”.
Identifying the Control
By default, the second argument in the subroutine contains the controls for the optimization. You may modify this by including the “coef=integer” option in the optimize command, where integer is the
argument identifier. For example, to identify the first argument of the subroutine as the control, you would use the option “coef=1”.
Starting Values
The values of the objects containing the control parameters at the onset of optimization are used as starting values for the optimization process. You should note that if any of the control
parameters contain missing values at the onset of optimization, or if the objective function, or any analytic gradients cannot be evaluated at the initial parameter values, EViews will error and the
optimization process will terminate.
Specifying Gradients
If included in the optimize command, the “grad=” option specifies which subroutine argument contains the analytic gradients for each of the coefficients. If you specify the “grad=” option, the
subroutine should fill out the elements of the gradient argument with values of the analytical gradients at the current coefficient values.
• If the objective argument is a scalar, the gradient argument should be a vector of length equal to the number of elements in the coefficient argument.
• If the objective argument is a series, the gradient argument should be a group object containing one series per element of the coefficient argument. The series observations should contain the
corresponding derivatives for each observation in the current workfile sample.
• For a vector objective, the gradient argument should be a matrix with number of rows equal to the length of the objective vector, and columns equal to the number of elements in the coefficient
• “grad=” may not be specified if the objective is a matrix.
If “grad=” is not specified, optimize will use numeric gradients. In general, we have found that using numerical gradients performs as well as analytic gradients. Since programming the calculation of
the analytic gradients into the subroutine can be complicated, omitting the “grad=” option should usually be one’s initial approach.
Calculating the Hessian
The “hess=” option tells EViews which Hessian approximation should be used in the estimation algorithm. You may employ numeric Hessian (“hess=numeric”), Broyden-Fletcher-Goldfarb-Shanno (“hess=
bfgs”), or outer-product of the gradients (“hess=opg”) approximations to the Hessian (see
“Hessian Approximation”
You may not specify an analytic Hessian, though all three approximations use information from the gradients, so that there will be slight differences in the Hessian calculation depending on whether
you use numeric versus analytical gradients.
The “finalh=” option allows you to save the Hessian matrix of the optimization problem at the final coefficient values as a matrix in the workfile. For least squares and maximum likelihood problems,
the Hessian is commonly used in the calculation of coefficient covariances.
For OPG and numeric Hessian approximations, the final Hessian will be the same as the Hessian approximation used during optimization. For BFGS, the final Hessian will be based on the numeric Hessian,
since the BFGS approximation need not converge to the true Hessian.
Numeric Derivatives
You can control the method of computing numeric derivatives for gradient or Hessian calculations using the “deriv=” option.
At the default setting of “deriv=auto”, EViews will change the number of numeric derivative evaluation points as the optimization routine progresses, switching to a larger number of points as it
approaches the optimum.
When you include the “deriv=high” option, EViews will always evaluate the objective function at a larger number of points.
Iteration and Convergence
The “m=” and “c=” options set the maximum number of iterations, and the convergence criterion respectively. Note that for optimization, the number of iterations is the number of successful steps that
take place, and that each iteration may involve many function evaluations, both to evaluate any required numeric derivatives and for backtracking in cases where a trial step fails to improve the
Reaching the maximum number of iterations will cause an error to occur (unless the “noerr” option is set).
Advanced Optimization Options
There are several advanced options which control different aspects of the optimization procedure. In general, you should not need to worry about these settings, but they may prove useful in cases
where you are experiencing estimation difficulties.
Trust Region
You may use the “trust=” option to set the initial trust region size as a proportion of the initial control values. The default trust region size is 0.25.
Smaller values of this parameter may be used to provide a more cautious start to the optimization in cases where larger steps immediately lead into an undesirable region of the objective.
Larger values may be used to reduce the iteration count in cases where the objective is well behaved but the initial values may be far from the optimum values.
“Technical Details”
for discussion.
Step Method
optimize offers several methods for determining the constrained step size which you may specify using the “step=” option. In additional to the default Marquardt method (“step=marquardt”), you may
specify dogleg steps (“step=dogleg”) or a line-search determined step (“step=linesearch”).
Note that in most cases the choice of step method is less important than the selection of Hessian approximation. See
“Step Method”
for additional detail.
By default, the optimization procedure automatically adjusts the scale of the objective and control variables using the square root of the maximum observed value of the second derivative (curvature)
of each control parameter. Scaling may be switched off using the “scale=none” option. See
for discussion.
Objective Accuracy
The “feps=” option may be used to specify the expected relative accuracy of the objective function. The default value is 2.2e-16.
The value indicates what fraction of the observed objective value should be considered to be random noise. You may wish to increase the “feps=” value if the calculation of your objective may be
relatively inaccurate.
Status Functions
To support the optimize command, EViews provides three functions that return information about the optimization process:
• @optstatus provides a status code for the optimizer, both during and post-optimization.
• @optiter returns the current number of iterations performed. If called post-optimization, it will return the number of iterations required for convergence.
• @optmessage returns a one line text message based on status and iteration information that summarizes the current state of an optimization.
All three of these functions may be used during optimization by including them inside the optimization subroutine, or post-optimization by calling them after the optimize command.
Error Handling
The “noerr” option may be used as an option to suppress any error messages created when the optimization fails. By default, the optimization procedure will generate an error whenever the results of
the optimization appear to be unreliable, such as if convergence was not met, or the gradients are non-zero at the final solution.
If noerr is specified, these errors will be suppressed. In this case, your EViews program may still test whether the optimization succeeded using the @optiter function. Note that the noerr option is
useful in cases where you are deliberately stopping optimization early using the m= maximum iterations option, since otherwise this will generate an error. | {"url":"https://help.eviews.com/content/optimize_cr-The_Optimize_Command.html","timestamp":"2024-11-07T02:50:54Z","content_type":"application/xhtml+xml","content_length":"24521","record_id":"<urn:uuid:732083db-a9ae-4aad-b8cc-8125f02e1c8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00190.warc.gz"} |
The motion of a particle along a straight line is described by equatio
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The elo Package
The elo package includes functions to address all kinds of Elo calculations.
• elo.prob(): calculate probabilities based on Elo scores
• elo.update(): calculate Elo updates
• elo.calc(): calculate post-update Elo values
• elo.run() and elo.run.multiteam(): calculate “running” Elo values for a series of matches
It also includes comparable models for accuracy (auc, MSE) benchmarking:
• elo.glm() which fits a logistic regression model
• elo.markovchain() which fits a Markov chain model
• elo.colley() for a method based on the Colley matrix.
• elo.winpct() which fits a model based on win percentage
Please see the vignettes for examples.
Naming Schema
Most functions begin with the prefix “elo.”, for easy autocompletion.
• Vectors or scalars of Elo scores are denoted “elo.A” or “elo.B”.
• Vectors or scalars of wins by team A are denoted by “wins.A”.
• Vectors or scalars of win probabilities are denoted by “p.A”.
• Vectors of team names are denoted “team.A” or “team.B”. | {"url":"https://cloud.r-project.org/web/packages/elo/readme/README.html","timestamp":"2024-11-13T08:38:39Z","content_type":"application/xhtml+xml","content_length":"3216","record_id":"<urn:uuid:92fe20e6-583b-4d48-a1f4-5e0fba55218d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00432.warc.gz"} |
Bell’s Theorem: A Nobel Prize For Metaphysics - 3 Quarks Daily
Bell’s Theorem: A Nobel Prize For Metaphysics
by Jochen Szangolies
Bells Theorem Crescent in Belfast, John Bell’s birth town.
There has been no shortage of articles on this year’s physics Nobel, which, just in case you’ve been living under a rock, was awarded to Alain Aspect, John Clauser, and Anton Zeilinger “for
experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”. Why, then, add more to the pile?
A justification is given be John Bell himself in his 1966 review article On the Problem of Hidden Variables in Quantum Mechanics: “[l]ike all authors of noncommissioned reviews [the writer] thinks
that he can restate the position with such clarity and simplicity that all previous discussions will be eclipsed”. While I like to think that I’m generally more modest in my ambitions than Bell
semi-seriously positions himself here, I feel that there is a lacuna in most of the recent coverage that ought to be addressed. That omission is that while there is much talk about what the
prize-winning research implies—from the possibility of groundbreaking new quantum technologies to the refutation of dearly held assumptions about physical reality—there is considerably less talk
about what it, and Bell’s theorem specifically, actually is, and why it has had enough impact beyond the scientific world to warrant the unique (to the best of my knowledge) distinction of having a
street named after it.
In part, this is certainly owed to the constraints of writing for an audience with a diverse background, and the fear of alienating one’s readers by delving too deeply into what might seem like
overly technical matters. Luckily (or not), I have no such scruples. However, I—perhaps foolishly—believe that there is a way to get the essential content of Bell’s theorem across without breaking
out its full machinery. Indeed, the bare statement of his result is quite simple. At its core, what Bell did was to derive an inequality—a bound on the magnitude of a certain quantity—such that, when
it holds, we can write down a joint probability distribution for the possible values of the inputs of the inequality, where these ‘inputs’ are given by measurement results.
Now let’s unpack what this means.
Flipping A Quantum Coin
The box with a possible result of checking coins 2 and 3.
Suppose you’re given a box that consists of three chambers, each of which contains a coin behind a small window. The view into each chamber is obstructed by a movable panel that can be slid away to
reveal the coin behind it. However, the mechanism of the box is designed such that you can only ever open at most two of the panels—the third is then locked tight. Furthermore, after having opened
two panels and closing them again, you can only reopen any panels after shaking the box again. Your task is to figure out the probability of all coins coming up heads.
So you shake the box, open up two panels at random—you figure you can just get things done more quickly by taking both values you have access to in each go—and note down the outcome of each coin
throw. Sure enough, as you tally the results, counting the number of times each coin has come up heads or tails, you find they all do so about half the time—in other words, the coins seem to be fair.
So you figure that the probability for all coins to come up heads together is equal to that of each coming up heads separately—hence, ½ · ½ · ½ = 1/8. That was easy!
However, going back over your notes, you see something curious. Whenever you have noted the result of checking the first and second coins, they agree—when the first comes up heads, so does the
second; likewise for tails. The same thing, you notice, holds true for the second and third coins, while for the first and third coins, if you have opened their respective panels together, they
always seem to disagree in their values—if the first comes up heads, the third comes up tails, and vice versa. Let’s note this down:
• When you look at the first and second coins together, the outcome is either ‘heads, heads’ or ‘tails, tails’ with equal probability
• Likewise, when you look at the second and third coins together, the outcome is also either ‘heads, heads’ or ‘tails, tails’ with equal probability
• However, when you look at the first and third coins together, the outcome is either ‘heads, tails’ or ‘tails, heads’, again with equal probability
This is not, in and of itself, anything particularly strange—what you’ve discovered is that the outcomes of the coin throws are not independent, but correlated. But correlations aren’t mysterious,
but quite common in everyday life. To illustrate the point, Bell tells the story of Bertlmann’s socks: Bertlmann, who had been Bell’s co-worker at CERN, was fond of wearing socks of different colors
on each foot. Thus, if you see one of his socks, and know that fact about Bertlmann, you can immediately conclude that the sock on the other foot must have a different color. Supposing that there are
only two different colors of socks in Bertlmann’s closet, say red and green, you could even immediately predict the color of the sock on the other foot, without ever having to see it!
The reason for this is that the combinations ‘green, green’ or ‘red, red’ are simply forbidden. Likewise, through whatever internal mechanism, the box forbids the combinations ‘heads, tails’ or
‘tails, heads’ for the first and second coin. But then, this screws up the above calculation: the probability of ‘heads, heads’ isn’t ¼, as surmised, but ½—if one of the coins comes up heads
(probability ½), then so must the other. Multiplying the probabilities only works for independent events!
Correlation: out of four nominal possibilities, two are forbidden.
Where does that leave you? Well, there still ought to be information in the probabilities of the pairs of outcomes you have collected to yield some conclusion about the distribution of outcomes for
all three coins together. So suppose that the first coin comes up heads. Then, so must the second. But if the second comes up heads, so must the third. So it seems, the whole set then must come up
‘heads, heads, heads’. But, and here’s the punchline: looking at the first and third coins together, you know they must always come up oppositely—so this state can never occur! Could that then be the
answer: the probability of ‘heads, heads, heads’ is zero?
But of course, going through the possible outcomes, any combination of outcomes for all three coins violates one of the rules set down above. Each, you find, can never occur—and thus, can’t be
assigned a valid probability. You can’t even set them all to zero, as the sum of probabilities of all possible outcomes must be one—something must always happen. This is what is meant if we say that
the three coins—more accurately, their values after flipping—can’t be assigned a joint probability distribution.
This should seem somewhat troubling: after all, it seems reasonable to assume, there must be some state for the three coins after a throw (a principle often called ‘value definiteness’); but any
possible such state is incompatible with the correlations we have observed.
Physics Goes Meta
On the other hand, the above discovery might leave you rather unimpressed. The coin we’re not looking at, you might surmise, doesn’t really matter. It’s no problem at all to write down a probability
distribution for the three coins if we require that only those we actually look at obey the rules above. If we look only at the first and third coins, we could assign to the possibilities (writing H
for ‘heads’ and T for ‘tails’ for brevity) ‘HHT’, ‘HTT’, ‘THH’, ‘TTH’ probabilities of ¼ each (in fact, any combinations such that the first and last two sum to ½ works). The second coin will violate
the correlations—but we don’t open the second panel, so we never see it do so. Similar assignments are possible for looking at the other pairs.
The trouble with this is just that usually, we would assume that the box doesn’t know beforehand which of the panels you will open. If you had looked instead at coins one and two, or two and three,
in the above setup, there would be a nonzero probability of observing a violation of the correlation rules. So supposing that you can open panels randomly and independently of the configuration
present in the box after shaking it, this strategy won’t work. (This assumption is often called ‘free will’ in the literature, but this sometimes invites confusion—what’s really needed is that the
choice is independent of the outcome of the coin throws, which need not necessarily imply anything about metaphysical free will; thus, I prefer the less ambiguous ‘free choice’. The violation of this
assumption is generally known as ‘superdeterminism’.)
But another option is that the box’ internal machinery is more sophisticated than you had assumed. After all, it already can’t merely throw each coin independently—then, there would be no way to
account for the correlations between pairs of coins. So suppose that the outcome of each coin toss is only selected once you open up the first panel. Say you decide to look at the third coin, and see
that it has come up ‘heads’—then, immediately, the internal machinery of the box sets up the first coin to display ‘tails’ and the second one to show ‘heads’. Thus, if checking on one coin disturbs
the value of the others, correlations such as those observed become possible. Call the assumed lack of such influence the ‘no disturbance principle’.
No matter what value is hidden behind the unopened panel, it must disagree with one of the constraints.
Summing it up, we find that if we assume that
• there is some state the three coins are in, after shaking the box (value definiteness),
• we can choose to check the values of any two coins, independent of those values (free choice), and
• looking at one coin does not disturb the values of the others (no disturbance),
correlations such as the ones we have observed should be impossible. That we have observed them, however, then tells us that one of those assumptions must be thrown out.
Now, for a box with some unknown internal mechanism, there seems a clear candidate: there is no good reason to assume that opening up any of the panels should not influence what’s behind the others.
It would be trivial to program a computer simulation of such a box, for instance.
However, there are three main aspects of Bell’s result that cement its importance—and that of its conclusive empirical validation by the recent Nobel laureates. First, in the above, we have nowhere
had to make any assumptions about the theory we use to describe the physical world. We’re not merely talking about how a given theory tells us the world is, but about the world in a
theory-independent way—which is why physicist Abner Shimony, whom we’ll shortly meet again, has coined the term ‘experimental metaphysics’ for such results: literally, they go beyond (any given
theory of) physics.
Additionally, while it might be reasonable to expect complex mechanical devices to show properties dependent on the manner of their observation, the same strikes us as profoundly counterintuitive
when extended to simple physical objects. Take, for instance, a brick: we consider it perfectly adequate to measure its size, mass, and color in isolation, without having to take account of how these
measurements might influence one another. If we measure its mass, we don’t expect that to influence the outcome of a measurement of its color; but with the coins, ‘measuring’ the outcome of one seems
to influence that of another. If we thus see such strange correlations in nature on simple objects—e. g., elementary particles—then we find we must let go of an assumption that seems obvious to us in
everyday physical objects, namely, that their properties don’t depend on the context of their measurement (what other measurements are carried out simultaneously). This assumption is accordingly
termed non-contextuality and is the core of a result intimately related to Bell’s, namely, the Kochen-Specker theorem.
For the final reason why Bell’s result, unlike the behavior of coins in mechanical contraptions, should shock us and forces us to reconsider some dearly-held assumptions, we must look at how
correlations like the above are actually realized in quantum mechanics.
Neither Here Nor There
Two pairs of two coins, such that one from each may be examined.
In quantum mechanics, the situation as described above is not exactly realized. However, there are closely related scenarios, and we can inch up our way to more realistic cases: suppose we have four
coins in two pairs, such that we can always only look at one of each pair. Let’s call them A[1], A[2], B[1], and B[2] for short. Doing some rounds of experiments with the box as before, you quickly
find the following:
□ The pairs (A[1], B[1]), (A[1], B[2]), and (A[2], B[1]), if observed together, always yield the same results—both heads, or both tails
□ The pair (A[2], B[2]) always yields opposing results—one heads, the other tails
While this situation is, again, not realized in quantum mechanics, it makes things a bit simpler, and the additional subtleties introduced by the correct quantum behavior won’t affect our
conclusions. For those willing to dig a little deeper, I give a somewhat more realistic treatment in the excursion below.
We can now reason as before: finding H for A[1], B[1] must be H, too, and consequently, so must A[2] and B[2]. But A[2] and B[2] must always yield opposing values: hence, no simultaneous assignment
of values to every coin is possible—except if, as before, looking at one coin has the potential to disturb the value of the others (or conversely, to take the superdeterministic option, the coin’s
values determine which ones we will look at). But this situation now allows us to conclude something more.
The coins in this setup come neatly packaged in two sets, and whether one panel can be opened is decided only by the state of the other. In quantum mechanics, this is a result of Heisenberg’s
uncertainty principle: there are properties of a given system—the canonical example being position and velocity/momentum—such that only ever one or the other can be known at any given time.
By now, you’re determined to get at the heart of this puzzle no matter what. So you whip out a saw, and with grim resolve, cut the box apart right through the middle. Opening one panel each of both
part A and B, the other still being blocked by the box’ mechanics, you find agreement with the previously established rules—but of course, that doesn’t tell you much. So you try to repeat the
experiment—however, to your dismay, you find that now, the mysterious correlation is broken: the outcomes of box A no longer tell you anything about box B. So, you surmise that whatever influence
there may be, something needs to be transferred between A and B to make sure each conforms to the observed distribution of outcomes.
But things are not quite that simple. Suppose you’re supplied with a large number of copies of the original box. Sure enough: shaking each box, then cutting it apart and opening a panel of each half
at random, again yields the paradoxical behavior—but only for the first values observed. Whatever ensures that the right values are present in box B after opening a panel on box A, or vice versa,
apparently works over a certain distance—if only once.
So, you decide to test the limits of this strange behavior—you prepare a large number of copies of the box, and give the A- and B-parts each to a friend of yours (who, contrary to popular belief,
don’t necessarily have to be called Alice and Bob), each marked with a number so that later, you can properly combine the results from the original four-coin boxes. Then, you instruct both to get as
far away from each other as possible, while taking the utmost care not to accidentally jostle any of their boxes. (In reality, the difficulties involved in both the ‘getting away from each other as
far as possible’ and the ‘not jostling the boxes’ were a large part of the reason for this year’s Nobel win.) Finally, each opens one panel of each box at random, noting down the number of the box,
and the time it was opened. Then, they are to return.
Once both are back, you tally up the results—and, sure enough, find the same results as before. Moreover, the relative distance of both does not seem to have any effect on this behavior—indeed,
examining the timing of the experiments, you find that not even a signal traveling at the speed of light could have crossed the distance in time!
This should give us pause, for it is here that we learn something about physical reality—reality as such, mind, not just reality according to quantum mechanics. If correlations like the ones observed
in the box-experiment exist in the world, then one of our assumptions must fail to hold. Moreover, carrying out the experiments in regions isolated from one another considerably strengthens the no
disturbance-assumption: to ensure it, one needs only hold that physics is local, that is, what happens right here does not instantaneously influence what happens over there—it first has to traverse
the distance between.
So what this all means, the true import of Bell’s theorem, is that one of our assumptions must go: either we are not free to choose which panel to open; or, there are no definite values associated
with certain quantities before looking; or, certain ghostly influences exist regardless of spatial separation.
Which option to take is, to a certain degree, a matter of taste. Some insist that there must be definite values to observable quantities even without observing them, often appealing to auxiliary
arguments—most notably the one due to Albert Einstein, Boris Podolsky, and Nathan Rosen—to bolster their case. But this reading is controversial, and should be taken with a grain of salt. Even
superdeterminism, for a long time at best an outsider option—for how should one do science if the properties of objects determine our observations thereof?—has recently reemerged as an option. But at
least one of these options must be taken: after the conclusive experimental violation of Bell’s inequality, there is no going back to a classical world with objects having definite properties
independently of observation.
Excursion: The CHSH-Inequality
As noted, quantum systems do not quite implement the above behavior. The actually observed values do not show the perfect (anti-)correlation posited above. However, with a few clever manipulations,
and just a tiny dash of math, we can still reach much the same conclusions.
Suppose we assign numerical values to the possible outcomes—say, ‘heads’ is 1, and ‘tails’ is -1. Then we can look at the following quantity:
This combination of outcomes can never exceed a total value of 2—if B[1] and B[2] are both equal to 1, the first term can maximally be 2, but the second will be 0; if B[1] is 1 and B[2] is -1, the
second term may be 2, but the first must vanish. Any assignment of values to all four coins at once thus leaves the above expression upper bounded by 2.
Unfortunately, we cannot directly evaluate it in this form. But suppose we multiply out the brackets, yielding:
This can still only at most equal 2. But each term within this equation is now something that can be evaluated, as it contains one coin from the A-pair, and one from the B-pair. So, you might shake
the box, open two panels, and note down the outcomes for A[2] and B[1], say. But how do we get from there to evaluating the expression above? Clearly, we can’t simply combine the results from
different runs: as the box is shaken in between, all the values will be reset.
However, since we know that the results in each run can’t produce a value exceeding two, we also know that they can’t do so in the average over many runs. So, we can simply run the box experiment a
large number of times, opening one set of two panels at random every time, tallying up the results, and divide it by the number of repetitions—and find that we can still not exceed a maximum value of
two. This yields an expression first written down by John Clauser, one of the co-laureates of this year’s Nobel prize, together with Michael Horne, Abner Shimony, and Richard Holt, called the ‘
CHSH-inequality’ after its originators’ initials:
Here, the bar above each term indicates taking the average. (Note that this argument is still a bit condensed; but a more careful treatment yields the same result.)
It’s important to realize that we are still beholden to our earlier assumptions, here. Thus, let’s return to the example presented in the main text. If the values in the first three terms always
agree, while those in the last one always disagree, the above expression takes a value of 4—the first three terms being equal to 1, and the last yielding -1. Violation of the above inequality thus
indicates violation of the assumptions listed above. In quantum mechanics, the maximum attainable value for the CHSH-expression is equal to 2√2, or about 2.83. Nobody knows why. | {"url":"https://3quarksdaily.com/3quarksdaily/2022/10/bells-theorem-a-nobel-prize-for-metaphysics.html","timestamp":"2024-11-07T16:48:49Z","content_type":"text/html","content_length":"80946","record_id":"<urn:uuid:d5ed091c-041a-4fd9-9120-61aa64f79d82>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00771.warc.gz"} |
The world we experience is governed by classical physics. How we move, where we are, and how fast we’re going are all determined by the classical assumption that we can only exist in one place at any
one moment in time.
But in the quantum world, the behavior of individual atoms is governed by the eerie principle that a particle’s location is a probability. An atom, for instance, has a certain chance of being in one
location and another chance of being at another location, at the same exact time.
When particles interact, purely as a consequence of these quantum effects, a host of odd phenomena should ensue. But observing such purely quantum mechanical behavior of interacting particles amid
the overwhelming noise of the classical world is a tricky undertaking.
Circa 2014
Scientists have come closer than ever before to creating a laboratory-scale imitation of a black hole that emits Hawking radiation, the particles predicted to escape black holes due to quantum
mechanical effects.
The black hole analogue, reported in Nature Physics^1, was created by trapping sound waves using an ultra cold fluid. Such objects could one day help resolve the so-called black hole ‘information
paradox’ — the question of whether information that falls into a black hole disappears forever.
The physicist Stephen Hawking stunned cosmologists 40 years ago when he announced that black holes are not totally black, calculating that a tiny amount of radiation would be able to escape the pull
of a black hole^2. This raised the tantalising question of whether information might escape too, encoded within the radiation.
The team of academician GUO Guangcan of University of Science and Technology of China (USTC) of the Chinese Academy of Sciences has made important progress in the research of cold atom.
An atom is the smallest component of an element. It is made up of protons and neutrons within the nucleus, and electrons circling the nucleus.
😮 circa 2021.
The fundamental forces of physics govern the matter comprising the Universe, yet exactly how these forces work together is still not fully understood. The existence of Hawking radiation — the
particle emission from near black holes — indicates that general relativity and quantum mechanics must cooperate. But directly observing Hawking radiation from a black hole is nearly impossible due
to the background noise of the Universe, so how can researchers study it to better understand how the forces interact and how they integrate into a “Theory of Everything”?
According to Haruna Katayama, a doctoral student in Hiroshima University’s Graduate School of Advanced Science and Engineering, since researchers cannot go to the Hawking radiation, Hawking radiation
must be brought to the researchers. She has proposed a quantum circuit that acts as a black hole laser, providing a lab-bench black hole equivalent with advantages over previously proposed versions.
The proposal was published on Sept. 27 Scientific Reports.
“In this study, we devised a quantum-circuit laser theory using an analogue black hole and a white hole as a resonator,” Katayama said.
The universe is governed by two sets of seemingly incompatible laws of physics – there’s the classical physics we’re used to on our scale, and the spooky world of quantum physics on the atomic scale.
MIT physicists have now observed the moment atoms switch from one to the other, as they form intriguing “quantum tornadoes.”
Things that seem impossible to our everyday understanding of the world are perfectly possible in quantum physics. Particles can essentially exist in multiple places at once, for instance, or tunnel
through barriers, or share information across vast distances instantly.
These and other odd phenomena can arise as particles interact with each other, but frustratingly the overarching world of classical physics can interfere and make it hard to study these fragile
interactions. One way to amplify quantum effects is to cool atoms right down to a fraction above absolute zero, creating a state of matter called a Bose-Einstein condensate (BEC) that can exhibit
quantum properties on a larger, visible scale.
For a few brief moments, the high-powered lasers generated 1.3 megajoules of fusion energy.
A breakthrough experiment last month at Lawrence Livermore National Laboratory’s (LLNL) National Ignition Facility (NIF) in California has turned up a whopping 1.3 megajoules of energy, or about
three percent of the energy contained in one kilogram of crude oil. The work, as outlined in the journal Physical Review E, puts physicists “at the threshold of fusion ignition,” according to the
lab’s press release.
Nuclear fusion, in the simplest terms, is a reaction in which atoms are smashed together to generate an abundance of energy. In some ways, it’s less dangerous than nuclear fission —a process that
involves splitting heavy, unstable atoms into two lighter ones—and has the potential to create a lot more energy.
All of today’s functional nuclear power plants currently use nuclear fission, and scientists have long been on the hunt for a way to make nuclear fusion a reality; consider it a kind of holy grail of
clean energy.
The uncertainty principle, first introduced by Werner Heisenberg in the late 1920’s, is a fundamental concept of quantum mechanics. In the quantum world, particles like the electrons that power all
electrical product can also behave like waves. As a result, particles cannot have a well-defined position and momentum simultaneously. For instance, measuring the momentum of a particle leads to a
disturbance of position, and therefore the position cannot be precisely defined.
Particle physics needs a new collider to supersede the Large Hadron Collider. Muons, not electrons or protons, might hold the key.
Roughly 13.8 billion years ago, our Universe was born in a massive explosion that gave rise to the first subatomic particles and the laws of physics as we know them.
About 370,000 years later, hydrogen had formed, the building block of stars, which fuse hydrogen and helium in their interiors to create all the heavier elements. While hydrogen remains the most
pervasive element in the Universe, it can be difficult to detect individual clouds of hydrogen gas in the interstellar medium (ISM).
This makes it difficult to research the early phases of star formation, which would offer clues about the evolution of galaxies and the cosmos.
Invisibility devices may soon no longer be the stuff of science fiction. A new study published in the De Gruyter journal Nanophotonics by lead authors Huanyang Chen at Xiamen University, China, and
Qiaoliang Bao, suggests the use of the material Molybdenum Trioxide (a-MoO3) to replace expensive and difficult to produce metamaterials in the emerging technology of novel optical devices.
The idea of an invisibility cloak may sound more like magic than science, but researchers are currently hard at work producing devices that can scatter and bend light in such a way that it creates
the effect of invisibility.
Thus far these devices have relied on metamaterials – a material that has been specially engineered to possess novel properties not found in naturally occurring substances or in the individual
particles of that material – but the study by Chen and co-authors suggests the use of a-MoO3 to create these invisibility devices. | {"url":"https://demo.lifeboat.com/blog/category/particle-physics/page/2","timestamp":"2024-11-06T18:19:15Z","content_type":"text/html","content_length":"146028","record_id":"<urn:uuid:8068b052-1e2a-4f9b-8afd-037a1e4ad3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00516.warc.gz"} |
On-line and off-line approximation algorithms for vector covering problems
This paper deals with vector covering problems in d-dimensional space. The input to a vector covering problem consists of a set X of d-dimensional vectors in [0, 1]^d. The goal is to partition X into
a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly
interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This
result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d ≥ 2,
no on-line algorithm can have a worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1/log d). For d = 2,
we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the area of compact vector summation can be used to
construct off-line approximation algorithms with worst case ratio 1/d for every d ≥ 2.
All Science Journal Classification (ASJC) codes
• General Computer Science
• Computer Science Applications
• Applied Mathematics
• Approximation algorithm
• Competitive analysis
• Covering problem
• On-line algorithm
• Packing problem
• Worst case ratio
Dive into the research topics of 'On-line and off-line approximation algorithms for vector covering problems'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/on-line-and-off-line-approximation-algorithms-for-vector-covering-2","timestamp":"2024-11-02T21:33:16Z","content_type":"text/html","content_length":"52907","record_id":"<urn:uuid:82d37d47-f8cf-4fa8-b9be-4534c0cdab3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00522.warc.gz"} |
Unit 1 – Ratios, Rate, Percent and Proportions - Review Guide - MathTeacherCoach
Unit 1 – Ratios, Rate, Percent and Proportions – Review Guide
Here is your Review Guide for Unit 1 – Ratios, Rate, Percent and Proportions!
(Click the links below to Download)
Unit 1 – Review Guide SE – Ratios and Proportional Relationships (PDF)
Unit 1 – Review Guide TE – Ratios and Proportional Relationships (PDF)
If you want access to Editable Version of the Review Guide and ALL of our Lessons
Simply click the image below to GET ALL OF OUR LESSONS! | {"url":"https://mathteachercoach.com/unit-1-ratios-rate-percent-and-proportions-review-guide/","timestamp":"2024-11-10T21:36:31Z","content_type":"text/html","content_length":"111837","record_id":"<urn:uuid:a35939ec-e48c-4607-8276-6e950a719ed2>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00161.warc.gz"} |
Multi-output microwave single-photon source using superconducting circuits with longitudinal and transverse couplings
Single-photon devices at microwave frequencies are important for applications in quantum information processing and communication in the microwave regime. In this work, we describe a proposal of a
multi-output single-photon device. We consider two superconducting resonators coupled to a gap-tunable qubit via both its longitudinal and transverse degrees of freedom. Thus, this qubit-resonator
coupling differs from the coupling in standard circuit quantum-electrodynamic systems described by the Jaynes-Cummings model. We demonstrate that an effective quadratic coupling between one of the
normal modes and the qubit can be induced, and this induced second-order nonlinearity is much larger than that for conventional Kerr-type systems exhibiting photon blockade. Assuming that a coupled
normal mode is resonantly driven, we observe that the output fields from the resonators exhibit strong sub-Poissonian photon-number statistics and photon antibunching. Contrary to previous studies on
resonant photon blockade, the first-excited state of our device is a pure single-photon Fock state rather than a polariton state, i.e., a highly hybridized qubit-photon state. In addition, it is
found that the optical state truncation caused by the strong qubit-induced nonlinearity can lead to an entanglement between the two resonators, even in their steady state under the Markov
leave comment | {"url":"https://circuitqed.net/publication/multi-output-microwave-single-photon-source-using-superconducting-circuits-with-longitudinal-and-transverse-couplings/","timestamp":"2024-11-12T20:49:16Z","content_type":"text/html","content_length":"19486","record_id":"<urn:uuid:22cf52f4-7d44-4d73-a4e0-61d3917e1091>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00111.warc.gz"} |
INVESTMENT APPRAISAL / CAPITAL BUDGETING – NPV AND IRR -
Net present value (NPV) of investment appraisal and Internal Rate of Return (IRR) of capital budgeting are the two methods of using the discounted cash flow (DCF) to evaluate capital investment. This
article is a continuation of two previous investment appraisal articles introduction to investment appraisal and investment appraisal- ROCE AND PAYBACK
WHAT IS DISCOUNTED CASH FLOW (DCF)?
DCF is an investment appraisal technique which takes into account both the timings of cash flows and total profitability over a project’s life.
• DCF is concerned with the cash flows of a project, not the accounting profits. The reason for this is that cash flows show the costs and benefits of a project when actually executed and ignore
notional costs like depreciation.
• Timing of cash flows- which is very important- is fully recognized by DCF by discounting the cash flows. The effect of discounting is to give a bigger value per $1 for each flow that occurs
earlier – time vale of money ($1 earned today is much better than $1 earned at the end of one year.)
For other terms used in investment appraisal, read up terms used in investment appraisal techniques.
For investment appraisals evaluation conducted using the NPV and IRR methods of capital budgeting to be meaningful, cash flows must be rightly and correctly timed. As a general rule, the following
guidelines may be applied – especially if you are writing accounting professional exams that require you to time initial investment, working capital, and tax cash flows, ACCA and CIMA are good
Threes ways to time cash flows:
• Any cash outflow to be incurred at the beginning of an investment project (‘now’) occurs in year 0. Note that the present value (PV) of $1 now, in year 0, is $1 irrespective of the value r. r=
compound rate of return per time period, expressed as a proportion.
• Any cash outflow, saving, and inflow that occurs during the course of a time frame (one year for instance) is assumed to have occurred all at once at the end of the time period. For example,
receipt of $12,500 during year 1 is taken to have occurred at the end of year 1.
• Any cash outlay or inflow that occurs at the beginning of a time period (at the beginning of year one for instance) is assumed to have occurred at the end of the year before i.e., previous year.
So, a cash outlay of $1,200 at the beginning of year 2 is seen as occurring at the end of year 1.
Please make sure you are at home with the timing conventions of these cash flows for good understanding of NPV and IRR!
Net present value (NPV)
NPV is the value we get by discounting all cash outflows and inflows of a capital investment project by a chosen cost of capital or target rate of return. This is to say that the NPV method of
investment appraisal compares present value (PV) of all cash inflows from an investment with the PV of all the cash outlays from an investment. Mathematically, NPV = PV of inflows – PV of outflows.
Example 1: example of NPV calculation
A company is considering capital investment, where the estimated cash flows for this project over a period of four years are as follows:
Years Cash Flows
0 (i.e. now) (100,000)
1 160,000
2 90,000
3 20,000
4 30,000
The company’s cost of capital is 15%. You are required to calculate the NPV of the project and assess whether it should be undertaken or not.
Years Cash flows Discount factor 15% Present value
$ $
0 (100,000) 1.000 (100,000)
1 160,000 1/ (1.15)^1 = 0.870 139,200
2 90,000 1/ (1.15)^2 = 0.756 68,040
3 20,000 1/ (1.15)^3 = 0.658 13,160
4 30,000 1/ (1.15)^4 = 0.572 17,160
NPV= 137,560
Note: remember that the discount factor for any cash flow in year 0 (now) is always = 1, regardless of what the cost of capital is.
There are three things that manager can do after calculating the NPV of a project. They are:
1. Accept the project when the NPV is positive. This is a situation where returns from investment’s cash inflow are in excess of the cost of capital.
2. Reject the project when the NPV is negative. You refuse to undertake a project when the return from investment’s cash inflow is below cost of capital.
3. Become indifferent from financial perspective. Managers consider other non-financial factors before deciding on whether to undertake a project or not when the return from investment’s cash inflow
is the same as cost of capital.
Based on the above investment decision criterion, we can accept the project as it is a good investment that has a positive NPV of $137,560.
Internal rate of return (IRR)
Unlike the NPV method of investment appraisal that is calculated by discounting streams of cash flows with a target rate of return or cost of capital, and the difference taken as the NPV, the IRR
method of capital appraisal is to calculate the exact DCF rate of return which a project or investment opportunity is expected to achieve; in other words, the rate at which the NPV is zero.
Investment decision under the IRR rule is to accept any project whose IRR or DCF yield exceeds a target rate or return. Without a computer, IRR is calculated using a trial and error, crude method
called interpolation method.
Three steps are involved in calculating IRR, they are:
Step 1, calculate the NPV of the project using a company’s cost of capital
Step 2, calculate the NPV of the project using another discount rate,
• If the NPV is positive, use a second rate that is greater than the first rate
• If the NPV is negative, use a second rate that is less than the first rate.
Step 3, use the two NPV values to estimate the IRR. The formula to apply is thus:
IRR= a + {[NPVa/ NPVa – NPVb] (b-a) } %
Where; a= the lower of the two rates of return used
b= the higher of the two rates of return used
NPVa = the NPV obtained using rate a
NPVb = the NPV obtained using rate b
Example 2: example of IRR calculation
A company ants to decide whether to buy a machine for $100,000 which will save costs of $25,000 per annum for 5 years and which will have a residual value of $15,000 at the end of year 5. If it is
the company’s policy to undertake projects only if they are expected to yield a DCF return of 10% or more, decide whether this project should be undertaken or not.
Step 1 calculate the NPV using the company’s cost of capital of 10%
Year Cash flow PV factor 10% PV of cash flow
$ $
0 (100,000) 1.000 (100,000)
1-5 25,000 3.791 94,775
5 15,000 0.621 9,315
NPV= 4090
Step 2, calculate the second NPV, using a rate that is greater than the first rate, as the first rate gave a positive answer.
Let us try 12%.
Year Cash flow PV factor 12% PV of cash flow
$ $
0 (100,000) 1.000 (100,000)
1-5 25,000 3.605 90,125
5 15,000 0.567 8,505
NPV= (1,370)
This figure is close to zero and is negative, but we haven’t got what we are looking for which is somewhere in-between.
Step 3, use the two values to interpolate.
Substituting the above values for our formula we have:
IRR= 10+ {[4,090/4,090+1,370] x (12-10)} %
= 10+ 4090/5460 x 2 /100
IRR= 10.10%
This project should be accepted since its IRR is marginally above the cost of capital.
I would like to bring to your attention that IRR and NPV as methods of investment appraisal are not without some disadvantages and limitations. I would have loved to discuss all those here but you
will agree with me that this article is already long, and will there be too long if I include that. Watch out for another article on that.
1. ALLY says
THE FIGURE THAT YOU HAVE OBTAINED IS WRONG IT SUPPOSE TOBE
IRR= 10.01498…….
2. chinweike says
Aziz, thanks for pointing that error out. you are right. The correct answer is 10.015%
3. gabe says
I have a finance homework question for you…..if you have a project that has a cost of 15,000, cost of capital is 14%, and the IRR is 17%, how would you figure out the capital budget?
4. chinweike says
Hi gabe,
could you please rephrase your question? Thanks for asking.
5. Kelvin says
This is helpfull. However here is my question.
I invest $2000 in a project which is expected to achieve cost saving as follows
year 1 to year 3 equals $400 per year, year 4 and year 5 equals $500, year 6,7 and 8 equals $450, year 9 and 10 equals $400.
project will last for 10years, at the end of which it will have zero residual value. The company will require a target ARR of 25 percent and a disired payback period of 4 years.
The company also requires a return on capital of 15 percent after tax.
Calculate the payback period, NPV and IRR.Also advise the management on which criteria to use.
6. Khanhan says
Thank you very much!
7. chinweike says
8. Moyotole Daniel Ezuem says
thanks 4 ur solving
9. Rhodah, says
this is some good work done, thax alot
10. chinweike says
You are welcome, Rhodah.
11. Mimiana says
This is quality work, it has really helped me, keep up with good work.
12. chinweike says
Thanks Mimiana. I am happy to hear that you did enjoy this article on capital budgeting and investment appraisal.
13. holden says
Two firms are identical in all respect except for their capital structure. Their NOI is as follows:
• -$ 500 with a probability of ¼
• $ 1 000 with a probability of ½
• $ 2 000 with a probability of ¼
The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .
a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?
b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.
c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.
d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?
assist me
14. Holden says
Two firms are identical in all respect except for their capital structure. Their NOI is as follows:
• -$ 500 with a probability of ¼
• $ 1 000 with a probability of ½
• $ 2 000 with a probability of ¼
The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .
a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?
b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.
c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.
d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?
15. Holden says
Two firms are identical in all respect except for their capital structure. Their NOI is as follows:
• -$ 500 with a probability of ¼
• $ 1 000 with a probability of ½
• $ 2 000 with a probability of ¼
The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .
a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?
b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.
c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.
d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?
please assist me
16. kassa says
it is nice | {"url":"https://accountantnextdoor.com/investment-appraisal-capital-budgeting-%E2%80%93-npv-and-irr/","timestamp":"2024-11-07T01:14:42Z","content_type":"text/html","content_length":"130318","record_id":"<urn:uuid:208900f4-b665-4b29-b8d2-55f8e6fd83e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00516.warc.gz"} |
Evaluating Algebraic Expressions Worksheet With Negative Numbers
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Worksheet On Evaluating Algebraic Expressions
Evaluating Expressions Worksheet Algebra 1
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Printable Calendars AT A GLANCE
Domain And Range Of Quadratic Function Worksheet
Domain And Range Of Quadratic Function Worksheet - If the parabola is opening downwards, i.e. Range is all real values of y for the given domain (real values values of x). Web domain and range of
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identify the appropriate domain for a function in context. It includes examples, exercises and answers for level 2 further maths students. Because, y is defined for all real values of x. Web given a
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Domain And Range Of A Function Graph Worksheet With Answers —
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Domain And Range Of Quadratic Function Worksheet Pdf
Web we can use this function to begin generalizing domains and ranges of quadratic functions. The general form a quadratic function is. Web domain and range of radical and rational functions. Web
learn how to find the domains and ranges of functions with this printable worksheet from corbettmaths. Exercise \(\pageindex{d}\) \( \bigstar \) use the vertex of the graph of.
Application Of Quadratic Functions Worksheet Function Worksheets
A < 0 , the range is y ≤ k. 1) domain is the collection of all input values (x values) 2) range is the collection of all output values (y values) domain: Web domain is all real values of x for which
the given quadratic function is defined. Given a quadratic function in the form of f(x) = ax2.
Range of a Quadratic Function YouTube
Y = ax2 + bx + c. Determine the maximum or minimum value of the parabola, \(k\) if the function is in the form \(f(x)=a(x−h)^2+k\), then the value of \(k\) is readily visible as one of the
parameters. Web domain is all real values of x for which the given quadratic function is defined. Create your own worksheets like this.
Domain And Range Of Quadratic Function Worksheet - The function is increasing for all values in which x > 2. Determine the maximum or minimum value of the parabola, \(k\) if the function is in the
form \(f(x)=a(x−h)^2+k\), then the value of \(k\) is readily visible as one of the parameters. Web domain is all real values of x for which the given quadratic function is defined. To determine the
domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist. A > 0 , the range is y ≥ k ; Web for every
polynomial function (such as quadratic functions for example), the domain is all real numbers. Web state the domain and range. From the graph of of a parabola with no endpoints (arrows on both ends);
Finding the domain and range of a quadratic function is as easy as looking at the graph. How to find the domain of a quadratic function.
Web these algebra 1 domain and range worksheets will produce problems for finding the domain and range of sets of ordered pairs. Determine the maximum or minimum value of the parabola, \(k\) if the
function is in the form \(f(x)=a(x−h)^2+k\), then the value of \(k\) is readily visible as one of the parameters. Y = ax2 + bx + c. Web we can use this function to begin generalizing domains and
ranges of quadratic functions. Exercise \(\pageindex{d}\) \( \bigstar \) use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the
Because parabolas have a maximum or a minimum point, the range is restricted. Walk through this collection of free printable worksheets and identify the domain and range of the functions from the
graphs. In the quadratic function, y = x 2 + 5x + 6, we can plug any real value for x. Any number can be the input value of a quadratic function.
Range Is All Real Values Of Y For The Given Domain (Real Values Values Of X).
Finding the domain and range of a quadratic function is as easy as looking at the graph. The general form a quadratic function is. Web we can use this function to begin generalizing domains and
ranges of quadratic functions. To determine the domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist.
Improve Your Math Knowledge With Free Questions In Domain And Range Of Quadratic Functions:
Graphs and thousands of other math skills. A > 0 , the range is y ≥ k ; Comparing the given quadratic function y = x 2 + 5x + 6 with y = ax 2 + bx + c It includes examples, exercises and answers for
level 2 further maths students.
Any Number Can Be The Input Value Of A Quadratic Function.
Therefore, the domain of any quadratic function is all real numbers. Web given a quadratic function, find the domain and range. The domain of any quadratic function in the above form is all real
values. You can select the range of numbers used in ordered pairs as well as whether the sheet should ask if each set of pairs is a.
Web State The Domain And Range.
Create your own worksheets like this one with infinite algebra 1. Web for every polynomial function (such as quadratic functions for example), the domain is all real numbers. Web topics in this unit
include: Web these printable quadratic function worksheets require algebra students to evaluate the quadratic functions, write the quadratic function in different form, complete function tables,
identify the vertex and intercepts based on formulae, identify the various properties of quadratic function and much more.
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Tue April 21
First : let's go over any of the homework that you would like. We covered quite a lot in this 2D rotation stuff:
• center of mass
• rotational kinetic energy
• torque
• angular momentum
Discuss, and practice as needed.
Depending on how this feels, this week I'd like you to look into some of how the 3D stuff works.
Second : what happens in the full 3D case?
The full answer is beyond the scope of this class. I am not sure yet how many problems we want to look at for this stuff. At least the basic precession equation, which is in the textbook.
The key point is that the torque, angular momentum, and spin vector all have both direction and length. And they can all point in different directions.
The definitions depend on what you decide is the coordinate system origin, similar to how the notion of potential energy depends on where you put zero.
For this class, the goal is to at least be exposed to a few specific iconic situations.
• moment of inertia tensor (google it for math & explanations) - we won't do more than peek at this.
I am used to the explanations in chapter 7 of Kleppner - Kolenkow, particularly the off-axis baton, and examining the motion of two opposite points on a spinning wheel, and how their motion is
changed by an applied force that tries to twist the wheel.
Discussion :
• Does a spinning marble falling towards the earth precess? Why or why not?
• The earth's axis precesses. What is causing the torque? (This one is not entirely obvious.)
• How can the spin axis and angular momentum axis be different?
• When a book is tossed into the air, it can "tumble" - the spin axis changes while in midair. Doesn't this violate some conservation law? | {"url":"https://cs.marlboro.college/cours/spring2020/mechanics/notes/apr21","timestamp":"2024-11-10T20:46:36Z","content_type":"text/html","content_length":"7708","record_id":"<urn:uuid:989fa076-28f6-4641-bfe5-b609fac9fd3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00143.warc.gz"} |
Fourth Grade
Number and Operations (NCTM)
Compute fluently and make reasonable estimates.
Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.
Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the
computation and use the selected method or tools.
Algebra (NCTM)
Use mathematical models to represent and understand quantitative relationships.
Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
Grade 4 Curriculum Focal Points (NCTM)
Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication
Students use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts. They apply their understanding of models for multiplication (i.e.,
equal-sized groups, arrays, area models, equal intervals on the number line), place value, and properties of operations (in particular, the distributive property) as they develop, discuss, and use
efficient, accurate, and generalizable methods to multiply multi-digit whole numbers. They select appropriate methods and apply them accurately to estimate products or calculate them mentally,
depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on
the basis of place value and properties of operations), and use them to solve problems. | {"url":"https://newpathworksheets.com/math/grade-4/more-multiplication?dictionary=subtraction&did=100","timestamp":"2024-11-13T05:57:57Z","content_type":"text/html","content_length":"40349","record_id":"<urn:uuid:1aca1ab6-2961-4f73-933b-54d5772e68d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00385.warc.gz"} |
Anonymity and governance
Commenter 'Belette' has
my attempt to connect anonymity with the failure of democracy. I thought this would be obvious, but fair enough: this is a logic blog and the argument I gave in the previous post was far from
demonstrable, being full of all sorts of missing assumptions, handwaving, enthymeme and absent premisses. Let's try and tidy it up and take it from the top with the following pseudo-syllogism:
Anonymity leads to conflict of interest
Conflict of interest leads to bad governance
, anonymity leads to bad governance
Even this is not deductively valid, but it will do for now. I will leave the minor (COI leads to bad governance) unless anyone objects, for while not self-evident, it can easily be made so by
supplying some missing assumptions. Note also that I have substituted the idea of governance for that of 'democracy'. Democracy does not always result in good governance, and it is governance we are
interested in. I also use 'governance' rather than 'government'. The latter almost always implies the action of a state, whereas it is governance in the widest sense that I am concerned with.
So let's focus on the major premiss: does anonymity lead to conflict of interest? Certainly. A conflict of interest arises when an individual under one description F1 has an interest which is opposed
to the the interest of that individual under the description F2. For example, let F1 be 'director of charity X' and let F2 be 'commercial supplier of provisions and services to charity X'. Clearly
the individual under the first description has an interest and a duty to procure the same service at the lowest possible price, whereas, under the second description, he has an interest in supplying
them at the highest possible price. Allowing the directory of a charity to be the same person as its chief commercial supplier is therefore a conflict of interest. But if we cannot identify the
individual satisfying each description, it will be practically impossible to prevent such a conflict. Let F1 be satisified by John Smith, a publicly identifiable individual. Let F2 be satisfied by
some individual under a fake user name. Unless we can connect the public name with the made up one, we will never know if there is a conflict of interest.
One recent example of such a conflict on Wikipedia was when the PR agency Bell Pottinger was found to be editing Wikipedia under the fake user name 'Biggleswiki'. Another less recent example was when
London Labour councillor David Boothroyd was not just an editor of Wikipedia under the pseudonym 'Sam Blacketer', but was a member of its important 'Arbitration Committee', probably the most senior
decision-making body on the project. Paul Williams, then director at Wikimedia UK, said: "Sock-puppeting is a very serious offence for anybody. But for someone on the Arbitration Committee it is even
more so. It can result in a lifetime ban. The problem with Wikipedia is that you can hide behind user names, but there is an expectation that you don't write for self-interest. In this case there is
a conflict of interest." There you go.
There are other instances of politicians anonymously acquiring important positions in Wikipedia, and who don't want to be publicly identified, but I will leave those for now, as they are for use in
the book I am writing on Wikipedia. I will leave you with an example from last weeks 'Blackout vote'. The full vote for the blackout was on
this page
. Over 700 members of the Wikipedia 'community' voted in support of an action that affected hundreds of millions of people across the world. Closer inspection reveals that many members of this
community may have voted
, or even more. About 150 of the support votes were from accounts opened specifically for voting, or re-used. There were a number of obvious 'sleeper' accounts, opened some time ago, almost certainly
by existing accounts who wanted to vote twice or more. Typical of the first kind is
*, who only edits twice, the
first time
to the article on Oboe, displaying knowledge of Wikipedia that is untypical of a first time user, the
second time
to vote. Typical of the second kind is
* who opened his or her account in 2006, made a handful of edits since then, and now
for the blackout. It is highly probable that all these votes were second or third votes from more established Wikipedia account names. There is no more evident and absurd form of conflict of interest
than pretending not to be yourself under a different, anonymous description.
Belette will probably argue that such conflict of interest only arises in the real world, because it has dishonest motives, whereas Wikipedians only act in the best motives. There is no reply to that
except to laugh or shrug, with a wink at the gallery.
*With apologies to any editor who was really acting in good faith. The point is, it is difficult to know under a system that almost guarantees anonymity in this way.
11 comments:
I'll look at your examples in a bit; thanks for posting them.
As a holding action: your syllogism is inadequate (i.e., just demonstrating, even if you could, that anon -> bad governance isn't good enough). Bad governance is everywhere; most countries in the
world are mis-governed, and whilst you could attribute some of that to a sort-of-not-really-anon civil service, rather a lot of it is attributable to known politicians being influenced to the
public ill (only slightly o/t: have you ever read http://en.wikipedia.org/wiki/The_Anome ?)
>>As a holding action: your syllogism is inadequate
I'm not sure 'inadequate' is in the lexicon of logic or not. I've already said it is not formally valid. What I think you mean, however, is that proving that anonymity leads to bad governance is
not what I want to prove.
I reply, proving that anonymity leads to bad governance is exactly what I want to prove.
It is a long time since I read any science fiction. I liked Jack Vance a lot when I was 16. I believe the eponymous Wikipedia editor took his handle from that story, and that editor is also a
strong proponent of anonymity.
The question is whether this kind of anonymity leads to conflict of interest. I would say the interest of a ruler includes the possibility of being deposed by his or her people. An anonymous
ruler cannot be thus deposed, thus, conflict of interest.
> Bell Pottinger
Yes, but that ended pretty quickly.
> David Boothroyd... Sam Blacketer
Not familiar with that. Since I endure endless allegations of COI, I'm not terribly favourable to the idea. What evil did he commit?
> Typical of the first kind is Korney654
Not a good example. He isn't on the final tally page. Any number of SPA's were tagged in the voting (http://en.wikipedia.org/w/index.php?title=Wikipedia:SOPA_initiative/Action&diff=471414983&
oldid=471414896, http://en.wikipedia.org/w/index.php?title=Wikipedia:SOPA_initiative/Action&diff=471414563&oldid=471414337, http://en.wikipedia.org/w/index.php?title=Wikipedia:SOPA_initiative/
Action&diff=471414298&oldid=471414083) and their votes are ignored. Everyone knows this happens, and it doesn't matter.
> Phauxcamus
Can't see him in the final list, either.
> Belette will probably argue...
Nope, or at least not yet, cos I don't need to.
> proving that anonymity leads to bad governance is exactly what I want to prove
What I was trying to say was that bad governance is everywhere. You could just as easily prove that people lead to bad governance. What you need to prove (or rather, demonstrate, since I can see
no way to prove it) is that anon leads to worse governance than known-identity.
For example:
1. known identity of leaders leads to concentrated lobbying
2. lobbying leads to decisions that favour the lobbyer and not the public at large
3. therefore, known-identity leads to bad governance.
>>Any number of SPA's were tagged in the voting
Thanks for pointing that out, I'll look at the tally page.
>>anon leads to worse governance than known-identity.
OK let's go for that case next.
>>Yes, but that [Bell Pottinger] ended pretty quickly.
So? The point was to demonstrate how anonymity implies conflict of interest, which it does. As soon as the anonymity ended, the coi was spotted. And did it end pretty quickly? I thought they were
editing for a long time? In any case, it wasn't Wikipedia systems and controls which ended it, but an external investigation. BP admitted to the undercover investigators that they were editing
Wikipedia anonymously.
>>Can't see [Phauxcamus] in the final list, either.
He is Phaux. As for the other one, a number of the accounts gave different signatures from the actual account name, which is yet another feature of the system which leads to bad governance.
> So? The point was to demonstrate how anonymity implies conflict of interest, which it does
No, the point was to demonstrate that anon -> bad governance. BP isn't a convincing example of that.
> As soon as the anonymity ended, the coi was spotted
I'm not convinced. Do yo have a good timeline for it? The WP article is ambiguous.
> He is Phaux
Good point. And Korney is Josh. But anyone with a redlink talkpage isn't going to get any notice.
>>But anyone with a redlink talkpage isn't going to get any notice.
Are you saying that there was an ‘official’ count of the votes and that the redlink votes weren’t counted? Where is this official count, please? What about redlink user pages? What about the
bluelink socks who bothered to write a few words on their talk and user pages? What about redlinks who have had a long editing history (yes, they exist).
>>I'm not convinced. Do yo have a good timeline for it? The WP article is ambiguous.
Biggleswiki began editing in November 2010. So, editing for a year before he was discovered by accident. The socking was crude and unsophisticated, too. For my research on the book I have been
talking to much more polished operators, who have been working for years without being uncovered.
The discovery of BP was the result of an undercover investigation by the Bureau of Investigative Journalism. Reporters from the Bureau posed as agents for various clients, including the
government of Uzbekistan, which has a poor human rights record, and its the cotton industry, which uses child labour, to investigate the ethical standards of the agency. Ten firms were contacted,
of which five were prepared to work for the 'Uzbekistan' agents. The discussions were secretly recorded. The Wikipedia sockfarm was mentioned, and the Bureau published its report, which was then
covered by the Independent. The rest is history.
>>No, the point was to demonstrate that anon -> bad governance. BP isn't a convincing example of that.
It’s a perfect example of governance failure, if we mean by that a failure of systems and controls. | {"url":"https://ocham.blogspot.com/2012/01/commenter-belette-has-questioned.html?showComment=1327310296230","timestamp":"2024-11-14T01:12:50Z","content_type":"text/html","content_length":"91897","record_id":"<urn:uuid:cbbf353e-3b23-4e8b-acd9-aa9835da12a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00261.warc.gz"} |
Updated: 2022/Sep/29
Please read Privacy Policy. It's for your privacy.
DIV(3) Library Functions Manual DIV(3)
div, ldiv, lldiv, imaxdiv - quotient and remainder from division
Standard C Library (libc, -lc)
#include <stdlib.h>
div(int num, int denom);
ldiv(long int num, long int denom);
lldiv(long long int num, long long int denom);
#include <inttypes.h>
imaxdiv(intmax_t num, intmax_t denom);
These functions compute the value of num / denom and return the quotient
and remainder in a specific division structure. The functions differ
only with respect to the type of the return value and the parameters.
The returned structure always contains two members named quot and rem,
denoting the quotient and the remainder. The type of these correspond
with the underlying type of the function.
The following example demonstrates the basic usage of the functions.
div_t d;
int a = 4321;
int b = 1234;
d = div(a, b);
(void)printf("%d %d\n", d.quot, d.rem);
fast_divide32(3), math(3), qdiv(3)
All described functions conform to ISO/IEC 9899:1999 ("ISO C99").
NetBSD 10.99 April 13, 2011 NetBSD 10.99 | {"url":"https://www.daemon-systems.org/man/lldiv.3.html","timestamp":"2024-11-09T06:03:16Z","content_type":"text/html","content_length":"4578","record_id":"<urn:uuid:080bf85c-594c-4222-b8af-ec43817a0fca>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00475.warc.gz"} |
Completely-reducible set
From Encyclopedia of Mathematics
A set $ M $ of linear operators on a topological vector space $ E $ with the following property: Any closed subspace in $ E $ that is invariant with respect to $ M $ has a complement in $ E $ that is
also invariant with respect to $ M $. In a Hilbert space $ E $ any set $ M $ that is symmetric with respect to Hermitian conjugation is completely reducible. In particular, any group of unitary
operators is a completely-reducible set. A representation $ \phi $ of an algebra (group, ring, etc.) $ A $ is called completely reducible if the set $ M = \{ {\phi (a) } : {a \in A } \} $ is
completely reducible. If $ A $ is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of $ A $ in a finite-dimensional vector space is completely reducible (the
principle of complete reducibility).
[1] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
The principle of complete reducibility is commonly referred to as Weyl's theorem (cf. [a1], Chapt. 2 Sect. 6).
[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
How to Cite This Entry:
Completely-reducible set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_set&oldid=44954
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/wiki/Completely-reducible_set","timestamp":"2024-11-14T08:31:07Z","content_type":"text/html","content_length":"15068","record_id":"<urn:uuid:79655bd8-3e04-4bd9-94c5-04cb99894370>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00066.warc.gz"} |
osting example
Introduction to this document
Activity Based Costing example
If your company wants to rationalise its range of products and services, you need to work out which ones to keep and which to bin. Activity Based Costing can help you to do this.
What’s Activity Based Costing?
You need to understand the cost of your company’s products and services in order to properly gauge the return made on each. Direct costs are normally easy to attribute; however, indirect costs or
overheads can be more difficult. Activity Based Costing (ABC) enables you to allocate costs based on the activities consuming indirect expenses. Use our Activity Based Costing Example to help you
calculate a more accurate unit cost and so identify the relative profitability of your product range.
Traditional indirect cost allocation methods divide costs across products by reference to a common denominator, for example, share of machine or labour hours. But this can result in one product
subsidising others leading to incorrect decision making. ABC calculates how much of each type of indirect cost is consumed by each product.
● start by identifying the resources and processes used up in the business, for example: R&D, marketing, selling and technical support
● then calculate the cost of these resources. For R&D, this could include staff, premises and the cost of any specialist materials used by that function; and
● finally for each product, estimate the usage of each type of resource, which should be directly identifiable to the product.
In our example
Our example assumes that you produce 90,000 units of product A and 10,000 of product B each year. Let’s say direct costs of labour and materials per unit are £10 for each product and that indirect
costs amount to £100,000. Each product takes the same number of machine hours per unit so 90% of machine hours are consumed by product A. To simplify, we’ll assume that the indirect costs are all R&
Traditional method. Under traditional costing, the indirect cost of each unit of product A and product B would be £1 (£100,000/100,000 units), calculated by reference to share of machine hours. The
total cost of each product would be calculated as £11 per unit (the £10 direct cost plus £1 for indirect cost allocation).
ABC method. Using ABC, we have now ascertained that it’s product B that uses all the R&D costs. So the indirect unit cost of products A and B would be £0 and £10 respectively. So under ABC, the total
unit cost for products A and B would be £10 and £20 respectively.
Different results. If products A and B had been priced based on traditional costing at, say, £15 per unit, both would appear to make a £4 profit whereas using ABC, product B would actually be losing
£5 per unit!
10 Dec 2012
File size: 29.50K
# Pages: 2
Credits: 1 | {"url":"https://docs4biz.indicator-flm.co.uk/tax_and_business/financial_management_and_accountancy/budgeting/activity_based_costing_example","timestamp":"2024-11-04T07:41:32Z","content_type":"application/xhtml+xml","content_length":"22023","record_id":"<urn:uuid:7f57f33c-1e99-4d46-802b-3ab67cb4af54>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00180.warc.gz"} |
A Golf Ball Strikes A Hard, Smooth Floor At An Angle Of 27.0 And, As The Drawing Shows, Rebounds At The
J = 3.564 N.s
From the given information:
angle θ = 27°
mass = 0.0200 kg
speed = 33.0 m/s
To determine the impulse applied using the equation:
J = m(2V cos θ)
J = 0.0200 (2 × cos (27.0))
J = 0.0200 (2 × 0.8910)
J = 0.03564
J = 3.564 N.s
The turntable's angular speed after the event is 28.687 revolutions per minute.
The system formed by the turntable and the two block are not under the effect of any external force, so we can apply the Principle of Conservation of Angular Momentum, which states that:
[tex]I_{T}\cdot \omega_{o} = (2\cdot r^{2}\cdot m +I_{T})\cdot \omega_{f}[/tex] (1)
[tex]I_{T}[/tex] - Moment of inertia of the turntable, in kilogram-square meters.
[tex]r[/tex] - Distance of the block regarding the center of the turntable, in meters.
[tex]m[/tex] - Mass of the object, in kilograms.
[tex]\omega_{o}[/tex] - Initial angular speed of the turntable, in radians per second.
[tex]\omega_{f}[/tex] - Final angular speed of the turntable-objects system, in radians per second.
In addition, the momentum of inertia of the turntable is determined by following formula:
[tex]I_{T} = \frac{1}{2}\cdot M\cdot r^{2}[/tex] (2)
Where [tex]M[/tex] is the mass of the turntable, in kilograms.
If we know that [tex]\omega_{o} \approx 7.330\,\frac{rad}{s}[/tex], [tex]M = 1.5\,kg[/tex], [tex]m = 0.54\,kg[/tex] and [tex]r = 0.1\,m[/tex], then the angular speed of the turntable after the event
[tex]I_{T} = \frac{1}{2}\cdot M\cdot r^{2}[/tex]
[tex]I_{T} = 7.5\times 10^{-3}\,kg\cdot m^{2}[/tex]
[tex]I_{T}\cdot \omega_{o} = (2\cdot r^{2}\cdot m +I_{T})\cdot \omega_{f}[/tex]
[tex]\omega_{f} = \frac{I_{T}\cdot \omega_{o}}{2\cdot r^{2}\cdot m +I_{T}}[/tex]
[tex]\omega_{T} = 3.004\,\frac{rad}{s}[/tex] ([tex]28.687\,\frac{rev}{min}[/tex])
The turntable's angular speed after the event is 28.687 revolutions per minute. | {"url":"https://www.cairokee.com/homework-solutions/a-golf-ball-strikes-a-hard-smooth-floor-at-an-angle-of-270-a-nnmn","timestamp":"2024-11-05T10:19:48Z","content_type":"text/html","content_length":"85990","record_id":"<urn:uuid:6cf195cf-5a35-4367-97f2-556eb4528c02>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00268.warc.gz"} |
Seminars: Silvia Ghilezan, Kripke-style semantics in typed lambda calculus, combinatory logic and more
Abstract: Kripke-style semantics have gained an important role and wide applicability in logic and computation since they were introduced by Saul Kripke in the late 1950s as semantics for modal
logics. In logic, these semantics were later adapted to intuitionistic logic and other non-classical logics. In computation, a class of Kripke-style models was defined for typed lambda calculus.
In this talk, we present a new approach to Kripke semantics for simply typed lambda calculus endowed with conjunction, disjunction and negation. We show soundness and completeness of this typed
lambda calculus w.r.t. the proposed semantics, [1]. This approach is extended to typed combinatory logic, [2].
Building on the previous results, we develop a classical propositional logic for reasoning about combinatory logic. We define its syntax, axiomatic system and semantics. The syntax and axiomatic
system are presented based on classical propositional logic, with typed combinatory terms as basic propositions, along with the semantics based on applicative structures extended with special
elements corresponding to primitive combinators. Both the equational theory of untyped combinatory logic and the proposed axiomatic system are proved to be sound and complete w.r.t. the given
semantics. In addition, we prove that combinatory logic is sound and complete w.r.t. the given semantics.
This is joint work with Simona Kasterovic.
Language: English
1. Simona Kasterovic, Silvia Ghilezan, “Kripke semantics and completeness for full simply typed lambda calculus”, Journal of Logic and Computation, 30:8 (2020), 1567–1608
2. Silvia Ghilezan, Simona Kasterovic, “Semantics for combinatory logic with intersection types”, Frontiers in Computer Science, 4 (2022) | {"url":"https://m.mathnet.ru/php/seminars.phtml?option_lang=eng&presentid=37270","timestamp":"2024-11-12T10:46:19Z","content_type":"text/html","content_length":"10505","record_id":"<urn:uuid:d76e59c8-90e2-4bd4-ad9c-2d2c22ba69d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00024.warc.gz"} |
Of Quantum Non-Locality And Green Cheese Moons In Many Worlds
The fourth FQXi essay contest “Which of Our Basic Physical Assumptions Are Wrong?” is underway.
Wittgenstein knew precisely which one; I am all about that one as you may know, but we are idiosyncratic, of German culture, and incomprehensible to many. Thus I humbly ask you to return the sweat I
have put into this already and please criticize the following constructively in order to help me to get the difficult message across. Tell me where the text starts to sound awkward / idiotic /
unintelligible/ hopelessly nonsensical; tell me in the comments or privately. All suggestions are welcome (Title stupid? Structure upside down? Figure suggestions? …).
The first half of the total essay “Direct Realism falling in Wittgenstein’s Silence: Accelerating the Paradigm Change that Renders Relativistic Quantum Mechanics Natural” has already been revised
with lots of helpful hints from readers (thank you all very much indeed). Here is the second half:
Spooky Non-Locality more Unreal than Modal Realism
All the above is crucially relevant to the advancement of QM, which can be didactically most easily clarified with the Einstein Podolsky Rosen (EPR)paradox [4]. EPR’s violation of John Bell’s famous
inequality [5] is usually presented as leaving two options, namely either so called non-locality or modifying realism. Non-locality is usually seen as less suspicious, because realism is widely
confused with, for example, the scientific method generally. This situation is aggravated by the difficult to fully grasp the profound nature of QM non-locality. Even after having studied Bell’s
proof [6] against hidden variables and all of that, many physicists still opine that the indicated non-locality is merely a ‘really complicated’ correlation, but in the end not profoundly different
from the correlation that ensures Alice getting the left sock of a pair if Bob gets the right sock. “Really complicated” feels not as ‘ghostly’ as anti-realism.
Einstein,although he could not find the solution in his lifetime, already understood the problem much better, and therefore he did not just say “well, so it is non-local and I am fine with that”. He
called it “spooky” {Footnote a} for good reasons. Einstein would not casually throw away relativistic micro causality{Footnote b}, the arguably most successful ingredient in all of modern physics
still today, just in order to prop up a kind of realism which then, via non-locality, becomes a ghost story nevertheless. QM non-locality destroys DR anyway, as shall become ever more obvious as you
read on. You are far more conservative if you accept modal realism – still a realism after all.
{Footnote a: In 1947, Einstein wrote to Max Born that he could not believe that quantum physics is complete "because it cannot be reconciled with the idea that physics should represent a reality in
time and space, free from spooky actions at a distance." (Emphasis added)}
{Footnote b: “Relativistic micro causality” is very roughly that stuff needs to bump against each other, that there is no mechanism for instantaneous interaction at a distance (in a direct realism
(DR)). This limit derives from how one can measure.}
Today it is known why QM non-local correlations are so spooky: They are correlations with non-actualized alternatives! It is the entanglement with counter factual possibilities, which has been
empirically verified for instance in interaction-free bomb-testing [7,8]. This very core of QM is precisely what makes QM correlations stronger than any classical hidden variables could possibly
provide. QM correlations can be profoundly more correlated than even complete classical correlations could ever deliver, because QM correlations are moreover/additionally fixed between the many
alternative worlds that classical physics ignores.
Variations of Bell’s inequality [9] have been violated by diverse experiments, most impressively by closing the so called “communication loophole” [10] and quite recently again by confirmation of the
Kochen-Specker theorem [11]. Desperate attempts at saving unmodified realism try to exploit the so called “detection loophole”, but they have by now retreated to claiming what Shimony called a
conspiracy {Footnote c} – one not much different from a creator god planting a fossil record to actively deceive us. As Einstein said: spooky!
{Footnote c: “… there is little that a determined advocate of local realistic theories can say except that, despite the space-like separation of the analysis-detection events involving particles 1
and 2, the backward light-cones of these two events overlap, and it is conceivable that some controlling factor in the overlap region is responsible for a conspiracy affecting their outcomes.” [Abner
Shimony, http://plato.stanford.edu/entries/bell-theorem]}
Two clarifications should be mentioned right away in order to understand QM non-locality and the confusion around it (although these clarifications will be much better understood later on):
1) If you consider a direct reality suitably misinterpreted as a lonely world being ‘really out there’, locality is crucially relevant. If space is a box with all objects ‘really’ at certain
locations inside, localism is implicit. Non-locality in a direct realism (where spatially separated events outside of a past light cone are merely unknown yet not fundamentally undetermined/
indeterminate) requires superluminal interaction. The huge success of relativistic micro causality argues against such, but is not as relevant as the following: Locality is implicitly assumed
precisely because non-locality observed in an internally relativistic, micro-causal billiard table, implies that there is something else, something spooky that interferes in the mere mechanical,
supposedly dynamically self-sufficient, independent ongoing of a classical box! Correlations that are proven to be faster than the fastest velocity inside the game board (its ‘velocity of light’) are
a sure sign of players (gods) messing around with the game pieces {Footnote d}.
If however you modify realism, you start undermining the tacit assumption about ‘space really being out there’. In a modal realism, localism is not implicit. Therefore, if for example Smerlak/
Rovelli, Bousso, Deutsch, Zeh, and so on state that QM is fundamentally local, do not confuse them with those who desperately cling to naïve directly real (DR) models and hidden variables.
{Footnote d: No ‘new atheism’ here, but a fundamental description of the totality of possible creators/creations (e.g. QM decoupling of ‘creations’ from ‘creators’) is beyond the scope of this work.
All I say here is that such ghostly issues afflict any non-local relativistic DR.}
2) The local/non-local distinction is almost precisely parallel to the determinism versus indeterminism one: QM is fundamentally a determinism (~ unitarity), and if it were not, then a more
fundamental theory would be anyway, because totality is totally determined as all that there is (including all times etc.). However, this very fact allows the phenomenal world that we perceive to
show in-determinism. Modal realism (MR) assures our experiencing QM in-determinism. The same goes through for locality: QM is fundamentally Einstein-local (micro causality, no correlations faster
than light), but MR allows the classical worlds that we seemingly find ourselves in to appear QM non-local. Admittedly, without an intuitive model, I would find this last statement suspiciously
mysterious. I therefore promote Many World (MW) models which greatly clarify these issues in visually intuitive ways.
Everett Relativity: Einstein could have Known
Quantum physics is our starting to find the mathematical description that does not neglect any possibilities. At first it was thought such encompasses only the alive and dead Schrödinger cats in our
particular universe, but this immediately includes all the ways a universe may be described to unfold internally as its own Schrödinger box, so it contains anyway all possible universes; everything
possibly phenomenal is included in the ultimate description, else it is not the ultimate description. The core insight is not quantum; already with classical determinism would an ultimate description
contain all possible worlds. "Many worlds" are tautologically true. The core of QM is the interference (entanglement) between alternatives. Ignorance about non-actualized alternatives is no longer an
option. But is it news?
Theoretical quantum gravity considerations have lead Gibbons and Hawking to propose an ‘anti-realistic’ observer dependent definition of particles already in the seventies, clearly endorsing “
something like the Everett-Wheeler interpretation of quantum mechanics” [12]. Zeilinger stresses anti-realism for a number of years, recently with a novel setup [13]. Black hole complementarity
(holographic horizons), and the cosmological measure problem all point to local, observer dependent ‘causal patch’ descriptions that crucially deny ‘real reality’ to what lies beyond horizons. The
horizons are wrapped ever closer around the observer. Why do we still largely ignore the paradigm change? Answering that is a different essay, but for now, the message is that perhaps the following
should not surprise:
Einstein could conceivably have anticipated QM long before even the Everett relative state description in 1957 [14]. Although EPR is commonly misunderstood as a clash between relativity and QM,
Everett relativity is only suspect without special relativity {Footnote e}. Special relativity (SR) is more than merely a ‘temporal modal realism’. SR already deconstructs the classical world into a
collection of past light cones, which each are an ‘observer’s’ individual determined past. Assuming otherwise immediately implies a fully pre-determined, directly real block universe where any
phenomenal indeterminism is divine pre-arrangement. SR already demanded MR to enter physics. Einstein locality and micro causality are yet more important than already widely recognized. They prepare
QM, which merely (see VI first part) adds correlations between alternative “determined pasts”.
{Footnote e: A branching MW model illustrates why: A non-relativistic universe would have to branch everywhere at once, into infinitely many different ones, all the time. SR is thus prerequisite for
understanding QM, because apparent “world branching” only occurs at observation events (interactions).}
In fact, I expect that the following argument will be made rigorous: SR and the demand that at least some possibilities should be unobservable, both together imply that there is some “mechanics” that
lets different alternatives interact: Quantum mechanics’ interference. In this way, Einstein could have conceivably understood QM and resolved the EPR paradox right away.
Many World Models – Just Models (!) – That Though Immensely Clarify
Relational Quantum Mechanics (RQM) [15] provided a resolution [2] of the EPR paradox. Its abstractness lacks an intuitive, didactic picture and also does not yet, so it seems to me, aim at deriving
the QM Born probabilities (which I believe can be done by considering the fact that Alice and Bob, the two famous participants in the EPR setup, must exchange information about the relative angle
setting every time, not just the outcome of the spin measurement). My own work [16] has contributed many world (MW) models which clarify vital aspects in an intuitive way – you can kind of sort of
“really see it” (again, it is modal realism). Allow me to convey a few lessons that such models taught me, as I constructed them somewhat by accident and they subsequently indeed taught me these
lessons with a clarity that I did not expect to be possible before.
However, I must first stress with the strongest possible emphasis that I do not imply any ontology, do not imply that MW ‘branch counting’ does not mess up normalizations in the measure problem, and
do certainly not claim that any particular model is on the same level as tautological truths. The model I refer to never claimed more than proving that 1) MR resolves EPR while preserving Einstein
locality and 2) that the QM probabilities become possible with the precise step that destroys the DR description. I suspect a certain way in which the model offers to derive the correct QM
probabilities instead of just allowing violations of Bell inequalities, but even that was never claimed yet.
It is far beyond the current work to introduce whole MW models, but I think it is not impossible to understand the following even without having a model, which look initially like sausages
incidentally, in front of one’s inner eyes.
1) One aspect is that MW and MR can be classical. The initially classical model already has many worlds, but they do not interfere. They are literally parallel worlds (but of course QM orthogonal
states). Therefore, a little arrow, literally a little arrow labeled “DR”, can pick out the one world that a direct realist believes to be the only real one.
2) The nature of the QM non-locality is illuminated: The mere classical non-locality in the form of the (anti)-correlations are already present in the classical model, even at all relative angles
between Alice’s and Bob’s measurement crystals. However, the Bell inequality cannot be violated, because it is still a classical model. The deeper QM non-locality arises at the very point, again
literally the very space-time point, at which the number of worlds in the MW model branches according to how different alternative worlds correlate (interact) locally. Let me refer to this vital step
as the “last local branching”. Thus, the QM non-locality is very clearly separated into its classical part and the QM component, the latter being necessarily correlations between alternative worlds.
3) QM indeterminism: Looking at the model, all possible alternative outcomes are already there {Footnote f} and thus everything is determined. You can point arrow DR to any of the resulting worlds
where certain outcomes are known. However, if the last local branching is QM, there is no preferred path that DR can follow from one initial world to a certain final one.
{Footnote f: Given a random angle setting w.l.o.g.; multiplying by all angle alternatives would be very bothersome. Also: those are not equally distributed. All aliens throughout the universe
experiment at the Bell angles (n Pi /8) where it makes most sense to do so.}
4) QM probability (empirical, Bayesian) is very different from the classical, circular “fair randomness” and its regress error: DR cannot, say via hidden variables, be initially rigged to end up in
any certain one of the new worlds (their numbers depend on the relative angle that is not yet determined in the beginning). It may, via randomness added from outside of the model, be made to point to
a particular new world, but such interference does not touch the classical probabilities. The classical measure stays the same while the empirical probabilities of observers violate the Bell
inequality. Inside of an MW model, observers have neither access to the classical probability measure nor can they directly count other worlds. But they remember and add to their records, and those
predominantly reflect the QM probabilities.
5) In a modified realism, QM non-locality comes for free: The expected phenomenal observations, i.e. the empirical data that the different worlds contain, depends primarily on the number of “new”
worlds (especially the “extra branching” {Footnote g} that Wallace so despises [17]). Therefore, once MW and thus MR is in place and you, the god of the model, add branching of worlds into new
worlds, you may change the model’s parameters to let the inhabitants find locality in their perceived worlds, or non-locality, or even, if you micromanage enough to change the numbers of worlds at
will at every measurement separately, that the moon is made from green cheese! If the realism is so modified that all is 'only in your head' instead of “out there” to such a degree that the moon may
as well become green cheese dancing over the mountains, obviously it does no longer matter whether the moon “is really at a certain locality”. The distinction between local and non-local lost its
relevance, because the model can adjust it arbitrarily; it can tune down the QM non-locality until it vanishes.
{Footnote g: “Extra growth” does not necessarily mean that the total number rises (new worlds), because the changing of microstates means that information is also “forgotten”, i.e. distinctions not
relevant to the measurement in question disappear.}
6) In nature (not a model on paper), you cannot change the parameters at will. Some consistency fixes the tuning so that we find non-locality instead of locality or cheese moons. But what
consistency? I suggest two ways to think about it:
6.1) The Born probabilities in the EPR setup are constrained by the consistency between the quantum (~ microcosm) and the classical (~ macro world) descriptions. In the former alone, photons may as
well just go 50-50 into the two different beam splitter channels without leading to Bell inequality violation, but in the latter, you know from playing with polarized sunglasses that polarization
filters act on the light’s electromagnetic vectors, thus they diminish intensity via a sine-squared law – nothing else would make sense in a macro world. (This point 6.1 should possibly be put before
mentioning MW models.)
6.2) The MW model shows that the numbers of new worlds are proportional to the dot product between Alice’s and Bob’s measurement axes in order to get probabilities consistent with experiments. This
suggests that the neglected microstates which carry the information about the relative angle setting are what distinguish these worlds (because the resolution of an angle measurement depends on the
total spin available).
What do you think will be the conclusion? Or do you think I should not add one and instead add another 2000 words to make the above clearer? Please let me know.
I am sincerely grateful for the many crucial questions and revision suggestions that readers of the internet draft have contributed in order to make this text as accessible and clear as possible.
[1] Ludwig Wittgenstein: “Tractatus Logico-Philosophicus”, Routledge and Kegan Paul, London (1922)
[2] Smerlak, Matteo, Rovelli, Carlo: “Relational EPR." Found. Phys. 37,427-445 (2007) http://lanl.arxiv.org/abs/quant-ph/0604064v3
[3] Lewis, David Kellogg: “On the Plurality of Worlds.” Blackwell (1986)
[4] Einstein, A., Podolsky, B., Rosen, N.: “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?” Phys. Rev. 47(10),777-780 (1935)
[5] Bell, J.S.: “On the Einstein Podolsky Rosen paradox.” Physics 1(3), 195–200 (1964); reprinted in Bell, J.S.Speakable and Unspeakable in Quantum Mechanics. 2nd ed., Cambridge: Cambridge University
Press, 2004; S. M. Blinder: “Introduction to Quantum Mechanics.” Amsterdam:Elsevier, 272-277 (2004)
[6] Bell, J. S.: “On the problem of hidden variables in quantum mechanics.” Rev. Mod. Phys. 38,447–452 (1966)
[7] Elitzur, A.C., Vaidman, L.: “Quantum mechanical interaction-free measurements.” Found. Phys. 23, 987-97 (1993)
[8] Paraoanu, G.S.: “Interaction-free Measurement.” Phys. Rev. Lett. 97(18),180406 (2006)
[9] Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: “Proposed Experiment to Test Local Hidden-Variables Theories.” Phys. Rev. Lett. 23,880-884 (1969)
[10] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger: “Violation of Bell’s inequality under strict Einstein locality condition.”Phys. Rev. Lett. 81, 5039-5043 (1998)
[11] G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A.Cabello, R. Blatt, C. F. Roos: “State-independent experimental test of quantum contextuality.” Nature 460, 494-497 (2009)
[12] Gibbons, G. W., Hawking, S. W.: “Cosmological event horizons,thermodynamics, and particle creation.” Physical Review D 15(10), 2738–2751(1977)
[13] Radek Lapkiewicz, Peizhe Li, Christoph Schaeff, Nathan K. Langford, Sven Ramelow, MarcinWiesniak, Anton Zeilinger: “Experimental non-classicality of an indivisible quantum system.” Nature 474,
490-493 (2011)
[14] Everett, Hugh: “‘Relative State’ Formulation of Quantum Mechanics.” Rev Mod Phys 29, 454-462(1957), reprinted in B. DeWitt and N. Graham (eds.), The Many-Worlds Interpretation of Quantum
Mechanics, Princeton University Press (1973)
[15] Rovelli, Carlo: “Relational Quantum Mechanics.” Int. J. Theor. Phys. 35,1637-1678 (1996)
[16] S. Vongehr: “Many Worlds Model resolving the Einstein Podolsky Rosen paradox via a Direct Realism to Modal Realism Transition that preserves Einstein Locality.” arXiv:1108.1674 [quant-ph]
[17] D. Wallace: “Quantum probability from subjective likelihood: Improving on Deutsch’s proof of the probability rule.” SHPMP 38, 311-332 (2007) | {"url":"https://www.science20.com/alpha_meme/quantum_nonlocality_and_green_cheese_moons_many_worlds-90860","timestamp":"2024-11-02T05:27:31Z","content_type":"text/html","content_length":"54158","record_id":"<urn:uuid:438e94e4-fabb-4f2e-bd17-3035b625814b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00386.warc.gz"} |
The CHOOSE Function | SumProduct are experts in Excel Training: Financial Modelling, Strategic Data Modelling, Model Auditing, Planning & Strategy, Training Courses, Tips & Online Knowledgebase
A to Z of Excel Functions: The CHOOSE Function
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the CHOOSE function.
The CHOOSE function
Do you choose to use CHOOSE? This function uses index_num to return a value from the list of value arguments. CHOOSE may be used to select one of up to 254 values based on the index number (index_num
). For example, if value1 through value7 are the days of the week, CHOOSE returns one of the days when a number between 1 and 7 is used as index_num.
The CHOOSE function employs the following syntax to operate:
CHOOSE(index_num, value1, [value2])
The CHOOSE function has the following arguments:
• index_num: this is required and is used to specify which value argument is to be selected. Index_num must be a number between 1 and 254, or a formula or reference to a cell containing a number
between 1 and 254.
□ if index_num is 1, CHOOSE returns value1; if it is 2, CHOOSE returns value2; and so on
□ if index_num is less than 1 or greater than the number of the last value in the list, CHOOSE returns the #VALUE! error value
□ if index_num is a fraction, it is truncated to the lowest integer before being used.
• value1, value2, ...: value 1 is required, but subsequent values are optional. There may be between 1 and 254 value arguments from which CHOOSE selects a value or an action to perform based on
index_num. The arguments can be numbers, cell references, defined names, formulas, functions, or text.
It should be further noted that:
• If index_num is an array, every value is evaluated when CHOOSE is evaluated
• The value arguments to CHOOSE can be range references as well as single values.
For example, the formula:
which then returns a value based on the values in the range B1:B10.
The CHOOSE function is evaluated first, returning the reference B1:B10. The SUM function is then evaluated using B1:B10, the result of the CHOOSE function, as its argument. A similar idea is also
expressed by the formula
which will return the result of =SUM(A1:A3).
Certainly it is a function used in modelling, but perhaps it is not used as regularly as some others. This is useful for non-contiguous references:
Just so that we are clear on jargon: a non-contiguous range (with reference to Excel) means a range that cannot be highlighted with the mouse alone. In the image above, to highlight the cells
coloured you would have to press down the CTRL key as well.
INDEX, LOOKUP, VLOOKUP and HLOOKUP all require contiguous references. They refer to lists, row vectors, column vectors and / or arrays. CHOOSE is different:
=CHOOSE(index_num, value1, [value2]…)
As explained above ,this function allows references to different calculations, workbook / worksheet references, etc. Try to use the function appropriately. For instance, a well-known Excel website
proposes the following formula for calculating the US Thanksgiving date. Assuming cell A1 has the year:
To understand this formula, note that DATE(Year,Month,Day) returns a date and WEEKDAY(Date) returns a number 1 (Sunday) through 7 (Saturday). But doesn’t this formula look horrible? It is full of
hard code and it contains an unnecessary number of arguments. The formula could exclude CHOOSE viz.
Now let me be clear here. I am not saying this is a simple, transparent formula. Test it. They both provide the same answer. CHOOSE – and plenty of additional hard code – has been used unnecessarily.
That’s not to say there isn’t a time and a place for CHOOSE. It is useful when you need to refer to cells on different worksheets or in other workbooks. Some argue that it is useful when a
calculation needs to be computed using different methods, e.g.
=CHOOSE(index_num, calculation1, calculation2, calculation3, calculation4)
I disagree. Let me explain. In the example below, I have created a lookup table in cells E10:E13 which I have called Data (I will explain how to create range names later). The calculations are all
visible on the worksheet, rather than hidden away in the formula bar. The index_num selection, here referred to as Selection_Number, is input in cell E2. The result?
It’s identical, but easier to follow
I have taught financial modelling to many gifted analysts over the years and a common mistake made by many is that they build models that are easy to build rather than models that are easy to
understand. The end user is the customer. It should be simple to use: taking shortcuts invariably only helps the modeller – and even then, more often than not, shortcuts will backfire.
CHOOSE can lead to opaque models that need to be rebuilt and are often less flexible to use. You have been warned!
We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every other business day.
A full page of the function articles can be found here. | {"url":"https://www.sumproduct.com/blog/article/a-to-z-of-excel-functions/the-choose-function","timestamp":"2024-11-09T15:28:42Z","content_type":"text/html","content_length":"25308","record_id":"<urn:uuid:2ad4b849-cbfe-4abf-831f-7bcbfe3198e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00774.warc.gz"} |
The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
The Open Mapping Theorem
We are finally going to prove the open mapping theorem in $F$-space. In this version, only metric and completeness are required. Therefore it contains the Banach space version naturally.
(Theorem 0) Suppose we have the following conditions:
1. $X$ is a $F$-space,
2. $Y$ is a topological space,
3. $\Lambda: X \to Y$ is continuous and linear, and
4. $\Lambda(X)$ is of the second category in $Y$.
Then $\Lambda$ is an open mapping.
Proof. Let $B$ be a neighborhood of $0$ in $X$. Let $d$ be an invariant metric on $X$ that is compatible with the $F$-topology of $X$. Define a sequence of balls by
where $r$ is picked in such a way that $B_0 \subset B$. To show that $\Lambda$ is an open mapping, we need to prove that there exists some neighborhood $W$ of $0$ in $Y$ such that
To do this however, we need an auxiliary set. In fact, we will show that there exists some $W$ such that
We need to prove the inclusions one by one.
The first inclusion requires BCT. Since $B_2 -B_2 \subset B_1$, and $Y$ is a topological space, we get
according to BCT, at least one $k\Lambda(B_2)$ is of the second category in $Y$. But scalar multiplication $y\mapsto ky$ is a homeomorphism of $Y$ onto $Y$, we see $k\Lambda(B_2)$ is of the second
category for all $k$, especially for $k=1$. Therefore $\overline{\Lambda(B_2)}$ has nonempty interior, which implies that there exists some open neighborhood $W$ of $0$ in $Y$ such that $W \subset \
overline{\Lambda(B_1)}$. By replacing the index, it’s easy to see this holds for all $n$. That is, for $n \geq 1$, there exists some neighborhood $W_n$ of $0$ in $Y$ such that $W_n \subset \overline
The second inclusion requires the completeness of $X$. Fix $y_1 \in \overline{\Lambda(B_1)}$, we will show that $y_1 \in \Lambda(B)$. Pick $y_n$ inductively. Assume $y_n$ has been chosen in $\
overline{\Lambda(B_n)}$. As stated before, there exists some neighborhood $W_{n+1}$ of $0$ in $Y$ such that $W_{n+1} \subset \overline{\Lambda(B_{n+1})}$. Hence
Therefore there exists some $x_n \in B_n$ such that
Put $y_{n+1}=y_n-\Lambda x_n$, we see $y_{n+1} \in W_{n+1} \subset \overline{\Lambda(B_{n+1})}$. Therefore we are able to pick $y_n$ naturally for all $n \geq 1$.
Since $d(x_n,0)<\frac{r}{2^n}$ for all $n \geq 0$, the sums $z_n=\sum_{k=1}^{n}x_k$ converges to some $z \in X$ since $X$ is a $F$-space. Notice we also have
we have $z \in B_0 \subset B$.
By the continuity of $\Lambda$, we see $\lim_{n \to \infty}y_n = 0$. Notice we also have
we see $y_1 = \Lambda z \in \Lambda(B)$.
The whole theorem is now proved, that is, $\Lambda$ is an open mapping. $\square$
You may think the following relation comes from nowhere:
But it’s not. We need to review some set-point topology definitions. Notice that $y_n$ is a limit point of $\Lambda(B_n)$, and $y_n-W_{n+1}$ is a open neighborhood of $y_n$. If $(y_n - W_{n+1}) \cap
\Lambda(B_{n})$ is empty, then $y_n$ cannot be a limit point.
The geometric series by
is widely used when sum is taken into account. It is a good idea to keep this technique in mind.
The formal proof will not be put down here, but they are quite easy to be done.
(Corollary 0) $\Lambda(X)=Y$.
This is an immediate consequence of the fact that $\Lambda$ is open. Since $Y$ is open, $\Lambda(X)$ is an open subspace of $Y$. But the only open subspace of $Y$ is $Y$ itself.
(Corollary 1) $Y$ is a $F$-space as well.
If you have already see the commutative diagram by quotient space (put $N=\ker\Lambda$), you know that the induced map $f$ is open and continuous. By treating topological spaces as groups, by
corollary 0 and the first isomorphism theorem, we have
Therefore $f$ is a isomorphism; hence one-to-one. Therefore $f$ is a homeomorphism as well. In this post we showed that $X/\ker{\Lambda}$ is a $F$-space, therefore $Y$ has to be a $F$-space as well.
(We are using the fact that $\ker{\Lambda}$ is a closed set. But why closed?)
(Corollary 2) If $\Lambda$ is a continuous linear mapping of an $F$-space $X$ onto a $F$-space $Y$, then $\Lambda$ is open.
This is a direct application of BCT and open mapping theorem. Notice that $Y$ is now of the second category.
(Corollary 3) If the linear map $\Lambda$ in Corollary 2 is injective, then $\Lambda^{-1}:Y \to X$ is continuous.
This comes from corollary 2 directly since $\Lambda$ is open.
(Corollary 4) If $X$ and $Y$ are Banach spaces, and if $\Lambda: X \to Y$ is a continuous linear bijective map, then there exist positive real numbers $a$ and $b$ such that
for every $x \in X$.
This comes from corollary 3 directly since both $\Lambda$ and $\Lambda^{-1}$ are bounded as they are continuous.
(Corollary 5) If $\tau_1 \subset \tau_2$ are vector topologies on a vector space $X$ and if both $(X,\tau_1)$ and $(X,\tau_2)$ are $F$-spaces, then $\tau_1 = \tau_2$.
This is obtained by applying corollary 3 to the identity mapping $\iota:(X,\tau_2) \to (X,\tau_1)$.
(Corollary 6) If $\lVert \cdot \rVert_1$ and $\lVert \cdot \rVert_2$ are two norms in a vector space $X$ such that
□ $\lVert\cdot\rVert_1 \leq K\lVert\cdot\rVert_2$.
□ $(X,\lVert\cdot\rVert_1)$ and $(X,\lVert\cdot\rVert_2)$ are Banach
Then $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent.
This is merely a more restrictive version of corollary 5.
The series
Since there is no strong reason to write more posts on this topic, i.e. the three fundamental theorems of linear functional analysis, I think it’s time to make a list of the series. It’s been around
half a year.
• The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
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Hybrid Rossby-Shelf Modes in a Laboratory Ocean
1. Introduction
There is evidence that planetary-scale Rossby waves have been generated off the western coast of the United States, either by unstable coastal boundary currents or by coastal waves matching in
frequency and scale (cf. Kelly et al. 1998). Bokhove and Johnson (1999), therefore, investigated the matching of planetary Rossby modes with coastal shelf modes in a cylindrical basin. Otherwise
said, linear free modes were calculated with so-called semianalytical “mode matching” techniques, as well as linear forced–dissipative finite-element methods, to find resonances. Two parameter
regimes were considered: an ocean one and a laboratory analog. These laboratory-scale hybrid Rossby-shelf modes had been considered with validating laboratory rotating tank experiments in mind.
Such an experimental validation is the topic of the present paper. Planetary barotropic Rossby modes have been shown before in the laboratory using the analogy between planetary β–plane Rossby modes
and topographic shelf modes for a uniform basin-scale north–south background topography—for example, in the classic book of Greenspan (1968). Rotating tank experiments geared toward enforcing
resonant hybrid coastal and planetary modes appear to be (relatively) new. In preparation of the rotating tank experiments, linear forced–dissipative finite-element calculations of barotropic
potential vorticity dynamics have revealed the resonant frequencies of two primary hybrid Rossby-shelf modes. These primary forcing frequencies were then used to drive the harmonic “wind” forcing
provided via the Ekman pumping and suction due to an oscillating rigid lid. Various forcing strengths have been imposed in which a match with the linear calculations is best suited by weak forcing,
whereas better visualization requires a larger signal-to-noise ratio and, consequently, stronger forcing. The latter promotes, however, the emergence of nonlinear effects. We, therefore, also compare
the experimental streamfunction fields with some nonlinear simulations of barotropic potential vorticity dynamics in an attempt to explain the differences between linear theory and the experimental
results under stronger forcing.
In ocean general circulation models (OGCMs), the coastal regions are often underresolved; furthermore, it is hypothesized that significant energy exchange takes place between the deeper ocean and the
shallower coastal zones, for example, Wunsch (2004). Vertical walls placed in the shallow seas are generally used in coastal zones as lateral boundaries in OGCMs, whereas in reality, mass, momentum,
and energy are transferred across these virtual vertical walls. As a consequence, such an exchange would not be resolved or modeled properly in the OGCMs. Suitable parameterizations of these
underresolved processes would therefore be required in the coastal zones of OGCMs. The laboratory experiments and finite element calculations we present aim to serve as an idealized barotropic system
to investigate this modal coupling between basin- and coastal-scale dynamics.
The outline of this paper is as follows: barotropic potential vorticity dynamics is introduced in section 2, and linear finite-element calculations are presented to find the relevant forcing
frequencies. These forcing frequencies are a building block in section 3, where the experimental setup and results are presented. Preliminary nonlinear simulations in section 4 indicate the effects
of strong forcing on the dynamics observed. A short conclusion is found in section 5.
2. Rigid-lid potential vorticity model
a. Nonlinear model
The forced–dissipative evolution of vertical vorticity
in a rigid-lid model is governed by the following dimensional system of equations:
where the horizontal coordinates are
= (
is time,
= (∂
, ∂
= (∂/∂
, ∂/∂
, the Jacobian
) = ∂
− ∂
for functions
) and
), the Coriolis parameter
), the transport streamfunction is Ψ, total depth
), forcing curl
, and Ekman layer depth
with (effective) viscosity
. For the laboratory case
= 2Ω
is constant with Ω
as the rotation rate of the domain, whereas for the planetary case a
-plane approximation is used (e.g.,
Pedlosky 1987
) and
for constant
. The forcing either relates to the wind stress or the differential velocity of the rigid lid, provided by Ekman suction or pumping, respectively (e.g.,
Pedlosky 1987
). Hereafter, unless otherwise indicated, we consider the laboratory case with
in which the vorticity at the top rigid lid
is related to the velocity
) = (
of the driven rigid lid. Ekman damping is assumed to be valid in the transition region between a quasigeostrophic deep ocean and the ageostrophic coastal zone, even though the topography is rapidly
changing from open ocean to coastal zone. For the rigid-lid case
= 1, it results in twice the amount of Ekman damping relative to the case with a free surface for which
= ½. The above system can be derived using classical methods (
Pedlosky 1987
Cenedese et al. 2007
The dimensional equations are scaled with radius
of the cylindrical domain, a typical depth
of the domain, and the Coriolis parameter
(in which starred variables are dimensional):
The potential vorticity of the fluid is defined by
The evolution of this potential vorticity weighted by the depth
follows by scaling with
and rewriting the nondimensionalized form of the system
with transport velocity
, two-dimensional curl operator ∇
= (−∂
, ∂
, and parameter
A cylindrical domain Ω is considered with Ψ = 0 at the boundary ∂Ω:
and initial conditions
, 0); we also used
≈ 1/
to simplify forcing and damping terms.
The numerical (dis)continuous Galerkin finite-element discretization used is based on formulation (6), by extending the inviscid formulation in Bernsen et al. (2006); it couples the hyperbolic
potential vorticity Eq. (6a) to the elliptic Eq. (6c) for the transport streamfunction and is advantageous for complex-shaped domains. Instead of a classical continuous finite-element method as used
for the streamfunction, potential vorticity is descretized discontinuously. For smooth profiles of potential vorticity, the numerical discontinuities between elements are negligible and scale with
the mesh size and order of accuracy. The numerical method conserves vorticity and energy for infinitesimally small time steps in the inviscid and unforced case, whereas enstrophy is slightly decaying
for the upwind flux used. The weak formulation of this finite-element method is given in the appendix. The method provides an alternative to classical numerical methods and is well suited for
complex-shaped domains, mesh, and order (h and p) refinement.
b. Linear model and hybrid Rossby-shelf modes
We have performed laboratory experiments to assess whether hybrid Rossby-shelf modes exist that couple planetary-scale Rossby modes with coastal-scale shelf modes. In the laboratory experiments, a
uniformly rotating tank is used with rotation frequency Ω[r]. Consequently, there is no planetary variation of the background rotation in the north–south direction. We consider the Northern
Hemisphere and define a north–south direction indirectly by introducing a background slope s = s(y) in the y direction, which is related mathematically to the β effect (e.g., Greenspan 1968). Hence,
the nondimensional β parameter β = s/H at leading order for a mean depth H. (Dimensionally, β* = β f[0]/R with β* ≈ df*/dy*, with starred dimensional variables and coordinates.) In addition, a step
shelf topography is chosen with a sudden change in depth at r = R[s] < R from the mean deep-ocean value H[2] to the mean shallow coastal value H[1] < H[2]. The topography in the laboratory domain
therefore consists of an interior deep ocean on a topographic β plane with slope s[2] = βH[2] and likewise, a shallow shelf ocean with s[1] = βH[1]. Hence, H = H[1] − s[1]y for r > R[s] on the
axisymmetric shelf and H = H[2] − s[2]y in the deep ocean for r < R[s]. A sketch of the domain is given in Fig. 1.
To assess at which forcing frequencies we expect resonant responses, we numerically solved the
counterpart of nonlinear dynamics
. The flow regime of interest concerns quasi-steady-state dynamics under harmonic forcing
with forcing frequency
, angle Δ
, and complex number
= −1. Because of this enforced harmonic behavior, the linear dynamics can be reduced to a spatial problem. A second-order Galerkin finite-element model (FEM) discretization in space was used with
piecewise linear basis and test functions. In a first set of simulations, we took 4671 nodes and 4590 quadrilateral elements on an unstructured mesh. In this FEM, a matrix system results of the form
, with Ψ
the values on the node with index
the result of the forcing, and matrix elements
the combination of the rotational advective and dissipative terms. To test the finite-element implementation, we successfully recovered the forced–dissipative analogs of the free planetary Rossby
modes and their frequencies [
appendix b(1)
The forced–dissipative response of the laboratory ocean, described above and sketched in Fig. 1, is displayed in Fig. 2 for κ = ν/Ω[r]/H[0] = 0.042 and Δθ = 2π (using laboratory values of the
viscosity of water ν = 10^−6 m^2 s^−1, Ω[r] = 2 s^−1, and H[0] = R* = 0.17 m). The streamfunction field at maximum response is shown in Fig. 3 at forcing frequency σ = 0.0613. When the experiments
were performed in late 1997, Onno Bokhove (O. B.) calculated a resonance at σ = 0.0612 for fewer and less accurate triangular elements and used that slightly different resonant frequency in the
laboratory. This small difference, however, falls within the accuracy to which the forcing frequency could be determined in the laboratory. Another resonant frequency resides at σ = 0.0878
(calculation FEM 2009), where O. B. found σ = 0.0871 earlier (calculation FEM 1997) with streamfunction fields shown in Fig. 4. The modal structure for either value—σ = 0.0878 (or 0.0871)—is the
same, and the mode relates most clearly to the m = 2 shelf mode. In these cases, the hybrid Rossby-shelf mode is a combination of an azimuthal mode number m = 0–Rossby mode in the deep ocean
absorbing into an m = 2–shelf mode at the western shelf, which after traversing counterclockwise along the southern shelf edge at R[s] = 0.8R radiates again into the deep-ocean “planetary” Rossby
mode. Such behavior is suggested directly from the dispersion relations (A6) and (A15) of the invisicid or free planetary Rossby wave and shelf modes, plotted separately in Fig. 5. The exact solution
of the planetary Rossby mode is given in (A5) and (A6) and one of the shelf modes in (A15). The modal structure of the free planetary Rossby mode clearly includes the east–west structure in its
explicit x dependence, whereas the free shelf mode is well described solely in terms of polar coordinates. The spatial structure of the hybrid-shelf mode, for the case with both β-plane topography
and a step shelf, is thus seen to correspond best with a combination of the m = 0 planetary free Rossby mode, with its explicit x and r dependence, and the m = 2 free shelf mode, with its explicit r
and θ dependence. In contrast, the modal structures of the m = 1 free Rossby mode and the m = 1 free shelf mode do not match that well, have more structure, and are therefore more affected by
damping. The resonant peak expected at around σ = 0.04 for this m = 1 mode appears to be wiped out in Fig. 2. [See also the relevant m = 1 modes displayed in Figs. 3 and 4 in Bokhove and Johnson
In the nonlinear numerical simulations described in the next section, instead of the abrupt step shelf break, a smoothed shelf break is introduced between R[s] − ϵ < r < R[s] + ϵ. The finite element
mesh contains regular nodes placed at the circles with radii R[s] ± ϵ and R; then random nodes are added, subject to a minimum distance criterion outside the shelf break; subsequent triangulation
yields a triangular mesh; and additional nodes are placed at the centroid and midpoints of the edges of each triangle to further divide the triangular mesh into a quadrilateral one. The shelf break
contains two elements across and upon mesh refinement, the value of ϵ decreases and hence the shelf break also becomes narrower. For such smoothed topography, the linear forced–dissipative response
and the streamfunction field at the maximum resonance are given in Figs. 6 and 7 for ϵ = 0.0314. Relative to the abrupt shelf topography with resonant forcing frequency σ = 0.0613, the resonant
frequency in the new calculation of the linearized system has lowered by 6% to σ = 0.0577, whereas the actual fields at resonance remain highly similar.
3. Laboratory experiments
a. Experimental setup
The laboratory tank had the following dimensions:
* = 16.7 cm,
= 0.8
, and
) =
= 0.3125 for
= 1, 2. The topography was cut out of foam, subsequently made smooth with paste, and fit tightly into the cylindrical tank. The average shelf and interior depths were
= 0.6
= 0.8
, respectively. The corresponding laboratory setup is sketched in
Fig. 8
. Dimensionless variables therein are defined in terms of the dimensional tank radius
* and
*. The glass plate on top of the water column was harmonically oscillating in a horizontal plane, its motion driven by a programmable stepping motor connected to the plate with a driving belt. The
nondimensional azimuthal velocity of the rigid lid is
, with Δ
the maximum angle of the lid reached over the forcing period
= 2
. A thin horizontal light sheet had been constructed with a tungsten lamp. including a linear filament and a lens. The black background permitted optimal reflection of light from whitish Pliolite
particles of diameter 150–250
m suspended in the flow. An analog photo camera mounted above the rotating tank tracked the movement of the particles. The whole configuration was placed on a uniformly rotating table with Ω = 1 s
. An approximate steadily oscillating state was achieved by spinning up the table for about 30 min (i.e., 30–50 forcing periods).
b. Coupled modes
Numerous experiments were carried out for a few forcing frequencies. For each experiment, streak photography was obtained with 2–4-s exposure time. The main forcing frequencies used in the laboratory
experiments were calculated with a finite-element model of the linearized equations, as explained in section 2b. We report here solely four sets of experiments, deemed best for their visual
resolution and the hybrid character of the Rossby-shelf mode.
The two sets of eight images in Figs. 9 and 10 give an impression of the flow during one forcing period of 51.3 s—that is, with dimensional frequency σ* = 0.1226 s^−1 (σ = 0.0612). It nearly
corresponds to a numerically calculated forced–dissipative resonance for a hybrid Rossby-shelf mode of the rigid-lid model (6), linearized around a state of rest (see Figs. 2 and 3). The underlying
Rossby mode has azimuthal mode number zero, whereas the underlying shelf mode has azimuthal number m = 1, 2. The shelf break is visible as a thin whitish line at r = 0.8R. In the time sequence from
top left to bottom right, a Rossby mode circulation cell in the deep interior “ocean” travels westward, where it absorbs onto the shelf and propagates counterclockwise (in the Northern Hemisphere) as
a trapped shelf mode circulation cell. On the eastern boundary, this shelf mode radiates into a planetary Rossby mode. Apart from the striking qualitative resemblance with linear forced–dissipative
calculations in Fig. 3, discrepancies occur in the northwest, presumably as a result of nonlinear effects, and in the east where the shelf mode disappears, presumably as a result of strong damping.
The rigid lid or glass plate has rotated from zero to a relatively large angle, 2π and π (Figs. 10 and 9, respectively), and back during a period. The nonlinear oscillations in the northwest corner
at t = 14, 42 s in Fig. 10 are larger under the greater forcing amplitude than at t = 0, 28 s in Fig. 9. These oscillations diminish even more when the forcing amplitude is reduced to π/2, which is
not shown here. The experimental dilemma is that a comparision with linearized modal solutions requires a weak forcing, whereas good visualization requires strong forcing for the streak photography
used. Particle image velocimetry techniques could have been used for weak forcing as well, but they were not available in 1997 at the rotating tank facilities in Woods Hole. Although the tank
dimensions are not shallow, the simplifying assumption has been that the rotation is sufficiently strong to render the flow to be nearly two-dimensional outside the thin Ekman top and bottom boundary
layers, the sidewall boundary layers, as well as the internal and boundary layers at the shelf break.
To compare the amplitudes observed and calculated, the streaks under 4-s exposure are compared with streak lengths in the calculation for σ = 0.006 13 and Δθ = π in Fig. 11. Note that the calculated
streaks are weaker, about 40%, than the observed streaks. Such a difference also occurs for forcing with Δθ = 2π in Fig. 12. The precise location of the mode around resonance, and hence its
amplitude, as well as nonlinear shifts, might cause this discrepancy. The dimensionless speed at (x, y) = (−0.11, 0.11) is about 0.0276; at the southern shelf, the maximum speed is about 0.0773 in
Fig. 11a. The speed at (x, y) = (0, 0) is about 0.0543; at the southern shelf, the maximum speed is about 0.1087 in Fig. 12a.
Similarly, the linear forced–dissipative mode calculated at σ = 0.0878 corresponds reasonably well with the observed mode for σ = 0.0871 (based on the FEM calculation in 1997) in Fig. 13 concerning
observations for stronger forcing with Δθ = 2π.
Finally, we conclude that hybrid Rossby-shelf modes exist and can be successfully visualized and measured in the laboratory; they correspond well with linear forced–dissipative calculations at a
similar frequency. Discrepancies between the experimental and numerical flow patterns are observed especially at the northwestern boundary. Additional nonlinear simulations aim to explain this
4. Laboratory results versus numerical simulations
Nonlinear simulations at resonance frequency σ = 0.0613 have been performed, starting from a state of rest and with sinusoidal forcing. The forcing period is thus 102.5 time units; in addition, κ =
0.0042, with a smoothed shelf break of width 2ϵ = 0.0628, and Δθ = 2π. The value of κ = 0.0042 implies that start-up transients disappear below 1% of their initial value within about 11 periods (cf.
the energy and enstrophy graphs versus time in Fig. 14). We note that the solution appears quasi periodic for t > 800. Shown is the solution over period 20 in Fig. 15 (i.e., from t = 1947.5 to
2050.0). A second-order spatial and third-order temporal discretization has been used; single- and double-resolution runs have been performed with 4671 and 10 363 nodes and 4590 and 10 242 elements;
the former are shown but agree well with the latter. These nonlinear simulations reveal the cause of the disturbances in the northwest corner of the domain at t = 14, 42 s in Fig. 10: a vortex starts
to roll up on the northern shelf once the cell of the southern shelf mode starts to radiate into a basin Rossby mode; subsequently, the vortex gets advected counterclockwise around the domain by the
basin Rossby mode and is dissipated once the new forcing cycle starts (see Figs. 15 and 16 in tandem). The simulated potential vorticity field displayed is less smooth than the streamfunction fields
and displays more structure, including a vortex shedding.
5. Conclusions
Hybrid Rossby–shelf modes were shown to exist analytically and numerically in Bokhove and Johnson (1999). Based on depth-averaged potential vorticity dynamics, we showed numerically that these hybrid
modes also emerged as linear forced–dissipative solutions on a laboratory scale. These hybrid modes matched the largest planetary-scale Rossby mode to a trapped shelf mode—the latter propagating
around the southern shelf. The calculated frequencies of the two dominant hybrid modes were used as driving frequency in laboratory tank experiments, based on the linearized calculations. Therein a
driven lid provided the Ekman forcing. The spatial structure of the streamfunction fields in the linear calculations and the observed streamfunction fields in streak photography agreed well or
reasonably well in the weaker and stronger forcing cases. Discrepancies in amplitude and structure were attributed to nonlinear effects, and nonlinear simulations of the depth-averaged flow suggested
observed differences to be due to a vortex generated and shed off the northern shelf by the large westward-propagating planetary Rossby mode in the deep basin. The minimum and maximum amplitudes in
the calculations of the linear and nonlinear models differed: we observed values of −0.005 and 0.009 in the former and values of −0.014 and 0.005 in the latter. The simulations provided extra
information on potential vorticity dynamics, which was unavailable from the laboratory observations.
Even though the topography used is still simple, the exhibited mode merging shows that the linear normal modes rapidly obtain a complicated structure. The distinction between trapped shelf modes and
planetary modes becomes less clear in complex domains and is a bit artificial as both modes emerge from the spatial structure in the background potential vorticity. Vortical normal modes have
recently been used to explain temporal variability in the Mascarene Basin (Weijer 2008) and in the Norwegian and Greenland gyres (LaCasce et al. 2008). Unexplored yet interesting aspects in the
idealized ocean basin used by us concern the effects of a midocean ridge (cf. Pedlosky 1996) on the communication between two separate deep-ocean half basins connected only via coastal shelves and
the numerical parameterization of underresolved shelf mode dynamics on the deep-ocean dynamics as a way to explore energy exchange through an effective, permeable boundary.
It is a great pleasure to acknowledge the assistance of John Salzig in the laboratory. Without his help, the experiments would have failed. The laboratory experiments were performed while O. B. was a
postdoctoral scholar at the Woods Hole Oceanographic Institution (1996–97); preliminary results were posted in Bokhove (1999). The encouragements and advice of Karl Helfrich, Joe Pedlosky, and Jack
Whitehead have, as always, been of great value. Jack Whitehead also kindly allowed O. B. to use the rotating table in his laboratory, and additional support came via Joe Pedlosky’s NSF Grant
OCE–9901654 in 1997.
• Bernsen, E., O. Bokhove, and J. van der Vegt, 2006: A (dis)continuous finite element model for generalized 2D vorticity dynamics. J. Comput. Phys., 211 , 719–747.
• Bokhove, O., 1999: Forced-dissipative response for coupled planetary Rossby and topographic shelf modes in homogeneous, cylindrical oceans. Preprints, 12th Conf. on Atmospheric and Oceanic Fluid
Dynamics, New York, NY, Amer. Meteor. Soc., 104–107. [Available online at http://ams.confex.com/ams/older/aofd12/abstracts/245.htm].
• Bokhove, O., and E. R. Johnson, 1999: Hybrid coastal and interior modes for two-dimensional flow in a cylindrical ocean. J. Phys. Oceanogr., 29 , 93–118.
• Cenedese, C., J. Whitehead, J. Pedlosky, and S. Lentz, 2007: 2007 Program of studies: Boundary layers. Tech. Rep. WHOI-2008-05, WHOI, 319 pp. [Available online at https://
• Greenspan, H., 1968: The Theory of Rotating Fluids. Cambridge University Press, 354 pp.
• Kelly, K., R. Beardsley, R. Limeburner, K. Brink, J. Paduan, and T. Chereskin, 1998: Variability of the near-surface eddy kinetic energy in the California Current based on altimetric, drifter,
and moored current data. J. Geophys. Res., 103 , 13067–13083.
• LaCasce, J., O. Nost, and P. Isachsen, 2008: Asymmetry of free circulations in closed ocean gyres. J. Phys. Oceanogr., 38 , 517–526.
• Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer-Verlag, 710 pp.
• Pedlosky, J., 1996: Ocean Circulation Theory. Springer, 453 pp.
• Weijer, W., 2008: Normal modes of the Mascarene Basin. Deep-Sea Res. I, 55 , 128–136.
• Wunsch, C., 2004: Vertical mixing, energy, and the general circulation. Annu. Rev. Fluid. Mech., 36 , 281–314.
(Dis)continuous Galerkin Finite Element Discretization
Weak formulation
A (dis)continuous Galerkin finite element method is used to discretize the following generalized system of equations:
) > 0,
) ≥ 0, and
) and for a scaling in which
= 1 in a relevant leading-order way. Here, only a singly-connected domain Ω is considered with boundary ∂Ω; it is cylindrical with radius
and Ψ = 0 at ∂Ω:
. In
, we have used the approximation
in the forcing and damping terms because
In Bernsen et al. (2006), a finite-element discretization is given and verified for the inviscid, unforced version of (A1) for complex-shaped, multiconnected domains. The generalized streamfunction
and vorticity formulation (A1) is advantageous because it unifies several systems into one, such as the barotropic quasigeostrophic, and rigid-lid equations. A third-order Runge–Kutta discretization
in time and second-, third-, or fourth-order discretizations in space are implemented and available for use. Without forcing and dissipation, discrete energy conservation is guaranteed in space,
whereas the discrete enstrophy decays for the upwind numerical flux and is conserved for the central flux for infinitesimal time steps. The latter central flux is stable but yields small oscillations
in combination with the third-order time integrator. When necessary, the circulation along the boundary can also be treated properly on the discrete level.
In extension to
Bernsen et al. (2006)
, a discontinuous Galerkin method is used to find the weak formulation of
. After we multiply
with an arbitrary test function
, integrate over the domain Ω, and use numerical fluxes between interior elements and at the boundary elements, over each element
the following weak formulation is obtained:
where the numerical flux
is replacing
, in which
is the component of the transport velocity
normal to an element face ∂
are the limit values of the vorticity just inside and outside an element face. Likewise,
is the limit value of the test function just inside the element and ∂
= ∂/∂
. Moreover, (·, ·)
is the
inner product over element domain
. In contrast, a continuous Galerkin finite element is used to find the weak formulation of
. We multiply
with an arbitrary test function
, integrate over the domain Ω, and use proper boundary conditions to obtain
with circulation
for the case
> 0 with
Γ as the line element and
as the unit tangential vector along ∂Ω. In case
= 0, the streamfunction Ψ
= 0 and then the circulation is not required, in the singly-connected domain used here. Subscripts (·)
denote that test functions and variables are approximations in the appropriate function spaces (see
Bernsen et al. 2006
). These spaces are chosen specifically to guarantee energy conservation; that is, the space of functions for the discontinuous part of the discretization must cover the space of functions of the
continuous part of the discretization. We have calculated several modal solutions as linear exact and asymptotic test solutions for the nonlinear numerical model.
Normal mode numerical tests
Linear free and forced–dissipative planetary Rossby modes
A free Rossby mode in a cylindrical domain of radius
satisfies the linearized version of system
with constant
= 1,
= 1 +
, and
= 0, or
= 1/
= 0,
= 1 +
, and
= 0. A linear modal solution reads
is the amplitude,
is the Bessel function of the first kind,
is the radius,
is the azimuthal angle, and frequency
The boundary condition is satisfied because
are the zeroes of
) = 0, given the azimuthal mode number
= 1, 2, … , ∞). A comparison between the linear solution
= 0 is made with the nonlinear numerical solution initialized by the exact
, 0), given the initial streamfunction Ψ(
, 0) = Ψ
, 0). An approximate Ψ
, 0) is then calculated numerically. Good agreement between the linear exact and the nonlinear numerical solutions is found. We also numerically calculated the linear forced–dissipative response of
planetary Rossby modes with
= 0.3125 as a representative laboratory value and
. Some of the first few frequencies of inviscid free modes are
and correspond to solutions
. The larger resonant frequencies match finely with the free-wave frequencies
Linear free shelf mode
In the nonlinear numerical simulations, we wish to avoid a discontinuous profile of
) or the depth
= 1/
. Instead of a discontinuous step in the depth, we consider the continuous axisymmetric depth profile
, the shelf break radius
, where the depth changes suddenly, and
≪ 1. A matched asymptotic solution (cf. Bokhove and Vanneste 2001, unpublished manuscript) to the linearized version of
is sought for
= 1,
= 0, and
= 1/
—that is, a solution of
Outer solutions in the regions
will satisfy the linearized Eq.
with depth
in the step shelf limit
→ 0. Inner solutions are valid in the transition region
, and a suitable sum of inner and outer solutions will provide the entire asymptotic solution.
Outer solutions satisfy, to all orders, ∇
Ψ = 0 with Ψ(
) = 0; hence, it concerns the real part of
with coefficients
, frequency
, and azimuthal wavenumber
For the inner solution, a stretched coordinate
= (
is introduced. Subsequently, the inner expansion
is substituted in
and evaluated at leading order in
, giving the leading-order equation
Because the outer expansion in the inner region satisfies
and the inner one
, we obtain together with the continuity requirement the following inner boundary conditions at
= ±1:
The integration of
= −1 to
= 1 using
yields the dispersion relation for the outer expansion
In rewritten form and by using Ψ
) = Ψ
) and
, the outer solutions therefore become
A first integration of
= −1 to
< 1 or
> −1 to
= 1, using
, followed by a second integration, gives
= −1, using
, we find
and, likewise, at
= 1,
where we used the dispersion relation
to find the relation at
= 1. Considering the inner and outer expansions in the inner region, we choose
= 0; it also explains why the outer solution is chosen to hold at all orders.
The uniformly valid matched asymptotic solution consists of the mean of inner and outer solutions
= 0 and
We initialized the nonlinear numerical simulation with the potential vorticity based on the asymptotic solution
of the streamfunction Ψ = Ψ
= 0. The agreement between the asymptotic linear solution and the nonlinear numerical solution of the streamfunction was quite reasonable.
Forced–dissipative hybrid-shelf modes
In the nonlinear numerical simulations, we use
) = 1/
) =
= 1. Instead of a discontinuous step in the depth, we consider a depth profile
, and
≪ 1 for
= 1, 2. The simulated streamfunction fields displayed in
Fig. 15
over forcing period 20 have been verified to equal the ones in forcing period 19. Parameter values are
= 0.0613,
= 0.0042,
= 0.0314, and Δ
= 2
Fig. 1.
Sketch of laboratory domain with abrupt shelf topography, and deep interior ocean and shallow-shelf slopes mimicking β (Bokhove and Johnson 1999).
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 2.
Linear forced–dissipative response for topographic Rossby-shelf modes displayed as the L[∞] − norm of |Ψ| against 500 forcing frequencies σ. Parameter values and κ = ν/Ω[r]/H[0] = 0.0042.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 3.
Streamfunction field for Rossby-shelf modes over one forcing period T nearby the maximum response σ = 0.0613. Parameter values and κ = ν/Ω[r]/H[0] = 0.0042.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 4.
Streamfunction field for Rossby-shelf modes over one forcing period T at the maximum response σ = 0.0878.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 5.
Dispersion relation of the free planetary Rossby mode of zeroth order in the radial direction; the coastal shelf mode for β = 0.3125; H[1] = 0.6R, H[2] = 0.8R, and R[s] = 0.8R.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 6.
Linear forced–dissipative response for topographic Rossby-shelf modes displayed as the L[∞] − norm of |Ψ| against 500 forcing frequencies σ for a smoothed shelf break of width 2ϵ = 0.0628 around r =
R[s]. Parameter values and κ = ν/Ω[r]/H[0] = 0.0042.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 7.
Streamfunction field for Rossby–shelf modes over one forcing period T at the maximum response σ = 0.0577 for a smoothed shelf break. Parameter values and κ = ν/Ω[r]/H[0] = 0.0042.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 8.
Sketch of the laboratory setup.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 9.
Streak photography of hybrid Rossby-shelf modes at t = 0, 7, 14, 28, 35, 42, and 49 s for a forcing period of 51.3 s (nondimensional σ = 0.0612), and maximum rigid-lid excursion Δθ = π. Exposure time
was 4 s.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 10.
Streak photography of hybrid Rossby–shelf modes at t = 0, 7, 14, 28, 35, 42, and 49 s for a forcing period of 51.3 s (nondimensional σ = 0.0612), and maximum rigid-lid excursion Δθ = 2π. Exposure
time was 2 s.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 11.
(a) Streak photography of hybrid Rossby–shelf mode observed at t = 0 s for a forcing period of 51.3 s (nondimensional σ = 0.0612), and maximum rigid-lid excursion Δθ = π. Exposure time was 4 s. (b)
Same as in (a), but for the calculated linear solution; phase shift adjusted semioptimally by eye.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 12.
Same as in Fig. 11, but with maximum rigid-lid excursion Δθ = 2π, and exposure time was 2 s.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 13.
Streak photography of hybrid Rossby-shelf modes at t = 0, 5, 10, 15, 20, 25, 30, and 35 s for a forcing period of 36.1 s (nondimensional σ = 0.0871), and maximum rigid-lid excursion Δθ = 2π.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 14.
Energy and enstrophy vs time.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1
Fig. 15.
Streamfunction over forcing period 20; σ = 0.0613, κ = 0.0042, ϵ = 0.0314, and Δθ = 2π.
Citation: Journal of Physical Oceanography 39, 10; 10.1175/2009JPO4101.1 | {"url":"https://journals.ametsoc.org/view/journals/phoc/39/10/2009jpo4101.1.xml","timestamp":"2024-11-02T15:51:31Z","content_type":"text/html","content_length":"785969","record_id":"<urn:uuid:a25c917b-1845-42e8-a9e9-389190ba7d5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00226.warc.gz"} |
Bitcoin and the claim of Total Turingness.
Craig Wright is a very controversial man — Bitcoin SV, Claims of being Satoshi Nakamoto, major plagiarism accusations- Overall, not the face of Blockchain you'd want. But Craig Wright is also a very
smart man, much more than the Twitter handles in the Crypto Community would have you believe. His degrees and years of experience are something you cannot ignore, or at least that's what the BSV
community and Wright himself want you to believe.
We give Dr. Wright the benefit of doubt today. In 2014, Dr. Wright came out with a paper titled “Bitcoin: A total Turing machine” [1] and today we talk will about it.
If you don’t know about Turing Machines, now is a good time to talk about it. Turing machines are theoretical models of computation that represent an abstract machine. Turing machines are tapes
divided into cells.
A Universal Turing Machine
Turing machine was put forth by the legendary Alan Turing in ’36 and it proved something very important: the uncomputability of the Entscheidungsproblem. Turing Machines cannot be implemented in real
life of course, because we don’t have infinite tapes in real life, and because these machines aren’t optimized for real-life implementations. For example. Real Life computers use RAMs, which TMs
don’t. But TMs can compute anything a real computer can, given duration of computation is not an issue.
Turing Completeness is the idea that any program that is Turing Complete will halt. And any program that halts will not tend to infinity. From here we deduce that a Turing Machine should run infinite
How does this all relate to Bitcoin?
Bitcoin’s Script is a stack-based, non-Turing Complete programming language. And there is a lot of conversation about this, especially attacking the Ethereum community for being Turing complete, and
the lack of need to be Turing complete because “Post’s theorem”.
Representation of a Stack Data Structure
Post’s theorem posits the connection between Arithmetical Hierarchy and Turing degrees [2]. It is often stated defending the lack of need of Turing Completeness, and safety of decidability as
Wright proposes Bitcoin has been what he proposes to call a “Probabilistic Total Turing machine”. Let me try to summarize Craig’s claims
• Wright proposes that Bitcoin is equivalent to the machine proposed in Wolfram’s conjecture, which states that a 2 state, 3 symbol Turing Machine is a Universal Turing Machine. By this, a
proposition establishes that Bitcoin Script must also be a Universal Turing Script.
• Wright states that Bitcoin is Turing Complete and proves so using the Ackermann functions.
Wright proposes that Bitcoin Script is a Probabilistic Total Turing Machine, which is something he proposes earlier. PTTM is different from PTMs. The alternate stack in the Script makes it Turing
Lack of looping in Bitcoin Script does not it non-Turing Incomplete. Since Bitcoin is formed using Primitive Recursion Functions, the script construct is Turing Complete.
Let’s try to dissect all this. Bitcoin does not support loops: This by itself does not make the script Turing Complete or Incomplete, but this does mean that the script lacks Control Structures.
Sure, if-else is there, but without For loops, there is a lack of Control. This is not a bug in the language, it is a feature. A small script such as Busy Beaver [3] will cause the Bitcoin network to
return DoS overtime at worst, and slow down the hash rate at best. This was implemented to increase decidability and safety.
Two stacks don’t make a Turing Complete Machine: To Simulate a Two-Stack PDA (Which is Turing complete), you need a control structure. As I said in, Bitcoin lacks For loops.
Bitcoin limits the number of non-Push operations you can do per script: If you look at Bitcoin’s Github, you will see the following piece of code
static const int MAX_OPS_PER_SCRIPT = 201;
This means that you can do 201 non-Push ops per script. This is to safeguard against undecidable problems. So even if you implemented a 2PDA, you would be practically limited by this.
Whether or not Bitcoin is Turing Complete is a futile discussion. Bitcoin Network is “more” Turing than other networks because of its size (That statement holds no scientific value, I said it as a
figure of speech). Bitcoin Script is better off Turing incomplete by design, even though there might be some “proof” of it being otherwise. P2P consensus networks need to implement some form of
decidability in their scripts. Ethereum, for example, has a Gas-system. It’s quasi-Turing Complete for that reason. Bitcoin Script, if used outside of the Bitcoin network, would have issues because
it is Turing Incomplete by design, but a few modifications, and you can build it to be Turing Complete. But in practicality, no system is truly Turing Complete. I would love your comments about this!
Further Reading
[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3265146
[2] https://youtu.be/TGE6jrVmt_I
[3] https://en.wikipedia.org/wiki/Busy_beaver
Top comments (6)
chimpyTheDev •
Hey, I'd love to try out the text.
abhinav the builder •
Hi, please share your mail!
William Boakye •
I’d definitely like to have a read!
abhinav the builder •
Thank You William!
Is there anywhere I can share prototypes with you fast?
Calvin •
That would be awesome!
abhinav the builder •
Cool, thanks!
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://practicaldev-herokuapp-com.global.ssl.fastly.net/abhinavmir/i-am-making-an-introductory-text-to-blockchain-and-making-your-own-decentralized-apps-let-me-know-if-anyone-wants-it-51l8","timestamp":"2024-11-11T00:45:34Z","content_type":"text/html","content_length":"137460","record_id":"<urn:uuid:66e219dd-d141-4ea6-9cd0-f36f308d5e3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00100.warc.gz"} |
A plane is a flat surface with no thickness.
Our world has three dimensions, but there are only two dimensions on a plane:
length and width make a plane
x and y also make a plane
A plane has no thickness, and goes on forever.
It is actually hard to give a real example!
When we draw something on a flat piece of paper we are drawing on a plane ...
... except that the paper itself is not a plane, because it has thickness! And it should extend forever, too.
So the very top of a perfect flat piece of paper
that goes on forever is the right idea!
Also, the top of a table, the floor and a whiteboard are all like a plane.
Here you can spin part of a plane:
A plane has 2 Dimensions (and is often called 2D):
Point, Line, Plane and Solid
A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)
Plane vs Plain
In geometry a "plane" is a flat surface with no thickness.
But a "plain" is a treeless mostly flat expanse of land ... it is also flat, but not in the pure sense we use in geometry.
Both words have other meanings too:
• Plane can also mean an airplane, a level, or a tool for cutting things flat
• Plain can also mean without special things, or well understood
Imagine you lived in a two-dimensional world. You could travel places, visit friends, but nothing in your world has height.
You could measure distances and angles.
You could go fast or slow. You could go forward, backwards or sideways. You could move in straight lines, circles, or anything, so long as you never go up or down.
What would life be like living on a plane? | {"url":"http://wegotthenumbers.org/plane.html","timestamp":"2024-11-03T15:39:36Z","content_type":"text/html","content_length":"5742","record_id":"<urn:uuid:396c8f8b-b86c-4bb0-bfb0-7df2a85f3158>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00204.warc.gz"} |
Is there any way I can make the attached contour plots more smooth ?
Accepted Answer
Answered: jacob faerman on 15 Jan 2018
I have attached two contour plots for pressure field outside an ellipse. The contour plots I got are jagged, may be due to the numerical integration I am using to calculate the pressure fields.The
two plots are with Gaussian points 15000 and 5000 respectively. It is observable that with more Gaussian points, the contour plots are becoming smooth, but it is also taking time. Is there any other
way to get smooth contour lines.
21 views (last 30 days)
Is there any way I can make the attached contour plots more smooth ?
Sorry. If you try to resize a matrix (using interpolation) that is not itself sufficiently smooth that the original matrix yields non-smooth contours, you will get more finely interpolated noise!
Interp2 is a waste of time here.
If you have a noisy matrix, then BEFORE you compute contours, you MUST smooth it FIRST if you want smooth contours. That smoothing may be done by removing the noise before you create the matrix, thus
noise reduction prior to creating the matrix of data. Or you can do smoothing by applying a smoothing tool to your matrix, after creation.
You are doing some form of numerical integration that creates a noisy result. So a higher precision numerical integration is an option. Noise in the integration, as well as your comments, suggests
you might be doing a Monte Carlo integration. I can't help if you don't say.
If you must do post smoothing on the matrix, then the classic solution is simply to apply a Gaussian blur to the matrix. You can do that using conv2. Or it should be possible to use methods like
Savitsky-Golay, in two dimensions.
4 Comments
Image Analyst on 7 Aug 2017
I agree with John. Smoothing array before finding contours is a good solution, otherwise Savitzky-Golay filter to smooth a jaggedy outline on a non-smoothed array. It's easy enough to try them both
and see which you like better. For what it's worth, see my attached Savitzky-Golay outline smoothing demo.
More Answers (4)
You should be plotting the contours from a matrix say Z..you can increase the dimensions of this matrix using imresize. If you have spatial data matrices X and Y also, you may go for interpolation
using interp2.
I like KSSV's option of using imresize, because it performs antialiasing before interpolation. To scale by 1/5,
sc = 1/5; % scaling factor
Alternatively, you can do a 2D gaussian lowpass filter pretty easily with filt2. Syntax would be
res = ?;
lambda = ?;
Zf = filt2(Z,res,lambda,'lp');
where you'll have to enter res, which is the pixel size in x,y; and you'll have to enter lambda, the approximate lowpass wavelength.
Edited: Teja Muppirala on 7 Aug 2017
Maybe a median filter? However ORDFILT2 needs the Image Processing Toolbox.
Z = peaks(501); % Sample data
Z = Z +0.1*randn(size(Z));
Z(abs(Z)<0.5) = 0;
rad = 5; %Filter radius
[x,y] = meshgrid(-rad:rad);
filtArea = x.^2+y.^2 <= rad.^2; % Use a round filter
Zfilt = ordfilt2(Z,ceil(nnz(filtArea)/2),filtArea);
The median filter worked nicely for me, and filter2 does not need the image processing toolbox. | {"url":"https://au.mathworks.com/matlabcentral/answers/351709-is-there-any-way-i-can-make-the-attached-contour-plots-more-smooth","timestamp":"2024-11-12T15:07:55Z","content_type":"text/html","content_length":"183627","record_id":"<urn:uuid:bb8941a7-08c3-49a4-94e4-979b21bb8bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00129.warc.gz"} |
Lesson: Using an outcome tree to display outcomes for two events | KS3 Maths | Oak National Academy
Using an outcome tree to display outcomes for two events
I can systematically find all the possible outcomes for two events by using an outcome tree diagram.
Using an outcome tree to display outcomes for two events
I can systematically find all the possible outcomes for two events by using an outcome tree diagram.
Lesson details
Key learning points
1. Possible outcomes for two events can be shown in an outcome tree diagram.
2. The outcome tree diagram can be used to generate either an outcome table or list.
3. An outcome tree can be generated from either an outcome table or list.
Common misconception
Every group of branches should have the same number of branches.
In a multi-staged trial, each stage may have a different number of outcomes. The number of branches in each group reflects the number of outcomes of a stage. This means each group of branches can be
made from a different number of branches.
• Tree diagrams - Tree diagrams are a representation used to model statistical/probability questions. Branches represent different possible events or outcomes.
• Outcome tree - Each branch of an outcome tree shows a possible outcome from an event or from a stage of a trial. The full outcome tree shows all possible outcomes.
• Sample space - A sample space is all the possible outcomes of a trial. A sample space diagram is a systematic way of producing a sample space.
This lesson will focus on outcome trees, which demonstrate one particular purpose of a tree diagram. These outcome trees will focus on an understanding of how to show outcomes of a 1-stage or 2-stage
trial on an outcome tree. There will be no probabilities attached to the branches this lesson.
Teacher tip
6 Questions
Which of these are equivalent to $$1\over5$$?
Correct answer: $$2\over10$$
Correct answer: $$0.2$$
Correct answer: $$20$$%
Andeep carries out the following trial: spinning this spinner once. Which of these is the sample space of distinct outcomes for Andeep's trial?
There are 4 distinct outcomes.
Correct answer: {A, B, C, D}
{A, B, C, D}
{A, A} is the repeated outcome.
Each sector in a spinner is of equal size. In which spinners are the outcomes of “win” and “lose” equally likely?
A fair, 6-sided dice has outcomes: {2, 3, 4, 5, 6, 7}. Which of these events is most likely to happen?
Correct answer: prime number
prime number
A fair 6-sided dice has outcomes: {1, 2, 3, 4, 5, 6}. Event: if the dice lands on a prime number, Aisha wins. Which of these outcome spinners most closely represents the likelihood of Aisha winning?
Trial: a spinner will be spun twice. The outcome of each spin will be multiplied together. The spinner has outcomes: {2, 5, 7, 11, 15}. Match the letters to their correct value in the outcome table.
6 Questions
The outcome tree represents all outcomes from spinning the spinner once. Which outcomes are missing from the outcome tree?
The outcome tree represents all outcomes from spinning spinner P and spinner Q once each. Match each letter to its value.
This outcome tree represents all outcomes from spinning spinner R twice. The sample space shows the outcomes when the results of each spin are added. Match each letter to its value.
Trial: spinning this spinner once. Event: the spinner landing on a prime number. Which group of outcomes should be written inside the box?
Correct answer: 4, 6, 9
4, 6, 9
This incomplete outcome tree represents all outcomes from spinning spinner X twice and adding the results of each spin. In the sample space, match the letter to its value.
Trial: spinning this spinner once. Event 1: the spinner landing on a even number. Event 2: the spinner landing on a square number. Which of these outcomes should be written inside the box? | {"url":"https://www.thenational.academy/teachers/programmes/maths-secondary-ks3/units/probability-possible-outcomes/lessons/using-an-outcome-tree-to-display-outcomes-for-two-events","timestamp":"2024-11-14T08:56:27Z","content_type":"text/html","content_length":"303646","record_id":"<urn:uuid:c3bc834c-9cfe-40b0-a508-2a840c9fc601>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00094.warc.gz"} |
Table of Contents
This screen lists all parameters that can be changed by the user.
For a normal deterministic simulation, you need to enter a nominal value for each parameter. For a probabilistic simulation, you need also to assign probability density functions to the uncertain
parameters that you wish to include in the analysis.
Export to Excel See importing and exporting data
Import from Excel See importing and exporting data
Help Shows this page
Entering parameter data
Select a parameter in the list to edit its data. A short presentation of the parameter is displayed on the right hand side. Under, in the Data section, a table with the data for the parameter is
When you click a cell in the table an editor will appear under the table which can assist you in setting a value for the parameter.
• PDF
- The
probability density function
defines the uncertainty distribution. It is recommended that you use the
, which is displayed as you click a PDF field, to set probability density functions.
^1) - Click the button to display this header.
Below the table are four buttons:
Transpose the table (switch rows for columns). Only available for objects with one or more dimensions.
Hide default rows/columns.
Hide the non-default rows/columns.
Show (or hide) more headers. Some headers are originally hidden, such as headers for comments and statistics
The editor appears when you click in a field in the table. When you click a field for a numeric value, an editor which can help you convert values from one unit to another. When you click a PDF the
PDF editor appears. You can hide the editor by clicking the Toggle button.
Number editor
The constants editing tool appears when a table cell is clicked that contains a number. It can be hidden by clicking the Toggle button, as shown in the screen shot below.
The editor contains two tools:
Keypad - Use the keypad with the mouse to enter a value or an operator without having to use the keyboard.
Unit conversion - This tool lets you convert a numerical value from one unit to another (compatible) unit.
• From - The original unit
• Value - The original value
• To - The target unit
• Result - The converted value
• Apply - Copies the results into the selected cell in the table.
PDF Editor
Choose a probability density function from the Distribution functions drop-down list. Input fields will appear depending on the number of arguments required for the selected function.
The Upper trunc and Lower trunc fields are always available and can be used to truncate the function. Lower trunc is often used to avoid negative values for parameters.
See probability density functions for a list of all PDFs.
The PDF Editor has four tools, which are accessed through the buttons next to the Editor button.
The chart tool displays a graph of the current function. By using the buttons on the left hand side you can choose to plot
• The PDF (Probability density function)
• The CDF (Cumulative distribution function)
• The survival function
• The hazard function
• The cumulative hazard function
Delimiters and calculated values from the calculation tool are displayed in the graph using markers.
Displays statistics on the currently selected function.
Allows you to calculate values and percentiles for the current function. These values are marked in the graph of the Chart tool. Information
Describes the currently selected function and it’s parameters.
See also | {"url":"https://wiki.merlin-expo.eu/doku.php?id=parameters_screen","timestamp":"2024-11-12T21:36:15Z","content_type":"text/html","content_length":"23498","record_id":"<urn:uuid:b665aad8-0cf7-4f96-b663-748529abcb1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00468.warc.gz"} |
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Settlement of DLCFD - Suredbits
On December 17th, 2020, Roman and I (Nadav) entered into a special kind of Discreet Log Contract (DLC) called a Contract for Difference (CFD), or Discreet Log Contract for Difference (DLCFD) for
short. Roman entered into the CFD with $22.80 worth of BTC, and exited with that same amount of USD even though the price of BTC moved. That is to say, if the price were to go down, then I would have
to pay Roman to make up the difference so that he would exit with $22.80. If the price were to go up on the other hand, as it did, then Roman would have to pay me to make up for the difference and
exit with that same fixed USD amount.
From Roman’s point of view, the CFD allows him to hold BTC without being exposed to bitcoin price volatility (assuming he is denominating his assets in USD). Whether the price moves up or down, the
Contract for Difference covers the difference for him, leaving his USD value fixed. Roman’s payout curve is shown above (with oracle outcomes denominated in BTC/USD). He entered with 100,000 satoshis
when the price was $22,800/BTC and you will notice that if the price does not move, then 100,000 satoshis will again be his payout. If, on the other hand, the price moves down, he is compensated in
BTC terms, and if the price moves up, then he loses satoshis to stay at a fixed USD value. Another way to see this is if we were to re-plot the above payout curve, denominating payout in USD instead
of sats, it would just be a flat line at $22.80 for any outcome.
From my point of view however (the upside-down version of the payout curve above), this is a type of long position on BTC relative to USD, meaning that I am speculating that the BTC/USD price will
rise resulting in Roman paying me the difference.
In some ways, the existence of DLCFDs allows for the decoupling of holding/using BTC from the price exposure that is usually associated with such activity. So long as there are those interested in
taking long positions on BTC/(Other Asset), then entities such as businesses can match with them to create CFDs which leave their funds fixed in (Other Asset) terms for any period of time. And when
this entity wishes to pay someone in BTC, they can do so atomically with exiting the CFD so that from their perspective, they were hardly ever exposed to BTC price volatility while still being able
to use BTC as a payment infrastructure.
Now that we know what a Contract for Difference is, and some of its uses, let’s dive into some of the details as to how we can execute DLCFDs on Bitcoin. A DLC consists of a single on-chain funding
transaction, and a set of off-chain transactions called Contract Execution Transactions (CETs). There is a single CET for every possible outcome and the outputs on the CET reflect each party’s payout
for that outcome. Each CET spends the on-chain funding transaction’s 2-of-2 multisignature output and the oracle contract is enforced by making this spending contingent on an oracle signature of a
specific message unique to that CET. To learn more about how DLCs work, check out our previous blog post series or the DLC work-in-progress specification which has additional resources.
As we discussed in the blog post about our previous (volatility) DLC, this scheme supports an arbitrary number of outcomes (and hence arbitrary oracle contracts) in theory. However, in practice we
need to compress our set of outcomes to a reasonably small number to accommodate communication and computational constraints. We discussed how continuous intervals of constant-valued outcomes can be
compressed into negligible sizes by having oracles sign each binary digit (aka bit) of the outcome individually. This allows us to ignore the least significant digits to construct transactions that
cover many outcomes, for example if the last 10 bits can be ignored (as no matter their value the payout is the same), then we can construct a single transaction which covers 2^{10} = 1024 outcomes
at once. This means we only need to create, send, and store a single adaptor signature in place of 1024!
Our oracle, Skrik, committed to signing the BTC/USD price as 17 binary digits (supporting all values between $0/BTC and $131,071/BTC).
In our volatility DLC, we used this compression trick to cover all cases that were not in our expected price range of $17k-$21k all the way up to $131,071 with fewer than 20 CETs! However, you’ll
notice that in the case of a CFD, only one side has a constant-valued collar, and there are far fewer cases that must be covered. This is where our first new tool comes into play: Rounding Intervals.
During contract negotiation, Roman and I agreed that we were comfortable rounding our payouts to the nearest 100 satoshis for all outcomes, and to the nearest 1,000 satoshis for any outcome beyond
$30,000 (noting that even if BTC is worth $130k, 1,000 satoshis is still only worth $1.30, a small portion of the locked up assets). Our willingness to do this rounding enables us to use our CET
compression algorithm everywhere, instead of only for collars. Specifically, it allows us to take advantage of any flatness (non-steepness) along the payout curve because if the curve is relatively
flat, then we can round large sections to the same nearest 1000 satoshis and compress that interval.
How effective is this scheme? Well, here are some numbers computed for our DLCFD:
• Without rounding: 86,718 CETs which requires significant computing time as well as ~65MBs of data over the wire.
• Rounding everything beyond $30k to the nearest 1,000 satoshis: 20,547 CETs which requires ~15MBs of data over the wire.
□ Note that this is about the number of values between 10k (where the collar is) and 30k. Essentially the mostly flat part of the curve after $30k has been compressed into a negligible number
of CETs!
• Rounding to the nearest 100 satoshis on $0 to $30k and the nearest 1,000 satoshis afterwards: 5,511 CETs which requires only 3.5MBs of data over the wire!
□ Note that even rounding to the nearest 100 satoshis (1-3 cents) on the steepest part of the payout curve resulted in a ~4x improvement in the number of CETs!
If you are interested in more details, rounding intervals are included in the numeric outcome proposal and implemented on an experimental branch of bitcoin-s.
The second novel tool that allowed us to execute this CFD was Antoine Riard’s Non-Interactive Protocol proposal. When Roman and I initially broadcasted our funding transaction, I immediately realized
that we had made a mistake. We had agreed on a fee rate of 50 sats/vbyte without checking to see if this was reasonable, and average fees at the time were well over 100 sats/vbyte! Rather than
requiring that we reconnect to each other and set up a new contract forcing us to re-sign thousands of CETs, I instead unilaterally bumped our fee using Child Pays for Parent (CPFP) which is simply
the act of spending on output on a too-low-fee transaction with a too-high fee transaction so that the average becomes reasonable and both transactions become desirable to miners (they cannot have
the child alone as it is only valid if the parent is confirmed as well). Thus, I broadcasted a CPFP transaction which caused our funding transaction to be confirmed, all without requiring any
interaction between me and Roman. If I had been foolish again and my CPFP transaction did not pay enough in fees to cover its parent, then I could have used Replace by Fee (RBF) which I enabled on my
unilateral transaction to replace the child with a higher-fee child.
We also used the Non-Interactive Protocol’s CPFP mechanism to confirm our CET which again, used 50 sats/vbyte so that Roman broadcasted a child transaction to cover the fees.
In the end, our oracle broadcasted signatures corresponding to the outcome $23,427/BTC and you will notice that in our CET, Roman’s payout is 0.000974 BTC corresponding to (23,427 * 0.000974) =
$22.82 which is within our rounding agreement (of 100 sats = $0.023) of his initial dollar amount of $22.80!
As was mentioned in our last post, in the near future there will be support for threshold oracle schemes, such as using 2 of 3 oracles to execute a CFD. I am currently working on a proposal for how
this should be done, which will be released on the specification repository very soon!
A more long-term, but still very important, additional improvement is support for DLCFDs on the Lightning Network. On-chain CFDs are fully supported by our current code, but they only really make
economic sense today for larger amounts due to high fees. Not only will Lightning DLCFDs allow contract execution to be virtually fee-less with near-instant confirmation, but Lightning DLCFDs can
even be used to enable trustless version of Rainbow-esque synthetic assets in channels which are liquid (i.e. spendable)!
Stay tuned for more updates and progress on Discreet Log Contract development and other cool stuff we’re working on at Suredbits!
Feel free to connect with us on Twitter @Suredbits or join our Suredbits Slack community.
If you are interested in setting up a bitcoin-s wallet, check out the bitcoin-s documentation. | {"url":"https://suredbits.com/settlement-of-dlcfd/","timestamp":"2024-11-13T01:21:01Z","content_type":"text/html","content_length":"87058","record_id":"<urn:uuid:fc002100-ab2a-4f40-b1cd-2351c81220a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00891.warc.gz"} |
ncl_wmap_params: This document briefly describes all the - Linux Manuals (3)
ncl_wmap_params (3) - Linux Manuals
ncl_wmap_params: This document briefly describes all the
Wmap_params - This document briefly describes all the internal parameters of Wmap.
The following shows all of the internal parameters that affect the behavior of Wmap. Each entry includes the name of a parameter, its Fortran type, its default value, and a short description of the
'ALO' - Integer - 0
A Flag to indicate whether a weather front is at the surface or aloft. ALO=0 is surface; ALO=1 is aloft.
'AOC' - Integer - -1
Color index for the outlines of arrows (outlines drawn only if AOC is non-negative).
'ARC' - Real - 0.
Length of current weather front line (retrieval only).
'ARD' - Real - 0.
Direction of arrows, expressed in degrees.
'ARL' - Real - 1.
Scale factor for the length of an arrow's tail, independent of the arrow's size.
'ARS' - Real - 0.035
Size of arrows, expressed as a fraction of the maximum screen height.
'ASC' - Integer - -1
Color index for the shadows of arrows (shadows are drawn only if ASC is non-negative).
'AWC' - Integer - 1
Color index for the interior of arrows.
'BEG' - Real - 0.015
Space, expressed in world coordinates, to leave along a front line before the first symbol is drawn.
'BET' - Real - 0.045
Space, expressed in world coordinates, to leave along a front line before the first symbol is drawn.
'CBC' - Integer - 1
The color index to be used for backgrounds of city and daily high/low labels.
'CC1' - Integer - 2
Color index for the interior of a cloud symbol.
'CC2' - Integer - 1
Color index for the outline of the cloud symbol.
'CC3' - Integer - 1
Color index for the shadow of the cloud symbol.
'CFC' - Integer - 1
Color index to use for cold front symbols.
'CHT' - Real - 0.0105
Height of characters, expressed as a fraction of the maximum screen width, of the city labels and daily high/low temperatures.
'CMG' - Real - 0.002
Size, expressed as a fraction of the maximum screen height, of the margins used for the city labels.
'COL' - Integer - 1
Color index to use for all objects that require only a single color setting.
'CS1' - Real - N/A
Slope of the left edge of a front line as calculated internally as measured in degrees from the X-axis (used when SLF=0,2, or 3). This parameter is for retrieval only.
'CS2' - Real - N/A
Slope of the right edge of a front line as calculated internally as measured in degrees from the X-axis (used when SLF=0,1, or 3). This parameter is for retrieval only.
'DBC' - Integer - 0
The color index to be used for the background shadow for the dots marking the city locations.
'DTC' - Integer - 1
The color index to use for the dots marking the city locations.
'DTS' - Real - 0.006
Size, expressed as a fraction of the maximum screen height, of the dots used to mark cities.
'DWD' - Real - 2.0
Line widths for front lines that do not have symbols along them (like tropical fronts and convergence lines).
'END' - Real - 0.015
Space, expressed in world coordinates, to leave along a front line after the last symbol has been drawn.
'FRO' - Character - 'WARM'
Front type (one of `WARM', `COLD', `OCCLUDED', `STATIONARY', `SQUALL', `TROPICAL', or `CONVERGENCE').
'HIB' - Integer - 0
Background color index for the "H" drawn for the high pressure symbols.
'HIC' - Integer - 1
Color index of the circumscribed circle for the "H" drawn for the high pressure symbols.
'HIF' - Integer - 1
Color index for the "H" drawn for the high pressure symbols.
'HIS' - Integer - 1
Color index for the shadow of the high pressure symbols
'LC1' - Integer - 2
Color index for the interior of the lightening bolt symbol
'LC2' - Integer - 1
Color index for the outline of the lightening bolt symbol.
'LC3' - Integer - 1
Color index for the shadow of the lightening bolt symbol
'LIN' - Real - 8.
Line widths, expressed as a fraction of the maximum screen width, for fronts having symbols along them.
'LOB' - Integer - 1
Color index for the background of the "L" drawn for the low pressure symbols.
'LOF' - Integer - 0
Color index for the "L" drawn for the low pressure symbols.
'LOS' - Integer - 0
Color index for the shadow of the low pressure symbols.
'LWD' - Real - 0.00275
Line width used when parameter WTY=1 (see below)
'MXS' - Integer - 300
Maximum number of symbols allowed along a weather front line.
'NBZ' - Integer - 51
The number of points to use in the Bezier curves for the symbols along the warm fronts.
'NMS' - Integer - internally calculated
Specifies precisely the number of symbols to appear along a weather front line (if this parameter has not been set by the user, then it is calculated internally).
'PAI' - Integer - 1
Current parameter array index used in specifying internal parameters that are arrays.
'RBS' - Integer - -1
The color index to use for the background of the regional temperature labels (plotted only if RBS is non-negative).
'RC1' - Integer - 1
Color index for the outline of the boxes drawn for the regional weather labels.
'RC2' - Integer - 0
Color index for the backgrounds of the boxes used for the regional weather labels.
'RC3' - Integer - 1
Color index for the shadow color of the boxes used for the regional weather labels.
'RC4' - Integer - 1
Color index for the text string used for regional weather labels.
'RC5' - Integer - 1
Color for the outlines of the characters in the text strings used for regional weather labels (plotted only if RC5 is non-negative).
'REV' - Integer - N/A
Reverses the current setting for the direction symbols will be drawn along front lines.
'RFC' - Integer - 1
The color index to be used for the foreground of regional temperature labels and cities.
'RHT' - Real - 0.008
Height, expressed as a fraction of the maximum screen width, of the characters used for the regional weather patterns (like rain, snow, etc.).
'RLS' - Integer - 1
The color index to use for shadows of regional temperature labels.
'RMG' - Real - 0.001
Size, expressed as a fraction of the maximum screen height, of the margins of the regional temperature labels.
'ROS' - Integer - -1
The color index to use for the outlines of the regional temperature labels (plotted only if ROS is non-negative).
'SC1' - Integer - 2
Color index to be used for the center of the sun symbol.
'SC2' - Integer - 3
Color index for the points of the sun symbol.
'SC3' - Integer - 1
Color index for the outline of the sun symbol.
'SC4' - Integer - 1
Color index for the shadow of the sun symbol.
'SHT' - Real - 0.02
Height of symbols, expressed as a fraction of the maximum screen width, for all the special symbols.
'SL1' - Real - 0.0
The slope of the beginning of a front line measured in degrees from the X-axis. This parameter is used in conjunction with the parameter SLF.
'SL2' - Real - 0.0
The slope of the end of a front line measured in degrees from the X-axis. This parameter is used in conjunction with the parameter SLF.
'SLF' - Integer - 3
Flag for indicating how the slopes at the end of a front line should be handled (0=use SL1 & SL2; 1=use SL1 only; 2=use SL2 only; 3=use neither SL1 or SL2). When either SL1 or SL2 is not used, it
is calculated internally.
'STY' - Integer Array - all 2s
An array for precisely specifying whether a warm front or cold front symbol is to be drawn at the specified position along a front line (1=cold; 2=warm). Use the internal parameter PAI for
defining this array.
'SWI' - Real - 0.0325
Width of a symbol along a weather front, expressed as a fraction of the maximum screen width.
'T1C' - Integer - 1
One color to use for the alternating colors of the dashes in the tropical fronts.
'T2C' - Integer - 1
A second color to use for the alternating colors of the dashes in the tropical fronts.
'THT' - Real - 0.0165
Height of characters, expressed as a fraction of the maximum screen width, for the regional temperature labels.
'UNT' - Integer - 0
Flags whether imperial units (the default) or metric units are used.
'VVC' - Integer - 0
Flags whether the raw SYNOP codes are plotted for surface visibility (default is not to).
'WBA' - Real - 62.
Angle (in degrees) that the wind barb ticks make with the wind barb shafts.
'WBC' - Real - 0.3
Diameter of sky cover circle at base of wind barb, expressed as a fraction of the shaft length.
'WBD' - Real - 0.1
Spacing between tick marks along a wind barb expressed as a fraction of the wind barb shaft length.
'WBF' - Integer - 0
Flag indicating whether the base of a wind barb should be drawn to allow for the sky cover circle at its base (WBF=1 means yes; WBF=0 means no).
'WBL' - Real - 0.17
Size of the text labels in the station model display, expressed as a fraction of the shaft length.
'WBR' - Real - 0.25
Radius of the larger circle drawn for calm, as a fraction of the wind barb shaft length.
'WBS' - Real - 0.035
The size, expressed as a fraction of the maximum screen height, of a wind barb shaft.
'WBT' - Real - 0.33
Length of wind barb full tic as a fraction of its shaft length.
'WFC' - Integer - 1
Color index to use for warm front symbols.
'WHT' - Real - 0.014
Height of characters, expressed as a fraction of the maximum screen width, of the characters used for regional weather labels (plotted with WMLABW or c_wmlabw).
'WTY' - Integer - 0
Flag indicating whether linewidths are to be implemented via GKS (WTY=0), or simulated internally (WTY=1).
Copyright (C) 1987-2009
University Corporation for Atmospheric Research
The use of this Software is governed by a License Agreement.
Online: wmap_params, wmbarb, wmdflt, wmdrft, wmdrrg, wmgetc, wmgeti, wmgetr, wmlabc, wmlabs, wmlabt, wmlabw, wmlgnd, wmsetc, wmseti, wmsetr, wmstnm, ncarg_cbind.
Hardcopy: WMAP - A Package for Producing Daily Weather Maps and Plotting Station Model Data | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/3-ncl_wmap_params/","timestamp":"2024-11-07T05:25:42Z","content_type":"text/html","content_length":"18338","record_id":"<urn:uuid:07bfd44f-b05d-4c50-97a5-14cb86a1e6e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00438.warc.gz"} |