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Multiple intelligence's - The 1% Club
Do you love a good challenge? Are you looking for a way to test your logic and problem-solving skills? If so, then you'll love our new line of logic puzzles! These puzzles are based on the popular
ITV show, The 1% Club, and they're sure to put your mind to the test.
Logic puzzles are a type of brain teaser that requires you to use your logical reasoning skills to solve a problem. These puzzles often involve finding patterns, making deductions, and applying
logic. Logic puzzles can be a great way to improve your problem-solving skills, and they can also be a lot of fun!
When you are trying to solve a logic puzzle, it is important to be creative and imaginative. Don't be afraid to think outside the box and to come up with new ideas. Sometimes, the most creative
solutions are the ones that work best.
Here are a few tips for being creative and imaginative when solving logic puzzles:
• Look for patterns. Logic puzzles often have patterns that can help you to solve them. For example, if you see that a certain number always appears in a certain column, you can use that
information to eliminate possible answers.
• Make assumptions. Sometimes, you need to make assumptions in order to solve a logic puzzle. For example, if you know that there are three red cars and two blue cars in a parking lot, you can
assume that the first car you see is red.
• Be flexible. Don't be afraid to change your mind if you get stuck. Sometimes, the best way to solve a logic puzzle is to start over from scratch.
With a little practice, you will be able to think more creatively and imaginatively when solving logic puzzles in your classroom.
So what are you waiting for? Challenge your mind and see how good you are at thinking outside of the box today.
PS, the answer to the last puzzle here is 3 (kite), but why? Answers are included with the resources.
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Quantitative Risk - MATHRH3946
Direct Find
Use this search only if you have an exact code for a Program, Stream, or Course, e.g. 3403, ACCTA13502, ACCT1501 or ACCT*.
Quantitative Risk - MATHRH3946
Stream Summary
Stream Outline
The School of Mathematics and Statistics (MathsStats) offers a number of Honours streams including in Quantitative Risk (QR). The Honours year introduces students to the investigative and research
aspects of knowledge and consists of advanced lecture courses, an Honours thesis and seminar participation. We offer expert supervision across a wide range of areas in modern mathematics and
statistics. Our Honours students are supervised in their Honours project by some of Australia's finest mathematicians.
Students who enrol in the Quantitative Risk Honours Stream are expected to have completed a mathematics or statistics major with a focus on financial and risk applications in an advanced
undergraduate science or other mathematically focused program. Students who have completed other cognate disciplines and who are completing a project within the usual concerns of this discipline may
also apply for entry to this stream.
Honours in Quantitative Risk is restricted to the advanced mathematics degree, or dual degrees with advanced mathematics, and can be completed full-time or part-time. Most students commence their
enrolment in semester 1 (S1) but mid-year entry is available subject to resources. Students should check the MathStats Honours webpages for application procedures and enrolment deadlines.
Entry Requirements
So that students have sufficient background to attempt the courses in the honours year, students must discuss their selection of Level III courses with the School of Mathematics Honours Coordinator
or another academic adviser.
To enter honours in QR, students must have completed Stage 3 of the quantitative risk plan in the Advanced Mathematics degree.
In addition, students will normally be required to have:
• An average above 70 in the level III mathematics courses and
• An average above 70 in the compulsory level III courses taken as part of their QR major.
The compulsory level III courses in the QR major includes MATH3901.
With the permission of the Head of School (or nominee), a student may be allowed into Quantitative Risk Honours without having satisfied the specific requirements, and instead have shown some
evidence of the ability to undertake independent study.
Stream Structure
The thesis component of Quantitative Risk Honours requires a student to undertake these two courses:
These courses form two parts of the same thesis/project. Students (full or part-time) must complete the honours thesis in any two consecutive semesters of their honours enrolment, preferably by
enrolling in
but there may be special cases where the School of MathsStats will allow the order to be reversed.
Students will also be required to participate in the weekly honours seminar, which will be timetabled as a joint class in the thesis courses. This seminar is intended to allow students to practise
their final honours seminar presentation, listen to presentations of other honours students and engage in other honours training activities. Students should also attend any appropriate seminars in
their thesis area.
The thesis will be assessed by at least two academic staff. The supervisor or supervisors of the thesis is expected to submit a report, but will not be a marker for the thesis. Students are required
to give a short seminar on their thesis and this will account for 10% of the final mark for the thesis, the remaining 90% coming from the written thesis report.
The 30uoc coursework component of Quantitative Risk Honours will consist of five 6uoc lecture courses selected from the list below, or others, with the approval by the Head of School or nominee and
taken with the advice of the honours thesis supervisor.
Approved Courses
Note: Enrolment in courses outside the School of Mathematics and Statistics may be subject to approval from the controlling School or Department.
Final Mark
The marks for the thesis and other honours courses will be combined to give a weighted average mark forming a final honours mark which will be used to decide the grade of honours the student will be
awarded as follows:
• Honours class 1 -- final mark of 85 or over
• Honours class 2, Division 1 -- final mark from 75 to 84
• Honours class 2, Division 2 -- final mark from 65 to 74
• Honours class 3 -- final mark from 50 to 64
Each student completing honours will be given an official document from the School of MathsStats which will list the courses completed, marks or grades awarded as well as the title and supervisor(s)
of the honours thesis.
Students who successfully complete Mathematics, Statistics or QR Honours are qualified to continue further in their research careers by applying to undertake postgraduate studies by PhD or Masters.
Students with successful honours are qualified to enrol in a PhD program at UNSW.
Students achieving a high Honours Grade (Class 1 or 2.1) may apply for an Australian Postgraduate Award (APA) PhD scholarship to support such studies. Further information can be obtained from
MathsStats postgraduate studies webpages:
Graduates of a QR honours plan are also well qualified to find employment in many sectors, but typically in the banking or finance sector. Past honours graduates in other maths and stats honours
programs have found employment in areas such as banking, computing, education, finance, government, medical research and meteorology.
The Australian Mathematical Society
Australian Mathematical Sciences Institute
maintain up-to-date information on career prospects in mathematics and statistics.
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Dr. V. C. KuriakosePhase synchronization in an array of driven Josephson junctionsCreation and Annihilation of Fluxons in ac‐driven Semi-annular Josephson JunctionPhase effects on synchronization by dynamical relaying in delay-coupled systemsDynamics of coupled Josephson junctions under the influence of applied fields
https://dyuthi.cusat.ac.in:443/xmlui/handle/purl/2905 2024-11-10T21:30:38Z 2024-11-10T21:30:38Z Chitra, R N Kuriakose, V C https://dyuthi.cusat.ac.in:443/xmlui/handle/purl/2909 2012-05-21T20:30:12Z
2008-01-01T00:00:00Z Phase synchronization in an array of driven Josephson junctions Chitra, R N; Kuriakose, V C We consider an array of N Josephson junctions connected in parallel and explore the
condition for chaotic synchronization. It is found that the outer junctions can be synchronized while they remain uncorrelated to the inner ones when an external biasing is applied. The stability of
the solution is found out for the outer junctions in the synchronization manifold. Symmetry considerations lead to a situation wherein the inner junctions can synchronize for certain values of the
parameter. In the presence of a phase difference between the applied fields, all the junctions exhibit phase synchronization. It is also found that chaotic motion changes to periodic in the presence
of phase differences. 2008-01-01T00:00:00Z Chitra, R N Kuriakose, V C https://dyuthi.cusat.ac.in:443/xmlui/handle/purl/2908 2012-05-21T20:30:12Z 2008-01-01T00:00:00Z Creation and Annihilation of
Fluxons in ac‐driven Semi-annular Josephson Junction Chitra, R N; Kuriakose, V C A new geometry (semiannular) for Josephson junction has been proposed and theoretical studies have shown that the new
geometry is useful for electronic applications [1, 2]. In this work we study the voltage‐current response of the junction with a periodic modulation. The fluxon experiences an oscillating potential
in the presence of the ac‐bias which increases the depinning current value. We show that in a system with periodic boundary conditions, average progressive motion of fluxon commences after the
amplitude of the ac drive exceeds a certain threshold value. The analytic studies are justified by simulating the equation using finite‐difference method. We observe creation and annihilation of
fluxons in semiannular Josephson junction with an ac‐bias in the presence of an external magnetic field. 2008-01-01T00:00:00Z Chitra, R N Kuriakose, V C https://dyuthi.cusat.ac.in:443/xmlui/handle/
purl/2907 2012-05-21T20:30:11Z 2008-01-01T00:00:00Z Phase effects on synchronization by dynamical relaying in delay-coupled systems Chitra, R N; Kuriakose, V C Synchronization in an array of mutually
coupled systems with a finite time delay in coupling is studied using the Josephson junction as a model system. The sum of the transverse Lyapunov exponents is evaluated as a function of the
parameters by linearizing the equation about the synchronization manifold. The dependence of synchronization on damping parameter, coupling constant, and time delay is studied numerically. The change
in the dynamics of the system due to time delay and phase difference between the applied fields is studied. The case where a small frequency detuning between the applied fields is also discussed.
2008-01-01T00:00:00Z Chitra, R Nayak Kuriakose, V C https://dyuthi.cusat.ac.in:443/xmlui/handle/purl/2906 2012-05-21T20:30:11Z 2007-06-04T00:00:00Z Dynamics of coupled Josephson junctions under the
influence of applied fields Chitra, R Nayak; Kuriakose, V C We investigate the effect of the phase difference of appliedfields on the dynamics of mutually coupledJosephsonjunctions. A phase
difference between the appliedfields desynchronizes the system. It is found that though the amplitudes of the output voltage values are uncorrelated, a phase correlation is found to exist for small
values of applied phase difference. The dynamics of the system is found to change from chaotic to periodic for certain values of phase difference. We report that by keeping the value of phase
difference as π, the system continues to be in periodic motion for a wide range of values of system parameters. This result may find applications in devices like voltage standards, detectors, SQUIDS,
etc., where chaos is least desired. 2007-06-04T00:00:00Z
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Difference between Volume and Mass in context of volume to mass
26 Aug 2024
Title: The Distinction between Volume and Mass: A Fundamental Concept in Physics
In the realm of physics, understanding the difference between volume and mass is crucial for grasping various physical phenomena. While both terms are often used interchangeably, they have distinct
meanings and units. This article aims to elucidate the distinction between volume and mass, highlighting their unique characteristics and relationships.
Volume (V) and mass (m) are two fundamental physical quantities that describe the size and amount of matter, respectively. Volume is a measure of the three-dimensional space occupied by an object or
substance, whereas mass is a measure of the amount of matter present in an object or substance. The distinction between these two concepts is essential for understanding various physical phenomena,
such as density, buoyancy, and pressure.
Definition and Units:
Volume (V) is defined as the three-dimensional space occupied by an object or substance, measured in units of cubic meters (m³), liters (L), or cubic centimeters (cm³). The formula for volume is:
V = length × width × height
Mass (m) is a measure of the amount of matter present in an object or substance, measured in units of kilograms (kg), grams (g), or milligrams (mg).
Key Differences:
1. Units: Volume is measured in cubic units (e.g., m³, L, cm³), whereas mass is measured in units of mass (e.g., kg, g, mg).
2. Physical Meaning: Volume describes the size and shape of an object or substance, while mass describes the amount of matter present.
3. Density: Density (ρ) is a measure of mass per unit volume. The formula for density is:
ρ = m / V
In conclusion, understanding the difference between volume and mass is essential for grasping various physical phenomena. While both terms are often used interchangeably, they have distinct meanings
and units. By recognizing the unique characteristics of each concept, physicists can better describe and analyze the behavior of objects and substances in the natural world.
• Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers. Cengage Learning.
Related articles for ‘volume to mass ‘ :
• Reading: **Difference between Volume and Mass in context of volume to mass **
Calculators for ‘volume to mass ‘
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NetLogo User Community Models
(back to the NetLogo User Community Models)
If clicking does not initiate a download, try right clicking or control clicking and choosing "Save" or "Download".(The run link is disabled because this model uses extensions.)
WHAT IS IT?
This is a 1 dimensional cellular automata explorer
Patches get new states from the states of their 2 closest neighbors and the state of themself. These values are composed into a single value by a rule.
Rules can be:
2)changed automatically and manually
3)gotten from examples
Rules are feedforward neural networks (http://en.wikipedia.org/wiki/Feedforward_neural_network) containing 1 - 3 hidden layers with each layer having 1 - 20 neurons + output layer consisting of one
neuron and input layer consisting of 3 neurons.
Activation function:
y = x / (0.1 + abs x)
Patch states, neuron output values and connection weight values are all in range [-1,1]
Each neuron operates as follows:
1)sum of all previous layer inputs times the incoming connection weight is calculated
2)previous value (1) is sent trough the activation function
3)previous value (2) is compared to the threshold value of neuron. if (2) > threshold value of neuron and output is produced and it has the value (2), otherwise the output value will be 0
The only special case are the neurons in the feedforward neural network input layer that only produce neuron output.
HOW TO USE IT
Press go to start. Perss new-rules for new rules. Switch world by changing the world slider. Mutate the most interesting world.
IN DEPTH:
start the program
Visualizes current row of the world with a line. Doing so may give valuable information about the nature of the observed rule.
no color updates, easier to see line
refresh the color values of the whole screen
Makes completely new rules
random patch values
random patch value for one patch
if its on then random-one will be used when switching/mutating worlds, otherwise random-values determine the states of the world
mutate the rules based on current rule. if the rule stays the same it is mutated up to 1000 times, if it is still the same a random rule will be generated
mutate previously mutated world
world slider:
switch active world and rule
Example rules. Push initiate-example to view some rules.
Restarts the model. There is no need to press this unless you encounter some crazy errors i think
Clears all turtles. This is useful when the neural network visualization failed for some reason.
get the current rule or set the current rule on the rule input field.
the current value of the patch where the mouse is located
continue or not when hit the bottom of the screen
Visualizes currently used neural network. The network is colored according to threshold and weight values. Values above 0 are green and below are blue with increasing whiteness as the absolute value
increases. Neurons are also shifted on y-axis according to their threshold values.
NetLogo colors explained:
NetLogo math explained:
vision-level and color-offset:
patches get their color from:
state * (70 + color-offset) * 10 ^ vision-level
Using a different vision level may reveal the true nature of some patterns. Also it may reveal patterns that emerge only on some vision-level and stay completely hidden on others. Similar patterns
may not be similar at all or vice versa depending on the vision level.
exponentially distributed absolute change value for a single neuron threshold or connection weight when mutating
exponentially distributed percentage value for a single layer in the neural network that determines the amount of units to change. In this case "single layer" refers to either weight layer or
threshold layer.
table:length data is a valuable input for an quick overview of a rule. Different values enable making some predictions on the nature of a rule
Table ditch point defines algorithm choice. If table:length data > ditch-table then tables will not be filled anymore. The remaining data will still be used if possible though. This results in a
minor slowdown, however the overall benefits gained seem to outweigh ditching table usage completely.
In this case a table with 1 million entries is equal to about 122 MB of memory space.
frames per second
all values on the rule field (neuron thresholds and connection weights) are sent trough the "formula". formula must be a valid (in netlogo language) math formula. Please see http://
ccl.northwestern.edu/netlogo/docs/dictionary.html#mathematicalgroup for some help.
In formula:
z is current value from the feedforward neural network
a,b and c are constants
All operations are done on the neural network that is in the rule field. After each operation all the rules will be replaced with the resulting rule and mutations of it.
in case omit-zero is "on", z is 0 and the formula is z + 0.1, value 0 will become 0. Values that are not zero will become z + 0.1
Patch states, neuron output values and connection weight values are all in range [-1,1]. Automatic operations will restrict the resulting values from operation-on-rule into that range when
restrict-to-range? is "on". If restrict-to-range? is "off" then the values will be looped in that range instead.
restrict-to-range input min max
observer> show restrict-to-range 1.01 -1 1
observer: 1
observer> show restrict-to-range 0.56347347 -1 1
observer: 0.56347347
observer> show restrict-to-range -50000000000 -1 1
observer: -1
loop-in-range input min max
observer> show loop-in-range 0.5 -1 1
observer: 0.5
observer> show loop-in-range 1.1 -1 1
observer: -0.8999999999999999
observer> show loop-in-range 1.2 -1 1
observer: -0.8
observer> show loop-in-range 2 -1 1
observer: 0
observer> show loop-in-range -400.335 -1 1
observer: -0.33499999999997954
If it is "off" all values will be restricted/looped in range [-1,1]. If it is on then negative values will be restricted/looped in range [-1,0] and positive in range [0,1].
Examples with operation-on-rule (after loading the model with default settings for formula and related fields):
1)initiate-example 25
2)start the model with go (in order to see the transition)
3)press operation-on-rule
1)initiate-example 3
2)start the model with go
3)press operation-on-rule 5 times
4)press operation-on-rule 6 more times
5)press operation-on-rule 3 more times
Notice how very simple neural networks can produce very complicated behaviour. Visualize-networks for example rules 12 and 13.
Play with color-offset and vision-level while the model is running to understand how exactly they work.
Each patch on the current row checks if there is a key in the table that corresponds to the states of their 2 closest neighbors and the state of themself in the form of a 3 element list (example
[0.1513513613681 -0.30628268623 0]) to avoid recalculating the values with the neural network. Tables grow very large, yet this operation does not become any slower.
2d totalistic CA explorer
Netlogo models library:
Cellular automata group
Artificial Neural Net
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Discrete k-nearest neighbor resampling for simulating multisite precipitation occurrence and model adaption to climate change
Articles | Volume 12, issue 3
© Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License.
Discrete k-nearest neighbor resampling for simulating multisite precipitation occurrence and model adaption to climate change
Stochastic weather simulation models are commonly employed in water resources management, agricultural applications, forest management, transportation management, and recreational activities.
Stochastic simulation of multisite precipitation occurrence is a challenge because of its intermittent characteristics as well as spatial and temporal cross-correlation. This study proposes a novel
simulation method for multisite precipitation occurrence employing a nonparametric technique, the discrete version of the k-nearest neighbor resampling (KNNR), and couples it with a genetic algorithm
(GA). Its modification for the study of climatic change adaptation is also tested. The datasets simulated from both the discrete KNNR (DKNNR) model and an existing traditional model were evaluated
using a number of statistics, such as occurrence and transition probabilities, as well as temporal and spatial cross-correlations. Results showed that the proposed DKNNR model with GA-simulated
multisite precipitation occurrence preserved the lagged cross-correlation between sites, while the existing conventional model was not able to reproduce lagged cross-correlation between stations, so
long stochastic simulation was required. Also, the GA mixing process provided a number of new patterns that were different from observations, which was not feasible with the sole DKNNR model. When
climate change was considered, the model performed satisfactorily, but further improvement is required to more accurately simulate specific variations of the occurrence probability.
Received: 16 Jul 2018 – Discussion started: 12 Oct 2018 – Revised: 25 Feb 2019 – Accepted: 18 Mar 2019 – Published: 28 Mar 2019
Stochastic simulation of weather variables has been employed for water resources management, hydrological design, agricultural irrigation, forest management, transportation planning and evacuation,
recreation activities, filling in missing historical data, simulating data, extending observed records, and simulating different weather conditions. Stochastic simulation models play a key role in
producing weather sequences, while preserving the statistical characteristics of observed data. A number of stochastic weather simulation models have been developed using parametric and nonparametric
approaches (Lee, 2017; Lee et al., 2012; Wilby et al., 2003; Wilks, 1999; Wilks and Wilby, 1999).
Parametric approaches simulate statistical characteristics of observed weather data with a set of parameters that are determined by fitting (Jeong et al., 2012; Lee, 2016; Zheng and Katz, 2008),
whereas in nonparametric approaches, historical analogs with current conditions are searched, following the weather simulation data (Buishand and Brandsma, 2001; Lee et al., 2012). Combinations of
parametric and nonparametric approaches have also been proposed (Apipattanavis et al., 2007; Frost et al., 2011).
Among weather variables, precipitation possesses intermittency and zero values between precipitation events, which make it difficult to properly reproduce the events (Beersma and Buishand, 2003;
Hughes et al., 1999; Katz and Zheng, 1999). To overcome the problem of intermittency and zero values, precipitation is simulated separately from other variables. The main method for reproducing
intermittency has been the multiplication of precipitation occurrence and an amount as $Z=X\cdot Y$, where X is the occurrence (binary as either 0 or 1) and Y is the amount (Jeong et al., 2013; Lee
and Park, 2017; Todorovic and Woolhiser, 1975). The spatial and temporal dependence in the occurrence and amount of precipitation introduce further complexity in multisite simulation.
Wilks (1998) presented a multisite simulation model for the occurrence process (i.e., X) using the standard normal variable that is spatially dependent, representing the relationship between the
occurrence variable and the standard normal variable with simulation data. Originally, the occurrence of precipitation had been simulated with a discrete Markov chain (MC) model (Katz, 1977).
Compared to the MC model that requires a significant number of parameters for generating multisite occurrence, the multisite occurrence model proposed by Wilks (1998) transforms the standard normal
variate and simulates the sequence with multivariate normal distribution, and then back-transforms the multivariate normal sequence to the original domain. The model is able to reproduce the
contemporaneous multisite dependence structure and lagged dependence only for the same site but it requires a complex simulation process to estimate parameters for each site and is unable to preserve
lagged dependence between sites. Also, a recent improvement has also been made, but the weakness of the model in Wilks (1998) was not significantly improved (Evin et al., 2018; Mehrotra et al., 2006;
Srikanthan and Pegram, 2009).
Lee et al. (2010) proposed a nonparametric-based stochastic simulation model for hydrometeorological variables. Their model overcame the shortcomings of a previous nonparametric simulation model
(Lall and Sharma, 1996), called k-nearest neighbor resampling (KNNR), but the simulated data do not produce patterns different from those of the observed data (Brandsma and Buishand, 1998; Mehrotra
et al., 2006; St-Hilaire et al., 2012). In addition to KNNR, Lee et al. (2010) used a metaheuristic genetic algorithm (GA) that led to the reproduction of similar populations by mixing the simulated
datasets. Note that the reproduction procedure of the GA allows to generate new patterns that are similar to observed patterns, but a small number of totally new patterns are simulated from the
mutation procedure of the GA.
While KNNR is employed to find historical analogues of multisite occurrence similar to the current status of a simulation series, GA is applied to use its skill to generate a new descendant from the
historical parent chosen with the KNNR. In this procedure, the multisite occurrence of precipitation can be simulated while preserving spatial and temporal correlations. Metaheuristic techniques,
such as GA, have been popularly employed in a number of hydrometeorological applications (Chau, 2017; Fotovatikhah et al., 2018; Taormina et al., 2015; Wang et al., 2013). Although a number of
variants of KNNR-GA have been applied (Lee et al., 2012; Lee and Park, 2017), none of them can simulate multisite occurrence of precipitation whose characteristics are binary and temporally and
spatially related.
Therefore, this study proposes a stochastic simulation method for multisite occurrence of precipitation with the KNNR-GA-based nonparametric approach that (1) simulates multisite occurrence with a
simple and direct procedure without parameterization of all the required occurrence probabilities, and (2) reproduces the complex temporal and spatial correlation between stations, as well as the
basic occurrence probabilities. The proposed nonparametric model is compared with the popular model proposed by Wilks (1998). Even though the multisite occurrence data generated from the Wilks model
preserves various statistical characteristics of the observed data well, significant underestimation of lagged cross-correlation still exists. Furthermore, the relation between standard normal
variable and occurrence variable relies on long stochastic simulation.
The paper is organized as follows. The next section presents the mathematical background of existing multisite occurrence modeling and section discusses the modeling procedure. The study area and
data are reported in Sect. 4. The model application is presented in Sect. 5. Results of the proposed model are discussed in Sect. 6, and summary and conclusions are presented in Sect. 7.
2.1Single site occurrence modeling
Let ${X}_{t}^{s}$ represent the occurrence of daily precipitation for a location s ($s=\mathrm{1},\mathrm{\dots },S$) on day t ($t=\mathrm{1},\mathrm{\dots },n$; n is the number observed days) and
let ${X}_{t}^{s}$ be either 0 for dry days or 1 for wet days. The first-order Markov chain model for ${X}_{t}^{s}$ is defined with the assumption that the occurrence probability of a wet day is fully
defined by the previous day as
$\begin{array}{}\text{(1)}& & \mathrm{Pr}\left\{{X}_{t}^{s}=\mathrm{1}\mathrm{|}{X}_{t-\mathrm{1}}^{s}=\mathrm{0}\right\}={p}_{\mathrm{01}}^{s}\text{(2)}& & \mathrm{Pr}\left\{{X}_{t}^{s}=\mathrm{1}\
Also, ${p}_{\mathrm{00}}^{s}=\mathrm{1}-{p}_{\mathrm{01}}^{s}$ and ${p}_{\mathrm{10}}^{s}=\mathrm{1}-{p}_{\mathrm{11}}^{s}$, since the summation of 0 and 1 should be unity with the same previous
condition. This consists of a transition probability matrix (TPM) as
$\begin{array}{}\text{(3)}& {\mathrm{TPM}}^{s}=\left[\begin{array}{cc}{p}_{\mathrm{00}}^{s}& {p}_{\mathrm{01}}^{s}\\ {p}_{\mathrm{10}}^{s}& {p}_{\mathrm{11}}^{s}\end{array}\right]=\left[\begin{array}
{cc}\mathrm{1}-{p}_{\mathrm{01}}^{s}& {p}_{\mathrm{01}}^{s}\\ \mathrm{1}-{p}_{\mathrm{11}}^{s}& {p}_{\mathrm{11}}^{s}\end{array}\right].\end{array}$
The marginal distributions of TPM (i.e., p[0] and p[1]) can be expressed with TPM and its condition of ${p}_{\mathrm{0}}+{p}_{\mathrm{1}}=\mathrm{1}$ as
$\begin{array}{}\text{(4)}& & {p}_{\mathrm{0}}^{s}=\frac{{p}_{\mathrm{01}}^{s}}{\mathrm{1}+{p}_{\mathrm{01}}^{s}-{p}_{\mathrm{11}}^{s}}\text{(5)}& & {p}_{\mathrm{1}}^{s}=\frac{\mathrm{1}-{p}_{\mathrm
Note that p[1] represents the probability of precipitation occurrence for a day, while p[0] represents non-occurrence. The lag-1 autocorrelation of precipitation occurrence is the combination of
transition probabilities as
$\begin{array}{}\text{(6)}& {\mathit{\rho }}_{\mathrm{1}}\left(s,s\right)={p}_{\mathrm{11}}^{s}-{p}_{\mathrm{01}}^{s}.\end{array}$
The simulation can be done by comparing TPM with a uniform random number (${u}_{t}^{s}$) as
where${p}_{i\mathrm{1}}^{s}$ is the selected probability from TPM regarding the previous condition i (i.e., either 0 or 1). Wilks (1998) suggested a different method using a standard normal random
number ${w}_{t}^{s}\sim N\left[\mathrm{0},\mathrm{1}\right]$ as
where Φ^−1indicates the inverse of the standard normal cumulative function Φ.
2.2Multisite occurrence modeling
Wilks (1998) suggested a multisite occurrence model using a standard normal random number (here denoted as MONR) that is spatially dependent but serially independent. The correlation of the standard
normal variate for a site pair of q and s can be expressed as
$\begin{array}{}\text{(9)}& \mathit{\tau }\left(q,s\right)=\mathrm{corr}\left[{w}_{t}^{q},{w}_{t}^{s}\right].\end{array}$
Also, the correlation of the original occurrence variate is
$\begin{array}{}\text{(10)}& \mathit{\rho }\left(q,s\right)=\mathrm{corr}\left[{X}_{t}^{q},{X}_{t}^{s}\right].\end{array}$
Once the correlation of the standard normal variate is known, the simulation of multisite precipitation occurrence is straightforward. Multivariate standard normal distribution is used with a
parameter set of [0, T], where 0 is the zero vector (S×1) and T is the correlation matrix with the elements of τ(q,s) for $q\in \mathit{\left\{}\mathrm{1},\mathrm{\dots },S\mathit{\right\}}$ and $s\
in \mathit{\left\{}\mathrm{1},\mathrm{\dots },S\mathit{\right\}}$.
Since direct estimation of τ(q,s) is not feasible, a simulation technique is used to estimate τ(q,s) from ρ(q,s). A long sequence of the occurrences is simulated with different values of τ(q,s) and
its corresponding correlation of the original domain ρ(q,s) is estimated with the simulated long sequence by the inverse standard normal cumulative function (i.e., Φ^−1). A curve between τ(q,s) and ρ
(q,s) is derived from this long simulation with the MONR model and is employed for parameter estimation for a real application.
3.1DKNNR modeling procedure
In the current study, a novel multisite simulation model for discrete occurrence of precipitation variable with the KNNR technique (Lall and Sharma, 1996; Lee and Ouarda, 2011; Lee et al., 2017) for
a discrete case (denoted as discrete KNNR; DKNNR) is proposed by combining a mixture mechanism with GA. Provided the number of nearest neighbors, k, is known, the discrete k-nearest neighbor
resampling with genetic algorithm is done as follows:
1. Estimate the distance between the current (i.e., time index: c) multisite occurrence ${X}_{c}^{s}$ and the observed multisite occurrence ${x}_{i}^{s}$. Here, the distance is measured for $i=\
mathrm{1},\mathrm{\dots },n-\mathrm{1}$ as
$\begin{array}{}\text{(11)}& {D}_{i}=\sum _{s=\mathrm{1}}^{S}\left|{X}_{c}^{s}-{x}_{i}^{s}\right|.\end{array}$
2. Arrange the estimated distances from step (1) in ascending order, select the first k distances (i.e., the smallest k values), and reserve the time indices of the smallest k distances.
3. Randomly select one of the stored k time indices with the weighting probability given by
$\begin{array}{}\text{(12)}& {w}_{m}=\frac{\mathrm{1}/m}{\sum _{j=\mathrm{1}}^{k}\mathrm{1}/j},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}m=\mathrm{1},\
mathrm{\dots },k.\end{array}$
4. Assume the selected time index from step (3) as p. Note that there are a number of values that have the same distance as the selected D[p], since D[p] is a natural number between 0 and S. For
example, if S=2 and ${X}_{c}^{\mathrm{1}}=\mathrm{0}$ and ${X}_{c}^{\mathrm{2}}=\mathrm{1}$, the two sequences have the same D=1 as [${x}_{i}^{\mathrm{1}}=\mathrm{0}$ and ${x}_{i}^{\mathrm{2}}=\
mathrm{0}$] and [${x}_{i}^{\mathrm{1}}=\mathrm{1}$ and ${x}_{i}^{\mathrm{2}}=\mathrm{1}$]. In this case, a random selection procedure is required to take into account the cases with the same
quantity. One particular time index is randomly selected with equal probabilities among the time indices of the same distances. Note that instead of the random selection, one can always use the
first one. In such a case, only one historical combination of multisite occurrences will be selected.
5. Assign the binary vector of the proceeding index of the selected time as ${\mathbit{x}}_{p+\mathrm{1}}=\left[{x}_{p+\mathrm{1}}^{s}{\right]}_{s\in \mathit{\left\{}\mathrm{1},S\mathit{\right\}}}$.
Here, p is the finally selected time index from step (4).
6. Execute the following steps for GA mixing if GA mixing is subjectively selected. Otherwise, skip this step.
□ a.
Reproduction: select one additional time index using steps (1) through (4) and denote this index as p^∗. Obtain the corresponding precipitation occurrence values, ${\mathbit{x}}_{{p}^{\ast }+
\mathrm{1}}=\left[{x}_{{p}^{\ast }+\mathrm{1}}^{s}{\right]}_{s\in \mathit{\left\{}\mathrm{1},\mathrm{\dots },S\mathit{\right\}}}$. The subsequent two GA operators employ the two selected
vectors, x[p] and ${\mathbit{x}}_{{p}^{\ast }}$. This reproduction process is a mating process by finding another individual that has characteristics similar to those of the current one (x[p
+1]). With this procedure, a vector similar to the current vector will be mated and will produce a new descendant.
□ b.
Crossover: replace each element ${x}_{p+\mathrm{1}}^{s}$ with ${x}_{{p}^{\ast }+\mathrm{1}}^{s}$ at probability P[cr], i.e.,
where ε is a uniform random number between 0 and 1. From this crossover, a new occurrence vector whose elements are similar to the historical ones is generated.
□ c.
Mutation: replace each element (i.e., each station, $s=\mathrm{1},\mathrm{\dots },S$) with one selected from all the observations of this element for $i=\mathrm{1},\mathrm{\dots },n$ with
probability P[m], i.e.,
where ${x}_{\mathit{\xi }+\mathrm{1}}^{s}$ is selected from $\left[{x}_{i}^{s}{\right]}_{i\in \mathit{\left\{}\mathrm{1},\mathrm{\dots },n\mathit{\right\}}}$ with equal probability for $i=\
mathrm{1},\mathrm{\dots },n$, and ε is a uniform random number between 0 and 1. This mutation procedure allows to generate a multisite occurrence combination that is totally different from
the historical records. Without this procedure, multisite occurrences always similar to historical combinations are generated, which is not feasible for a simulation purpose.
7. Repeat steps (1)–(6) until the required data are generated.
The selection of the number of nearest neighbors (k) has been investigated by Lall and Sharma (1996) and Lee and Ouarda (2011). A simple selection method was applied in the current study as $k=\sqrt
{n}$. For hydrometeorological stochastic simulation, this heuristic approach of the k selection has been employed (Lall and Sharma, 1996; Lee and Ouarda, 2012; Lee et al., 2010; Prairie et al., 2006;
Rajagopalan and Lall, 1999). One can use generalized cross-validation (GCV) as shown in Lall and Sharma (1996) and Lee and Ouarda (2011) by treating this simulation as a prediction problem. However,
the current multisite occurrence simulation does not necessarily require an accurate value prediction and not much difference in simulation using the simple heuristic approach has been reported.
Also, this heuristic approach of the k selection has been popularly employed for hydrometeorological stochastic simulations (Lall and Sharma, 1996; Lee and Ouarda, 2012; Lee et al., 2010; Prairie et
al., 2006; Rajagopalan and Lall, 1999).
In Appendix A, an example of the DKNNR simulation procedure is explained in detail.
3.2Adaptation to climate change
The capability of model to take climate change into account is critical. For example, the marginal distributions and transition probabilities in Eqs. (5) and (3) can change in future climate
scenarios. It is known that nonparametric simulation models have difficulty adapting to climate change, since the models employ in general the current observation sequences. However, the proposed
model in the current study possesses the capability to adapt to the variations of probabilities by tuning the crossover and mutation probabilities in P[cr] (Eq. 13) and P[m] (Eq. 14), adding the
condition when applied.
For example, the probability of P[11] can be increased with the crossover probability P[cr] by adding the condition to increase the probability of P[11] as
It is obviously possible to increase the probability of P[1] by removing the condition of ${X}_{c}^{s}=\mathrm{1}$.
In addition, further adjustment can be made with the mutation process in Eq. (14) as
This adjustment of adding the condition ${x}_{\mathit{\xi }+\mathrm{1}}^{s}=\mathrm{1}$ can increase the marginal distribution as much as P[m]×P[1]. This has been tested in a case study.
4Study area and data description
For testing the occurrence model, 12 weather stations were selected from the Yeongnam province, which is located in the southeastern part of South Korea, as shown in Fig. 1. Information on longitude
and latitude (fourth and fifth columns), as well as order index and the identification number (first and second columns) of these stations operated by Korea Meteorological Administration with the
area name (third column), is shown in Table 1. The employed precipitation dataset presents strong seasonality, since this area is dry from late fall to early autumn and humid and rainy during the
remaining seasons, especially in summer. The employed stations are not far from each other, at most 100km apart, and not many high mountains are located in the current study area. Therefore, this
region can be considered as a homogeneous region (Lee et al., 2007).
Figure 1 illustrates the locations of the selected weather stations. All the stations are inside Yeongnam province, which consists of two different regions (north and south Gyeongsang), as well as
the self-governing cities of Busan, Daegu, and Ulsan. Most of the Yeongnam region is drained to the Nakdong River. To validate the proposed model appropriately, test sites must be highly correlated
with each other as well as have significant temporal relation. The stations inside the Yeongnam area cover one of the most important watersheds, the Nakdong River basin, where the Nakdong River
passes through the entire basin, and its hydrological assessments for agriculture and climate change have a particular value in flood control and water resources management such as floods and
^∗ The station number indicates the identification number operated by Korea Meteorological Administration (KMA).
It is important to analyze the impact of weather conditions for planning agricultural operations and water resources management, especially during the summer season, because around 50%–60% of the
annual precipitation occurs during the summer season from June to September. The length of daily precipitation data record ranges from 1976 to 2015 and the summer season record was employed, since a
large number of rainy days occur during summer and it is important to preserve these characteristics. Also, the whole-year dataset was tested and other seasons were further applied but the
correlation coefficient was relatively high and its estimated correlation matrix was not a positive semi-definite matrix for the MONR model.
To analyze the performance of the proposed DKNNR model, the occurrence of precipitation was simulated. The DKNNR simulation was compared with that of the MONR model. For each model, 100 series of
daily occurrence with the same record length were simulated. The key statistics of observed data and each generated series, such as transition probabilities (P[11], P[01], and P[1]) and
cross-correlation (see Eq. 10), were determined. The MONR model underestimated the lag-1 cross-correlation, as indicated by Wilks (1998). In the current study, this statistic was analyzed, since a
synoptic-scale weather system often results in lagged cross-correlation for daily precipitation data (Wilks, 1998). It was formulated as
$\begin{array}{}\text{(17)}& {\mathit{\rho }}_{\mathrm{1}}\left(q,s\right)=\mathrm{corr}\left[{X}_{t-\mathrm{1}}^{q},{X}_{t}^{s}\right].\end{array}$
Statistics from 100 generated series were evaluated by the root mean square error (RMSE), expressed as
$\begin{array}{}\text{(18)}& \mathrm{RMSE}={\left(\frac{\mathrm{1}}{N}\sum _{m=\mathrm{1}}^{N}{\left({\mathrm{\Gamma }}_{m}^{G}-{\mathrm{\Gamma }}^{h}\right)}^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm
where N is the number of series (here 100), ${\mathrm{\Gamma }}_{m}^{G}$ is the statistic estimated from the mth generated series, while Γ^h is the statistic for the observed data. Note that lower
RMSE indicates better performance, represented by the summarized error of a given statistic of generated series from the statistic of the observed data.
The 100 simulated statistic values were illustrated with box plots to show their variability as shown in Figs. 5–7. The box of the box plot represents the interquartile range (IQR), ranging from the
25th percentile to the 75th percentile. The whiskers extend up and down to 1.5 times the IQR. The data beyond the whiskers (1.5×IQR) are indicated by a plus sign (+). The horizontal line inside the
box represents the median of the data. The statistics of the observed data are denoted by a cross (×). The closer a cross is to the horizontal line inside the box, the better the simulated data from
a model reproduce the statistical characteristics of the observed data.
6.1GA mixing and its probability selection
The roles of crossover probability P[cr] (Eq. 13) and mutation probability P[m] (Eq. 14) were studied by Lee et al. (2010). In the current study, we further tested these by selecting an appropriate
parameter set of these two parameters with the simulated data from the DKNNR model and the record length of 100000. RMSE (Eq. 18) of the three transition and limiting probabilities (P[11], P[01],
and P[1]) between the simulated data and the observed was used, since those probabilities are key statistics that the simulated data must match the observed data, and no parameterization of these
probabilities was made for the current DKNNR model. Results are shown in Figs. 2 and 3 for P[cr] and P[m], respectively. For P[cr] in Fig. 2, the probability of 0.02 shows the smallest RMSE in all
transition and limiting probabilities. The RMSE of P[m] in Fig. 3 shows a slight fluctuation along with P[m]. However, all three probabilities (P[11], P[01], and P[1]) have relatively small RMSEs in
P[m]=0.003. Therefore, the parameter set 0.02 and 0.003 was chosen for P[cr] and P[m], respectively, and employed in the current study. We also tested the simulation without the GA mixing procedure
(results not shown). The results showed that no better result could be found from the simulation without GA mixing. The necessity of the GA mixing is further discussed in the following.
We further tested and discuss why the GA mixing is necessary in the proposed DKNNR model as follows. For example, assume that three weather stations are considered and observed data only have the
occurrence cases of 000, 001, 011, 010, 011, 100, and 111, among 2^3=8 possible cases. In other words, no patterns for 110 and 101 are found in the observed data. Note that 0 indicates dry days and 1
indicates rainy (or wet) days. The KNNR is a resampling process in which the simulation data are resampled from observations. Therefore, no new patterns such as 110 and 101 can be found in the
simulated data.
This can be problematic for the simulation purpose in that one of the major simulation purposes is to simulate sequences that might possibly happen in the future. The wet (1) or dry (0) for multisite
precipitation occurrence is decided by the spatial distribution of a precipitation weather system. A humid air mass can be distributed randomly, relying on wind velocity and direction, as well as the
surrounding air pressure. In general, any combinations of wet and dry stations can be possible, especially when the simulation continues infinitely. Therefore, the patterns of simulated data must be
allowed to have any possible combinations (here 4096), even if they have not been observed from the historical records. Also, the probability to have this new pattern must not be high, since it has
not been observed in the historical records, and this can be taken into account by low probability of the crossover and mutation.
This drawback of the KNNR model frequently happens in multisite occurrence as the number of stations increases. Note that the number of patterns increases as 2^n, where n is the number of stations.
If n=12, then 4096 cases must be observed. However, among 4096 cases, observed cases are limited, since the amount of data is limited. The GA process can mix two candidate patterns to produce new
patterns. For example, in the three-station case, a new pattern of 101 can be produced from two observed occurrence candidates of 001 and 100 by the crossover of the first value of 001 with the first
value of 100 (i.e., 001→101), which is not in the observed data.
Note that the data employed in the case study are 40 years and 122 days (summer months) in each year. The total number of the observed data is 4880 and the number of possible cases is 4096. We
checked the number of possible cases that were not found in the observed data. The result shows that 3379 cases were not observed at all for the entire cases as shown in Fig. 4.
We further investigated the number of new patterns that were generated with the probabilities P[cr]=0.02, P[m]=0.001 by the proposed GA mixing. The generated data for 100 sequences from DKNNR with
the GA mixing show that the number 3379 was reduced to 1200, which is not in the dataset among the 4096 possible patterns. Therefore, more than 2000 new patterns were simulated with the GA mixing
process. The KNNR model without the GA mixing did not produce any new patterns in the 100 sequences with the same length of the historical data.
6.2Occurrence and transition probabilities
The data simulated from the proposed DKNNR model and the existing MONR model were analyzed. The estimated transition probabilities (P[11] and P[01] in Eq. 3) as well as the occurrence probability (P
[1] in Eq. 5) are shown in Table 2 and Figs. 5–7 for the observed data and the data generated from the DKNNR and MONR models. In Table 2, the observed statistic shows that P[11] is always higher than
P[01], and P[1] is between P[11] and P[01]. Site 6 shows the lowest P[11] and P[1], and site 12 shows the highest P[11].
As shown in Fig. 5, the probability P[11] of the observed data shows that sites 6, 7, 8, and 9 located in the northern part of the region exhibited lower consistency (i.e., consecutive rainy days)
than did the other sites, while sites 5 and 12 had higher probability of P[11] than did other sites. Both models preserved well the observed P[11] statistic. It seems that the MONR model had a
slightly better performance, since this statistic is parameterized in the model, as shown in Sect. 2.2, and that is the same for P[01] and P[1], as shown in Figs. 6 and 7. Note that the MONR model
employed the transition probabilities in simulating rainfall occurrence, while the DKNNR model did not. The occurrence probability P[1] can be described with the combination of transition
probabilities as in Eq. (5). Even though the transition probabilities were not employed in simulating rainfall occurrence, the DKNNR model preserved this statistic fairly well.
In the DKNNR modeling procedure, the simple distance measurement in Eq. (11) allows to preserve transition probabilities in that the following multisite occurrence is resampled from the historical
data whose previous states of multisite occurrence (${x}_{i}^{s}$) are similar to the current simulation multisite occurrence (${X}_{c}^{s}$). This summarized distance (D[i]) is an essential tool in
the proposed DKNNR modeling. The condition of the current weather system is memorized and the system is conditioned on simulating the following multisite occurrence with the distance measurement like
a precipitation weather system dynamically changes, but often it impacts the system of the following day.
As shown in Fig. 6, the P[01] probability showed a slightly different behavior such that sites 1, 2, and 3 located in the middle part of the Yeongnam province showed a higher probability than did
other sites. A slight underestimation was seen for sites 2 and 11 but it was not critical, since its observed value with a cross mark was close to the upper IQR representing the 75th percentile.
The behavior of P[1] was found to be the same as that of the P[11] probability. It can be seen in Fig. 7a that no significant underestimation is seen for the DKNNR model. The P[1] statistic was
fairly preserved by both DKNNR and MONR models. Note that the MONR model parameterized the P[1] statistic through the transition probabilities as in Eq. (5), while the DKNNR model did not. Although
the DKNNR model used almost no parameters for simulation, the P[1] statistic was preserved fairly well.
Cross-correlation is a measure of the relationship between sites. The preservation of cross-correlation is important for the simulation of precipitation occurrence and is required in the regional
analysis for water resources management or agricultural applications. Furthermore, lagged cross-correlation is also as essential as cross-correlation (i.e., contemporaneous correlation). For example,
the amount of streamflow for a watershed from a certain precipitation event is highly related to lagged cross-correlation.
Daily precipitation occurrence, in general, shows the strongest serial correlation at lag-1 and its correlation decays as the lag gets longer. This is because a precipitation weather system moves
according to the surrounding pressure and wind direction that dynamically change within a day or week. Therefore, we analyzed the lag-1 cross-correlation in the current study as the representative
lagged correlation structure.
Note that no negative value can be found, implying that the DKNNR model preserves the cross-correlation better than the MONR model.
^∗ Values in bold font represent lag-1 autocorrelation (i.e., the one lagged correlation for the same site).
The cross-correlation of observed data is shown in Table 3. High cross-correlation among grouped sites, such as sites 6, 7, and 8 (northern part) and sites 3, 4, and 5, as well as 12 (southeast
coastal area, 0.68–0.87), was found. As expected, sites 5 and 12 had the highest cross-correlation (0.87) due to proximity. The northern sites and coastal sites showed low cross-correlation. This
observed cross-correlation was well preserved in the data generated from both DKNNR and MONR models, as shown in Fig. 8 as well as Tables 4 and 5. However, consistently slight but significant
underestimation of cross-correlation was seen for the data generated by the MONR model (see Fig. 8b). Note that the error bars are extended to upper and lower lines of the circles to 1.95 times the
standard deviation. The difference of RMSE in Table 6 showed this characteristic, as most of the values were positive, indicating that the proposed DKNNR model performed better for cross-correlation.
The lag-1 cross-correlation of observed data, as shown in Table 7, ranged from 0.22 to 0.35. The lag-1 cross-correlation for the same site (i.e., ρ[1](q,s), q=s) was autocorrelation and was highly
related to P[01] and P[11] as in Eq. (6). All the lag-1 cross-correlations exhibited similar magnitudes even for autocorrelation. This implies that the lag-1 cross-correlation among the selected
sites was as strong as the autocorrelation and as much as the transition probabilities P[01] and P[11] thereof.
The observed lag-1 cross-correlations were well preserved in the data generated by the DKNNR model, as shown in Fig. 9a, while the MONR model showed significant underestimation, as seen in Fig. 9b.
The difference of RMSE shown in Table 8 reflects this behavior. In Fig. 9b, some of the lag-1 cross-correlations were well preserved, that were aligned with the baseline. From Table 8, the MONR model
reproduced the autocorrelations well with the shaded values. It is because the lag-1 autocorrelation was indirectly parameterized with the transition probabilities of P[11] and P[01] as in Eq. (6).
Other than this autocorrelation, the lag-1 cross-correlation was not reproduced well with the MONR model. This shortcoming was mentioned by Wilks (1998). Meanwhile, the proposed DKNNR model preserved
this statistic without any parameterization.
We further tested the performance measurements of mean absolute error (MAE) and bias whose estimates showed that MAE had no difference from RMSE. In addition, bias of lag-1 correlation presented
significant negative values, implying its underestimation for the simulated data of the MONR model as shown in Table 9, while Table 10 of the DKNNR model showed a much smaller bias.
Also, the whole-year data instead of the summer season data were tested for model fitting. Note that all the results presented above were for the summer season data (June–September), as mentioned in
Sect. 4 in the data description. The lag-1 cross-correlation is shown in Fig. 10, which indicates that the same characteristic was observed as for the summer season, such that the proposed DKNNR
model preserved better the lagged cross-correlation than did the existing MONR model. Other statistics, such as correlation matrix and transition probabilities, exhibited the same results (not
shown). Also, other seasons were tried but the estimated correlation matrix was not a positive semi-definite matrix and its inverse cannot be made for multivariate normal distribution in the MONR
model. It was because the selected stations were close to each other (around 50–100km) and produced high cross-correlation, especially in the occurrence during dry seasons. Special remedy for the
existing MONR model should be applied, such as decreasing cross-correlation by force, but further remedy was not applied in the current study since it was not within the current scope and focus.
6.4Adaptation to climate change
Model adaptability to climate change in hydro-meteorological simulation models is a critical factor, since one of the major applications of the models is to assess the impact of climate change.
Therefore, we tested the capability of the proposed model in the current study by adjusting the probabilities of crossover and mutation as in Eqs. (15) and (16). A number of variations can be made
with different conditions.
In Fig. 11, the changes of transition and marginal probabilities are shown, along with the increase of crossover probability P[cr] from 0.01 to 0.2 with the condition that the candidate value is 1
and the previous value is also 1 as in Eq. (15) for the selected 5 stations among the 12 stations (from station 1 to station 5; see Table 1 for details). The stations were limited in this analysis
due to computational time. In each case, 100 series were simulated. The average value of the simulated statistics is presented in the figure. It is obvious that the transition probability P[11]
increased as intended along with the increase of P[cr]. As expected from Eq. (5), P[1] presents that the change of P[1] is highly related to P[11]. However, the probability P[01] fluctuated along
with the increase of P[cr]. Elaborate work to adjust all the probabilities is however required.
The changes in transition and marginal probabilities are presented in Fig. 12 with increasing mutation probability P[m] from 0.01 to 0.2 under the condition that the candidate value is 1, so that the
marginal probability P[1] increased. P[01] also increased along with increasing P[1]. The change of P[11] was not related to other probabilities. The combination of the adjustment of P[cr] and P[m]
with a certain condition to the previous state will allow the specific adaptation for simulating future climatic scenarios.
As an example, assume that the occurrence probability (P[1]) of the control period is 0.26 (see the dotted line with a cross in the bottom panels of Figs. 11 and 12) and global circulation model
(GCM) output indicates that the occurrence probability (P[1]) increases up to 0.27. This can be achieved with increasing either the crossover probability to 0.1 or the mutation probability to 0.05.
Note that the crossover probabilities might cause the stations to affect each other, while the mutation probabilities do not.
Climate change, however, may refer to a larger phenomenon, which cannot be addressed directly through modifying only the marginal and transition probabilities as in the current study. Further model
development on systematically varying temporal and spatial cross-correlations is required to properly address climate change of the regional precipitation system.
In the current study, the discrete version of a nonparametric simulation model, based on KNNR, is proposed to overcome the shortcomings of the existing MONR model, such as long stochastic simulation
for parameter estimation and underestimation of the lagged cross-correlation between sites, as well as testing the adaptability for climatic change. Occurrence and transition probabilities and
cross-correlation, as well as lag-1 cross-correlation are estimated for both models. Better preservation of cross-correlation and lag-1 cross-correlation with the DKNNR model than the MONR model is
observed. For some cases (i.e., the whole-year data and seasons other than the summer season), the estimated cross-correlation matrix is not a positive semi-definite matrix, so the multivariate
normal simulation is not applicable for the MONR model, because the tested sites are close to each other with high cross-correlation.
Results of this study indicate that the proposed DKNNR model reproduces the occurrence and transition probabilities satisfactorily and preserves the cross-correlations better than the existing MONR
model. Furthermore, not much effort is required to estimate the parameters in the DKNNR model, while the MONR model requires a long stochastic simulation just to estimate each parameter. Thus, the
proposed DKNNR model can be a good alternative for simulating multisite precipitation occurrence.
We further tested the enhancement of the proposed model for adapting to climate change by modifying the mutation and crossover probabilities (P[m] and P[cr]). Results show that the proposed DKNNR
model has the capability to adapt to the climate change scenarios, but further elaborate work is required to find the best probability estimation for climate change. Also, only the marginal and
transition probabilities cannot address the climate change of regional precipitation. The variation of temporal and spatial cross-correlation structure must be considered to properly address the
climate change of the regional precipitation system. Further study on improving the model adaptability to climate change will follow in the near future. Also, the simulated multisite occurrence can
be coupled with a multisite amount model to produce precipitation events, including zero values. Further development can be made for multisite amount models with a nonparametric technique, such as
KNNR and bootstrapping.
Code and data availability
DKNNR code is written in Matlab and is available as a Supplement.
The precipitation data employed in the current study are downloadable through https://data.kma.go.kr/cmmn/main.do (Korea Meteorological Administration, 2019).
TL and VPS conceived of the presented idea. TL developed the theory and programming. VPS supervised the findings of the current work and the writing of the manuscript.
The authors declare that they have no conflict of interest.
This work was supported by the National Research Foundation of Korea (NRF) grant (NRF-2018R1A2B6001799) funded by the South Korean government (MEST).
This paper was edited by Jeffrey Neal and reviewed by two anonymous referees.
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What Do Food and Research Have in Common? More Than You Might Think
Laureates of mathematics and computer science meet the next generation
A common German saying is that “love goes through the stomach” – but perhaps the same could be said of research?
At the 10th Heidelberg Laureate Forum in 2023, an annual networking conference bringing together some of the brightest minds in mathematics and computer science, we asked twelve researchers:
If you had to choose a meal to describe your research – which one would you choose and why?
Here is what they said:
Jana Brunátová
PhD student in Mathematics at Charles University in Prague & University of Groningen
Research field: Mathematical Modeling
“My research is about modeling how blood flows through vessels in a human brain. I would pick spaghetti with ketchup to describe my research because the vessels are as thin as spaghetti and the blood
is a red non-Newtonian fluid just like ketchup (remember how tricky is it to pour the ketchup from the bottle). I am interested in mathematical models that describe such behavior. The reason for
studying this topic is to help understand and treat cardiovascular diseases such as brain aneurysms.”
Thomas Jiralerspong
Master’s student in Computer Science at the University of Montreal
Research field: Latent Disentanglement and Reinforcement Learning
“I’d say spaghetti ice cream, although I’ve never tried it myself. It’s essentially ice cream designed to resemble spaghetti with tomato sauce and cheese. So this draws inspiration from spaghetti to
build ice cream. Similarly, in my research, I take inspiration from the human brain to build AI models. Also, I think, spaghetti kind of looks like a brain.”
Mark Colley
PhD student and Research Associate in Computer Science at Ulm University
Research field: Human-Computer Interaction
“I’d pick peanut butter and jelly as my meal to symbolize my field of Human-Computer Interaction. I don’t individually enjoy peanut butter or jelly, but combined they create something wonderful.
Similarly, in my field, computers and humans, while capable on their own, achieve greater results together. Additionally, the white bread, which is fluffy just like an airbag, connects with my work
involving vehicles.”
Pêdra Andrade
Postdoc in Mathematics at Instituto Superior Técnico, Universidade de Lisboa
Research field: Partial Differential Equations (PDE) in Mathematics
“My research concerns the study of regularity theory for Partial Differential Equations (PDEs). A great example to illustrate regularity properties is a very popular Brazilian snack called “Coxinha”
which you can find especially at children’s birthday parties and cafeterias in Brazil.
This is why:
In regularity theory, the ideal scenario is when the solution of an equation is smooth – in other words, it is a continuous function without sharp corners and jumps. For example, sine and cosine are
smooth functions. By cutting the coxinha horizontally into two pieces, we can obtain two smooth functions. (Please note: Whether we can find smooth functions depends on the shape of the coxinha – a
coxinha with a spiky top would not allow that, for example).
Additionally, when looking at the regions formed by the coxinha filling and the dough, we encounter what’s known as a Transmission Problem. This means there’s an interface where the solutions of the
equations suddenly change. The main goal here is to understand how these solutions act at the boundary between the coxinha filling and the dough.
Another interesting problem appears when we deep fry the coxinha in oil. In this case, we obtain two heat equations with different diffusion coefficients (meaning that heat spreads differently): one
inside of the coxinha and the other on its surface. This effect characterizes what is called a Free Boundary Problem, where we do not know what happens at the interface created by these equations.”
Theo McKenzie
Postdoc in Mathematics at Stanford University
Research field: Probability Theory / Mathematical Physics
“As I study randomness and chaos, I think about mixing two things like coffee with milk. You have two separate things which interact with each other randomly, until they find some kind of equilibrium
state – i.e., when coffee and milk are equally mixed and stirring doesn’t help anymore. My research focuses on the question: How long does it take to get to an equilibrium? In the coffee example,
this would mean: How many stirs of your spoon does it take to get there, considering factors like cup size and ingredient quantities?”
Thiago Holleben
PhD student in Mathematics at Dalhousie University
Research field: Combinatorial Commutative Algebra
“There is this Brazilian food called pastel – you have a specific dough and you have some sort of flavour like cheese or meat inside. Let’s imagine you are hosting a party and you want to make one
pastel for each person. The question I am dealing with in my research is: If you have a fixed number of pastéis of one flavour, for example ten pastéis of cheese (pastéis de queijo) – in how many
ways can you distribute them among your guests? However, there are some restrictions. For example, in some cases, only one person from a group of friends can have a pastel of that particular flavor
because you don’t want too many of the same kind on one table. These kinds of problems appear in combinatorics and algebra.”
Silvia Sellán
PhD student in Computer Science at the University of Toronto
Research field: Computer Graphics and Geometry Processing
“My research (in Computer Graphics and Geometry) is like a poké bowl. If you’re the kind of person (like me) that quickly gets bored of a single flavour, a poké bowl is great because you can have a
little bit of salmon, followed by edamame, corn, rice, avocado, onion, etc. This is what I love about poké bowls and also about my research: there’s a bit of math followed by some programming, as
well as art and creative writing. In both, you can mix it up and eat it all at once, or take it one at a time, depending on how you’re feeling that day.”
“I would choose Chinese Hot Pot to describe one aspect of my research because it embodies the idea of a shared collaborative environment. In a hot pot setting, people gather around a table with a
central pot of boiling broth and various raw ingredients. It’s like a shared virtual 3D environment where you encourage conversation and collaboration by sharing resources. I work on designing
innovative tools to manage this space effectively, just as you have chopsticks and special spoons for the hot pot. This includes developing ways to efficiently select and manipulate objects in
cluttered environments, such as working on 3D models, and ways to foster coordination and collaboration, such as in hybrid meetings. It’s about facilitating seamless interactions in a complex shared
Ishika Ghosh
PhD student in Mathematics at Michigan State University
Research field: Computational Topology
“For me, the food would be a donut and a cup of coffee, because I do computational topology. Topology is the study of shapes and to us, a coffee cup and donut are the same thing.”
Michal Jex
Assistant Professor in Mathematics at Czech Technical University in Prague
Research field: Quantum Mechanics, Operator Theory
“I’d compare my research to a soufflé. I focus on weakly bounded states in quantum mechanics, which are states that are stable but at the same time can be easily disturbed. This mirrors a soufflé’s
delicate nature – it’s wonderful when handled gently but can quickly deflate if mishandled.”
Victoria Kaial
Master’s student in Mathematics at Universidad Nacional de Rosario
Research field: Combinatorial Optimization and Structural Graph Theory
The following text describes the research she did during an internship at the Laboratory of Informatics, Modelling and Optimization of the Systems (LIMOS).
“If my research topic was a meal, it would probably be a so-called “pionono”: a salty and delicious pile of layers of thin sponge cake and different ingredients (like roasted bell pepper, beet [or
beet paste], ham, sliced cheese, lettuce, olives, boiled [and afterwards grated or sliced] eggs, etc.) intercalated. Mayonnaise is present in every layer and covers the outside (at least the top).
I study graph classes that arise from an application: the Routing and Spectrum Assignment Problem (RSA), where the goal is to assign a path and a channel to each demand over an optical fiber network.
The optical spectrum is divided into narrow frequency slots and a channel is basically an interval of slots. Some data demands need wider channels than others. For example, text is lighter than
video, just like lettuce leaves are thinner than bell peppers.
One of the conditions that the channel assignment must satisfy is “slots continuity”, which means that the channel assigned to each demand is fixed along the whole path, i.e. you cannot change the
channel through which certain data travels. The same happens in the pionono: Once one has chosen a position for the cheese layer for example, it cannot be changed in the middle of the path from one
extreme to the other of the pionono. We want all our guests to have the same experience when eating it.
Choosing the order of the layers in the pionono feels like a rigorous optimization problem. In particular, like the spectrum (channel) assignment part of the RSA.”
Sascha Gaudlitz
PhD student in Mathematics at Humboldt-Universität zu Berlin
Research field: Statistical Inference for Stochastic Partial Differential Equations
“Imagine having this delicious cake, but your friend won’t share the recipe – it’s their secret. However, it’s so delightful that you want to uncover how it’s made. In this analogy, the cake
represents real-world data, like microscope images of cells, and the recipe is a mathematical model we use to understand and describe the data. Essentially, we’re trying to reverse engineer the
recipe from the cake, which is like reconstructing the model from the data.”
How would you describe your research with a meal? Let us know in the comments below or tweet it @HLForum with the hashtag #WhatsCookingHLF.
Illustrations by Sina Loriani:
Sina Loriani is a German climate researcher and illustrator (https://sciconaut.de/). In his current science journalism fellowship at the MIP.labor he is working on a comic project to communicate the
math and physics behind climate tipping points.
2 comments
1. Nina Beier wrote (24. Jan 2024):
> […] How would you describe your research with a meal?
In culinary terms, my research on spacetime geometry reminds me of »Wildhase königlicher Art« (»Hare à la royale«, a.k.a. „Lièvre à la Royale“):
An old recipe for a fabulous but only very occasionally prepared dish (for me, Einstein’s command: »All our spatio-temporal determinations boil down to determinations of coincidence.«) has to be
painstakingly tried out to come up the optimal preperation techniques and to identify the very best ingredients. The goal is to reliably prepare a delicious meal living up to its proud name and
history, while satisfying highest expectations of today’s tastes.
This resembles my absorbing tinkering with applications of coincidence determinations; progressing from the obvious Comstock-Einstein simultaneity, through intricate constructions such as
tetrahedral-octahedral (a.k.a. “octet strut”) ping coincidence lattices, with participants (as vertices of the lattice) in several intersecting photon sheets, with my sights set (so far) on
eventually letting the notion of a spacetime region being “conformally flat” explicitly boil down to coincidence determinations between participants in that region.
2. Sehr guter Artikel, danke fürs Teilen
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Research Data - Pure Mathematics and Mathematical Statistics (DPMMS)Code/software supporting 'Explicit moduli spaces for curves of genus 1 and 2'Concept Lab: Precomputed Associations for Shared Lexis Tool and Associated Files (Public)Numerical verification of Beilinson's conjecture on Fermat quotientsResearch Data Supporting "Tests for separability in nonparametric covariance operators of random surfaces"Research Data Supporting "Semimartingale detection and goodness of fit tests"Research data supporting 'Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics'.Software for "Near-optimal estimation of jump activity in semimartingales"
No Descriptionhttps://www.repository.cam.ac.uk/handle/1810/2489582024-11-04T18:23:04Z2024-11-04T18:23:04Z71Frengley, Samuelhttps://www.repository.cam.ac.uk/handle/1810/
3743252024-10-01T10:45:51Zdc.title: Code/software supporting 'Explicit moduli spaces for curves of genus 1 and 2' dc.contributor.author: Frengley, Samuel dc.description: This code supports my PhD
thesis 'Explicit moduli spaces for curves of genus 1 and 2'. In particular, it contains: (1) code which computes the numerical invariants of the surfaces $Z_{N,r}$, $W_{N,r}$ and those birational to
them, (2) code for studying congruences of quadratic-twist type (in particular models and $j$-maps for the modular curves $X^+(H_1, H_2)$), (3) models for the surfaces $Z_{N,r}$ and $W_{N,r}$ for $N
= 12, 14$ and for $(N,r) = (15,2), (15,7)$ computed in my thesis (and some others), (4) examples of genus $2$ Jacobians with $7$-torsion in their Tate--Shafarevich groups. Please read the READMEs in
the various folders for more information. Recchia, GabrielJones, EwanNulty, Paulde Bolla, PeterRegan, Johnhttps://www.repository.cam.ac.uk/handle/1810/3360542023-05-04T01:48:35Zdc.title: Concept Lab:
Precomputed Associations for Shared Lexis Tool and Associated Files (Public) dc.contributor.author: Recchia, Gabriel; Jones, Ewan; Nulty, Paul; de Bolla, Peter; Regan, John dc.description: This
dataset consists of: I. Source code and documentation for the "Shared Lexis Tool", a Windows desktop application that provides a means of exploring all of the words that are statistically associated
with a word provided by the user, in a given corpus of text (for certain predefined corpora), over a given date range. II. Source code and documentation for the "Coassociation Grapher", a Windows
desktop application. Given a particular word of interest (a “focal token”) in a particular corpus of text, the Coassociation Grapher allows you to view the relative probability of observing other
terms (“bound tokens”) before or after the focal token. III. Numerous precomputed files that need to be hosted on a webserver in order for the Shared Lexis Tool to function properly; IV. Files that
were created in the course of conducting the research described in "Tracing shifting conceptual vocabularies through time" and "The idea of liberty" (full citations in above section 'SHARING/ACCESS
INFORMATION'), including "cliques" (https://en.wikipedia.org/wiki/Clique_(graph_theory)) of words that frequently appear together; V. Source code of text-processing scripts developed by the Concept
Lab, primarily for the purpose of generating precomputed files described in section III, and associated data. The Shared Lexis Tool and Coassociation Grapher (and the required precomputed files) are
also being hosted at https://concept-lab.lib.cam.ac.uk/ from 2018 to 2023, and therefore those who are merely interested in using the tools within this time frame will have no use for the present
dataset. However, these files may be useful for individuals who wish to host the files on their own webserver, for example, in order to use the Shared Lexis tool past 2023. See README.txt for more
information. Cain, CBhttps://www.repository.cam.ac.uk/handle/1810/2622842022-08-24T14:34:11Zdc.title: Numerical verification of Beilinson's conjecture on Fermat quotients dc.contributor.author: Cain,
CB dc.description: As described in my PhD thesis K-Theory of Fermat Curves I give PARI/GP scripts and programs written in C that numerically verify Beilinson's conjecture on certain quotients of
Fermat curves. Aston, JohnPigoli, DavideTavakoli, Shahinhttps://www.repository.cam.ac.uk/handle/1810/2564892019-01-12T17:03:07Zdc.title: Research Data Supporting "Tests for separability in
nonparametric covariance operators of random surfaces" dc.contributor.author: Aston, John; Pigoli, Davide; Tavakoli, Shahin dc.description: #### Software - covsep-1.0.1.tar.gz : R package
implementing the methodology of the paper. #### Data for reproducing the Numerical simulations: - reproduce-sims.R: R script to reproduce the simulation studies; WARNING: the simulations take a lot
of time. - SIM2016a-SIM12feb.RData: results of the simulation studies, and parameters to reproduce them. - grid-sims2.RData: results of the simulation studies (Figure 5 and Figures S2 & S3 of the
supplement), and parameters to reproduce them. - sim_function.R: function performing the simulation studies #### Data application: - Acoustic_Data_part1.RData: R workspace containing the preprocessed
log-spectrograms considered in the phonetic application. (part 1) - Acoustic_Data_part2.RData: R workspace containing the preprocessed log-spectrograms considered in the phonetic application. (part
2) - Info_Acoustic_Data.txt: text file with information about the phonetic sounds. This includes the language, the word being pronounced and the gender of the speaker. - separability_Acoustic_Data.R:
script to replicate the test for separability for the phonetic data as described in the manuscript.; This record is embargoed until publication. Bull, Adam D.https://www.repository.cam.ac.uk/handle/
1810/2561842019-01-12T16:56:43Zdc.title: Research Data Supporting "Semimartingale detection and goodness of fit tests" dc.contributor.author: Bull, Adam D. dc.description: Code generating the
research data in the paper 'Semimartingale detection and goodness-of-fit tests'. Tavakoli, ShahinPanaretos, Victor. M.https://www.repository.cam.ac.uk/handle/1810/
2536952021-04-30T13:14:26Z2016-02-10T00:00:00Zdc.title: Research data supporting 'Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics'.
dc.contributor.author: Tavakoli, Shahin; Panaretos, Victor. M. dc.description: - dna-dynamics.RData: results of the analysis conducted in the paper. Contains sample spectral the spectral densities of
the data (cap.spec, tata.spec), the differences between the spectral densities localized in frequencies (spec.difference.on.frequencies) and localized in frequencies and along curvelength
(spec.difference.freq.curvelength). - simulations-level-power.RData: results of the simulation studies presented in the paper to assess the finite sample level and power of the procedure for
detecting differences between spectral densities at the level of frequencies. - simulations-grid-density.RData: results of the simulation studies presented in the paper to determine the effect of the
density of the frequency grid on the power. An R package for reproducing the results, as well as the simulation studies, available on https://zenodo.org/badge/latestdoi/20339/stavakol/ftsspec
2016-02-10T00:00:00ZBull, Adamhttps://www.repository.cam.ac.uk/handle/1810/2489592021-04-29T12:03:56Z2015-07-14T00:00:00Zdc.title: Software for "Near-optimal estimation of jump activity in
semimartingales" dc.contributor.author: Bull, Adam dc.description: An R script generating the data in the paper "Near-optimal estimation of jump activity in semimartingales." 2015-07-14T00:00:00Z
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The inversion matrix and error estimation in data inversion: Application to diffusion battery measurements for Journal of Aerosol Science
Journal of Aerosol Science
The inversion matrix and error estimation in data inversion: Application to diffusion battery measurements
View publication
Judicious selection of the measurement conditions and analysis methods to be used can make it less difficult to produce accurate data inversions in the presence of experimental error. The response
(data) vector (b) of a multi-channel instrument, such as an optical particle counter or multi-stage impactor or diffusion battery, to an input distribution vector (x) can be modelled as a set of
linear equations given by the vector-matrix equation b = Ax. For low resolution instruments, one of several methods of data inversion is usefully employed: simple inversion, least-squares inversion,
various smoothing inversions and various non-linear approaches. One non-linear approach [Twomey, S. (1975) J. Comput. Phys. 18, 188-200.] we found to be sensitive to starting conditions and to show
cycling during iteration, similar to equations leading to 'chaos' [Wu, J. J. et al. (1989) J. Aerosol Sci. 20, 477-482.]. Simple inversion, least-squares inversion and smoothing are alike in that
they produce their solutions from x = Zb, where Z(i, j) are the elements of what one could call the 'inversion matrix', Z, a kind of transfer function. Z gives the sensitivity of the inferred values
to changes (or errors) in the data values. A criterion for the best measurement instrument or measurement conditions could be the minimum largest absolute Z(i, j) or the mean absolute value or some
other weighting. Propagation of error analysis indicates that another measure of Z(i, j) that would be useful would be its root mean square. The 'condition number' is another measure that has also
been suggested [Cooper, D. W (1974) Ph.D. dissertation. Division of Engineering and Applied Science, Harvard University, Cambridge, MA, (1975) 68th Annual Meeting of the Air Pollution Control Assoc.,
Boston, MA; Yu, P.-Y. (1983) Ph.D. dissertation. Department of Chemical and Nuclear Engineering, College Park, MD; Farzanah, F. F. et al. (1984) Environ. Sci. Technol. 19, 121-126; Hirleman, E. D.
(1987) 1st Intl. Conf. on Particle Sizing, Rouen, France.]. Some comparisons of these measures are made. The inversion matrix gives the clearest indication of the relationship between the data and
the results of inversion. We recommend that proposed experimental conditions should be adjusted based on inversion matrix studies in order to lessen ill-conditioning and the reliance on various data
analysis methods to cope with ill-conditioned systems. © 1990.
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seminars - Pointwise convergence of noncommutative Fourier series
I will talk about pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as
summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, I will introduce the main theorem: a general criterion of maximal inequalities for
approximate identities of noncommutative Fourier multipliers. By using this criterion, for any countable discrete amenable group, there exists a sequence of finitely supported positive definite
functions tending to 1 pointwise, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence for all p > 1. In a similar fashion, I will show a large
subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. I will also talk about the Pointwise convergence of Fejér and Bochner-Riesz means in the
noncommutative setting. Finally, I will mention a byproduct-- the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.
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ID : 356 501 3138
PW : 471247
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David Herzog : Supports of Degenerate Diffusion Processes: The Case of Polynomial Drift and Additive Noise
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David Herzog : Supports of Degenerate Diffusion Processes: The Case of Polynomial Drift and Additive Noise
We discuss methods for computing supports of degenerate diffusion processes. We assume throughout that the diffusion satisfies a stochastic differential equation on R^d whose drift vector field X[0]
is ``polynomial'' and whose noise coefficients are constant. The case when each component of X[0] is of odd degree is well understood. Hence we focus our efforts on X[0] having at least one or more
components of even degree. After developing methods to handle such cases, we shall apply them to specific examples, e.g. the Galerkin truncations of the Stochastic Navier-Stokes equation, to help
establish ergodic properties of the resulting diffusion. One benefit to our approach is that, to prove such consequences, all we must do is compute certain Lie brackets.
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Chapter 98: Methods For Solving Market Problems By Scientists In Medieval Central Asia
The development of mathematics in Central Asia is widely reflected in scientific research and popular science literature on the history of mathematics, and in periodicals, but there are no studies
focused on the interaction between mathematical methods and market problems in medieval Central Asia. In order to reveal the evolution of the interaction between mathematical methods and market
problems in medieval Central Asia and its conditions, historical analysis and theoretical analysis of philosophical, mathematical and methodological literature were employed in the study. The object
of the study is the interaction between mathematical methods and market problems in the works by scientists of medieval Central Asia. The subject of the study is mathematical methods used to solve
market problems. The periodization of the development of science in medieval Central Asia defined by the orientalist G. Sieter revealed the evolution of the interaction between mathematical methods
and market problems and its conditions in medieval Central Asia. In the Middle Ages, the algebraic treatise by- Short book on the calculus of algebra and almukabala was a breakthrough scientific work
in Central Asia. First,-, then al-Buzjani and al-Biruni used mathematical methods in their works to solve market problems. The results of the study reveal the evolution of the interaction between
mathematical methods and market problems in medieval Central Asia and its conditions. The results are of relevance since they help fill the gap in the study of development of economic and
mathematical methods in medieval Central Asia.
Keywords: Algebra, medieval Central Asia, mathematical methods, market problems, scientific works
There is now an extensive literature on the history of the development of mathematics in Central Asia. The necessary information is provided in a bio-bibliographic reference book (Matvievskaya &
Rosenfeld, 1983) about mathematicians and astronomers of Muslim countries who lived in the 8th–17th centuries. The edition consists of three books. The first book includes an introductory paper and a
reference section. The authors of the paper are Rosenfeld, Yushkevich, and Matvievskaya write: 'the purpose of this book is to provide bio-bibliographic information about mathematicians and
astronomers of a vast region – Central Asia, Azerbaijan and the North Caucasus, Iran and Afghanistan, Turkey and North India, the Arab countries of Western Asia, North Africa and the Iberian
Peninsula – over 8th–17th centuries with due regard to all available literature on the issue and data from the manuscripts' (Matvievskaya & Rosenfeld, 1983). The book states that the mathematicians
of the Middle Ages-, al-Buzjani, al-Biruni, and al-Kashi used mathematical methods in their treatises to solve market problems. A market problem is defined as a problem of market relations, the
solution of which requires the use of mathematical methods. The paper (Luther, 2018) describes the establishment of the Soviet school of the history of mathematical sciences in the medieval Near and
Middle East. 'The Soviet school of the history of mathematical sciences in the medieval Near and Middle East was established and further developed by Yushkevich and Rosenfeld, the greatest historians
of mathematics of the 20th century' (Luther, 2018). The study (Akhmedov, 1962), which dates back to the second half of the 20th century, emphasizes that the contribution of medieval scientists in
Central Asia to mathematics has not been fully investigated. At present, the works of mathematicians of the Middle Ages in Central Asia, as well as studies into the natural-scientific work of
scientists of that time, are being investigated (Komilov, 2000). A book was published about scientific discoveries, cultural achievements, progress and other events that took place in Central Asia (
Starr, 2017).
The study explores the works of scientists of medieval Central Asia investigated by the scientists of the Golden Horde (Fazlyoglu, 2014; Fazlyoglu, 2015). Three papers by Fazlioglu are three parts of
one work. These papers are devoted to the work, which appeared in the Golden Horde state and was transferred during the reign of Uzbek Khan (1313–1342) to the ruler of the Crimean ulus of the Golden
Horde Abul-Muzaffer Giyaseddin Tuluktemir Bey. The author of the treatise is unknown. The work allows conclusion that representation of mathematical issues adheres to the tradition of-, al-Kashi and
other mathematicians of medieval Central Asia, but the treatise does not consider the interaction between mathematical methods and market problems.
Problem Statement
This paper deals with the interaction between mathematical methods and market problems in medieval Central Asia, which has not been considered for a long time. First, this is due to the fact that
historians of mathematics studied the development and teaching of arithmetic, algebra and other mathematical disciplines, without any focus on their interaction with market problems, and second, the
works by Central Asian scientists of the Middle Ages, which consider market problems solved by mathematical methods, began to be translated much later. For example, the second part of theof the
algebraic treatise (al-Khwarizmi, 1983) was absent in medieval Latin translations. 'This issue has not been covered for a long time in the historical and scientific literature. It was believed that
from a mathematical point of view, it is not of interest, and the restrictions set in the condition were considered arbitrary. For the first time thewas investigated in 1917 by Rushka, and then by
Vileitner and Gandz' (al-Khwarizmi, 1983).
Research Questions
The object of the study is mathematical methods used to solve market problems. To solve market problems, scientists of medieval Central Asia employed the 'triple rule' method, the 'rule of five
quantities' method, a linear equation, and a geometric progression. Theby- contains more problems on the application of a linear equation, a number of problems on the application of an indefinite
equation and one problem on the application of a system of two linear equations with two unknowns.
Purpose of the Study
The purpose of the study was to find out the conditions and evolution of the interaction between mathematical methods and market problems in the works by scientists of medieval Central Asia. To
achieve the purpose, general scientific and special methods were used, including the historical method. The periodization of the development of science in medieval Central Asia was determined by the
well-known historian of science and orientalist Ziter. He identified three periods: 1) from 750 to 900, 2) from 900 to 1275, and 3) from 1275 to 1600. The second period is divided into two
sub-periods: a) from 900 to 1100, and b) from 1100 to 1275. (Komilov, 2000). The first period was called by him the period of translation activity and the development of the scientific heritage of
the ancient Greek and ancient Indian peoples. Zuter claims that during the first sub-period (900–1100) science developed at the courts of the Arab Caliphs, Samanids, Ghaznavids, Bunds, Fatimids,
Seljukids, etc. The second sub-period (1100–1275) is associated with the Maragha observatory and the scientific works by Nasir ad-Din at-Tusi (1201–1274) and his students and followers. The third
period is mainly associated with the works of the Samarkand scientific school (Komilov, 2000).
Research Methods
The study employed narrative, historical, systemic and comparative methods. The narrative method was used to describe the interaction between mathematical methods and market problems in the works by
scientists of medieval Central Asia based on the following: chronological scale, personalities, scientific works. The historical method based on Suther's periodization considers the evolution of the
interaction between mathematical methods and market problems over three periods. Systemic and comparative methods were used to clarify the conditions for this interaction. The analysis of
philosophical, mathematical literature and literature on the history of mathematics was carried out.
The interaction between mathematical methods and market problems in Central Asia in 750–900.
Muhammad ibn Musa al- (783–850) was the leading mathematician and astronomer among scientists in Baghdad and was the first to compose his work in Arabic (Baki, 1992) and used mathematical methods to
solve market problems. His was born in the state. During the Middle Ages, was a 'powerful economic and cultural center in the East' (al-Khwarizmi, 1983). This is evidenced by materials from
archaeological excavations and written sources. The Silk Road from China passed through. In addition, was engaged in lively trade with India and Iran at that time. Scientists from these states moved
to together with trade caravans. In turn, scientists from visited other countries. This involved the exchange of scientific ideas between scientists. Due to economic ties, scientists of the trading
states exchanged works on philosophy, mathematics, astronomy, and medicine. In these conditions, the formation of al-Khwarizmi as a scientist took place. According to Matviyevskaya, a scientist
al-Khwarizmi worked under the patronage of al-Mamun when he moved to the city of Merv, which in the Middle Ages was 'the largest center of economic and cultural life' in Central Asia. Al-Mamun was
the governor of the caliph, and then became the caliph in 813. He lived in Merv from 809 to 818. Later, Caliph al-Mamun moved to Baghdad with a group of Central Asian scientists, including
al-Khwarizmi. With al-Mamun, science continued to develop in Baghdad, the capital of the Caliphate. The House of Wisdom was set up, which turned into the center of a large scientific school.
Outstanding scientists from various areas of the Caliphate were brought to Baghdad. Due to his scholarship, al-Khwarizmi occupied a prominent position in the House of Wisdom.
In Baghdad, the scientific works of Greek and Indian scientists were translated into Arabic. These scientific works were brought to Baghdad not only through trade, but international relations also
facilitated the exchange of scientific ideas.
The algebraic treatise consists of two parts. The first part of the algebraic treatise by al-Khwarizmi contains the dedicated to the triple rule. Al-Khwarizmi notes that human transactions 'deal with
four numbers set by the questioner – measure, price, quantity and value. The number equal to the measure stands against the number equal to the value, and the number equal to the price stands against
the number equal to the quantity. The rule states that among three known numbers there are necessarily two, each of which stands against the other. Multiply each of the two known numbers standing
against each other by the other, and divide the product by another known number standing against the unknown. If you have this quotient, it is an unknown number that the questioner asks about, it
stands against the number which you divided by' (al-Khwarizmi, 1983). As an example, al-Khwarizmi offers a market problem and explains how to solve it: 'If the questioner asks: an employee whose
monthly salary is ten dirhams worked six days, what is his share, then you know that six days is one fifth of the month and that his share of dirhams is the same as his share of the working time. The
rule is as follows: a month that includes thirty days is a measure, ten dirhams is a price, six days is a quantity and [it is asked] what is the share, that is a cost; multiply the price (ten) by the
amount that stands against it (six), the product is sixty, then divide it by thirty (known number), that is a measure, and you get two dirhams, that is a value' (al-Khwarizmi, 1983).
The second part of the algebraic treatise considers many problems about the division of inheritance in accordance with the norms of Islamic law. Theis also interesting in terms of market relations.
At the same time, solving such problems, al- Khwarizmi operates with the concepts of the market: property, debt, price, redemption, monetary unit (dirhem), remuneration, etc.
In the book, the medieval thinker al-Farabi (870–950), who was the 'Second Teacher' after Aristotle in the East, reveals the meaning of 'the science of skillful methods.' As an example of the science
of skillful methods, he cites the science called 'aljabr wa muqabala', that is, algebra, in which, for the first time in the history of science, the great mathematician al-Khwarizmi included not only
a systematic presentation of algebra, but also methods and techniques of using mathematics in trade and in transactions of people with each other, or science, which contains the methods of practical
arithmetic. This is evidenced by the following: 'Practical arithmetic (‘ilm al-‘adad al-‘amali), according to al-Farabi, studies the number of specific items to be counted and is used in commercial
and civil cases. Civil cases (mu’amalat madaniyah) include problems of division of inheritance. Thus, the calculation of inheritance shares using al-Farabi's arithmetic is considered as a branch of
mathematics' (Luther, 2019). The first page of al-Khwarizmi's manuscript states: 'This is the first book written about algebra and almukabal in (countries) of Islam' (al-Khwarizmi, 1983). The same is
written in the notes to the treatise of al-Khwarizmi: 'A large volume of the indicates the importance that al-Khwarizmi attached to these problems. Since earlier Arabic sources are unknown, he was
likely the first to apply algebraic rules to the solution of these problems' (al-Khwarizmi, 1983). Algebraic treatise by al-Khwarizmi is considered as a breakthrough work: 'Technologies of irrigation
management, agriculture based on seasonal fluctuations in the amount of water, paper work in a complex tax system and ensuring the stability of the national currency – all this required knowledge of
mathematics and technology. The followers of al-Khwarizmi developed these fields, especially mathematics and astronomy, to an astonishingly high level. It can be seen that al-Khwarizmi proceeded from
local traditions in these fields when writing his breakthrough work' (Starr, 2017).
The use of mathematical methods in solving market problems in Central Asia in 900–1275.
An outstanding scientist and teacher Abu-l-Wafa al-Buzjani (940–998) (Hashemipour, 2007) worked under the patronage of the Caliph in Baghdad. After the death of the caliph, he was invited by the
Khwarizmshah to the Khwarizm state. In new conditions, Abu-l-Wafa al-Buzjani continued to apply mathematical methods proposed by al-Khwarizmi in solving market problems and wrote a work devoted to
methods for calculating land taxes and monetary investments. It was one of his books on mathematics written at the time (Starr, 2017). Medovoy studied the first part of the treatise and,
characterizing the second (applied) part of the treatise, notes that it deals with the issues of 'charging for work, construction calculations, exchange and purchase of various types of grain, etc.'
Al-Buzjani was the mentor of Abu Nasr Mansur Ibn Iraq (960–1036), the heir to the ruling house of Khwarizm, who later became a talented astronomer, mathematician, and expert in trigonometry.
A famous scientist-encyclopedist of Central Asia Muhammad al-Biruni (973–1048) (Farmonova & Sultanov, 2020) was born in a wealthy family in Kyat, the capital of Khwarizm (now the city of Beruni in
Uzbekistan). He was left without parents early. Abu Nasr Mansur Ibn Iraq, who knew al-Biruni's parents well, adopted him and raised him together with his son Ibn Iraq. Al-Biruni has a penchant for
mathematics and astronomy, and, like Ibn Iraq, studied these fields. He studied and worked in the large scientific center Kyat. The Afghan Sultan Mahmud, who occupied Khwarizm in 1017, transferred
him to the capital Ghazni, where al-Biruni headed a group of scientists gathered by Mahmud from the conquered countries. For several years al-Biruni lived in North India conquered by the Sultan,
where he deeply studied scientific works in Sanskrit. Al-Biruni writes his, which is similar to the treatises by al-Khwarizmi and al-Buzjani. He is the author of the following works: (Rosenfeld et
al., 1973).
Rosenfeld, Rozhanskaya, and Sokolovskaya write about the works by al-Biruni: 'A special treatise, (Rashiki), Biruni dedicated to one of the methods of practical arithmetic widely used in the Middle
and Near East and later in Europe – the 'triple rule'. ... Brahmagupta generalized this rule to 5, 7, 9 and 11 quantities that require the combined application of respectively 2, 3, 4 and 5 triple
rules. Beruni explains in detail the direct and inverse rules and generalizes them to any odd number of quantities, giving tasks for 13, 15, 17 quantities' (Rosenfeld et al., 1973).
Consider the problem proposed by al-Biruni, which he solves using the 'rule of five quantities': 'If 10 dirhams bring income 5 dirhams in 2 months, how much income will 8 dirhams bring in 3 months?'
To solve this problem, the quantities are placed according to the scheme.
$10 8 5 3 5 x$
Al-Biruni explains the 'rule of five quantities' as follows: 'To determine the unknown, five is transferred to an empty space, multiplied by 3, and then the product is multiplied by 8; it gives 120;
it is remembered. Next, multiply 2 by 10, it gives twenty. What is remembered is divided by 20, in the quotient it will be 6; this is the income of 8 dirhams in 3 months.' The formula of the 'rule of
five' is as follows:
$x = 5 * 3 * 8 2 * 10 = 6$
The 'chessboard problem' associated with an ancient Indian legend can be considered as practical content: it is required to find the total number of wheat grains if 1 grain is placed on the first
field of the board, 2 on the second, 4 on the third, etc., doubling the number of grains in each next field (Rosenfeld et al., 1973). To solve this problem, al-Biruni uses a geometric progression.
Mathematical methods for solving market problems in Central Asia in 1275–1600.
The major scientific center of the medieval East was Samarkand. Timur's grandson, Ulugbek (1394–1449), the ruler, was seriously engaged in astronomy and mathematics. Ulugbek set up a scientific
center as part of the Samarkand observatory. The observatory building was built in 1420. The scientist Dzhemshid al-Kashi (1380–1429) came to Samarkand from Iran. Djemshid al-Kashi headed the
observatory in the first years (1420–1429). In his work (1427), he outlined methods for extracting roots using a formula (expressed verbally), which was later called Newton's binomial; proposed the
approximate solution of the equation of the third degree and in the (1427) calculated π with 17 correct decimal places. 'In mathematics, he continued the long work initiated by al-Khwarizmi to
introduce the decimal system and provided a systematic method for calculating decimal fractions' (Starr, 2017). The method for calculation with decimal fractions was used to solve market problems.
The book by Bavrina and Fribus includes the problem of Jemshid al-Kashi on the application of the triple rule: 'The salary of an employee per month (30 days) is 10 dinars and a dress. He worked for 3
days and earned a dress. What is the cost of a dress?' To solve it, compose the proportion according to al-Khwarizmi.
30 days – x
10 days + x – 3,
where is the cost of a dress, and, applying the rule formulated by al-Khwarizmi, we obtain an equation with one unknown:
$10 + x * 3 30 = x$
Solve the equation and obtain the result that the cost of the dress is (1 + 1/9) dinars.
Thus, the Zuther's periodization yields the conclusion that the algebraic treatise by al-Khwarizmi (750–900) for the first time in medieval Central Asia considers mathematical methods for solving
market problems. After al-Khwarizmi, the interaction between mathematical methods and market problems continued due to scientific continuity in the works by al-Buzjani and al-Biruni in the period
from 900 to 1275. The 'triple rule' method is developed in the works by al-Biruni; he generalizes this rule to any odd number of quantities, giving tasks for 13, 15, 17 quantities. In the 18th–19th
centuries, the works by al-Khwarizmi, al-Buzjani, al-Biruni, and al-Kashi were not ignored, since at that time 'mathematics in madrasahs was taught using a certain system based on the works by
Central Asian mathematicians. For example, Khwarizmi, Tusi, Kashi and others' (Akhmedov, 1962). 'In the last part of mathematics, complicated problems are solved on the distribution of property,
according to Muslim traditions, between the heirs, which requires knowledge of mathematics' (Akhmedov, 1962).
In medieval Central Asia, the conditions for the evolutionary interaction between mathematical methods and market problems were as follows: lively trade of Central Asia with China, India, Iran;
al-Khwarizmi, al-Buzjani, al-Biruni, al-Kashi were under the patronage of the caliphs and their wealthy confidants; mathematicians worked in the cities (scientific centers) of Kyat, Merv, Baghdad,
Samarkand, where they studied mathematics, were engaged in mathematical research and translation of Greek and Indian works in mathematics.
• Akhmedov, S. A. (1962). Teaching mathematics and the stages of its development in Central Asia [Cand. dissertation thesis]. Tashkent.
• Al-Khwarizmi, Muhammad ibn Musa. (1983). Mathematical treatises. Tashkent, FAN.
• Baki, A. (1992). Al Khwarizmi’s contributions to the science of mathematics: Al Kitab Al Jabr Wa’l Mugabalah. Med. J. Islamic World Acad. Sci., 5(3), 225.
• Farmonova, M. O., & Sultanov, J. S. (2020). Some historical information about the life and mathematical heritage of Beruni. International Journal on Orange Technologies, 2(1), 43.
• Fazlyoglu, I. (2014). The first mathematical book in the Golden Horde state is a masterpiece in computational mathematics (at-Tuhfe fi‘ ilm al-hisab) (1). Golden Horde Review, 4(6), 57.
• Fazlyoglu, I. (2015). The first mathematical book in the Golden Horde state is a masterpiece in computational mathematics (at-Tuhfe fi‘ ilm al-hisab) (2). Golden Horde Review, 1, 106.
• Hashemipour, B. (2007). Buzjani: Abu al-Wafa’Muhammad ibn Muhammad ibn Yahya al-Buzjani. In: Thomas Hockey et al. (Eds.). Biographical Encyclopedia of Astronomers. Springer.
• Komilov, A. Sh. (2000). Physics of Central Asia in the 9th–13th centuries [Doct. Dissertation].
• Luther, I. O. (2018). Formation of the Soviet School of the History of Arab Mathematical Science: 1940–1960. Issues of the history of natural science and technology, 3, 421.
• Luther, I. O. (2019). Religious and legal aspect of al-Khwarizmi's algebra and its status in the hierarchy of sciences of al-Farabi. Chebyshev collection, 20(1), 391.
• Matvievskaya, G. P., & Rosenfeld, B. A. (1983). Mathematicians and astronomers of the Muslim Middle Ages and their works (8–17 centuries) (Book 1). Nauka.
• Rosenfeld, B. A., Rozhanskaya, M. M., & Sokolovskaya, Z. K. (1973). Abu-r-Raikhan al-Biruni. Nauka.
• Starr, S. F. (2017). Lost enlightenment: the golden age of Central Asia from the Arab conquest to the time of Tamerlane. Alpina Publisher.
About this article
Publication Date
29 November 2021
Article Doi
eBook ISBN
European Publisher
Edition Number
1st Edition
Cultural development, technological development, socio-political transformations, globalization
Cite this article as:
Issin, M. (2021). Methods For Solving Market Problems By Scientists In Medieval Central Asia. In D. K. Bataev, S. A. Gapurov, A. D. Osmaev, V. K. Akaev, L. M. Idigova, M. R. Ovhadov, A. R. Salgiriev,
& M. M. Betilmerzaeva (Eds.), Social and Cultural Transformations in The Context of Modern Globalism, vol 117. European Proceedings of Social and Behavioural Sciences (pp. 735-742). European
Publisher. https://doi.org/10.15405/epsbs.2021.11.98
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Do Parallelograms Have Right Angles - Reader Ship
Do Parallelograms Have Right Angles
Parallelograms are a kind of quadrilateral (a 4-sided shape with two identical sides) with opposite angles that don’t have the same measure. Parallelograms come in different flavors, but all of them
share the same uniqueness. If you look at any parallelogram, you can see that it has two sets of parallel sides. These parallel sets of sides are known as “adjacent” and “non-adjacent” sides. Each
side is either adjacent or non-adjacent to another side on the other end. In this article, we will be discussing whether or not parallelograms have right angles and why? Read on to know more.
Do Parallelograms Have Right Angles?
In geometry, the term “parallel” means that two or more lines are in the same plane. A line can be parallel to a straight line, a curve, or another line. Two lines may be in the same plane regardless
of their length but only if they are coplanar. If two lines do not intersect or touch each other, then they are said to be parallel.
What Is A Parallelogram?
A parallelogram is a quadrilateral with two sides that are parallel to each other. A parallelogram is a 2D shape. It has no height. A parallelogram has four sides. Just like a rectangle, a
parallelogram also has two pairs of parallel sides. A parallelogram is also a rectangle, but with one side longer than the other. A parallelogram can also be a rhombus, but with both pairs of sides
having the same length. A parallelogram can also be a square, but with both pairs of sides having the same length. A parallelogram can also be a kite, but with both pairs of sides having the same
length. Parallelograms have many uses in geometry. You can use them to solve many geometry problems.
Why Don’t Parallelograms Have Right Angles?
Two Lines are Parallel
Parallelograms can have more than one pair of parallel sides. The diagram below shows a parallelogram with two pairs of parallel sides. The lines AB and CD are both parallel to line AB. Lines BC and
CD are also both parallel to the line AC. Since two lines are parallel, they will always have the same angle between them. For example, in the diagram below, if you were to draw a line from point A
to point B and another line from point C to point D, then the angles between these lines will always be equal (all angles will be equal).
Two Lines Have an Angle
The angles between two lines are always different when they intersect at some point (see diagram above). If you were to draw a line from point A to point B and another line from point C to point D,
then this intersection would not be in a straight line (see diagram below). This means that there is no the same angle.
Two Lines are Coplanar
Parallelograms can also have two pairs of parallel sides that are not coplanar. The lines AB and AC do not intersect or touch each other, but they are still parallel. The diagram below shows a
parallelogram with two pairs of non-coplanar parallel sides.
Parallelograms Have No Height
Parallelograms can have no height, like the following diagram the same angle between them. Line AB will always have the same angle as line AC.
Two Lines are Parallel, but they are not Coplanar
Parallelograms can also have more than one pair of parallel sides, but they may not be coplanar. The above diagram shows a parallelogram with two pairs of parallel sides that are not coplanar. The
lines AB and AC are both parallel to line BC, but they do not intersect or touch each other. Since two lines are parallel, these lines will always have the same angle between them; however, these
lines cannot be coplanar since they do not touch each other or intersect.
Two Lines have different lengths and different angles
Parallelograms can also have more than one pair of parallel sides that differ in length and/or angle from each other. In this case, the lines may intersect at some point(s). In this case, both e the
same angle between them.
Two Lines are Coplanar
Parallelograms can have two pairs of parallel sides that intersect or touch each other (a point is considered to be a pair of parallel sides if it touches another pair). Consider the parallelogram
below. The lines AB and AC are coplanar, but they do not intersect or touch each other. Since two lines are coplanar, they will always have the same angle between them.
One Line is Parallel to Another Line
Parallelograms can also have one side that is parallel to another line (a line is considered to be a pair of parallel sides if it touches another line). The diagram below shows a parallelogram with
one side parallel to line AB. Since one line is parallel to another, it will always have the same angle between them.
When Do Parallelograms Have Right Angles?
• If a right angle is formed by two lines that are parallel to each other.
• If the angles of the parallelograms formed by two lines are equal.
• If the angles of the parallelograms formed by two lines are not equal but opposite (180 degrees).
• If a line is parallel to another line and both of them are in the same plane, then they will have right angles, because they intersect at a single point.
• If a line is parallel to another line and one of them is in the same plane as the other, then they will not have right angles because they do not intersect at a single point.
• Two lines can only be parallel if all of their points lie on or within one plane or all lie on or within two planes that are perpendicular to each other (i.e., when one goes up and down, then so
does the other).
• Two lines can only be parallel if they do not intersect at any point.
• Two lines can only be parallel if the sum of their lengths is equal to the length of a third line (if one goes up and down, then so does the other).
• Two lines can only be parallel if the angles formed by them are equal (180 degrees).
• Two lines can only be parallel if they do not intersect at any point.
• If a line is perpendicular to another line, then they will have right angles because they intersect at a single point.
• If two lines are perpendicular to each other, then they will have right angles because they intersect at a single point.
Now, you know what a parallelogram is, why parallelograms do not have right angles when parallelograms have right angles and the reason why parallelograms do not have right angles. All you have to do
now is to practice these concepts and make them part of your daily geometry routine Gmapros. Once you have mastered these concepts, you can move on to the next level of geometry.
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Max Veloz
Max Veloz is a news blogger who has a passion for writing about current events and sharing his insights on the latest happenings in the world. He is always up for a good debate, and he loves to learn
new things. When he's not writing or reading the news, Max likes to spend time with his friends and family.
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How To Create Charts With Multiple Groups Of Stacked Bars 2024 - Multiplication Chart Printable
How To Create Charts With Multiple Groups Of Stacked Bars
How To Create Charts With Multiple Groups Of Stacked Bars – You can create a Multiplication Chart Nightclub by marking the columns. The left column ought to say “1” and signify the total amount
multiplied by one. On the right hand side from the kitchen table, brand the columns as “2, 4, 8 and 6 and 9”. How To Create Charts With Multiple Groups Of Stacked Bars.
Tips to find out the 9 instances multiplication dinner table
Learning the nine occasions multiplication dinner table is not an easy task. There are several ways to memorize it, but counting down is one of the easiest. Within this technique, you place the hands
around the desk and number your fingers one by one from a to 15. Fold your seventh finger to be able to start to see the ones and tens on it. Then add up the volume of hands on the left and right of
your own flattened finger.
When understanding the table, kids could be intimidated by bigger phone numbers. The reason being adding larger figures frequently gets to be a laborious task. However, you can exploit the hidden
patterns to make learning the nine times table easy. One of many ways is usually to create the 9 periods kitchen table with a cheat page, study it all out noisy, or exercise producing it straight
down regularly. This technique is likely to make the kitchen table a lot more memorable.
Patterns to consider on the multiplication graph
Multiplication graph or chart cafes are ideal for memorizing multiplication facts. You can find the merchandise of two numbers by looking at the rows and columns of your multiplication graph or
chart. As an example, a column that may be all twos plus a row that’s all eights ought to fulfill at 56. Designs to find on the multiplication graph pub are exactly like those who work in a
multiplication dinner table.
A pattern to search for with a multiplication graph or chart will be the distributive property. This house can be noticed in most columns. As an example, a product or service x two is the same as
several (periods) c. This same home pertains to any column; the sum of two posts equates to the value of one other line. Therefore, a strange amount times an even amount is surely an even amount. A
similar applies to the items of two strange amounts.
Building a multiplication graph or chart from recollection
Building a multiplication chart from memory may help kids find out the various phone numbers within the instances furniture. This simple exercise enables your kids to commit to memory the figures and
discover the way to multiply them, which can help them later whenever they learn more complicated math concepts. For a enjoyable and great way to memorize the figures, you are able to organize shaded
control keys to ensure that every one corresponds to a particular periods dinner table variety. Ensure that you content label each row “1” and “” to be able to quickly recognize which variety comes
When young children have perfected the multiplication graph or chart pub from memory space, they must dedicate on their own towards the project. That is why it is better to use a worksheet instead of
a standard notebook to practice. Colorful and computer animated personality layouts can interest the senses of your own young children. Before they move on to the next step, let them color every
correct answer. Then, display the chart in their study region or sleeping rooms to function as a memory.
Employing a multiplication graph or chart in your everyday living
A multiplication chart demonstrates how to grow phone numbers, someone to twenty. Furthermore, it displays the product of two numbers. It can be useful in everyday life, such as when dividing funds
or getting info on folks. The next are some of the methods you can use a multiplication graph. Rely on them to aid your child understand the principle. We certainly have mentioned just a few of the
most prevalent uses of multiplication desks.
You can use a multiplication graph or chart to aid your son or daughter learn how to reduce fractions. The secret is to keep to the denominator and numerator on the left. In this way, they may see
that a portion like 4/6 could be lowered to a fraction of 2/3. Multiplication maps are especially helpful for youngsters since they help them to understand amount patterns. You can get Totally free
computer models of multiplication chart bars on the internet.
Gallery of How To Create Charts With Multiple Groups Of Stacked Bars
Stacked Bar Chart Js Example Free Table Bar Chart
Clustered Stacked Bar Chart Template Free Table Bar Chart
Bar Chart R Barplot With N Groups Which Stacks 2 Values Stack Overflow
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Comparison of Frictional Torque with Other Types of Torque (Inertial, Elastic) in context of frictional torque
31 Aug 2024
Title: A Comparative Analysis of Frictional Torque with Inertial and Elastic Torques: Theoretical Framework and Implications
Frictional torque, a fundamental concept in mechanics, plays a crucial role in various engineering applications. However, its comparison with other types of torques, such as inertial and elastic
torques, has received limited attention. This article aims to bridge this gap by providing a comprehensive theoretical framework for comparing frictional torque with inertial and elastic torques. The
analysis is conducted within the context of rotational dynamics, and the results are presented in a mathematical format.
Frictional torque (τ_f) arises from the interaction between two surfaces in contact, resulting in a force that opposes motion or rotation. Inertial torque (τ_i), on the other hand, is a consequence
of an object’s inertia, causing it to resist changes in its rotational state. Elastic torque (τ_e), related to the elastic properties of materials, can also contribute to rotational dynamics.
Theoretical Framework:
Let’s consider a rotating system with a moment of inertia (I) and angular velocity (ω). The frictional torque (τ_f) is given by:
τ_f = μ * F * r
where μ is the coefficient of friction, F is the normal force, and r is the radius of rotation.
Inertial torque (τ_i), related to an object’s moment of inertia, can be expressed as:
τ_i = I * α
where α is the angular acceleration.
Elastic torque (τ_e), associated with elastic properties, can be represented by:
τ_e = k * θ^2
where k is a spring constant and θ is the angular displacement.
Comparison of Torques:
To compare these torques, we need to consider their rotational dynamics. The net torque (τ_net) acting on the system is given by:
τ_net = τ_f + τ_i + τ_e
Substituting the expressions for each torque, we get:
τ_net = μ * F * r + I * α + k * θ^2
This equation highlights the interplay between frictional, inertial, and elastic torques in rotational dynamics.
In conclusion, this article has provided a theoretical framework for comparing frictional torque with inertial and elastic torques. The analysis demonstrates that these torques interact within the
context of rotational dynamics, influencing the overall behavior of rotating systems. Further research is needed to explore the implications of this comparison in various engineering applications.
[1] Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics. Addison-Wesley Publishing Company.
[2] Landau, L. D., & Lifshitz, E. M. (1976). Mechanics. Butterworth-Heinemann.
[3] Thornton, S. C., & Marion, J. B. (2007). Classical Dynamics of Particles and Systems. Brooks Cole.
Related articles for ‘frictional torque’ :
Calculators for ‘frictional torque’
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What is Regularization in Machine Learning - Techno Grade
Machine learning regularization is a vital concept that can help you improve model performance and robustness when you’re training with complex datasets. It addresses one of the most common
challenges faced by machine learning models: also known as overfitting: training a model that does very well on the training data, but doesn’t generalize to new, unseen data. In the following
article, we’ll discuss what regularization is in machine learning, why it’s needed, and some most popular ways to apply regularization.
In this article let’s deep dive into What Is Regularization in Machine Learning.
In a nutshell, regularization adds some extra constraints or some type of penalty on a machine learning model’s parameters (weights), in order to make it ‘become’ simpler. Regularization, by doing
that, encourages the model to attend less on the training data and to generalize better to unseen data.
In layman’s terms, regularization is to prevent models from being ‘too optimistic’ about the training data. If a model is too complex, it might latch onto the noise in the data and start fitting
‘patterns’ which can’t be applied to new data and in fact perform poorly on unseen data. To avoid this regularization penalizes large weights or coefficients which could cause overfit.
Why is this important?
1. Prevents Overfitting
When a model learns the noise along with the genuine patterns in the data that’s called overfitting. Therefore, the model is too specific to training data and works badly on the new data. By adding a
penalty to complex models, this prevents that by essentially making them lose the ability to memorize the training data.
2. Improves Generalization
The primary reason to use a machine learning model is to generalize well to unseen data. Regularization is good in case of models to read in a trend of the data instead of unworthy noise.
Regularization controls model complexity allowing the model to generalize well on datasets other than the training dataset.
3. Simplifies Models
Regularization also helps making the model more simple by decreasing the coefficients or weights magnitude. The model stays interpretable, and this is absolutely important, especially when we are
working with real-life problems where we have to understand the relationships between the variables.
Different kinds of Regularization Techniques.
One way to apply regularization to a machine learning model is L1 regularization, L2 regularization, Elastic Net regularization and so on. Now let’s get into details of each.
1. Ridge regression – L2 Regularization
L2 regularization also is known as Ridge Regression and is one of the most common forms of regularization in machine learning. Instead, it adds onto the cost function, a penalty term which is
squared, multiplied by the magnitude of the coefficients or weights. This penalty term end up shrinking the weights, so that model does not favor one particular feature too much. The formula for L2
regularization is:
• MSE is the mean squared error
• λ (lambda) is the regularization parameter (controls the strength of regularization)
• w are the model weights
In Ridge Regression smaller λ means that weights get penalized more and the model becomes more simple.
2. Lasso Regression (L1 Regularization)
L1 (or Lasso Regression (Least Absolute Shrinkage and Selection Operator}}, adds a penalty that’s proportional to the absolute value of the magnitude of the coefficients. While Ridge Regression does
not produce sparse models, Lasso is able to do so, where some feature coefficients become precisely zero. Due to this, Lasso is a good feature selection technique as it chooses to leave out
unimportant or less important features from the model.
The cost function for L1 regularization is:
Lasso produces a more interpretable model because it encourages sparsity, i.e., it gives us the ability to select only some features.
3. Elastic Net Regularization
Lasso and Ridge are both types of regularization; Elastic Net is a combination of both. It combines the pros of the two approaches, and also neutralizes their drawbacks, by bringing in two
hyperparameters, one for each type of regularisation. Due to this, Elastic Net works best when there are more features than the amount of data (High-dimensional data), and Ridge or Lasso can ’t be
applied independently because of the risk of Multicollinearity and a weak feature selection, and that’s where Ridge and Lasso come to the rescue.
The Elastic Net cost function is:
• λ1 controls L1 regularization
• λ2 controls L2 regularization
How to Select the Most Suitable Regularization Technique?
When it comes to regularization, whether L1 or L2 or Elastic Net, which one to go for depends on the data and the purpose of the modeling:
• L1 Regularization (Lasso) is handy when you want to consider only important features and get rid of all irrelevant ones.
• L2 Regularization (Ridge) is applicable in situations when all features are likely to play a role in a model, however, it is necessary to curb extreme weight values to reduce overfitting.
• Elastic Net Regularization is helpful, when one wish to have both Lasso and Ridge benefits over the other particularly in high-dimensional data which is prone to multicollinearity.
Other regularization techniques and settings can also be evaluated by cross-validation techniques to enhance model performance.
Regularization is very important in machine learning since it prevents models from overfitting, increases their generalization capability, and reduces their complexity. L1, L2 or Elastic Net
Regularization makes no difference in how your model will be able to ‘see’ only the true data patterns and become even more robust for different unseen data. Understanding and applying regularization
approaches will help you create better and more explainable machine learning models that will perform better in the real world.
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Puzzles on graphs
The morbid infatuation with Sam Loyd's Fifteen that followed its introduction in the late 1870's ended with an article in the American Journal of Mathematics [Ref 2]. However, interest in what's
known as Combinatorial Games continues to this day. [Ref 1] offers a compendium of known results. This is where I have discovered a reference to R.M.Wilson's paper in which by abstracting the 15
puzzle as a puzzle on graphs he, in one blow, classified a whole family of puzzles. I shall not be able to go into the details of the proof but shall provide a background information that may make
the formulation of the main result in the paper more intuitive. The paper serves as a fine example of the power and utility of generalization in Mathematics. But first I need a few definitions.
1. A graph is called simple if no two edges are equal as sets. In other words, a graph is simple if at most one edge connects any two nodes.
2. A graph is called bipartite if the set of its vertices can be represented as a union of two disjoint sets such that no two nodes of the same set are connected by an edge. I.e., edges only go from
the nodes of one set to the nodes of another.
3. A subgraph of a graph is a graph whose vertices and edges form subsets of the sets of vertices and edges, respectively, of the given graph that may be called a supergraph.
4. A connected component of a node is the biggest subgraph of a graph that consists of all the nodes that serve as endpoints of walks starting at the given node. A connected component of one node is
a connected component of all its nodes.
5. A graph that consists of a single connected component is said to be connected. It's disconnected, otherwise.
6. A connected graph with no articulation nodes is called nonseparable.
1. The graph of the Konigsberg Bridges is not simple.
2. Obviously, a graph is connected if and only if it consists of a single connected component.
3. Bipartite graphs often appear as a Description of mapping (or matches) between two sets. If the definition seems to you a little contrived consider representation of the following real world
matches: resumes and job openings, boys and girls in a dance class, seats and passengers.
This is more or less obvious that the board of the 15 puzzle might be abstracted to a 4x4 graph every counter position corresponding to a node. Edges indicate possible puzzle moves, i.e. moves of the
empty square. Less obvious is that the graph is bipartite. There is nothing to prepare you for this fact. Even as I write this I feel a tinge of surprise. How could one think of separating a solid
puzzle into two parts without damaging the board? Do you feel the power of abstraction? Power may be a wrong word or be wasted fruitlessly unless a benefit is gained by looking at the puzzle as a
game on a graph. And a benefit there is as I plan to demonstrate shortly. But abstracting the board is obviously not enough. We still have to shift the counters as stipulated by the rules.
1. The set of vertices of a graph G is denoted as V(G).
2. The set of edges of a graph G is denoted as E(G).
3. |A| stands for the number of elements in a set A.
4. For a graph G with |G|=n a labeling is a 1-1 correspondence from V(G) onto the set {0,1,2,...,n-1}.
Labeling a graph means placing a unique number at every node. Let's agree that 0 (quite appropriately) corresponds to the empty square. Instead of physically sliding the counters we shall just modify
a labeling - clean and simple. What I enjoy about this example is that, as you may have rightly noticed, so far there was not much discussion and nothing was proven either. Bear a little longer
(there is going to be another definition) and you'll see the power of finding the right language to describe a problem. As is well known, a problem well described is a problem half solved.
1. For a given graph G, puz(G) is a graph whose nodes are all the labelings of G: V(puz(G))={f: f is a labeling of G}.
2. Two nodes of puz(G) are connected by an edge iff there is a legal move that transforms one labeling into the other.
First of all, since any legal move changes labelings, a labeling can't be transformed into itself by a legal move. Therefore, puz(G) is a simple graph. Secondly, a connected component of puz(G)
consists of those labelings that can be transformed into one another by a sequence of legal moves. If puz(G) is proven to be connected then the puzzle on G can be solved from any starting
configuration. Otherwise, moving from one labeling to another one is bound to stay inside a connected component of puz(G). Following is a theorem from [Ref 4].
Let G be a simple nonseparable graph other than a polygon or the graph shown at right. Then puz(G) is connected unless G is bipartite, in which case puz(G) has exactly two components. In the latter
case, labelings f and g on G having unoccupied vertices at even (respectively, odd) distance in G are in the same component of puz(G) iff fg^-1 is an even (respectively, odd) permutation of V(G). For
the exceptional case of the graph on the right, puz(G) has exactly 6 components.
The proof gets quite technical but the formulation itself has a very intuitive appeal. The distance between two nodes is of course the length of the shortest walk from one node to another. So that,
except for the notion of permutation which I am going to define on the next page, we are all set up to interpret the result. But note right away that the theorem completely resolves the problem for
the 15 puzzle and its generalizations to the nxn board with n≠4. The board's graph is always bipartite. Hence puz(G) has two components. Swapping two adjacent counters makes the labelings
incompatible - belonging to different connected components of puz(G). The same applies to the Lucky 7 puzzle with n = 7, 11, 15, etc.
Swapping two counters means applying a transposition to a labeling. Moving the empty square is equivalent to applying a sequence of transpositions, the number of transpositions being equal to the
number of moves or to the distance between the starting and final locations of the empty square. The theorem then states that if two labelings have the same parity they belong to the same component
of puz(G) iff the distance between empty squares in the two labelings is even. If two labelings have different parities they belong to the same component of puz(G) iff the distance between empty
squares in the two labelings is odd.
1. E.R.Berlekamp, J.H.Conway, R.K.Guy, Winning Ways for Your Mathematical Plays, Academic Press, 1982.
2. W.E.Story, Note on the '15' puzzle, Amer. J. Math., 2 (1879), 399-404.
3. R.J.Wilson, Graphs And Their Uses, MAA, New Math Library, 1990.
4. R.M.Wilson, Graph Puzzles, Homotopy, and the Alternating Group, J. of Combinatorial Theory, Ser B 16, 86-96 (1974).
|Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
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Creating continuous percentile value for all observations in data
I'd like to calculate percentile of BMI of all observation (person) in data. Below is my attempt. I'm puzzled with resulting histogram on percentile (rank_BMI) . People with higher BMI percentile are
at higher risk of being obese et.c. However, how is it possible that all percentiles are uniformly distributed 😞 Correct and expected "percentile of BMI for age" is also shown in the image below
proc rank data=Mydata groups=100 out=ranked;
var BMI;
ranks rank_BMI;
proc univariate data=ranked noprint;
histogram rank_BMI/
label rank_BMI="pctl of BMI";
05-30-2018 02:24 PM
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filter Metrics Operator | Sumo Logic Docs
The functionality provided by the filter operator has been incorporated into the where operator. We recommend the use of where over filter, because filter will be deprecated in the future. For more
information, see where Metrics Operator
You can use the filter operator to limit the results returned by a metric query. There are several ways you can restrict results. You can apply an aggregation function, such as avg, to a time series.
You can also filter based on how many times the value of individual data points meet a value condition over a particular duration.
There are two supported syntaxes for the filter operator.
filter [REDUCER BOOLEAN EXPRESSION]
filter _value [VALUE BOOLEAN EXPRESSION] [all | atleast n] [first | any | last] [duration]
Syntax 1​
The first variant filters based on a function (usually an aggregation function) applied to the time series.
filter [REDUCER BOOLEAN EXPRESSION]
[REDUCER BOOLEAN EXPRESSION] is an expression that takes all the values of a given time series, uses a function to reduce them to a single value, and evaluates that value.Â
The supported functions are:
• avg. Returns the average of the time series.
• min. Returns the minimum value in the time series.
• max. Returns the maximum value in the time series.
• sum. Returns the sum of the values in the time series.
• count. Returns the count of data points in the time series.
• pct(n). Returns the nth percentile of the values in the time series.
• latest. Returns the last data point in the time series.
Syntax 1 examples​
Example 1
Return the time series in which the average value of the CPU_User metric is greater than 95:
metric=CPU_User | filter avg > 95`
Example 2
Return the time series in which the latest value of the CPU_User metric is greater than 50:
metric=CPU_User | filter latest > 50
Syntax 2​
The second variant filters based on how many times the values of individual data points of a time series meet a value condition over a particular duration.
filter _value [VALUE BOOLEAN EXPRESSION] [all | atleast n] [first | any | last] [duration]
• [VALUE BOOLEAN EXPRESSION] is a value expression that operates on individual data points of a time series. For example, > 3
• Use all to specify that all data points within the duration must meet the value condition, or atleast n, where n is a count, to specify how many data points must meet the value condition.
• Use first, any, or last to specify what part of the time range that duration applies to: the start of the time range, any part of the time range, or the end of the time range.
• Use duration to specify the length of time to consider in the query in minutes (m), hours (h), or days (d). For example, 5m, 6h, or 1d.
Syntax 2 examples​
Example 1
Return only the time series in which all data points during the last 5 minutes of the query time range have a value greater than 3.Â
There must be a least one data point in the last 5 minutes of the time range for this to be valid.
metric=CPU_User | filter _value > 3 all last 5m
Example 2
Return only the time series that have at least 1 data point greater than 3 for the last 5 minutes of the query time range.Â
metric=CPU_User | filter _value > 3 atleast 1 last 5m
Example 3
Return only the time series that have only values greater than 3 for any consecutive 5 minutes of the time range.
metric=CPU_User | filter _value > 3 all any 5m
Example 4
Return only the time series that have only values greater than 3 for the first 5 minutes of the query time range.Â
There must be a least one data point in the first 5 minutes of the time range for this to be valid.
metric=CPU_User | filter _value > 3 all first 5m
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Nagel & Newman’s
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might
affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions,
conundrums, and contentious issues in modern philosophy and mathematics.
Site Mission
• To promulgate the understanding that the validity of a logical argument is not necessarily independent of the way in which language is used by that argument.
• To rid the fields of philosophy and mathematics of arcane and irrational notions which have resulted in numerous contradictions.
• To ensure that future generations of young people will not be put off the study of mathematics and philosophy by the mystical and illogical notions that are currently widespread in those
Please see the menu for numerous articles of interest. Please leave a comment or send an email if you are interested in the material on this site.
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Nagel & Newman’s Book: Gödel’s Proof
Page last updated 16 Feb 2024
This page discusses Nagel & Newman’s book on Gödel’s incompleteness proof, entitled Gödel’s Proof. (Footnote: PDF E Nagel and J Newman: Gödel’s Proof. New York University Press, revised edition,
2001. ISBN: 0814758169.) To follow this page, you should preferably have a copy of the book at hand. For convenience, we will refer to Nagel-Newman as though they are a singular person.
It should be noted at this point that Nagel-Newman’s book is an informal exposition. It does not claim to be a proof, rather it is an overview of the main thrust of Gödel’s argument. Most of the book
is in the form of a general discussion, rather than a detailed logical argument.
This page was written as a response to the many people who have asked where there is a flaw in Nagel-Newman’s book. One response to that might be that Nagel-Newman’s account is not a detailed logical
argument, and hence cannot be said to be a proof at all. However, rather than use that as a convenient cop-out, I have tried to give an explanation of the flawed argument in Nagel-Newman’s account.
It is also worth pointing out that Nagel-Newman’s erroneous proof has spawned a plethora of copies, claiming to be proofs of incompleteness while glossing over Nagel-Newman’s fudge in exactly the
same way as Nagel-Newman does, see for example: How Gödel’s Proof Works.
Note: Before dealing with Nagel-Newman’s overview, it might be pointed out that while most people take a ‘formula’ to be any mathematical expression, Nagel-Newman sometimes considers that a formula
of the formal system can only be a symbol combination of the formal system that states a proposition, but at other times he uses the term more freely. It is worth bearing in mind the two different
connotations that Nagel-Newman attaches to the term.
Number-theoretic expressions
Note: the term ‘number-theoretic’ is used below. Many people are put off by this term, which sounds more complex than it actually is - it simply indicates that a number-theoretic expression is an
expression only about numbers, not about any other things. See also number-theoretic.
Nagel-Newman’s proof of incompleteness
It might be noted that Douglas Hofstadter’s book, ‘Gödel, Escher, Bach’ (Footnote: Douglas Hofstadter. Gödel, Escher, Bach. Basic Books, 1999. ISBN‑13: 978‑0465026562 Gödel, Escher, Bach -
Hofstadter: Details.) gives a similar incompleteness proof to that in Nagel & Newman’s book, although Nagel & Newman can claim priority, as their book was published prior to Hofstadter’s. The proof
in Hofstadter’s book is dealt with in detail on another webpage: Gödel, Escher, Bach. The argument presented on that page could equally well be applied to Nagel & Newman’s proof, and similarly, the
argument below could be applied to Hofstadter’s proof. They are simply different ways of demonstrating the confusion of language that is inherent in the proofs, which is a common feature of many
incompleteness proofs.
A language that makes statements about another language is called a meta-language, while a language that a meta-language makes statements about is called an object or sub-language. In a discussion of
a proof that involves a language making statements about another language, you might expect that the distinctions between any languages that are involved would be made absolutely clear. But
Nagel-Newman, as with Gödel’s own proof, manages to confuse the language systems involved. There is a failure to ensure a clear delineation of the different systems in the proof, and in
Nagel-Newman’s account, in common with Gödel’s, there is a consequent confusion of language systems. (Footnote: This aspect of the number-theoretic system being an object language to the
meta-language is dealt with in more detail in the paper on Gödel’s proof, see The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.) The part of the book where this confusion becomes
most evident is in the section VII, B ‘The arithmetization of meta-mathematics’, where Nagel-Newman introduces a function called sub(x, 17, x).
Nagel-Newman’s ‘Substitution’ Function
Before reading the rest of this section, the reader might like to first read the webpage Gödel’s Substitution Function which describes the substitution function that Gödel uses in his proof.
The confusion of language in Nagel-Newman’s account can be seen to center around the same functions as in Gödel’s proof, and the correspondences defined by the Gödel numbering system. The Gödel
numbering system is a function takes any expression of the formal language and outputs a number that corresponds uniquely to that expression, for example if we call the function GN, then GN[w] gives
a unique number for w as some expression of the formal system.
Although Nagel-Newman’s book follows most of Gödel’s proof, the last part involving the use of the substitution function is somewhat different to Gödel’s proof. Nagel-Newman refers to a function sub(
x, 17, x), with only one free variable, x, but does not give a precise definition of the function. He also says that there is a function Sub(x, 17, x) in the formal system that functions in exactly
the same way as the function sub(x, 17, x).
Nagel-Newman states that the function gives: the Gödel number of the formula obtained by taking the formula with Gödel number x and, wherever there are occurrences of the variable ‘y’ in that
formula, replacing them by the numeral for x (where he means numeral to mean the number x in the format of the formal system).
Flawed Assumptions
So, what Nagel-Newman is referring to by his use of Sub(x, 17, x) is the combination of two functions. And in exactly the same way as Gödel does in his proof, Nagel-Newman simply assumes that the
function Sub(x, 17, x) contains within itself a purely number-theoretic function that exactly replicates the Gödel numbering function, and that this function, even though it is within a purely
number-theoretic system, is able to assert that a number is the Godel number of an expression of the formal system - i.e, Nagel-Newman assumes that there is a purely number-theoretic function that we
can call Z(x), and that Z(x) = GN[x], provided that x is a number, and which is contained within his Sub(x, 17, x) function.
But this is absurd, since the two functions Z(x) and GN[x] belong to two different language systems. The function GN[x] is defined as being a function that is defined outside of the formal system,
whereas the function Z(x) id defined within the formal system itself.
Previously, I have gone into detail regarding this absurdity on different pages, but since I have already demonstrated it in detail elsewhere, it seems more sensible to direct the reader to that page
rather than trying to maintain several different pages saying essentially the same thing.
And since Nagel-Newman is simply following Gödel’s account (while leaving out many details), the reader is therefore directed to Gödel’s Substitution Function which gives a detailed explanation of
the absurdity of the assumption of equivalence of the two functions Z and GN.
With regard to logical analysis, it is a somewhat unfortunate consequence of human evolution that the human mind almost invariably attempts to attach a meaning to an expression, rather than subject
it to precise logical analysis. Until the last few thousand years, all expressions were spoken, and so the human mind evolved to assume that all expressions are intended to convey a meaning, rather
than logically analyze them. And so we have evolved to feel the need to attach a meaning to all expressions, even though there may be no logical justification for such a meaning. Similarly, people
almost invariably attempt to attach a meaning to Nagel-Newman’s statements that have no logical justification.
Nagel-Newman’s assumptions are a demonstration of a nonsensical confusion of language systems because of a misapplication of the encoding correspondence given by the Gödel numbering system, a
confusion which is made possible by the use of some symbols that are the same for the formal system and for the system of number-theoretic relations.
Finally, you might be interested that Nagel-Newman discusses Richard’s paradox in detail, and points out the linguistic confusion that results in the paradox. He also observes that:
“The importance… of recognizing the distinction between mathematics and meta-mathematics cannot be overemphasized. Failure to respect it has produced paradoxes and confusion.”
Indeed - it is rather ironic that Nagel and Newman’s explanation of Gödel’s proof is itself an instance of the failure to observe that distinction, as is Gödel’s original proof.
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not
include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Derogatory comments.
Long-winded comments.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical
terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous user names for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously.
After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Based on HashOver Comment System by Jacob Barkdull
The Lighter Side
When a statistician passed through the airport security checkpoint, officials discovered a bomb in his bag. He explained:
“According to statistics, the probability of a bomb being on an airplane is 1/1000. Consequently, the chance that there are two bombs on one plane is 1/1,000,000, so I feel much safer if I bring one
James R Meyer
Recently added pages
A new section on set theory
How to setup Dark mode for a web-site
I have set up this website to allow a user to switch to a dark mode, but which also allows the user to revert back to the browser/system setting. The details of how to implement this on a website are
given at How to setup Dark mode on a web-site.
Decreasing intervals, limits, infinity and Lebesgue measure
The page Understanding sets of decreasing intervals explains why certain definitions of sets of decreasing intervals are inherently contradictory unless limiting conditions are included, and the page
Understanding Limits and Infinity explains how the correct application of limiting conditions can eliminate such contradictions. The paper PDF On Smith-Volterra-Cantor sets and their measure has
additional material which gives a more formal version.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
strict rigor.
For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
strict rigor.
For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material.
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A Fast Recursive Algorithm to Calculate the Reliability of a Communication Network for IEEE Transactions on Communications
IEEE Transactions on Communications
A Fast Recursive Algorithm to Calculate the Reliability of a Communication Network
View publication
This paper describes a recursive algorithm to calculate the probability that all paths between two nodes in a given network are interrupted. It is assumed that all links are undirected and that links
and nodes fail with given probabilities. These failures are assumed to be statistically independent. The probability that two nodes are disconnected is expressed in terms of the probability that
pairs of nodes are disconnected in subnetworks smaller than the original one. The advantage of the algorithm given in this paper compared to other known procedures results from the fact that, in most
cases, these subnetworks can be considerably simplified. These simplifications lead to considerable savings in computing time. Copyright © 1972 by The Institute of Electrical and Electronics
Engineers, Inc.
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Math is in the Air
Introduction and the 7 dwarfs It was October when one of my professors entered in the lesson room and said: “Once upon a time, Snow White wished to prepare cookies for the seven dwarfs.
Obviously…with some conditions. She wanted to prepare the least number of cookies such that dividing them into 2 dwarfs there would […]
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CHSPE Math Practice Workbook
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Virtual population analysis (VPA) — VPA
Virtual population analysis (VPA)
A VPA model that back-calculates abundance-at-age assuming that the catch-at-age is known without error and tuned to an index. The population dynamics equations are primarily drawn from VPA-2BOX
(Porch 2018). MSY reference points and per-recruit quantities are then calculated from the VPA output.
x = 1,
AddInd = "B",
expanded = FALSE,
SR = c("BH", "Ricker"),
vulnerability = c("logistic", "dome", "free"),
start = list(),
fix_h = TRUE,
fix_Fratio = TRUE,
fix_Fterm = FALSE,
LWT = NULL,
shrinkage = list(),
n_itF = 5L,
min_age = "auto",
max_age = "auto",
refpt = list(),
silent = TRUE,
opt_hess = FALSE,
n_restart = ifelse(opt_hess, 0, 1),
control = list(iter.max = 2e+05, eval.max = 4e+05),
A position in the Data object (by default, equal to one for assessments).
An object of class Data
A vector of integers or character strings indicating the indices to be used in the model. Integers assign the index to the corresponding index in Data@AddInd, "B" (or 0) represents total biomass
in Data@Ind, "VB" represents vulnerable biomass in Data@VInd, and "SSB" represents spawning stock biomass in Data@SpInd.
Whether the catch at age in Data has been expanded. If FALSE, then the catch in weight should be provided in Data@Cat so that the function can calculate annual expansion factors.
Stock-recruit function (either "BH" for Beverton-Holt or "Ricker") for calculating MSY reference points.
Whether the terminal year vulnerability is "logistic" or "dome" (double-normal). If "free", independent F's are calculated in the terminal year (subject to the assumed ratio of F of the
plus-group to the previous age class). See details for parameterization.
Optional list of starting values. Entries can be expressions that are evaluated in the function. See details.
Logical, whether to fix steepness to value in Data@steep. This only affects calculation of MSY and unfished reference points.
Logical, whether the ratio of F of the plus-group to the previous age class is fixed in the model.
Logical, whether to fix the value of the terminal F.
A vector of likelihood weights for each survey.
A named list of up to length 2 to constrain parameters:
□ vul - a length two vector that constrains the vulnerability-at-age in the most recent years. The first number is the number of years in which vulnerability will be constrained (as a random
walk in log space), the second number is the standard deviation of the random walk. The default
□ R - a length two vector that constrains the recruitment estimates in the most recent years. The first number is the number of years in which recruitment will be constrained (as a random walk
in log space), the second number is the standard deviation of the random walk.
The number of iterations for solving F in the model (via Newton's method).
An integer to specify the smallest age class in the VPA. By default, the youngest age with non-zero CAA in the terminal year is used.
An integer to specify the oldest age class in the VPA. By default, the oldest age with non-zero CAA for all years is used.
A named list of how many years to average parameters for calculating reference points, yield per recruit, and spawning potential ratio:
□ vul An integer for the number of most recent years to average the vulnerability schedule (default is 3).
□ R A length two for the quantile used to calculate recruitment in the year following the terminal year and the number of years from which that quantile is used, i.e., c(0.5, 5) is the default
that calculates median recruitment from the most recent 5 years of the model.
Logical, passed to TMB::MakeADFun(), whether TMB will print trace information during optimization. Used for diagnostics for model convergence.
Logical, whether the hessian function will be passed to stats::nlminb() during optimization (this generally reduces the number of iterations to convergence, but is memory and time intensive and
does not guarantee an increase in convergence rate). Ignored if integrate = TRUE.
The number of restarts (calls to stats::nlminb()) in the optimization procedure, so long as the model hasn't converged. The optimization continues from the parameters from the previous (re)start.
A named list of arguments for optimization to be passed to stats::nlminb().
Other arguments to be passed.
An object of class Assessment. The F vector is the apical fishing mortality experienced by any age class in a given year.
The VPA is initialized by estimating the terminal F-at-age. Parameter Fterm is the apical terminal F if a functional form for vulnerability is used in the terminal year, i.e., when vulnerability =
"logistic" or "free". If the terminal F-at-age are otherwise independent parameters, Fterm is the F for the reference age which is half the maximum age. Once terminal-year abundance is estimated, the
abundance in historical years can be back-calculated. The oldest age group is a plus-group, and requires an assumption regarding the ratio of F's between the plus-group and the next youngest age
class. The F-ratio can be fixed (default) or estimated.
For start (optional), a named list of starting values of estimates can be provided for:
• Fterm The terminal year fishing mortality. This is the apical F when vulnerability = "logistic" or "free".
• Fratio The ratio of F in the plus-group to the next youngest age. If not provided, a value of 1 is used.
• vul_par Vulnerability parameters in the terminal year. This will be of length 2 vector for "logistic" or length 4 for "dome", see SCA for further documentation on parameterization. For option
"free", this will be a vector of length A-2 where A is the number of age classes in the model. To estimate parameters, vulnerability is initially set to one at half the max age (and subsequently
re-calculated relative to the maximum F experienced in that year). Vulnerability in the plus-group is also constrained by the Fratio.
MSY and depletion reference points are calculated by fitting the stock recruit relationship to the recruitment and SSB estimates. Per-recruit quantities are also calculated, which may be used in
harvest control rules.
Additional considerations
The VPA tends to be finicky to implement straight out of the box. For example, zeros in plusgroup age in the catch-at-age model will crash the model, as well as if the catch-at-age values are close
to zero. The model sets F-at-age to 1e-4 if any catch-at-age value < 1e-4.
It is recommended to do some preliminary fits with the VPA before running simulations en masse. See example below.
Shrinkage, penalty functions that stabilize model estimates of recruitment and selectivity year-over-year near the end of the time series, alters the behavior of the model. This is something to
tinker with in your initial model fits, and worth evaluating in closed-loop simulation.
Online Documentation
Model description and equations are available on the openMSE website.
# \donttest{
OM <- MSEtool::testOM
# Simulate logistic normal age comps with CV = 0.1
# (set CAA_ESS < 1, which is interpreted as a CV)
OM@CAA_ESS <- c(0.1, 0.1)
Hist <- MSEtool::Simulate(OM, silent = TRUE)
# VPA max age is 15 (Hist@Data@MaxAge)
m <- VPA(x = 2, Data = Hist@Data, vulnerability = "dome")
# Use age-9 as the VPA max age instead
m9 <- VPA(x = 2, Data = Hist@Data, vulnerability = "dome", max_age = 9)
compare_models(m, m9)
# }
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8- How to make an analysis of steel beams? Solved problems.
Last Updated on September 7, 2024 by Maged kamel
How do we analyze steel beams? Solved problems.
The difference between analysis and design of steel beam.
The difference between analysis and design problems is that for the analysis, the section is given, and a check of stress is needed, while for the design, the loads are provided, and it is required
to find the section.
Analysis of steel beam at zone-1.
The solved example is 5-3 from Prof. Segui’s handbook. The beam shown in Figure 5.11 is W16x31 of A992 steel, for which 16″is the overall height or nominal depth, while 31 is the weight in lbs per
linear ft of A992 steel, where Fy=50 ksi. It supports a reinforced concrete floor slab that provides continuous lateral support of the compression flange.ย
Here, the compression flange is mentioned as being supported continuously, which means we are dealing with a plastic range or zone -1.
Our lambda ฮป between 0 and ฮปp, and Mn=Mp=Fy*Zx.
The beam shown in Figure 5.11 is W16x31 of A992 steel. W16x31, 16″ is the overall height, while 31 is the weight in lbs per linear ft of A992 steel, where Fy=50 ksi.
How do we estimate the Ultimate load and Ultimate Moment?
The weight of the beam, which equals 31 lb/ft, is added to the given uniform dead load; The ultimate load equals 1.2*Wd+1.6*wl=1.2*(0.45+0.031)+1.6*(0.55)=1.4572 k/Ft. We estimate the ultimate moment
as equal to Wu*L^2/8.The span length equals 30 feet. The final value of the Ultimate moment equals 164 Ft.kips.Please refer to the Next slide image.
How to check the compactness of the flange and the web of a beam?
We use Table 1 as the first step in analyzing the steel beam. For W16x31 A=9.13 inch2, the overall depth=15.90 inch, the web thickness =0.275 inches, bf =5.53 inch.
The thickness of the t flange is 7/16 inches.
The controlling factor is 3.76*sqrt(E/Fy) for the web=90.55. The ฮปF, which is 6.28 for the flange, is<ฮปp, which is 9.15, then the section will be in the compact zone for the flange, while ฮปw,
which is 51.60, is also <90.55. The section will be in the compact zone for the web, the first zone.
The first step of the steel beam analysis is to get the nominal moment value Mn=Mp=Fy*Zx. From table 1-1, we can get the Zx value, Zx=54.0 inch3.
To get the Mn=Mp=50*54=2700 inch kips. To convert into ft. kips, we will divide by 12.
LRFD Design-Beam Moment capacity.
Mp=225.0 kips ft. For the LRFD, Our phi is ฮฆb=0.90, then ฮฆb*Mn=0.90*225.0=202.50ย ft. Kips.
The Ultimate moment, acting on the section, should be <= ฮฆb*Mn.
As estimated earlier, the section can carry 202.50 Ft.kips. This section is adequate for analyzing the steel beam for the LRFD design. This is the end of the analysis of the steel beam.
Check the ฮฆb*Mn for the steel beam section is bigger than the M-ultimate.
ASD design.
While for the ASD Design, Wd= uniform load+own weight=450+31, Wd=481 lb/ft, and WL=550 lb/ft, adding together for Wt, the Mt=(480+550)*30^2/8/1000=116.0 ft. kips.
This is the total Moment. For Mn =225.0 ft. kips, divide by the omega ฮฉb, 1.67. M all =225/1.67=1350.0 ft.kips. The section can carry 135.0 ft. kips and is only subjected to 116.0 ft. kips; the
section is safe.
This section is adequate for analyzing steel beams for the ASD design. The idea of the example is that the section of the beam carries a slab with studs to provide the continuous bracing, then a
section was selected Bf/2tf, lambda ฮปF was< lambda ฮปp flange also, ฮปw is<ฮปp for the web.
A second problem is to check the compactness of a given section.
This is an example from Lindeburg. Establish whether a W21x55ย beam of A992 steel is compact or non-compact. An analysis of steel beams is required.
There are four options. First is the A992 steel,ย with Fy=50 ksi.
For W21x55, the overall depth is 20.80″, which is highlighted. The web has a Thickness of 3/8″ or 0.375″ t-web. I draw the section, Bf=8.22″, and its thickness =0.522″.
To estimate the controlling lambda, first, we need to find the value of Tf/2Tf for flange, which is 8.22/2*0.22=7.87, ฮปp for flange =64.70/sqrt(50)=9.15. Then, ฮปf<ฮปp for the flange.
For the second part, we will correct the web thickness as 0.375″, h, as estimate =(20.80-2*0.522)/0.375, where tweb=0.375. If we divide 20.80-2*0.522 by the calculator, we get hw, hw=19.756″/0.375 =
Check against ฮปp, which isย 640/sqrt(50)=90.55, then 52.55 is <90.55; since bf/2tf< ฮปp f and hw/tw is< ฮปp t, then option A is correct. This is the end of the analysis of the steel beam. Option A
is the proper selection.
For bending members, please refer to this link from Profย T. Bart Quimby, P.E., Ph.D. F.ASCE site.
For the next post, A Solved problem-4-7-1, how to design a steel beam.
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Microsoft® JScript Language Reference
- Operator Version 1
Used to find the difference between two numbers or to indicate the negative value of a numeric expression.
Syntax 1
result = number1 - number2
Syntax 2
The - operator syntax has these parts:
│ Part │ Description │
│ result │ Any numeric variable. │
│ number │ Any numeric expression. │
│ number1 │ Any numeric expression. │
│ number2 │ Any numeric expression. │
In Syntax 1, the - operator is the arithmetic subtraction operator used to find the difference between two numbers. In Syntax 2, the - operator is used as the unary negation operator to indicate
the negative value of an expression.
For information on when a run-time error is generated by Syntax 1, see the Operator Behavior table.
For Syntax 2, as for all unary operators, expressions are evaluated as follows:
□ If applied to undefined or null expressions, a run-time error is raised.
□ Objects are converted to strings.
□ Strings are converted to numbers if possible. If not, a run-time error is raised.
□ Boolean values are treated as numbers (0 if false, 1 if true).
The operator is applied to the resulting number. In Syntax 2, if the resulting number is nonzero, result is equal to the resulting number with its sign reversed. If the resulting number is zero,
result is zero.
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January 2014 – Kate S. Owens
I spent some time over the last several days trying to track down documentation about SBG/SBL. I wanted to find something to pass along to my students to address some of their questions or concerns,
like, “What’s this SBG thing?” or “How will this work in our course?” or “How is this going to be beneficial?”
Thankfully, Joshua Bowman came to the rescue and sent me something he gives out to his students. It addressed some of his students’ frequently asked questions and it was a great launchpad to write my
own. I’ll post it below. I [DEL:kept:DEL] stole his format and questions, but re-wrote (most of?) the answers as the apply to my own course.
Introduction to Standards-Based Grading
How is standards-based grading different from traditional grading?
You are probably accustomed to the following system: You do an assignment (like for homework, a quiz, or a test) and give it to your instructor to grade. After grading, it is returned to you with a
score like “14/15” or “93%”. In our course, I won’t keep track of how you do on particular assignments; instead, I will keep track of how well you master specific mathematical tasks or concepts that
are called standards. Once I see your work, my goal is to give you meaningful feedback: I want my feedback to tell you what you have mastered, what you should practice, and how what you have mastered
relates to the goals of our course.
There are three major advantages to this system:
• First, it rewards mastery instead of a “hunt for partial-credit” strategy. On an assignment with five problems, I believe it is better to do three problems extremely well (and leave two problems
blank) than to just write stuff down on every page hoping you’ll earn enough points.
• Second, I hope that it will allow you to see how to improve your knowledge of our course material. This system will allow us to track what topics you understand well, and also what topics you
should spend more time working on. This way, if you seek additional help, you will know exactly what you need help with! Since your grade on a standard is not a fixed number — it changes over
time — it is always advantageous to go back and fill in any gaps in your knowledge.
• Third, it allows us to be clear about what the expectations of the course are (namely, demonstrating an understanding of topics in Calculus II) and how well you are meeting (or exceeding!) those
How will I know how well I did on a test?
Each assignment will probably look similar to those you have seen in prior courses. When I return them to you, you will be provided with a rubric. The rubric will give you two kinds of information.
First, it will outline what standards correspond to each problem you solved. Second, it will outline the level of mastery you demonstrated on that problem, using a scale of 0-4. Apart from the
rubric, my hope is to offer additional feedback on your solutions that will help you toward your goal of continued mastery.
How do I know which standards will be tested?
On each quiz, you can expect to see material we covered in the previous week. However, as you know, mathematics tends to build on itself. So although maybe we didn’t talk about the Quotient Rule last
week, you will probably still have to know how to use it this week! Before each test, I will provide a list of all of the standards the test will cover. Since our course is cumulative, although a
particular test might focus on recent standards, you might encounter problems that require knowledge of previous standards from earlier in our semester — or even prior mathematics courses.
How often will each standard be assessed?
It will depend on the particular standard. Standards that appear early in our course will be assessed multiple times, since we will be using them (either implicitly or explicitly) to solve problems
later on. Toward the end of our course, you might only encounter a particular standard once or twice.
Why can my score on a standard go down?
It’s important that your score shows your current level of mastery. Your score on a standard may go down because you’ve forgotten some of the material, or you were unable to apply earlier techniques
in solving problems later on.
In addition, some of our standards are quite broad: For instance, one of them deals with “techniques of integration.” We will see many of these techniques in our course. So your score may go down if
you show mastery of the earlier techniques, but aren’t comfortable with techniques that show up later on.
How can I raise my score on a standard?
There are two ways to have a score on a standard raised.
First, you can wait for that standard to be re-assessed later on. For example, some standards assessed on quiz questions will be re-assessed on test problems. Especially early in the course, when
there will be many opportunities to reassess standards, this may be the easiest way to raise your scores.
Second, it will be possible to “retest” a particular standard by making an appointment to meet with me. At this meeting, you will demonstrate your understanding by trying new problems and then
answering questions I pose to you. You can make appointments to retest up to two standards each week. You choose which standards you would like to retest and when. You can retest any given standard
more than once, as long as you only retest up to two each week. Each “retest” will take 10-15 minutes. Please request an appointment for re-assessment at least one class day in advance; this will
allow me to prepare materials for you. You can request an appointment simply by e-mailing me and letting me know which standard you have chosen.
How many times can I ask for a standard to be reassessed?
You can ask for any standard to be reassessed as many times as you want, subject to the limitation that you may only retest two standards each week. If you require multiple attempts on a particular
standard, I might ask you to work on some additional problems first (potentially with my help) so we can clear up any knowledge gap more quickly.
What about the final exam?
Our final exam will be cumulative and will have problems reflecting standards we have encountered throughout the course. Not every standard will be directly assessed on the final exam (after all, we
don’t want to make it too lengthy!). Also, by the nature of final exams, you cannot re-assess any standard after the final exam. Your course score on each standard will be decided as follows:
• If a standard does not appear on the final exam, your course score for that standard will be your score as of Reading Day. For Spring 2014, the date is Thursday, April 24th.
• If a standard does appear on the final exam, your course score for that standard will be the average of [your score as of Reading Day] and [your score for that standard on the final exam].
How will my final grade be computed from my scores?
Your midterm grade and course grade will be the usual sorts of letter grades you are accustomed to. Here is how I will convert your mastery of the course standards into letter grades:
• In order to guarantee a grade of A, you should attain 4s (or 5s) on 85% of course standards and have no scores below 3.
• In order to guarantee a grade of B, you should attain 3s on 85% of course standards and have no scores below 2.
• In order to guarantee a grade of C, you should attain 2s on at least 85% of course standards.
Plus and minus grades will be given based on how closely your performance is to a full letter grade. (For example, if you earn 3s on only 80% of course standards, and 2s on the other 20% of course
standards, a grade of “B-” may be more appropriate than a grade of “B.”)
If I don’t like this method of grading, can I tell you about it?
Please! This is my first time using standards-based grading, and there are bound to be hiccups. However, I truly believe it will provide more helpful feedback and give you a better chance to prove
your mastery of the material, so I ask that you at least give it a try, even if it seems strange at first.
If I have questions about how I’m doing in the class, can I ask you about it?
Absolutely! One drawback of this system of assessment is that you may have questions about your performance in the class. If you have questions or concerns about this, feel free to come talk with me
and I will try my best to give you an accurate picture of your progress with our course material.
An Adventure in Standards Based Calculus
Today was the first day of our new semester. This spring, I’ll be teaching two sections of “Calculus I” and one section of “Calculus II.” I feel like “Calculus I” is basically on autopilot; I’ve
taught the class every semester for the last couple years and so I’m very comfortable with the course content. But this will be my first time teaching “Calculus II” in many years. (I think the last
time I taught it was 2006 or so, at the University of South Carolina, using an entirely different textbook.) I’ve decided that I want to try something different & I am embarking on my first attempt
at Standards Based Grading (SBG) — or as someone suggested today on twitter, maybe Standards Based Learning (SBL) is more appropriate?
Why Am I Doing This?
For the last few years, I’ve noticed a few things about traditional grading (TG) that I did not like. One thing that has bothered me is that a student can go the entire semester without ever solving
a problem 100% correctly, yet still do very well in the course. For example, it is entirely possible to earn a “B+” grade, by performing pretty well on everything, but never really and truly
mastering a single topic or problem type. I hope that Standards Based Grading helps me motivate my students to really try to master specific sorts of problems, rather than try to bounce around,
hoping they can earn enough “partial credit” points to propel them to success. Really, I want to reward a student who gets four problems absolutely correct (and skips two problems) more than a
student who just writes jumbled stuff down on every page. I think SBG will allow me to do this.
Another (related) thing that has bothered me: The point of calculus class is not to earn as many points as possible, doing the least effort possible. I will admit that I have used a TG scheme for
years and years; I have no idea how many college-level courses I’ve taught. And I am pretty sure that I can look at a calculus quiz question, assign it a score between 0 and 10, and accurately give a
number close to what my colleagues would give for that same problem. We might all agree, “Okay, this solution is worth 7 out of 10 points for these reasons.” But I think this gives the students the
idea that the reason they should study is to earn points on the quiz — after all, 9 points is better than 7 points! Instead, I think the reason they should study is to understand the material deeper
than they presently do now, and I think by assigning X points out of 100 sends them the wrong message.
Something that has really bothered me recently is that when a student is struggling with the course, I am never entirely sure what to tell them. I look up their grades in my gradebook; I see that
they have an average of 62%; and then I try to give them advice. But what advice should I give? The 62% in my gradebook does not tell me very much: I do not know if this student is struggling because
they need more practice in trigonometry. Or maybe they were doing very well, but bombed our last test because they got some bad news the night before. Or maybe they got L’Hopital’s Rule confused with
the Quotient Rule. I want to be able to tell a student exactly what they can do to improve their understanding. By tracking each student’s mastery of particular standards, if a student comes to my
office for extra help, I can tell that student, “Okay, it looks like you need extra help with [insert specific topic].”
Lastly, I would like to give students more low-stakes feedback about their understanding: That is, feedback without the worry that it will negatively affect their grade in the class. I will be giving
a weekly quiz, and I will grade it, offer feedback, and return it to my students; then (eventually) their score on that standard can be replaced with a newer [hopefully better!] score. I will
constantly replace their previous score on a standard with their current score on a standard. This way, if they are really struggling with (say) Taylor polynomials, I can communicate this to them
early, they can seek extra help and resources, and then they can be re-assessed without penalty for their original lack of understanding.
What Worries Me?
I have lots of different things worrying me about this system! For example, since this is my first time teaching Calculus II in many years, I don’t know all the “common pitfalls” that my students
will encounter, so I don’t feel like I’m going to see them coming until they’re already here. Also, I am worried that students will struggle to understand this method of assessment & won’t really
“get it” about how they are doing in the course — or won’t take the opportunity to re-assess when they need it. Lastly, despite reading online that “before a course begins, start by making a list of
what you want them to master (a.k.a, the standards)” I was unable to do this. I have the first half (or so), but I don’t know how good they are. Am I being too vague? Am I being too specific? Do I
have too many? Too few? How difficult will they be to assess?
Some Resources
In my own course planning, here are links to resources I found helpful:
Wish me luck!
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Calculate the nth root of a number in JavaScript - 30 seconds of code
Calculate the nth root of a number in JavaScript
The nth root of a number x is a value that, when multiplied by itself n times, gives x. The nth root can also be expressed as a power of x, where x ^ (1/n) is equal to the nth root of x.
Given this, we can use Math.pow() to calculate the nth root of a given number. Simply pass it the number x and a power of 1 / n, and you'll get the nth root of x.
const nthRoot = (x, n) => Math.pow(x, 1 / n);
nthRoot(32, 5); // 2
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CourseNana | CS465/CS565: Introduction to Artificial Intelligence - Project 1: Search
Project 1: Search
All those colored walls,
Mazes give Pacman the blues,
So teach him to search.
In this project, your Pacman agent will find paths through his maze world, both to reach a particular location and to collect food efficiently. You will build general search algorithms and apply them
to Pacman scenarios.
As in Project 0, this project includes an autograder for you to grade your answers on your machine. This can be run with the command:
See the autograder tutorial in Project 0 for more information about using the autograder.
The code for this project consists of several Python files, some of which you will need to read and understand in order to complete the assignment, and some of which you can ignore. You can download
all the code and supporting files as a zip archive.
Files you'll edit:
search.py Where all of your search algorithms will reside.
searchAgents.py Where all of your search-based agents will reside.
Files you might want to look at:
pacman.py The main file that runs Pacman games. This file describes a Pacman GameState type, which you use in this project.
game.py The logic behind how the Pacman world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid.
util.py Useful data structures for implementing search algorithms.
Supporting files you can ignore:
graphicsDisplay.py Graphics for Pacman
graphicsUtils.py Support for Pacman graphics
textDisplay.py ASCII graphics for Pacman
ghostAgents.py Agents to control ghosts
keyboardAgents.py Keyboard interfaces to control Pacman
layout.py Code for reading layout files and storing their contents
autograder.py Project autograder
test_cases/ Directory containing the test cases for each question
searchTestClasses.py Project 1 specific autograding test classes
Files to Edit and Submit: You will fill in portions of search.py and searchAgents.py during the assignment. Once you have completed the assignment, you will submit a token generated by
submission_autograder.py. Please do not change the other files in this distribution or submit any of our original files other than these files.
Evaluation: Your code will be autograded for technical correctness. Please do not change the names of any provided functions or classes within the code, or you will wreak havoc on the autograder.
However, the correctness of your implementation – not the autograder’s judgements – will be the final judge of your score. If necessary, we will review and grade assignments individually to ensure
that you receive due credit for your work.
Academic Dishonesty: We will be checking your code against other submissions in the class for logical redundancy. If you copy someone else’s code and submit it with minor changes, we will know. These
cheat detectors are quite hard to fool, so please don’t try. We trust you all to submit your own work only; please don’t let us down. If you do, we will pursue the strongest consequences available to
Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for help. Office hours, section, and the discussion forum are there for your support; please use
them. If you can’t make our office hours, let us know and we will schedule more. We want these projects to be rewarding and instructional, not frustrating and demoralizing. But, we don’t know when or
how to help unless you ask.
Discussion: Please be careful not to post spoilers.
Welcome to Pacman
After downloading the code (search.zip), unzipping it, and changing to the directory, you should be able to play a game of Pacman by typing the following at the command line:
Pacman lives in a shiny blue world of twisting corridors and tasty round treats. Navigating this world efficiently will be Pacman’s first step in mastering his domain.
The simplest agent in searchAgents.py is called the GoWestAgent, which always goes West (a trivial reflex agent). This agent can occasionally win:
python pacman.py --layout testMaze --pacman GoWestAgent
But, things get ugly for this agent when turning is required:
python pacman.py --layout tinyMaze --pacman GoWestAgent
Also, all of the commands that appear in this project also appear in commands.txt, for easy copying and pasting. In UNIX/Mac OS X, you can even run all these commands in order with bash commands.txt.
New Syntax
You may not have seen this syntax before:
def my_function(a: int, b: Tuple[int, int], c: List[List], d: Any, e: float=1.0):
This is annotating the type of the arguments that Python should expect for this function. In the example below, a should be an int -- integer, b should be a tuple of 2 ints, c should be a List of
Lists of anything -- therefore a 2D array of anything, d is essentially the same as not annotated and can by anything, and e should be a float. e is also set to 1.0 if nothing is passed in for it,
my_function(1, (2, 3), [['a', 'b'], [None, my_class], [[]]], ('h', 1))
The above call fits the type annotations, and doesn't pass anything in for e. Type annotations are meant to be an adddition to the docstrings to help you know what the functions are working with.
Python itself doesn't enforce these. When writing your own functions, it is up to you if you want to annotate your types; they may be helpful to keep organized or not something you want to spend time
Question 1 (3 points): Finding a Fixed Food Dot using Depth First Search
In searchAgents.py, you’ll find a fully implemented SearchAgent, which plans out a path through Pacman’s world and then executes that path step-by-step. The search algorithms for formulating a plan
are not implemented – that’s your job.
First, test that the SearchAgent is working correctly by running:
python pacman.py -l tinyMaze -p SearchAgent -a fn=tinyMazeSearch
The command above tells the SearchAgent to use tinyMazeSearch as its search algorithm, which is implemented in search.py. Pacman should navigate the maze successfully.
Now it’s time to write full-fledged generic search functions to help Pacman plan routes! Pseudocode for the search algorithms you’ll write can be found in the lecture slides. Remember that a search
node must contain not only a state but also the information necessary to reconstruct the path (plan) which gets to that state.Hint: Each algorithm is very similar. Algorithms for DFS, BFS, UCS, and
A* differ only in the details of how the fringe is managed. So, concentrate on getting DFS right and the rest should be relatively straightforward. Indeed, one possible implementation requires only a
single generic search method which is configured with an algorithm-specific queuing strategy. (Your implementation need not be of this form to receive full credit).
Implement the depth-first search (DFS) algorithm in the depthFirstSearch function in search.py. To make your algorithm complete, write the graph search version of DFS, which avoids expanding any
already visited states.
Your code should quickly find a solution for:
python pacman.py -l tinyMaze -p SearchAgent
python pacman.py -l mediumMaze -p SearchAgent
python pacman.py -l bigMaze -z .5 -p SearchAgent
The Pacman board will show an overlay of the states explored, and the order in which they were explored (brighter red means earlier exploration). Is the exploration order what you would have
expected? Does Pacman actually go to all the explored squares on his way to the goal?
Hint: If you use a Stack as your data structure, the solution found by your DFS algorithm for mediumMaze should have a length of 130 (provided you push successors onto the fringe in the order
provided by getSuccessors; you might get 246 if you push them in the reverse order). Is this a least cost solution? If not, think about what depth-first search is doing wrong.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q1
Question 2 (3 points): Breadth First Search
Implement the breadth-first search (BFS) algorithm in the breadthFirstSearch function in search.py. Again, write a graph search algorithm that avoids expanding any already visited states. Test your
code the same way you did for depth-first search.
python pacman.py -l mediumMaze -p SearchAgent -a fn=bfs
python pacman.py -l bigMaze -p SearchAgent -a fn=bfs -z .5
Does BFS find a least cost solution? If not, check your implementation.
Hint: If Pacman moves too slowly for you, try the option --frameTime 0.
Note: If you’ve written your search code generically, your code should work equally well for the eight-puzzle search problem without any changes.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q2
Question 3 (3 points): Varying the Cost Function
While BFS will find a fewest-actions path to the goal, we might want to find paths that are “best” in other senses. Consider mediumDottedMaze and mediumScaryMaze.
By changing the cost function, we can encourage Pacman to find different paths. For example, we can charge more for dangerous steps in ghost-ridden areas or less for steps in food-rich areas, and a
rational Pacman agent should adjust its behavior in response.
Implement the uniform-cost graph search algorithm in the uniformCostSearch fuctation. You should now observe successful behavior in all three of the following layouts, where the agents below are all
UCS agents that differ only in the cost function they use (the agents and cost functions are written for you):
python pacman.py -l mediumMaze -p SearchAgent -a fn=ucs
python pacman.py -l mediumDottedMaze -p StayEastSearchAgent
python pacman.py -l mediumScaryMaze -p StayWestSearchAgent
Note: You should get very low and very high path costs for the StayEastSearchAgent and StayWestSearchAgent respectively, due to their exponential cost functions (see searchAgents.py for details).
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q3
Question 4 (3 points): A* search
Implement A* graph search in the empty function aStarSearch in search.py. A* takes a heuristic function as an argument. Heuristics take two arguments: a state in the search problem (the main
argument), and the problem itself (for reference information). The nullHeuristic heuristic function in search.py is a trivial example.
You can test your A* implementation on the original problem of finding a path through a maze to a fixed position using the Manhattan distance heuristic (implemented already as manhattanHeuristic in
python pacman.py -l bigMaze -z .5 -p SearchAgent -a fn=astar,heuristic=manhattanHeuristic
You should see that A* finds the optimal solution slightly faster than uniform cost search (about 549 vs. 620 search nodes expanded in our implementation, but ties in priority may make your numbers
differ slightly). What happens on openMaze for the various search strategies?
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q4
Question 5 (3 points): Finding All the Corners
The real power of A* will only be apparent with a more challenging search problem. Now, it’s time to formulate a new problem and design a heuristic for it.
In corner mazes, there are four dots, one in each corner. Note that for some mazes like tinyCorners, the shortest path does not always go to the closest food first! Hint: the shortest path through
tinyCorners takes 28 steps.
Note: Make sure to complete Question 2 before working on Question 5, because Question 5 builds upon your answer for Question 2.
Implement the CornersProblem search problem in searchAgents.py. You will need to choose a state representation that encodes all the information necessary to detect whether all four corners have been
reached. Now, your search agent should solve:
python pacman.py -l tinyCorners -p SearchAgent -a fn=bfs,prob=CornersProblem
python pacman.py -l mediumCorners -p SearchAgent -a fn=bfs,prob=CornersProblem
To receive full credit, you need to define an abstract state representation that does not encode irrelevant information (like the position of ghosts, where extra food is, etc.). In particular, do not
use a Pacman GameState as a search state. Your code will be very, very slow if you do (and also wrong).
Hint 1: The only parts of the game state you need to reference in your implementation are the starting Pacman position and the location of the four corners.
Hint 2: When coding up getSuccessors, make sure to add children to your successors list with a cost of 1.
Our implementation of breadthFirstSearch expands just under 2000 search nodes on mediumCorners. However, heuristics (used with A* search) can reduce the amount of searching required.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q5
Question 6 (3 points): Corners Problem: Heuristic
Note: Make sure to complete Question 4 before working on Question 6, because Question 6 builds upon your answer for Question 4.
Implement a non-trivial, consistent heuristic for the CornersProblem in cornersHeuristic.
python pacman.py -l mediumCorners -p AStarCornersAgent -z 0.5
Note: AStarCornersAgent is a shortcut for
-p SearchAgent -a fn=aStarSearch,prob=CornersProblem,heuristic=cornersHeuristic
Admissibility vs. Consistency: Remember, heuristics are just functions that take search states and return numbers that estimate the cost to a nearest goal. More effective heuristics will return
values closer to the actual goal costs. To be admissible, the heuristic values must be lower bounds on the actual shortest path cost to the nearest goal (and non-negative). To be consistent, it must
additionally hold that if an action has cost c, then taking that action can only cause a drop in heuristic of at most c.
Remember that admissibility isn’t enough to guarantee correctness in graph search – you need the stronger condition of consistency. However, admissible heuristics are usually also consistent,
especially if they are derived from problem relaxations. Therefore it is usually easiest to start out by brainstorming admissible heuristics. Once you have an admissible heuristic that works well,
you can check whether it is indeed consistent, too. The only way to guarantee consistency is with a proof. However, inconsistency can often be detected by verifying that for each node you expand, its
successor nodes are equal or higher in in f-value. Moreover, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent. This stuff is tricky!
Non-Trivial Heuristics: The trivial heuristics are the ones that return zero everywhere (UCS) and the heuristic which computes the true completion cost. The former won’t save you any time, while the
latter will timeout the autograder. You want a heuristic which reduces total compute time, though for this assignment the autograder will only check node counts (aside from enforcing a reasonable
time limit).
Grading: Your heuristic must be a non-trivial non-negative consistent heuristic to receive any points. Make sure that your heuristic returns 0 at every goal state and never returns a negative value.
Depending on how few nodes your heuristic expands, you’ll be graded:
Number of nodes expanded Grade
more than 2000 0/3
at most 2000 1/3
at most 1600 2/3
at most 1200 3/3
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q6
[Optional] Question 7 (4 points): Eating All The Dots
Now we’ll solve a hard search problem: eating all the Pacman food in as few steps as possible. For this, we’ll need a new search problem definition which formalizes the food-clearing problem:
FoodSearchProblem in searchAgents.py (implemented for you). A solution is defined to be a path that collects all of the food in the Pacman world. For the present project, solutions do not take into
account any ghosts or power pellets; solutions only depend on the placement of walls, regular food and Pacman. (Of course ghosts can ruin the execution of a solution! We’ll get to that in the next
project.) If you have written your general search methods correctly, A* with a null heuristic (equivalent to uniform-cost search) should quickly find an optimal solution to testSearch with no code
change on your part (total cost of 7).
python pacman.py -l testSearch -p AStarFoodSearchAgent
Note: AStarFoodSearchAgent is a shortcut for
-p SearchAgent -a fn=astar,prob=FoodSearchProblem,heuristic=foodHeuristic
You should find that UCS starts to slow down even for the seemingly simple tinySearch. As a reference, our implementation takes 2.5 seconds to find a path of length 27 after expanding 5057 search
Note: Make sure to complete Question 4 before working on Question 7, because Question 7 builds upon your answer for Question 4.
Fill in foodHeuristic in searchAgents.py with a consistent heuristic for the FoodSearchProblem. Try your agent on the trickySearch board:
python pacman.py -l trickySearch -p AStarFoodSearchAgent
Our UCS agent finds the optimal solution in about 13 seconds, exploring over 16,000 nodes.
Any non-trivial non-negative consistent heuristic will receive 1 point. Make sure that your heuristic returns 0 at every goal state and never returns a negative value. Depending on how few nodes your
heuristic expands, you’ll get additional points:
Number of nodes expanded Grade
more than 15000 1/4
at most 15000 2/4
at most 12000 3/4
at most 9000 4/4 (full credit; medium)
at most 7000 5/4 (optional extra credit; hard)
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful! Can you solve mediumSearch in a short time? If so, we’re either very, very impressed, or your heuristic is
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q7
[Optional] Question 8 (3 points): Suboptimal Search
Sometimes, even with A* and a good heuristic, finding the optimal path through all the dots is hard. In these cases, we’d still like to find a reasonably good path, quickly. In this section, you’ll
write an agent that always greedily eats the closest dot. ClosestDotSearchAgent is implemented for you in searchAgents.py, but it’s missing a key function that finds a path to the closest dot.
Implement the function findPathToClosestDot in searchAgents.py. Our agent solves this maze (suboptimally!) in under a second with a path cost of 350:
python pacman.py -l bigSearch -p ClosestDotSearchAgent -z .5
Hint: The quickest way to complete findPathToClosestDot is to fill in the AnyFoodSearchProblem, which is missing its goal test. Then, solve that problem with an appropriate search function. The
solution should be very short!
Your ClosestDotSearchAgent won’t always find the shortest possible path through the maze. Make sure you understand why and try to come up with a small example where repeatedly going to the closest
dot does not result in finding the shortest path for eating all the dots.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q8
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Review of Short Phrases and Links
This Review contains major "Planetmath"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. PlanetMath is a free, collaborative, online mathematics encyclopedia. (Web site)
2. PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible.
1. PlanetMath: order topology (more info) Math for the people, by the people.
1. PlanetMath: Grothendieck spectral sequence (more info) Math for the people, by the people.
1. This entry is a collection of links to entries on algebraic number theory in Planetmath (therefore bound to be always under construction).
1. PlanetMath: metric space (more info) Math for the people, by the people.
1. PlanetMath: monoidal category (more info) Math for the people, by the people.
1. PlanetMath: axiom of pairing (more info) Math for the people, by the people.
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1. External links Infinitely-differentiable function that is not analytic on PlanetMath The original article is from Wikipedia.
1. PlanetMath: topological space (more info) Math for the people, by the people.
1. PlanetMath: isomorphism (more info) Math for the people, by the people.
1. When written, this section will include links to entries on differential geometry on PlanetMath.
1. He goes to PlanetMath, and find no entry for "Hausdorff space".
1. PlanetMath: free group (more info) Math for the people, by the people.
2. PlanetMath: derived functor (more info) Math for the people, by the people.
3. PlanetMath: groupoid (category theoretic) (more info) Math for the people, by the people.
1. This article incorporates material from the following PlanetMath articles: Signed measure, Hahn decomposition theorem, and Jordan decomposition. (Web site)
2. The PlanetMath articles are licensed under the GFDL and are thus compatible with the Wikipedia.
1. Articles that cite this source should add {{ Planetmath }} to the article.
2. If you copy or merge an article from PlanetMath, please update the WP and Status fields for that entry. (Web site)
3. This article is based on PlanetMath 's article on examples of initial and terminal objects.
1. This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the GFDL. (Web site)
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3. The mathematics articles in this category incorporate material from PlanetMath.
1. This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the GFDL.
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1. PlanetMath: examples of Cauchy-Riemann equations (more info) Math for the people, by the people.
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3. PlanetMath: proof of the Cauchy-Riemann equations (more info) Math for the people, by the people.
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Understanding Constant Mass Flow
Understanding Constant Mass Flow (CMF)
Paul Raymaeker s
Draft 14/10/2010 V0.2
Abstract: this text explains the principles of constant mass flow, how it can be achieved, why we use it in rebreather design
Disclaimer: this text is a simplified explanation of the CMF principle, only to make it understandable related to the use of CMF in rebreathers the author of this article does not take any
responsibility in case people use the information of this article for building/modifying rebreathers
Terms : CMF, orifice, nozzle, sonic speed, shocked flow,
Constant Volume Flow
When we discuss CMF, there is one law in physics we specifically focus on: when a gas is pushed trough a small hole, (also called orifice, or nozzle) the speed of that gas is limited: it can never be
higher than a certain maximum speed: the sonic speed (Vmax) (1)
So, when the conditions to reach maximum or sonic speed are achieved, what ever you do, increasing the pressure at the entrance of the hole, decreasing, or even vacuuming on the exit side of the
hole, the speed of the gas travelling trough the hole will not increase anymore, but stay constant at Vmax.
This means also, as the speed of the gas is limited, for an orifice with a fixed diameter, that the flow (l/min) of gas trough the orifice is also limited, and can never increase once sonic speed is
achieved: we have 'Constant Volume Flow' (because flow = speed of the gas X surface of the opening in the orifice)
Now when do we reach the maximum speed, or sonic speed? We can apply a simple rule: sonic speed is reached when the inlet pressure P1 is at least twice the outlet pressure P2 or P1 >= 2 x P2
P1 = 10 bar P2 = 1 bar P1 must be minimum twice P2, or > 2 bar: we have sonic speed
P1 = 10 bar P2 = 7 bar P1 is less then twice P2 or 14 bar, we have no sonic speed
This means also, with a fixed diameter orifice, as long as P1 >= 2 x P2, we have constant volume flow.
Constant Mass Flow
Now we must be careful: I wrote the 'volume flow' is constant once sonic speed is reached. Please note that the 'volume flow' is not the same as the amount of gas, the amount of molecules of that
gas, or grams of gas: when diving rebreathers we are mostly not interested in what volume of O2 goes through our orifice, but how many molecules of oxygen, (similar to how many grams / minute) flows
into our system.
When we know the maximum volume of gas that can go through an orifice, and we want to know the MASS, or the number of grams/minute (gr/min) that flows through the orifice, we have to add another
factor: the DENSITY of the gas (kg/m³ of gram/litre): ρ
So when we multiply the volume flow (l/min) by the density of the gas (gram/l), we get
gram/min, and now here it comes:
Although the speed, and so the volume flow (l/min) is limited always to the maximum volume flow at sonic speed, if we want, we can get more molecules, more grams/min gas trough the orifice ... by
increasing the density of the gas (the denser the gas, the more molecules/litre, so even with a fixed volume flow, but having a 'denser' gas, the number of molecules/minute, or the 'grams/minute' ( =
the MASS flow) can increase) (thinking about rebreathers already, but we will come back to this later, when we talk about an oxygen flow of 0.8litre/min, we actually mean 0.8 litre/minute measured at
1 bar, at the surface, which gives us roughly 1.14 grams/minute, as the density of oxygen at 1 bar is around 1.43 gram/litre)
Now how can we increase the density of the gas? Simple, by compressing it, or increasing the pressure of the gas
The density of the gas can be increased, by compressing the gas, (=by increasing P1). So when P1 increase, the density of the gas ρ will increase, and the MASS flow, (density x volume flow) will
Example: when we meet the conditions of sonic flow (Vmax), so that we have Constant Volume Flow, but we double the inlet pressure at the orifice, so we double the density of the gas, we will double
the MASS flow through the orifice.
... ok enough theory, let's go to some practical examples
The 'normal' scuba regulator
For the test we take a normal scuba regulator, the first stage, and connect it to an oxygen tank. The pressure at the outlet of the first stage, the intermediate pressure (IP) is set at 10 bar
absolute pressure. To this outlet we connect an orifice with a fixed diameter, and the outlet of the orifice flows in the breathing bag of our rebreather
'Normal' scuba regulators are made this way that they 'sense' the pressure of the surrounding water, and adjust there IP, so that the difference between the IP and the ambient pressure of the water
(the pressure in the water when diving) stays constant: this is needed for the second stage, normally attached to this first stage, to operate correctly while diving. This means that a normal scuba
regulator will always increase it's IP with the water pressure
it is sensing.
Example: at surface (1 bar absolute) the IP of our regulator is 10 bars absolute, so the pressure difference is 9 bars. When we go diving, the water pressure increases with 1 bar every 10meters we
descent, so the IP of the normal scuba regulator will also increase by 1 bar for every 10 meters of depth. So with an IP set at surface at 10 bars absolute, when diving at 20 meters depth the IP will
have increased by 2 bars till 12 bars absolute, or at 50 m till 15 bars absolute... and so on.
Now let's go back to our regulator/orifice/breathing bag
Suppose we choose the diameter of the orifice this way, that at surface, with the IP, or P1, set at 10 bars, we have a flow of 1 l/min (measured at the surface!!, so +/1.43 grams/min). P1 is 10 bars,
the outlet pressure (P2) is 1 bar (at surface), so P1 is more then twice P2, we have 'sonic speed', so we reach maximum volume flow.
Now we go diving with our rebreathers and we go to 20meters depth. The absolute pressure in the water is 3 bars now, so P2 has increased 2 bars, from 1 to 3, so the regulator has also adjusted
(increased) it's IP with 2 bars, and so now P1 = 12 bars. We still have maximum volume flow (as P1 = 12 is still more then twice P2 = 3), so the volume flow did not change. But what happened: as P1
increased from 10 to 12 bars, the gas (oxygen) at the inlet has been compressed by 20% (from 10>12 bars), and so has become 20% more dense, so our MASS flow (volume flow multiplied by density of the
gas) has increased by 20%!
So now +/1.72 grams/min flows trough our orifice! If we could now measure the flow at surface again, we would measure 1.2 l/min!
Now we dive to 50m: P1 = 15 bars, P2 = 6 bars, we still have maximum flow, (15 > 2 x 6), but the density has increased by 50%, and so has the mass flow, and we now have +/2.14grams/ liter (equivalent
to a flow of 1.5l/min at the surface)
We notice the deeper we go, using our system with a normal scuba regulator connected to the oxygen tank, the mass flow of oxygen (and so the volume flow measured at 1 bar), increases when we go down:
we do not have a constant mass flow of oxygen.
Do we want this?....
The 'absolute pressure regulator
Let's now look at a different system: we modify our standard scuba regulator, so that it does not sense the water pressure anymore, so that it does not increase the IP when we descent in the water.
(this can be done by mounting a special cap on the regulator, so that the water pressure does not get in contact with the 'sensingmembrane' of the first stage)
At surface the IP is set at 10 bars absolute, and using the same orifice, at surface we have the same volume flow of 1l/min. (+/1.43gr/min). Again we dive to 20m. The absolute pressure in the water
is now 3 bars, but, since the regulator does not sense the water pressure, and so does not adjust it's IP, the IP, so P1, still stays at 10 bars absolute. Do we still have maximum volume flow? Yes,
as P1 (10 bars) is still more then twice P2 (3 bars): so the volume flow has not changed.
What happened to the mass flow? Since P1 did not change when descending from surface to 20m, the density of the gas flowing through the orifice did not change, so the mass flow through the orifice
has not changed: we still have our 1.43 gr/min, (or 1l/min if measured at surface)
We have a CONSTANT mass flow.
Now we descent further to 40m, the absolute pressure in the water is now 5 bars. P1 is still 10 bars, as our regulator is 'blind', and we still have maximum volume flow (as P1 (10) is twice P2 (5))
The density of the gas at the inlet of the orifice has not changed, (P1 always constant at 10 bars), so the mass flow has not changed, we still have constant mass flow (CMF)
And this is what we want when we have oxygen flowing into a rebreather: we inject an amount of oxygen into the breathing bag, so that it just compensates for the basic oxygen consumption of our body
(metabolic consumption) when we are at rest, or moving very slowly. And since our metabolic consumption does not change with depth, we don't want the oxygen mass flow into the breathing bag change
when we descent or ascent: we want it to stay constant. The extra amount of oxygen our body needs when we do more exertion can then be added manually in an mCCR, or electronically in the hCCR.
If the oxygen mass flow would increase when we go down, the mass flow would quickly become higher then our metabolic consumption, and so the PPO2 of the gas would rise, and become hyperoxic, so we
would have to 'flush it down' all the time with diluent.
Now what happens when we continue to go deeper with our absolute pressure regulator system: we descent to 60m. The pressure in the water is now 7 bars, the IP is still 10 bars, now P1 is not minimum
twice P2 anymore.. (10 and 7): the condition to have sonic flow, and so maximum volume flow, is not achieved any more! The speed of the gas trough the orifice has become less then max, so the volume
flow has dropped, and since the density (determined by P1) has not changed, our mass flow has dropped! We don't have CMF any more.
And while we go deeper, 70m, 80m ... the pressure in the water P2 will increase and come closer to P1 (the IP), the velocity of the gas will keep dropping and so will the mass flow, until we reach
90m. At that point the pressure in the water equals the intermediate pressure (both 10 bars), there is no pressure difference over the orifice anymore, so no gas will flow through it, the mass flow
has dropped to zero.
Now what does this mean when we dive a rebreather with an 'absolute pressure regulator' (so that we can use an orifice that delivers us a constant mass flow of oxygen over a range of depths): it
means that the maximum operation depth of the rebreather is limited to the depth, where the IP equals the pressure in the water. (because at that depth or deeper, no oxygen can be added to the system
anymore, as also the manual add valves (MAV) will not give anymore flow when activated: the inlet pressure of the MAV is also the IP).
For the rEvo in mCCR or hCCR mode, we even limit the maximum operation depth to 20m less then the depth where the IP equals the water pressure: as the graph shows, at that depth there is still a
reasonable mass flow through the orifice, and still enough differential pressure over the MAV to inject extra oxygen. In the normal rEvo setup, the IP is set at +/11 bars absolute (10 bars
overpressure) and the flow at surface is +/0.8l/min. At a depth of 100m, the water pressure equals the IP, so the maximum recommended working depth of the rEvo in mCCR of hCCR mode is 80m.
Lowering the IP, to decrease the mass flow to mach the flow to a lower metabolic consumption, will also decrease the maximum operation depth of the unit: every bar the IP is decreased, the maximum
recommended operation depth decreases with 10m. In case we do not want to have any depth limitation, the first stage has to be a 'normal' depth compensated scuba regulater, but in that case a fixed
orifice can not be used: so if there is an orifice in the system, and we want to use a normal regulator to go deeper, the orifice has to be blocked.
This is not practical for mCCR diving, but possible for the hybrid CCR version: by removing the 'cap' from the first stage, so turning it into a 'normal' scuba regulator, AND blocking off the orifice
with a plug, the hCCR rEvo is changed into a 'pure' eCCR. In that case the maximum recommended working depth is 100m.
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Analysis of Long-Term Water Level Variation in Dongting Lake, China
College of Hydrology and Water Resources, Hohai University, Nanjing 210098, China
State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China
Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China
State Key Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining 810016, China
Department of Water Resources and Flood Control, School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China
Author to whom correspondence should be addressed.
Submission received: 28 March 2016 / Revised: 1 July 2016 / Accepted: 7 July 2016 / Published: 21 July 2016
The water level of Dongting Lake has changed because of the combined impact of climatic change and anthropogenic activities. A study of the long-term statistical properties of water level variations
at Chenglingji station will help with the management of water resources in Dongting Lake. In this case, 54 years of water level data for Dongting Lake were analyzed with the non-parametric
Mann–Kendall trend test, Sen’s slope test, and the Pettitt test. The results showed the following: (1) Trends in annual maximum lake water level (WLM), annual mean lake water level (WL), and annual
minimum lake water level (WLm) increased from 1961 to 2014; however, the three variables showed different trends from 1981 to 2014; (2) The annual change trends in Dongting Lake between 1961–2014 and
1981–2014 were found to be from approximately 0.90 cm/year to −2.27 cm/year, 1.65 cm/year to −0.79 cm/year, and 4.58 cm/year to 2.56 cm/year for WLM, WL, and WLm, respectively; (3) A greater degree
of increase in water level during the dry season (November–April) was found from 2003 to 2014 than from 1981 to 2002, but a smaller degree of increase, even to the point of decreasing, was found
during the wet season (May–October); (4) The measured discharge data and numerical modeling results showed the operation of Three Gorge Reservoir (TGR) pushed to influence partly the recent
inter-annual variation of water level in Dongting Lake region, especially in the flood and dry seasons. The analysis indicated that the water level of Dongting Lake has changed in the long term with
decreasing of range between WLM and WLm, and may decrease the probability of future drought and flood events. These results can provide useful information for the management of Dongting Lake.
1. Introduction
Lakes provide valuable economic resources for human beings and play an important role in environmental and ecosystem services, such as hydrological cycles, the regulation of runoff, and maintenance
for abundant biodiversity [
]. However, because of the combined impact of climatic change and anthropogenic activities, many lakes around the world are changing [
]. Climate warming and anthropogenic activities (e.g., overexploitation, dams and diversions) will adversely affect water quality and quantity. In some regions, wetlands area will decrease or
disappear and water tables will decline. Habitats for wetland species will be changed in some lakes [
]. These changes may affect the availability of fresh water and regional eco-environments, thus having a critical influence on regional sustainable development. Lakes may display different responses
to external impacts and have different influences on their littoral zones and habitats [
]. In temperate zone lakes with enough rainfall, these changes may increase ground water input, surface water flux, and cause water storage-flooding during wet seasons that may decrease or eliminate
the most emergent species and submerged plant cover and affect both the abundance and species diversity [
]. Changes in the lakes in semi-arid and arid climate zone may lead to buried underwater salt exposure, salinization, a large expansion of the littoral zone [
], desertification, and the reestablishment of dominant vegetation during the dry season, all of which will influence the abundance, biomass and community structure of organisms [
]. As a consequence of climatic change and anthropogenic activities during the past 50 years, Dongting Lake, the second largest freshwater lake in China, has changed in the hydrological regimes
(e.g., wetland area and water table) [
], associated lake patterns [
], the habitats for important species [
], etc. Some of these changes resulted in many environmental and ecological problems, such as a decline of biodiversity and decline/extinction of some species [
]. Therefore, the first step is to determine the current hydrologic status and trends in Dongting Lake. Water level, a sensitive marker of change, is an important driving factor for lakes, affecting
the lake’s physical environment, biota and ecosystem [
]. The duration, timing, magnitude, rate, and frequency of these changes influence the ecological processes and patterns of lakes [
]. Any significant change to the water level in a lake will affect its physical processes and biological productivity [
]. However, it needs to be stressed that changes in water levels follow natural patterns that are necessary for the survival of many species [
]. Only extreme or untimely floods and droughts have adverse effects for biota and humans [
]. Thus, determining and understanding long-term changes in water levels are necessary to protect and restore lakes’ ecological services. A significant challenge in this field is existed because of
limited of measured data, multiple scales of the object variables, uncertainty of impacted factors, etc.
Numerous studies of the spatial and temporal changes in lake water level around the world have been conducted. Yin et al. [
] used annual maximum water level data to explore the changes in water levels in Hongze lake. Pasquini et al. [
] selected the mean, maximum, and minimum monthly water levels to analyze the fluctuation of water level in proglacial lakes in Patagonia. Motiee and McBean [
] analyzed the change trend in water levels at Lake Superior and tested the slope magnitude using the water balance model. Haghighi and Kløve [
] used the area-volume-depth curve and the water balance model to test the change in water level for lakes. Li et al. [
] selected the daily mean, upper and lower quartiles, and maximum and minimum water levels from June to September for a statistical analysis of intra-annual flood distributions in Poyang Lake. Using
the water level and rainfall data during the period of 1961–2010, the analysis by Yuan et al. [
] showed the water level at Dongting Lake decreased significantly during the period of 1961–1980 and there were longer duration in dry season where the water level was below 24 m during 2003–2010.
Compared other sub-periods 1961–1980 and 1981–2002, dam construction was regard as the main factor for the period of 2003–2010. The Three Gorge Reservoir (TGR) started to operate according to 145–175
m scheme after 2009, which brought larger alteration than the 135–156 m scheme before. Therefore, a longer series of measured data are required. Moreover, the quantitative evaluation on the water
level variation in Dongting Lake region driven by TGR and other dams may require measured water level, inflow and outflow data at Dongting Lake boundaries, and hydrologic modeling to analyze this
kind of problem.
Here we selected seventeen hydrological variables that would reveal statistical characteristics of water level regimes to detect temporal trends in terms of frequency, timing, duration, magnitude,
and rate in Dongting Lake over a 54-year period. This article provides an overall view of the long-term changes and sub-period changes in water level regimes through the investigation of the
differences in variables in Dongting Lake from 1961 to 2014. The main objectives of this study are: (1) to reveal the long-term trends in water level; (2) to determine the possible change point in
the long-term water level data; (3) to estimate the changes in sub-periods; and (4) to perform qualitative and quantitative evaluation the water level variation at Dongting Lake driven by TGR and GZB
using the measured inflow and outflow discharge data from 2003 to 2014 and two scenarios simulated by our numerical model.
2. Materials and Methods
2.1. Study Area and Data
Dongting Lake (
Figure 1
) is located in a basin on the alluvial plain of the Yangtze (110°40′–113°10′ E, 28°30′–30°20′ N) [
]. It is fed by four tributaries (Xiang, Zi, Yuan, and Li) and by the Yangtze River’s three outlets (Songzi, Hudu, and Ouchi) on the south bank of the Jingjiang reach, and its outflow returns into
the Yangtze River from Chenglingji (the sole outlet of the Dongting Lake). Dongting Lake consists of three sub-lakes (east, south, and west), all those sub-lakes include permanent water areas and
larger periodically-inundated area. During the wet season, Dongting Lake expands to a large water surface and during the dry season, the periodically-inundated area become lake bottom land. The lake
usually reaches its maximum water level in the flood season from June to September and reduces to its yearly minimum surface starting in October. It is located in the subtropical monsoon climate zone
and has ample sunshine and abundant rainfall. The annual mean temperature of the lake water is about 16.4–17.0 degrees. Annual total rainfall is about 1200–1400 mm [
]. The wet season is from May to October and dry season is from November to April. The mean annual runoff volume is approximately 3.13 × 10
], 37% and 55% of the inlet comes from three outlets and from four rivers, respectively [
].Together with Poyang lake, it is one of two large lakes still linked with the Yangtze River. Dongting Lake is also an important international wetland, with a capacity of more than 1.7 × 10
]. The wetland helps regulate flooding in the Yangtze River as well as the local climate and serves as a water source for industry, agriculture, domestic use and entertainment. The inflows and the
water level at the outlet have an important impact on the evolution of the lake. Because Chenglingji station is located at the confluence of the Yangtze and the outlet of Dongting Lake (see
Figure 1
) and the small longitudinal slopes of the lake’s bathymetry and water level, water level changes at Chenglingji station can reflect changes in all of Dongting Lake’s water level, although the
intense unsteady flow states in the four tributaries and the hybrid river networks linked to Jinjiang may briefly cause some temporary changes along the thalweg of Dongting Lake. Therefore, for this
study, we selected daily water level data at the Chenglingji station from 1961 to 2014, as measured by Changjiang Water Resources Commission.
2.2. Methodology
In this study, we selected non-parametric methods to detect and confirm trends with confidence, namely the Mann–Kendall (MK) [
] test. In addition, we applied the Pettitt test [
] to locate the start of a trend and Sen’s slope test [
] to estimate the slope magnitude when a linear trend was present in a time series. All tests were conducted in R i386 3.2.1 [
MK test: The MK test has been widely used to test stationary statistics against trend statistics in hydrology and climatology [
]. The MK trend test is begun by computing the statistic
using Equation (1)
$S = ∑ i = 1 n − 1 ∑ j = i + 1 n sgn ( x j − x i )$
is the number of data;
$x j$
$x i$
are the
th and the
th observation, respectively; and sgn (.) is the sign function, which can be calculated by the following Equation (2):
$sgn ( x j − x i ) = { 1 x j − x i > 0 0 x j − x i = 0 − 1 x j − x i < 0$
The statistic
is approximately normally distributed when
≥ 10. The mean of
is zero and the variance can be calculated by the following Equation (3) [
$V a r ( S ) = n ( n − 1 ) ( 2 n + 5 ) − ∑ i = 1 m t i ( t i − 1 ) ( 2 t i + 5 ) 18$
is the number of tied groups, each with
$t i$
tied observations. A set of data that has the same value is a tied group. The test statistic
can be calculated by the following Equation (4):
$Z = { S − 1 V a r ( S ) S > 0 0 S = 0 S + 1 V a r ( S ) S < 0$
Thus, in a two-sided trends test, the null hypothesis should be accepted if at the level of significance (α). The positive value of Z indicates an upward trend, and the negative value of Z indicates
a downward trend. The absolute of critical value at α 0.10, 0.05, and 0.01 significance levels of the trend test are 1.64, 1.96, and 2.58, respectively.
The MK trend test requires the data series are independent and the presence of autocorrelation would affect the effectiveness of this method. Here we used the “pre-whitened” method [
] to eliminate the autocorrelation in data series and then the MK trend test is carried out based on the “pre-whitened” time series.
Pettitt test: The Pettitt test [
] is designed to detect the changing point by testing the change in the mean in the time series, using the statistic
$u j , N$
from the Mann–Whitney test to test two samples from the same sample. The statistic
$u j , N$
can be calculated by the following Equation (5):
$u j , N = u j , N − 1 + ∑ sgn ( x j − x i ) j = 2 , 3 , … N$
$u j , N$
$u j , N − 1$
are the U-Statistics at the time of
−1, respectively; and sgn (.) is the sign function, which can be calculated using Equation (2).
The statistic
$k j , N$
can be calculated using Equation (6):
$k j , N = max | U j , N | 1 ≤ j ≤ N$
The statistic
can be calculated using Equation (7):
$p ≅ 2 exp ( − 6 k j , N N 3 + N 2 )$
Usually, approximate probability is good for
≤ 0.05 [
]. Pettitt test is used to detect the changing point, and then the original sequence can be divided into two sub-sequences. For the two sub-sequences, we can use the test to detect both the changing
point and the multi-level changing point.
Sen’s slope test
: The magnitude of the trend slope is computed using the approach developed by Theil [
] and Sen [
]. First, a set of linear trend slopes is calculated for (1 ≤
≤ l), as follows (Equation (8)):
$β k$
) is the slope; and
is the number of data.
$β k$
is ranked from smallest to largest and the Sen’s slope estimator is calculated using Equation (9):
$β = { β M + 1 2 M i s o d d β M 2 + β M + 2 2 2 M i s e v e n$
is the Sen’s slope.
2.3. Determining the Hydrological Variables
For this study, we chose seventeen hydrological variables to describe long-term changes in the water level of the Dongting Lake: Annual maximum lake water level (WLM); Annual mean lake water level
(WL); Annual minimum lake water level (WLm); Range between maximum and minimum water level (RA); Coefficient of variation (Cv); and Mean monthly lake water levels in January (JAN), February (FEB),
March (MAR), April (APR), May (MAY), June (JUN), July (JUL), August (AUG), September (SEP), October (OCT), November (NOV), and December (DEC). Cv is used to measure the dispersion of a data series
around its mean and to measure the inter-annual variability of the annual mean water level. The seventeen hydrological variables and their measurement units are shown in
Table 1
3. Results of Water Level Variation at Dongting Lake
3.1. The Results of Lake Water Level from 1961 to 2014
We gained an overall view of long-term changes in the water level by investigating the differences in yearly time scale variables in Dongting Lake from 1961 to 2014. We analyzed the trends of WLM,
WL, and WLm using the MK trend detection technique and estimated their slope magnitude in three time series using Sen’s slope test.
Figure 2
demonstrates the three time series of WLM, WL, and WLm. The three time series showed an increasing trend. The respective variations in significant trends of the lake water level regime using slope
values, which were calculated by Sen’s slope method, are shown in
Table 2
. The MK test showed a significant increasing trend in WL and WLm on an annual time scale, with a confidence level of 95%. The increasing trend of WLM is not significant (
= 0.6). Although
Figure 2
reveals changes in the three time series and it is noted that the changes in WLm suggest a greater degree of increase compared to the WLM and WL.
Table 2
shows the increasing trend line slope was 0.90 cm/year, 1.65 cm/year, and 4.58 cm/year for WLM, WL, and WLm, respectively.
Significant positive trends (confidence level > 95%) were identified in water levels for January, February, March, and June; a negative trend (confidence level > 95%) was identified in October;
negligible negative trends were observed in September and November; and the remaining months demonstrated positive trends without significance (
Table 2
Table 2
also shows the slope magnitude of monthly water levels, illustrating similar change properties in water levels when compared with the results of the MK trend test.
Figure 3
Table 2
show the result of the time series for RA and Cv from 1961 to 2014. There is a significant decreasing trend (confidence level > 95%) in the two time series that describes the dispersion of annual
water level data.
In this study, abrupt behaviors in water level variables were investigated using Pettitt-test techniques. From 1961 to 2014, the change points were in 1987 (
= 0.79), 1979 (
p $≪$
0.05), and 1980 (p
0.05) for WLM, WL, and WLm, respectively. The approximate probability for the change point for WLM in 1987 is not significant, but the approximate probability for the change points for WL and WLm in
1979 and 1980, respectively, are significant. When we compare the change points according to the Pettitt test and the real situation concerning the construction of the Gezhouba project, which is
located on the upper reach of Dongting Lake and may have had some impact on water levels at the Chenglingji station (see
Figure 1
), we determined that the change point was in the year 1980.
3.2. The Results of Lake Water Levels from 1981 to 2014
Figure 4
Table 3
illustrate the trends in water levels from 1981 to 2014.
Table 3
shows a significant increasing trend annually in WLm according to the MK test. The confidence level was 95%. The decreasing trends of WL and WLM were not significant (z values are −1.0 and −0.7).
Table 3
shows a trend line slope of −2.27 cm/year, −0.79 cm/year, and 2.56 cm/year for WLM, WL, and WLm, respectively.
Significant positive trends (confidence level > 90%) can be identified in water levels for January and May; significant negative trends (confidence level > 90%) can be identified in water levels for
October and November; a negligible negative trend was observed in April, July, September, and December; and the remaining months demonstrated positive trends without significance (
Table 3
Table 3
also shows the slope magnitude of monthly water levels, illustrating similar change properties in water levels when compared with the results of the MK trend test.
Figure 5
Table 3
show the result of the time series for RA and Cv from 1981 to 2014. There was a negligible decreasing trend in the two variables for the time series describing the dispersion of annual water level
From 1981 to 2014, the change points are in 2003 (p = 0.31), 2005 (p = 0.39), and 1999 (p = 0.04) for WLM, WL, and WLm, respectively. The approximate probability for the change point for WLM and WL
in 2003 and 2005 are not significant, but the approximate probability for the change points for WLm in 1999 is significant. The approximate probability of the change point was in 1999.
3.3. Comparisons of Water Levels between Three Time Periods
The variable changes from 1961 to 2014 and 1981 to 2014 require further study of sub-periodic changes in water level. Considering the change points discovered using the Pettitt test, we divided the
time series into three sub-periods: 1961–1980, 1981–1999, and 2000–2012. Considering the operation of Three Gorges Reservoir (TGR), the time series are divided the time series into three different
sub-periods: 1961–1980, 1981–2002, and 2003–2014.
After the normal distribution test and homogeneity test of variance using Shapiro–Wilk [
] and Bartlett test [
], respectively, analysis of variance (ANOVA) [
] was performed to investigate the variations of the water level.
Table 4
shows that no significant differences in mean water level among three sub-periods divided by the change points discovered using the Pettitt test and the operation of TGR, but there were some
differences in mean water level among different months.
Figure 6
shows the comparison between the three sub-periods. To calculate the water level variations quantitatively between the three sub-periods, we first calculated the annual WLM, WL, WLm, and the monthly
average water level for the three sub-periods. Then, we compared the first sub-period (1961–1980) with relative differences from the subsequent two sub-periods. The results are shown in
Table 5
. Clearly, WL and WLm increased in the last two sub-periods but WLM increased over the period 2000–2014 and decreased over the period 2003–2014. However, for the sub-periods obtained via the Pettitt
test, the mean monthly water level from 1981 to 2002 increased except for May (decreased) and October (not changed), and the water level from 2003 to 2014 increased except for May (decreased) and
September–November (decreased). For the sub-periods obtained via the operation of TGR, the mean monthly water level from 1981 to 1999 increased except for May (decreased), and the water level from
2000 to 2014 increased except for May (not changed) and September–November (decreased).
4. Impacted Factors Analysis for the Water Level Variations
4.1. Impacted Factors Identification and Analysis for the Inter-Annual Variations of Water Level
The water level in Dongting Lake may be mainly controlled by the water level at Chenglingji station and inflows from four branched rivers and three outlets around Dongting Lake. Due to the backwater
effect, the water level at Chenglingji station may partly impacted by the flow discharge and water level at Luoshan station in the Yangtze main stem, which is about 30 km downstream to the
Chenglingji station (see
Figure 1
). Based on the qualitative analysis above, the daily measured inflow discharge and water level at 12 hydrological stations around Dongting Lake at the period of 1961–2014 are collected and processed
to identify and analyze the key driving factors for the water level variation. By comparison with inter-annual variations of flow discharge at three sub-periods, the following observations can be
made: (1) In flood season from June to September, there are obvious declines in flow discharge at Yichang and Luoshan stations in Yangtze main stem (
Figure 7
a,b). There is also obvious reduction in the total inflows (
Figure 7
c), in which, generally reduction from four branched rivers (
Figure 7
d) and seriously decline of the inflow from three outlets (
Figure 7
e) during the period 2003–2014 (black line in the
Figure 7
). The inflow decline from three outlets may attribute to the inflow reduction to the TGR, operation function of TGR and TGR-induced variation in the river–lake relationship, which is occupied 68.9%,
13.9% and 17.2%, respectively [
]. Compared to the other sub-periods, the 12-year flow declines may mainly be driven by the smaller rainfalls in the controlled catchments. For example, a dry hydrological year appeared with 50-year
frequency in 2011. Moreover, the peak flood reduction may partly attribute to flood detention by the large reservoirs at upper reach of Yangtze River and four branched rivers; (2) In dry season from
November to March, the additional released flow from large reservoirs, such as TGR, GZB, etc., for hydropower generation, navigation and eco-environmental basic flow, will increase about 1000 m
/s and 2200 m
/s of flow discharge at Yichang and Luoshan, respectively. Among them, flow increment of 1200 m
/s at Luoshan may be mainly produced by early larger rainfall and the releasing flow from dams in the four rivers at the start of flood season (April–June); (3) In early flood season from April to
June, additional flow discharge from reservoir releasing for flood control will increase the flow discharge at some degree, especially on the stations at the downstream of TGR in Yangtze main
channel; (4) At the end of the flood season, there is an obvious reduction in flow discharge because of the rapid water storage by the large dams. The comparison between the shapes for inter-annual
variations at three sub-periods can explain qualitatively the possible combination effects driven by the net rainfall variation from climate changes and dams’ operation at some degree. However, only
using combined in situ data, it is not easy to evaluate the effects caused by the operations of dams separately and quantitatively. Therefore, in subsequent sub-section, a refined numerical modeling
is applied to further identify the water level and flow discharge variation linked to the operation of TGR and GZB.
4.2. Quantitive Identification on the Hydrological Variation Linked to TGD and GZB
The operation of TGD and GZB may cause the variation in water level and flow discharge at river networks and lakes system from Zhutuo to Yangtze estuary. Considering the main impacted region, our
hydrodynamic model domain included the complex Yangtze river–lake–reservoir system from Zhutuo to Datong station, in which, Yangtze main channel, main branches, lakes and inner river networks linked
to lake and Yangtze stem are included. There are more than 5700 cross-sections and 220 sub-channels in the model system, and related measured data are used in the model. This indicates the model has
a good representativeness in the distribution, linkage and bathymetry of complex river–lake–reservoir system. The verifications showed Nash–Sutcliffe coefficient of efficiency (CE) at 94 controlled
stations were large than 0.90. The details about the model theory, parameters setup, validation and calibration in the Yangtze River were given in the references [
]. In order to identify quantitatively the water level and discharge variation at Dongting Lake driven by TGP and GZB, two scenarios are designed and setup in the model system for the third
sub-period (2003–2014) including S1. There are no construction and operation of TGP and GZB, the water can flow freely from Zhutuo to Datong as same as the natural state before (S2). The two dams
operated according to the real situations. After 2009, TGP is operated according to 145 and 175 m scheme in flood seasons and dry seasons, respectively. The numerical results (2009–2012) in the given
stations are presented in
Figure 8
. After the operation of TGP and GZB, the main variations at Yichang station, which is about 4 km downstream to the outlet of GZB, are summarized as follows: (1) Because of additional releasing flow
from TGR, there is an obvious increasing in flow discharge in pre-floods from May to June and in dry seasons from December to May respectively. The averaged increment during 2009–2012 arrived at 6000
/s and 1500 m
/s, respectively; (2) Because of flood detention by reservoirs, there is some reduction during the flooding peaks usually from July to August; (3) At the end of flood from the mid of September to
November, in order to guarantee the enough hydropower generation in dry season, the inflows storage by a reservoir will obviously decrease outflow from the reservoir. For example, the reduced flow
discharge at Yichang station was 20,000 m
/s on 25 September 2011; (4) In general, the operations of TGR and GZB make the flood period earlier than before. Compared to the flow discharge at Yichang, the discharge process at Luoshan, located
about 400 km downstream of Yichang, showed the similar variation trend with smaller variation magnitudes because of hydrodynamic attenuation in the Jingjiang–Dongting systems and flow mixture with
inflows from four rivers and three outlets (
Figure 8
b). The discharge variation in Luoshan will cause a similar variation in the water level at Luoshan and Chenglingji station (
Figure 8
The water storage by TGP at the end of the flood altered the flow and water level situations from Yangtze to Dongting Lake based on our modeling results (
Figure 8
). For example, maximum decrease of the water level at Chenglingji station on 25 September 2011 were about 2.2 m, the duration of this obvious reduction was more than one month till the TGR was
filled (
Figure 8
d). Meanwhile, this reduced water level weakened the backwater effect and even decreased the returned flow from Yangtze stem to Dongting Lake (
Figure 8
Based on the numerical modeling results, the inter-annual variations with and without operation of TGR and GZB during the sub-period 2003–2014 are presented in
Figure 9
. After the operation of two large reservoirs, the water level had an averaged about 0.5 m increment in dry season and 0.3 m reduction in the flood season. Because of the larger water releasing and
storing at the start and end of the flood season compared to other periods, there were relative larger variations in the water level process, and the increasing and reducing magnitudes arrived 0.7 m
and 1.0 m in May and October, respectively. The 145–175 m operation scheme was put into practice in 2009, and the altered magnitude amounted to about 1.2 m and 2.0 m, respectively (see
Figure 8
d). Totally, the operation of TGR and GZB, especially the 145–175 m operation of TGR, will alter the Dongting Lake’s flood storage functions in floods from four rivers and early floods from upper
Yangtze River, respectively, speeding up the flow releasing processes from Dongting to Yangtze River at the end of Yangtze floods. These alterations may bring some impacts on the flood controls,
lakebed deformation, aquatic ecosystems, etc. in Dongting Lake and Yangtze River.
4.3. Relationships between the Hydrological Processes and Net Rainfall around the Dongting Lake Region
The daily measured rainfall and evaporation data measured by 20 cm-diameter evaporating dish at 23 stations are collected from China Meteorological Data Sharing Service System (CMDSS,
). The inverse distance weight (IDW) interpolations including four adjacent stations are used to interpolate daily rainfall and evaporation in Dongting basin with 1000 m cell size. From
Figure 10
, the annual rainfall in the Dongting region had a small reduction during 2003–2014 compared to the sub-periods 1961–1980 and 1981–2003, while the annual evaporation increased to some degree.
However, the water level at Dongting Lake was increasing in dry season, which means the reduced local flow driven by the net rainfall in the Dongting Lake region did not have a significant dominated
to the water level in Dongting Lake. Further statistics showed this local flow component only occupied 8% of total inflows. Moreover, the analysis showed there was slight erosion [
] in the thalweg region of East Dongting Lake and a large deepening in the lower reach of Xiangjiang channel [
] because of the sand mining and water way regulation in the recent ten years. Considering the reasons above, the operations of the large dams were main driving factors for the increasing WL trend at
Dongting Lake. With regard to the other aspects, the net rainfall processes in Dongting region at three sub-periods had a well representativeness in flow processes from four branched rivers because
of the region adjacency (See
Figure 11
5. Discussion
5.1. The Analysis on Water Level Changing Trend
The analysis revealed changes in the mean annual water level throughout the entire time series, as presented in
Figure 2
Table 2
. WLM generally increased, especially during the 1990s, WLM progressively increased, but during the 2000s, the water level began to decrease. The change in water level in the last two sub-periods has
some relationship with the change in net precipitation around Dongting Lake (
Figure 10
Figure 11
). Considering the change trend in water levels (
Figure 2
Table 2
) and the reduce inflow into the lake during the wet season (
Figure 7
c–e), future flood periods may occur less often. Although WL progressively increased, we noticed that WLm increased more significantly, particularly from the 1980s on. From December until June, the
reservoirs discharged more water to the downstream, increasing the water level in Dongting Lake [
]. Obviously, low water levels increased substantially from 1980s, perhaps reducing the hydrological conditions that would allow droughts to occur. As a direct result, drought events will probably
happen less frequently in the future.
The analyses in
Figure 3
Figure 5
indicate that the range between WLM and WLm decreases over time. These two figures also show a continuous decrease in Cv over time, but during the 1990s, this hydrological variable was very high.
Clearly, the dispersion of annual water level around the mean becomes very low in later sub-periods, but during the 1990s, the range between WLM and WLm is high. These analyses prove the presence of
large changes in the annual water level of the Dongting Lake and may decrease the probability of high and low water levels occurring in the future. However, during the 1990s, the high water level was
increasing, mainly driven by the serious rainfall in the flood seasons, especially in 1998 and 1999.
Generally, the changes of water level over the sub-periods obtained via the Pettitt test are similar to the changes over the sub-periods obtained via the operation of TGR. It is evident that the
water level in the dry season was subject to greater degrees of increase during the second sub-period compared to the wet season, and the water level in the dry season was subject to greater degrees
of increase during the third sub-period (
Figure 6
Table 5
). Greater degrees of increase in the dry season can be found during the third sub-period than during the second sub-period, but fewer degrees of increase, even a decreasing trend, can be found in
the wet season. For the monthly scale, the maximum monthly average water level over the period 2000–2014 have increased compared to the period 2003–2014 and the minimum monthly average water level
over the period 2000–2014 have decreased compared to the period 2003–2014. For the annual scale, WLM, WL, and WLm have increased, but the increasing magnitude of WLM and WL were less compared to WLm.
All those shows that the operation of TGR playing an important role to decrease the probability of future drought and flood events. Thus, we can conclude tentatively that the Dongting Lake is
dominated by less increasing WLM and significantly increasing Wlm with decreasing RA.
5.2. Identification on the Hydrological Variation Linked to TGR and GZB
Based on this case study, anthropogenic activities, especially the construction and operation of large reservoirs is one of main driving factor for changes in lake water levels during sub-period
2003–2014, because the following reasons: (1) there is an obvious declining trend in the water level before the start of the dry season in August and November, which is driven by water storage in
large reservoirs on the upper reach of the Yangtze River and four branched rivers around Dongting Lake; (2) in the recent several years, low water levels in the dry season increase despite the slight
erosion of the lake bed, especially at the narrow navigation region near Chenglingji station [
], indicated that the rise in water level is mainly driven by backwater effect from the increasing flow discharge and water level at Luoshan station in the Yangtze main channel from the release of
surplus water from reservoirs at the start of the flood season (
Figure 7
a,b) together with moderate inflow increase at four rivers (
Figure 7
c,d); and (3) three sub-periods showed a large variation in high water levels during the flood season, directly and mainly caused by the rainfall, and flood dispatches from an increasing number of
large reservoirs in the river basins around Dongting Lake shaped flow discharge and water level processes at some degree. It is not easy to separate quantitatively the variations from rainfall and
reservoir operations only using limited measured data. Our modeling scenario results indicated quantitatively the operations of TGR played an important role on the water level changing in Dongting
Lake (
Figure 8
Figure 9
). The operation of TGR and GZB, especially the 145–175 m operation of TGR, had shaped the water level processes and pushed hydrological process earlier with small RA than before. It may prevent the
floods transport from four rivers, speeding up the flow releasing processes from Dongting Lake to Yangtze River at the end of the flood season. Modeling results in the references [
] showed that other existing large reservoirs in the upper Yangtze River could decrease flood water levels by as much as 1.0–3.0 m at the Chenglingji station and furthermore strengthened the water
level shaping at Dongting Lake.
5.3. Further Evaluation on the Hydrological Variation Linked to Anthropogenic and Climate Factors
The detailed driving factors for the water levels in worldwide lakes are quite different and there may be combined effects for a specific lake. The causes for water level reduction in individual
lakes and wetlands are usually poorly understood and most assuredly are multi-faceted and additive [
]. And, it is difficult to separate the influence of human alterations from climate change and other factors [
]. For example, in China, there are quite different trend for the lakes in Qinghai–Tibet Plateau, Inner-Mongolia, Xinjiang, Northeast Plain of China and East China Plain because of the differences in
key controling factors related to the snow or glacier melts, intensified evaporation, intensified western wind in the winter, desertification and aggravated soil erosion, water consumption for
agricultural use, etc. [
]. The mutiple forces by precipitation, inflow, the air temperature and evaporation resulted in a decline in water levels in Lake Superior in North America at the rate of approximately 1.0 cm per
year [
]. The water level variation of Poyang Lake in China may be a combination effect by the operation of TGR [
], serious sand mining and waterway regulation [
]; The precipitation in the Huai River basin decreased since the mid-1960s, while the WLM increased significantly. WLM of Hongze Lake has been changed more and more by anthropogenic activities
(especially hydraulic engineering) from 1950s [
]. The operations of dams in upper reach of the Yellow river altered the hydrological processes in the middle and lower Yellow river reach [
]. The decline of water level in Urmia Lake in Iran is mainly driven by flow diversion and recent drought in river basin [
Our results indicated that water level in Dongting Lake has an increasing trend from 1961 to 2014. However, from 1981 to 2014, this balance changed, perhaps driven by the change of precipitation and
the operation of reservoirs. Reservoirs may play an important role on weakening the flood peaks and maintaining the ecological water level of the lake. In particular, TGR has resulted in a
significant decline in flood water level in the wet season and a significant increase in the dry season, a result that coincides with the reference [
]. Moreover, the related measured flow discharge at period 1961–2014 are collected and processed to identify and analyze qualitatively the key driving factors for the water level variation in
Dongting Lake. The inflows around Dongting Lake or rainfall at upper reach of Yangtze rivers and four rivers are main driving force to the annual variation of the water level in Donting Lake. Our
numerical modeling results indicated quantitatively the operation of TGR shaped the water level processes especially when the 145–175 scheme was used since 2009. The operation of reservoirs will
decrease the RA, and this may decrease the probability of high and low water levels occurring in future. In addition, the hydrological variation in Dongting Lake may also be caused by long-term lake
sedimentation, historic reclaimation and river–lake system alteration around Dongting Lake. Futhermore, in recent one decade, the TGR and other large existing reservoirs in the upper Yangtze River
caused serious sediment declines in the lower reach of these dams and obvoius erosion appeared especially in the Jingjiang reach, this might decrease water level in Yangtze main channel at some
degree. The combined effects of these factors are required to be seperated further and evaluated using more historic measured bathemetry data and an integrated modeling system at next step.
6. Conclusions
Long-term statistical analysis of water level variations in Dongting Lake will definitely help to manage the water resources in this large freshwater lake. In this study, we analyzed the trends for
seventeen hydrological variables using the MK method. Then, we averaged change points and classified sub-periods using the Pettitt test. Finally, we used Sen’s slope test to calculate the magnitude
of the trend slope for the key variables in the sub-periods. According to these results, the main conclusions are as follows.
(1) WLM, WL, and WLm show significantly increasing trends from 1961 to 2014; however, different trends appear from 1981 to 2014. For example, in this sub-period, WLm shows a statistically significant
increasing trend, but WLM and WL show non-significant decreasing trends.
(2) The annual changing trends of WLM in Dongting Lake occurred at a rate of approximately 0.90 cm/year and −2.27 cm/year from 1961 to 2014 and 1981 to 2014, respectively. The annual changing trends
of WL in Dongting Lake occurred at a rate of approximately 1.65 cm/year and −0.79 cm/year from 1961 to 2014 and 1981 to 2014, respectively. The annual increasing trends of WLm in Dongting Lake
occurred at a rate of approximately 4.58 cm/year and 2.56 cm/year from 1961 to 2014 and 1981 to 2014, respectively.
(3) The water levels in the dry season are subject to greater degrees of increase during the last two sub-periods when compared to the wet season. However, the water level in the wet season is
subject to greater degrees of decrease when compared to the first sub-period. Greater degrees of increase in the dry season were found during the third sub-period than during the second sub-period,
but smaller degrees of increase, even a decreasing trend, was found in the wet season, which may be driven by the monthly variation of precipitation in wet season, together with the reservoirs’
(4) Our numerical modeling results indicated that the rise in water level at Dongting Lake in dry season especially since 2009 was mainly driven by backwater effect from the increasing flow discharge
and related higher water level at Luoshan station by 145–175 m operation of the TGR.
Long-term changes in the water levels of Dongting Lake may decrease the probability of future drought and flood events. The finding of this study is fundamental and can be used to apply the effective
water resource management and ecological protection in Dongting Lake. In next step, using combined methods including measured data mining and scenario simulation by integrated numerical model,
further studies will be conducted to quantitatively to identify and predict the long-term and accumulative changing trends of the lake’s water level under the dynamical changing river–lake system in
the middle Yangtze River and the operations of all the large dams in the upper and middle Yangtze River.
The research reported herein is funded by the National Basic Research Program of China (the NBRP 973 program) (Grant No. 2013CB036406), the Key Project of National Natural Science Foundation of China
(Grant No. 91325201) and the National Science Foundation of China (51479010).
Author Contributions
Guoxian Huang and Shuanghu Zhang developed the original ideas. Qiaoqian Han and Rui Zhang undertook the studies and further developed and improved on the original ideas as reported herein. Qiaoqian
Han and Guoxian Huang drafted the manuscript, which was revised substantially by all authors.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Diagram of the river–lake system in the Dongting Lake. Note: (1) F1, Shimen station at Lishui River; F2, Taoyuan station at Yuanjiang River; F3, Taojiang station at Zishui River; F4,
Xiangtan station at Xiangjiang River. Inflow from Four rivers is equal to summarization of F1–F4; (2) T1, Xinjiangkou station at west Songzi River; T2, Shadaoguan station at east Songzi River; T3,
Mituosi station at Hudu River; T4, Guanjiapu station at west Ouchi River; T5, Kangjiagang station at east Ouchi River. The inflow from three outlets of Jingjiang is equal to summarization inflow of
T1–T5; (3) West, South, and East in Dongting Lake region mean West, South, and East Dongting Lakes.
Figure 2. Trends for WLM, WL, and WLm from 1961 to 2014 (note: the black dotted line denotes the linear trend of water levels).
Figure 3. The changes in hydrological variables (RA and Cv) from 1961 to 2014 (note: the black dotted line denotes the linear trend of hydrological variables).
Figure 4. Trends for WLM, WL, and WLm from 1981 to 2014 (note: the black dotted line denotes the linear trend in water level).
Figure 5. The changes in hydrological variables (RA and Cv) from 1981 to 2014 (note: the black dotted line denotes the linear trend of hydrological variables).
Figure 7. Inter-annual variations in flow discharge during three sub-periods (1961–1980, 1981–2002, and 2003–2014). (note: a–f is the variations of Yicheng discharge, Luoshan discharge, Dongting
inflow, Four Rivers inflow, Three outlets inflow, and Chenglingji outflow, respectively).
Figure 8. Water level and discharge comparison with and without operation of TGP and GZB. (note: a, b, and e is the discharge comparison at Yicheng, Luoshan, and Chenglingji, respectively; c and d is
the water level comparison at Luoshan and Chenglingji , respectively).
Figure 9. Inner annual water level variation during 2003–2014 at Chenglingji with and without the operation of TGR and GZB.
Figure 11. Comparison between total inflow from four rivers and net rainfall induced inflow around Dongting Lake.
Table 1. The hydrological variables abbreviations and their measurement units used in this manuscript.
Table 1. The hydrological variables abbreviations and their
measurement units used in this manuscript.
Number Variable Definition Unit
1 WLM Annual maximum lake water level m
2 WL Annual mean lake water level m
3 WLm Annual minimum lake water level m
4 RA Range between maximum and minimum water level m
5 Cv Coefficient of variation %
6 JAN Monthly lake water level in January m
7 FEB Monthly lake water level in February m
8 MAR Monthly lake water level in March m
9 APR Monthly lake water level in April m
10 MAY Monthly lake water level in May m
11 JUN Monthly lake water level in June m
12 JUL Monthly lake water level in July m
13 AUG Monthly lake water level in August m
14 SEP Monthly lake water level in September m
15 OCT Monthly lake water level in October m
16 NOV Monthly lake water level in November m
17 DEC Monthly lake water level in December m
Table 2. Mann–Kendall statistics and trend
slopes of the lake water levels (1961–2014).
Variable Z (MK Test) Sen’s Slope (cm/year)
WL 2.1 ^★ 1.65
WLM 0.6 0.90
WLm 2.5 ^★ 4.58
RA −2.3 ^★ −0.37
Cv −3.5 ^★
JAN 4.1 ^★ 3.53
FEB 3.5 ^★ 3.07
MAR 2.9 ^★ 4.29
APR 1.6
MAY 0.6
JUN 2.1 ^★ 2.68
JUL 0.8
AUG 1.3
SEP −0.1
OCT −1.7
NOV −1.4
DEC 1.5
Note: Numbers with ^★ are significant at >95% confidence level. Positive values indicate increasing trends, and negative values indicate decreasing trends.
Table 3. Mann–Kendall statistics and trend
slopes of lake water levels (1981–2014).
Variable Z (MK Test) Sen’s Slope (cm/year)
WL −1.0 −0.79
WLM −0.7 −2.27
WLm 2.6 ^★ 2.56
RA −1.3
SD −0.7
Cv −0.6
HR −1.5
JAN 2.3 ^★ 2.62
FEB 0.8
MAR 0.3
APR −0.9
MAY 1.7
JUN 0.8
JUL −0.5
AUG 0.0
SEP −0.9
OCT −3.8 ^★ −9.39
NOV −1.8
DEC −0.7
Note: Numbers with ^★ are significant at >95% confidence level (positive values indicate increasing trends, and negative values indicate decreasing trends).
Table 4. ANOVA results for the mean values of water level.
Parameters Element Sum of Squares df Mean Square F p Value
1 Months 346.08 11 31.46 9.71 4.18 × 10^−6
Sub-periods 0.61 2 0.31 0.09 0.91
2 Months 349.04 11 31.73 9.66 4.36 × 10^−6
Sub-periods 0.75 2 0.38 0.11 0.89
Notes: 1 is the three sub-periods divided by the operation of TGR (1961–1980, 1981–2002, and 2003–2014); 2 is the three sub-periods divided by the change points discovered using the Pettitt test
(1961–1980, 1981–1999, and 2000–2014).
Table 5. The mean water level for all sub-periods and percentage changes from the first sub-period to the two subsequent sub-periods.
Table 5. The mean water level for all sub-periods and percentage changes
from the first sub-period to the two subsequent sub-periods.
1961–1980 1981–2002 2003–2014 1981–1999 2000–2014
Mean (m) Mean (m) % Mean (m) % Mean (m) % Mean (m) %
WLM 31.72 32.76 3.29 31.66 −0.19 32.85 3.56 31.77 0.16
WL 24.44 25.32 3.58 24.91 1.91 25.30 3.52 25.02 2.37
WLm 18.45 19.81 7.38 20.28 9.92 19.73 6.94 20.29 9.97
JAN 19.34 20.75 7.29 21.18 9.52 20.73 7.16 21.13 9.26
FEB 19.23 20.90 8.68 21.21 10.30 20.89 8.66 21.15 10.00
MAR 20.22 22.06 9.10 22.46 11.08 21.98 8.69 22.48 11.20
APR 22.87 24.07 5.25 23.60 3.18 24.10 5.37 23.65 3.42
MAY 26.23 25.88 −1.33 26.13 −0.36 25.76 −1.79 26.23 0.00
JUN 27.14 27.94 2.95 28.09 3.48 27.84 2.58 28.19 3.85
JUL 29.77 30.85 3.63 29.84 0.23 30.97 4.03 29.89 0.41
AUG 28.59 29.70 3.88 29.02 1.51 29.77 4.13 29.06 1.66
SEP 28.24 28.83 2.09 27.99 −0.88 28.87 2.23 28.11 −0.45
OCT 26.87 26.87 0.00 24.80 −7.69 26.90 0.12 25.18 −6.30
NOV 23.90 24.19 1.21 23.22 −2.84 24.08 0.76 23.55 −1.48
DEC 20.93 21.78 4.06 21.40 2.23 21.71 3.71 21.56 3.00
© 2016 by the authors; 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:/
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Han, Q.; Zhang, S.; Huang, G.; Zhang, R. Analysis of Long-Term Water Level Variation in Dongting Lake, China. Water 2016, 8, 306. https://doi.org/10.3390/w8070306
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Han Q, Zhang S, Huang G, Zhang R. Analysis of Long-Term Water Level Variation in Dongting Lake, China. Water. 2016; 8(7):306. https://doi.org/10.3390/w8070306
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Han, Qiaoqian, Shuanghu Zhang, Guoxian Huang, and Rui Zhang. 2016. "Analysis of Long-Term Water Level Variation in Dongting Lake, China" Water 8, no. 7: 306. https://doi.org/10.3390/w8070306
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On Application of Taper Windows for Sidelobe Suppression in LFM Pulse Compression
Progress In Electromagnetics Research C, Vol. 107, pp.259-271 (2021)
The efficiency of the standard tapered windows as applied to sidelobe suppression in compressed pulses with linear frequency modulation (LFM) or chirp pulses corresponds to the literature data only
in the case of rather great values of the pulse duration-bandwidth product B ≥ 100. With comparatively small values of B (several dozens or so), the side-lobe levels prove to be essentially greater
than those announced in the literature. In the paper, the output signal of the chirp-pulse compression filter is analyzed in order to look into causes of the discrepancy between the sidelobe level
obtainable using standard tapered windows and the literature data. Expressions are derived for estimating the maximum number of zeros and maxima in the response of the optimum filter of chirp-pulse
compression and separation between adjacent and ?like? (with the same numbers) zeros and maxima in dependence on the signal duration-bandwidth product. The amount of loss in the signal-to-noise ratio
due to the application of smoothing functions is determined. The case of applying window functions in the form of cosine harmonics of the Fourier series, which describes a rather great number of the
standard windows, is analyzed in detail. Analytical expressions are presented for the output signal of the chirppulse compression filter on the basis of such windows and the amount of loss in the
signal-to-noise ratio. A comparative analysis of the Hamming and Blackman windows is made in dependence on B. It is shown that application of the Hamming window is more efficient up to B ≈ 80. For
greater values of B, the Blackman window shows a higher efficiency. As B increases, the efficiency of both windows steadily increases asymptotically approaching the figure declared in the literature.
Coefficients of window functions containing 2 cosine harmonics of the Fourier series have empirically been selected which made it possible to reduce the sidelobe level by approximately 0.34 dB for B
= 21 and by more than 1 dB for B = 7 as compared with the Hamming window. The obtained results allow concluding that the optimization problem for the window function parameters in the case of small
values of the pulse duration-bandwidth product should be solved individually for each specific value of B. Most likely it would be impossible to obtain the extremely low sidelobe level; however, a
certain improvement of the characteristics of the chirp-pulse compression filter seems quite possible.
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What are the differences in cryptographic algorithms for PKI?
What are RSA Keys
RSA (Rivest Shamir Adleman) has been the standard for asymmetric cryptography for 3 decades. RSA uses the prime factorization method for one-way encryption of a message. While certain attacks have
been found due to poorly implemented random number generation, this method has passed the test of time by being a trusted cryptographic method for over 3 decades with no significant findings. While
the algorithm is very secure, with the advances on compute, factorizing large numbers has become easier for large computers, forcing the industry to move from 512 bit keys, to 1024 bit keys, and now
2048 bit keys and some governments recommending 4096 bit keys.
Since the RSA algorithm has been the standard since the 90s, most computers will be capable of encrypting and decrypting data using RSA. Making RSA the perfect default cryptographic algorithm for new
PKI implementations that must be compatible with older devices or IoT devices that might not have ECC capabilities.
Mathematical Overview of RSA
In this method, two large prime numbers are multiplied together to create a larger number. This makes it very easy to create the large number but very hard to find the two prime numbers that created
the number. Learn more about the mathematics behind RSA. If the mathematics for cryptography excite you, also check our careers page.
What are ECDSA Elliptic Curve Digital Signature Algorithm
ECDSA (Elliptic Curve Digital Signature Algorithm) was standardized in 2005. Due to its ability to use smaller keys to achieve the same security as RSA, it makes it a faster and more efficient
algorithm. However, due to its complexity and relatively shorter age, ECDSA has not been widely adopted by the web community. Meaning that if your PKI needs to be validated by older devices/browsers
you might run into compatibility issues and would better off with RSA. However, if you do not have that restriction we highly recommend using ECDSA for more efficient cryptographic operations.
Fun fact: Bitcoin relies on ECDSA for security. Every Bitcoin address is a cryptographic hash of an ECDSA public key and all Bitcoin transactions are signed by the ECDSA private key of the user.
Mathematical Overview of ECDSA Elliptic Curve Digital Signature Algorithm
It is hard to explain ECDSA mathematics without getting very deep into the mathematics behind elliptic curves over finite fields. For sake of simplicity, we will say that ECDSA works by having a
mathematical equation which draws a curve on a graph, then the computer chooses a random point on that curve. This is your private key. Then some mathematical magic using the curve and your origin
point creates the public key. The computational complexity of this discrete log problem allows ECDSA to achieve the same level of security as RSA with significantly smaller keys. Learn more about the
mathematics behind ECDSA. If the mathematics for cryptography excite you, also check our careers page.
RSA vs ECDSA Key Size Comparison
As mentioned in the ECDSA section, ECDSA can achieve the same level of theoretical security with smaller keys. The table bellow shows the equivalent key size for each algorithm.
RSA Key bit length ECDSA Key bit length Equivalent Maximum Validity Period
1024 160 not greater than 6-12 months (and shouldn’t be used)
2048 256 not greater than 2 years
4096 N/A not greater than 16 years
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Dimension drop of the harmonic measure of some hyperbolic random walks
We consider the simple random walk on two types of tilings of the hyperbolic plane. The first by 2π⁄q-angled regular polygons, and the second by the Voronoi tiling associated to a random discrete set
of the hyperbolic plane, the Poisson point process. In the second case, we assume that there are on average λ points per unit area.
In both cases the random walk (almost surely) escapes to infinity with positive speed, and thus converges to a point on the circle. The distribution of this limit point is called the harmonic measure
of the walk.
I will show that the Hausdorff dimension of the harmonic measure is strictly smaller than 1, for q sufficiently large in the Fuchsian case, and for λ sufficiently small in the Poisson case. In
particular, the harmonic measure is singular with respect to the Lebesgue measure on the circle in these two cases.
The proof is based on a Furstenberg type formula for the speed together with an upper bound for the Hausdorff dimension by the ratio between the entropy and the speed of the walk.
This is joint work with P. Lessa and E. Paquette.
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The Kelly Formula and Event-Driven Investing
The Kelly Formula, and its application to investing has been discussed by Charlie Munger, Monish Pabrai, and other legends in the value investing community. A few months ago, we presented commentary
from Peter Lupoff, founder of Tiburon Capital Management. We saw some of his new commentary on the Kelly Formula and Event-Driven Investing and he allowed me to share it on the site. Enjoy.
Edge/Odds – The Kelly Formula and Maximizing Returns
Professional money managers, particularly in our strategies (event-driven, deep value and the like) seek to distinguish themselves with the construction of portfolios spiced with unique calls where
they perceive that they have an “edge.” It is what should differentiate us from the market, or beta, and from each other in attracting capital. However, it is my view that very few managers spend any
time attempting to define the appropriate sizing of positions to mitigate downside and maximize returns. We at Tiburon use our own proprietary variation on what is known as “The Kelly Formula.” It is
part of every decision to put a position in our portfolio. While the Kelly Formula has some well-known advocates in the investing world, such as Warren Buffett, Charlie Munger and Bill Gross, Kelly
requires some meaningful modification in order to be an effective investment sizing tool. We will discuss here, the Kelly Formula, its flaws and modifications we make at Tiburon in order to use it
effectively to optimize position sizing.
John Kelly and an Edge
John Kelly, a scientist who worked at Bell Labs in the 1950’s, is best known for formulating what has become known as the “Kelly Formula” or “Kelly Criterion.”[1] It is an algorithm for maximizing
winnings in bets. Kelly’s early work was based on sizing bets when the gambler had an “edge.” The issue was in what circumstances would a gambler have an “edge” in games of chance? Dancing around the
matter of morality, Kelly’s examples were mostly rigged horse races and game shows. Privately he’d mentioned the logical application to investing as well. By analogy, some had suggested that the
“edge” necessary to effectively use Kelly to size investments was inside information.[2] However, we would argue its justifiable use predicated on the real edge we and others obtain due to our unique
investment process and accurate interpretation of information available in the public domain.
The Kelly Formula Explained
For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:
f* is the fraction of the current bankroll to wager;
b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
p is the probability of winning;
q is the probability of losing, which is 1 - p.
For example, you have $1,000 and are offered 2-1 on coin flips – you win $2 if it comes up heads; you’ll lose $1 if it comes up tails. With a 50% chance of winning (p = 0.50, q = 0.50), you receive
2-to-1 odds on a winning bet (b = 2), then you should bet 25% of the bankroll at each opportunity (f* = 0.25), in order to maximize the long-run growth rate of the bankroll.
If the edge is negative the formula gives a negative result, indicating that the gambler should take the other side of the bet.
Why Kelly Needs Modification to Apply to Investments
Distinctions between Games and Investments
Kelly assumes sequential bets that are independent.[3] That may be a good model for some gambling games, but generally does not apply in investing. The roll of dice is not influenced by the price of
oil, the outbreak of war, the failure of financial systems, but securities prices are. Kelly requires lack of correlation between bets – this is a difficult task when applied to a portfolio. A game
of Poker starts with a hand dealt and ends with players displaying cards. The game then starts anew. Investing professionally usually means there’s a portfolio. Even if a portfolio is concentrated,
there are a variety of “bets.” Considering the bets one at a time, let’s say Kelly says to bet 10% of wealth on each, which means the investor's entire wealth is at risk. That risks ruin, especially
if the payoffs of the bets are correlated. If, as an investor, you put on 10 positions in this way, there would need to be zero correlation in order for Kelly to be effective (where correlation is
defined as a dependent statistical relationship). Further, the portfolio has these 10 bets on simultaneously. The sequential nature of Kelly is more applicable to gambling games than to investing.
Gambling game results are statistical whereas investments have an idiosyncratic nature. Winning and losing in games of chance leave no room for second guessing or changes in assumptions based on
qualitative matters. Security prices are impacted not only by market recognition of intrinsic value and exogenous macro events, but by the behaviors of rational interested parties and irrational
uninformed ones.[4] Determining “edge” when considering investments is most often qualitative and based on the analyst or portfolio manager’s personal perspective.
In the Long Run We All Die
The Kelly Formula works out "in the long run" (that is, it is an asymptotic property). Kelly considers long-term wealth solely. This is one of the reasons that value investors discuss using the Kelly
Formula to size investments. However, for many us, the near term matters as well. Sizing trades using Kelly leads to highly volatile short-term outcomes regardless of what might happen in the long
Do Not Bet Thy Whole Wad
One of the most unrealistic assumptions of the Kelly Formula’s application to investments is that wealth is both the goal and the limit to what one can bet. Most people cannot bet their entire
fortune. As a manager with absolute return criteria, we still afford our investors with quarterly liquidity. How do we reconcile sizing positions fully with the prospect of loss (no matter what edge
we might have) and/or volatility in market value when the sources of capital are varied and with short term liquidity rights? So clearly Kelly, if applied to investments, has relevance regarding
personal investment choices, less so with professional investment managers, particularly when the sources of capital also value liquidity. One last comment here: the answer isn’t locking up investor
capital for long periods. Look at 2008. Many deep value investors lost copious amounts of money as they sized up positions based on a sense of edge (over odds or not). My point is why not seek to
make money over the long term while not losing money in the short term?
A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount
decreases the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since over
betting is worse than under betting, it makes sense to err on the side of caution and bet less than the Kelly amount.
The Tiburon/Kelly Formula Variance
The Kelly Formula is an essential tool we use to size positions that enter our portfolio. However, as noted above, there are a number of flaws in the strict adherence to Kelly. Just to innumerate
them once more succinctly:
• Kelly applies to sequential bets with, therefore, no correlation. Professionally run pools of capital are done so in a portfolio with underlying “bets” influenced by macro factors, markets and
each other. There is some inherent correlation.
• Gambling games are won and lost in ways that can be statistically derived. Movement and terminal value of investments have idiosyncratic properties.
• Determining the “edge” in gambling is quantitative and precise. Determining the “edge” in investments is most often qualitative and based on personal perspective, and therefore hard to define
• Professionally run investment pools rarely have the ability to place highly concentrated bets with no care for short term volatility, while seeking long term absolute returns. In the real world,
there are competing objectives.
While Marty Whitman, my old mentor, has said, “Diversification is a surrogate, and usually a damn poor surrogate, for knowledge, control, and price consciousness,” we deal, in part, with the varying
goals of performance, liquidity for our investors and longevity as a manager by diversifying across approximately 40-50 positions. This amount of line items is small enough to have an “edge” in each
investment, know them intimately and yet still moderate volatility.
We do correlation analysis on the portfolio to identify correlation between positions and between the portfolio broadly, and markets, interest rates, commodity prices, etc. We wring out correlation
among the investments in portfolio via hedges that mitigate risk exogenous to the thesis of each investment.
Getting a handle on “edge,” we create “base,” “best,” and “worst” case scenarios, probability weight them, of course, after processing via our Five Prong Methodology.
Tiburon Variance A - Sizing
Our variation of the Kelly Formula solves two issues we have with Kelly when applying his methodology to portfolio allocation: 1. Tiburon risk rules and investment prudence limits us to 10% at market
in any given position, and; 2. We won’t invest in a situation where we don’t see at least 40% potential upside. If we weight 40-50 portfolio positions by the traditional Kelly Formula, it would
suggest that we use approximately 6x leverage. Therefore, we common size[5] the Kelly derived sizing recommendation. This enables us to assemble and appropriately size our portfolio without leverage.
Moderating the Kelly derived position size has some interesting mathematical properties. It cuts our risk of temporary loss (i.e., volatility) by a large amount while reducing our return expectation
only a little. A 50% of Kelly bet, for example, gives a large margin of safety in the risk estimate. If you are off by a factor of two on your risk of loss estimate, a full Kelly bet will reduce your
return expectation to zero. But a half Kelly bet will leave you with 2/3 of the return expectation.
With the full Kelly bet, your probability of temporary loss is a linear function of the amount of loss. For example, you stand a 90% chance of losing 10%, an 80% chance of losing 20%, a 50% chance of
losing 50%, etc. Not many investors are comfortable with the prospect of a 50% probability of losing 50% of their money. With a reduced Kelly bet, your probability of temporary loss is a quadratic
function of the amount of loss. For example, at half the Kelly bet you stand an 81% chance of losing 10%, a 64% chance of losing 20%, a 25% chance of losing 50%, etc.
Your expected gain with the half Kelly bet is reduced by 25%. For example, if your expected gain is 40% with the full Kelly, it is 30% with the half Kelly, and if your expected gain is 10% with the
full Kelly, it is 7.5% with the half Kelly.
Tiburon Variance B – Beta Correlation
Every new trade is evaluated for its correlation to the rest of the portfolio. For the Kelly Formula to be effective in sizing positions, positions need to have little to no correlation to each
other. Tiburon evaluates sizing trades as a function of the correlation between the new investment and the existing portfolio. The beta of the portfolio to the market and macro events is another
matter, reviewed and potentially hedged as part of portfolio considerations and does not weigh on the sizing of the prospective new investment.
Therefore, the Tiburon/Kelly Formula Variance is:
For any investment reviewed, subject to the trade’s correlation to the Portfolio being less than 1[6],
· Ts is Tiburon position sizing;
· Ts cannot exceed 10% of Tiburon portfolio at market;
b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
p is the probability of winning;
q is the probability of losing, which is 1 – p;
? is the summation of Kelly allocations across our portfolio.
For any investment professional building a portfolio, they first need to identify positions that meet their criteria and are part and parcel of their investment strategies. Given this, sizing the
position is a function of the manager’s edge in the trade and the odds of the favorable outcome. At Tiburon, every trade idea passes through our Five Pronged Methodology. As a function of this work,
we probability weight the outcomes. As every trade idea requires a Revaluation Catalyst, and sizing is in part, a function of our conviction level about the catalyst (or the odds of its occurrence
and impact on securities price), sizing naturally become edge/odds. We use the Tiburon/Kelly Formula Variance to, as best possible, accurately size positions to maximize profitability while
minimizing downside.
Peter M. Lupoff March, 2010 Tiburon Capital Management, LLC
[1] See “A New Interpretation of Information Rate” J.L. Kelly, March 21, 1956 (ATT) http://www.racing.saratoga.ny.us/kelly.pdf
[2] Claude Shannon, Kelly’s Bell Labs mentor and collaborator. See “Fortune’s Formula”, William Poundstone, Hill & Wang 2005.
[3] In probability theory, independent means that the occurrence of one event makes others no more or less probable.
[4] See discussion of Tiburon’s Five Pronged Methodology – “Rational Actors Assessment.” http://www.distressed-debt-investing.com/2009/09/exclusive-interview-with-hedge-fund.html
[5] Common sizing is the expression of items as percentages rather than as dollar amounts.
[6] Tiburon may reduce ß via hedges to meet these criteria.
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4.11: Quantitative Units of Concentration
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Learning Objective
• Determine specific concentrations with several common units.
Rather than qualitative terms (Section 11.2 - Definitions) we need quantitative ways to express the amount of solute in a solution; that is, we need specific units of concentration. In this section,
we will introduce several common and useful units of concentration.
Molarity (M) is defined as the number of moles of solute divided by the number of liters of solution:
\[molarity \: =\: \frac{moles\: of\: solute}{liters\: of\: solution}\nonumber \]
which can be simplified as
\[M\: =\: \frac{mol}{L},\; or\; mol/L\nonumber \]
As with any mathematical equation, if you know any two quantities, you can calculate the third, unknown, quantity.
For example, suppose you have 0.500 L of solution that has 0.24 mol of NaOH dissolved in it. The concentration of the solution can be calculated as follows:
\[molarity \: =\: \frac{0.24\: mol\: NaOH}{0.500L}=0.48\, M\; NaOH\nonumber \]
The concentration of the solution is 0.48 M, which is spoken as "zero point forty-eight molarity" or "zero point forty-eight molar." If the quantity of the solute is given in mass units, you must
convert mass units to mole units before using the definition of molarity to calculate concentration. For example, what is the molar concentration of a solution of 22.4 g of HCl dissolved in 1.56 L?
First, convert the mass of solute to moles using the molar mass of HCl (36.5 g/mol):
\[22.4\cancel{gHCl}\times \frac{1\: mol\: HCl}{36.5\cancel{gHCl}}=0.614\, mol\; HCl\nonumber \]
Now we can use the definition of molarity to determine a concentration:
\[M \: =\: \frac{0.614\: mol\: HCl}{1.56L}=0.394\, M\nonumber \]
Example \(\PageIndex{1}\):
What is the molarity of a solution made when 32.7 g of NaOH are dissolved to make 445 mL of solution?
To use the definition of molarity, both quantities must be converted to the proper units. First, convert the volume units from milliliters to liters:
\[445\cancel{mL}\times \frac{1\: L}{1000\cancel{mL}}=0.445\, L\nonumber \]
Now we convert the amount of solute to moles, using the molar mass of NaOH, which is 40.0 g/mol:
\[32.7\cancel{gNaOH}\times \frac{1\: mol\: NaOH}{40.0\cancel{gNaOH}}=0.818\, mol\: NaOH\nonumber \]
Now we can use the definition of molarity to determine the molar concentration:
\[M \: =\: \frac{0.818\: mol\: NaOH}{0.445L}=1.84\, M\: NaOH\nonumber \]
Exercise \(\PageIndex{1}\)
What is the molarity of a solution made when 66.2 g of C[6]H[12]O[6] are dissolved to make 235 mL of solution?
1.57 M
The definition of molarity can be used to determine the amount of solute or the volume of solution, if the other information is given. Example 4 illustrates this situation.
Example \(\PageIndex{1}\):
How many moles of solute are present in 0.108 L of a 0.887 M NaCl solution?
We know the volume and the molarity; we can use the definition of molarity to mathematically solve for the amount in moles. Substituting the quantities into the definition of molarity:
\[0.887\, M \: =\: \frac{mol\: NaCl}{0.108L}\nonumber \]
We multiply the 0.108 L over to the other side of the equation and multiply the units together; "molarity × liters" equals moles, according to the definition of molarity. So
mol NaCl = (0.887 M)(0.108 L) = 0.0958 mol
Exercise \(\PageIndex{1}\)
How many moles of solute are present in 225 mL of a 1.44 M CaCl[2] solution?
0.324 mol
If you need to determine volume, remember the rule that the unknown quantity must be by itself and in the numerator to determine the correct answer. Thus rearrangement of the definition of molarity
is required.
Example \(\PageIndex{1}\):
What volume of a 2.33 M NaNO[3] solution is needed to obtain 0.222 mol of solute?
Using the definition of molarity, we have
\[2.33\, M \: =\: \frac{0.222mol}{L}\nonumber \]
To solve for the number of liters, we bring the 2.33 M over to the right into the denominator, and the number of liters over to the left in the numerator. We now have
\[L \: =\: \frac{0.222mol}{2.33\, M}\nonumber \]
Dividing, the volume is 0.0953 L = 95.3 mL.
Exercise \(\PageIndex{1}\)
What volume of a 0.570 M K[2]SO[4] solution is needed to obtain 0.872 mol of solute?
1.53 L
A similar unit of concentration is molality (m), which is defined as the number of moles of solute per kilogram of solvent, not per liter of solution:
\[molality\: =\: \frac{moles\: solute}{kilograms\: solvent}\nonumber \]
Mathematical manipulation of molality is the same as with molarity.
Another way to specify an amount is percentage composition by mass (or mass percentage, % m/m). It is defined as follows:
\[\%m/m\: =\: \frac{mass\: of\: solute}{mass\: of\: entire\: sample}\times 100\%\nonumber \]
It is not uncommon to see this unit used on commercial products (Figure \(\PageIndex{1}\) - Concentration in Commercial Applications)
Figure \(\PageIndex{1}\) Concentration in Commercial Applications © Thinkstock
The percentage of urea in this package is 5% m/m, meaning that there are 5 g of urea per 100 g of product.
Example \(\PageIndex{1}\):
What is the mass percentage of Fe in a piece of metal with 87.9 g of Fe in a 113 g sample?
Using the definition of mass percentage, we have
\[\%m/m\: =\: \frac{87.9\, g\, Fe}{113\, g\, sample}\times 100\%=77.8\%\, Fe\nonumber \]
Related concentration units are parts per thousand (ppth), parts per million (ppm) and parts per billion (ppb). Parts per thousand is defined as follows:
\[ppth\: =\: \frac{mass\: of\: solute}{mass\: of\: sample}\times 1000\nonumber \]
There are similar definitions for parts per million and parts per billion:
\[ppm\: =\: \frac{mass\: of\: solute}{mass\: of\: sample}\times 1,000,000\nonumber \]
\[ppb\: =\: \frac{mass\: of\: solute}{mass\: of\: sample}\times 1,000,000,000\nonumber \]
Each unit is used for progressively lower and lower concentrations. The two masses must be expressed in the same unit of mass, so conversions may be necessary.
Example \(\PageIndex{1}\):
If there is 0.6 g of Pb present in 277 g of solution, what is the Pb concentration in parts per thousand?
Use the definition of parts per thousand to determine the concentration. Substituting
\[\frac{0.6g\, Pb}{277g\, solution}\times 1000=2.17\, ppth\nonumber \]
Exercise \(\PageIndex{1}\)
If there is 0.551 mg of As in 348 g of solution, what is the As concentration in ppm?
1.58 ppm
As with molarity and molality, algebraic rearrangements may be necessary to answer certain questions.
Example \(\PageIndex{1}\):
The concentration of Cl^– ion in a sample of H[2]O is 15.0 ppm. What mass of Cl^– ion is present in 240.0 mL of H[2]O, which has a density of 1.00 g/mL?
First, use the density of H[2]O to determine the mass of the sample:
\[240.0\cancel{mL}\times \frac{1.00\: g}{\cancel{mL}}=240.0\, g\nonumber \]
Now we can use the definition of ppm:
\[15.0\, ppm\: =\: \frac{mass\: of\: solute}{240.0\: g\: solution}\times 1,000,000\nonumber \]
Rearranging to solve for the mass of solute,
\[mass\: solute =\: \frac{(15.0\, ppm)(240.0\: g\: solution)}{1,000,000}=0.0036g=3.6\, mg\nonumber \]
Exercise \(\PageIndex{1}\)
The concentration of Fe^3+ ion in a sample of H[2]O is 335.0 ppm. What mass of Fe^3+ ion is present in 3,450 mL of H[2]O, which has a density of 1.00 g/mL?
1.16 g
For ionic solutions, we need to differentiate between the concentration of the salt versus the concentration of each individual ion. Because the ions in ionic compounds go their own way when a
compound is dissolved in a solution, the resulting concentration of the ion may be different from the concentration of the complete salt. For example, if 1 M NaCl were prepared, the solution could
also be described as a solution of 1 M Na^+(aq) and 1 M Cl^−(aq) because there is one Na^+ ion and one Cl^− ion per formula unit of the salt. However, if the solution were 1 M CaCl[2], there are two
Cl^−(aq) ions for every formula unit dissolved, so the concentration of Cl^−(aq) would be 2 M, not 1 M.
In addition, the total ion concentration is the sum of the individual ion concentrations. Thus for the 1 M NaCl, the total ion concentration is 2 M; for the 1 M CaCl[2], the total ion concentration
is 3 M.
Key Takeaway
• Quantitative units of concentration include molarity, molality, mass percentage, parts per thousand, parts per million, and parts per billion.
Exercise \(\PageIndex{1}\)
1. Differentiate between molarity and molality.
2. Differentiate between mass percentage and parts per thousand.
3. What is the molarity of a solution made by dissolving 13.4 g of NaNO[3] in 345 mL of solution?
4. What is the molarity of a solution made by dissolving 332 g of C[6]H[12]O[6] in 4.66 L of solution?
5. How many moles of MgCl[2] are present in 0.0331 L of a 2.55 M solution?
6. How many moles of NH[4]Br are present in 88.9 mL of a 0.228 M solution?
7. What volume of 0.556 M NaCl is needed to obtain 0.882 mol of NaCl?
8. What volume of 3.99 M H[2]SO[4] is needed to obtain 4.61 mol of H[2]SO[4]?
9. What volume of 0.333 M Al(NO[3])[3] is needed to obtain 26.7 g of Al(NO[3])[3]?
10. What volume of 1.772 M BaCl[2] is needed to obtain 123 g of BaCl[2]?
11. What are the individual ion concentrations and the total ion concentration in 0.66 M Mg(NO[3])[2]?
12. What are the individual ion concentrations and the total ion concentration in 1.04 M Al[2](SO[4])[3]?
13. If the C[2]H[3]O[2]^– ion concentration in a solution is 0.554 M, what is the concentration of Ca(C[2]H[3]O[2])[2]?
14. If the Cl^− ion concentration in a solution is 2.61 M, what is the concentration of FeCl[3]?
1. Molarity is moles per liter, whereas molality is moles per kilogram of solvent.
11. Mg^2+ = 0.66 M; NO[3]^− = 1.32 M; total: 1.98 M
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How to confirm the accuracy of statistical analysis in my assignment? | Hire Some To Take My Statistics Exam
How to confirm the accuracy of statistical analysis in my assignment? Thanks to very big comments in my last essay. What I’m still doing is testing for and comparing the accuracy of the correct
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observations. And now I’m getting a variety of results. I keep putting up some blog posts on random sample for you’ll face if I understand what I’m saying. This is a test for whether a class I have
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can effectively apply linear discriminant analysis to your data and get a classification of data for your task. My analysis work in this model just gets it. This is how an “equidistant class” looks
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my case would prove results are correct: I cannot. I have been given the following articles or maybe if i understood these you have to explain and give me another way for checking the accuracy.
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what are some more on Google and how to utilize that toolHow to confirm the accuracy of statistical analysis in my assignment? Thank you very much to everyone who will be making ready to answer this.
– What is the difference Our site the two methods of data representation? Also, what are some examples from different data generation projects on online learning concepts, such as “Visual learning
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CIPM Exam Tips & Tricks
At the Expert Level, CIPM candidates are required to learn a few of the more commonly used linking methodologies for smoothing the attribution effects that come from arithmetic models. The required
methodologies include:
Of these methods, the GRAP method is my personal favorite because of it's simplicity... which may not be apparent looking at the formula! Upon first glance, the GRAP formula, which appears above, may
seem intimidating for a couple of reasons:
• It uses three "series" one summation series a two multiplication series.
• The three series have different indexes (T=1 through N, t = 1 through T-1, and t = T+1 through N).
When teaching the GRAP method, I find that it is easier to explain what it does, effectively, rather than explaining the formula.
Consider a situation where we are linking together monthly attribution effects for the first six months of the calendar year (January through June). Let's consider how we obtain the "G" factor that
we will use to smooth the attribution effects for the month of April. Basically, what the GRAP formula tells us is that:
• The excess return is the sum of the "smoothed" attribution effects
• We obtain the smoothed attribution effects by multiplying the original attribution effects by the "G" factor
• The G factor for a particular month is a combination of portfolio returns and benchmark returns for the periods being linked together.
• In our example, we are smoothing the attribution effects for April. This will be done by multiplying together:
1. the unitized portfolio returns for the months preceding April (January, February, March)
2. the unitized benchmark returns for the months coming after April (May, June)
Thus, the following formula yields the G factor that we can use to smooth the April attribution effects:
Thus, the GRAP formula is very easy to remember, and much simpler to use than the Menchero or Cariño methods.
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How To Calculate Linear Magnification
Magnification is the process of appearing to enlarge an object for purposes of visual inspection and analysis. Microscopes, binoculars and telescopes all magnify things using the special tricks
embedded in the nature of light-transducing lenses in a variety of shapes.
Linear magnification refers to one of the properties of convex lenses, or those that show an outward curvature, like a sphere that has been severely flattened. Their counterparts in the optical world
are concave lenses, or those that are curved inward and bend light rays differently than convex lenses.
Principles of Image Magnification
When light rays traveling in parallel are bent as they pass through a convex lens, they are bent toward, and thus become focused on, a common point on the opposite side of the lens. This point, F, is
called the focal point, and the distance to F from the center of the lens, denoted f, is called the focal length.
The power of a magnifying lens is just the inverse of its focal length: P = 1 / f. This means that lenses that have short focal lengths have strong magnification capabilities, whereas a higher value
of f implies lower magnifying power.
Linear Magnification Defined
Linear magnification, also called lateral magnification or transverse magnification, is just the ratio of size of the image of an object created by a lens to the object's true size. If the image and
the object are both in the same physical medium (e.g., water, air or outer space), then the lateral magnification formula is the size of the image divided by the size of the object:
\(M = \frac{-i}{o}\)
Here M is the magnification, i is the image height and o is the object height. The minus sign (sometimes omitted) is a reminder that images of objects formed by convex mirrors appear inverted, or
The Lens Formula
The lens formula in physics relates the focal length of an image formed by a thin lens, the distance of the image from the center of the lens, and the distance of the object from the center of the
lens. The equation is
Say you position a tube of lipstick 10 cm from a convex lens with a focal length of 6 cm. How far away will the image appear on the other side of the lens?
For d[o]= 10 and f = 4, you have:
&\frac{1}{10}+\frac{1}{d_i}=\frac{1}{4} \
&\frac{1}{d_i}=0.15 \
You can experiment with different numbers here to gain a sense of how altering the physical set-up affects the optical results in this type of problem.
Note that this is another way to express the concept of linear magnification. The ratio d[i] to d[o] is the same as the ratio of i to o. That is, the ratio of the height of the object to the height
of its image is the same as the ratio of the length of the object to the length of its image.
Magnification Tidbits
The negative sign as applied to an image that appears on the opposite side of the lens from the object indicates that the image is "real," i.e., that it can be projected onto a screen or some other
medium. A virtual image, on the other hand, appears on the same side of the lens as the object and is not associated with a negative sign in pertinent equations.
Although such topics lie beyond the scope of the present discussion, a variety of lens equations pertaining to a host of real-life situations, many of them involving changes in media (e.g., from air
to water), can be uncovered with ease on the internet.
Cite This Article
Beck, Kevin. "How To Calculate Linear Magnification" sciencing.com, https://www.sciencing.com/calculate-linear-magnification-6148080/. 21 December 2020.
Beck, Kevin. (2020, December 21). How To Calculate Linear Magnification. sciencing.com. Retrieved from https://www.sciencing.com/calculate-linear-magnification-6148080/
Beck, Kevin. How To Calculate Linear Magnification last modified March 24, 2022. https://www.sciencing.com/calculate-linear-magnification-6148080/
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Does size matter in Pull Requests: Analysis on 30k Developers
At one point or another you might have found yourself putting a Pull Request up for review that was significantly bigger than what you were expecting it to be. And you found yourself wondering:
“How big should it really be? Is there a sweet spot for the size of a review? If we could theoretically always fully control it, how big should we make it?”
You googled around, and you found a lot of resources, sites, and articles like this one, analysing the subject and ending up with something along the lines of:
“Too few lines might not offer a comprehensive view of the changes, while an excessively large PR can overwhelm reviewers and make it challenging to identify issues or provide meaningful
And although you understood the sentiment of the writer, you also understood that the theoretical answer could only be vague, as there is not a silver bullet. As always life is more complicated than
What we are gonna be doing in this article is something different however:
“We will analyze the PRs of ~30k developers to see how the size of PRs correlates with lead time, comments received and change failure, to try and find what statistically is the best size, as
well as examine what affects it.”
[Disclaimer: For anyone who has played around with data, and especially if you did any courses/training in data, the above might bring back some memories of this phrase “Correlation does not mean
causation”. First of all hello to you my fellow scholar, and secondly you are absolutely right. We will try to look at it from various angles to see how this correlation varies by company, developer,
per developer and amount of code committed, and any other angles which might help us understand what others values, for any reason, follow relevant patterns. However, these are “only” numbers and
correlations, they do not explain the reason behind them, so any assumptions for causes that we make are more anecdotal and less scientifically backed.]
Lead Time
In this case we use as lead time the time between the earliest PR event (either 1st commit, or PR open), and when the PR gets merged in.
Data Preparation
Data that are removed as outliers:
1. PRs that had a lead time of more that 6 months
2. PRs that had a lead time of less than 5 minutes
3. File changes of more than 20k lines
4. PRs with more than 20k line changes
After we have done that we have a few hundreds of thousands of merged Pull Requests that are used to produce the below analysis.
All Correlations have been done using the kendall tau method, which should be able to better estimate the correlation in the case of non-linear relationships.
How does Lead Time relate to PR size
Before we go more deeply, intuitively we expect that the size of a PR should correlate in one way or another with the lead time, but is it actually the case? Running correlation between the two
variables for the whole dataset, gives us as a result the below correlation matrix.
PR size to Lead Time correlation
From these numbers we could say that there seems to be some correlation between the two variables but it seems to be a bit above the limit of statistical insignificance, meaning that:
Their correlation is there, but is not very strong, maybe less than one would have expected.
Seems like we’ll have to dig deeper to see why this correlation appears to be so weak, and unfortunately, plotting the graph of total line changes to lead time if anything makes things less clear, as
although the trend seems to suggest that the ones with the higher lead time had slightly bigger size on average, we see that any link between them is not so clear to see.
Now, if we change this chart a bit, by grouping the data points by day, and taking the median of the total changes by day, we start to see a bit more clearly how they relate and potentially an
explanation for why their correlation is not that high.
Mean total PR size to daily Lead Time
So this suggests that at fast lead times the PRs are consistently low in lines changed, and as they get bigger there is a linear increase on the lead time. However, higher lead times can be produced
by any size of PR and the correlation is very low between them.
What is the best size
To try and answer this question, we’d first have to ask ourselves what is it that matters to us, ie what are we trying to optimize for. Now, that is a question with endless possibilities. For our
purposes however, we will examine what is the largest size of PR that statistically works better given these 3 wants:
1. Low lead time (aka be done fast)
2. High number of comments (not too big to review properly)
3. Low defects/reverts (aka we are not breaking things)
If we plot in a heatmap the probability of a PR getting done in a number of weeks to the size we get the below.
Heatmap of probability of PR of a size (x axis) getting done in a number of weeks (y axis)
Meaning that a PR of less than 100 lines of code has ~80% chance of getting done within the first week
A similar heatmap for the amount of comments gives us the below.
Heatmap of probability of a PR of a size (x axis) getting an amount of comments (y axis)
Which means that a PR of 6000 lines of code has the same probability of getting 0 review comments as much as a review of less than 50 lines of code.
And finally doing the same for the probability of reverts gives us the below heatmap, and depicting the probability of no commits from that PR getting reverted gives us the below.
Probability of a PR of a size (x axis) not having to be reverted (0 reverts)
Which means that generally larger size PRs have a larger probability of being having some parts of their code reverted (ie faulty)
From the above if we plot on the same graph the probabilities of completing an PR within the 1st week, the probability of getting at least 1 or more comments, and the probability to not have to
revert a commit from that PR, we get the below.
Probability (y axes) of a PR getting done in a week (blue), to have comments (green) and to be reverted (red) over lines of code
Therefore, statistically, below ~400 lines of code per PR gives a good probability of getting some review comments, completing it within the first week and not having issues with the code.
Of course that is only “statistically” the case. It surely depends on a lot of things. Let examine some potential ones.
Does it depend on the user
We would potentially expect it to vary per user, but how different it could be per user, either that being the author or the reviewer, could be more interesting. After removing all users that have
one or more of the below:
• Less than 10 merged PRs
• Less than 10 commits
• Less than 100 lines of code changed
And all reviewers that have:
• Less than 10 approved and merged PRs
We perform correlation analysis between Lead Time and PR size per user. If we then put the result of the analysis on a histogram showing how many users had each of the correlation value, we get the
below charts:
Histograms of Lead Time to PR size correlation per amount of unique PR authors (left) and PR reviewers (right)
The correlation between Lead Time and PR size heavily depends on the PR author as well as the PR reviewer
There are a wide range of reasons why that could potentially happen, like level of seniority, company/team process, coding language, review tool, etc.
Below we plot the relation of the correlation depending on the amount of lines of code a developer has written throughout the last 6 months. Although that instinctively could lead us to think that
that means a more “experienced” developer, it is not necessarily true, as it may be also affected by multiple factors, such as eg amount of meetings, mentoring, collaboration per day, which could
vary on seniority, the tasks each one took up, etc., and so on and so forth.
Nonetheless we depict it here for anyone that might find it interesting. Also keep in mind that the difference in the correlation between a user with many PRs merged and few is not a very large one.
Lead Time to PR size correlation value for developers per code committed within the last 6 months
The more lines one has written the more correlated the PR size is with the lead time. This could also mean that lead time becomes more predictable in this case, and it depends more heavily on the
size of the PR and not other parameters (e.g. complexity). However, more analysis would be required to establish that.
Does it depend on the company
We mentioned earlier that there are various potential reasons for a correlation between Lead Time and PR size, and we also said that the cause of the strength of the correlation would also be
multivariate. One of the potential causes being company/team processes. If that would be the case we’d expect to see the correlation varies by the company.
Taking a small sample of companies and examining the strength of that correlation seems to suggest that that is a valid assumption as well, as we can see here it varies from 0.1 suggesting that the
two metrics are not related at all for the specific company to almost 0.7 suggesting a relatively strong correlation between the two.
Lead Time to PR size mean correlation per company (sample)
How much PR size relates to Lead Time seems to depend heavily on the specific company
Does it change over time
It absolutely does, and massively so! Unfortunately, it’s rather hard to depict that for everyone in a single chart. However, I’m putting here my own correlation over time that I got from our free
analytics platform so you can get an image of how much it can vary.
Lead Time to PR Size Correlation over time chart for myself
We examined the correlation between Lead Time and PR size to try and see if we can draw some conclusions about what is the size we should be aiming for. We found that statistically there are some
generalisations we can do and estimate an optimal size. However we also came to the conclusion that the link between them heavily depends on the company, the team and even the individual developer.
Which seems to suggest in the end that:
Each developer works in unique ways, and only you, if anyone, knows what is the optimal for you, and your team.
Now If you would like to check where you or your team/company stands wrt this correlation between Lead Time and PR size, we created a simple way for developers and teams to get insight on how this
correlation changes over and see where they stand, either individual, team, or as a whole company. If you are curious about it, feel free to check it out.
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EE1C1 Linear Circuits A
Topics: Circuit theory course for first year EE students, Part 1
This course deals with the calculation of voltages, currents and power in electric circuits with current and voltage sources, resistors, inductors and capacitors. The basic components, different
calculation methods and first order circuits are introduced in the first part, EE1C1.
An important part of the first part of Linear Circuits (EE1C1) is a practicum. Here the student learns to read a circuit diagram, to build and engineer an electrical circuit, and to test an electric
circuit dealing with real measurements.
After following this course the student should know:
• The basic concepts of electrical circuits (current, voltage, charge, energy, power);
• The basic components as independent sources, resistors, inductors, capacitors, and dependent sources;
• The laws of Ohm's, Kirchhoff's theorem, Norton and Thevenin equivalents, to calculate currents and voltages in circuits;
• Serial and parallel circuits to calculate currents and voltages;
• The nodal method, the mesh method to calculate currents and voltages in circuits;
• How to solve first order differential equations to analyse transients in electric circuits.
• How to build circuits making use of soldering techniques;
• How to read and build a circuit given the diagram of an electrical circuit;
• To interpret data of components in the data sheets;
• How to test an electrical circuit making use of a signal-generator, a meter and / or an oscilloscope
dr. Ioan Lager
Computational electromagnetics, antenna engineering
dr. Francesco Fioranelli
Bistatic/multistatic radar systems; micro-Doppler signature characterization and classification.
prof.dr.ir. Leon Abelmann
Magnetism, Nanotechnology, MEMS
Last modified: 2024-10-01
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Chris (Evans) R SAFAQ: Bonferroni correction
[This post cross-links to entries in the glossary for our OMbook (Evans, C., & Carlyle, J. (2021). Outcome measures and evaluation in counselling and psychotherapy (1st ed.). SAGE Publishing.),
particularly the one on the Bonferroni correction!]
The correction
The Bonferroni correction is a very simple and easy to understand “correction” for the multiple tests problem. That problem is that when we do more than one null hypothesis test the risk of at least
one false positive result climbs above the nominal criterion (usually .05 or 1 in 20). See the glossary entry here.
The Bonferroni correction is to reset the alpha criterion, that conventional .05, to a smaller number that exactly cancels out the increased risk of any false positives across your tests. If k is the
number of tests and \(\alpha\) is your criterion the Bonferroni correction is actually to use a criterion of \(\alpha/k\) so if you are using the conventional \(\alpha\) of .05 and you were doing two
tests you would only consider either test statistically significant if the observed p value were \(\leqslant .025\).
So staying with an \(\alpha\) of .05 we get this table of Bonferroni corrected \(\alpha\) levels (correctedAlpha) to use for different numbers of tests.
Show code
maxK <- 10
overallAlpha <- .05
tibble(k = 1:maxK,
alphaDesired = rep(overallAlpha, maxK),
correctedAlpha = overallAlpha / k) -> tibDat
tibDat %>%
flextable() %>%
colformat_double(digits = 4)
k alphaDesired correctedAlpha
1 0.0500 0.0500
2 0.0500 0.0250
3 0.0500 0.0167
4 0.0500 0.0125
5 0.0500 0.0100
6 0.0500 0.0083
7 0.0500 0.0071
8 0.0500 0.0063
9 0.0500 0.0056
10 0.0500 0.0050
Why does this work?
Under the general null hypothesis of no population effect for any of the k tests the probability of any one test giving a false positive is always your chosen \(\alpha\) i.e. .05 usually. The
probability of all of tests not giving a false positive must be \(.95^{k}\) and the risk of at least one false positive must be one minus that risk, i.e. \(1 - .95^{k}\), hence the multiple tests
Show code
k alphaDesired riskFalsePositives
1 0.05 0.05
2 0.05 0.10
3 0.05 0.14
4 0.05 0.19
5 0.05 0.23
6 0.05 0.26
7 0.05 0.30
8 0.05 0.34
9 0.05 0.37
10 0.05 0.40
That’s the problem: the risk of at least one false positive is going up rapidly with the number of tests despite the general null hypothesis, i.e. that nothing is going on for any of the tests, being
true of the population.
The Bonferroni correction works because it is replacing the \(\alpha\) for a single test with \(\alpha/k\) so the risk of at least one false positive is \(1-(\alpha/k)^{k}\). This next table shows
that using the Bonferroni correction, i.e. testing against \(\alpha/k\) keeps the overall risk of at least one false positive (\(1-(\alpha/k)^{k}\)) at .05.
Show code
k alphaDesired correctedAlpha BonfRiskFalsePositives
1 0.05 0.05 0.05
2 0.05 0.03 0.05
3 0.05 0.02 0.05
4 0.05 0.01 0.05
5 0.05 0.01 0.05
6 0.05 0.01 0.05
7 0.05 0.01 0.05
8 0.05 0.01 0.05
9 0.05 0.01 0.05
10 0.05 0.01 0.05
What’s the catch?
The correction is fine to keep the overall (experimentwise or studywise) false positive risk at \(\alpha\) given the general null hypothesis being true for the population from which it is assumed the
sample data came. The problem is that the correction is inevitably costing a lot of statistical power so unless you can easily increase the size of your samples your risk of failing to detect as
statistically significant one or more effects that may be non-null in the population.
This plot shows power for a simple two group t-test with a population effect size of .5 against sample size and how using the Bonferroni correction with different numbers of tests drops the power
below that for a single test (using \(\alpha = .05\)).
Show code
getPower <- function(n, d, alpha){
pwr::pwr.t.test(n = n, d = d, sig.level = alpha)$power
# getPower(100, .5, .05)
tibble(n = 10:100,
k = list(1:10)) %>%
unnest_longer(k) %>%
mutate(power = getPower(n, d = .5, alpha = .05 / k),
k = factor(k,
levels = 1:10)) -> tibPower
ggplot(data = tibPower,
aes(x = n, y = power, colour = k, group = k)) +
geom_point() +
geom_line() +
# geom_hline(yintercept = .05,
# linetype = 3) +
# geom_hline(yintercept = getPower(100, .5, .05),
# colour = "red") +
# ggtitle("Power for two-group t-test and effect size .5 against n",
# paste0("Separate lines for different numbers of tests (k) with Bonferroni correction",
# "\nDotted reference line at .05 and red line at power for single test and n = 100"))
ggtitle("Power for two-group t-test and effect size .5 against n",
"Separate lines for different numbers of tests (k) with Bonferroni correction")
Another way to show this for the same model is show the effect of increasing numbers of tests for a few selected sample sizes (in facets).
Show code
tibPower %>%
filter(n %in% c(10, 25, 50, 100)) %>%
mutate(doubleK = as.double(k)) -> tibPower2
ggplot(data = tibPower2,
aes(x = doubleK, y = power)) +
facet_wrap(facets = vars(n),
nrow = 2) +
geom_line(linewidth = 1) +
geom_point(aes(colour = k),
size = 3) +
breaks = 1:10) +
ggtitle("Power against number of tests",
subtitle = "Facets for samples sizes")
So the catch is that if the general null model is incorrect and you do have non-null effects in the population then you are losing real statistical power to detect these effects using the Bonferroni
correction as the number of tests you apply goes up. There are alternatives to the Bonferroni correction that can balance this inevitable trade off of power for retention of the experimentwise false
positive rate involving different population models but the basic trade off is inescapable. This leads into real issues about when we might take what approach to the multiple tests problem including
accepting that the overall, experimentwise, reportwise, false positive rate may be well above your conventional \(\alpha\) but that you will accept that because you are more worried about failing to
detect individual effects as significant than about the rising overall false positive risk. Given that most papers report more than one test the issue is probably too often either ignored completely,
or dealt with simply by using the Bonferroni correction without much or any discussion of the cost to statistical power. Sadly, I confess that I’ve done both of those wriggles. One problem is that
discussing the issues properly in the discussion section of a paper is not easy to do clearly for all levels of statistical knowledge in the readers and, perhaps more sadly, probably pulls away some
of the pretence that we have clear and easy answers to these trade offs. I have created a shiny app, using the t-test example, to show the trade off between correction of the false positive rate
against loss of statistical power, see here.
Though it’s easy to show the multiple tests problem for NHSTs (Null Hypothesis Significance Tests) the issues apply equally when using confidence intervals(CIs)/estimation rather than the NHST
paradigm though perhaps the sense of definite implications in the move from the maths to the English language is a bit less savage when using CIs/estimation.
Text and figures are licensed under Creative Commons Attribution CC BY-SA 4.0. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in
their caption: "Figure from ...".
For attribution, please cite this work as
Evans (2023, Dec. 23). Chris (Evans) R SAFAQ: Bonferroni correction. Retrieved from https://www.psyctc.org/R_blog/posts/2023-12-23-bonferroni-correction/
BibTeX citation
author = {Evans, Chris},
title = {Chris (Evans) R SAFAQ: Bonferroni correction},
url = {https://www.psyctc.org/R_blog/posts/2023-12-23-bonferroni-correction/},
year = {2023}
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To avoid the large matrices of Sec. 7, consider just the sixteen ancestors of the eight configurations of three cells, for which in principle a
which gives rise to the following moments (the residuals remain when the powers of the dominant frequency are subtracted; the weights, to be used in Eq. 22, were approximated from Abramowitz and
Stegun [34]):
Note how the residuals are eventually dominated by the second largest datum; of course the phenomenon would repeat if an attempt were made to separate more and more of the largest data from the
rest. Even in this restricted example, at least ten moments are required to approximate
Harold V. McIntosh
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Corn ethanol yields continue to improve - U.S. Energy Information Administration (EIA)
May 13, 2015
Source: U.S. Energy Information Administration, Monthly Energy Review
Note: Ethanol volumes do not include gasoline that must be added to denature ethanol, rendering it undrinkable and therefore taxable as fuel rather than as beverage alcohol.
In 2014, U.S. fuel ethanol production reached 14.3 billion gallons of ethanol fuel, the highest level ever. The growth in U.S. fuel ethanol production has outpaced growth in corn consumed as
feedstock—as the industry has grown, it has become more efficient, using fewer bushels of corn to produce a gallon of ethanol.
If ethanol plant yields per bushel of corn in 2014 had remained at 1997 levels (when ethanol made up just 1% of the total U.S. motor gasoline supply), the ethanol industry would have needed to grind
an additional 343 million bushels, or 7% more corn, to produce the same volume of fuel. To supply this incremental quantity of corn without withdrawing bushels from other uses would have required 2.2
million additional acres of corn to be cultivated, an area roughly equivalent to half the land area of New Jersey.
Several factors contributed to the yield increases from a bushel of corn. Increased scale has allowed producers to incorporate better process technology, such as finer grinding of corn to increase
starch conversion and improved temperature control of fermentation to optimize yeast productivity. The growth of the corn ethanol industry also enabled the development of better enzymes and yeast
strains for improved output per bushel of corn.
This growth in ethanol production has been made possible by a rise in demand for ethanol to increase octane levels as MTBE (methyl tert-butyl ether, a gasoline additive) has been phased out of
gasoline, and to meet Renewable Fuel Standard (RFS) targets enacted in 2005 and expanded by subsequent legislation in 2007. RFS requirements effectively placed a floor under ethanol demand. Recently,
ethanol's volumetric share of total U.S. motor gasoline supply has been just below 10%, reaching 9.8% in 2014.
Source: U.S. Energy Information Administration, Short-Term Energy Outlook, May 2015 edition
Principal contributor: Tony Radich
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proxied: Make functions consume Proxy instead of undefined
proxied is a simple library that exports a function to convert constant functions to ones that take a proxy value in the Data.Proxied module. This is useful for retrofiting typeclasses that have
functions that return a constant value for any value of a particular type (but still need to consume some value, since one of the parameterized types must appear in a typeclass function). Often,
these functions are given undefined as an argument, which might be considered poor design.
Proxy, however, does not carry any of the error-throwing risks of undefined, so it is much more preferable to take Proxy as an argument to a constant function instead of undefined. Unfortunately,
Proxy wasn't included in base until GHC 7.8, so many of base's typeclasses still contain constant functions that aren't amenable to passing Proxy. proxied addresses this issue by providing variants
of those typeclass functions that take an explicit proxy value.
This library also contains the Data.Proxyless module, which works in the opposite direction. That is, one can take functions which take Proxy (or undefined) as an argument and convert them to
functions which take no arguments. This trick relies on the -XTypeApplications extension, so it is only available with GHC 8.0 or later. This library also offers Data.Proxyless.RequiredTypeArguments,
a variant of Data.Proxyless that uses -XRequiredTypeArguments to make type arguments explicit, which is only available with GHC 9.10 or later.
Skip to Readme
Maintainer's Corner
For package maintainers and hackage trustees
Versions [RSS] 0.1, 0.1.1, 0.2, 0.3, 0.3.1, 0.3.2
Change log CHANGELOG.md
Dependencies base (>=4.3 && <5), generic-deriving (>=1.10.1 && <2), tagged (>=0.4.4 && <1) [details]
Tested with ghc ==7.0.4, ghc ==7.2.2, ghc ==7.4.2, ghc ==7.6.3, ghc ==7.8.4, ghc ==7.10.3, ghc ==8.0.2, ghc ==8.2.2, ghc ==8.4.4, ghc ==8.6.5, ghc ==8.8.4, ghc ==8.10.7, ghc ==9.0.2, ghc ==
9.2.8, ghc ==9.4.8, ghc ==9.6.5, ghc ==9.8.2, ghc ==9.10.1
License BSD-3-Clause
Copyright (C) 2016-2017 Ryan Scott
Author Ryan Scott
Maintainer Ryan Scott <ryan.gl.scott@gmail.com>
Category Data
Home page https://github.com/RyanGlScott/proxied
Bug tracker https://github.com/RyanGlScott/proxied/issues
Source repo head: git clone https://github.com/RyanGlScott/proxied
Uploaded by ryanglscott at 2024-04-20T11:56:13Z
Distributions LTSHaskell:0.3.2, NixOS:0.3.2, Stackage:0.3.2
Reverse 2 direct, 0 indirect [details]
Downloads 4043 total (43 in the last 30 days)
Rating (no votes yet) [estimated by Bayesian average]
Your Rating
Status Docs available [build log]
Last success reported on 2024-04-20 [all 1 reports]
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Relationship of Line and Phase Voltages & Currents in a Star Connected System
To derive the relations between line and phase currents and voltages of a star connected system, we have first to draw a balanced star connected system.
Suppose due to load impedance the current lags the applied voltage in each phase of the system by an angle ϕ. As we have considered that the system is perfectly balanced, the magnitude of current and
voltage of each phase is the same. Let us say, the magnitude of the voltage across the red phase i.e. magnitude of the voltage between neutral point (N) and red phase terminal (R) is V[R].
Similarly, the magnitude of the voltage across yellow phase is V[Y] and the magnitude of the voltage across blue phase is V[B]. In the balanced star system, magnitude of phase voltage in each phase
is V[ph].
∴ V[R] = V[Y] = V[B] = V[ph]
We know in the star connection; line current is same as phase current. The magnitude of this current is same in all three phases and say it is I[L].
∴ I[R] = I[Y] = I[B] = I[L], Where, I[R] is line current of R phase, I[Y] is line current of Y phase and I[B] is line current of B phase. Again, phase current, I[ph] of each phase is same as line
current IL in star connected system.
∴ I[R] = I[Y] = I[B] = I[L] = I[ph].
Now, let us say, the voltage across R and Y terminal of the star connected circuit is V[RY].
The voltage across Y and B terminal of the star connected circuit is V[YB The voltage ]across B and R terminal of the star connected circuit is VBR.
From the diagram, it is found that
VRY = VR + (− VY)
Similarly, VYB = VY + (− VB)
And, VBR = VB + (− VR)
Now, as angle between VR and VY is 120o(electrical), the angle between VR and – VY is 180o – 120o = 60o(electrical).
Thus, for the star-connected system line voltage = √3 × phase voltage.
Line current = Phase current
As, the angle between voltage and current per phase is φ, the electric power per phase is
So the total power of three phase system is
1.Bunty B. Bommera
2.Dakshata U. Kamble
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Three team trade with Cards moving to #5?
I would ask more, but like this
Nice idea but the Cards need more from the Vikings
Go for it
Supporting Member
Sep 5, 2005
Reaction score
As long as the Vikes take a QB we still get MHJ if he's still there
But only moving up 4 spots in the bottom of the first doesn't seem like enough
Mar 7, 2003
Reaction score
Would like to see all the details as it sounds like the Chargers would go to 11 and get 27. We get 5 & 23. Think Minnesota would have to pay more. Maybe Jefferson is going to LA? If that is the case
then I think the Cards should get a 2nd as well this year. MHJ @ 5 then BPA at 23, 2 2nds and 3 3rds help you get alot of talent fast.
If verified and true. I assume that this is the Vikings opening idea and their opening offer..
If you look at the draft value chart, The Chargers get a bit extra in terms of points for the move, but Cardinals are not.
Cards give up 2480 and gain 2460 = -20
Vikings give up 2010 and gain 1800 = -210
Chargers give up 1700 and gain 1930 = +230
There should be more suitors for #4 and I would ask more. And then the Franchise QB tax on top. I would ask for 2025 1st and settle for 2025 2nd or swapping our 2025 2nd for their 2025 1st
Last edited:
Sep 12, 2002
Reaction score
I would ask more, but like this
I’m assuming that is if a QB is available to the Vikings and MJH for us? I like the early 20s pick.
May 15, 2002
Reaction score
This trade makes quite a bit of sense for all 3 teams. I think the Cardinals ultimately get an additional pick. #5 still get’s us Harrison, and #23 get’s us in range for Newton.
Dec 9, 2019
Reaction score
As the deal currently stands, hate it.
You gift MIN their QBOF to move up 4 spots at the end of the first? (without knowing who will be available at #23 or #27)
This only works in a world where MHJ is the only option at WR and #4 isn't going to be 'the last QB'.
You could stick and pick at #4 and be in no worse of a situation. You could trade with the Giants and pick up a 1st/2nd/more. MIN's season is F'd if they don't make this deal work......we hold the
freaking pick.
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Mar 10, 2004
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I was going to post this Tweet yesterday, then I checked his Twitter/Profile and asked myself, what would
Mar 10, 2004
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Jul 16, 2004
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I have seen that kid. Not a crazy as most. Is a Cardinalfan for sure.
Jul 21, 2002
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I would ask more, but like this
Why not if you still get Harrison?
Jul 21, 2002
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If verified and true. I assume that this is the Vikings opening idea and their opening offer..
If you look at the draft value chart, The Chargers get a bit extra in terms of points for the move, but Cardinals are not.
Cards give up 2480 and gain 2460 = -20
Vikings give up 2010 and gain 1800 = -210
Chargers give up 1700 and gain 1930 = +230
There should be more suitors for #4 and I would ask more. And then the Franchise QB tax on top. I would ask for 2025 1st and settle for 2025 2nd or swapping our 2025 2nd for their 2025 1st
Eh screw the value chart. If you still get MHJ, then losing the value of 4 is only math, it's not real.
But the gain from 27 to 23 is real.
Eh screw the value chart. If you still get MHJ, then losing the value of 4 is only math, it's not real.
But the gain from 27 to 23 is real.
I disagree. The chart is Almost always on point in non QB trades
Mar 8, 2015
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Nice idea but the Cards need more from the Vikings
They don't actually need more.
I would think everyone hyperventilating over the thought of not drafting MHJ would love this. We still get him, and we move up 4 spots with our second pick.
How good this deal is completely depends on how much you think of Harrison.
Aug 24, 2002
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At dinner last night and ESPN was on one TV. I look up and it said “Cardinals trade Kyler Murray to Vikings for 11 & 23” and I yelled “what!”… guess is was an ESPN prediction…
Apr 26, 2005
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I want a day 3 pick, just moving up four places ain't enough. Vikings are loaded with day 3 picks, cough up a 4 or a 5.
Jan 13, 2023
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The problem is Minny is the one proposing these trades, but they dont have much to give unless they are trading players.
id either ask for Minnys 3rd next year....or I would throw them #90 and ask for Addison.
Apr 26, 2005
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Here is my counter. Instead of coughing up #27 we give the Vikings/Chargers our second round pick #35 overall AND we also receive a 2025 day 3 pick from Minnesota.
Michael snuggles the cap space
Jul 22, 2002
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the net / net / net
for moving back one spot from #4 to #5, the Cards move up three spots from #27 to #23
the "value" per the trade value chart works -- about 100 points of diff for both moves, or the equiv of a higher 4th round pick
Chargers are the real winners here -- giving up 1700 points and getting 1930 back.
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Jul 21, 2002
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The QB tax has to be applied imo. Vikings need us or we trade with the Giants.
They don't actually need more.
I would think everyone hyperventilating over the thought of not drafting MHJ would love this. We still get him, and we move up 4 spots with our second pick.
How good this deal is completely depends on how much you think of Harrison.
A potential franchise quarterback is far more valuable.
The only upside for the Cardinals versus just staying at number four is they get to move up four spots at the end of the first round
Meanwhile Minnesota gets their quarterback of the future
the net / net / net
for moving back one spot from #4 to #5, the Cards move up three spots from #27 to #23
the "value" per the trade value chart works -- about 100 points of diff for both moves, or the equiv of a higher 4th round pick
Chargers are the real winners here -- giving up 1700 points and getting 1930 back.
Which is why this trade is not the best for the Cardinals.
The Cardinals should be the ones that get the most value
Jan 10, 2020
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Nice idea but the Cards need more from the Vikings
Do they though?
They are just moving back one spot, still getting the player they want and gaining 4 spots in the 20's. Why would you turn it down? It's already no risk and all reward.
It's like ordering lunch and being offered a free apple pie to move one spot over at the counter so a couple can sit together.
Comin for you!
Super Moderator
Supporting Member
May 13, 2002
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Why even discuss? This is a made up rumor from a random internet guy?
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Nambu-Jona-Lasinio model
Nambu-Jona-Lasinio model
From Scholarpedia
Giovanni Jona-Lasinio and Yoichiro Nambu (2010), Scholarpedia, 5(12):7487. doi:10.4249/scholarpedia.7487 revision #148546 [link to/cite this article]
The Nambu-Jona-Lasinio model is an effective chiral field theory realizing the spontaneous symmetry breaking mechanism.
Spontaneous breakdown of symmetry (SSB) is a concept that is applicable only to systems with infinitely many degrees of freedom. Although it pervaded the physics of condensed matter for a very long
time, magnetism is a prominent example, its formalization and the recognition of its importance has been an achievement of the second half of the 20th century. Strangely enough the name was adopted
only after its introduction in particle physics: it is due to Baker and Glashow (1962).
What is SSB? In condensed matter physics it means that the lowest energy state of a system can have a lower symmetry than the forces acting among its constituents and on the system as a whole. As an
example consider a long elastic bar on top of which we apply a compression force directed along its axis. Clearly there is rotational symmetry around the bar which is maintained as long as the force
is not too strong: there is simply a shortening according to the Hooke's law. However when the force reaches a critical value the bar bends and we have an infinite number of equivalent lowest energy
states which differ by a rotation.
Heisenberg (1959, 1960) was probably the first to consider SSB as a possibly relevant concept in particle physics but his proposal was not the physically right one. The theory of superconductivity of
Bardeen, Cooper and Schrieffer (1957) provided the key paradigm for the introduction of SSB in relativistic quantum field theory and particle physics on the basis of an analogy proposed by Nambu
To appreciate the innovative character of this concept in particle physics one should consider the strict dogmas which constituted the foundation of relativistic quantum field theory at the time. One
of the dogmas stated that all the symmetries of the theory, implemented by unitary operators, must leave the lowest energy state, the vacuum, invariant. This property does not hold in presence of SSB
and degenerate vacua. These vacua cannot be connected by local operations and are orthogonal to each other giving rise to different Hilbert spaces. If we live in one of them SSB will be manifested by
its consequences, in particular the particle spectrum.
The BCS theory of superconductivity, immediately after its appearance, was reformulated and developed by several authors including Bogolyubov, Valatin, Anderson, Rickayzen and Nambu. The following
facts were emphasized
1. The ground state proposed by BCS is not invariant under gauge transformations.
2. The elementary fermionic excitations (quasi-particles) are not eigenstates of the charge as they appear as a superposition of an electron and a hole.
3. In order to restore charge conservation these excitations must be the source of bosonic excitations described by a long range (zero mass) field. In this way the original gauge invariance of the
theory is restored.
The peculiarity of the paper of Nambu (1960b), was that he used a language akin to quantum field theory, that is the Green's functions formalism, and the role of gauge invariance was discussed in
terms of vertex functions and the associated Ward identities. The search for analogies in particle physics became quite natural. In particular, following the suggestion of Nambu (1960a), the study of
chiral symmetry breaking was developed in detail in two papers by Nambu and Jona-Lasinio (1961a,1961b) which had a considerable influence on the evolution of elementary particle theories.
The analogy with superconductivity
Let us illustrate the elements of the analogy.
Electrons near the Fermi surface are described by the following equation \[ E \psi_{p,+} &= \epsilon_p \psi_{p,+} + \phi \psi_{-p,-}^{\dagger} \\ E \psi_{-p,-}^{\dagger} &= -\epsilon_p \psi_{-p,-}^{\
dagger} + \phi \psi_{p,+} , \] with eigenvalues \[ E = \pm \sqrt{\epsilon_p^2 + \phi^2}. \]
Here, \(\psi_{p,+}\) and \(\psi_{-p,-}^{\dagger}\) are the wavefunctions for an electron and a hole of momentum \(p\) and spin \(+\ ;\) \(\phi\) is the gap.
In the Weyl representation, the Dirac equation reads \[ E \psi_{1} &= \mathbf{\sigma} \cdot \mathbf{p} \psi_{1} + m \psi_{2} \\ E \psi_{2} &= -\mathbf{\sigma} \cdot \mathbf{p} \psi_{2} + m \psi_{1},
\] with eigenvalues \[ E = \pm \sqrt{p^2 + m^2}. \] Here, \(\psi_{1}\) and \(\psi_{2}\) are the eigenstates of the chirality operator \(\gamma_5\ .\) Particles with mass are superpositions of states
of opposite chirality. The similarity is obvious.
The bosonic excitations necessary to restore gauge invariance in a superconductor appear in the approximate expressions for the charge density and the current in a BCS superconductor (Nambu Y, 1960b)
, \[ \rho(x,t) &\simeq \rho_0 + \frac{1}{\alpha^2} \partial_t f \\ \mathbf{j}(x,t) &\simeq \mathbf{j}_0 - \mathbf{\nabla} f , \] where \(\rho_0 = e \Psi^{\dagger} \sigma_3 Z \Psi\) and \(\mathbf{j}_0
= e \Psi^{\dagger} (\mathbf{p}/m) Y \Psi\) are the contributions of the quasi-particles, \(Y\ ,\) \(Z\ ,\) \(\alpha\) are constants and \(f\) satisfies the wave equation \[ \left( \nabla^2 - \frac{1}
{\alpha^2} {\partial_t}^2 \right) f \simeq -2 e \Psi^{\dagger} \sigma_2 \phi \Psi. \] Here, \(\Psi^{\dagger} = (\psi^{\dagger}_1, \psi_2)\ .\)
In the elementary particle context the axial current \(\bar{\psi} \gamma_5 \gamma_{\mu} \psi\) is the analog of the electromagnetic current in BCS theory. In the hypothesis of exact conservation, the
matrix elements of the axial current between nucleon states of four-momentum \(p\) and \(p'\) have the form \[ \Gamma_{\mu}^A (p', p) = \left( i \gamma_5 \gamma_{\mu} - 2m \gamma_5 q_{\mu} / q^2 \
right) F(q^2), \qquad q = p' - p . \] Exact conservation is compatible with a finite nucleon mass \(m\) provided there exists a massless pseudoscalar particle.
Assuming exact conservation of the chiral current, a picture of chiral SSB may consist in a vacuum of a massless Dirac field viewed as a sea of occupied negative energy states, and an attractive
force between particles and antiparticles having the effect of producing a finite mass, the counterpart of the gap. The pseudoscalar massless particle, which may be interpreted as a forerunner of the
pion, corresponds to the bosonic field associated to the fermionic quasi-particles in a superconductor.
To implement this picture the construction of a relativistic field theoretic model was required. At that time Heisenberg and his collaborators had developed a comprehensive theory of elementary
particles based on a non linear spinor interaction: the physical principle was that spin ½ fermions could provide the building blocks of all known elementary particles. Heisenberg was however very
ambitious and wanted at the same time to solve in a consistent way the dynamical problem of a non renormalizable theory. This made their approach very complicated and not transparent.
The Nambu-Jona-Lasinio model
A Heisenberg type Lagrangian was adopted without pretending to solve the non-renormalizability problem and introducing a relativistic cut-off to cure the divergences. This model is known in the
literature with the acronym NJL (Nambu-Jona-Lasinio model). The energy scale of interest was of the order of the nucleon mass and one hoped that higher energy effects would not change substantially
the picture.
The Lagrangian of the NJL model is \[ \mathcal {L} = -\bar{\psi} \gamma_{\mu} \partial_{\mu} \psi + g \left[ \left( \bar{\psi} \psi \right)^2 - \left( \bar{\psi} \gamma_5 \psi \right)^2 \right] . \]
It is invariant under ordinary and chiral gauge transformations \[ \psi &\to e^{i \alpha} \psi, \qquad &\bar{\psi} &\to \bar{\psi} e^{-i \alpha} \\ \psi &\to e^{i \alpha \gamma_5} \psi, \qquad &\bar
{\psi} &\to \bar{\psi} e^{i \alpha \gamma_5} . \]
To investigate the content of the model a simple mean field approximation for the mass was adopted \[ m &= -2g \left[ <\bar{\psi} \psi> - \gamma_5 <\bar{\psi} \gamma_5 \psi> \right] \\ &= 2g \left[
tr S^{(m)}(0) - tr \gamma_5 S^{(m)}(0) \right], \] where \(S^{(m)}\) is the propagator of the Dirac field of mass \(m\ ,\) or more explicitly, \[ \frac{2 \pi^2}{g \Lambda^2} = 1 - \frac{m^2}{\Lambda^
2} \ln \left(1 + \frac{\Lambda^2}{m^2} \right), \] where \(\Lambda\) is the invariant cut-off. This equation is very similar to the gap equation in BCS theory. If \(\frac{2\pi^2}{g\Lambda^2} < 1\)
there exists a solution \(m>0\ .\)
From this relationship a rich spectrum of bound states follows, \[ \begin{array}{cccc} \hline \hline \text{nucleon} & \text{mass} \mu & \text{spin-parity} & \text{spectroscopic} \\ \text{number} & &
& \text{notation} \\ \hline 0 & 0 & 0^- & ^1S_0 \\ 0 & 2m & 0^+ & ^3P_0 \\ 0 & \mu^2 > \frac{8}{3} m^2 & 1^- & ^3P_1 \\ \pm 2 & \mu^2 > 2 m^2 & 0^+ & ^1S_0 \\ \hline \hline \end{array} \]
The bosonic field in the superconductor and the pseudoscalar particle in the NJL model are special cases of a general proposition formulated by Goldstone (1961).
Whenever the original Lagrangian has a continuous symmetry group, which does not leave the ground state invariant, massless bosons appear in the spectrum of the theory.
Other examples are:
│ physical system │ broken symmetry │ massless bosons │
│ ferromagnets │ rotational invariance │ spin waves │
│ crystals │ translational and rotational invariance │ phonons │
These massless bosons are now known in the literature as Nambu-Goldstone bosons. In nature, however, the axial current is only approximately conserved. The model made contact with the real world
under the hypothesis that the small violation of axial current conservation gives a mass to the massless boson, which is then identified with the \(\pi\) meson. Actually the NJL model, reinterpreted
in terms of quarks, has been the starting point of a successful effective theory of low energy QCD, see e.g. the review by Hatsuda and Kunihiro (1994).
After the NJL model, SSB became a key concept in elementary particle physics, in particular electroweak unification (Weinberg S, 1967) requires SSB.
• Baker(1962). Spontaneous Breakdown of Elementary Particle Symmetries. Physical Review 128(5): 2462-2471. doi:10.1103/physrev.128.2462.
• Heisenberg, W; Dürr, H P; Mitter, H; Schlieder, S and Yamazaki, K (1959). Zur Theorie der Elementarteilchen. Zeitschrift für Naturforschung 14a: 441-485.
• Heisenberg, W (1960). Recent research on the nonlinear spinor theory of elementary particles. Proc. 1960 Annual Intern. Conf. on High Energy Physics, Rochester : 851-858.
• Bardeen, J; Cooper, L N and Schrieffer, J R (1957). Microscopic Theory of Superconductivity. Physical Review 106(1): 162-164. doi:10.1103/physrev.106.162.
• Nambu, Y (1960a). Axial Vector Current Conservation in Weak Interactions. Physical Review Letters 4: 380-382. doi:10.1103/physrevlett.4.380.
• Nambu, Y (1960b). Quasi-Particles and Gauge Invariance in the Theory of Superconductivity. Physical Review 117: 648-663. doi:10.1103/physrev.117.648.
• Nambu(1961a). Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I. Physical Review 122(1): 345-358. doi:10.1103/physrev.122.345.
• Nambu(1961b). Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II. Physical Review 124(1): 246-254. doi:10.1103/physrev.124.246.
• Goldstone, J (1961). Field theories with Superconductor solutions. Nuovo Cimento 19: 154-164. doi:10.1007/bf02812722.
• Hatsuda(1994). QCD phenomenology based on a chiral effective Lagrangian. Physics Reports 247(5-6): 221-367. doi:10.1016/0370-1573(94)90022-1.
• Weinberg, S (1967). A Model of Leptons. Physical Review Letters 19(21): 1264-1266. doi:10.1103/physrevlett.19.1264.
Further reading
See also
Bardeen-Cooper-Schrieffer theory, Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism, Gauge invariance, Gauge theories, Nambu-Jona-Lasinio model (Particle physics, Nuclear physics), Symmetry
breaking in quantum systems
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The DGA of Ranbyus
Table of Contents
The DGA in this blog post has been implemented by the DGArchive project.
Ranbyus is a trojan that steals banking information — among other personal data. End of April 2015 I first noticed samples of Ranbyus that use a Domain Generation Algorithm (DGA) to generate the
domains for its command and control (C2) traffic:
In this post I show how the DGA works by reversing the following sample:
PE32 executable (GUI) Intel 80386, for MS Windows
I focused my efforts exclusively on the domain generation part of Ranbyus. Refer to the blog posts of Aleksandr Matrosov here and here for an in-depth analysis of Ranbyus.
This section shows the algorithm behind the domains of Ranbyus and its seeding and parametrization.
Callback Loop
The next image represents the part of the Ranbyus that tries to find a valid C2 target. It consists of an outer loop (month_loop) and an inner loop. The register edi holds the index of the outer
loop. It runs from 0 down to -30. The number of iterations for the inner loop is specified by a parameter of the DGA (set to 40 in all analysed samples):
The first act of the outer loop is to get the current time:
Ranbyus then subtracts days from the current date according to the index of the outer loop:
The resulting date will be used to seed the DGA with a granularity of one day. In the first iteration, the DGA uses the current date. In the next iteration — when the index is -1 — yesterday’s date
is used. This continues up to 30 days in the past if need be. So even though the DGA generates a fresh set of domains every day, it also checks the domains of past days. This gives the DGA the
benefit of fast changing domains in case domains get blocked or sinkholed, while at the same time enabling older domains to be used for up to one month if they still work.
The inner loop generates the domains for the day with the_dga and makes the callback. In case of failure, Ranbyus sleeps for wait_time milliseconds (500 in my samples) and retries up to nr_of_domains
(40) different domains.
DGA Parameters and Seed
Apart from the current date, the DGA is seeded with a hardcoded magic number:
The number of domains per day is hardcoded to 40:
The wait time after a failed callback attempt is set to 500 ms:
Ranbyus also uses a hard-coded list of top level domains:
The top level domains are: .in, .me, .cc, .su, .tw, .net, .com, .pw, and .org. The last domain .org is never used due to a bug of the DGA. The top level domains are tested one after another (except
the last one), starting at a day-dependent offset:
The error of subtracting 1 from the modulus is repeated also when picking the letters of the second level domain.
The DGA
This is the disassembly of the DGA routine:
The subroutine generates domains in two independent parts:
1. the top level domain is picked from the hardcoded list shown above
2. the second level domain is generated.
The following disassembly shows how the top level domain is picked:
Starting at the day dependent offset determined earlier, the algorithm picks the domains in a circular fashion, omitting the last domain because the DGA wrongly subtracts one from the modulus.
[".in", ".me", ".cc", ".su", ".tw", ".net", ".com", ".pw", ".org"][offset % (9-1)]
The disassembly for the second level domain looks as follows. It generates 14 different letters based on the DGA’s seed, and the value of day, month and year. Note that these names are misleading:
although theses values are initialized with the current or past dates, the values are modified by each call to the routine.
This pseudo-code summarizes the algorithm:
FOR i = 0 TO 13
day = (day >> 15) ^ 16 * (day & 0x1FFF ^ 4 * (seed ^ day))
year = ((year & 0xFFFFFFF0) << 17) ^ ((year ^ (7 * year)) >> 11)
month = 14 * (month & 0xFFFFFFFE) ^ ((month ^ (4 * month)) >> 8)
seed = (seed >> 6) ^ ((day + 8 * seed) << 8) & 0x3FFFF00
int x = ((day ^ month ^ year) % 25) + 'a'
domain[i] = x;
The malware authors repeated their modulus error: like for the tld, the modulus needed to be increased by one. As it stands, ‘z’ is no reachable. Ranbyus shares this bug with the DGAs of Ramnig and
Seed the end of this blog post for a C-implementation of the DGA.
Observed Seeds
The following tables lists some of the samples from malwr.com that are Ranbyus with the described DGA. All samples use the same parametrization, only the seed varies.
md5 seed
4b04f6baaf967e9c534e962c98496497 65BA0743
087b19ce441295207052a610d1435b03 65BA0743
28474761f28538a05453375635a53982 65BA0743
b309eab0277af32d7a344b8a8b91bd73 C5F128F3
4c7057ce783b2e4fb5d1662a5cb1312a C5F128F3
b309eab0277af32d7a344b8a8b91bd73 C5F128F3
7cbc671bcb97122e0ec5b448f0251dc0 C5F128F3
437028f94ceea4cab0d302d9ac6973eb C5F128F3
b309eab0277af32d7a344b8a8b91bd73 C5F128F3
6378b7af643e87c063f69ddfb498d852 B6354BC3
fa57f601402aab8168dea94c7c5f029f B6354BC3
9f2c89ad17e9b6cf386028a8c9189264 0478620C
DGA Characteristics
The characteristics of Ranbyus’ DGA are:
property value
seed magic number and current date
granularity 1 day, with a 31 day sliding window
domains per seed and day 40
domains per sliding window 1240
sequence sequential
wait time between domains 500 ms
top level domains .in, .me, .cc, .su, .tw, .net, .com, .pw
second level characters lower case letters except ‘z’
second level domain length 14 letters
Decompiled Code
The following C code generates the domains for a given day and seed. In order to generate all domains that the malware can generate for any given seed and date, one would also need to run the code
for all dates going 30 days in the past.
Edit 23.5.2015: The following code had contained a bug that led to a wrong sequence of top level domains, thanks to Anthony Kasza for sharing that with me.
#include <stdio.h>
#include <stdlib.h>
char* dga(unsigned int day, unsigned int month, unsigned int year,
unsigned int seed, unsigned int nr)
char *tlds[] = {"in", "me", "cc", "su", "tw", "net", "com", "pw", "org"};
char domain[15];
int d;
int tld_index = day;
for(d = 0; d < nr; d++)
unsigned int i;
for(i = 0; i < 14; i++)
day = (day >> 15) ^ 16 * (day & 0x1FFF ^ 4 * (seed ^ day));
year = ((year & 0xFFFFFFF0) << 17) ^ ((year ^ (7 * year)) >> 11);
month = 14 * (month & 0xFFFFFFFE) ^ ((month ^ (4 * month)) >> 8);
seed = (seed >> 6) ^ ((day + 8 * seed) << 8) & 0x3FFFF00;
int x = ((day ^ month ^ year) % 25) + 97;
domain[i] = x;
printf("%s.%s\n", domain, tlds[tld_index++ % 8]);
main (int argc, char *argv[])
if(argc != 5) {
printf("Usage: dga <day> <month> <year> <seed>\n");
printf("Example: dga 14 5 2015 b6354bc3\n");
dga(atoi(argv[1]), atoi(argv[2]), atoi(argv[3]),
strtoul(argv[4], NULL, 16), 40);
Archived Comments
Note: I removed the Disqus integration in an effort to cut down on bloat. The following comments were retrieved with the export functionality of Disqus. If you have comments, please reach out to me
by Twitter or email.
Aug 26, 2015 07:42:01 UTC
In the C code the strtoul function should be used instead as strtol will limit the seed to 0x7fffffff.
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How to explain ensemble models to your grandma - QuantSense
An ensemble model can significantly improve the reliability of your decisions. No wonder this powerful concept drives common machine learning models like random forest and gradient boost. But how can
you explain to your grandma why ensemble models are so useful?
Let's assume you have just arrived at a new place you have never been before. You make it to this T-junction not being sure whether you should either turn left or turn right to get to your hotel. You
decide to ask a local. He tells you to be 80% certain that you should turn left. What should you do? If you follow his advice, your chances of going into the wrong direction are still 20%. That's why
you decide to ask two other locals who are passing by. They both recommend you (again with 80% certainty) that you should turn right. This contradictory information may seem confusing at first, but
then you realize that the majority (2 out of 3 locals) recommends you to turn right. How can this reduce your risk of still going into the wrong direction? Well, let's calculate the probability that
X locals (X = 3, 2, 1, 0) happen to send you into the wrong direction:
• P[X=3] = 20% x 20% x 20% = 0.8%
• P[X=2] = 20% x 80% x 80% x 3 = 9.6%
• P[X=1] = 80% x 80% x 20% x 3 = 38.4%
• P[X=0] = 80% x 80% x 80% = 51.2%
Note that these probablities add up to 100%, as you would expect.
As you decide to use the majority vote to make up your mind, you will only arrive late at your hotel for dinner if at least 2 out of 3 locals give you the wrong information. The probability that this
happens is only 0.8% + 9.6% = 10.4%.
Conclusion: by asking three different individuals and taking a decision based on the majority, you almost decreased your error rate by half!
From this simplified example we can conclude the following:
• using several sources of information for decision making can reduce the error rate;
• this approach assumes the sources of information to be independent.
Congratulations! You did your best explaining an ensemble model to your grandma. Feel free to drop me a line for any suggestions you (or your grandma) may have.
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AI Search Algorithms Explained
Understanding Search Algorithms in AI
Artificial Intelligence (AI) systems often need to navigate through complex problem spaces to find optimal solutions. Whether it's solving puzzles, routing problems, or playing games, search
algorithms are fundamental tools that guide AI in exploring possible solutions. Two widely-used search algorithms in AI are Depth First Search (DFS) and Breadth First Search (BFS). These algorithms
provide a structured way for AI systems to solve problems that can be represented as graphs or trees.
What Are Search Algorithms?
In AI, search algorithms are strategies used to find paths from an initial state (or starting point) to a goal state (or solution). These paths represent a sequence of decisions or actions that an AI
must take to solve a problem.
For example, imagine an AI trying to solve a maze. The AI's task is to navigate from the start of the maze to the exit. Every turn it takes represents a decision, and the goal is to find a sequence
of turns that leads to the exit. Search algorithms help AI explore these possibilities systematically.
In AI, search problems are typically framed as:
• Initial state: Where the AI starts.
• Goal state: The desired outcome or solution.
• Actions: The possible decisions or moves the AI can make.
• Path cost: The cost of taking specific actions (in some cases).
• Search space: All possible states or configurations the AI can encounter.
The challenge lies in exploring the search space efficiently to find a solution while minimizing resources like time and memory.
Depth First Search (DFS)
Depth First Search (DFS) is a search algorithm that explores a problem space by going as deep as possible into the search tree or graph before backtracking. DFS explores one path fully before
considering alternative paths, making it a last in, first out (LIFO) approach.
How DFS Works:
1. Start at the initial state: DFS begins by considering the initial state.
2. Choose a path and explore: It picks one of the available paths and follows it to the next state.
3. Go as deep as possible: DFS continues to follow this path, exploring deeper and deeper into the search space, until it reaches a dead end (no further states to explore).
4. Backtrack if necessary: If DFS hits a dead end or the goal state is not reached, it backtracks to the previous decision point and explores a different path.
5. Continue until goal: This process repeats until the AI finds a goal state or exhausts all possible paths.
Example: DFS in a Maze
Consider an AI tasked with navigating a maze. Using DFS, the AI will pick one path and follow it to the end. If it encounters a wall or dead end, it backtracks to the last fork in the road and tries
a different path.
In a large maze, DFS may explore deep into one section before realizing it has hit a dead end and needs to start over, which can be inefficient if the solution lies closer to the starting point.
DFS Characteristics:
• Memory-efficient: DFS only needs to remember the current path, making it relatively memory-efficient, especially in large search spaces.
• May not find optimal solutions: Since DFS explores paths one by one, it can miss shorter or more optimal paths if those are not explored first.
• Prone to getting stuck in loops: If not implemented carefully, DFS can get stuck exploring loops or revisiting the same states repeatedly.
Breadth First Search (BFS)
Breadth First Search (BFS) is a search algorithm that explores the search space level by level, considering all the neighbors of the current state before moving deeper into the search tree. BFS uses
a first in, first out (FIFO) approach, exploring shallow paths first.
How BFS Works:
1. Start at the initial state: BFS begins by considering the initial state.
2. Explore all immediate neighbors: It then explores all states that can be reached directly from the initial state (i.e., all neighboring states).
3. Move to the next level: After exploring all immediate neighbors, BFS moves on to explore the neighbors of those neighbors, continuing this pattern level by level.
4. Continue until goal: BFS repeats this process until it finds the goal state.
Example: BFS in a Maze
In the context of solving a maze, BFS will first explore all possible moves that are one step away from the starting point. If it doesn’t find the exit, it will then explore all states that are two
steps away, then three steps away, and so on. This guarantees that BFS will find the shortest path to the exit, as it explores all shorter paths first.
BFS Characteristics:
• Guaranteed to find the optimal solution: BFS is guaranteed to find the shortest path to the goal, as it explores all possible paths level by level.
• Memory-intensive: Since BFS must remember all nodes at the current level of the search tree, it can require significant memory, especially in large search spaces.
• Slower for deep searches: In cases where the solution is deep within the search tree, BFS can take a long time to explore all the shallow levels first, making it slower than DFS in certain
Comparing DFS and BFS
1. Search Strategy:
• DFS: Explores paths deeply first, backtracking when necessary.
• BFS: Explores paths level by level, ensuring that shallow paths are explored before deep ones.
2. Optimality:
• DFS: May not find the optimal solution, especially in cases where shorter paths are deeper in the tree.
• BFS: Guarantees the shortest path if one exists, making it optimal for problems where the shortest or least-cost path is desired.
3. Memory Usage:
• DFS: Memory-efficient, as it only needs to remember the current path.
• BFS: Can consume a lot of memory, as it needs to store all nodes at the current level.
4. Efficiency:
• DFS: Can be faster in finding solutions, especially in deep trees, but may waste time exploring non-optimal paths.
• BFS: Slower in deep searches, but guaranteed to find the shortest path.
Applications of DFS and BFS in AI
1. Solving Puzzles:
□ Search algorithms like DFS and BFS are used to solve puzzles like the 15-puzzle, where the goal is to arrange tiles in a specific order by sliding them. DFS may explore one path of moves
deeply, while BFS guarantees finding the shortest sequence of moves.
2. Maze Solving:
□ In maze-solving problems, BFS is often preferred because it finds the shortest path to the goal. DFS might get lost in deep paths or take longer routes.
3. Pathfinding in Games:
□ AI systems in video games often use BFS or DFS to move non-player characters (NPCs) through the environment. For example, BFS ensures that the NPC takes the shortest route to chase a player,
while DFS might be useful for exploring areas of the map deeply.
4. Routing Problems:
□ BFS and DFS are employed in network routing, where the AI must find efficient paths for data packets to travel across the internet. BFS is often used when finding the shortest route is
When to Use DFS or BFS
• Use DFS when:
□ Memory is limited, and you need a memory-efficient solution.
□ The solution is likely to be deep in the search tree.
□ The search space is very large, but you can tolerate non-optimal solutions.
• Use BFS when:
□ You need to find the shortest or optimal path.
□ Memory usage is not a primary concern.
□ The search space is small or the solution is expected to be closer to the starting point.
Search algorithms like Depth First Search (DFS) and Breadth First Search (BFS) form the backbone of many AI problem-solving strategies. Whether an AI is navigating through a maze, solving a puzzle,
or determining the best route in a network, these algorithms enable it to explore possible solutions systematically. While DFS is more memory-efficient and faster for deep searches, BFS guarantees
finding the shortest path. Understanding when and how to apply these algorithms is crucial for solving complex AI problems effectively.
In the world of AI, where problem-solving is key, mastering search algorithms is a vital step in developing intelligent, efficient systems. As AI continues to evolve, new algorithms will build on the
foundations laid by DFS and BFS, helping machines solve ever more complex problems.
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(The) Boring Investor | Quint Essenceny
(The) Boring Investor
Some visitors might know that I run a parallel blog at (The) Boring Investor’s Statistics that shows some of the investment figures which i monitor on a regular basis. Among these statistics is a
forecast of the rates of interest of the Singapore Savings Bonds (SSBs) to be announced in the upcoming month.
The rates of interest for the SSB to be announced in the next month is dependant on the average yield (i.e. interest rates) of the Singapore Government Securities (SGS) benchmark bonds in today’s
month. There is, however, a small issue. Applications for SSBs close on the 4th last working day of the month. For my (The) Boring Investor’s Statistics blog, I usually blog on weekends only, hence,
the forecast is carried and posted even earlier out, on the weekend before the close of application. Which means that the post can sometimes be as many as 8 business days from the finish of the
month. The forecast error can be bigger.
As a good example, for Oct, the application form shut on 26 Oct, but my forecast was published last Sun on 23 Oct. Which means that I’ve 3 fewer times of data to handle my forecast. Fig. 1 below
shows the forecast errors for both methods. The number above shows that the average mistakes for the end-weighted forecasts are smaller than that of the equal-weighted forecasts for all time periods
except for the 10-yr interest levels.
However, on closer inspection, the equal-weighted forecast errors are of higher magnitude and are sometimes postive and sometimes negative, producing a smaller mistake when averaged. When compared
using standard deviation, which considers only the total value of the mistakes, the end-weighted forecasts have smaller variance than the equal-weighted forecasts for all right time intervals. Thus,
end-weighted forecasts provide better estimates of the SSB interest levels. Fig. 2 below shows the forecast SSB interest rates superimposed on the SGS produces for the prior month. When SGS yields
are relatively constant for the month, both forecasts yield positive results. However, when SGS produces are either dropping or increasing, the errors become larger.
The equal-weighted forecasts have bigger errors than the end-weighted forecasts because they don’t take into consideration the path of the SGS yields. End-weighted forecasts are more accurate as they
provide more excess weight to the SGS yields close to the end of the month as referred to above. Finally, the most effective lessons that I learned from forecasting SSB rates of interest over the
past 12 months is this: the near future cannot be predicted.
Although I could enhance my forecast technique as well as perhaps provide good estimations of SSB interest rates, the best I possibly could forecast is 1 month in advance. Month I cannot forecast the
SSB rates of interest beyond 1. When the first tranche of SSB was announced in Sep 2015 with a 10-year interest rate of 2.63%, I forecasted that another tranche of SSB would have a higher interest
and didn’t make an application for it. When the second tranche was announced with a 10-yr interest rate of 2.78%, I used to be proven right!
• 5% (31 December 2016 est.)
• Short gamma
• Of the next steps of the accounting cycle, which step should be completed first
• 27-07-2019, 11:47 AM #182
• Isoquant for two inputs that are used in set proportion
• Quick funding for real estate investment loans
At that point, there is even an outcry among the SSB traders who had applied for the first tranche as their SSBs were now less valuable. Little did I expect that the SSB interest rates for all
following tranches would drop below that of the first tranche! The 10-12 months interest rate for the current tranche is 1.79%, which is a lot lower than the two 2.63% for the first tranche.
Investors who bought in to the first tranche of SSBs got a good deal in the end.
Smart alecs like me can only just watch the SSB interest rates going lower. If you’re thinking about forecasts of the SSB interest rates, you can refer to SSB INTEREST Estimates within my (The)
Boring Investor’s Statistics blog. Just bear in mind the lesson I above learnt, which is that the near future cannot be forecasted.
The Rectors keep properties for at least five to a decade. Jason will sell if he will get a buyer prepared to pay more than market value or if he can profit from home-price gratitude to buy a more
substantial property with more units. “My approach to real estate investing isn’t get-rich-quick,” he says.
Jason, who still works 52 hours a week as a firefighter and lately launched a structure company, used to control and maintain the properties himself, and he says he never evicted a tenant. Now he
employs 20 people as well as his two brothers and his mother. “I get to do the fun stuff, concentrating on acquisitions and starting new businesses,” he says. “I’ve found that to be successful, you
must have a why, and the bigger the why, the more successful you will be,” says Jason.
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Tanks, goats and buses- Anne Watson - Oxford Education Blog
Tanks, goats and buses- Anne Watson
Consistent use of images supports the understanding of mathematical structures. The outstanding examples of this in mathematics have become ‘canonical’, that is part of the mathematical canon. At
school level these canonical images are: number line; function graphs (thankyou Descartes); 2-dimensional combination grids (thankyou Omar Khayyam and Cayley); Venn diagrams (thankyou Cantor and
Charles Dodgson). You could say our systems of numerical and algebraic notation are also canonical, but these do not carry meaning in quite the same way as the images do – the symbols need
translating and the grammar of combining them needs to be learned.
Then there are the images we use that are not part of the canon but connect mathematical meaning to quantity or structure in ways that school learners can recognise. Examples of this include:
• balances for equations,
• rods or bars for quantitative statements and relationships,
• tables of values,
• function machines,
• number base apparatus etc.
The best images also model mathematical meaning and show how elements combine. Learning is well-supported when teachers find out what images their students are familiar with and build on those. That
is why good textbooks use a few images consistently throughout the school years, and also why one teacher’s ‘good way to teach’ does not automatically ‘work’ for another teacher.
There is another role for consistent images in teaching and learning, and that is creating situations that become familiar and can be extended and complexified as learners become more competent and
knowledgeable about them. The world of puzzles has many of these. For example:
Tank problems
If you Google ‘tank problems’ you will find all kinds of ways to mend your toilet flush, clean out your hot water system etc. but you will also find families of mathematics problems. The simplest is
to consider two connected water tanks, one higher than the other, and describe what changes and what stays the same as water flows between (height, total volume, partitioned volume, mass,
cross-sectional area, range of possible values etc.) Some puzzles have networks of several water tanks asking which tanks will fill in which order and how far. Others offer a way in to using and
reading graphs, (e.g. http://www.pmtheta.com/animated-situations.html). Then we have increasing difficulty by introducing taps that control flow, so that the rate of flow is no longer a matter for
gravity alone. So I can imagine learners throughout a school having water tank problems offered as a way to think about a new-to-them mathematical concept, such as equality of volumes, rate of
change, volumes by integration, building on what is already known.
Goat grazing problems
I know goat grazing problems from Dudeney’s collections of puzzles but they have been around since the 18^th century, having been published in the Ladies’ Journal. They may be older. A goat is
tethered to a stake, what area can it graze? (Spot the locus content here). Suppose the tether is a ring that can slide along a straight rod; what area can it graze? Suppose a fixed stake is in the
corner of a square field; what area can it graze? What about fields that are right angled triangles, or circular fields? (This last suggestion is not casual). And so on. Again, I can imagine these
problems being posed to generate various geometric, mensuration and computation ideas throughout school.
Bus problems
When I visited Prague I had the privilege of spending time with Milan Hejny, who is one of those mathematics educators who spend a lot of time in schools working with learners, but also manages to be
internationally known. He has spent about 40 years writing a textbook series that still isn’t finished, and every time he uses one of the tasks in classrooms he listens to learners and often rewrites
elements to take account of something new he has learnt. The series is used in about 25% of Czech schools. At this point you might want to be looking up the relevant PISA results to see where the
Czech Republic sits, and you may not want to read on, but I encourage you to think differently because of the 40 year development, the close attention to what learners actually do and say in
classrooms, and that – as Milan says himself – many teachers do not use the teachers’ guides so do not fully use the potential of the series. That is true for all textbooks, and that is why teacher
training in authoritarian countries often hinges on learning how to use the authorised scheme well, and the PD provided in relation to schemes in this country also focuses on use.
In Milan Hejny’s textbooks there are several consistent images that crop up year after year in increasingly complex ways. I was taken with the idea of his use of bus problems, which through
imagination, role play, modelling, and arithmetical problem solving during the primary years introduce the additive relation, linear structures, and simultaneous linear equations in ways that enable
learners to construct mathematical meaning and devise, adapt and extend robust methods of solving additive problems. Eventually learners need algebra to express what they can already do through
reasoning. I know these appear elsewhere and are not exclusive to his books, but seeing them prompted me to think about the ways in which images can extend and even accelerate learning of early
algebra. Algebraic representation of the unknowns and variables follows from wanting to describe the situations.
The basic model is about people getting on and off buses at bus stops, and comparing these quantities with the numbers on the bus. This may be about buses but it has NOTHING to do with ‘the bus stop
method’ for division (or does it?). Diagrammatically it looks like this:
stops A B C D E F
off 2 3 2 p
on 1 2 q
Start 5 4 7 r s
At stop A the bus arrives with 5 people on board, two people get off, one gets on, this means there are now 4 people on the bus. You can see that there are many possibilities for missing number
problems that use various transformations of the additive relationship.
Cutting to the chase for early secondary work, there are various ways in which the relationships between the shaded cells can be expressed: r – p + q = s describes the actions at each stop;
s – r = q – p is relatively easy to explain but might involve negative numbers; s + p = r + q is less easy to explain and might prompt discussions about proof, since numerical demonstration from one
or two cases does not guarantee that it will work for all cases. It is even worth discussing whether the third equation is better written as s + p = q + r as that reflects the algebraic manipulation,
where the former expresses the status of the variables in the problem. The consistent structure of the situation at each stop creates both a need and the support for algebra to generalise
demonstration of particular cases. How do the totals change if the starting number is increased /decreased by 3 but nothing else changes?
Playing with the givens creates interesting extensions. For example, supposed we do not know any of values for totals on the bus except for the starting value, or the finishing value: how could these
be reconstructed?
There is also a related programming problem: what is the minimum amount of information needed to model the usage of one bus route?
Suppose two of the passengers are mathematicians (as often happens on my regular bus route in Oxford). One says to the other: ‘there are three times as many people after stop D as there were before
stop B’. The other replies: ‘there were four times as many before stop D and there were after stop B’. What possible numbers could there be?
What problems could be posed if we change the bus stop labels to numbers?
As with any context, there are natural restrictions on the domain. We cannot use negative numbers for people, and bus capacity also sets constraints, but fortunately the whole problem type can also
work for trains whose capacity is much greater.
It is not too fanciful to imagine that learners who have developed strong familiarity with the situation might find related problems more accessible. I have also posed problems for myself using the
bus model where I found that using subscripts made the situation clearer for me; for example, suppose that during the rush hour the average number boarding increases by 10% at each stop and the
average number leaving decreases by 20%.
You might have started thinking about similarities with bag arithmetic, in which teachers use bags containing unknown numbers of counters and do various things like adding more counters or taking
counters out to nudge students into expressing algebraic relationships.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for
thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.
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Find value sports bets - First Floor English
Following your analysis, you will try to estimate a probability that the bet will pass. The trick is that bookmakers also estimate the probability that a bet will pass.
Well yes, when a bookmaker sets an odds, he estimates the probability that a bet has to pass. Basically, the smaller the odds (closer to 1) the more likely the bet is to pass.
A value bet is when you feel that the odds offered by the bookmaker are too high for you. At this point you think the bet has a better chance of going through than the bookmaker thinks.
A concrete example ? Ok imagine you want to bet on Bayern Munich winning at 1.50, and you estimate their chances of winning at 70%. Is this odd a value bet?
With a table like this you see that for a probability of 70%, your odds must be greater than 1.43. Our odds at 1.50 for Bayern is therefore a value bet.
If you want to calculate a valuebet more precisely, I refer you to the linked article, which does the job very well. The goal is for you to understand the principle of value betting.
So of course it is difficult to best estimate the probability that a bet will pass, not to mention the risk of making a mistake in its analysis. But betting on valuebets is the best technique,
because you take advantage of the bookmakers’ small mistakes.
Don’t dream, bookmakers make few rating errors. There is therefore little value betting, and when there is, the betting sites do not hesitate to readjust the odds.
The value bet is rare, but when you find it it’s worth it.
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Transactions Online
Kyosuke YAMASHITA, Ryu ISHII, Yusuke SAKAI, Tadanori TERUYA, Takahiro MATSUDA, Goichiro HANAOKA, Kanta MATSUURA, Tsutomu MATSUMOTO, "Fault-Tolerant Aggregate Signature Schemes against Bandwidth
Consumption Attack" in IEICE TRANSACTIONS on Fundamentals, vol. E106-A, no. 9, pp. 1177-1188, September 2023, doi: 10.1587/transfun.2022DMP0005.
Abstract: A fault-tolerant aggregate signature (FT-AS) scheme is a variant of an aggregate signature scheme with the additional functionality to trace signers that create invalid signatures in case
an aggregate signature is invalid. Several FT-AS schemes have been proposed so far, and some of them trace such rogue signers in multi-rounds, i.e., the setting where the signers repeatedly send
their individual signatures. However, it has been overlooked that there exists a potential attack on the efficiency of bandwidth consumption in a multi-round FT-AS scheme. Since one of the merits of
aggregate signature schemes is the efficiency of bandwidth consumption, such an attack might be critical for multi-round FT-AS schemes. In this paper, we propose a new multi-round FT-AS scheme that
is tolerant of such an attack. We implement our scheme and experimentally show that it is more efficient than the existing multi-round FT-AS scheme if rogue signers randomly create invalid signatures
with low probability, which for example captures spontaneous failures of devices in IoT systems.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2022DMP0005/_p
author={Kyosuke YAMASHITA, Ryu ISHII, Yusuke SAKAI, Tadanori TERUYA, Takahiro MATSUDA, Goichiro HANAOKA, Kanta MATSUURA, Tsutomu MATSUMOTO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Fault-Tolerant Aggregate Signature Schemes against Bandwidth Consumption Attack},
abstract={A fault-tolerant aggregate signature (FT-AS) scheme is a variant of an aggregate signature scheme with the additional functionality to trace signers that create invalid signatures in case
an aggregate signature is invalid. Several FT-AS schemes have been proposed so far, and some of them trace such rogue signers in multi-rounds, i.e., the setting where the signers repeatedly send
their individual signatures. However, it has been overlooked that there exists a potential attack on the efficiency of bandwidth consumption in a multi-round FT-AS scheme. Since one of the merits of
aggregate signature schemes is the efficiency of bandwidth consumption, such an attack might be critical for multi-round FT-AS schemes. In this paper, we propose a new multi-round FT-AS scheme that
is tolerant of such an attack. We implement our scheme and experimentally show that it is more efficient than the existing multi-round FT-AS scheme if rogue signers randomly create invalid signatures
with low probability, which for example captures spontaneous failures of devices in IoT systems.},
TY - JOUR
TI - Fault-Tolerant Aggregate Signature Schemes against Bandwidth Consumption Attack
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1177
EP - 1188
AU - Kyosuke YAMASHITA
AU - Ryu ISHII
AU - Yusuke SAKAI
AU - Tadanori TERUYA
AU - Takahiro MATSUDA
AU - Goichiro HANAOKA
AU - Kanta MATSUURA
AU - Tsutomu MATSUMOTO
PY - 2023
DO - 10.1587/transfun.2022DMP0005
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E106-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2023
AB - A fault-tolerant aggregate signature (FT-AS) scheme is a variant of an aggregate signature scheme with the additional functionality to trace signers that create invalid signatures in case an
aggregate signature is invalid. Several FT-AS schemes have been proposed so far, and some of them trace such rogue signers in multi-rounds, i.e., the setting where the signers repeatedly send their
individual signatures. However, it has been overlooked that there exists a potential attack on the efficiency of bandwidth consumption in a multi-round FT-AS scheme. Since one of the merits of
aggregate signature schemes is the efficiency of bandwidth consumption, such an attack might be critical for multi-round FT-AS schemes. In this paper, we propose a new multi-round FT-AS scheme that
is tolerant of such an attack. We implement our scheme and experimentally show that it is more efficient than the existing multi-round FT-AS scheme if rogue signers randomly create invalid signatures
with low probability, which for example captures spontaneous failures of devices in IoT systems.
ER -
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Cave Kame | Blablawriting.com
The whole doc is available only for registered users
• Pages: 6
• Word count: 1262
• Category: Cash
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If a project with conventional cash flows has a payback period less than its life, can you definitely state the algebraic sign of the NPV? Why or why not? No you cannot because you don’t know wether
there is a cutoff time or how long it will take to pay off the amount. Just because it will pay itself back does not mean it will be a positive investment to take in. Suppose a project has
conventional cash flows and a positive NPV. What do you know about its payback? Its profitability index. Its IRR? If a project has a positive NPV it means it will cover the initial investment so
there will be an eventual payback period and some type of IRR. Concerning payback:
a) Describe how the payback period is calculated and describe the information this measure provides about a sequence of cash flows. What is the payback criterion decision rule? The payback period is
calculated by an investment that is acceptable based on the calculation period that is less than prespecified on a number of years. The measure of the sequence of cash flows gives us some insight
into what our numbers will look like for the following years and we can make a decision based on these predictions. b)What are the problems associated with using the payback period as a means of
evaluating cash flows? It ignores the cash flows beyond the cutoff date. c)What are the advantages of using the rule to evaluate cash flows? Are there any circumstances under which using payback
might be appropriate? The advantage of the rule with cash flows is that it adjusts for uncertainty of later cash flows. Using the payback method may be appropriate if you just want an idea on how
long it will take to get your bait back. Concerning AAR:
a) Describe how the average accounting return is usually calculated and describe the information it provides about a sequence of cash flows. What is AAR criterion decision rule? The AAR method is
calculated by an investment’s average net income divided by its average book value. It does not provide any information on cash flows because it uses net and book value instead.
b) What are the problems associated with using the AAR as a means of evaluating a project’s cash flows? What underlying feature of AAR is most troubling to your from a financial perspective? Does the
AAR have any redeeming qualities? AAR is not efficient for evaluating a projects cash flow because we are not looking at the same things. The most troubling factor of this from a financial
perspective is that AAR does not tell us what the effect on share price will be from taking an investment, so it doesn’t tell us what we need to know. The redeeming quality of AAR is that it almost
always can be computed.
Chapter 8 Questions and Problems:
3.Offshore Drilling Products Inc. imposes a payback cutoff of three years for its international investment projects. If the company has the following two projects available, should it accept either
of them? It should reject project A because it does not meet the 2 year rule. Even though project B does not follow the 2 year rule in year 4 they will have a cash flow of 250,000 so I would advise
them to take it.
5.A firm evaluates all of its projects by applying the IRR rule. If the required return is 13% should the firm accept the following project? No I would not take the project because after calculations
it comes out to 16%.
9. For the cash flows in the previous problem, what is the NPV at a discount rate of 0%? What is the discounted rate is 10%? If its 20%? 30%? 0%:
Chapter 9: CTACR
9.7 In evaluating the Cayenne, would you consider the possible damage to Porsche’s reputation? I remember when the Cayenne came out and the reactions people had. I think it did not damage the
reputation, if anything it stayed the same or gotten a little bit better. Now, people who wanted a Porsche but had kids and could not get one, now do. It is still at a different luxury level than the
other SUV’s and the cherry on top was the engineering and speediness to it.
9.9 In evaluating the Cayenne, what do you think Porsche needs to assume regarding the substantial profit margins that exist in this market? Is it likely they will be maintained as the market becomes
more competitive, or will Porsche be able to maintain the profit margin because of its image and the performance of the Cayenne? I think Porsche needed to assume that not only is their car more
expensive than the other luxury SUV’s but that not everyone who can afford it will jump into it because a fast SUV is not necessarily family friendly. I believe that the Porsche will remain somewhat
neutral in the market, they have not had a huge impact on the car market and no new versions of SUV’s have came out from Porsche.
Questions & Problems:
Winnebagel Corp. currently sells 28,000 motor homes per year at $73,000 each and 7,000 luxury motor coaches per year at $115,000 each. The company wants to introduce a new portable camper to fill out
its product line. It hope to sell 23,000 of these campers per year at $19,000 each. An independent consultant has determined that if Winnebagel introduces the new campers, it should boost the sales
of its existing motor homes by 2,600 units per year and reduce the sales of its motor coaches by 850 units per year.
What is the amount to use as the annual sales figure when evaluating this project? Why? The annual sales figure they should for this project is an increase of profits by $529,050,000. This number is
used because it is the estimate of the units sold of the new camper and the additional 2,600 motor homes that would sell. The depreciation of the motor coaches (almost $98 million) has already been
subtracted from the $529 billion.
After calculating, the IRR of this project is: 18.47%
Chapter 10 CTACR
10.5 A stock market analysis is able to identify mispriced stocks by comparing the average price for the last 10 days to the average price for the last 60 days. If this is true, what do you know
about the market? The market is on in a equilibrium.
10.7 What are the implications of the efficient markets hypothesis for investors who buy and sell stocks in an attempt to “beat the market”? They are probably not going to be as successful as
they want because the market has been very efficient and there is so much competition for these mistakes that it is difficult to find since they are now more than ever avoided.
Questions & Problems:
Suppose you bought a 7% coupon bond one year ago for $893. The bond sells for $918 today. a. Assuming a $1,000 face value, what was your total dollar return on this investment over the past year?
1.75 per dollar. b. What was your total nominal rate of return on this investment over the past year? 6.25% c. If the inflation rate last year was 4%, what was your total rate of return on this
7. Using the following returns, calculate the average returns, the variances, and the standard deviations for X and Y
Related Topics
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Faraday's law. Maxwell's equations | JustToThePointFaraday's law. Maxwell's equations
All things are difficult before they are easy, Thomas Fuller.
When you have eliminated the impossible, whatever remains, however improbable, must be the truth, Sherlock Holmes.
A vector field is an assignment of a vector $\vec{F}$ to each point (x, y) in a space, i.e., $\vec{F} = M\vec{i}+N\vec{j}$ where M and N are functions of x and y.
A vector field on a plane can be visualized as a collection of arrows, each attached to a point on the plane. These arrows represent vectors with specific magnitudes and directions.
Work is defined as the energy transferred when a force acts on an object and displaces it along a path. In the context of vector fields, we calculate the work done by a force field along a curve or
trajectory C using a line integral. The work done by a force field $\vec{F}$ along a curve C is: W = $\int_{C} \vec{F}·d\vec{r} = \int_{C} Mdx + Ndy = \int_{C} \vec{F}·\hat{\mathbf{T}}ds$, where $\
hat{\mathbf{T}}$ is the unit tangent vector.
A vector field is conservative if there exist a scalar function such that $\vec{F}$ = ∇f (the vector field is its gradient). This scalar function is known or referred to as the potential function
associated with the vector field.
Theorem. Fundamental theorem of calculus for line integral. If $\vec{F}$ is a conservative vector field in a simply connected region of space (i.e., a region with no holes), and if f is a scalar
potential function for $\vec{F}$ in that region, then $\int_{C} \vec{F}·d\vec{r} = \int_{C} ∇f·d\vec{r} = f(P_1)-f(P_0)$ where P[0] and P[1] are the initial and final points of the curve C,
The line integral of the vector field $\vec{F}$ along the curve C is defined as: $\int_{C} \vec{F}d\vec{r}$ where $d\vec{r}$ is an infinitesimal vector tangent to the curve, given by: $d\vec{r} =
⟨dx, dy, dz⟩$.
$\int_{C} \vec{F}d\vec{r} = \int_{a}^{b} (P\frac{dx}{dt} + Q\frac{dy}{dt} + R\frac{dz}{dt})dt = \int_{a}^{b} (P(x(t), y(t), z(t))\frac{dx}{dt} + Q(x(t), y(t), z(t))\frac{dy}{dt} + R(x(t), y(t), z(t))
Let our vector field $\vec{F} = P\hat{\mathbf{i}}+ Q\hat{\mathbf{j}}+R\hat{\mathbf{k}}$, we define the curl of $\vec{F}$ as $curl(\vec{F})=(R_y-Q_z)\hat{\mathbf{i}} + (P_z-R_x)\hat{\mathbf{j}}+
$curl(\vec{F}) = ∇ x \vec{F} = |\begin{smallmatrix}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}}\\ \frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\ P & Q & R\end{smallmatrix}| = (\frac{∂R}{∂y}-\
frac{∂Q}{∂z})\hat{\mathbf{i}}-(\frac{∂R}{∂x}-\frac{∂P}{∂z})\hat{\mathbf{j}} + (\frac{∂Q}{∂x}-\frac{∂P}{∂i})\hat{\mathbf{k}}$
Theorem. The curl of a conservative field is $\vec{0}$. Conversely, if $curl(\vec{F})=\vec{0}$, the components P, Q, and R have continuous first-order partial derivatives, and its domain is open and
simply-connected, the vector field is conservative.
Given a vector field $\vec{F}$ and a surface S with boundary curve C, Stokes’ Theorem states: $\oint_C \vec{F} \cdot d\vec{r} = \int \int_{S} (∇x\vec{F})·\hat{\mathbf{n}}dS$
Maxwell’s equations
Maxwell’s equations are a set of four fundamental laws that describe how electric and magnetic fields behave and interact with each other. These equations form the foundation of classical
electromagnetism, optics, and electric circuits.
An electric field ($\vec{E}$) is an invisible force field generated by electric charge. It describes how a charged particle would be pushed (attracted) or pulled (repelled) by other charges around
it. The strength and direction of the electric force, per unit charge, an electrically charged particle would feel at any point in space, are given by the electric field. Mathematically, this is
expressed as: $\vec{F} = q\vec{E}$ where $\vec{F}$ is the force experienced by a charge q in the electric field $\vec{E}$.
The unit of the electric field is newtons per coulomb (N/C), which measures the force per unit charge. A positive charge creates an electric field that radiates outward, while a negative charge
creates an electric field that points inward toward the charge.
A magnetic field $\vec{B}$ is another type of invisible physical field that is generated by moving electric charges (such as current-carrying wires) or by magnetic materials (as in permanent
magnets). The force a charged particle q experiences in a magnetic field $\vec{B}$ depends on its velocity $\vec{v}$ and is given by the equation: $\vec{F} = q\vec{v}x\vec{B}$. It describes the force
experienced by a charged particle q moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$. This force is perpendicular to both the velocity of the charged particle $\vec{v}$ and the magnetic
field $\vec{B}$. The magnetic field is measured in teslas (T).
Gauss-Coulomb law (Gauss’ Law) relates the behaviour of electric fields to the distribution of electric charge. It states that the divergence of the electric field ($\vec{E}$) is proportional to the
charge density ρ, which is the amount of electric charge per unit volume. The mathematical expression is: $\vec{∇}·\vec{E} = \frac{ρ}{\epsilon_0}$ where $\epsilon_0$ is a physical constant known as
the permittivity of free space. It is a constant that characterizes the strength of the electric field in a vacuum.
To understand the implications of Gauss’s Law, let’s consider an arbitrary closed surface S. We want to study the flux of the electric field through this closed surface. Flux is a measure of how much
of the (electric) field passes through the surface. According to Gauss’s Law, the total flux of the electric field through the surface is proportional to the total charge enclosed within the volume D
enclosed by the surface S. (Figure iii):
$\oint_S \vec{E} \cdot d\vec{S} =$[Using the Divergence Theorem, which connects the flux through a closed surface to the divergence of the field within the volume D enclosed by the surface S] $\int \
int \int_{D} div \vec{E}dV = \int \int \int_{D} \vec{∇}·\vec{E}dV$= [Substituting Gauss’s Law into this equation gives:] = $\frac{1}{\epsilon_0} \int \int \int_{D} ρdV$ =[The right-hand side of this
equation represents the total charge Q enclosed within the volume D] $\frac{Q}{\epsilon_0}$.
This means that the total electric flux through any closed surface is equal to the total charge enclosed within the volume D divided by ϵ[0].
Faraday’s law
Faraday’s Law is one of the four Maxwell’s equations, which form the foundation of electromagnetism. It describes how a changing magnetic field can create an electric field. This phenomenon is the
principle behind many modern technologies such as electric generators, transformers, and induction motors.
Imagine a magnetic field $\vec{B}$ as an invisible force field that can exert forces on electrically charged particles. Now, if this magnetic field changes over time, it will create or induce an
electric field. An electric field, in turn, is a force field that exerts forces on electric charges, causing them to move. This movement of electric charges due to this induced electric field is what
we call an electromotive force (EMF) or voltage.
Faraday’s law of electromagnetic induction states that a changing magnetic field over time creates a rotating (or curling) electric field. It creates an electric field that loops around in circles.
This circular electric field can create a voltage if there’s a conducting path (like a wire loop) for electric charges to move through. Mathematically, it is represented as: $∇×\vec{E} = -\frac{∂\vec
• $∇×\vec{E}$ is called the “curl” of the electric field $\vec{E}$. It measures how much the electric field “curls”, “rotates” or “circulates” around a point.
• $-\frac{∂\vec{B}}{∂t}$ represents how the magnetic field ($\vec{B}$) is changing over time t, i.e., the rate of change of the magnetic field over time. The negative sign reflects Lenz’s Law. It
indicates the direction of the induced electric field is such that it opposes the change in the magnetic field.
According to Stokes’ Theorem, the circulation (looping) of an electric field around a closed path C can be related to the rate of change of the magnetic field ($\vec{B}$) over a surface S bounded by
that path (Figure iv):
$\oint_C \vec{E} \cdot d\vec{r} = \int \int_{S}(∇ x \vec{E})·d\vec{S} = $
The line integral on the left-hand side of Stoke’s theorem represents the circulation of the electric field around the closed loop C. It is the work done per unit charge in moving a charge around the
closed loop in the presence of the induced electric field. The surface integral on the right-hand side of Stoke’s theorem involves the curl of the electric field ($∇ x \vec{E}$) integrated over the
surface S bounded by the loop.
By Faraday’s law, $\int \int_{S}(∇ x \vec{E})·d\vec{S} = \int \int_{S} (-\frac{∂\vec{B}}{∂t})·d\vec{S}$. It allows us to express the voltage (or EMF) around the loop ($\oint_C \vec{E} \cdot d\vec{r}
$) as the negative rate of change of the magnetic field ($\frac{∂\vec{B}}{∂t}$) passing through the surface: $\oint_C \vec{E} \cdot d\vec{r} = \int \int_{S} (-\frac{∂\vec{B}}{∂t})·d\vec{S}$
It’s helpful to mention Gauss’s Law for Magnetism, which states that magnetic fields always have a north and a south pole, and why isolated “magnetic charges”, analogous to electric charges, do not
exist, and therefore the divergence of the magnetic field is always zero, $∇ x \vec{B} = 0$. In other words, magnetic field lines always form closed loops and do not begin or end at a point.
The final piece of the puzzle is the Maxwell-Ampère Law, which describes how both electric currents and changing electric fields generate magnetic fields. It’s expressed as: $∇ x \vec{B} = μ_0\vec{J}
+ε_0μ_0\frac{∂\vec{E}}{∂t}$ where:
• Curl of the Magnetic Field, $∇ x \vec{B}$ measures the tendency of the magnetic field to rotate or circulate around a point.
• $μ_0\vec{J}$ is the contribution from the current density $\vec{J}$, describing how electric currents create magnetic fields around them.
• $ε_0μ_0\frac{∂\vec{E}}{∂t}$. A changing (time-varying) electric field also generates a magnetic field, even in the absence of a current.
This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts Mathematics, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
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A comprehensive guide to the Relative Strength Index (RSI) | Scientia News
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A comprehensive guide to the Relative Strength Index (RSI)
The maths behind trading
In this piece, we will delve into the essential concepts surrounding the Relative Strength
Index (RSI). The RSI serves as a gauge for assessing the strength of price momentum and
offers insights into whether a particular stock is in an overbought or oversold condition.
Throughout this exploration, we will demystify the underlying calculations of RSI, explore its
significance in evaluating market momentum, and unveil its practical applications for traders.
From discerning opportune moments to buy or sell based on RSI values to identifying
potential shifts in market trends, we will unravel the mathematical intricacies that underpin
this critical trading indicator.
Please note that none of the below content should be used as financial advice, but for
educational purposes only. This article does not recommend that investors base their
decisions on technical analysis alone.
As indicated in the name, RSI measures the strength of a stock's momentum and can be used to show when a stock can be considered over- or under-bought, allowing us to make a more informed decision as
to whether we should enter a position or hold off until a bit longer.
It’s all very well and good to know that ‘you should buy when RSI is under 30 and sell when
RSI is over 70', but in this article, I will attempt to explain why this is the case and what RSI is really measuring.
The calculations
The relative strength index is an index of the relative strength of momentum in a market. This
means that its values range from 0 to 100 and are simply a normalised relative strength. But
what is the relative strength of momentum?
Initial Average Gain = Sum of gains over the past 14 days / 14
Initial Average Loss = Sum of losses over the past 14 days / 14
Relative strength is the ratio of higher closes to lower closes. Over a fixed period of usually
14 days (but sometimes 21), we measure how much the price of the stock has increased in
each trading day and find the mean average between them.
We then repeat and do the same to find the average loss.
The subsequent average gains and losses can then be calculated:
Average Gain =Â [(Previous Avg. Gain * 13) + Current Day's Gain] / 14
Average Loss =Â [(Previous Avg. Loss * 13) + Current Day's Loss] / 14
With this, we can now calculate relative strength!
Therefore, if our stock gained more than it lost in the past 14 days, then our RS value would
be >1. On the other hand, if we lost more than we gained, then our RS value would be <1.
Relative strength tells us whether buyers or sellers are in control of the price. If buyers were
in control, then the average gain would be greater than the average loss, so the relative
strength would be greater than 1. In a bearish market, if this begins to happen, we can say that there is an increase in buyers’ momentum; the momentum is strengthening.
We can normalise relative strength into an index using the following equation:
Relative Strength=Average Gain / Average Loss
Traders then use the RSI in combination with other techniques to assess whether to buy or
sell. When a market is ranging, which means that price is bouncing between support and
resistance (has the same highs and lows for a period), we can use the RSI to see when we
may be entering a trend.
When the RSI is reaching 70, it is an indication that the price is being overbought, and in a
ranging market, there is likely to be a correction and the price will fall so that the RSI stays at
around 50. The opposite is likely to happen when the RSI dips to 30. Price action is deemed
to be extreme, and a correction is likely.
It should, however, be noted that this type of behaviour is only likely in assets presenting
mean-reversion characteristics.
In a trending market, RSI can be used to indicate a possible change in momentum. If prices
are falling and the RSI reaches a low and then, a few days later, it reaches a higher low
(therefore, the low is not as low as the first), it indicates a possible change in momentum; we
say there is a bullish divergence.
Divergences are rare when a stock is in a long-term trend but is nonetheless a powerful
In conclusion, the relative strength index aims to describe changes in momentum in price
action through analysing and comparing previous day's highs and lows. From this, a value is generated, and at the extremes, a change in momentum may take place. RSI is not supposed to be predictive
but is very helpful in confirming trends indicated by other techniques.
Written by George Chant
Project Gallery
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Resources for mathematics competitions
The Art of Problem Solving hosts this AoPSWiki as well as many other online resources for students interested in mathematics competitions. Look around the AoPSWiki. Individual articles often have
sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.
Free AMC 8 Bootcamp covering all the essential concepts: https://thepuzzlr.com/courses/amc-8-bootcamp/
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.
Art of Problem Solving maintains a very large database of math contest problems. Many math contest websites include archives of past problems. The List of mathematics competitions leads to links for
many of these competition homepages. Here are a few examples:
Problem and Solutions: AMC 8 Problems in the AoPS wiki
How preparing for the AIME will help AMC 10/12 Score
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Label-based parameters in increasing trees
Label-based parameters in increasing treesArticle
• 1 Institut für Diskrete Mathematik und Geometrie [Wien]
Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of
increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the
analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node
labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the
children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting
distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.
Volume: DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities
Section: Proceedings
Published on: January 1, 2006
Imported on: May 10, 2017
Keywords: increasing trees,label-based parameters,limiting distribution,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete
Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]
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Introduction to Computer Science: Midterm Exam
Introduction to Computer Science: Exam 1
1. (10 points)Convert each of the following base ten representations to its equivalent two’s complement representation in which each value is represented in 7 bits.
1. (15 points)Describe two drawbacks of “sign and magnitude”representation of signed integers. When we use N bits for “sign and magnitude”representation of signed integers, what are the largest and
smallest numbers?
1. (10 points)Suppose that we use 5 bits for excess notation of signed integers. Convert each of the following excess representations to its equivalent base ten representation.
1. (15 points)Decode the following bit patterns using the floating-point format given below.
1. (15 points)Encode the following values into the floating-point format given above. Indicate the occurrence of truncation errors.
1. (20 points)Suppose the memory cells at addresses 00 through 0D in the machine described in Appendix C contain the following bit patterns.
Address / Contents
00 / 20
01 / 06
02 / 21
03 / 01
04 / 40
05 / 12
06 / 51
07 / 12
08 / B1
09 / 0C
0A / B0
0B / 06
0C / C0
0D / 00
Assume that the machine starts with its program counter equal to 00.
①What bit pattern will be in register 0 when the machine halts?
②What bit pattern will be in register 1 when the machine halts?
③What bit pattern is in the program counter when the machine halts?
1. (15 points) Suppose we want to complement the 4 middle bits of a byte while leaving the other 4 bits undisturbed. What mask must we use together with what operations?
APPENDIX C: Virtual Machine Language
Op-code / Operand / Description
C / RXY
000 / LOAD the register R with the bit pattern found in the memory cell whose address is XY
LOAD the register R with the bit pattern XY.
STORE the bit pattern found in register R in the memory cell whose address is XY.
MOVE the bit pattern found in register R to register S.
ADD the bit patterns in registers S and T as though they were two’s complement representations and leave the result in register R.
ADD the bit patterns in registers S and T as though there represented values in floating-point notation and leave the floating-point result in register R.
OR the bit patterns in registers S and T and place the result in register R.
AND the bit patterns in registers S and T and place the result in register R.
EXCULSIVE OR the bit patterns in registers S and T and place the result in register R.
ROTATE the bit pattern in register R one bit to the right X times. Each time place the bit that started at the low-order end at the high-order end.
JUMP to the instruction located in the memory cell at address XY if the bit pattern in register R is equal to the bit pattern in register 0. Otherwise, continue with the normal sequence of execution.
HALT execution.
------END ------
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Machine Theory
This unique compendium discusses some core ideas for the development and implementation of machine learning from three different perspectives — the statistical perspective, the artificial neural
network perspective and the deep learning methodology.The useful reference text represents a solid foundation in machine learning and should prepare readers to apply and understand machine learning
algorithms as well as to invent new machine learning methods. It tells a story outgoing from a perceptron to deep learning highlighted with concrete examples, including exercises and answers for the
Machine Learning with Amazon SageMaker Cookbook: 80 proven recipes for data scientists and developers to perform machine learning experiments and deployments 1st Edition
80 proven recipes for data scientists and developers to perform machine learning experiments and deployments; A step-by-step solution-based guide to preparing building, training, and deploying
high-quality machine learning models with Amazon SageMaker
AI and Machine Learning for Coders: A Programmer’s Guide to Artificial Intelligence
If you’re looking to make a career move from programmer to AI specialist, this is the ideal place to start. Based on Laurence Moroney’s extremely successful AI courses, […]
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Direct observation of the neural computations underlying a single decision
Neurobiological investigations of perceptual decision-making have furnished the first glimpse of a flexible cognitive process at the level of single neurons (Shadlen and Newsome, 1996; Shadlen and
Kiani, 2013). Neurons in the parietal and prefrontal cortex (Kim and Shadlen, 1999; Romo et al., 2004; Hernández et al., 2002; Ding and Gold, 2012) are thought to represent the accumulation of noisy
evidence, acquired over time, leading to a decision. Neural recordings averaged over many decisions have provided support for the deterministic rise in activity to a termination bound (Roitman and
Shadlen, 2002). Critically, it is the unobserved stochastic component that is thought to confer variability in both choice and decision time (Gold and Shadlen, 2007). Here, we elucidate this
drift-diffusion signal on individual decisions. We recorded simultaneously from hundreds of neurons in the lateral intraparietal cortex (LIP) of monkeys while they made decisions about the direction
of random dot motion. We show that a single scalar quantity, derived from the weighted sum of the population activity, represents a combination of deterministic drift and stochastic diffusion.
Moreover, we provide direct support for the hypothesis that this drift-diffusion signal approximates the quantity responsible for the variability in choice and reaction times. The population-derived
signals rely on a small subset of neurons with response fields that overlap the choice targets. These neurons represent the integral of noisy evidence. Another subset of direction-selective neurons
with response fields that overlap the motion stimulus appear to represent the integrand. This parsimonious architecture would escape detection by state-space analyses, absent a clear hypothesis.
Neural signals in the mammalian cortex are notoriously noisy. They manifest as a sequence of action potentials (spikes) that approximate non-stationary Poisson point processes. Therefore, to
characterize the signal produced by a neuron, electrophysiologists typically combine the spike times from many repetitions, or trials, relative to the time of an event (e.g., stimulus onset), to
yield the average firing rate of the neuron as a function of time. Such trial-averaged firing rates are the main staple of systems neuroscience. They are the source of knowledge about spatial
selectivity (e.g., receptive fields), feature selectivity (e.g., direction of motion, faces vs. other objects), and even cognitive signals associated with working memory, anticipation, attention,
motor planning, and decision making. But there is an important limitation.
Trial averages suppress signals that vary independently across trials (here and throughout, trial average and across-trial average refer to the mean of signal values, over all specified trials at the
same time, t, relative to a trial event [e.g., motion onset]).. In many cognitive tasks, such as difficult decisions, the variable component of the signal is the most interesting, because it is this
component that is thought to explain the variable choice and response time. This variability is thought to arise from a decision process that accumulates noisy evidence in favor of the alternatives
and terminates when the accumulated evidence for one alternative, termed the decision variable (DV), reaches a terminating bound. The DV is stochastic because the integral of noisy samples of
evidence is biased Brownian motion (or drift-diffusion) and this leads to a stochastic choice and response time on each decision. However, the stochastic part of this signal is suppressed by
averaging across trials. We will use the term drift-diffusion because it is the expression most commonly applied in models of decision making (Ratcliff and Rouder, 1998; Gold and Shadlen, 2007), and
we will consider the noise part—that is, diffusion—as the signal of interest.
In the setting of difficult perceptual decisions, studied here, bounded drift-diffusion reconciles the relationship between decision speed and accuracy. It also explains the trial-averaged firing
rates of neurons in the lateral intraparietal area (LIP) that represent the action used by monkeys to indicate their choice. These firing rate averages show motion-dependent, ramping activity that
reflects the direction and strength of motion, consistent with the drift component of drift-diffusion (Roitman and Shadlen, 2002). Up to now, however, the diffusion component has not been observed,
owing to averaging.
There is thus a missing link between the mathematical characterization of the decision process and its realization in neural circuits, leaving open the possibility that drift-diffusion dynamics do
not underlie LIP activity (e.g. Latimer et al., 2015), or emerge only at the level of the population, without explicit representation by single neurons. We reasoned that these and other alternatives
to drift-diffusion could be adjudicated if it were possible to resolve the DV giving rise to a single decision.
This stratagem is now feasible, owing to the development of high-density Neuropixels probes, which are capable of recording from deep sulci in the primate brain. Here we provide the first direct
evidence for a drift-diffusion process underlying single decisions. We recorded simultaneously from up to 203 neurons in the lateral intraparietal area (LIP) while monkeys made perceptual decisions
about the direction of dynamic random dot motion (Newsome et al., 1989; Gold and Shadlen, 2007). Using a variety of dimensionality reduction techniques, we show that a drift-diffusion signal can be
detected in such populations on individual trials. Moreover, this signal satisfies the criteria for a DV that controls the choice and reaction time. Notably, the signal of interest is dominated by a
small subpopulation of neurons with response fields that overlap one of the choice targets, consistent with earlier single neuron studies (e.g., Shadlen and Newsome, 1996; Roitman and Shadlen, 2002;
Churchland et al., 2011; Gold and Shadlen, 2007)
Two monkeys made perceptual decisions, reported by an eye movement, about the net direction of dynamic random dot motion (Fig. 1a). We measured the speed and accuracy of these decisions as a function
of motion strength (Fig. 1b, circles). The choice probabilities and the distribution of reaction times (RT) are well described (Fig. 1b, traces) by a bounded drift-diffusion model (Fig. 1c). On 50%
of the trials, a brief (100 ms) pulse of weak leftward or rightward motion was presented at a random time. The influence of these pulses on choice and RT further supports the assertion that the
choices and RTs arose through a process of integration of noisy samples of evidence to a stopping bound (Fig. S1; Stine et al., 2020, 2023; Hyafil et al., 2023). In addition to the main task, the
monkeys performed two control tasks: instructed, delayed saccades to peripheral targets and passive viewing of random dot motion (see Methods). These control tasks served to identify, post hoc,
neurons with response fields that overlap the choice target in the hemifield contralateral to the recording site [in]; Table 2).
Perceptual decisions are explained by the accumulation of noisy evidence to a stopping bound.
a, Random dot motion discrimination task. The monkey fixates a central point. After a delay, two peripheral targets appear, followed by the random dot motion. When ready, the monkey reports the net
direction of motion by making an eye movement to the corresponding target. Yellow shading indicates the response fields of a subset of neurons in area LIP that we refer to as T[in] neurons (target in
response field). b, Mean reaction times (top) and proportion of leftward choices (bottom) plotted as a function of motion strength and direction, indicated by the sign of the coherence: positive is
leftward. Data (circles) are from all sessions from monkey M (black, 9684 trials) and monkey J (brown, 8142 trials). Solid lines are fits of a bounded drift-diffusion model. c, Drift-diffusion model.
The decision process is depicted as a race between two accumulators: one integrating momentary evidence for left; the other for right. The momentary samples of evidence are sequential samples from a
pair negatively correlated Normal distributions with opposite means (ρ = −0.71). The decision is terminated when one accumulator reaches its positive bound. The example depicts leftward motion
leading to a leftward decision.
We recorded simultaneously from populations of neurons in area LIP, using newly developed macaque Neuropixels probes while monkeys performed these tasks. The data set comprises eight sessions from
two monkeys (1696– 2894 trials per session; Table 2). Our primary goal was to identify activity in LIP that relates to the decision variable, a theoretical quantity that determines the choice and
reaction time on each trial. To achieve this, we formed weighted averages from all neurons in the sample population, including those with response fields that do not overlap a choice target or the
motion stimulus. We used several strategies to assign this vector of weights, which we refer to as a coding direction in the neuronal state space. The projection of the spiking activity from the
population of neurons onto the vector of weights gives rise to a scalar function of time, S^x(t), where the superscript x labels the strategy. We focus on such one-dimensional projections because of
the long-standing hypothesis that the DV is drift-diffusion, which is a scalar function of time.
We first developed a targeted strategy that would reproduce the well-known coherence-dependent ramping activity evident in the across-trial averages. This strategy applies regression to best
approximate a linear ramp, on each trial, i, that terminates with a saccade to the choice-target contralateral to the hemisphere of the LIP recordings. The ramps are defined on the epoch spanning the
decision time: from t[0] = 0.2 s after motion onset to t[1] = 0.05 s before saccade initiation (black lines in Fig. S2). The epoch is motivated by many previous studies (e.g., Gold and Shadlen, 2007,
and see below). Each ramp begins at f[i](t[0]) = −1 and ends at f[i](t[1]) = 1. The ramp approximates the expectation—conditional on the choice and response time—of the deterministic components of
the drift-diffusion signal which can incorporate (i) the deterministic drift, (ii) a time-dependent but evidence-independent urgency signal (Drugowitsch et al., 2012), and (iii) a dynamic bias signal
(Hanks et al., 2011). It can also be viewed as an approximation to firing rates averaged across trials and grouped by contraversive choice and RT quantile (e.g., Roitman and Shadlen, 2002, see also
Fig. S3). Importantly, the fit is not guided by an assumption of an underlying diffusion process. That is, the ramp coding direction is agnostic to the underlying processes whose averages approximate
ramps. The weights derived from these regression fits specify a ramp-coding direction in the state space defined by the population of neurons in the session. The single-trial signal, S^ramp(t), is
rendered as the projection of the population firing rates onto this coding direction.
The left side of Fig. 2a shows single-trial activity rendered by this strategy. The right side of the figure shows the averages of the single-trial responses grouped by signed coherence and aligned
to motion onset or response time (saccade initiation). These averaged traces exhibit features of the firing rate averages in previous studies of single neurons in LIP (e.g., see Roitman and Shadlen,
2002; Gold and Shadlen, 2007, and Fig. S3). They begin to diverge as a function of the direction and strength of motion approximately 200 ms after the onset of motion. The traces converge near the
time of saccadic response to the contralateral choice-target such that the coherence dependence is absent or greatly diminished. Coherence dependence remains evident through the initiation of
saccades to the right (ipsilateral) target, consistent with a race architecture—between negatively correlated accumulators—depicted in Fig. 1c.
Population responses from LIP approximate drift-diffusion.
Rows show three types of population signals. The left columns show representative single-trial firing rates during the first 300 ms of evidence accumulation using two motion strengths: 0% and 25.6%
coherence toward the left (contralateral) choice target. For visualization, single-trial traces were baseline corrected by subtracting the activity in a 50 ms window around 200 ms. We highlight
several trials with thick traces (same trials in a–c). The right columns show the across-trial average responses for each coherence and direction. Motion strength and direction are indicated by color
(legend) and aligned to motion onset (left) or saccade initiation (right). The gray bars under the motion-aligned averages indicate the 300 ms epoch used in the display of the single-trial responses
(left panels). The epoch begins when LIP first registers a signal related to the strength and direction of motion. Except for saccade-aligned response, trials are cut off 100 ms before saccade
initiation. Error trials are excluded from the saccade-aligned averages, only. a, Ramp coding direction. The weight vector is established by regression to ramps from −1 to + 1 over the period of
putative integration, from 200 ms after motion onset to 100 ms before saccade initiation (see Fig. S2). Only trials ending in left (contralateral) choices are used in the regression. b, First
principal component (PC1) coding direction. c, Average firing rates of the subset of neurons that represent the left (contralateral) target. The weight vector consists of N
We complemented this regression strategy with Principal Component Analysis (PCA) and use the first PC (PC1), which explains 44 ± 7% of the variance (mean ± s.e. across sessions) of the activity
between 200 and 600 ms from motion onset (see Methods). This coding direction renders single trial signals, S^PC1(t) (Fig. 2b). In a third strategy, we consider the mean activity of neurons with
response fields that overlapped the contralateral choice target (Shadlen and Newsome, 1996; Platt and Glimcher, 1999; Roitman and Shadlen, 2002). In those studies, the task was modified so that one
of the choice targets was placed in the neural response field, whereas here we identify neurons post hoc with response fields that happen to overlap the contralateral choice target. This difference
probably accounts for the lower firing rates of the Fig. 2c shows single-trial and across-trial averages from these Fig. 2 left correspond to the same trials rendered by the three coding directions.
It is not difficult to tell which are the corresponding traces, an observation that speaks to their similarity, and the same is true for the averages. We will expand on this observation in what
The averages show the deterministic drift component of the hypothesized drift-diffusion process, with the slope varying monotonically with the signed motion strength (Fig. 2 right). The rise begins
to saturate as a consequence of the putative termination bound—a combination of dropout of trials that are about to terminate and the effect on the distribution of possible diffusion paths imposed by
the very existence of a stopping bound. This saturation is evident earlier on trials with stronger motion, hence shorter RT, on average. The positive buildup rate on the 0% coherence motion
represents the time-dependent, evidence-independent signal that is thought to reflect the cost of time. It leads to termination even if the evidence is weak, equivalent to collapsing stopping bounds
in traditional, symmetric drift-diffusion models (Drugowitsch et al., 2012). Removal of this urgency signal, u(t), from the non-zero coherence traces renders the positive and negative coherence
averages symmetric relative to zero on the ordinate (Fig. S4).
The single-trial responses in Fig. 2 do not look like the averages but instead approximate drift-diffusion. We focus on the epoch from 200 to 500 (or 600) ms from motion onset—that is, the first 300
(or 400) ms of the period in which the averages reflect the integration of evidence. Some traces are cut off before the end of the epoch because a saccade occurred 100 ms later on the trial. However,
most 0% coherence trials continue beyond 500 ms (median RT > 600 ms). The single-trial traces do not rise monotonically as a function of time but meander and tend to spread apart from each other
vertically. For unbounded diffusion, the variance would increase linearly, but as just mentioned, the existence of an upper stopping bound and the limited range of firing rates (e.g., non-negative)
renders the function sublinear at later times (Fig. 3a). The autocorrelation between an early and a later sample from the same diffusion trace is also clearly specified for unbounded diffusion. The
theoretical values shown in Fig. 3b & c are the autocorrelations of unbounded diffusion processes that are smoothed identically to the neural signals (see Methods and Appendix). The autocorrelations
in the data follow a strikingly similar pattern. These observations support the assertion that the coherence-dependent (ramp-like) firing rate averages observed in previous studies of area LIP are
composed of stochastic drift-diffusion processes on single trials.
Variance and autocorrelation of the single trial signals.
The analyses here are based on samples of S^ramp at six time points during the first 300 ms of putative integration, using all 0% and ±3.2% coherence trials (N = 5927). Samples are separated by the
width of the boxcar filter (51 ms), beginning at t[1] = 226 ms. a, Variance increases as a function of time. The measure of variance is normalized so that it is 1 for the first sample. Error bars are
s.e. (bootstrap). b, Autocorrelation of samples as a function of time and lag approximate the values expected from diffusion. The upper triangular portion of the 6 × 6 correlation matrix for
unbounded diffusion (r[i,j] is represented by brightness). The values from the data (S^ramp) are similar (right). c, Nine of the 15 autocorrelation terms in b permit a more direct comparison of
theory and data. The lower limb of the C-shaped function shows the decay in r[i,j] as a function of lag (j − i). This is the top row of b. The upper limb shows the increase in r[i,j] as a function of
time (for fixed lag). This is the lower diagonal in b. Error bars are s.e. (bootstrap). Note that the autocorrelations incorporate a free parameter, ϕ ≤ 1, that serves to correct for an unknown
fraction of the measured variance that is not explained by diffusion (see Methods).
Single-trial drift-diffusion signals approximate the decision variable
We next evaluate the hypothesis that the drift-diffusion signal, S^x(t), is the decision variable that controls the choice and response time. We have identified several coding directions that produce
candidate DVs, and as we will see below, there are also other coding directions of interest that can be derived from the population. Additionally, PCA indicates that the dimensionality of the data is
low, but greater than one (participation ratio = 4.4 ±1.3; Mazzucato et al., 2016; Gao et al., 2017). Therefore, one might wonder whether it is sensible to assume that the DV can be approximated by a
scalar measure arising from a single coding direction as opposed to a higher dimensional representation. Two decoding exercises are adduced to support the assumption.
We constructed a logistic decoder of choice using each neuron’s spike counts in 50 ms bins between 100 and 500 ms after motion onset. As shown in Fig. 4a, this What-decoder (orange) predicts choice
as accurately as a decoder of simulated data from a drift-diffusion model (black) using parameters derived from fits to the monkeys’ choice and RT data (see Methods). The simulation establishes a
rough estimate of the decoding accuracy that can be achieved, given the stochastic nature of the choice, were we granted access to the drift-diffusion signal that actually determines the decision. In
this analysis, the decoder can use a different vector of weights at each point in time (time-dependent coding directions; see Peixoto et al., 2021). However, if the representation of the decision
variable in LIP is one-dimensional, then a decoder trained at one time should perform well when tested at a different time. The red curve in Fig. 4a shows the performance of a What-decoder with a
fixed training-time (450 ms after motion onset; red arrow). This decoder performs nearly as well as the decoder trained at each time bin. The heatmap (Fig. 4b) generalizes this observation. It shows
two main features for all times 300 < t < 500 ms (dashed box). First, unsurprisingly, for a What choice decoder trained on data at one time t = x, the predictions improve as the testing time advances
(the decoding accuracy increases along any vertical) as more evidence is accrued. Second, and more importantly, decoders tested at time t = y perform similarly, independent of when they were trained
(there is little variation in decoding accuracy along any horizontal). This observation suggests that a single vector of weights may suffice to decode the choice from the population response.
The population signal predictive of choice and RT is approximately one-dimensional.
Two binary decoders were trained to predict the choice (What-decoder) and its time (When-decoder) using the population responses in each session. The When-decoder predicts whether a saccadic response
to the contralateral target will occur in the next 150 ms, but critically, its accuracy is evaluated based on its ability to predict choice. a, Choice decoding accuracy plotted as a function of time
from motion onset (left) and time to saccadic choice (right). Values are averages across sessions. The What-decoder is either trained at the time point at which it is evaluated (time-dependent
decoder, orange) or at the single time point indicated by the red arrow (t = 450 ms after motion onset; fixed training-time decoder, red). Both training procedures achieve high levels of accuracy.
The When-decoder is trained to capture the time of response only on trials terminating with a left (contraversive) choice. The coding direction identified by this approach nonetheless predicts choice
(green) nearly as well as the fixed training-time What-decoder. The black trace shows the accuracy of a What-decoder trained on simulated signals using a drift-diffusion model that approximates the
behavioral data in Fig. 1. Error bars signify s.e.m. across sessions. The gray bar shows the epoch depicted in the next panel. b, The heat map shows the accuracy of a decoder trained at times along
the abscissa and tested at times along the ordinate. Time is relative to motion onset (gray shading in a). In addition to training at t = 450 ms, the decoder can be trained at any time from 300 < t <
500 ms (dashed box) and achieve the same level of accuracy when tested at any single test time. The orange and red traces in a correspond to the main diagonal (x = y) and the column marked by the red
arrow, respectively. c, Trial-averaged activity rendered by the projection of the population responses along the When coding direction, S^When. Same conventions as in Fig. 2. d, Cosine similarity of
five coding directions. The heatmap shows the mean values across the sessions, arranged like the lower triangular portion of a correlation matrix. Cosine similarities are significantly greater than
zero for all comparisons (all p < 0.001, t-test). e, Correlation of single-trial diffusion traces. The Pearson correlations are calculated from ordered pairs of S[i](t) are the detrended signals
rendered by coding directions, x and y, on trial i. The detrending removes trial-averaged means for each signed coherence, leaving only the diffusion component. Reported correlations are
significantly greater than zero for all pairs of coding directions and sessions (all p < 10^−23, t-test, see Methods). The variability in cosine similarity and within-trial correlation across
sessions is portrayed in Fig. S8.
The second decoder is trained to predict whether a saccade to the contralateral choice target will be initiated in the next 150 ms. This When-decoder is trained by logistic regression to predict a
binary output: 1 at all time-points that are within 150 ms of an upcoming saccade and 0 elsewhere (Fig. S6). We validated the When-decoder by computing the area under an ROC (AUC) using the held-out
(odd) trials (mean AUC over all time points: 0.84), but this is tangential to our goal. Although the When-decoder was trained only to predict the time of saccades, our rationale for developing this
decoder was to test whether the When coding direction can be used to predict the choice. The green trace in Fig. 4a shows the accuracy of S^When(t) to predict the choice. The performance is almost
identical to the choice decoder, despite being trained on a temporal feature of trials ending in the same left choice. This feat is explained by the similarity of signals produced by the When- and
other coding directions. Note the similarity of the trial averaged S^When signals displayed in Fig. 4c to those in Fig. 2 (see also Fig. S7a, right). Indeed, the cosine similarity between the When
and Ramp coding directions is 0.67 ± 0.03 (Fig. 4d). In light of this, it is not surprising that the weighting vectors derived from both the What- and When-decoders also render single-trial
drift-diffusion traces that resemble each other and those rendered by other coding directions (Fig. 4e). Together these analyses support the assertion that the DV is likely to be captured by a single
dimension, consistent with Ganguli et al. (2008).
If the one-dimensional signals, S^x(t), approximate the decision variable, they should explain the variability of choice and reaction time for trials sharing the same direction and motion strength.
Specifically, (i) early samples of S^x(t) should be predictive of choice and correlate inversely with the RT on trials that result in contraversive (leftward) choices, (ii) later samples ought to
predict choice better and correlate more strongly (negatively) with RT than earlier samples, and (iii) later samples should contain the information present in the earlier samples and thus mediate (
i.e., reduce the leverage) of the earlier samples on choice and RT. Each of these predictions is borne out by the data.
The analyses depicted in Fig. 5 allow us to visualize the influence of the single trial signals, S^x(t), on the choice and RT on that trial. We focus on the early epoch of evidence accumulation
(200–550 ms after random dot motion onset) and restrict the analyses to decisions with RT ≥ 670 ms and coherence ≤ 6.4%. The RT restriction eliminates 17% of the eligible trials. Larger values of S^x
(t) are associated with a larger probability of a left (contraversive) choice and a shorter RT, hence negative correlation between S^x(t) and RT. We use the term, leverage, to describe the strength
of both of these associations. The leverage on choice (Fig. 5a, black traces) is the contribution of S^x(t) to the log odds of a left choice, after accounting for the motion strength and direction (
i.e., the coefficients, β[1](t) in Eq. 8). The leverage on RT (Fig. 5b) is the Pearson correlation between S^x(t) and the RT on that trial, after accounting for the effect of motion strength and
direction on S^x and RT (See Methods). The leverage is evident from the earliest sign of evidence accumulation, 200 ms after motion onset, and its magnitude increases as a function of time, as
evidence accrues (Fig. 5, top). The filled circle to the right of the traces in each graph shows the leverage of S^x at t = 550 ms, which is 120 ms before any of the included trials have terminated.
Both observations are consistent with the hypothesis that S^x represents the integral of noisy evidence used to form and terminate the decision. Two control analyses demonstrate that the degree of
leverage on choice and RT do not arise by chance: (i) random coding directions in state space produce negligible leverage (Fig. S10, top), and (ii) breaking the trial-by-trial correspondence between
neural activity and behavior eliminates all leverage (see Reviewer Figure 1 in reply to peer review).
The drift-diffusion signal approximates the decision variable.
The graphs show the leverage of single-trial drift-diffusion signals on choice and RT using only trials with RT ≥ 0.67 s. Rows correspond to the same coding directions as in Fig. 2. The graphs also
demonstrate a reduction of the leverage of the samples at t ≤ 0.5 s by a later sample of the signal at t = 0.55 s. Error bars are s.e.m. across sessions. a, Leverage of single-trial drift-diffusion
signals on choice. Leverage is the value of β[1], the coefficient that multiplies S^x(t) in Eq. 8. The black traces show the increase in leverage as a function of time. The dashed linestyle at the
left end of three of the traces indicate values that are not statistically significant (p>0.05, bootstrap shuffle test, see Methods) Filled symbols show the leverage at t = 0.55 s. The blue curve (
mediated) shows the leverage when the later sample is included in the regression (Eq. 9). Open symbols show the leverage of t = 0.55 s (same value as the filled symbol in bottom row). The yellow
curves (top and middle rows) show the leverage of the cross-mediated signal by S^x(t), leverage at t=0.4 s is significantly mediated by S^x(0.55) and cross-mediated by p < 10^−5, Paired Samples
t-test). b, Leverage of single-trial drift-diffusion signals on response time. Same conventions as in a. Leverage is the correlation between S^x(t) and RT. The mediated leverage is the partial
correlation, given the later sample. For all three signals, S^x(t), leverage at t=0.4 s is significantly mediated by S^x(0.55) and cross-mediated by p < 10^−8, Paired Samples t-test).
Importantly, the leverage at earlier times is mediated by the later sample at t = 550 ms. The blue traces in all graphs show the remaining leverage, once this later sample is allowed to explain the
choice and RT—by including, respectively, an additional term in the logistic regression (Eq. 9) and calculating the partial correlation, conditional on S^x(t = 0.55). We assessed statistical
significance of the mediation statistics, ξ^Ch and ξ^RT (Eqs. 7 and 10) in each session for the three signals shown in Fig. 5 using a bootstrap procedure (see Methods, below Eq. 10). Mediation is
significant in 47 of the 48 comparisons (all p<0.023, median p < 10^−317). The one non-significant comparison is ξ^Ch for p = 0.73). The mediation is significant when this comparison is included in
the combined data (p < 10^−9, Paired Samples t-test). The stark decrease in leverage is consistent with one-dimensional diffusion in which later values of the signal contain the information in the
earlier samples plus what has accrued in the interim. Had we recorded from all the neurons that represent the DV, we would expect the mediation to be complete (e.g., partial correlation = 0).
However, our recorded population is only a fraction of the entire population. Indeed, the observed degree of mediation is similar to values obtained from simulations of weakly correlated, noisy
neurons (Fig. S10, bottom).
There is one additional noteworthy observation in Fig. 5 that highlights the importance of the S^ramp and S^PC1) contain a second, open symbol, which is simply a copy of the filled symbol from the
bottom rowS^ramp and S^PC1 by the sample, p < 0.05; median p < 10^−268; bootstrap as above). This signal, carried by 9–21% of the neurons, mediates signals produced by the full population of 54–203
neurons nearly as strongly as S^ramp and S^PC1 mediate themselves. The observation suggests that minimal leverage is gained by sophisticated analyses of the full neuronal state space compared to a
simple average of Paré and Wurtz, 1998; Ferraina et al., 2002); disquieting because the functional relevance of these neurons is not revealed by the other coding directions. The weights assigned to
the ^st percentile, AUC = 0.74±0.05) in the ramp coding direction. They contribute disproportionately to PC1 and the What- and When-decoders but not enough to stand out based on their weights.
Indeed, the ability to predict that a neuron is Fig. S9).
These observations support the idea that the single-trial signals, S^ramp, S^PC1 and Fig. S11 we show that the S^What and S^When coding directions achieve qualitatively similar results. Moreover, a
late sample from S^x(t) mediates the earlier correlation with RT and choice of signals rendered by other coding directions, S^y(t), at earlier times (Fig. S11). Such cross-mediation is consistent
with the high degree of cosine similarity between the coding directions (Fig. 4d). The observation suggests that the decision variable is a prominent signal in LIP, discoverable by a variety of
strategies, and consistent with the idea that it is one-dimensional. In Fig. S12, we show that linear and nonlinear decoders achieve similar performance, which argues against a non-linear embedding
of the decision variable in the population activity.
Activity of direction-selective neurons in area LIP resembles momentary evidence
Up to now, we have focused our analyses on resolving the DV on single trials, paying little attention to how it is computed or to other signals that may be present in the LIP population. The
drift-diffusion signal approximates the accumulation, or integral, of the noisy momentary evidence—a signal approximating the difference in the firing rates of direction-selective (DS) neurons with
opposing direction preferences (e.g, in area MT; Britten et al., 1996). DS neurons, with properties similar to neurons in MT, have also been identified in area LIP (Freedman and Assad, 2006;
Shushruth et al., 2018; Fanini and Assad, 2009; Bollimunta and Ditterich, 2012), where they are proposed to play a role in motion categorization (Freedman and Assad, 2011). We hypothesize that such
neurons might participate in routing information from DS neurons in MT/MST to those in LIP that contain a choice-target in their response fields.
We identified such DS neurons using a passive motion viewing task (Fig. 6a-b, left). Neurons preferring leftward or rightward motion constitute 5–10% of the neurons in our sample populations Table 2.
Fig. 6 shows the average firing rates of 51 leftward-preferring neurons (Fig. 6a) and 26 rightward-preferring neurons (Fig. 6b) under passive motion viewing and decision-making. The separation of the
two traces in the passive viewing task is guaranteed because we used this task to identify the DS neurons. It is notable, however, that the DS is first evident about 100 ms after the onset of random
dot motion, and this latency is also apparent in mean firing rates grouped by signed coherence during decision making (Fig. 6a,b right). The activity of DS neurons is modulated by both the direction
and strength of motion. However, unlike the T[in] neurons, the traces associated with different motion strengths are mostly parallel to one another and do not reach a common level of activity before
the saccadic eye movement (i.e., they do not signal decision termination).
The representation of momentary evidence in area LIP.
a, Leftward preferring neurons. left, Response to strong leftward (blue) and rightward (brown) motion during passive viewing. Traces are averages over neurons and trials. The neurons were selected
for analysis based on this task, hence the stronger response to leftward is guaranteed. Note the short-latency visual response to motion onset followed by the direction-selective (DS) response
beginning ∼100 ms after motion onset. right, Responses during decision-making, aligned to motion onset and the saccadic response. Response averages are grouped by direction and strength of motion
(color legend). The neurons retain the same direction preference during passive viewing and decision-making. The responses are also graded as a function of motion strength. b, Rightward preferring
neurons. Same conventions as a. c, Cumulative distribution of the times at which individual neurons start showing evidence-dependent activity. Evidence dependence emerges earlier in d, left, Leverage
of neural activity on choice for right, Same as left, for the correlation between neural activity and reaction time. The absence of negative correlation is explained by insufficient power (see
Methods). e, Correlation between the neural representation of motion evidence—the difference in activity of neurons selective for leftward and rightward motion
In addition to their shorter onset latency, the direction-selectivity of M[in] neurons precedes the choice-selectivity of Fig. 6c). The responses bear similarity to DS neurons in area MT. Such
neurons are known to exhibit choice-dependent activity insofar as they furnish the noisy evidence that is integrated to form the decision (Britten et al., 1996; Shadlen et al., 1996). We computed
putative single trial direction signals by averaging the responses from the left- and right-preferring DS neurons, respectively. The resulting signals, Fig. 6d, left). This is what would be expected
if the M[in] neurons represent the noisy momentary evidence as opposed to the accumulation thereof (Mazurek et al., 2003). We failed to detect a correlation between RT and either S[Min] signal (Fig.
6d, right). This is surprising, but it could be explained by lack of power—a combination of small numbers of M[in] neurons, narrow sample windows (50 ms boxcar) and the focus on the long RT trials.
Indeed, we found a weak but statistically significant negative correlation between RT and the difference in leftward vs. rightward signals, averaged over the epoch 0.1 ≤ t ≤ 0.4 s from motion onset (
p = 0.0004; ℋ[0] : ρ ≥ 0, see Methods).
We considered the hypothesis that these DS signals are integrated by the Fig. 6e supports this hypothesis. On each trial, we formed the ordered pairs, {x, y}, where t[y] > t[x] when t[x] > 100 ms and
t[y] > 200 ms, and if the operation approximates integration, the level of correlation should be consistent at all lags, t[y] −t[x] > 100 ms. The correlations are significant in the epoch of
interest, and they differ significantly from the average correlations in the rest of the graph (i.e., t[x] < 100, t[y] < 200, or t[y] < t[x]; p < 0.0001, permutation test). Although correlative, the
observation is consistent with the idea that evidence integration occurs within area LIP, rather than inherited from another brain area (Zhang et al., 2022; Bollimunta and Ditterich, 2012).
We have observed a neural representation of the stochastic process that gives rise to a single decision. This is the elusive drift-diffusion signal that has long been thought to determine the
variable choice and response time in the perceptual task studied here. The signal was elusive because it is the integral of noisy momentary evidence, hence stochastic, and undetectable in the firing
rates when they are computed as averages over trials. The averages preserve the ramp-like drift component, leaving open the possibility that the averages are composed of other stochastic processes (
e.g., Latimer et al., 2015; Cisek et al., 2009). By providing access to populations of neurons in LIP, macaque Neuropixels probes (Trautmann et al., 2023) allowed us to resolve, for the first time,
the evolution of LIP activity during a single decision.
The present findings establish that the ramp-like averages arise from drift-diffusion on single trials, and this drift-diffusion signal approximates the decision variable that arbitrates the choice
and RT on that trial. We used a variety of strategies to assign a weight to each neuron in the population such that the vector of weights defines a coding direction in neuronal state space. The
weighted averages render population firing rate signals on single trials. Our experience is that any method of assigning the weights that captures a known feature of evidence accumulation (or its
termination in a saccadic choice) reveals drift-diffusion on single trials, and this also holds for data-driven, hypothesis-free methods such as PCA. This is because the actual dimensionality of the
DV is effectively one— a scalar function of time that connects the visual evidence to a saccadic choice (Ganguli et al., 2008). Thus a weighting established by training a decoder at time t = τ to
predict the monkey’s choice performs nearly as well when tested at times other than the time the decoder was trained on (i.e., t ≠ τ; Fig. 4).
The different strategies for deriving coding directions lead to different weight assignments, but the coding directions are linearly dependent (Fig. 4d). They produce traces, S(t), that are similar (
Fig. 4e) and suggestive of drift-diffusion. Traces accompanying trials with the same motion coherence meander and spread apart at a rate similar to diffusion (i.e., standard deviation proportional to
Fig. 3). The calculations applied in the present study improve upon previous applications (e.g. Churchland et al., 2011; de Lafuente et al., 2015; Shushruth et al., 2018) by incorporating the
contribution of the smoothing to the autocorrelations. The departures from theory are explained by the fact that the accumulations are bounded. The upper bound and the fact that spike rates must be
non-negative (a de facto lower reflecting bound) limits the spread of the single-trial traces.
The single-trial signals, i (Fig. 5 and Fig. S11). Support for this assertion is obtained using a conservative assay, which quantifies the leverage of the first 300 ms of the signal’s evolution on
decision outcomes—choice and RT—occurring at least 670 ms after motion onset. Naturally, the signals do not explain all the variance of these outcomes. The sample size is limited to N randomly
selected, often weakly correlated neurons. The sample size and correlation are especially limiting for the r) = 0.067 ± 0.0036). Control analyses show that the degree of leverage on behavior and
mediation of these relationships by later activity is on par with that obtained from simulated, weakly-correlated neurons (Fig. S10, bottom). In addition, because they are identified post hoc, many
have response fields that barely overlap the choice target. Presumably, that is why their responses are weak compared to previous single-neuron studies in which the choice targets were centered in
the response field by the experimenter. Yet even this noisy signal, Fig. 5).
The Shadlen and Newsome, 1996; Platt and Glimcher, 1999). This neural type is also representative of the LIP projection to the region of the superior colliculus (SC) that represents the saccadic
vector required to center the gaze on the choice target. (Paré and Wurtz, 1998). In a companion study by Stine et al. (2023) we show that the SC is responsible for cessation of integration in LIP.
Features of the drift-diffusion signal from the Stine et al. (2023) also show that inactivation of
Previous studies of LIP using the random dot motion task focused primarily on the cf. Meister et al., 2013). It was thus unknown whether and how other neurons contribute to the decision process. The
Neuropixels probes used in the present study yield a large and unbiased sample of neurons. Many of these neurons have response fields that overlap one of the two choice targets, but the majority have
response fields that overlap neither the choice targets nor the random dot motion. Our screening procedures (delayed saccades and passive motion viewing tasks) do not supply a quantitative estimate
of their spatial distribution. It is worth noting that neurons with response fields that overlap neither of the two choice targets were assigned nonzero weights by the What- and When-decoders, and
yet, removal of the task-related neurons that represent the choice targets and motion (i.e., T[in] and M[in]) decreases decoding accuracy more substantially than removing all but the T[in] neurons,
especially when decoders are not retrained at every time point (Fig. S13). The accuracy the decoder achieves is likely explained by neurons with weak responses that simply failed to meet our
criterion for inclusion in the T[in] and M[in] categories (e.g., neurons with response fields that barely overlap the choice targets or RDM). Some neurons outside these groups might reflect
normalization signals from the T[in] and M[in] neurons (Shushruth et al., 2018; Carandini and Heeger, 2012), imbuing broad, decision-related co-variability across the population. It thus seems
possible that higher dimensional tasks (e.g., four choices instead of two) could decrease correlations among groups of neurons with different response fields.
The fact that the raw averages from a small number of weakly correlated T[in] neurons furnish a DV on par with that furnished by the full population underscores the importance of this functional
class. The role of the M[in] neurons is less well understood. Freedman and colleagues described direction selective neurons in LIP, similar to our M[in] neurons (Freedman and Assad, 2011; Fanini and
Assad, 2009; Sarma et al., 2016). They showed that the neurons represent both the direction of motion and the decision in their task. In contrast, we do not observe the evolution of the decision (i.e
., DV) by the M[in] neurons (Fig. 6). The latency of the direction and coherence-dependent signal as well as its dynamics resemble properties of DS neurons in area MT. The delayed correlation between
M[in] and T[in] responses evokes the intriguing possibility that M[in] neurons supply the momentary evidence, which is integrated within LIP itself (Zhang et al., 2022). Future experiments that
better optimize the yield of M[in] neurons will be informative, and direct, causal support will require perturbations of functionally identified M[in] neurons, which is not yet feasible. A natural
question is why LIP would contain a copy of the DS signals that are already present in area MT. We suspect it simplifies the routing of momentary evidence from neurons in MT/MST to the appropriate T
[in] neurons. This interpretation leads to the prediction that DS M[in] neurons would be absent in LIP of monkeys that are naïve to saccadic decisions informed by random dot motion, as has been
observed in the SC (Horwitz et al., 2004). Further, when motion is not the feature that informs the saccadic response—for example in a color categorization task (e.g. Kang et al., 2021)—LIP might
contain a representation of momentary evidence for color (Toth and Assad, 2002; Sereno and Maunsell, 1998).
The capacity to record from many neurons simultaneously invites characterization of the population in neuronal state space (NSS), in which the activity of each neuron defines a dimension. Often,
population activity is confined to a low-dimensional subspace or manifold within the NSS (Vyas et al., 2020). An ever-more-popular viewpoint is that representations within these subspaces are
emergent properties of the population—that is, distributed, rather than coded directly by single neurons—a dichotomy that has its roots in Barlow’s neuron doctrine (as updated in Barlow, 1994).
Indeed, it is tempting to conclude that the drift-diffusion signal in LIP is similarly emergent based on our NSS analyses—the identified subspaces (i.e., coding directions) combine neurons with
highly diverse activity profiles. In contrast, grouping neurons by the location of their spatial response field reveals a direct coding scheme: T[in] neurons directly represent the accumulated
evidence for making a particular saccade and M[in] neurons represent the momentary evidence. We argue that this explanation is more parsimonious and, importantly, more principled. Grouping neurons
based on spatial selectivity rests on the principle that neurons with similar RFs have similar projections, which is the basis for topographic maps in the visual and oculomotor systems (Schall, 1995;
Silver and Kastner, 2009; Kremkow et al., 2016; Felleman and Van Essen, 1991). In contrast, there are no principles that guide the grouping of neurons in state space analyses, as the idea is that
they may comprise as many dimensions as there are neurons that happen to be sampled by the recording device.
The present finding invites both hope and caution. It may be useful to consider a counterfactual state of the scientific literature that lacks knowledge of the properties of LIP T[in] neurons—a world
without Gnadt and Andersen (1988) and no knowledge of LIP neurons with spatially selective persistent activity. In this world we have no reason to entertain the hypothesis that decisions would
involve neurons that represent the choice targets. We do know about DS neurons in area MT and their causal role in decisions about the direction of random dot motion (Salzman et al., 1992; Fetsch et
al., 2018; Ditterich et al., 2003; Liu and Pack, 2017). We also know that drift-diffusion models explain the choice-response time behavior. Guided by no particular hypothesis, we obtain population
neural recordings in the random dot motion task. We do not perform the saccade and passive viewing control experiments. What might we learn from such a dataset? We might apply PCA and/or train a
choice decoder or possibly a When-decoder. If so, we could discover the drift-diffusion signal and we might also infer that the dimensionality of the signal is low. However, we would not discover the
T[in] neurons without a hypothesis and a test thereof. We might notice that the coding directions that reveal drift-diffusion often render a response at the onset of the choice targets as well as
increased activity at the time of saccades to the contralateral choice target. These facts might lead us to hypothesize that the population might contain neurons with visual receptive fields and some
relationship to saccadic eye movements. We might then query individual neurons, post hoc, for these features, and ask if they render the drift-diffusion signal too. The inferences could then be
tested experimentally by including simple delayed saccades in the next experiment. The hope in this counterfactual is that data-driven, hypothesis-free methods can inspire hypotheses about the
mechanism. The caution is to avoid the natural tendency to stop before the hypotheses and tests, thus accepting as an endpoint the characterization of population dynamics in high dimensions or a
lower dimensional manifold. If LIP is representative, these mathematically accurate characterizations may fail to illuminate the neurobiological parsimony.
Ethical approval declarations
Two adult male rhesus monkeys (Macacca mulatta) were used in the experiments. All training, surgery, and experimental procedures complied with guidelines from the National Institutes of Health and
were approved by the Institutional Animal Care and Use Committee at Columbia University. A head post and two recording chambers were implanted under general anaesthesia using sterile surgical
procedures (for additional details see So and Shadlen, 2022). One recording chamber allowed access to area LIP in the right hemisphere. The other was placed on the midline, allowing access to the
superior colliculus. Those recordings are described in Stine et al. (2023). Here we report only on the neural recordings from LIP, focusing on the epoch of decision formation.
Behavioral tasks
The monkeys were trained to interact with visual stimuli presented on a CRT video monitor (Vision Master 1451, Iiyama; viewing distance 57 cm; frame rate 75 Hz) using the psychophysics toolbox (
Brainard and Vision, 1997; Pelli, 1997; Kleiner et al., 2007). Task events were controlled by Rex software (Hays et al., 1982). The monkeys were trained to control their gaze and make saccadic eye
movements to peripheral targets to receive a liquid reward (juice). The direction of gaze was monitored by an infrared camera (EyeLink 1000; SR Research, Ottawa, Canada; 1 kHz sampling rate). The
tasks involve stages separated by random delays, distributed as truncated exponential distributions
where t[min] and t[max] define the range, λ is the time constant, and α is chosen to ensure the total probability is unity. Below, we provide the range (t[min] to t[max]) and the exponential
parameter λ for all variable delays. Note that because of truncation, the expectation 𝔼(t) < t[min] + λ.
In the main task (Fig. 1a) the monkey must decide the net direction of random dot motion and indicate its decision when ready by making a saccadic eye movement to the corresponding choice target.
After acquiring a central fixation point and a random delay (0.25–0.7 s, λ =0.15), two red choice-targets (diameter 1 dva) appear in the left and right visual fields. The random dot motion is then
displayed after a random delay (0.25 – 0.7 s, λ =0.4 s) and continues until the monkey breaks fixation. The dots are confined to a circular aperture (diameter 5 dva; degrees visual angle) centered on
the fixation point (dot density 16.7 dots⋅dva^−2⋅s^−1). The direction and strength of motion are determined pseudorandomly from ±{0, 3.2, 6.4, 12.6, 25.6, 51.2}% coherence. The sign of the coherence
indicates direction (positive for leftward, which is contraversive with respect to the recorded hemisphere). The absolute value of coherence determines the probability that a dot plotted on frame n
will be displaced by Δx on frame n + 3 (Δt =40 ms), as opposed to randomly replaced, where ^−1 speed of apparent motion (see also Roitman and Shadlen, 2002). The monkey is rewarded for making a
saccadic eye movement to the appropriate choice target. On trials with 0% coherence motion, either saccadic choice is rewarded with probability Stine et al., 2023, for additional details). On
approximately half of the trials, a 100 ms pulse of weak motion (±4% coherence) is added to the random dot motion stimulus at a random time (0.1–0.8 s, λ =0.4) relative to motion onset (similar to
Kiani et al., 2008). Monkey M performed 9684 trials (5 sessions); monkey J performed 8142 trials (3 sessions). The data are also analyzed in a companion paper that focuses on the termination of the
decision (Stine et al., 2023).
In the visually instructed delayed saccade task (Hikosaka and Wurtz, 1983), one target is displayed at a pseudo-random location in the visual field. After a variable delay (monkey M: 0.4–1.1 s, λ =
0.3; monkey J: 0.5–1.5 s, λ =0.2) the fixation point is extinguished, signalling ‘go’. The monkey is rewarded for making a saccade to within ±2.5 dva of the location of the target. In a memory-guided
variant of the task (Gnadt and Andersen, 1988; Funahashi et al., 1989), the target is flashed briefly (200 ms) and the monkey is required to make a saccade to the remembered target location when the
fixation point is extinguished. These tasks provide a rough characterization of the neural response fields during the visual, perisaccadic and delay epochs. Neurons are designated post hoc, after
spike sorting.
The passive motion-viewing task is identical to the main task, except there are no choice targets and only the strongest motion strength (±51.2% coherence) is displayed for 500 ms (1 s on a small
fraction of trials in session 1). The direction is left or right, determined randomly on each trial
Behavioral analyses
We fit a neurally inspired variant of the drift-diffusion model (Fig. 1c) to the choice-RT data from each session. The model constructs the decision process as a race between two accumulators: one
accumulating evidence for left and against right (e.g., left minus right) and one accumulating evidence for right and against left (e.g., right minus left). The decision is determined by the
accumulator that first exceeds its positive decision bound, at which point the decision is terminated. The races are negatively correlated with one another, owing to the common source of noisy
evidence. We assume they share half the variance,
We used the method of images van Den Berg et al. (2016); Shan et al. (2019) to compute the probability density of the accumulated evidence for each accumulator (which both start at zero at t = 0) as
a function of time (t) using a time-step of 1 ms. The decision time distributions rendered by the model were convolved with a Gaussian distribution of the non-decision times, t[nd], which combines
sensory and motor delays, to generate the predicted RT distributions. The model has six parameters: κ, B[0], α, μ[nd], σ[nd], and C[0], where κ determines the scaling of motion strength to drift
rate, C[0] implements bias in units of signed coherence (Hanks et al., 2011), μ[nd] is the mean non-decision time and σ[nd] is its standard deviation (Table 1). Additional details about the model and
the fitting procedure are described in van Den Berg et al. (2016).
Model fit parameters.
κ: scaling of motion strength to drift rate, B[0]: bound height, α: linear urgency component, μnd: mean of the non-decision time, σnd: standard deviation of the non-decision time, and C[0]: bias
Information about individual experimental sessions.
Simulated decision variables
We fit the race model described above to the combined behavioral data across all sessions (separately for each monkeys) and used the best-fitting parameters for monkey M (see Table 1) to simulate a
total of 60,000 trials representing all signed coherences of the motion discrimination task. Each simulated trial yields a time series for two decision variables, one for each accumulator in the
race. We assume that the model-derived non-decision time (t[nd] = 317 ms; Fig. 1b) comprises visual and motor processing times at the beginning and end of the decision: 200 ms from motion onset to
the beginning of evidence integration, and the remaining 117 ms after termination. The latter approximates the variability observed in the saccadic latencies in the delayed saccade task and is
simulated using a Normal distribution, 𝒩 (μ, σ), where μ = 117 ms and σ = 39 ms (Stine et al., 2023). In this variable time period between decision termination and the response (saccade), the
simulated DVs were assigned the values they had attained at the start of this epoch. For all analyses that employ these simulations, we use the decision variable for the left-choice accumulator,
because the neural recordings were from LIP in the right hemisphere.
We used prototype “alpha” version Neuropixels1.0-NHP45 probes (IMEC/HHMI-Janelia) to record the activity of multiple isolated single-units from the ventral subdivision of area LIP (LIP[v]; Lewis and
Van Essen 2000). We used anatomical MRI to identify LIP[v] and confirmed its physiological hallmarks with single-neuron recordings (Thomas Recording GmbH) before proceeding to multi-neuron
recordings. Neuropixels probes enable recording from 384 out of 4416 total electrical contacts distributed along the 45 mm long shank. All data presented here were recorded using the 384 contacts
closest to the tip of the probe (Bank 0), spanning 3.84 mm. Reference and ground signals were directly connected to each other and to the monkey’s head post. A total of 1084 neurons were recorded
over eight sessions (54–203 neurons per session). (Table 2).
The Neuropixels 1.0-NHP45 probe uses a standard Neuropixels 1.0 headstage and is connected via the standard Neuropixels1.0 5m cable to the PCI eXtensions for Instrumentation (PXIe) hardware
(PXIe-1071 chassis and PXI-6141 and PXIe-8381 I/O modules, National Instruments). Raw data were acquired using the SpikeGLX software (http://billkarsh.github.io/SpikeGLX/), and single-units were
identified offline using the Kilosort 2.0 algorithm (Pachitariu et al., 2016; Pachitariu, 2021), followed by manual curation using Phy (https://github.com/cortex-lab/phy).
Neural data analysis
The spike times from each neuron are represented as delta functions of discrete time, s[i,n](t) on each trial i and each neuron n (dt = 1 ms). The weighted sum of these s[i,n](t) give rise to the
single trial population signals:
where the superscript, x, identifies the method or source that establishes the weights —that is, the coding direction in neuronal state space or the neuron type contributing to a pooled average
(e.g.,gausswin (width = 80 ms, width-factor = 1.5, σ ≈ 26 ms). Unless otherwise specified, all other analyses employ a 50 ms boxcar (rectangular) filter; values plotted at time t include data from t
− 24 to t + 25 ms.
We used several methods to define coding directions in the neuronal state space defined by the population of neurons in each session. For PCA and choice-decoding, we standardized the single trial
firing rates for each neuron using the mean and standard deviation of its firing rate at in the epoch 200 ≤ t ≤ 600 ms after motion onset. This practice led to the exclusion of two neurons (session
1) that did not produce any spikes in the normalization window. Those neurons were assigned zero weight.
T[in] neurons. Neurons were classified post hoc as T[in] by visual-inspection of spatial heatmaps of neural activity acquired in the delayed saccade task. We inspected activity in the visual, delay,
and perisaccadic epochs of the task. The distribution of target locations was guided by the spatial selectivity of simultaneously recorded neurons in the superior colliculus (SC; see Stine et al.,
2023, for details). Briefly, after identifying the location of the SC response fields, we randomly presented saccade targets within this location and seven other, equally spaced locations at the same
eccentricity. In monkey J we also included 1–3 additional eccentricities, spanning 5–16 degrees. Neurons were classified as T[in] if they displayed a clear, spatially-selective response in at least
one epoch to one of the two locations occupied by the choice targets in the main task. Neurons that switched their spatial selectivity in different epochs were not classified as T[in]. The
classification was conducted before the analyses of activity in the motion discrimination task. The procedure was meant to mimic those used in earlier single-neuron studies of LIP (e.g., Roitman and
Shadlen, 2002) in which the location of the choice targets was determined online by the qualitative spatial selectivity of the neuron under study. The p < 0.05 for 97% of neurons, Wilcoxon rank sum
test). Given the sparse sampling of saccade target locations, we are unable to supply a quantitative estimate of the center and spatial extent of the RFs. We next describe the methods to establish
the coding directions.
Ramp direction
We applied linear regression to generate a signal that best approximates a linear ramp, on each trial, i, that terminates with a saccade to the choice-target contralateral to the hemisphere of the
LIP recordings. The ramps are defined in the epoch spanning the decision time: each ramp begins at f[i](t[0]) = −1, where t[0] = 0.2 s after motion onset, and ends at f[i](t[1]) = 1, where t[1] = t
[sac] − 0.05 s (i.e., 50 ms before saccade initiation). The ramps are sampled every 25 ms and concatenated using all eligible trials to construct a long saw-tooth function (see Fig. S2). The
regression solves for the weights assigned to each neuron such that the weighted sum of the activity of all neurons best approximates the saw-tooth. We constructed a time series of standardized
neural activity, sampled identically to the saw-tooth. The spike times from each neuron are represented as delta functions (rasters) and convolved with a non-causal 25 ms boxcar filter. The mean and
standard deviation of all sampled values of activity were used to standardize the activity for each neuron (i.e., Z-transform). The coefficients derived from the regression establish the vector of
weights that define S^ramp. The algorithm ensures that the population signal S^ramp(t), but not necessarily individual neurons, have amplitudes ranging from approximately −1 to 1.
We employed a lasso linear regression with λ = 0.005. The vector of weights assigned across the neurons defines a direction in neuronal state space, S^ramp, which we use to render the signal S^ramp(t
) on single trials by projecting the data onto this direction. To determine the effect of the regularization term in the lasso regression, we recomputed single-trial signals using standard linear
regression, without regularization. We then calculated the Pearson correlation between single-trial traces generated by projecting neural data onto the two coding directions (i.e., with and without
regularization). The high correlation between single trial traces (mean r = 0.99, across sessions) indicates that the findings are not a result of the regularization applied. Here and elsewhere we
compute the mean r using the Fisher-z transform, such that
where Z^inv is the inverse Fisher-z.
Principal Component Analysis (PCA)
We applied standard PCA to the firing rate averages for each neuron using all trials sharing the same signed motion coherence in the shorter of two epochs: 200 ms to either 600 ms after motion onset
or 100 ms before the median RT for the signed coherence, whichever produces the shorter interval. The results of the PCA indicate that the dimensionality of the data is low, but greater than one. The
participation ratio is 4.4 ±1.3 (Mazzucato et al., 2016; Gao et al., 2017) and the first 3 PCs explain 67.1 ±3.1 % of the variance on average (both mean ±s.e.m. across sessions). As in all other
analyses of neural activity aligned to motion onset, we exclude data in the 100 ms epoch ending at saccade initiation on each trial. We projected the neural data onto the first PC to generate the
signal S^PC1.
Choice decoder
For each experimental session, we trained logistic choice decoders with lasso regularization (λ = 0.01) on the population activity in 50 ms time bins spanning the first 500 ms after motion onset and
the 300 ms epoch ending at saccade initiation, respectively. Each of the decoders was trained on the even-numbered trials. Decoder accuracy was cross-validated using the activity of held-out, odd
trials at the same time point (Fig. 4a). For the time bins aligned to motion onset, we also assessed the accuracy of the decoders trained on each of the time bins to predict the choice on time bins
on which they were not trained (Fig. 4b; King and Dehaene, 2014). We use the decoder trained on the bin centered on t = 450 ms to define the What coding direction. We refer to this as the fixed
training-time decoder to distinguish it from the standard machine-learning decoder, which assigns a potentially distinct vector of weights at each time point. We applied a similar analysis to
simulated data (see Simulated decision variables) to generate the black curve in Fig. 4a. Assuming a stochastic drift-diffusion process giving rise to choice and response times, the exercise
establishes a rough upper bound on decoder accuracy, were the actual drift-diffusion process known precisely.
When decoder
This decoder is trained to predict whether a saccade to the left (contralateral) choice target will occur within the next 150 ms. We applied logistic regression with lasso regularization (λ = 0.01)
to spike counts from each neuron in discrete bins of 25 ms, from 200 ms before motion onset to 50 ms before the saccade. We used only trials ending in a left choice (including errors) and trained the
decoder on the even numbered half of those trials. The concatenation of these trials forms a sequence of step functions which are set to 1 if a saccade occurred within 150 ms of the start of the 25
ms time bin and 0 otherwise (Fig. S6).
The spike counts were also concatenated across these trials to construct column vectors (one per neuron) that match the vector of concatenated step functions. These concatenated vectors, one per
neuron, plus an offset (β[0]), serve as the independent variables of the regression model (one β term per neuron). The proportion of β weights equal to zero, controlled by the lasso parameter, λ, was
0.8 ± 0.02 across sessions. The weights define the S^When coding direction, which yields single-trial signals, S^When(t). The When-decoder signal is S^When(t) + β[0]. We validated the When-decoder by
computing the area under an ROC (AUC) using the held-out (odd-numbered) trials ending in left choices (mean AUC over all time points and sessions: 0.84 ± 0.024, mean ±s.e.).
Our motivation, however, was to ascertain whether the When coding direction also predicts the monkey’s choices on all trials—that is, to perform as a What decoder. To this end, we predicted the
choice using the sign of the detrended S^When(t), formed by subtracting the average of the signal using all trials:
where ⟨⋯⟩[i] denotes expectation across all trials contributing values at time t. The choice accuracy is
The green trace in Fig. 4a shows ⟨A[i](t)⟩[i].
Aggregation of data across experimental sessions
To combine single trial data across sessions (e.g. S^x(t)), we first normalize activity within each session as follows. Using all trials ending in the same choice, we construct the trial-averaged
activity aligned to both motion onset (0 ≤ t[motion] ≤ 0.6 s) and saccade onset (−0.6 ≤ t[sacc] ≤ 0 s). This produces four traces. The minimum and maximum values (a[min] and a[max]) over all four
traces establish the range, zero to one, of the normalized signal:
where lower-case i and time t in an individual session.
Cosine similarity
We computed the cosine similarity between the weight vectors that define coding directions S^ramp, S^PC1, S^What and S^When. Mean cosine similarities are portrayed in the heatmap in Fig. 4d and also
in Fig. S8 top, where they are accompanied by error bars. We evaluated the null hypothesis that the mean cosine similarity is ≤ 0 with t-tests. We also performed two control analyses that deploy
random coding directions in neuronal state space. For each of the original coding directions we obtained 1000 random coding directions as random permutations of the weight assignments. The cosine
similarities between pairs of such random directions in state space are shown in Fig. S8, top. The cumulative distribution of cosine similarities under permutation supports p-values less than 1/1000.
In a second control analysis, we used random unit vectors as random coding directions (Normal distribution with mean 0 and scaled to unit length).
Similarity of single-trial signals
We calculated Pearson correlation to quantify the similarity of the signals generated by pairs of coding directions, x and y. For each trial, i, the detrended signals, j indexes successive 50 ms bins
between 200 ms after motion onset and 100 ms before saccade initiation. We excluded trials comprising less than four such bins. Each trial gives rise to a correlation coefficient, r[i]. We report the
mean r using Eq. 4. The r-values for comparisons across all pairs of coding directions are summarized in Fig. 4e, and variability across sessions is portrayed in Fig. S8, bottom. For each pair of CDs
and session, we evaluated the null hypothesis,
We also performed two control analyses that deploy random (CDs) in neuronal state space. These analyses control for the possibility that the correlations observed in the signals are explained by
pairwise correlations between the neurons, regardless of the signals produced by the weighted sums. (1) We generated sets of single-trial traces S^rand(t) by projecting the neural responses onto
random CDs, defined by permuting the weights of each coding direction (S^ramp, S^PC1, S^What and S^When). For each pair of CDs, we compute within-trial correlations between ordered pairs of trials
using the same method applied to the original signals. We repeat this process for a total of 1,000 random permutations per pair of CDs, per session. (2) We sample a pair of random weight vectors from
standard Normal distributions. Each weight vector has a dimension equal to the number of recorded neurons in the session. The weight vectors are normalized to sum to 1. We generate random CDs using
these weights and compute within-trial correlations using the same method applied to the original signals. We repeat this process 1,000 times per session. For both analyses, we evaluated the null
hypothesis that the observed correlations are not greater than those produced by the random projections (t-tests using the Fisher-z transformed correlations). Mean ± stdev of the mean r-values
between ordered pairs for both control analyses are summarized in Fig. S8, bottom.
Leverage of single-trial activity on behavior
The leverage of single-trial signals, S^x(t), on choice and RT was assessed using the earliest 300 ms epoch of putative integration (0.2 < t < 0.5 s from motion onset), restricting analyses to trials
with RTs outside this range (0.67 < RT < 2 s). The single-trial signals are smoothed with a 50 ms boxcar filter and detrended by subtracting the mean S(t) for trials sharing the same motion strength
and direction (i.e., signed coherence). The reaction times are also expressed as residuals relative to the mean RT, using trials sharing the same signed coherence and choice. We include trials with |
coh| ≤ 0.064 that result in choices of the left response target in this analysis. Including trials of |coh| ≤ 0.128 produced comparable results. The leverage on RT is the Pearson correlation between
the residual signals t ≤ 0.5, and Eq. 4). We also show the correlation at t=0.55 s, using the ordered pairs, S^x by computing partial correlations Fig. 5. We show this mediation at all time points.
We also report a mediation statistic (ξ^RT) using the time point 200 ms after the beginning of putative integration (i.e., S(t = 0.4)):
The rationale for using the 400 ms time point is (i) to allow the process to have achieved enough leverage on RT so that a reduction is meaningful and (ii) to preserve a substantial gap between this
time and the sample at t = 0.55 s (e.g., to avoid autocorrelations imposed by smoothing). The rare cases in which there was no negative correlation between S(t = 0.4) and RT were excluded from this
summary statistic, ξ^RT, because no mediation is possible (Session 2: S^ramp & S^PC1). When combining values of ξ^RT across sessions, we rectify any
We compute the leverage on choice, ξ^Ch, using trials with |coh| ≤ 0.064 and the same time points as for ξ^RT. Instead of an R-squared measure, we based ξ^Ch on coefficients derived from logistic
where β[1](t) is the simple leverage of S(t) on choice, analogous to simple correlation. The regression analysis was performed separately for each session. The coefficients β[1](t) were divided by
their standard error and then averaged across sessions. This normalization step was implemented to control for potential variation in the magnitude of S(t) (and therefore of β[1](t)) across sessions.
Analogous to partial correlation, we include the later time point and fit
where t given the S(0.55). The regression coefficients β[1](t) coefficients obtained from Eq. 8. That is, the same normalization factors were used for the mediated and unmediated leverage. The
summary statistic for choice mediation is defined by
For both types of mediation, we also test whether the earlier S(t) is mediated by the S(0.55) in Eq. 9 and in the expression for partial correlations.
Because the mediation statistics, ξ^RT and ξ^Ch, are, by their definition, non-negative, we assess statistical significance by bootstrapping. For each session, we construct 1000 surrogate data sets
equal in size to the original data, by sampling with replacement. The standard deviation of the leverage and mediation values at each time approximates the standard error. We compare the distribution
of the mediation statistics, ζ, to their distribution under the null hypothesis, ℋ[0], that the values arise by chance, instantiated by breaking the correspondence with the trial giving rise to the
later sample, S^x(t = 0.55s). The permutation maintains correspondence in signed motion coherence. We compare the distributions of mediation from the bootstrap and ℋ[0] using the Wilcoxin Ranked Sum
To test whether the observed leverage of neural activity on choice and RT is achieved by projections onto arbitrary coding directions, we generated random weight vectors by permuting the weights
associated with the first PC for each session. We projected activity onto this random coding direction, applied the mediation analyses described above to this signal, and repeated this process 1000
times to produce a null distribution at each time point. The reported p-values represent the probability that the observed leverage was generated from this null distribution.
We performed a similar analysis to test whether the observed leverage depends on the trial-to-trial correspondence between neural activity and behavior. Here, the null distribution at each time point
was generated by randomly permuting the trial indices associated with the neural activity and those associated with the behavioral measures.
Finally, we used the simulated data from the racing accumulator model to test the degree of leverage and mediation expected had we known the ground-truth decision variable on each trial (Fig. S10).
Because the simulated DV is noiseless (and the process is Markovian), the mediation is expected to be complete for all time points tested in the analyses. We therefore took two steps to make the
simulated data more comparable to neural data: (i) We sub-sampled the simulated data to match the number of trials in each session. (ii) We generated N noisy instantiations of the signal for each of
the sub-sampled, simulated trials, where N is the number of N × (N − 1)/2 neuron pairs (r ≈ 0.09). We then applied the mediation analyses to the mean of these signals and repeated this process 1000
We performed three control analyses to determine whether the results of the mediation analysis were specific, meaningful and comparable to results obtained for the DV of a race model. To showcase
that the results are specific, we generated random weight vectors by permuting the weights of the PC1 coding direction. We repeated this procedure 1000 times per session and projected the data along
these directions in neuronal state space to generate S^rand. We then computed the mediation analyses detailed above on these signals and determined significance by comparing the leverage on choice
and correlation with RT of S^ramp, S^PC1, and S^rand at each time point.
To estimate an upper limit for the degree of possible leverage and mediation, we simulated 60,000 trials using the race model that best fits the behavioral data of monkey M (see Simulated decision
variables.). For any noise-free representation of a Markovian integration process, the leverage of an early sample of the DV on behavior would be mediated completely by later activity as the latter
sample by definition encompasses all variability captured by the earlier sample. We, therefore, took two steps to make the simulated decision variables more comparable to real neural data. (1) For
each session, we first subsampled the simulated data to match the each session. (2) To evaluate a DV approximated from the activity of n n noisy instantiations of the signal for each simulated trial.
The added noise independent across time points and weakly correlated across neurons (r ≈ 0.09). We then computed the measured DV S^sim as the mean activity of these n simulated neurons. We repeated
this procedure 1000 times per session and Fig. S10, bottom displays the mean and standard deviation across permutations of the leverage of S^sim(t) on behavior. The simulation results highlight that
we would not expect the mediation of the leverage on behavior by a later sample to be complete (blue traces in Fig. 5 and Fig. S10 not aligned with zero).
Noise correlation between neurons
The mean pairwise correlation between t ≤ 0.4. These scalar values are converted to residuals by subtracting the mean (for each neuron) across all trials sharing the same signed motion coherence. The
residuals from all eligible trials are concatenated for each neuron to support the calculation of N × (N − 1) Pearson r values, where N is the number of Eq. 4.
Direction selective neurons
We identified direction-selective (DS) neurons (M[in] neurons) using the passive motion-viewing task (described above). We classified a neuron as M[in] if it achieved two criterion. The first
criterion is a short-latency response to the onset of random dot motion, which we defined as a 5-fold increase in firing rate relative to baseline in the first 80 ms following motion onset or a
steeper increase in activity in the 80 ms after than the 200 ms before motion onset. The second criterion was direction selectivity. We calculated the area under the ROC (AUC) comparing leftward
versus rightward for two separate epochs:(i) 0.15 ≤ t < 0.3 s and (ii) 0.3 ≤ t < 0.5 s. Neurons were determined to be DS if the AUC in either epoch exceeded 0.6. We excluded one neuron from this
analysis because it switched its direction preference in the two epochs. We also excluded neurons that had previously been classified as T[in].
Latency analysis
We estimated the latency of direction-selective responses using the CUSUM method (Ellaway, 1978; Lorteije et al., 2015). We employed a receiver operating characteristic (ROC) analysis to estimate the
selectivity of each M[in] neuron to motion direction. The AUC reflects the separation of the distributions of spike counts (100 to 400 ms after motion onset) on single trials of leftward and
rightward motion, respectively. We included only correct trials with response times greater than 450 ms and motion strengths above 10% coherence. For each neuron with AUC > 0.6, we computed the
difference in spike counts (25 ms bins) between correct trials featuring leftward and rightward motion. Subsequently, we accumulated these differences over time, following the CUSUM method. The
resulting difference is approximately zero before the onset of direction selectivity and then either increases or decreases monotonically, depending on the preferred motion direction. To identify the
transition between these two regimes, we fit a dog leg function to the cumulative sum of spikes: a flat line starting at t[0] = 0 followed by a linearly increasing component beginning at t[1] > t[0].
The time of the end of the flat portion (between 0 and 500 ms from motion onset) of the fit was taken as the latency. Estimating latencies based on cumulative sums of spikes helps mitigate the effect
of neuronal noise. The fitting step reduces the effect of the number of trials on latency estimates compared to traditional methods that rely on t-tests in moving windows.
Correlations between M[in] and
The analysis of the correlations shown in Fig. 6e is based on the spike counts of the Fig. 6e). Statistical significance was assessed using permutation tests, as follows. Two regions of interest
(ROI) were defined based on the time from stimulus onset for the M[in] (x) and T[in] (y) dimensions. The first region of interest, ROI[1], is characterized by t[x] > 100, t[y] > 200, and t[y] > t[x].
According to our hypothesis that the M[in] neurons represent the momentary evidence integrated by [2], is defined by t[x] > 100, t[y] > 200, and t[y] < t[x]. If, contrary to our hypothesis, M[in]and
T[in] signals were influencing each other bidirectionally, we would expect high correlations in this region. We calculated the difference in correlations between these two groups, ⟨ρ[ROI1]⟩ − ⟨ρ
[ROI2]⟩, where the expectation is over the time bins within each region of interest. This difference was compared to those obtained after randomly shuffling the order of the trials for one of the
dimensions before calculating the pairwise correlations (N[shuffles] = 200). We assess significance with a z-test given the mean and standard deviation of the values obtained under shuffling. The
analysis was repeated with an alternative ROI[2] defined by t[x] < 100 and t[y] < 200, representing the times before direction selectivity is present in at least one of the two dimensions.
Correlations between M[in] signals and behavior
To assess the leverage of M[in] signals on choice and RT (Fig. 6d) we performed the same logistic regression and pairwise correlation analyses as in Fig. 5, substituting the S^x. The leverage on
choice is not mediated by a later sample of either M[in] signal (ξ^Ch ≤ 9.6%; not shown), and there is negligible leverage on RT to mediate. We suspect the failure to detect leverage of M[in] is
explained by a lack of power, owing to the focus on long RT trials, narrow sample windows (50 ms boxcar), and the small number of [in] signals (standardized as in the previous paragraph):
on the interval 0.1 ≤ t ≤ 0.4 s from motion onset, on each trial, k, including trials with contraversive choices and RT ≥ 500 ms. We calculated the Pearson correlation coefficient between ψ[k] and
RT. Response times were z-scored independently for each signed motion strength and session. We evaluated the null hypothesis that the correlation coefficient is non-negative. The reported p-value is
based on a one-tailed t-statistic.
Variance and autocorrelation of smoothed diffusion signals
The analyses in Fig. 3 compare the variance and autocorrelation of the single-trial signals, S^ramp(t), to those expected from unbounded drift-diffusion. To mitigate the effect of the bound, we focus
on the earliest epoch of putative integration (200 to 506 ms after motion onset; six 51 ms counting windows) and the weakest motion strengths (|coh| ≤ 3.2%). The single trial signals are detrended by
the mean across trials sharing the same signed motion coherence and baseline corrected by subtraction of i.
The variance as a function of time and the autocorrelation as a function of time and lag are well specified for the cumulative sum of discrete iid random samples, but the autocorrelation is affected
by the boxcar filter we applied to render the signals. We incorporated the correction in our characterization of unbounded diffusion. The derivation is summarized in Appendix 1, and we provide Matlab
code in the GitHub repository. The theoretical values shown in Fig. 3 assume a 1 kHz sampling rate and standard Wiener process (i.e., samples drawn from a normal distribution with Fig. 3a is that the
variance of mean single-trial signals, S[i](t) over the epoch 26 ± 25 ms should double in the epoch (26 + 51) ± 25 ms, and triple in the epoch (26 + 2 × 51) ± 25 ms), and so on for each successive
non-overlapping running mean. We therefore use arbitrary units, normalized to the measured variance of the first point. We do not know the variance of the drift-diffusion signal that S(t) is thought
to approximate, but we assume it can be decomposed—by the law of total variance—to a component given by drift-diffusion and components associated with spiking and other nuisance factors. We therefore
search for a scalar non-negative factor ϕ ≤ 1 that multiplies all terms in the diagonal of the empirical covariance matrix (i.e., the variance) before normalizing to produce the autocorrelation
matrix. We search for the value of ϕ that minimizes the sum of squares between Fisher-z transformed correlation coefficients in the theoretical and empirical autocorrelation matrices (Fig. 3b,c).
Standard errors of the variance and autocorrelations in (Fig. 3a & c are estimated by a bootstrap procedure respecting the composition of motion strength and direction (the s.e. is standard deviation
of each variance and autocorrelation term across 500 repetitions of the procedure.
We consider a discrete time (sampling interval dt) Wiener process with independent random increments ϵ[k] on time step k that are zero-mean noise with variance σ^2(ϵ[k]) = dt (i.e. unit variance per
second). The accumulated evidence (i.e. decision variable, DV) on time step p is
For such a Wiener process,
As the increments ϵ[k] are independent across time, Cov(ϵ[j], ϵ[k]) is 0 for j ≠ k and dt for j = k
We define the mean DV over a window of ±n points as
We consider two time points p < q with window n such that there is no overlap and hence p + n < q − n
Given that
Therefore the correlation
In contrast for the point estimates at p and q
It is useful to re-express the above two equations in terms of actual time t[p] and t[q] and window size t[n]. Substituting for p, q, and n with t[p]/dt, t[q] /dt and t[n]/dt
We thank Shushruth, NaYoung So, and David Gruskin for comments on the manuscript, Cornel Duhaney and Brian Madeira for their assistance in the planning and execution of surgeries, animal training and
general support, and we thank Columbia University’s ICM for the quality of care they provide for our animals, especially during the pandemic and lockdown. We would further like to thank Tanya
Tabachnik and her team at the Zuckerman Institute Advanced Instrumentation Core and Tim Harris, Wei-lung Sun, Jennifer Colonell, and Bill Karsh at HHMI Janelia for their continued support with
Neuropixels1.0-NHP45 probes development and testing. This research was supported by the Howard Hughes Medical Institute; an R01 grant from the NIH Brain Initiative (M.N.S., R01NS113113); a T32 and
F31 grant from the National Eye Institute (G.M.S, T32 EY013933, F31 EY032791); the Grossman center; and the Brain and Behavior Research Foundation.
Data availability statement
Data and code are available upon request.
Extended Data
Effect of motion pulses on behavior (adapted from Stine et al., 2023).
a, Choice (bottom) and mean reaction time (top) as a function of motion strength (combined data from the two monkeys; otherwise same conventions as Fig. 1b). The two traces show trials in which a
leftward (grey) and rightward (black) motion pulse occurred during motion viewing. The pulses (100 ms) had a biasing effect on RT and choice equivalent to shifting the functions left or right by ±1.4
% coh (p < 0.001, likelihood ratio test). b, Effect of motion pulses on choices as a function of time from the response. Pulses had a persistent effect on choices, consistent with temporal
integration of motion evidence. Shading is ±1 s.e.
Derivation of a ramp coding-direction in neuronal state space.
Weights are assigned to each of the N simultaneously recorded neurons in each session using simple least squares regression to approximate a ramp from −1 to +1 on the interval from 200 ms after
motion onset to 100 ms before saccade initiation. Only trials ending in left (contraversive) choices are included. The graph shows the quality of the regression on 6 trials from Session 1. Projection
of the population firing rates on the vector of weights renders the single trial signal, S^ramp(t).
Trial-averaged activity grouped by RT quantile.
Rows show the averages of single-trial responses of three signals. Choice and RT quantile are indicated by color (legend) and aligned to motion onset (left) and saccade initiation (right). a, Ramp
coding direction. b, First principal component from PCA. c, Averages of the subset of neurons that have the left (contralateral) target in their response field. Correct and error trials are included
in both motion and saccade aligned averages.
Trial-averaged activity after subtracting the urgency component.
The urgency signal, u(t), is a time-dependent, evidence-independent component of the neural activity that is thought to implement the equivalent of a collapsing bound in the race-model architecture
of drift-diffusion shown in (Fig. 1c). We estimate u(t) for each signal, S^x, as the average S^x(t), aligned to motion onset, using only the 0% coherence motion trials (gray traces in the third
column of Fig. 2). a, Ramp coding signal, S^ramp(t), with urgency subtracted. b, PC1 signal, S^PC1(t), with urgency subtracted. c,
Bounds induce sublinear increase in variance of diffusion paths.
For unbounded diffusion (black) the variance across diffusion paths increases linearly with unity slope, and this holds under our smoothing procedure too. Symbols mark the median sample times of the
first six non-overlapping t ± 25 ms running means from the beginning of the epoch of integration, as in Fig. 3a. The red trace shows the values produced by stimulating the model illustrated in Fig.
1c (combined residuals from 20,000 trials per coherence -.032, 0, & +.032, as in Fig. 3a).
Derivation of a When direction in neuronal state space.
Weights are assigned to each of the N simultaneously recorded neurons in each session using logistic regression to approximate a step that takes a value of 0 from 200 ms before motion onset to 150 ms
before saccade initiation, and a value of 1 for the following 100 ms (the last 50 ms before the saccade are discarded). Only trials ending in left (contraversive) choices are included. The graph
shows the quality of the regression on 8 example trials. The traces are shown from 200 ms after motion onset to 50 ms before the saccade. Projection of the population firing rates on the vector of
weights renders the single trial signal, S^When(t).
Single-trials and trial averaged signals furnished by the When- and What-decoders.
Same convention as Fig. 2, using the same single-trial examples.
Variability in cosine similarity and within-trial correlation across sessions.
Top. Cosine similarity (CS) of five coding directions. Black markers portray the same mean CS as in Fig. 4d. Here, we also show the variability in each measure across sessions (s.e.m.). CS are
significantly greater than zero between all pairs of CDs (all p < 0.001, t-test). Brown markers portray the same measure when the assignment between neurons and weights are permuted. Error bars
reflect the standard deviation across 1000 such permutations. The purple marker in the rightmost column reflects the CS between random directions in state space. These vectors were generated by
drawing from a Normal distribution, 𝒩 (0, 1) and scaled to length 1. The CS in the data (black symbols) are significantly greater than the CS obtained from the two control analyses, for every
pairwise combination of coding directions (permuted weights: all p < 10^−156, t-test; random unit vectors: all p < 10^−63). Bottom. Same conventions as Top for the within-trial correlations portrayed
in Fig. 4e. For each pair of coding directions, the within-trial correlations in the data are significantly greater than zero (all p < 10^−23). The correlations are also significantly greater than
those between pairs of signals generated by projections of the data onto pairs of (i) random vectors established by permutations of the weights defining each coding direction (brown, all pairwise
comparisons p < 0.009, z-test) and (ii) random unit vectors (purple, all pairwise comparisons p < 10^−14, z-test). This control serves mainly to refute the possibility that the correlations are
explained by correlated variability in the neural population regardless of the signals produced by the weighted sums.
The T[in] neurons are not discoverable by their weight assignments.
The ’s weight might identify it asa, Distribution of weights assigned to b, Same as a for weight percentiles (computed within session). c, The graphs are logistic fitsI = 1 if neuron k isI[k] = 0
otherwise. Neurons with stronger positive or negative weights are more likely to be
Control analyses bearing on the leverage and mediation results in Fig. 5
a, Leverage of signals generated by projection of the neural data on random coding directions in neuronal state space (permutations of the PC1 weights). Same conventions and ordinate scale as in Fig.
5. b, Leverage of drift-diffusion signals derived from simulations of the racing drift-diffusion model fit to behavior (Fig. 1). The simulated signals establish the expectation of N weakly correlated
noisy neurons. The traces agree qualitatively with the leverage and degree of mediation by Leverage of single-trial activity on behavior). Same conventions as in Fig. 5.
Cross-mediation of Single-trial correlations with behavior.
The figure extends the observations in Fig. 5 that a sample of t = 550 ms after motion onset (i) reduces the leverage of earlier samples of S^x on S^x) and (ii) also reduces the leverage of earlier
samples of other signals, S^PC1 and S^ramp (cross-mediation of S^x on S^y). The The heatmaps are 5 × 5 matrices of the mediation indices, ξ^Ch and ξ^RT, (Eqs. 7 and 10), that is how a signal at t =
550ms mediates a signal at t = 400 ms after motion onset. The index is zero if there is no mediation by the later sample; one of the mediation is complete. The main diagonal (top left to bottom
right) shows the mediation of S^x (0.55) on S^x (0.4). Values below the diagonal show cross-mediation of S^y by S^x; values above the diagonal show mediation of S^x by S^y. a, Leverage on choice. b,
Leverage on RT. Notice that the matrices are not symmetric. For example, S^ramp mediates the leverage of S^ramp, and S^When on RT more than S^When mediates the leverage of
Comparison of linear and non-linear choice decoders.
The figure depicts the cross-validated classification accuracy of two decoders that use population activity at t = 450 ms from motion onset to predict choice at the same time point on held out
trials. The first is a linear decoder (logistic classifier, abscissa), as used for the What decoder in Fig. 4. The second is a non-linear decoder (ordinate), which takes the form of a neural network
with two hidden layers (N[1] = 100, N[2] = 50, units with sigmoid activation function). The two decoders perform similarly, with the Neural network outperforming the logistic decoder in only one of
eight sessions. The analysis suggests that the assumption of linear embedding of the DV is justified.
Decoding choice from subsets of neurons.
Mean accuracy (across sessions) of four choice decoders, plotted as a function of time from motion onset (left) and time to saccadic choice (right). The decoders are trained on the neural activity
between 425 and 475 ms from motion onset (gray arrow), and applied to all other time points (same method as Fig. 4a, fixed training-time). Colors indicate whether the decoders were trained using the
activity of all neurons (red, same as in Fig. 4a), only [in] neurons (purple), or all but T[in] and M[in] neurons (blue). Decoding accuracy is diminished without the contribution of T[in] neurons,
which constitute only 21.7 ± 2.1% of the population (mean ± s.e.).
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Article and author information
Author information
1. Natalie A Steinemann
2. Gabriel M Stine
3. Eric M Trautmann
4. Ariel Zylberberg
5. Daniel M Wolpert
6. Michael N Shadlen
Views, downloads and citations are aggregated across all versions of this paper published by eLife.
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Commit 2020-08-13 09:22 cc6528e7
View on Github →
feat(analysis/calculus/fderiv): multiplication by a complex respects real differentiability (#3731) If f takes values in a complex vector space and is real-differentiable, then c f is also
real-differentiable when c is a complex number. This PR proves this fact and the obvious variations, in the general case of two fields where one is a normed algebra over the other one. Along the way,
some material on module.restrict_scalars is added, notably in a normed space context.
Estimated changes
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SDL Containment Proof
Recall SDL:
A1: All tautologous wffs of the language (TAUT)
A2: OB(p → q) → (OBp → OBq) (OB-K)
A3: OBp → ~OB~p (OB-NC)
R1: If ⊢ p and ⊢ p → q then ⊢ q (MP)
R2: If ⊢ p then ⊢ OBp (OB-NEC)
We have already shown OB-NC is derivable in Kd above, and TAUT and MP are given, since they hold for all formulas of Kd. So we need only derive OB-K and OB-NEC of SDL, which we will do in reverse
order. Note that RM, if ⊢ r → s, then ⊢ □r → □s), is derivable in Kd, and so we rely on it in the second proof.^[1]
Show: If ⊢ p then ⊢ OBp. (OB-NEC)
Proof: Assume ⊢ p. It follows by PC that ⊢ d → p. So by NEC for □, we get ⊢ □(d → p), that is, OBp.
Show: ⊢ OB(p → q) → (OBp → OBq). (K of SDL)
Proof: Assume OB(p → q) and OBp. From PC alone, ⊢ (d → (p → q)) → [(d → p) → (d → q)]. So by RM for □, we have ⊢ □(d → (p → q)) → □[(d → p) → (d → q)]. But the antecedent of this is just, OB(p →
q) in disguise, which is our first assumption. So we have □[(d → p) → (d → q)] by MP. Applying K for □ to this, we get □(d → p) → □(d → q). But the antecedent to this is just our second
assumption, OBp. So by MP, we get □(d → q), that is, OBq.
Metatheorem: SDL is derivable in Kd.
Note that showing that the pure deontic fragment of Kd contains no more than SDL is a more complex matter. The proof relies on already having semantic metatheorems available. An excellent source for
this is Åqvist 2002 [1984].^[2]
Return to Deontic Logic.
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(Not recommended) Histogram bin counts
bincounts = histc(x,binranges) counts the number of values in x that are within each specified bin range. The input, binranges, determines the endpoints for each bin. The output, bincounts, contains
the number of elements from x in each bin.
• If x is a vector, then histc returns bincounts as a vector of histogram bin counts.
• If x is a matrix, then histc operates along each column of x and returns bincounts as a matrix of histogram bin counts for each column.
To plot the histogram, use bar(binranges,bincounts,'histc').
[bincounts,ind] = histc(___) returns ind, an array the same size as x indicating the bin number that each entry in x sorts into. Use this syntax with any of the previous input argument combinations.
Create Histogram Plot
Initialize the random number generator to make the output of randn repeatable.
Define x as 100 normally distributed random numbers. Define bin ranges between -4 and 4. Determine the number of values in x that are within each specified bin range. Return the number of elements in
each bin in bincounts.
x = randn(100,1);
binranges = -4:4;
[bincounts] = histc(x,binranges)
bincounts = 9×1
To plot the histogram, use the bar function.
Return Bin Numbers for Histogram
Defined ages as a vector of ages. Sort ages into bins with varying ranges between 0 and 75.
ages = [3,12,24,15,5,74,23,54,31,23,64,75];
binranges = [0,10,25,50,75];
[bincounts,ind] = histc(ages,binranges)
bincounts = 1×5
ind = 1×12
bincounts contains the number of values in each bin. ind indicates the bin numbers.
Input Arguments
x — Values to be sorted
vector | matrix
Values to be sorted, specified as a vector or a matrix. The bin counts do not include values in x that are NaN or that lie outside the specified bin ranges. If x contains complex values, then histc
ignores the imaginary parts and uses only the real parts.
Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32
binranges — Bin ranges
vector | matrix
Bin ranges, specified as a vector of monotonically nondecreasing values or a matrix of monotonically nondecreasing values running down each successive column. The values in binranges determine the
left and right endpoints for each bin. If binranges contains complex values, then histc ignores the imaginary parts and uses only the real parts.
If binranges is a matrix, then histc determines the bin ranges by using values running down successive columns. Each bin includes the left endpoint, but does not include the right endpoint. The last
bin consists of the scalar value equal to last value in binranges.
For example, if binranges equals the vector [0,5,10,13], then histc creates four bins. The first bin includes values greater than or equal to 0 and strictly less than 5. The second bin includes
values greater than or equal to 5 and less than 10, and so on. The last bin contains the scalar value 13.
Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32
dim — Dimension along which to operate
Dimension along which to operate, specified as a scalar.
Output Arguments
bincounts — Number of elements in each bin
vector | matrix
Number of elements in each bin, returned as a vector or a matrix. The last entry in bincounts is the number of values in x that equal the last entry in binranges.
ind — Bin index numbers
vector | matrix
Bin index numbers, returned as a vector or a matrix that is the same size as x.
• If values in x lie outside the specified bin ranges, then histc does not include these values in the bin counts. Start and end the binranges vector with -inf and inf to ensure that all values in
x are included in the bin counts.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• The output of a variable-size array that becomes a column vector at run time is a column-vector, not a row-vector.
• If supplied, dim must be a constant.
• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
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Why so gloomy? A Bayesian explanation of human pessimism bias in the multi-armed bandit task
Why so gloomy? A Bayesian explanation of human pessimism bias in the multi-armed bandit task
Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)
Dalin Guo, Angela J. Yu
How humans make repeated choices among options with imperfectly known reward outcomes is an important problem in psychology and neuroscience. This is often studied using multi-armed bandits, which is
also frequently studied in machine learning. We present data from a human stationary bandit experiment, in which we vary the average abundance and variability of reward availability (mean and
variance of reward rate distributions). Surprisingly, we find subjects significantly underestimate prior mean of reward rates -- based on their self-report, at the end of a game, on their reward
expectation of non-chosen arms. Previously, human learning in the bandit task was found to be well captured by a Bayesian ideal learning model, the Dynamic Belief Model (DBM), albeit under an
incorrect generative assumption of the temporal structure - humans assume reward rates can change over time even though they are actually fixed. We find that the "pessimism bias" in the bandit task
is well captured by the prior mean of DBM when fitted to human choices; but it is poorly captured by the prior mean of the Fixed Belief Model (FBM), an alternative Bayesian model that (correctly)
assumes reward rates to be constants. This pessimism bias is also incompletely captured by a simple reinforcement learning model (RL) commonly used in neuroscience and psychology, in terms of fitted
initial Q-values. While it seems sub-optimal, and thus mysterious, that humans have an underestimated prior reward expectation, our simulations show that an underestimated prior mean helps to
maximize long-term gain, if the observer assumes volatility when reward rates are stable and utilizes a softmax decision policy instead of the optimal one (obtainable by dynamic programming). This
raises the intriguing possibility that the brain underestimates reward rates to compensate for the incorrect non-stationarity assumption in the generative model and a simplified decision policy.
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f(x)=x3−4x and g(x)=2x+5,f[g(2)]=... | Filo
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Quantum Tunneling | Brilliant Math & Science Wiki
Quantum tunneling refers to the nonzero probability that a particle in quantum mechanics can be measured to be in a state that is forbidden in classical mechanics. Quantum tunneling occurs because
there exists a nontrivial solution to the Schrödinger equation in a classically forbidden region, which corresponds to the exponential decay of the magnitude of the wavefunction.
To illustrate the concept of tunneling, consider trying to confine an electron in a box. One could try to pin down the location of the particle by shrinking the walls of the box, which will result in
the electron wavefunction acquiring greater momentum uncertainty by the Heisenberg uncertainty principle. As the box gets smaller and smaller, the probability of measuring the location of the
electron to be outside the box increases towards one, despite the fact that classically the electron is confined inside the box.
The easiest solvable example of quantum tunneling is in one dimension. However, tunneling is responsible for a wide range of physical phenomena in three dimensions such as radioactive decay, the
behavior of semiconductors and superconductors, and scanning tunneling microscopy.
Scattering from a Potential Barrier in One Dimension
Scattering particles off of a potential barrier in one dimension look like this:
Suppose that the height of the potential barrier is \(V_0\) and the width is \(L\) and that scattering particles have energy \(E <V_0\). Then the picture can be divided into three regions:
1. Region 1 \((-\infty < x<0)\): \(E > V(x)\)
2. Region 2 \((0 \le x \le L)\): \(E <V(x)\)
3. Region 3 \((L < x< \infty)\): \(E > V(x),\) where \(V(x)=0; x<0\), \(V(x)=V_0;x=0, 0<x<L\) and \(V(x)=0, x>L.\)
In Region 1, the potential is zero. A moving wave thus has energy greater than the potential. This is also true in Region 3. However, in Region 2, the energy of the wave is less than the potential.
Therefore, the Schrödinger equation yields two different differential equations depending on the region:
1. Region 1 and Region 3 \[\frac{{d}^{2}\psi}{d{x}^{2}}={k}^{2}\psi, \quad k=\sqrt{\frac{-2mE}{{\hbar}^{2}}}\]
2. Region 2 \[\frac{{d}^{2}\psi}{d{x}^{2}}={\kappa}^{2}\psi, \quad \kappa=\sqrt{\frac{2m(V-E)}{{\hbar}^{2}}}.\]
The general solutions can be written as linear combinations of oscillatory terms in Regions 1 and 3, and as linear combinations of growing and decaying exponentials in Region 2:
\[\psi (x)=\begin{cases} { Ae }^{ ikx }+{ Be }^{ -ikx } \quad &\text{: Region 1} \\ { Ce }^{ \kappa x }+{ De }^{ -\kappa x } \quad &\text{: Region 2} \\ { Fe }^{ ikx } &\text{: Region 3}. \end{cases}
Note that plane-waves that travel to the right are of the form \({e}^{ikx}\) and plane-waves that travel to the left are of the form \({e}^{-ikx}\). In this experiment, a particle (plane-wave) enters
from the left and will partially transmit and partially reflect. However, no particle enters from the right heading towards the left; therefore, there is no \(Ge^{-ikx}\) term above in Region 3.
The coefficients above are fixed by the continuity of the wavefunction and its derivative at each point where the potential changes. One obtains two conditions from continuity at \(x=0\) and \(x=L\):
• 1) \(A+B=C+D\)
• 2) \({Ce}^{\kappa L}+{De}^{-\kappa L}= {Fe}^{ikL}\)
and two conditions from continuity of the derivative at \(x=0\) and \(x=L\):
• 3) \(Aik-Bik=C\kappa-D\kappa\)
• 4) \({C\kappa e}^{\kappa L}-{D\kappa e}^{-\kappa L}= {Fike}^{ikL}.\)
Dividing 3) by \(ik\) and adding to 1) obtains
• 5) \(2A=\left(1+\frac{\kappa}{ik}\right)C+\left(1-\frac{\kappa}{ik}\right)D.\)
Similarly, dividing 4) by \(\kappa\) and adding or subtracting from 2) obtains
• 6) \(2C{e}^{\kappa L}=\left(1+\frac{ik}{\kappa}\right)F{e}^{ikL}\)
• 7) \(2D{e}^{-\kappa L}=\left(1-\frac{ik}{\kappa}\right)F{e}^{ikL}.\)
Combining 5), 6), and 7) yields an equation for \(A\) in terms of \(F:\)
\[2A=\left(1+\frac{\kappa}{ik}\right)\left(1+\frac{ik}{\kappa}\right)\frac{F{e}^{ikL}{e}^{-\kappa L}}{2}+\left(1-\frac{\kappa}{ik}\right)\left(1-\frac{ik}{\kappa}\right)\frac{F{e}^{ikL}{e}^{\kappa
which can be rearranged to
\[\frac{A{e}^{-ikL}}{F}=\cosh{(\kappa L)}+i\left(\frac{{\kappa}^{2}-{k}^{2}}{2k\kappa}\right)\sinh{(\kappa L)}.\]
Now the probability of a wave to tunnel through the barrier is equal to the probability of the wavefunction in Region 3 divided by the probability of the wavefunction in Region 1. Multiplying the
above equation by its conjugate and taking the inverse, the probability of transmission is therefore quantified by
\[T=\frac{|F|^2}{|A|^2}={\left[{{\text{cosh}}^{2}(\kappa L)+\left(\frac{{\kappa}^{2}-{k}^{2}}{2k\kappa}\right)}^{2}{\text{sinh}}^{2}(\kappa L)\right]}^{-1}.\]
Since \({\text{cosh}}^{2}(x)-{\text{sinh}}^{2}(x)=1\),
\[T={\left[1+\left(\frac{{k}^{2}+{\kappa}^{2}}{2k\kappa}\right){\text{sinh}}^{2}(\kappa L)\right]}^{-1}.\]
Defining \(\beta=\left(\frac{{k}^{2}+{\kappa}^{2}}{2k\kappa}\right)\) makes the solution more compact:
\[T=\frac{1}{1+{\beta}^{2}{\text{sinh}}^{2}(\kappa L)}.\]
This can also be rewritten in terms of the energies:
\[T = \Bigg(1+ \frac{V_0^2}{4E(V_0 -E)} \sinh^2 \left(\frac{L}{\hbar} \sqrt{2m(V_0 - E)}\right)\Bigg)^{-1} .\]
Naturally, the probability of reflection is \(1-T\); hence,
\[R=\frac{{\beta}^{2}{\text{sinh}}^{2}(\kappa L)}{1+{\beta}^{2}{\text{sinh}}^{2}(\kappa L)}.\]
Macroscopically, objects colliding against a wall will be deflected. This is analogous to the reflection probability being 100% and transmission probability being 0%. The above example shows that it
is possible for matter waves to "go through walls" with some probability, given that a matter wave has sufficient energy or the barrier being sufficiently narrow \((\)small \(L).\) Note that, for a
very wide or tall barrier \((L\) very large\()\) or \(V_0 \gg E,\) the \(\sinh\) term in the expression for \(T\) goes to \(\infty\), yielding \(T \approx 0:\) for a very wide or tall barrier, there
is almost no transmission.
The below animation shows a localized wavefunction tunneling through the one-dimensional barrier by evolving the time-dependent Schrödinger equation:
Challenge problem: derive the transmission coefficient for the rectangular potential well
\[V(x) = \begin{cases} -V_0 \quad & 0<x<L \\ 0 \quad & \text{ otherwise}. \end{cases}\]
We have
\[T = \Bigg(1+ \frac{V_0^2}{4E(V_0 +E)} \sin^2 \left(\frac{L}{\hbar} \sqrt{2m(V_0 + E)}\right)\Bigg)^{-1}.\ _\square\]
\[0\] \[1.81 \times 10^{-6}\] \[7.24 \times 10^{-4}\] \[3.62 \times 10^{-3}\]
Suppose an electron with energy \(1 \text{ eV}\) encounters a one-dimensional barrier potential of height \(3 \text{ eV}\) and width \(1 \text{ nm}\).
What is the probability that the electron tunnels through this barrier?
Gamow Model of Radioactive Decay
One of the first applications of quantum tunneling was to explain alpha decay, the radioactive decay of a nucleus leading to emission of an alpha particle (helium nucleus). The relevant model is
called the Gamow model after its creator George Gamow [4]. Gamow modeled the potential experienced by an alpha particle in the nucleus as a finite square well in the nuclear region and Coulombic
repulsion outside the nucleus, as displayed in the diagram:
The corresponding potential can be written formally as below:
\[V = \begin{cases} -V_0 \quad &r < r_0\\\\ \frac{1}{4\pi \epsilon_0} \frac{2Z e^2}{r} \quad & r>r_0. \end{cases}\]
A fact in advanced quantum mechanics is that the transmission probability \(T\) is well approximated by
\[T = e^{-2\gamma},\]
where \(\gamma\) is defined via
\[\gamma = \frac{1}{\hbar} \int_{r_0}^{r_1} \sqrt{2m \big(V(r) - E\big)}\, dr,\]
and \(r_0, r_1\) are the points where the energy \(E\) intersects the potential \(V(r)\).
Performing the integration for the Gamow model under the assumption that \(r_1 \gg r_0\), one obtains the result
\[\gamma = \frac{\sqrt{2mE}}{\hbar} \left(\frac{\pi}{2} r_1 - 2\sqrt{r_0 r_1} \right).\]
Although the radii \(r_0\) and \(r_1\) are not known a priori, the important part is the dependence of the logarithm of the lifetime on \(E^{-1/2}\), which can be confirmed experimentally. The energy
of emitted alpha particles can be computed using
\[E = (m_p - m_d - m_{\alpha}) c^2,\]
where \(m_p\) is the mass of the nucleus before decay, \(m_d\) is the mass of the nucleus of the decay product, and \(m_{\alpha}\) is the alpha particle mass.
\[\gamma = \dfrac{1}{\hbar} \int_{r_0}^{r_1} \sqrt{2m(V-E)}\, dr\]
Suppose that the charge of a nucleus and the energy \(E\) of particles in the nucleus change in such a way that the equation above is shifted to \(\gamma - \frac{\ln 2}{2}\), where \(r_0\) and \(r_1
\) are the points where the energy \(E\) intersects the potential \(V\).
By what factor does the probability of alpha decay in a given amount of time (transmission coefficient) change?
More Applications of Quantum Tunneling
Quantum tunneling is responsible for many physical phenomena that baffled scientists in the early \(20^\text{th}\) century. One of the first was radioactivity, both via Gamow's model of alpha decay
discussed above as well as via electrons tunneling into the nucleus to be captured by protons. Another wide area of applicability of quantum tunneling has been to the dynamics of electrons in
materials, such as microscopy, semiconductors, and superconductors.
Scanning Tunneling Microscopy
A scanning tunneling microscope is an incredibly sensitive device used to map the topography of materials at the atomic level. It works by running an extremely sharp tip of only a single-atom-thick
over the surface of the material, with the tip at a higher voltage than the material. This voltage allows a non-negligible tunneling current to flow from electrons that tunnel from the surface of the
material, through the potential barrier represented by the air, to the tip of the microscope, completing a circuit. By measuring the amount of current that flows at a given distance, the microscope
can resolve where the atoms are on the surface of the material.
Tunnel Diodes
In a tunnel diode, two p-type and n-type semiconductors are separated by a thin insulating region called the depletion region. Recall that a p-type semiconductor is one that has been doped with
impurity atoms that carry one less valence electron, while an n-type semiconductor has been doped with impurities carrying one more valence electron; both allow conduction to occur more easily due to
the extra electrons or "holes" provided by the dopant. In the depletion region, there are no conduction electrons; the electrons have been depleted to other regions. The main effect of a tunnel diode
is that an applied voltage can make electrons from the n-type semiconductor tunnel through the depletion region, causing a unidirectional current towards the p-type semiconductor at low voltages. As
voltage increases, the current drops as the depletion region widens and then increases again at high voltages to function as a normal diode. The ability of tunnel diodes to direct current at low
voltages due to tunneling allows them to operate at very high AC frequencies.
Josephson Junctions
Some semiconducting materials are superconductors, meaning that in certain temperature ranges a current can flow indefinitely without resistive heating occurring. In Josephson junctions, two
superconducting semiconductors are separated by a thin insulating barrier. In the Josephson effect, superconducting pairs of electrons (Cooper pairs) can tunnel through this barrier to carry the
superconducting current through the junction.
[1] By The original uploader was Jean-Christophe BENOIST at French Wikipedia - Transferred from fr.wikipedia to Commons., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=653747.
[2] Image from https://upload.wikimedia.org/wikipedia/commons/1/1d/TunnelEffektKling1.png under Creative Commons licensing for reuse and modification.
[3] By Yuvalr (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
[4] Griffiths, David J. Introduction to Quantum Mechanics. Second Edition. Pearson: Upper Saddle River, NJ, 2006.
[5] Illustration by Kristian Molhave for the Opensource Handbook of Nanoscience and Nanotechnology.
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A336702 - OEIS
Apart from missing 2, this sequence gives all numbers k such that the binary expansion of
(k) is a prefix of that of
(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in
-tree. This follows because of the two transitions x ->
(x) (doubling) and x ->
(x) (prime shift) used to generate descendants in
-tree, using
at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.
Proof: Any left child in
(k) for k) is larger than sigma(k), for any k > 2 [see
for a proof], and
(n) > n for all n > 1. Thus, apart from
(2) = 3 = sigma(2),
^t(k) > sigma(k), where
^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps
will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.
The first terms of
not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.
Odd terms of this sequence are given by the intersection of
applied to the odd terms of this sequence gives the fixed points of
, i.e., the positions of zeros in
, and a subset of the positions of ones in
Odd terms of this sequence form a subsequence of
, but should occur neither in
nor in
For 30240, sigma(30240) = 120960 = 4*30240, therefore, as sigma(k)/k = 2^2, a power of two, 30240 is present.
(PARI) isA336702(n) = { my(r=sigma(n)/n); (1==denominator(r)&&!bitand(r, r-1)); }; \\ (Corrected) -
Antti Karttunen
, Aug 31 2021
Union with {2} gives the positions of zeros in
Cf. also
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Minimal models of second-order set theories
This is a talk at the New York Graduate Student Logic Conference on 13 May 2016.
A well-known result is that there is a minimal transitive model of ZFC. In this talk I look at the analogous question for second-order set theories. That there is a minimal transitive model of GBC
follows immediately from the result from ZFC but the KM case is more difficult. The main result I will present is that the question for KM has a negative answer: there is no least transitive model of
KM. Along the way, we will look at another notion of minimality for models of second-order set theory and see that KM does not have minimal models in this other respect.
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Integral of xe^x/cos^2(xe^x-e^x) dx | Class 12 Anti derivatives ,Maths
Integral of xe^x/cos^2(xe^x-e^x) dx
Here we have provided you the solution of the integral of the question "xe^x/cos^2(xe^x-e^x)". The question is very simple and solution is also.
We have provided you the image solution of the integral of xe^x/cos^2(xe^x-e^x).
Image Solution:
Integral of xe^x/cos^2(xe^x-e^x) dx | Class 12 Anti derivatives ,Math
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and so many.
Post a Comment
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High strength, high stiffness, curved composite member
High strength, high stiffness, curved composite member
The invention provides a tubular composite member which has a plurality of plies and a generally elongate and selectively curved axis. The plies can include helically oriented fiber components, a
matrix material, and thermoplastic tubes. An intermediate layer preferably has two tri-axially braided plies arranged at selective angles relative to the member's axis and as a function of the
curvature of the member. Interior and exterior plies of circumferential windings can be part of the member such that the intermediate ply is contiguous and between the interior and exterior ply. In
one practice, the member can be formed into a selective shape and then cured. Alternatively, the member is formed between a thermoplastic core and a thermoplastic sheath to facilitate a secondary
process of heating and bending the member into a selective axially-curved shape.
Latest Composite Development Corporation Patents:
The invention relates generally to a non-linear composite structural member which has a plurality of plies and a matrix material. The invention particularly relates to the structure and method of
manufacture of a curved tubular composite wishbone boom which is suitable for use in sailing and wind surfing applications and which has high strength and light weight.
Wishbone-shaped booms are used in many applications of sailing and wind surfing. Traditionally the booms are made from curved wood or tubular aluminum elements which are bent and then joined with
fittings at both the leading and trailing edges of the sail mast.
Such traditional curved booms, however, have several problems. For example, aluminum is subject to corrosion and fatigue failures, and it is further weakened by the joining of the tubes at the most
heavily-loaded leading edge, where the boom attaches via the fittings to the sail mast.
Improved designs over the wood and aluminum booms utilize curved composite tubes, made from glass and carbon fiber with reinforced thermoset resins, which are fastened to an aluminum or plastic
fitting at the boom front. Although this design is an improvement over a boom made with aluminum tubing, booms made from curved composite tubes are still particularly subject to failure at the
heavily-loaded front end where there is a joint between the front end fitting and the composite tubes.
Thus, further improvements are desired in the performance and construction of wishbone-shaped booms, and in curved composite members generally. Accordingly, an object of this invention is to provide
a composite tubular member of high stiffness which can be formed into a continuous selectively-shaped tube thereby eliminating the joints necessary affix the mast to the boom.
A further object of this invention is to provide a composite tubular member which has variable bending stiffness along it's length to maximize the overall resistance of the wishbone boom to bending
loads which result from wind forces on the sail.
Yet another object of the invention is to provide a composite structural member which may be formed into a predetermined shape, e.g., a sailboard boom, via a secondary processing of heating and
These and other objects of the invention will become apparent in the description which follows.
As used herein, "composite member" includes curved articles of manufacture such as a "wind surfing boom" and a "helical spring". Likewise, "fiber component" includes the terms "fiber", "preform" and
"yarn", and is used to describe any fiber-like material which can be interwoven, matted, stitched, wound and/or spooled in a selected or a random orientation. Additionally, "ply" includes the terms
"laminate", "layer" and "sub-ply", and sometimes denotes a layered composition with layers, sub-plies, a plurality of fiber components, or a thermoplastic sheath or core.
In one aspect, the invention provides a composite member which has a selected curvature along an axis of elongation. The member has at least one interior ply with a first fiber component and a
polymer matrix material. This first fiber component is helically-oriented relative to the axis and is generally selected from materials such as glass, carbon, aramid, and mixtures thereof.
The composite member of this aspect also includes at least one intermediate ply with a clockwise helically oriented fiber component, a counter-clockwise helically oriented fiber component, and a
polymer matrix material. The clockwise helically oriented fiber component has a first angle of orientation relative to the axis which is substantially between zero and forty-five degrees, and the
counter-clockwise helically oriented fiber component has a second angle of orientation which is equal and opposite to the first angle. Preferably, the clockwise and counter-clockwise helically
oriented fiber components are selected from materials such as carbon, aramid, glass, and mixtures thereof.
This aspect of the invention also includes at least one exterior ply which has a second fiber component and a polymer matrix material. This second fiber component is helically-oriented relative to
the axis and is generally selected from materials such as glass, carbon, and aramid, and mixtures.
In a preferred aspect according to the invention, the first angle, and hence the second angle, has selected different values along the length of the composite member. These selected different values
of the first angle are a function of the curvature of the composite member and are greater at axial locations of increased curvature and less at axial locations of decreased curvature.
One primary reason for varying the first angle between zero and forty-five degrees is to facilitate bending of the composite member. Helically oriented fiber components are wrapped around the
member's circumference, as opposed to axially oriented or "zero degree" fiber components; and thus these helically oriented fiber components tend to function like a coiled-spring and cancel out the
tension and compressive strains imposed on the fiber components as a result of bending. If, on the other hand, the fiber components were not helically oriented, the compressed fiber components on the
inside of the bend would tend to buckle, and the fiber components on the outside of the radius of the bend would tend to slide because of tension. Therefore, the selected different values of the
first angle are chosen so that when a previously straight composite tube is bent, the compressed fiber components on the inside of the bend do not tend to buckle, and the fiber components on the
outside of the radius of the bend do not tend to slide.
It is nevertheless difficult to achieve this. Theoretically, it would take very shallow helical angle to balance out the tensile and compressive strains caused by bending on the fiber. However, the
friction of the resin and the compaction tape which is applied to the outside of the structure during manufacture prevents these tensile and compressive strains from canceling out completely.
Accordingly, a larger first angle is required to overcome the friction with increased curvature; and a smaller first angle is required with decreased curvature. An angle greater than forty-five
degrees is generally not practical because of the resulting loss in bending stiffness of the finished composite member. Thus, the preferred range of values according to the invention is from between
zero and forty-five degrees and is functionally dependent upon the curvature of the composite member: the first and second angles of the helically oriented fiber components are approximately zero at
axial locations along the composite member which are straight; and the first and second angles have a helical orientation of up to forty-five degrees at axial locations along the composite member
which have increased radial curvature.
In accord with another aspect of the invention, at least one of the first and second fiber components has a helical angle substantially between seventy-five and ninety degrees relative to the axis.
Preferably, both the interior and exterior plies are helically wound about the axis at this high angular orientation.
In accord with another aspect of the invention, the composite member has a bending stiffness which is substantially along the axis and the intermediate ply is arranged to provide at least 80% of the
bending stiffness. Thus, the intermediate ply provides the primary load-carrying capability of the composite member.
A composite member according to the invention can also have a variable bending stiffness along the axis. This bending stiffness is selectable by selecting the first angle substantially between zero
and forty-five degrees at axial locations along the composite member to select the bending stiffness at those axial locations.
In a preferred aspect, the invention also provides at least one pair of interlace fibers that are interlaced with at least one of the helically oriented fiber components of the aforementioned
three-ply composite member. This pair includes (i) a first interlace fiber component oriented at an angle substantially between ten and sixty degrees relative to the angle of the one helically
oriented fiber component, and (ii) a second interlace fiber component oriented relative to the angle of the one helically oriented fiber component with an angle that is equal and opposite to the
first interlace fiber component. These interlace fiber components function to bind the primary helically oriented fiber components together, and are thus made from materials such as glass, carbon,
aramid, polyethylene, polyester, and mixtures thereof.
Accordingly, and in another aspect, the intermediate ply can include a first interlace fiber component, a second interlace fiber component, a third interlace fiber component, and a fourth interlace
fiber component, each of which is made from aramid, glass, linear polyethylene, polyethylene, polyester, carbon, or mixtures thereof. The interlace fibers are interwoven with the helically oriented
fibers as follows: the first interlace fiber component is interwoven with the clockwise helically oriented fiber component at a first interlace angle substantially between ten and sixty degrees
relative to the first angle; the second interlace fiber component is interwoven with the clockwise helically oriented fiber component at a second interlace angle relative to the first angle that is
equal and opposite in sign to the first interlace angle; the third interlace fiber component is interwoven with the counter-clockwise helically oriented fiber component with a third interlace angle
between substantially ten and sixty degrees relative to the second angle; and the fourth interlace fiber component is interwoven with the counter-clockwise helically oriented fiber component with a
fourth interlace angle relative to the second angle that is equal and opposite in sign to the third interlace angle.
A composite member thus manufactured can have a variable bending stiffness along the axis by making the first angle variable along the axis and adjusting the first angle, and hence the second angle,
to attain a selected minimum value of the bending stiffness at selected different locations along the composite member.
A suitable polymer matrix material according to the invention includes B-staged thermoset, nylon-6 thermoplastic, polyether-ether-ketone, polyphenylene sulfide, polyethylene, polypropylene,
thermoplastic urethanes, epoxy, vinyl-ester, and polyester. B-staged thermoset is advantageous because of its relatively low melting temperature, which allows a secondary process of heating and
bending of the composite member, as discussed in greater detail below.
In another aspect, the invention provides at least one stitching fiber that is interwoven with itself and with at least one of the fiber components within the composite member. Such a stitching fiber
is made from polyester, glass, carbon, aramid, and mixtures thereof, and is particularly well-suited for binding together one of the fiber components from the intermediate ply, adding stability to
the pre-cured composite member and adding strength to the post-cured composite member. Similarly, the composite member according to the invention can include, in the intermediate ply, a first
clockwise helically oriented braiding yarn component and a second counter-clockwise oriented braiding yarn component which are interwoven with at least one of the fiber components of the intermediate
Preferably, the clockwise helically oriented fiber component of the intermediate ply is interwoven with the counter-clockwise helically oriented fiber component of the intermediate ply.
A composite member according to the invention can also include a fiber component which includes a plurality of interwoven fibers.
To facilitate a secondary process of forming the composite member into a shape with selected axial curvature, by heating and bending, the invention provides, in another aspect, a composite member
which includes (i) an outer sheath of thermoplastic that is disposed exterior to the exterior ply, and (ii) an inner core of thermoplastic disposed interior to the interior ply. The thermoplastic
outer sheath and inner core have higher melting temperatures than the matrix material within the plies so that the composite member is capable of reformation at selected locations along the axis by
heating and bending the composite member at the selected locations. This inner core of thermoplastic may be solid; but it may also be tubular to provide a conduit within the composite member for
transferring fluids such as air through the composite member. Accordingly, when the core is tubular and when the composite member is reformed into a shape with selected curvature, the composite
member can function as pneumatic tubing with high strength and high stiffness.
In another aspect, the invention provides a composite member with selected curvature along an axis of elongation. Such a member includes at least one interior ply with a first fiber component and a
polymer matrix material. The first fiber component is helically-oriented relative to the axis.
Such a member also includes a first intermediate ply with a first axially extending fiber component, first and second interlace fiber components interlaced with the first axially extending fiber
component, and a polymer matrix material. The first axially extending fiber component has a first angle of orientation relative to the axis of substantially between zero and forty-five degrees. The
first interlace fiber component is oriented at an angle substantially between ten and sixty degrees relative to the first angle of orientation. The second interlace fiber component is oriented
relative to the first angle of orientation at an angle that is equal and opposite to the angle of the first interlace fiber component.
Such a member further includes a second intermediate ply with a second axially extending fiber component, third and fourth interlace fiber components interlaced with the second axially extending
fiber component, and a polymer matrix material. The second axially extending fiber component has a second angle of orientation relative to the axis that is equal and opposite to the first angle of
orientation. The third interlace fiber component is oriented at an angle substantially between ten and sixty degrees relative to the second angle of orientation. The fourth interlace fiber component
is oriented relative to the second angle of orientation at an angle that is equal and opposite to the third interlace fiber component.
Finally, such a member includes at least one exterior ply with a second fiber component and the matrix material. Like the first fiber component, the second fiber component is also helically-oriented
relative to the axis.
The preferred materials for manufacturing the above plies is as follows: the first and second fiber components can be selected from glass, carbon, aramid, and mixtures thereof; the first and second
axially extending fiber components can be selected from made from carbon, aramid, glass, and mixtures thereof; and the first, second, third, and fourth interlace fibers can be selected from glass,
carbon, aramid, polyethylene, polyester, and mixtures thereof.
In a preferred aspect according to the invention, the composite member has a variable first angle of orientation that has selected different values along the length of the member. These values of the
first angle of orientation are a function of the curvature such that the first angle is greater at axial locations of increased curvature and less at axial locations of decreased curvature.
In yet another aspect, the invention provides a tubular composite member which has an initial axial shape that can be formed into a selected axially-curved shape. A thermoplastic inner core is the
interior most component, or "ply", of this tubular member.
The tubular member includes at least one intermediate ply, disposed exterior to the thermoplastic inner core, which has a clockwise helically oriented fiber component, a counter-clockwise helically
oriented fiber component, and a polymer matrix material. The clockwise helically oriented fiber component has a first angle of orientation substantially between zero and forty-five degrees relative
to the axis of the tubular composite member. The counter-clockwise helically oriented fiber component has a second angle of orientation relative to the axis which is equal and opposite to the first
angle. The clockwise and counter-clockwise helically oriented fiber components are preferably made from carbon, aramid, glass, and mixtures thereof.
The tubular member further includes an exterior sheath of thermoplastic disposed exterior to the intermediate ply. The thermoplastic inner core, which may be solid or tubular, and the exterior sheath
have a melting temperature which is higher than the matrix material of the intermediate ply so that the inner core, exterior sheath and the intermediate ply may be reformed from the initial axial
shape into the selected axially-curved shape by bending the composite member when heated.
Preferably, the first angle of the tubular member has selected different values along the length of the member. These values are a function of the selected axially-curved shape of the composite
member such that the first angle is greater at axial locations of increased curvature and less at axial locations of decreased curvature.
The intermediate ply of the tubular member can further include a first interlace fiber component, a second interlace fiber component, a third interlace fiber component, and a fourth interlace fiber
component, each of which is selected from materials such as aramid, glass, linear polyethylene, polyethylene, polyester, carbon, and mixtures thereof. The first interlace fiber component is
interwoven with the clockwise helically oriented fiber component with a first interlace angle of ten to sixty degrees relative to the first angle. The second interlace fiber component is interwoven
with the clockwise helically oriented fiber component with a second interlace angle relative to the first angle that is equal but opposite in sign to the first interlace angle. The third interlace
fiber component is interwoven with the counter-clockwise helically oriented fiber component with a third interlace angle of ten to sixty degrees relative to the second angle. And the fourth interlace
fiber component is interwoven with the counter-clockwise helically oriented fiber component with a fourth interlace angle relative to the second angle that is equal but opposite in sign to the third
interlace angle.
In another aspect, the tubular member includes at least one pair of interlace fibers interlaced with at least one of the helically oriented fiber components. The pair thus includes (i) a first
interlace fiber component oriented at an angle of between ten to sixty degrees relative to the angle of the one helically oriented fiber component, and (ii) a second interlace fiber component
oriented relative to the angle of the one helically oriented fiber component with an angle that is equal but opposite in sign to the first interlace fiber component.
In still another aspect of the invention, there is provided a tubular composite member that has an initial axial shape and that may be formed into a selected axially-curved shape. The tubular member
includes a thermoplastic inner core which may be solid or tubular.
The tubular member further includes a first intermediate ply with a first axially extending fiber component, first and second interlace fiber components interlaced with the first axially extending
fiber component, and a polymer matrix material. The first axially extending fiber component has a first angle of orientation substantially between zero and forty-five degrees relative to the axis of
the composite member. The first interlace fiber component is oriented at an angle substantially between ten and sixty degrees relative to the first angle of orientation. The second interlace fiber
component is oriented relative to the first angle of orientation at an angle that is equal and opposite to the first interlace fiber component.
The tubular member further includes a second intermediate ply with a second axially extending fiber component, third and fourth interlace fiber components interlaced with the second axially extending
fiber component, and a polymer matrix material. The second axially extending fiber component has a second angle of orientation relative to the axis that is equal and opposite to the first angle of
orientation. The third interlace fiber component is oriented at an angle substantially between ten and sixty degrees relative to the second angle of orientation. The fourth interlace fiber component
is oriented relative to the second angle of orientation at an angle that is equal and opposite to the third interlace fiber component.
The tubular member further includes an exterior sheath of thermoplastic disposed exterior to the intermediate ply.
In combination, the thermoplastic inner core and the exterior sheath have a melting temperature which is higher than the matrix material so that the inner core, exterior sheath and the intermediate
ply are formable from the initial axial shape into the selected axially-curved shape by heating and then bending the composite member.
There are many materials which can form the several layers of this tubular member. For example, the first and second axially extending fiber components can be made from materials such as carbon,
aramid, glass, and mixtures thereof. The first, second, third, and fourth interlace fibers can be made from materials such as glass, carbon, aramid, polyethylene, polyester, and mixtures thereof.
The invention also provides, in another aspect, a method of manufacturing a composite member which has a selected curvature along an axis of elongation and which has a plurality of radially
contiguous plies, including the steps of: (i) forming at least one interior ply exterior to an elongate flexible mandrel by helically orienting a first fiber component about the elongate mandrel and
wetting the first fiber component with a liquid matrix material; (ii) forming a first intermediate ply exterior to the interior ply by applying onto the interior ply a clockwise helically oriented
fiber component and a counter-clockwise helically oriented fiber component with the matrix material, including the step of helically orienting the clockwise helically oriented fiber component at a
first angle selected from substantially between zero and forty-five degrees relative to the axis, including the further step of helically orienting the counter-clockwise helically oriented fiber
component at a second angle of orientation relative to the axis which is equal and opposite to the first angle, (iii) forming at least one exterior ply onto the intermediate ply by helically
orienting a second fiber component about the intermediate ply and wetting the second fiber component with the matrix material, (iv) bending the composite member into a the selected curvature, and (v)
curing the composite member and removing the mandrel.
The method can additionally include the step of orienting the helically oriented fiber components with selected different values along the axial length of the composite member as a function of the
subsequently formed curvature of the composite member such that the first angle is greater at axial locations of increased curvature and less at axial locations of decreased curvature.
Further, the method can include the step of orienting the first and second fiber components at a helical angle selected from between 75-90 degrees relative to the axis.
In another aspect, the method can include the additional step of providing the intermediate ply with at least 80% of the bending stiffness of the selectively curved composite member by selecting (i)
the tensile modulus at least one of the fiber components and (ii) by selecting the angles of orientation. More particularly, the bending stiffness is increased by selecting one or more fiber
components with a higher tensile modulus, or by decreasing the angle of orientation of the helically oriented fiber component in the intermediate ply.
In another aspect, the method includes the step of varying the first angle selectively along the axis to attain a selected minimal bending stiffness of the composite member at different locations
along the axis.
In another aspect according to the invention, a method is provided for manufacturing a composite member with selected curvature along an axis of elongation and with a plurality of radially contiguous
plies, including the generally successive steps of: (A) forming at least one interior ply exterior to an elongate flexible mandrel by helically orienting a first fiber component about the elongate
mandrel and wetting the first fiber component with a liquid matrix material; (B) forming a first intermediate ply exterior to the interior ply by applying onto the interior ply a first axially
extending fiber component, and first and second interlace fiber components with the matrix material, including the steps of (i) interlacing the first axially extending fiber with the first and second
interlace fiber components, (ii) helically orienting the first axially extending fiber component at a first angle selected from substantially between zero and forty-five degrees relative to the axis,
(iii) helically orienting the first interlace fiber component at an angle from substantially between ten and sixty degrees relative to the first angle, and (iv) helically orienting the second
interlace fiber component relative to the first angle at an angle that is equal and opposite to the angle of the first interlace fiber component; (C) forming a second intermediate ply exterior to the
first intermediate ply by applying onto the first intermediate ply a second axially extending fiber component, and third and fourth interlace fiber components with the matrix material, including the
steps of (i) interlacing the second axially extending fiber with the third and fourth interlace fiber components, (ii) helically orienting the second axially extending fiber component at a second
angle selected from substantially between zero and forty-five degrees relative to the axis, (iii) helically orienting the third interlace fiber component at an angle from substantially between ten
and sixty degrees relative to the second angle, and (iv) helically orienting the fourth interlace fiber component relative to the second angle at an angle that is equal and opposite to the angle of
the third interlace fiber component; (D) forming at least one exterior ply onto the intermediate ply by helically orienting a second fiber component about the second intermediate ply and wetting the
second fiber component with the matrix material; (E) bending the composite member into the selected curvature, and (F) curing the composite member and removing the mandrel.
This method preferably includes the step of orienting the helically oriented fiber components with selected different values along the axial length of the composite member as a function of the
subsequently formed curvature of the composite member such that the first angle is greater at axial locations of increased curvature and less at axial locations of decreased curvature.
In still another aspect according to the invention, a method is provided for manufacturing a composite member with selected curvature along an axis of elongation, including the steps of: (A) forming
at least one intermediate ply exterior to an elongate core of thermoplastic by applying onto the core a clockwise helically oriented fiber component and a counter-clockwise helically oriented fiber
component with a liquid matrix material, and such that the matrix material has a lower melting temperature than the thermoplastic core, including the step of helically orienting the clockwise
helically oriented fiber component at a first angle selected from substantially between zero and forty-five degrees relative to the axis, including the further step of helically orienting the
counter-clockwise helically oriented fiber component at a second angle of orientation relative to the axis which is equal and opposite to the first angle; (B) polymerizing the intermediate ply; (C)
forming an outer sheath of thermoplastic onto the intermediate ply after the step of polymerizing the intermediate ply, and such that the outer sheath has a higher melting temperature than the matrix
material; (D) heating the core, intermediate ply, outer sheath and matrix material to approximately 450 degrees F.; and (E) bending the composite member, when so heated, into a shape having the
selected curvature.
In still another aspect, a method is provided for manufacturing a composite member with selected curvature along an axis of elongation, including the steps of: (A) forming a first intermediate ply
with a liquid matrix material exterior to an elongate core of thermoplastic by applying onto the core a first axially extending fiber component interlaced with first and second interlace fiber
components, such that the matrix material has a lower melting temperature than the thermoplastic core, including the step of helically orienting the first axially extending fiber component at a first
angle selected from substantially between zero and forty-five degrees relative to the axis, including the further steps of helically orienting the first interlace fiber component at an angle from
between ten and sixty degrees relative to the first angle and orienting the second interlace fiber component at an angle relative to the first angle that is equal and opposite to the angle of the
first interlace fiber component; (B) forming a second intermediate ply with the liquid matrix material exterior to the first intermediate ply by applying onto the first intermediate ply a second
axially extending fiber component interlaced with third and fourth interlace fiber components, including the step of helically orienting the second axially extending fiber component at a second angle
relative to the axis that is equal and opposite to the first angle, including the further steps of helically orienting the third interlace fiber component at an angle from between ten and sixty
degrees relative to the second angle and orienting the fourth interlace fiber component at an angle relative to the second angle that is equal and opposite to the angle of the third interlace fiber
component; (C) polymerizing the intermediate plies; (D) forming an outer sheath of thermoplastic onto the second intermediate ply after the step of polymerizing the intermediate plies, and such that
the outer sheath has a higher melting temperature than the matrix material; (E) heating the core, intermediate plies, outer sheath and matrix material to approximately 450 degrees F.; and (F) bending
the composite member, when so heated, into a shape with the selected curvature.
In each of the afore-mentioned methods, the selected curvature of the composite member may be, for example, be selected with the shape of a helical coil spring, an automobile spring, a leaf spring,
and a wind surfing sail board boom.
The method of the invention thus constructs a curved composite member that has several advantages. First, it has relatively high strength and high stiffness. Secondly, it provides a continuous member
with no intermediate joints which can fail. Thus, for example, the invention is particularly useful in constructing a one-piece wind surfing boom. In addition, the curved composite member of the
invention provides in certain aspects a high strength pneumatic tubing which can be formed into a desired and selected shape by a secondary process of heating and bending.
The invention is next described further in connection with preferred embodiments, and it will be apparent that various additions, subtractions, and modifications can be made by those skilled in the
art without departing from the scope of the invention.
A more complete understanding of the invention may be obtained by reference to the drawings, in which:
FIG. 1 is a top view, partially broken away, of a three-ply composite member constructed according to the invention and which is shaped into a wind surfing boom;
FIG. 1A shows a cross-sectional view of the composite member of FIG. 1;
FIG. 2A is a side view of a ply, constructed according to the invention, that has unidirectional fiber components and which is suitable for constructing the intermediate ply of the composite member
shown in FIGS. 1-1A;
FIG. 2B is a side view of a ply constructed according to the invention which has tri-axially braided fiber components and which is suitable for constructing the intermediate ply of the composite
member shown in FIGS. 1-1A;
FIG. 2C is a side view of a variably stiff ply, constructed according to the invention, that has tri-axially braided fiber components at differing angles relative to a longitudinal axis and which is
suitable for constructing the intermediate ply of the composite member shown in FIGS. 1-1A;
FIG. 2D is a side view of a ply, constructed according to the invention, that has tri-axially braided fiber components at an off-axis angle relative to a longitudinal axis and which is suitable for
constructing the intermediate ply of the composite member shown in FIGS. 1-1A;
FIG. 2E is a cross-sectional view of a four-ply laminate, constructed according to the invention, that has two sub-plies forming an intermediate load-carrying ply and which is suitable for
constructing the composite member shown in FIGS. 1-1A;
FIG. 2F is an enlargement of the wall section of the laminate depicted in FIG. 2E
FIG. 3 is a perspective view, partially broken away, of a four-ply composite member constructed according to the invention and which has a selected axially-curved shape;
FIG. 3A shows a cross-sectional view of the composite member shown in FIG. 3;
FIG. 4 is a perspective view, partially broken away, of a three-ply composite member constructed according to the invention and which permits secondary formation of the member into a selected
axially-curved shape;
FIG. 4A shows a cross-sectional view of the composite member shown in FIG. 4;
FIG. 5 is a cross-sectional view of a four-ply laminate, constructed according to the invention, that has two sub-plies forming an intermediate load-carrying ply and which is sandwiched between a
thermoplastic inner core and outer sheath to facilitate a secondary processing;
FIG. 5A is an enlargement of the wall section of the laminate depicted in FIG. 5
FIG. 6 shows a top view of a sailboard wind surfing boom constructed according to the invention;
FIG. 6A shows a cross-sectional view of the boom of FIG. 6;
FIG. 7 is schematic side view of apparatus which includes a heated die and which is suitable for manufacturing a composite member such as the boom of FIG. 6;
FIG. 8 is schematic side view of the heated die of FIG. 7;
FIG. 9 is a perspective view of a helical spring constructed according to the invention; and
FIG. 10 is a perspective view of pneumatic tubing constructed according to the invention.
A composite member according to the invention is generally a continuous, elongate shaft which has an axis with curvature and which can have a variety of tubular cross-sectional shapes, including
circular, oval, rectangular, square, polygonal, and the like. It is particularly well suited for constructing a continuous and curved tubular wind surfing boom. It is also well-suited for
constructing a helical spring, for replacing the iron spring in automobiles, and for constructing pneumatic tubing.
In one embodiment of the invention, the composite member is constructed with a plurality of plies, each having a fiber component disposed with a matrix material, e.g., a polymer resin. In one other
embodiment of the invention, the composite member is constructed with one ply, having a fiber component disposed with a matrix material, that is sandwiched between thermoplastic material such that
the member may be reformed into a predetermined shape by subsequent heating and bending. In each embodiment of the invention, the fiber and matrix materials, together with the fiber component
orientations, are selected in combination to provide high stiffness and strength and a desired curvature in a single, continuous curved composite structure.
FIGS. 1 and 1A show a hollow composite member 10 constructed in accordance with the invention which has an interior ply 12, an intermediate ply 14, and an exterior ply 16. The plies 12, 14 and 16
form a hollow center interior 17; and thus form a tubular composite member 10. FIG. 1 is broken away at portions 18 and 19 to illustrate the plies 12 and 14 which are interior to the exterior layer
The illustrated member 10 (not to scale) normally has an axis of elongation 20 through the center 17 and resists bending from its normal profile with a selected stiffness along the axis 22. The
illustrated interior ply 12 has a hollow, generally cylindrical shape. The intermediate ply 14 likewise has a generally cylindrical shape and covers the interior ply and is contiguous with it, and
the outer ply 16 likewise contiguously covers the intermediate ply.
Each ply 12, 14 and 16 has one or more fiber components disposed or embedded with a polymer matrix material 21, e.g., a thermoplastic resin. The material and the orientation for the fiber component
in each ply is selected to achieve the desired strength and operational characteristics for that ply. The interior ply 12 has a first fiber component 22 that is helically oriented relative to the
axis 20. Preferably, the helically oriented fiber component 22 has an angle of approximately seventy-five to ninety degrees relative to the axis 20 at each axial location along the member 10.
The intermediate ply 14 has a clockwise helically-oriented fiber component 24 and a counter-clockwise helically-oriented fiber component 24' The clockwise helically-oriented fiber component 24 has a
first angle .alpha. of orientation relative to the axis 20 of approximately zero to forty five degrees. The counter-clockwise helically-oriented fiber component 24' has a second angle of orientation
relative to the axis 20 which is equal and opposite to the first angle, i.e., -.alpha.. Accordingly, the intermediate ply 14 with the fiber components 24, 24' is appropriately titled a "bi-axially
braided ply".
Similar to the interior ply, the exterior ply 16 has a second fiber component 26 that is helically oriented relative to the axis 20. Preferably, the helically oriented fiber component 26 has an angle
of approximately seventy-five to ninety degrees relative to the axis 20 at each axial location along the member 10.
The first and second angles of the intermediate ply 14 preferably vary along the length of the composite member 10 with selected different values as a function of the curvature of the composite
member 10 such that the first and second angles are greater at axial locations of increased curvature, e.g., at portion 18, and are less at axial locations of decreased curvature, e.g., at portion
19. Accordingly, and as illustrated in FIG. 1, the first angle .alpha. at portion 19 is approximately zero, and the first angle .alpha. at portion 18 is greater, here shown as a maximum of forty-five
The selection of the angle .alpha. facilitates the bending of the composite member 10 during its manufacture. If the fiber components 24, 24' are not helically oriented at axial locations with
curvature, those fiber components would tend to compress and buckle in the inner parts of the bend, and they would also tend to slip on the outer parts of the bend.
However, it should be noted that a single angular orientation for the fiber components 24, 24' along the length of the member 10 falls within the scope of the invention. For example, one embodiment
of the invention includes a clockwise helically oriented fiber component which has a constant first angle .alpha. of fifteen degrees at each axial location along the axis 20. Moreover, the
counter-clockwise helically oriented fiber component within the intermediate ply has a constant second angle -.alpha. of minus fifteen degrees along the axial length of the composite member 10. These
constant first and second fiber angles facilitate the manufacturing process, as compared to the variably selected angles, and provide an acceptable fiber orientation to reduce the compression,
buckling, and strain on the fiber components at the curved portions along the composite member 10. Other acceptable first and second angles fall between the stated range of zero to forty-five
The plies 12, 14 and 16 provide, respectively, a portion of the entire binding stiffness of the composite member 10. The intermediate ply 14, however, preferably provides at least 80% of the total
bending stiffness. One way the intermediate ply 14 provides such overall stiffness is by selecting the appropriate fiber materials for the clockwise and counter-clockwise helically oriented fiber
components. For example, glass, aramid, and carbon fibers each provide a high tensile modulus, as for instance compared to polyester, and thus these materials provide relatively high axial stiffness.
One other way to increase the bending stiffness of the composite member 10 is to orient the helically oriented fiber components 24, 24' within the intermediate ply 14 at appropriate and selected
angles: the lower the angles of orientation, the greater the bending stiffness, and vice versa. This relationship is approximately illustrated as a cosine-to-the-fourth function at every axial
location along the member 10. That is, the bending stiffness decreases at approximately a cosine-to-the-fourth fall-off as a function of off-axis angle at axial locations along the member. For
example, a zero degree angle for the intermediate fiber components at one axial location means that approximately 100% of the intermediate ply's bending stiffness is achieved at that location, since
the cosine of zero is one. However, the cosine of forty-five degrees is 70%, and if the fiber components are oriented at forty-five degrees, the cosine-to-the-fourth fall off means that only 25% of
the intermediate ply's bending stiffness is achieved at that axial location. It is not very practical to have an intermediate ply 14 with less than 25% of the possible bending stiffness in that
layer; and hence the fiber orientations of the intermediate ply 14 are generally no more than forty-five degrees relative to the axis 20 and at each axial location along the member 10.
The fiber component 22 of the interior ply 12 is preferably an aramid, e.g., Kevlar by Dupont; glass; carbon; linear polyethylene, e.g., Allied Spectra fiber; polyethylene; polyester; or mixtures
thereof. The second fiber component 24, 24' of the intermediate ply 14 is preferably made from carbon, glass, aramid, or mixtures thereof. And the second fiber component 26 of the exterior ply 16 is
preferably each made from glass, carbon, aramid, polyester, or mixtures thereof.
The preferred matrix material 21 according to the invention is a high elongation, high strength, impact resistant thermoplastic material such as Nylon-6 or B-staged thermoset. Alternatively, a
thermosetting material such as epoxy, vinyl ester, or polyester is used. The matrix material is distributed substantially uniformly among the fiber components 22, 24, 24' and 26 of FIG. 1. In one
application, the matrix material is generously applied in viscous form to the fiber components during manufacture such that upon hardening or polymerization, e.g., through a bake or heating cycle,
the matrix material hardens to stabilize the fiber components. The hardened matrix and fiber components form the solid composite member 10.
More particularly, a preferred method for distributing a Nylon-6 matrix material with the fiber components 22, 24, 24', 26 of FIG. 1, for example, is to impregnate the "dry" fiber components, i.e.,
the fiber preform comprising un-wetted, interwoven, or braided fibers, with a low viscosity nylon monomer, e.g., .epsilon.-caprolactam, blended with a suitable catalyst, e.g., sodium, and a promoter.
Polymerization of the catalyzed and promoted .epsilon.-caprolactam occurs shortly after impregnation, yielding a high molecular weight Nylon-6 matrix material. The resulting composite member 10 is
extremely durable because of the high elongation to failure of the matrix as well as its abrasion resistance and ability to prevent crack propagation.
Alternatively, the matrix material is supplied in a dry, fiber or powdered form which is "co-mingled" with the fiber components 22, 24, 24', and 26. By applying heat and pressure to the co-mingled
fiber components and dry matrix material, the matrix melts and impregnates the fiber preform components 22, 24, 24', and 26 to yield the final composite member 10.
While Nylon-6 is deemed a preferred matrix material for use with the invention, thermosetting resins in the epoxy, vinyl-ester or polyester groups represent alternative matrix materials suitable for
the above-described high strength shaft-like composite members. For example, Polyether-Ether-Ketone (Peek), Polyphenylene Sulfide (PPS), Polyethylene, Polypropylene and Thermoplastic Urethanes
(TPU's) all provide a level of damage tolerance for the composite member 10 that is similar to Nylon-6.
It is to be understood that any of the plies 12, 14, and 16 can be constructed of a plurality of plies which combine to create the above-described characteristics. For example, and as described in
more detail below, a stitching fiber may be interwoven with any of the fiber components 22, 24, 24', and 26 to provide additional strength and binding of the member 10. Additionally, any of the fiber
components 22, 24, 24', and 26 can include a plurality of fibers to operate in substantially the same way.
However, it is the intermediate ply which can generally include the most variations within its ply and fiber construction and geometry. For example, a plurality of layers or plies can operate in
combination to form the ply 14 shown in FIGS. 1 and 1A as a compound ply. Illustrative details concerning additions or alternate orientations and geometries of the fiber components within the
intermediate ply 14 are described below, particularly in connection with FIGS. 2A-2B.
Each of FIGS. 2A-2D illustrate a "flattened out" view of a ply 40 in accordance with the invention and is suitable for constructing a ply 14 of a composite member 10. Each drawing thus shows a planar
projection of a ply 40 that has a generally cylindrical shape. The ply 40 provides a bending stiffness 43 that is substantially along an axis 41 of that cylindrical shape. Each ply 40 has at least
one fiber component 42 that is representative, for example, of the fiber components 24 and 24' described above.
FIG. 2A shows a ply 40a that has a fiber component 42a arranged axially and unidirectionally along the elongate axis 41a, such as similarly shown in the intermediate ply 14, portion 19, of FIG. 1.
The matrix material 44a is disposed with the fiber component 42a in a combination that can be cured to form a solid ply, as above. The axial fiber geometry of FIG. 2A provides high bending strength
43a as discussed previously with respect to the cosine-to-the-fourth fall-off. Nevertheless, the ply 14 of FIG. 1 can be formed in part with a unidirectional fiber component such as shown in FIG. 2A
and can further vary the fiber material and fiber volume to adjust the ply's strength.
FIG. 2B shows a ply 40b wherein the fiber component 42b is axially extending and interwoven, i.e., helically braided, with a plurality of like or different fiber components, here shown as first
interlace fiber component 46b and second interlace fiber component 48b. The interlace fiber components 46b and 48b are also preferably interwoven amongst themselves. To this end, successive crossings
of two fiber components 46b and 48b have successive "over" and "under" geometries. In this configuration, the ply fiber geometry of the ply 40b of FIG. 2B, appropriately denoted as a "tri-axially"
braided ply, is stronger than the axial geometry of FIG. 2A. The helically oriented interlace fiber components 46b and 48b tend to tightly bind the axially extending fiber component 42b with the
matrix material 44b, in addition to providing increased bending stiffness 43b.
In one embodiment of the invention, the intermediate ply 14 of FIGS. 1 and 1B includes at least one sub-ply with a tri-axial braided fiber geometry such as shown in FIG. 2B. In such a ply 14, the
axially extending fiber 42c is similar to the helically oriented fiber components 24, 24' of FIG. 1 oriented at first and second angles of approximately zero degrees, such as shown in ply 14, portion
Binding the fiber components together within a given ply, such as within intermediate ply 14 of FIG. 1, is a preferred practice of the invention. Each interlace fiber can therefore be considered as a
stitching fiber. In certain aspects of the invention, a single stitching fiber, such as the fiber 46b of FIG. 2B, binds the fiber component of a given ply together by interweaving the stitching fiber
with itself and with the fiber component 42. A fiber is interwoven with itself, for example, by successively wrapping the fiber about the member and looping the fiber with itself at each wrap.
The fibers 46b and 48b of FIG. 2B may be of different materials, although it generally is preferred that they be the same material. The group of fiber materials suitable to form the interlace fiber
components 46b and 48b include carbon, glass, polyester, aramid, and mixtures thereof.
The angles of orientation for the fibers 46b and 48b, denoted as .beta. and .alpha. respectively, relative to the longitudinal axis 41b, are also preferably equal, although they are not required to
be. They are oriented relative to the fiber orientation of the axially extending fiber component 42b and at an angle between approximately ten and sixty degrees. Because the bending stiffness 43b
changes when the angles .beta. and .alpha. change, it is a feature of the invention that one of more of the angles .beta. and .alpha. are adjusted to modify or change the bending stiffness 43b of the
composite member to the desired magnitude, or to a selected minimum value. The variable angles .beta. and .alpha. are introduced during the manufacturing process.
Thus, for example, if the composite member 10 of FIG. 1A is to have increased bending stiffness at a selected axial location along the axis 20, a ply 40b from FIG. 2B may be used instead of ply 14,
FIG. 1. Such a ply can have the variably selected angles .beta. and .alpha. decrease towards their minimum of ten degrees in the corresponding portion of length along axis 41b. On the other hand, to
decrease the bending stiffness of the composite member 10 at another selected portion of its length, the variably selected angles .beta. and .alpha. are increased towards their maximum of sixty
degrees in the portion of length along axis 41b corresponding to that portion of the member 10.
A composite member in accordance with the invention thus has a further feature relating to selectively variable angle geometry, as shown with the ply 40c of FIG. 2C. In FIG. 2C, the helically
oriented fiber components 46c and 48c of the ply have at least two separate and distinct angles relative to the axis 41 c, at different axial locations within the ply 40c such that a single composite
member has differing bending stiffness along its axial length. More particularly, and with reference to FIG. 2C, the ply 40c has two distinct sections m and n which denote regions of different fiber
angles. In section m, the interlace fiber components 46c and 48c have a greater angle relative to the longitudinal axis 20, and thus provide lesser bending stiffness 43 cm than provided in section n.
Similarly, in section n, the interlace fiber components 46c and 48c have a lesser angle relative to the longitudinal axis 41c, and thus provide greater bending stiffness 43 cm than provided in
section m.
It should be noted that the fiber components 42c, 46c and 48c preferably run continuously through the ply 40 of FIG. 2C. Therefore, the angles at which the helically oriented fibers are interwoven
with the axially extending fiber component 42c are most easily changed in a real-time fashion during the manufacturing process.
A composite member constructed in accordance with the invention also includes a ply 40 that contains only helically oriented fiber components. For example, if the unidirectional fiber component 42c
from FIG. 2C is removed, there remains a ply 40c having only the helically oriented fiber components 46c and 48c. Such a ply 40c is for example shown in ply 14, portion 18, of FIG. 1, and is
appropriately denoted "bi-axial", like before. Such a ply 40c can also have one or more selectively variable angles to adjust the bending stiffness along the length of the composite member. A
"bi-axial ply" can additionally include fiber components which are braided together or two sub-plys which are helically braided or filament wound.
In a preferred embodiment according to the invention, ply 14 of FIG. 1 is formed with two sub-plies, each with a tri-axially braided structure such as illustrated in FIG. 2D. Unlike the fiber
component 42 of FIGS. 2A-2C, the fiber component 42d of FIG. 2D is helically oriented relative to the longitudinal axis 41d at an angle .gamma., typically between zero and forty-five degrees, and the
interlace fiber components 46d and 48d are helically oriented at an angle between ten and sixty degrees relative to the angle .gamma.. The fiber component 42d preferably is selected from those fiber
materials which provide significant stiffness strength with an increased tensile modulus, e.g., carbon, aramid, and glass, while the fibers 46d and 48d are more appropriately selected from flexible
yarn components which have a decreased tensile modulus.
As earlier mentioned, it is preferred that the intermediate ply 14 of FIGS. 1-1A is formed with two sub-plies, each with a tri-axial braided ply such as shown in FIGS. 2D. However, it is most
preferred that these two tri-axially braided plies, which are radially contiguous with one another, have varying angles .gamma. relative to the axis 20 and as a function of the curvature of the
composite member. That is, the helcially oriented fiber component 42d of one tri-axially braided ply is selected with a first angle between zero and forty-five degrees relative to the axis 41d; while
the helically oriented fiber component 42d of the second tri-axially braided ply is oriented at a second angle that is equal and opposite to the first angle. The first and second angles are selected
according to the curvature of the member. For example, the angles are substantially zero for straight portions of the member; and the angles are greater for curved regions of the member. In each ply,
the interlace fiber components, e.g., the fibers 46d and 48d of FIG. 2D, are oriented with angles between ten and sixty degrees relative to the first and second angles, e.g., the angle .gamma..
FIG. 2E shows a cross-sectional views of a composite structure 49 constructed according to the invention and employing many of the ply geometries previously described. Such a composite structure is
preferably manufactured with a flexible mandrel, which may be removed after forming the plies, and with a matrix material which can be cured while the boom is in a selectively curved geometry.
Composite structure 49 has an interior ply 49a, a composite intermediate ply 49b, and an exterior ply 49c. The intermediate ply 49b is constructed with a plurality of plies, 49b1 and 49b2, that are
preferably helically braided, such as illustrated in FIGS. 2B-2D, or bi-axially braided, such as shown within intermediate ply 14, portion 18, of FIG. 1. The interior ply 49a and exterior ply 49c are
helically wound with a circumferentially extending fiber that subtends an angle of between approximately seventy-five and ninety degrees relative to the axis of the composite member, e.g., axis 20 of
FIG. 1. Representative dimensions include an inner tubular interior of 2.67 cm, and an outer diameter of 3.1 cm.
FIGS. 3 and 3A, which are not to scale for illustrative purposes, show a tubular composite member 50 that is partially broken away at portions 52 and 54 to illustrate the interior plies. The tubular
member 50 has an elongate axis 56 which has selected curvature: in region 58, the member 50 is substantially straight; and in region 60, the member 50 is curved. The axis 56 runs concentrically with
the hollow central interior 62.
The member 10 has an interior ply 64, which is substantially the same as interior ply 12 of FIG. 1, a first intermediate ply 66, a second intermediate ply 68, and an exterior ply 70, which is
substantially the same as exterior ply 16 of FIG. 1.
Interior ply 64 has a first helically oriented fiber component 72 that is preferably oriented at an angle of between seventy-five and ninety degrees relative to the axis 56. The fiber component 72 is
preferably made from carbon, glass, or aramid materials, or mixtures thereof.
Exterior ply 70 likewise has a second helically oriented fiber component 74 that is preferably oriented at an angle of between seventy-five and ninety degrees relative to the axis 56. The fiber
component 74 is preferably made from carbon, glass, or aramid materials, or mixtures thereof.
Intermediate plies 66 and 68 each have a tri-axially braided ply, with varying off-axis angle, such as described in connection with FIGS. 1 and 2D. Ply 66 has a first axially extending fiber
component 76, with interlace fiber components 77, 77'; and ply 68 has a second axially extending fiber component 78, with interlace fiber components 79, 79'. It should be noted that intermediate
plies show only two axially extending fiber components within each ply, for illustrative purposes, when in reality the axially extending fiber components, each with a pair of interlace fiber
components, are radially spaced about the circumference of the ply selectively.
Shown in portion 52, the orientation of the tri-axially braided ply 66 within region 58 is substantially axial, i.e., with an off axis angle of approximately zero degrees. Thus, the first axially
extending fiber component 76 has a first angle with respect to the axis 56 of approximately zero degrees. The interlace fiber 77 is oriented relative to the first angle and is between ten and sixty
degrees; and the interlace fiber 77' is oriented relative to the first angle with an angle which is equal and opposite to the angle of interlace fiber 77.
The orientation of the tri-axially braided ply 68 is similar within region 58. Thus, the second axially extending fiber component 78 has a second angle with respect to the axis 56 of approximately
zero degrees. The interlace fiber 79 is oriented relative to the second angle and is between ten and sixty degrees; and the interlace fiber 79' is oriented relative to the first angle with an angle
which is equal and opposite to the angle of interlace fiber 79.
Shown in portion 54, the orientations of the tri-axially braided plies change because of the curvature of the member 50 in region 60. The first axially extending fiber component 76 has a first angle
with respect to the axis 56 of greater than zero degrees, here shown as approximately forty-five degrees. The interlace fiber 77 remains oriented relative to the first angle with an angle between ten
and sixty degrees; and the interlace fiber 77' likewise remains oriented relative to the first angle with an angle which is equal and opposite to the angle of interlace fiber 77.
The orientation of the tri-axially braided ply 68 is similar within region 60. The second axially extending fiber component 78 has a second angle with respect to the axis 56 which is equal and
opposite to the first angle of fiber component 76 within the same region 60. Thus, fiber component 78 has an angle of approximately minus forty-five degrees illustrated within portion 54. The
interlace fiber 79 remains oriented relative to the second angle with an angle between ten and sixty degrees; and the interlace fiber 79' likewise remains oriented relative to the second angle with
an angle which is equal and opposite to the angle of interlace fiber 79.
Each of the fiber components within the plies 64,66,68,70 are disposed with a matrix material 80, such as earlier described. For example, a preferred matrix material is a high elongation, high
strength, impact resistant material such as Nylon-6; and one suitable matrix material is B-staged thermoset. Other acceptable resins include thermosetting material such as epoxy, vinyl ester, or
polyester. The axially extending fiber components 76, 78 are preferably made from carbon, glass, aramid, or mixtures thereof; and the interlace fiber components 77,77',79,79' are preferably made from
glass, carbon, aramid, polyester, polyethylene, or mixtures thereof.
It should again be noted that although the variation of the first and second angles is preferred, it is not required according to the invention. For example, the tri-axially braided plies 66,68 can
be constructed with constant, and opposite, off axis angles. One illustrated practice according to the invention, for example, is where the first angle is a constant angle selected between
approximately ten and thirty degrees; and the second angle is a constant angle selected to be equal and opposite in sign to the first angle. Such a constant off-axis orientation alleviates the
buckling and strain on the fiber components caused by bending within the region 60.
FIG. 4 illustrates another embodiment of the invention and which permits secondary reforming of the composite member into a selected axially-curved shape. More specifically, FIG. 4 shows an elongate
composite member 90 which includes in inner core 92 of thermoplastic, at least one intermediate ply 94, and an external sheath 96 of thermoplastic. The layers 92, 94 and 96 are radially contiguous
with each other, such as shown in FIG. 4A, and extend generally along the axis 98. The core 92 can be solid, or tubular, depending on the selected use of the member 90: for example, a tubular core
may be used as pneumatic tubing or as a wind surfing boom; and a solid core may be used also as a boom or for other applications requiring increased strength, such as a helical spring. The member 90
is partially broken away in region 99 to illustrate the inner layers.
The intermediate ply is preferably constructed like the intermediate ply 14 of FIG. 1, or as any of the plies described in connection with 2A-2D, or with multiple sub-plies with a tri-axially braided
ply configuration such as intermediate plies 66,68 of FIG. 3. However, the matrix material, e.g., the matrix material 80 of FIG. 3, which binds the intermediate ply together must have a melting
temperature which is less than the core 92 or sheath 96. This permits secondary processing of the member 90 so that the member 90 may be reheated and formed into a selected axially-curved shape. The
inner core 92 and outer sheath 96 can be heated into a state where the core and sheath are soft and pliable, while the melted intermediate ply is contained between the core and sheath. In this way,
once the matrix material reaches a liquid state, the whole of the composite member 90 may be reformed by bending and without losing the cross-sectional form of the member, such as shown in FIG. 4A.
Accordingly, the composite member 90 of FIG. 4 is generally manufactured in an axially straight fashion, although not required, and whereafter the member 90 is heated and formed into a desired shape.
The member 90 can be constructed with varying strength, as discussed earlier, by selecting the materials and the orientations of the fiber components within the intermediate ply 94.
Ply 94 may alternatively contain plurality of plies, such as the three-ply construction shown in FIG. 1, or the four-ply construction shown in FIG. 3. Accordingly, the helically oriented inner and
outer plies, e.g., the interior ply 12 and exterior ply 16 of FIG. 1, include fiber components which are helically wound about the axis of the member and at an angle of between seventy-five and
ninety degrees. In such a member, the multiple plies are sandwiched between the inner core 92 and outer sheath 96 to facilitate the secondary processing of the member 90. A member 90 constructed with
the inner core and outer sheath further facilitates the manufacture of the member 90 through pultrusion processing, described in more detail below, so that the circumferential interior ply or plies
may be "pulled" through the process.
FIG. 5 shows a cross-sectional view of a composite structure 100 constructed according to the invention which is preferably manufactured with a thermoplastic matrix, or a thermoset cured to an
intermediate B stage, and which is formed straight by a pultrusion process, and then reheated and bent into a desired axially-curved geometry in a secondary processing operation. With particular
reference to FIG. 5, the composite structure 100 includes an interior thermoplastic core 102, a composite intermediate ply 104, with sub-plies 104a and 104b, and an exterior thermoplastic sheath 106.
Similar to the intermediate ply structure of FIG. 2E, the intermediate plies 104 can be made from the plies described in connection with FIGS. 2A-2D, or described and shown in connection with the
intermediate ply 14 of FIG. 1. The sub-plies 104a and 104b preferably incorporate a helically braided ply structure such as illustrated in FIGS. 2B-2D. The interior core and outer sheath of
thermoplastic are similar to the layers 92 and 96 illustrated in FIG. 4.
The manufacturing of other composite members according to the invention are best described by way of the following non-limiting examples. Further understanding of these manufacturing techniques may
be obtained by reviewing U.S. Pat. No. 5,188,872, which is incorporated herein by reference, and which is particularly important for understanding Examples I and II below.
EXAMPLE I Sailboard Boom Geometry
The following details illustrate a design and method of manufacture of a an axially curved composite member, such a wind surfing boom 110 shown in FIG. 6. The length of the composite tubular member
110 is 340 cm, and the shape and dimensions of the tubular cross-section 112 are shown in FIG. 6A.
In FIG. 6, the fiber angles of the helically braided plies vary depending upon the selected curvature of the tubular structure illustrated in FIG. 6. The series of contiguous plies which form the
member 110 are shown substantially in FIG. 3, or alternatively in FIG. 1. That is, there are a plurality of intermediate plies, or alternatively one intermediate ply, sandwiched between (i) an
interior helically braided ply, with a first fiber component wound about the axis of the member 110 at an angle between seventy-five and ninety degrees, and (ii) an exterior ply, with a second fiber
component wound about the axis of the member 110 at an angle between seventy-five and ninety degrees.
One suitable apparatus used to apply the several plies of FIGS. 6, 6A is shown and described in U.S. Pat. No. 5,188,872. In particular, the following description summarizes rotation rates of this
apparatus in terms of "RPM", or revolutions per minute, for the intermediate ply, FIG. 1, or plies, FIG. 3.
In the straight regions of I & V, FIG. 6, the rate of rotation for the helical braiding is applied at zero RPMs. Accordingly, the helical braiding in those regions is substantially axial, such as
illustrated in the intermediate plies in the region 58 of FIG. 3; and thus the first angle of the primary load-carrying fiber component, e.g., the fiber component 24,24' of FIG. 1 or the fiber
components 76,78 of FIG. 3, is zero. Additionally, one or more pairs of interlace fiber components, e.g., the fibers 77,77' and 79,79' of FIG. 3, can be interlaced with the primary load-carrying
fiber component.
In the slightly curved regions of II & IV, FIG. 6, the rate of rotation for the helical braid is applied at 3.9 RPMs. Accordingly, the helical braiding in those regions is provided with an off-axis
angle, relative to the central axis of the member 110, of +/- fifteen degrees. Such a configuration of the fiber components is illustrated, for example, in the region 60 of FIG. 3, although the angle
of orientation is reduced in this Example. Accordingly, the first and second angles of the primary load-carrying fiber components are plus and minus fifteen degrees, respectively.
In the curved region III, FIG. 6, the rate of rotation for the helical braid is the greatest, at 8.4 RPMs. Accordingly, the helical braiding in those regions is provided with an off-axis angle,
relative to the central axis of the member 110, of +/- thirty degrees. Such a configuration of the fiber components is illustrated, for example, in the region 60 of FIG. 3, although, again, the angle
of orientation is reduced in this Example. Accordingly, the first and second angles of the primary load-carrying fiber components are plus and minus thirty degrees, respectively.
It should be noted that the boom 110 is constructed by way of a flexible mandrel. That is, the multiple plies are formed onto an elongate, flexible mandrel with a matrix material, whereinafter the
member is cured and the flexible mandrel is removed. The mandrel is flexible so that the uncured member may be curved into a selective axially-curved shape. The mandrel is also preferably formed by
two joined components so that the mandrel may be removed from the interior of the boom 110 from both ends.
EXAMPLE II Curved Tubular Member with Thermoplastic Matrix and Primary and Secondary Processing
The following example, and associated FIGS. 7 and 8, illustrate a method of manufacture for constructing a continuous composite tubular member such as shown in FIG. 4 and which supports secondary
processing to reform the member selectively into an axially-curved shape. The member is constructed, in part, with a thermoplastic matrix that is formed with the pultrusion process into a straight
tube, and then reformed via a secondary process into a u-shaped or selectively curved tubular member.
The materials used to manufacture the tubular member of this example are listed in Table I.
TABLE I
Exterior Sheath:
Copolyamide 6/66
Supplier: Du Pont and Nemours & Co.
Properties: T melt = 500 Degree. F.
Exterior Ply:
Component: Exterior Ply
Material: E-Glass
Type: 675 Type 30 Roving
Supplier: Owens-Corning
Properties: Fiber Modulus = 10.5 .times. 10.sup.6 psi,
Density = 2.5 g/cm.sup.3
Intermediate Tri-Axial Helically Braided Ply:
Component: Braiding Yarn
Material: S2-Glass
Type: S2CG1501/3
Supplier: Owens-Corning
Properties: Fiber Modulus = 12 .times. 10.sup.6 psi,
Density = 2.48 g/cm.sup.3
Component: Axial Component in H-Braid Plies
Material: Carbon Fiber
Type: 12k G30500
Supplier: BASF
Properties: Fiber Modulus = 34 .times. 10.sup.6 psi,
Density = 1.77 g/cm.sup.3
Interior Ply:
Component: Interior Ply
Material: E-Glass
Type: 675 Type 30 Roving
Supplier: Owens-Corning
Properties: Fiber Modulus = 10.5 .times. 10.sup.6 psi,
Density = 2.5 g/cm.sup.3
Interior Tubular Core:
Copolyamide 6/66
Supplier: Du Pont and Nemours & Co.
Properties: T melt = 500 Degree. F.
Matrix Material:
Component: Matrix Material
Material: Nylon-6 Thermoplastic Resin
anionic polymerization of
.epsilon.-caprolactam using sodium
caprolactamate catalyst and
activator (100%:2%:2%)
Type: Bruggolen C 10/Bruggolen C 20
Supplier: L. Bruggemann Chemical
Properties: T melt = 400 Degree. F.
FIG. 7 illustrates an apparatus 120 suitable for use in constructing the straight composite tube of this Example, such as the tubular member 90 of FIG. 4. The process begins with a continuous length
of thermoplastic core, shown here as a tubular core 122, which is either pre-manufactured and supplied from a reel or extruded in-line during fabrication. Materials are drawn through the heated die
124 by a rotating or reciprocating pulling mechanism 126.
The pulling mechanism 126 draws the interior thermoplastic core 122 at a rate of approximately twelve inches per minute. The core 122 proceeds downstream into the first orbital winder 128. The
orbital winder 128 applies circumferential glass windings at an angle of approximately 85.degree. by rotating at 42 RPMs.
The intermediate load carrying fiber components enter the process at the helical braider 130, which includes a counter-clockwise rotating plate "A" (not shown), with guide rings and warp posts, and a
clockwise rotating plate "B" (also not shown), with guide rings and warp posts. Seventy-two ends of carbon fiber are loaded onto plate "A" and passed over the guide rings and warp posts of braider
"A" and rotated about the core 122 in a counter-clockwise direction at a speed of 1.18 RPMs to produce a helical angle of the carbon yarns of +15 degrees. Plate "A" also includes ends of fibers for
the interlace fibers, and the braiding speed of plate "A" is thus 2.6 RPMs to provide an interlacing fiber with an interlacing angle of 37.degree. in relation to the carbon fibers. The carbon fiber
is stabilized and the fiber orientation maintained by the fine denier S2 glass braider yarns.
The fiber materials applied onto the thermoplastic core travel downstream from rotating plate "A" and into rotating plate "B" on the helical braider 130. Seventy-two ends of carbon fiber are loaded
onto plate "B" and passed over the guide rings and warp posts of braider "B" and rotated about the core 122 in a clockwise direction at a speed of 1.18 RPMs to produce a helical angle of the carbon
yarns of -15 degrees. Plate "B" also includes ends of fibers for the interlace fibers, and the braiding speed of plate "B" is thus 2.6 RPMs to provide an interlacing fiber with an interlacing angle
of -37.degree. in relation to the carbon fibers. The carbon fiber is stabilized and the fiber orientation maintained by the fine denier S2 glass braider yarns.
The interior and intermediate plies travel from the helical braider 130 into a second orbital winder 132. E-glass reinforcement on the orbital winder 132 is applied at a rate of 36 RPM to produce an
exterior circumferentially wound fiber angle of -85.degree. relative to the axis of the tube, e.g., the axis 98 of FIG. 4.
The completed preform 134 from the winder 132 is then pulled by the pulling mechanism 126 into a heated steel die 124 illustrated in FIG. 8. The steel die 124 is fabricated from 4140 tool steel,
known to those skilled in the art, and consists of two halves forming a split cavity. It is machined and ground to form the external profile of the tube and the molded surfaces are plated with a
0.0015" thick layer of hard chrome. The die 124 is thirty-six inches long and is uniformly heated to 160.degree. C. The entrance of the die 124 includes a machined injection port 136.
The matrix material is pumped into the die 124 and disposed with the formed fiber components in liquid form from two separate reservoirs (not shown): one reservoir contains a sodium caprolactamate
and caprolactam (the catalyst side); and the other reservoir contains the activator and caprolactam. The two sides 138, 139 are blended at the mixhead 140 in equal proportions and pumped at
approximately ten pounds-per-square-inch into the injection port 136 of the steel die 124. The low viscosity of the nylon matrix monomer impregnates the fiber preform 134. The elevated temperature of
the die 124, created by the heaters 142 and monitored by the thermocouples 144, accelerates the polymerization of the caprolactam as the now-wetted preform travels through the die 124. The reaction
and cure are completed before the composite member exits the die 124 resulting in a finished tube 146 which includes a high impact resistant fiber architecture that is preferably impregnated by
thermoplastic Nylon-6 matrix.
With further reference to FIG. 7, the fully polymerized composite member 146 exits the die 124 so that it cools in ambient conditions. The member 146 is drawn by the pulling mechanism 126 and through
the cross-head extruder 147 where a layer of Copolyamide 6/66 is extruded to provide an exterior sheath of thermoplastic. The part is then pulled further through the traveling cut-off saw 148. The
cut-off saw 148 separates the composite member into units 150 that are 340 cm long, which are thereafter used in manufacturing the selectively curved composite member in a secondary process described
To form the tube into a u-shaped curved tubular composite, or into other selectively curved shapes, it is necessary to heat the straight tube to about 450 degrees F. At that point, the interior Nylon
matrix material is molten while the interior core and exterior sheath remain malleable but contain the fluid interior matrix material. The tube is then bent into it's desired shape and held in a jig
until it cools to about 250 degrees F. when it can be removed and allowed to cool to ambient temperature.
The resulting curved tubular member has the following properties:
weight of the tube=1197 g average laminate density=1.81 g/cm.sup.3 Bending Stiffness of Tube=562.5 N-m.sup.2
A completed tube manufactured as in Example II can be used in various applications. FIG. 9 for example shows a helical spring 160 which can be used to replace metal automobile springs. FIG. 10 shows
pneumatic tubing 170 which is formed into an s-shape via secondary processing and which is hollow to provide a fluid conduit 172.
It should be apparent to those skilled in the art that the spring 160 and tubing 170 of FIGS. 9 and 10, respectively, can be manufactured with a variety of ply goemetries and thicknesses to select
the strength and characteristics for curving in a secondary process. For example, the ply geometries of FIGS. 2-3 may be used to strengthen the tubular member and to selectively alter the axial
curvature for assorted applications.
The invention thus attains the objects set forth above, in addition to those apparent from preceding description. Since certain changes may be made in the above composite member structures without
departing from the scope of the invention, it is intended that all matter contained in the above description or shown in the accompanying drawing be interpreted as illustrative and not in a limiting
It is also to be understood that the following claims are to cover all generic and specific features of the invention described herein, and all statements of the scope of the invention which, as a
matter of language, might be the to fall there between.
1. A composite member having selected curvature along an axis of elongation, comprising
at least one interior ply having a first fiber component and a polymer matrix material, said first fiber component being helically-oriented relative to said axis,
at least one exterior ply having a second fiber component and a polymer matrix material, said second fiber component being helically-oriented relative to said axis, and
at least one intermediate ply having a clockwise helically oriented fiber component, a counter-clockwise helically oriented fiber component, and a polymer matrix material, said clockwise
helically oriented fiber component having a first angle of orientation relative to said axis substantially between zero and forty-five degrees, said counter-clockwise helically oriented fiber
component having a second angle of orientation which is equal and opposite to said first angle,
said first angle having selected different values along the length of said composite member such that said composite member has a bending stiffness that varies along said axis at selected
different locations.
2. A composite member according to claim 1, wherein said first angle has selected different values along the length of said member, said value of said first angle being a function of said curvature
wherein said first angle is greater at axial locations of increased curvature and lesser at axial locations of decreased curvature.
3. A composite member according to claim 1, wherein at least one of said first and second fiber components has a helical angle substantially between seventy-five and ninety degrees relative to said
4. A composite member according to claim 1 having a bending stiffness along said axis and wherein said intermediate ply provides at least 80% of said bending stiffness.
5. A composite member according to claim 1 having a bending stiffness along said axis and wherein said first angle is variable substantially between zero and forty-five degrees to select said bending
6. A composite member according to claim 1, further comprising at least one pair of interlace fibers interlaced with at least one of said helically oriented fiber components, said pair including a
first interlace fiber component oriented at an angle substantially between ten and sixty degrees relative to the angle of said one helically oriented fiber component, said pair including a second
interlace fiber component oriented relative to the angle of said one helically oriented fiber component with an angle that is equal and opposite to said first interlace fiber component, said
interlace fibers being selected from the group consisting of glass, carbon, aramid, polyethylene, polyester, and mixtures thereof.
7. A composite member according to claim 1, wherein said intermediate ply further comprises a first interlace fiber component, a second interlace fiber component, a third interlace fiber component,
and a fourth interlace fiber component, each interlace fiber components being selected from the group consisting of aramid, glass, linear polyethylene, polyethylene, polyester, carbon, and mixtures
thereof, said first interlace fiber component being interwoven with said clockwise helically oriented fiber component and having a first interlace angle substantially between ten and sixty degrees
relative to said first angle, said second interlace fiber component being interwoven with said clockwise helically oriented fiber component and having a second interlace angle that is equal and
opposite in sign to said first interlace angle, said third interlace fiber component being interwoven with said counter-clockwise helically oriented fiber component and having a third interlace angle
between substantially ten and sixty degrees relative to said second angle, said fourth interlace fiber component being interwoven with said counter-clockwise helically oriented fiber component and
having a fourth interlace angle that is equal and opposite in sign to said third interlace angle.
8. A composite member according to claim 1 wherein said first fiber component and said clockwise helically oriented fiber component and said counterclockwise helically oriented fiber component are
each selected from the group consisting of glass, carbon, aramid, and mixtures thereof.
9. A composite member according to claim 1, wherein said polymer matrix material is selected from the group of resin-based materials consisting of B-staged thermoset, nylon-6 thermoplastic,
polyether-ether-ketone, polyphenylene sulfide, polyethylene, polypropylene, thermoplastic urethanes, epoxy, vinyl-ester, and polyester.
10. A composite member according to claim 1, further comprising at least one stitching fiber, said stitching fiber being interwoven with itself and with at least one of said fiber components.
11. A composite member according to claim 10, wherein said stitching fiber is selected from the group of fiber materials consisting of polyester, glass, carbon, aramid, and mixtures thereof.
12. A composite member according to claim 1, wherein said clockwise helically oriented fiber component is interwoven with said counter-clockwise helically oriented fiber component.
13. A composite member according to claim 1, wherein at least one of said fiber components comprises a plurality of interwoven fibers.
14. A composite member according to claim 1, further comprising, in said intermediate ply, a first clockwise helically oriented braiding yarn component and a second counter-clockwise oriented
braiding yarn component, said first and second yarn components being interwoven with at least one of said fiber components of said intermediate ply.
15. A composite member according to claim 1, further comprising an outer sheath of thermoplastic disposed exterior to said exterior ply and an inner core of thermoplastic disposed interior to said
interior ply, said thermoplastic outer sheath and inner core having a higher melting temperature than said matrix material, said composite member being capable of reformation at selected locations
along said axis by heating and bending said composite member at said selected locations.
16. A composite member according to claim 15, wherein said inner core of thermoplastic is tubular.
17. A composite member according to claim 1, further comprising an inner core of thermoplastic disposed interior to said interior ply and wherein said exterior ply comprises a thermoplastic matrix
material, said inner ply and said exterior ply both having a melting temperature higher than said polymer matrix material of said intermediate ply such that said composite member is formable into
said selected axial curved shape by bending said composite member when heated.
18. A tubular composite member according to claim 17, wherein said first angle has selected different values along the length of said member, said value of said first angle being a function of the
selected axially-curved shape of said composite member wherein said first angle is greater at axial locations of increased curvature and lesser at axial locations of decreased curvature.
19. A tubular composite member according to claim 17, further comprising at least one further ply disposed exterior to said inner ply and interior to said intermediate ply, said further ply having a
first fiber component and a polymer matrix material, said first fiber component thereof being helically-oriented substantially between seventy-five and ninety degrees relative to the axis of said
composite member and being selected from the group consisting of glass, carbon, aramid, and mixtures thereof.
20. A tubular composite member according to claim 17, further comprising at least one further ply disposed exterior to said intermediate ply and interior to said outer ply, said further ply having a
second fiber component and said matrix material, said second fiber component thereof being helically-oriented substantially between seventy-five and ninety degrees relative to the axis of said
composite member and being selected from the group consisting of glass, carbon, and aramid, and mixtures thereof.
21. A composite member according to claim 17, wherein said intermediate ply further comprises a first interlace fiber component, a second interlace fiber component, a third interlace fiber component,
and a fourth interlace fiber component, each interlace fiber components being selected from the group consisting of aramid, glass, linear polyethylene, polyethylene, polyester, carbon, and mixtures
thereof, said first interlace fiber component being interwoven with said clockwise helically oriented fiber component and having a first interlace angle of ten to sixty degrees relative to said first
angle, said second interlace fiber component being interwoven with said clockwise helically oriented fiber component and having a second interlace angle relative to said first angle that is equal but
opposite in sign to said first interlace angle, said third interlace fiber component being interwoven with said counter-clockwise helically oriented fiber component and having a third interlace angle
of ten to sixty degrees relative to said second angle, said fourth interlace fiber component being interwoven with said counter-clockwise helically oriented fiber component and having a fourth
interlace angle relative to said second angle that is equal but opposite in sign to said third interlace angle.
22. A composite member according to claim 17, further comprising at least one pair of interlace fibers interlaced with at least one of said helically oriented fiber components, said pair including a
first interlace fiber component oriented at an angle of between ten to sixty degrees relative to the angle of said one helically oriented fiber component, said pair including a second interlace fiber
component oriented relative to the angle of said one helically oriented fiber component with an angle that is equal but opposite in sign to said first interlace fiber component.
23. A composite member according to claim 17, wherein said intermediate ply comprises
a first intermediate ply having a first axially extending fiber component, first and second interlace fiber components interlaced with said first axially extending fiber component, and a polymer
matrix material, said first axially extending fiber component having a first angle of orientation substantially between zero and forty-five degrees relative to the axis of said composite member,
said first interlace fiber component oriented at an angle substantially between ten and sixty degrees relative to said first angle of orientation, said second interlace fiber component oriented
relative to said first angle of orientation at an angle that is equal and opposite to said first interlace fiber component, said first axially extending fiber component being selected from the
group consisting of carbon, aramid, glass, and mixtures thereof, said first and second interlace fibers being selected from the group consisting of glass, carbon, aramid, polyethylene, polyester,
and mixtures thereof, and
a second intermediate ply having a second axially extending fiber component, third and fourth interlace fiber components interlaced with said second axially extending fiber component, and a
polymer matrix material, said second axially extending fiber component having a second angle of orientation relative to said axis that is equal and opposite to said first angle of orientation,
said third interlace fiber component oriented at an angle substantially between ten and sixty degrees relative to said second angle of orientation, said fourth interlace fiber component oriented
relative to said second angle of orientation at an angle that is equal and opposite to said third interlace fiber component, said second axially extending fiber component being selected from the
group consisting of carbon, aramid, glass, and mixtures thereof, said third and fourth interlace fibers being selected from the group consisting of glass, carbon, aramid, polyethylene, polyester,
and mixtures thereof.
24. A composite member having selected curvature along an axis of elongation, comprising
at least one interior ply having a first fiber component and a polymer matrix material, said first fiber component being helically-oriented relative to said axis,
a first intermediate ply having a first axially extending fiber component, first and second interlace fiber components interlaced with said first axially extending fiber component, and a polymer
matrix material, said first axially extending fiber component having a first angle of orientation relative to said axis substantially between zero and forty-five degrees, said first interlace
fiber component oriented at an angle substantially between ten and sixty degrees relative to said first angle of orientation, said second interlace fiber component oriented relative to said first
angle of orientation at an angle that is equal and opposite to said first interlace fiber component,
a second intermediate ply having a second axially extending fiber component, third and fourth interlace fiber components interlaced with said second axially extending fiber component, and a
polymer matrix material, said second axially extending fiber component having a second angle of orientation relative to said axis that is equal and opposite to said first angle of orientation,
said third interlace fiber component oriented at an angle substantially between ten and sixty degrees relative to said second angle of orientation, said fourth interlace fiber component oriented
relative to said second angle of orientation at an angle that is equal and opposite to said third interlace fiber component,
at least one exterior ply having a second fiber component and said matrix material, said second fiber component being helically-oriented relative to said axis, and
said first and second angles having selected different values along the length of said composite member such that said composite member has a bending stiffness that varies along said axis at
selected different locations.
25. A composite member according to claim 24, wherein said first angle of orientation has selected different values along the length of said member, said value of said first angle of orientation
being a function of said curvature wherein said first angle is greater at axial locations of increased curvature and less at axial locations of decreased curvature.
26. A composite member according to claim 24, wherein at least one of said first and second fiber components has a helical angle substantially between seventy-five and ninety degrees relative to said
27. A composite member according to claim 24 wherein
said first and second fiber components and said first and second axially extending fiber components are selected from the group consisting essentially of glass, carbon, aramid, and mixtures
thereof, and
said first, second, third, and fourth interlace fibers are selected from the group consisting essentially of glass, carbon, aramid, polyethylene, polyester, and mixtures thereof.
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Inc. V.2-No. 4, Jul. 1992 (1 page). "TPI Tips, News and Tips for Pultruded Thermoplastic Composites", Thermoplastic Pultrusions, Inc. V.2-No. 5, Sep. 1992 (1 page). "TPI Tips, News and Tips for
Pultruded Thermoplastic Composites", Thermoplastic Pultrusions, Inc. V.1-No. 2, Nov. 1991 (1 page). "A New Generation of High-Strength Engineered, Composite Structural Shapes", The technology
exists today at Alcoa/Goldsworthy Engineering, ALCOA/Goldsworthy Engineering, 23930 Madison St. Torrance, CA. (no date available). Sanders, K. J., (1988) Organic Polymer Chemistry, 2nd Edition,
Chapman & Hill, pp. 203. Norton Company, Plastics & Synthetics Division, "Tygon Tubing", Bulletin T-104, Norton Performance Plastics, Akron, Ohio, pp. 3-26. (no date available). Rose, (1966), The
Condensed Chemical Dictionary, 7th Edition, Reinhold Publishing Corporation, pp. 684, 759, 760. "Advanced Production Systems for Composites", The Shape of Things to Come, Goldsworthy Engineering,
Inc. (no date available). Thermoplastic Pultrusions, Inc. publication, not dated, citing New Developments (8 pages). European Search Report mailed Jan. 4, 1994 during prosecution of EP 93/111
187. International Search Report mailed Sep. 22, 1995 during prosecution of PCT/US95/05083. European Search Report mailed Oct. 2, 1995 during prosecution of EP 100159.3.
Patent History
Patent number
: 5580626
: Aug 18, 1995
Date of Patent
: Dec 3, 1996
Composite Development Corporation
(West Wareham, MA)
Peter A. Quigley
(Pocasset, MA),
Stephen Briggi
(Wareham, MA),
Steven C. Nolet
(Leominster, MA),
James L. Gallagher
(Tiverton, RI)
Primary Examiner
Christopher Raimund Law Firm
Lahive & Cockfield Application Number
: 8/516,650
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Changjian Su, University of Toronto - Geometric Methods in Representation Theory Seminar - Department of Mathematics
Changjian Su, University of Toronto – Geometric Methods in Representation Theory Seminar
November 9, 2018 @ 3:30 pm - 4:30 pm
Title: Motivic Chern classes, K-theoretic stable basis and Iwahori invariants of principal series
Abstract: Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis.
In this talk, we will prove a conjecture of Bump–Nakasuji–Naruse about certain transition matrix between these two bases. We will first relate the Iwahori invariants to Maulik–Okounkov’s stable
envelopes and Brasselet–Schurmann–Yokura’s motivic Chern classes for the Langlands dual groups. Then the conjecture follows from a K-theoretic generalization of Kumar’s smoothness criterion for the
Schubert varieties. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.
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TULLY-FISHER Law Demonstrated by General Relativity and Dark Matter
TULLY-FISHER Law Demonstrated by General Relativity and Dark Matter ()
1. Introduction
One of the most important scaling laws is the empirical Tully-Fisher relation [1] , between the stellar mass or luminosity of a galaxy and its rotation velocity v. The stellar Tully-Fisher is a power
law $M\propto {v}^{\alpha }$ with α ~ 4 - 5 depending on the method used to estimate stellar masses [2] [3] and depending on how the rotation velocities are defined [4] [5] . When the baryonic mass M
[b] (stars + cold gas) is used instead of the stellar mass, the baryonic Tully-Fisher relation [6] becomes an extremely tight power law $M\propto {v}^{\alpha }$ , with α ~ 4 [5] [7] [8] .
In our study, we are going to demonstrate the Tully-Fisher law with the solution of DM explained without exotic matter but with a uniform gravitic field (the 2^nd component of GR similar to the
magnetic field in EM) as proposed by the author [9] . This field would be generated by galaxy clusters [10] and would embed large areas of the Universe (and then the galaxies) explaining this excess
of gravitation misnamed, in this explanation, DM. We will first remind you how Linearized General Relativity (LGR) is obtained from GR, how LGR equations can explain DM and the expected values of the
uniform gravitic field required to explain DM component. We will secondly verify that the measured coefficients of the Tully-Fisher law $M=a{v}^{\alpha }$ allow retrieving the expected gravitic field
explaining the DM. Third, the main goal of this study, we will demonstrate the expression of Tully-Fisher law in our explanation of DM, making this theoretical DM explanation extremely consistent
with the observations.
2. Dark Matter Explained by General Relativity
2.1. From General Relativity to Linearized General Relativity
From GR, one deduces the LGR in the approximation of a quasi-flat Minkowski space ( ${g}^{\mu u }={\eta }^{\mu u }+{h}^{\mu u }$ ; $|{h}^{\mu u }|\ll 1$ ). With the following Lorentz gauge, it gives
the following field equations as in [11] (with $\square \text{\hspace{0.17em}}=\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}-\Delta$ and $\Delta ={abla }^{2}$ ):
${\partial }_{\mu }{\stackrel{¯}{h}}^{\mu u }=0;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\square {\stackrel{¯}{h}}^{\mu u }=-2\frac{8\pi G}{{c}^{4}}{T}^{\mu u }$(1)
${\stackrel{¯}{h}}^{\mu u }={h}^{\mu u }-\frac{1}{2}{\eta }^{\mu u }h;\text{\hspace{0.17em}}\text{\hspace{0.17em}}h\equiv {h}_{\sigma }^{\sigma };\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{u }^
{\mu }={\eta }^{\mu \sigma }{h}_{\sigma u };\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{¯}{h}=-h$(2)
The general solution of these equations is:
${\stackrel{¯}{h}}^{\mu u }\left(ct,x\right)=-\frac{4G}{{c}^{4}}{\int }^{\text{}}\frac{{T}^{\mu u }\left(ct-|x-y|,y\right)}{|x-y|}{\text{d}}^{3}y$(3)
In the approximation of a source with low speed, one has:
${T}^{00}=\rho {c}^{2};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{0i}=c\rho {u}^{i};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{ij}=\rho {u}^{i}{u}^{j}$(4)
And for a stationary solution, one has:
${\stackrel{¯}{h}}^{\mu u }\left(x\right)=-\frac{4G}{{c}^{4}}{\int }^{\text{}}\frac{{T}^{\mu u }\left(y\right)}{|x-y|}{\text{d}}^{3}y$(5)
At this step, by proximity with electromagnetism, one traditionally defines a scalar potential $\phi$ and a vector potential ${H}^{i}$ . There are in the literature several definitions as in [12] for
the vector potential ${H}^{i}$ . In our study, we are going to define:
${\stackrel{¯}{h}}^{00}=\frac{4\phi }{{c}^{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{h}}^{0i}=\frac{4{H}^{i}}{c};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{h}}^{ij}
with gravitational scalar potential $\phi$ and gravitational vector potential ${H}^{i}$ :
$\phi \left(x\right)\equiv -G{\int }^{\text{}}\frac{\rho \left(y\right)}{|x-y|}{\text{d}}^{3}y$
${H}^{i}\left(x\right)\equiv -\frac{G}{{c}^{2}}{\int }^{\text{}}\frac{\rho \left(y\right){u}^{i}\left(y\right)}{x-y}{\text{d}}^{3}y=-{K}^{-1}{\int }^{\text{}}\frac{\rho \left(y\right){u}^{i}\left(y
with K (determined in [9] ) a new constant defined by:
This definition is ${K}^{-1}~7.4×{10}^{-28}\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{-1}$ very small compared to G.
The field Equations (1) can be then written (Poisson equations):
$\Delta \phi =4\pi G\rho ;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{H}^{i}=\frac{4\pi G}{{c}^{2}}\rho {u}^{i}=4\pi {K}^{-1}\rho {u}^{i}$(9)
with the following definitions of $g$ (gravity field) and $k$ (gravitic field), those relations can be obtained from the following equations (also called gravitomagnetism) with the differential
operators “ $rot=abla \wedge$ ”, “ $grad=abla$ ” and “ $div=abla \cdot$ ”:
$\begin{array}{l}g=-grad\text{\hspace{0.17em}}\phi ;\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=rot\text{\hspace{0.17em}}H\\ rot\text{\hspace{0.17em}}g=0;\text{\hspace{0.17em}}\text{\hspace
{0.17em}}div\text{\hspace{0.17em}}k=0;\\ div\text{\hspace{0.17em}}g=-4\pi G\rho ;\text{\hspace{0.17em}}\text{\hspace{0.17em}}rot\text{\hspace{0.17em}}k=-4\pi {K}^{-1}{j}_{p}\end{array}$(10)
with the Equations (2), one has:
${h}^{00}={h}^{11}={h}^{22}={h}^{33}=\frac{2\phi }{{c}^{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}^{0i}=\frac{4{H}^{i}}{c};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}^{ij}=0$(11)
The equations of geodesics in the linear approximation give:
$\frac{{\text{d}}^{2}{x}^{i}}{\text{d}{t}^{2}}\text{~}-\frac{1}{2}{c}^{2}{\delta }^{ij}{\partial }_{j}{h}_{00}-c{\delta }^{ik}\left({\partial }_{k}{h}_{0j}-{\partial }_{j}{h}_{0k}\right){v}^{j}$(12)
It then leads to the movement equations:
$\frac{{\text{d}}^{2}x}{\text{d}{t}^{2}}~-grad\text{\hspace{0.17em}}\phi +4v\wedge \left(rot\text{\hspace{0.17em}}H\right)=g+4v\wedge k$(13)
Remark: All previous relations can be retrieved starting with the parameterized post-Newtonian (PPN) formalism and with the traditional gravitomagnetic field ${B}_{g}$ . From [13] one has:
${g}_{0i}=-\frac{1}{2}\left(4\gamma +4+{\alpha }_{1}\right){V}_{i};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{i}\left(x\right)=\frac{G}{{c}^{2}}{\int }^{\text{}}\frac{\rho \left(y\right){u}_
The traditional gravitomagnetic field and its acceleration contribution are:
${B}_{g}=abla \wedge \left({g}_{0i}{e}^{i}\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{g}=v\wedge {B}_{g}$(15)
And in the case of GR (that is our case):
$\gamma =1;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{1}=0$(16)
It then gives:
${g}_{0i}=-4{V}_{i};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{g}=abla \wedge \left(-4{V}_{i}{e}^{i}\right)$(17)
And with our definition:
${H}_{i}=-{\delta }_{ij}{H}^{j}=\frac{G}{{c}^{2}}{\int }^{\text{}}\frac{\rho \left(y\right){\delta }_{ij}{u}^{j}\left(y\right)}{|x-y|}{\text{d}}^{3}y={V}_{i}\left(x\right)$(18)
One then has:
${g}_{0i}=-4{H}_{i};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{g}=abla \wedge \left(-4{H}_{i}{e}^{i}\right)=abla \wedge \left(4{\delta }_{ij}{H}^{j}{e}^{i}\right)=4abla \wedge H$(19)
with the following definition of gravitic field:
One then retrieves our previous relations:
$k=rot\text{\hspace{0.17em}}H;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{g}=v\wedge {B}_{g}=4v\wedge k$(21)
The interest of our notation ( $k$ instead of ${B}_{g}$ ) is that the field equations are strictly equivalent to Maxwell’s idealization, in particular, the speed of the gravitational wave obtained
from these equations is the light celerity ${c}^{2}=GK$ just like in EM ${c}^{2}=1/{\mu }_{0}{\epsilon }_{0}$ . Only the movement equations are different with the factor “4”. But of course, all the
results of our study can be obtained in the traditional notation of gravitomagnetism with the relation $k=\frac{{B}_{g}}{4}$ .
2.2. From Linearized General Relativity to DM
In the classical approximation ( $‖v‖\ll c$ ), the linearized general relativity gives the following movement equations from (13) with ${m}_{i}$ the inertial mass and ${m}_{p}$ the gravitational
${m}_{i}\frac{\text{d}v}{\text{d}t}={m}_{p}\left[g+4v\wedge k\right]$(22)
The traditional computation of rotation speeds of galaxies consists of obtaining the force equilibrium from the three following components: the disk, the bugle, and the halo of dark matter. More
precisely, one has [14] :
Then the total speed squared can be written as the sum of squares of each of the three-speed components:
Disk and bulge components are obtained from gravity field. They are not modified in our solution. So, our goal is now to obtain only the traditional dark matter halo component from the linearized
general relativity. According to this idealization, the force due to the gravitic field
Our idealization means that:
The equation of dark matter (gravitic field in our explanation) is then:
This equation gives us the curve of rotation speeds of the galaxies as we wanted. Because we know the curves of speeds that one wishes to have for DM component, one can then deduce the curve of the
gravitic field
2.3. Dark Matter as the 2^nd Component of the Gravitational Field
This solution of DM as the gravitic field has been studied in [9] for 16 galaxies (Table 1). It shows that this solution is mathematically possible but with two physical mandatory unexpected behavior
From these data (Table 1), one can deduce a mean value of
The position
Table 1. Distance r[0] to the center of the galaxy where the internal gravitic field [0] generated by the galaxies’ cluster. k[0] dominates for
3. TULLY-FISHER Law Obtained from a Uniform Gravitic Filed
We will first verify that this theoretical solution of DM is consistent with the Tully-Fisher law by retrieving our value of DM
Let’s note
And more explicitly with M the galaxy’s mass:
which gives:
3.1. From the TULLY-FISHER Law to the Uniform Gravitic Field
The Tully-Fisher law is written
Let’s rewrite our expression (33):
which gives:
In order to get an expression that looks like the Tully-Fisher law, we are going to use the following approximation for large r (in the flat part of the rotation speed curve):
Furthermore, in our explanation of DM,
If one has:
The expression becomes:
And finally:
The couples ^−^1 and the masses in solar mass (
We want to verify that the values of
For these calculations, one will use:
This value of r justify the previous approximation (36)
In [2] the observations give the following couple:
In [15] , one has the 4 following couples:
Let’s verify the previous approximation (38) with the smallest value of
These calculations show that the expected values of
3.2. From the Uniform Gravitic Field
As we noticed in the previous paragraph, in our relation, there is the parameter position r which appears while the Tully-Fisher relation does not explicitly depend on it. To no longer explicitly
depend on this parameter, we need to define a procedure to determine a characteristic position r. It should be noted that we are in the same situation for the TULLY-FISHER law to define the
characteristic speed of rotation to be considered. Several methods are used [5] and the mean velocity along the flat part of the rotation curve seems to minimize the scatter of the relation [4] . The
question in our context is somehow to find which method was adopted to define the characteristic position r corresponding to the characteristic velocity on the curve of the rotational velocities of
the galaxies. This characteristic speed is linked to the flat zone of the speed curve, the characteristic position will certainly be a characteristic position in this zone. We can imagine 2 simple
A 1st would consist of finding the position of the beginning of the flat zone
A 2nd method would consist of finding a threshold for the value of the gravitational forces
3.2.1. 1^st Method to Demonstrate the Tully-Fisher Law
The beginning of the flat zone of the rotational speed curve
We then write (to be somewhere in the flat zone):
Our relation (32) gives:
with Figure 1) one obtains a curve that is very close to the Tully-Fisher law, passing through the cloud of expected measured points. But the slope of the curve is unsatisfactory. This 1st method
informs us that the position
3.2.2. 2^nd Method to Demonstrate the Tully-Fisher Law
Let’s define a threshold value of the intensity of the force for which, we hope to find the position on the rotation curve corresponding to the characteristic speed considered for the Tully-Fisher
Figure 1. The red curve representing our relation obtained from our 1^st method is superposed on graph from [15] .
This position is denoted
We then obtain our expression of Tully-Fisher law (red curve in Figure 2):
An approximation of this relation in the form “Figure 2. Furthermore, the obtention
As calculated previously, if we take the average of the beginnings of the flat zones and the mean of the values of the gravitic field explaining DM (30) and by taking the characteristic mass
previously used for our calculations (43), one has:
Figure 2. The red curve representing our relation obtained from our 2^nd method is superposed on the graph from [15] .
For this value of mass, on the graphs (Figure 1 and Figure 2) the corresponding characteristic rotational speed is around:
All these characteristic values allow defining our characteristic force threshold:
The red curve in Figure 2 representing the relation (68) is obtained with
This time, compared to the 1st method, not only does the curve pass well through the cloud of measured points, but the slope of the curve is also excellent. Add to this that the characteristic value
4. Discussion
In the same way that there are several methods for defining the characteristic velocity used in the Tully-Fisher law giving more or less tight values of its coefficients, the role of
The fact of obtaining a value of
Another point seems important. The Tully-Fisher law in its form “
5. Conclusion
In this study, we show that the explanation of dark matter in the form of a uniform gravitic field ^nd component of GR similar to the magnetic field in EM giving the Lense-Thirring effect) makes it
possible to obtain the Tully-Fisher law. Obtaining this law is based on two important characteristics of this solution, one theoretical, namely the shape of this field, “
In addition to providing the correct values (passing through the measured points) and the correct slope of the Tully-Fisher law, this relationship goes further by showing a systematic break in the
curve for mass values roughly around
This study finally leads to 3 major results, the demonstration of the Tully-Fisher law, a justification of a maximal mass of the galaxies but perhaps even more important a validation of the
explanation of the DM in the form of a uniform gravitic field embedding the galaxies.
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How can I convert a log axis scale with a negative offset?
I have a puzzler problem. I have a left axis set from 2 x 10^11 to 1 x 10^12. It runs on a log scale view. I have a function that returns values of -0.31 (top) and 0.97 (bottom) for the two extrema
of the left axis. I want to add those conversion calibration values on a right axis. The right axis also has to run on a log scale.
How can I do this?
I am wrapping my head around the complexities of using log or loglinear axis settings, offsetting the scales on the right axis, creating manual ticks and labels, or using some variation of transform
Suppose that the left axis value is LAV. The conversion equation to obtain the right axis value RAV is
RAV = (LAVo - LAV)/LAVo
where LAVo is around 7.6 x 10^11. My data is nicely displayed within the lower to upper bounds given above. So, I cannot just shift the left axis to avoid the negative values on the transformation
for the right axis.
What is LAV? Is it log(some quantity)? Or is it distance along the axis in some sort of units (which would mean it is proportional to log(value))?
If the inputs to the equation are log values, the RAV seems like it is also a log value. If you want to use Igor's log axis, you may need to plot 10^RAV.
Or perhaps I have misunderstood something...
I resolved it. Suppose that you have y-values that range from 2E10 to 8E11. Suppose that you have an equation as:
f = (7E11 - y)/7E11
Here is the basic approach.
* The left axis is set from 10E10 to 10E12 on a log scale
* The right axis is set form 1 to 100 also on a log scale
* Use wave/N=10 for user defined tick marks and labels.
* Set the tick labels as 0, 0.1, 0.2, ... 0.9
* Set the tick location values to (70 - 70*(p/10))
Here is my graph. Note that this graph is NOT using the numbers above, just the approach. Also, by good fortune, when I got around to re-scaling my graph, the problem with negative values on the log
axis disappeared.
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C Subroutine file 'apvar.for': Applications and processing of the
C results of complete ray tracing -- Part2: Travel-time variations
C Date: 2005, April 24
C Coded by Ludek Klimes
C This file consists of the following external procedures:
C AP29... Subroutine designed to evaluate the variations of the
C travel time with respect to the model coefficients.
C It has to be called once at each point along the ray at
C which the computed quantities are stored, i.e. after each
C invocation of the subroutine AP00 which reads the
C quantities into the common block /POINTC/.
C The subroutine can also optionally calculate the
C variations of the line integral of the density for the
C gravimetric inversion. This option is indicated by NY=8
C in include file
C pointc.inc.
C AP29
C AP29A...Auxiliary subroutine to AP29.
C AP29A
SUBROUTINE AP29(NSUM,SUM)
INTEGER NSUM
REAL SUM(NSUM)
C This subroutine evaluates variations of the travel time with respect
C to the model coefficients. It has to be called once at each point
C along the ray at which the computed quantities are stored, i.e. after
C each invocation of the subroutine AP00 which reads the quantities into
C the common block /POINTC/.
C The subroutine can also optionally calculate the variations of the
C line integral of the density for the gravimetric inversion. This
C option is indicated by NY=8 in common block /POINTC/ declared in
C include file pointc.inc.
C Subroutine PARM2 is called to evaluate the material parameters at the
C current point and, at a structural interface, also subroutine SRFC2 is
C called to evaluate the function describing the interface. After the
C invocation of PARM2 or SRFC2, respectively, subroutine VAR6 is called
C to recall the variations of the model parameters or of the interface,
C with respect to the model coefficients. If the user replaces the
C subroutine file 'parm.for' or 'srfc.for' by his own version, it is his
C own responsibility to call subroutines VAR1 to VAR5 (see the file
C 'var.for') in such a way that the required variations are stored when
C returning from his own subroutine PARM2 or SRFC2.
C Input:
C NSUM... Total number of the coefficients describing the model.
C SUM... Array of dimension at least NSUM, in which the variations
C of the travel time with respect to the model coefficients
C are accumulated. Its elements are set to zeros at the
C initial point of the ray by this subroutine.
C Output:
C SUM... Variations of the travel time (from the initial point of
C the ray to the current point along the ray) with respect
C to the model coefficients.
C Common block /POINTC/:
INCLUDE 'pointc.inc'
C pointc.inc
C To calculate the variations of the line integral of the density for
C the gravimetric inversion, the folowing variables should be defined:
C NYF=0
C NY=8
C IPT,ICB1I,ICB1,ISRF... The same meaning as for ray tracing.
C YI(1),Y(1)... Arclength along a straight line.
C YI(3:5),Y(3:5)... Coordinates, possibly curvilinear.
C YI(6:8),Y(6:8)... Derivatives of the coordinates with respect to
C the arclength.
C None of the storage locations of the common block are altered.
C Subroutines and external functions required:
INTEGER KOOR
EXTERNAL KOOR,METRIC,SRFC2,VAR6,AP28
C SMVPRD
C PARM2,VELOC
C Date: 2005, April 24
C Coded by Ludek Klimes
C Auxiliary storage locations for local model parameters: FAUX(10),
C G(12),GAMMA(18),GSQRD, UP(10),US(10),RO,QP,QS, VP,VS,VD(10),QL:
INCLUDE 'auxmod.inc'
C auxmod.inc
C Auxiliary storage locations:
INTEGER MFUN
PARAMETER (MFUN=64)
INTEGER NFUN1,IFUN1(MFUN),NFUN2,IFUN2(MFUN),ISRF1,II,IBI
REAL B0I,B1I,B2I,B3I,FUN1(2*MFUN),FUN2(2*MFUN)
REAL X1,PIN1,PIN2,PIN3,C(3),P1,P2,P3,AUX0
SAVE NFUN1,IFUN1,FUN1,ISRF1,X1,PIN1,PIN2,PIN3
C NFUN1...Number of functions having nonzero values or nonzero first
C derivatives at the previous point along the ray.
C IFUN1...IFUN(1:NFUN1)... Indices in the array SUM corresponding to
C the functions having nonzero values or nonzero first
C derivatives at the previous point along the ray.
C FUN1... FUN(1:NFUN1)... Values of the functions having nonzero
C values or nonzero first derivatives at the previous
C point along the ray.
C FUN(NFUN1+1:2*NFUN1)... First derivatives with respect to
C the independent variable along the ray at the previous
C point along the ray, of the functions having nonzero
C values or nonzero first derivatives.
C At the first point after the initial point of the ray,
C the values of NFUN1, IFUN and FUN1 correspond to the
C initial (zero) point of the ray.
C NFUN2...Number of functions having nonzero values or nonzero
C first derivatives at the current point along the ray.
C IFUN2...Indices in the array SUM corresponding to the functions
C having nonzero values or nonzero first derivatives at the
C current point along the ray.
C FUN2... FUN(1:NFUN2)... Values of the functions having nonzero
C values or nonzero first derivatives at the current point
C along the ray.
C FUN(NFUN2+1:2*NFUN2)... First derivatives with respect to
C the independent variable along the ray at the previous
C point along the ray, of the functions having nonzero
C values or nonzero first derivatives.
C ISRF1...Index of the surface covering the interface, updated by
C subroutine AP28.
C II... Loop variable (sequential number of the required
C variation).
C IBI... Absolute index of the function coefficient.
C B0I,B1I,B2I,B3I... Variation of the functional value and the three
C first derivatives, with respect to the IBI-th coefficient
C of the model.
C X1... Independent variable along the ray, updated by AP28.
C PIN1,PIN2,PIN3... Contravariant components of the slowness vector
C at the point of incidence.
C C... Coordinates.
C P1,P2,P3... Contravariant components of the slowness vector.
C AUX0... Temporary storage location.
C Initial point of the ray:
IF(IPT.LE.1) THEN
CALL AP29A(NY,ICB1I,YI,C,P1,P2,P3,MFUN,NFUN1,IFUN1,FUN1)
END IF
C Another point of the ray:
IF(NYF.GT.0) THEN
CALL AP29A
* (NY,ICB1F,YF,C,P1,P2,P3,MFUN,NFUN2,IFUN2,FUN2)
CALL AP29A
* (NY,ICB1 ,Y ,C,P1,P2,P3,MFUN,NFUN2,IFUN2,FUN2)
END IF
C Numerical quadrature:
CALL AP28(NSUM,SUM,1,2,0.,X1,ISRF1,
* NFUN1,IFUN1,FUN1,NFUN2,IFUN2,FUN2)
C Structural interface:
IF(ISRF1.NE.0.AND.ISRF1.LE.100) THEN
IF(PIN1.EQ.0..AND.PIN2.EQ.0..AND.PIN3.EQ.0.) THEN
C incident ray:
C Reflected/transmitted ray:
C Including the variation of the travel time with respect to the
C structural interface
CALL SRFC2(IABS(ISRF1),C,VD)
IF(KOOR().NE.0) THEN
CALL METRIC(C,GSQRD,G,GAMMA)
AUX0=VD(2)*(G(7)*VD(2)+2.*(G(8)*VD(3)+G(10)*VD(4))) +
* VD(3)*(G(9)*VD(3)+2.*G(11)*VD(4)) + VD(4)*G(12)*VD(4)
END IF
AUX0=( VD(2)*(P1-PIN1)+VD(3)*(P2-PIN2)+VD(4)*(P3-PIN3) )/AUX0
30 CONTINUE
CALL VAR6(1,II,NFUN2,IBI,B0I,B1I,B2I,B3I)
IF(II.LE.NFUN2) THEN
IF(IBI.GT.NSUM) THEN
C 729
CALL ERROR('729 in AP29: Too small input array SUM')
C Dimension NSUM of input array SUM should be increased.
END IF
END IF
IF(II.LT.NFUN2) GO TO 30
END IF
END IF
SUBROUTINE AP29A(NY,ICB1,Y,C,P1,P2,P3,MFUN,NFUN,IFUN,FUN)
INTEGER NY,ICB1,MFUN,NFUN,IFUN(MFUN)
REAL Y(8),C(3),P1,P2,P3,FUN(2*MFUN)
C Auxiliary subroutine to AP29.
C Input:
C NY... If NY=8, line integral of the density for the gravimetric
C inversion is considered.
C Otherwise, line integral of slowness for the travel-time
C inversion is considered.
C ICB1... Index of the complex block.
C Y... Quantities computed along a ray.
C MFUN... Array dimension.
C Output:
C C... Coordinates.
C P1,P2,P3... Contravariant components of the slowness vector.
C NFUN... Number of variations.
C IFUN... Indices of variations.
C FUN... FUN(1:NFUN)... Values of variations.
C FUN(NFUN+1:2*NFUN)... First derivatives of variations
C with respect to the independent variable along the ray.
C Subroutines and external functions required:
INTEGER KOOR
EXTERNAL KOOR,METRIC,SMVPRD,PARM2,VELOC,VAR6
C Auxiliary storage locations for local model parameters: FAUX(10),
C G(12),GAMMA(18),GSQRD, UP(10),US(10),RO,QP,QS, VP,VS,VD(10),QL:
INCLUDE 'auxmod.inc'
C auxmod.inc
REAL AUX0,AUX1,AUX2,AUX3,AUX4
INTEGER NEXPS,IVAL,II
PARAMETER (NEXPS=0)
REAL B0I,B1I,B2I,B3I
C AUX0,AUX1,AUX2,AUX3,AUX4... Auxiliary storage locations
C for local model parameters or temporary variables.
C IVAL... Index of the function describing the model.
C IVAL=1 for P-wave,
C IVAL=2 for S-wave.
C II... Loop variable (sequential number of the required
C variation).
C B0I,B1I,B2I,B3I... Variation of the functional value and the three
C first derivatives, with respect to the IBI-th coefficient
C of the model.
C Assignments:
IF(NY.EQ.8) THEN
C Calculating the variations for gravimetric inversion:
IF(ICB1.EQ.0) THEN
END IF
CALL PARM2(IABS(ICB1),Y(3),UP,US,AUX0,AUX1,AUX2)
C Calculating the variations for travel-time inversion:
C Contravariant components of the slowness vector:
IF(KOOR().NE.0) THEN
CALL METRIC(Y(3),GSQRD,G,GAMMA)
CALL SMVPRD(G(7),Y(6),Y(7),Y(8),P1,P2,P3)
END IF
C Material parameters:
CALL PARM2(IABS(ICB1),Y(3),UP,US,AUX0,AUX1,AUX2)
CALL VELOC(ICB1,UP,US,AUX1,AUX2,AUX3,AUX4,VD,AUX0)
C Material parameters and their variations are defined.
IF(ICB1.GT.0) THEN
C P-wave:
C S-wave:
END IF
END IF
C Recalling the variations:
20 CONTINUE
CALL VAR6(IVAL,II,NFUN,IFUN(II),B0I,B1I,B2I,B3I)
IF(II.LE.NFUN) THEN
IF(NFUN.GT.MFUN) THEN
C 730
CALL ERROR('730 in AP29: Array index out of range')
C Dimension MFUN of arrays IFUN1, FUN1, IFUN2, FUN2 should
C be increased.
END IF
END IF
IF(II.LT.NFUN) GO TO 20
|
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|
Physics | NEB Grade 12 Model Question 2079-2023 - Dhan Raj GurungPhysics NEB Grade 12 Model Question 2079-2023
Physics | NEB Grade 12 Model Question 2079-2023
Dear Students of grade XII-12! Today, we’ve brought you Physics NEB Grade 12 Model Question 2079-2023 with subject code: 1021 and you can download the neb grade xii (12) Physics subject question
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Physics NEB Grade 12 Model Question 2079-2023.
NATIONAL EXAMINATIONS BOARD [NEB]
NEB Grade XII-12
Model Question
Subject Code – 1021
Time – 3 hrs
Full Marks – 75
The candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.
Check this …
Physics NEB Class 12 Question Paper 2080-2023 (Technical Stream) (Sub. Code: 9131) (New Course) | PDF Download.
Physics Exam Question Paper Class 12 of 2080-2023 (New Course) | PDF Download.
NEB Grade 12-XII Exam Model Question Papers 2077-2020 for All Subjects | Download in PDF.
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Instructional Pedagogy and Evaluation | NEB Grade 12 Model Question 2079-2023 Subject Code: 1181 | Download in PDF.
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Attempt all the questions
Group – ‘A’
Rewrite the correct options of each question in your answer sheet. (11×1=11)
1. The product of moment of inertia and angular velocity gives
(A) force
(B) torque
(C) linear momentum
(D) angular momentum
2. The bob of a simple pendulum has a mass of 0.40 kg. The pendulum oscillates with a period of 2.0 s and an amplitude of 0.15m. At an extreme point in its cycle, it has a potential energy of 0.044J.
What is the kinetic energy of the pendulum bob at its mean point?
(A) 0.022 J
(B) 0.044 J
(C) 0.011 J
(D) 0.033 J
3. What causes earthquakes?
(A) The flow of magma
(B) The expansion of the earth’s crust
(C) The rubbing together of earth’s plates
(D) Tsunami
4. What percentage of original radioactive atoms is left after 4 half-lives?
(A) 1%
(B) 6%
(C) 10%
(D) 20%
5. Two wave pulses travel toward each other as shown in the diagram below.
Which of the following diagrams represents the superposition of the pulses when they meet?
6. Which one of the following properties of sound is affected by the change in air temperature?
(A) amplitude
(B) frequency
(C) wavelength
(D) intensity
7. Internal energy of an ideal gas depends on
(A) volume only
(B) pressure only
(C) temperature only
(D) both pressure and volume
8. In which of the following processes of the gas, work done is the maximum?
(A) Isothermal
(B) Isobaric
(C) adiabatic
(D) Isochoric
9. The neutral temperature of a thermocouple is equal to 500°C when the temperature of cold junction is 0°C. Percentage change in the temperature of inversion when temperature of cold junction is
equal to 20°C is
(A) 2%
(B) 3%
(C) 4%
(D) 5%
10. In which of the following circuits the maximum power dissipation is observed?
(A) a circuit having inductor and resistor in series
(B) pure resistive circuit
(C) pure inductive circuit
(D) pure capacitive circuit
11. Why laminated cores are placed in transformers?
(A) to reduce hysteresis loss
(B) to reduce eddy current
(C) to reduce the magnetic effect
(D) to increase coercivity
Group – ‘B’ (8×5 = 40)
(i) Define simple harmonic motion. [1]
(ii) Derive an expression for the time period of oscillation for a mass m attached to a vertical spring of force constant k. [3]
(iii) What will be the time period of this system if it is taken inside the satellite? [1]
(a) State Bernoulli’s principle. [1]
(b) Figure below shows a liquid of density 1200kgm-3 flowing steadily in a tube of varying cross-sections. The cross-section at point A is 1.0 cm2 and that at B is 20 mm2, points A and B are in the
same horizontal plane. The speed of the liquid at A is 10 cm/s. Calculate
(i) the speed at B. [2]
(ii) the difference in pressure at A and B. [2]
(a) Draw a PV diagram of a petrol engine and explain its working based on its PV diagram. [3]
(b) Compare the efficiency of petrol engine with that of diesel engine based on their compression ratios. [2]
(a) When the wire of a sonometer is 75 cm long, it is in resonance with a tuning fork. On shortening the wire by 0.5 cm it makes 3 beats with the same fork. The beat is the difference in frequencies.
Calculate the frequency of the tuning fork. [ 3]
(b) The diagram below shows an experiment to measure the speed of a sound in a string. The frequency of the vibrator is adjusted until the standing wave shown in the diagram is formed. The frequency
of the vibrator is 120Hz. Calculate the speed at which a progressive wave would travel along the string. [2]
(a) State Lenz law in electromagnetism. Justify this law is in favor of the principle of conservation of energy. [1+2 =3]
(b) A magnet is quickly moved in the direction indicated by an arrow between two coils C1 and C2 as shown in the figure. What will be the direction of induced current in each coil as indicated by the
movement of magnet? Explain. [2]
(a) State the principle of the Potentiometer. A potentiometer isalso called a voltmeter of infinite resistance, why?
(b) In the meter bridge experiment, the balance point was observed at J with l=20cm.
(i) The values of R and X were doubled and then interchanged. What would be the new position of balance point? [2]
(ii) If the galvanometer and battery are interchanged at the balance position, how will the balance point get affected? [1]
(a) State the two Kirchhoff’s laws for electrical circuits. [2]
(b) In Meter Bridge shown below, the null point is found at a distance of 60.0 cm from A. If now a resistance of 5 ohm is connected in series with S, the null point occurs at 50 cm. Determine the
values of R and S. [3]
18. The graph below shows the maximum kinetic energy of the emitted photoelectrons as the frequency of the incident radiation on a sodium plate is varied.
(a) From the graph determine the maximum frequency of incident radiation that can cause a photoelectric effect? [1]
(b) Calculate the work function for sodium. [2]
(c) Use the graph to calculate the value of the Planck constant in Js. [2]
(a) Figure below shows the experimental setup of Millikan’s oil drop experiment. Find the expression for a charge of an oil drop of radius r moving with constant velocity v in a downward direction
using a free body diagram. [2]
(b)What will be the expression for the charge of an oil drop if the electric force is greater than its weight? [1]
(c) Determine the electric field supplied when the electric force applied between the two horizontal plates just balances an oil drop with 4 electrons attached to it and mass of oil drop is 1.3×10-14
kg. [2]
(a) A diode can be used as a rectifier. What characteristic of a diode is used in rectification? [1]
(b) Draw a circuit diagram of full wave rectifier. [1]
(c) A NOR gate ‘opens’ and gives an output only if both inputs are ‘low’, but an OR gate ‘closes’. An AND gate ‘opens’ only if both inputs are ‘high’, but a NAND gate ‘closes’. Construct a truth
table for the circuit shown below including the states at E, F and G. [3]
Group – ‘C’ (3×8 = 24)
(a) What is the significance of the negative energy of the electron in an orbit? [1]
(b) The energy levels of an atom are shown in Fig. below. Which one of these transitions will result in the emission of a photon of wavelength 275 nm? Explain with calculation. ( h= 6.64 x10-34Js, C=
3.0×108 ms-1) [3]
(c) Find the expression for the wavelength of radiation emitted from a hydrogen atom when an electron jumps from higher energy level n2 to the lower energy level n1. [2]
(d) Calculate the magnitude of the wavelength of the second Balmer series?(R =1.09x107m-1) [2]
(a) A student is trying to make an accurate measurement of the wavelength of green light from a mercury lamp (λ = 546 nm). Using a double slit of separation of 0.50 mm, he finds he can see ten clear
fringes on a screen at a distance of 0.80 m from the slits. He then tries an alternative experiment using a diffraction grating that has 3000 lines/cm.
(i) What will be the width of the ten fringes that he can measure in the first experiment? [2]
(ii) What will be the angle of the second-order maximum in the second experiment? [2]
(iii) Suggest which experiment you think will give the more accurate measurement of wavelength (). [1]
(b) A physics student went to buy polaroid sunglasses. The shopkeeper gave him two similar-looking sunglasses. In what way he can differentiate between polaroid sunglasses and non-polaroid
sunglasses? [1]
(c) At what angle of incidence will the light reflected from water ( = 1.3) be completely polarized? [2]
a) In what way does the intensity of sound heard by an observer change if the distance with the source changes by four times? [1]
b) A train is traveling at 30m/s in still air. The frequency of the note emitted by the train whistle is 262Hz. What frequency is heard by a passenger on the other train moving in the opposite
direction to the first at 18m/s (i) when approaches the first and (ii) when receding from the first?( velocity of sound= 340 m/s) [2+2]
c) In a sinusoidal sound wave of moderate loudness, the maximum pressure variations are of the order of 3.0×10-2 Pa above and below the atmospheric pressure. Find the corresponding maximum
displacement if the frequency is 1000Hz.in air at normal atmospheric pressure and density. The speed of sound is 344m/s and the bulk modulus of the medium is 1.42×105 Pa. [3]
(a) What is choke coil? [1]
(b) Why is it preferred over a resistor in ac circuit? [2]
(c) In figures (a), (b) and (c), three ac circuits with equal currents have been shown.
(i) If the frequency of e.m.f.be increased, then what will be effect on the currents flowing in them? Explain. [3]
(ii) What difference do you expect in the opposition provided by circuits for the current flow in figure (a) and (b) if given a.c. e.m.f. is replaced by its equivalent d.c. e.m.f. ? [2]
(a) Define 1 Ampere of current in terms of force between two parallel current-carrying conductors. [1]
(b) How do you explain the contraction of the solenoidal coil while the current is passed through it? [2]
(c) A conductor of linear mass density 0.2 g m-1 suspended by two flexible wires as shown in the figure. Suppose the tension in the supporting wires is zero when it is kept inside the magnetic field
of 1T whose direction is into the page.
(i) Compute the current inside the conductor. [3]
(ii) If the current determined in part (i) is passed through a 100-turn coil of 100 cm2 area with its axis held perpendicular to the magnetic field of flux density 10 T and plane of coil parallel to
the field, how much torque is produced? [2]
View and Download Physics NEB Grade 12 Model Question 2079-2023.
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Chap6.6: A Light in the Darkness | CollatzResearch.org
top of page
Chapter 6.6:
A Light in the Darkness
For the first time ever we may actually be holding in our hands a legitimate attack on one half of the Collatz Conjecture!
A sketch of a proof...
Suppose for a moment the my conjecture about "The Wiggle" is true. Suppose that indeed in x3+1 space every non-trivial branch's % membership function %(n) has LimSup > 0 aka all non-trivial branches
hold their own, as n goes to infinity. Then we get the following:
Suppose additionally that x3+1 space contains divergence. The existence of a divergent element implies that there is a divergent tree. As the orbit of this divergent element goes to infinity, we know
it must utilize steps to the right, x3+1 steps, to reach infinity. But with every x3+1 step to the right, we know that the Collatz tree splits right there into two branches, as you could also reach
that same value by a ÷2 step. Thus the divergent tree necessarily contains non-trivial branches.
These non-trivial branches have % membership LimSup > 0, so the overall divergent tree has % membership LimSup > 0. Thus the collection of all divergent trees put together has % membership LimSup >
0. Thus the long term fraction of hailstones which are divergent does not go to zero as n goes to infinity.
Which violates Terras Theorem! >>> Contradiction!
Thus if I can show my conjecture to be true about non-trivial branches holding their own, I'll have proven one half of the Collatz Conjecture!
But how do we prove that non-trivial branches hold their own and "wiggle"? This is now paramount! When I first realized the importance of this statement to the Collatz Conjecture, I set out with a
maniacal burst of enthusiasm to prove it.
... Instead it proved itself ... difficult.
Ultimately, who is surprised? This is still the Collatz Conjecture we're talking about. It's a notoriously very hard math problem.
My recent life has been dedicated (amongst other things) to a very active attempt to prove this interesting new version of the divergence half of the Collatz Conjecture. In the next chapter I'll walk
you through my main two plans of attack on the problem.
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Math 307: Introduction to Abstract Mathematics - Fall 2016
Instructor Contact and General Information
Instructor: Luís Finotti
Office: Ayres Hall 251
Phone: 974-1321 (don't leave messages! -- e-mail me if I don't answer!)
e-mail: lfinotti@utk.edu
Office Hours: MW 9-10 or by appointment.
Textbook: D. J. Velleman, "How to Prove It: A Structured Approach", 2dn Edition, Cambridge University Press, 2006.
Prerequisite: 142 or 148 (and consent from the Math Department).
Class Meeting Time: MWF 10:10am to 11:00am. (Section 001.)
Exams: Midterm 1 (Chapter 1): 09/07.
Midterm 2 (Chapter 2): 09/21.
Midterm 3 (Chapter 3): 10/12.
Midterm 4 (Chapter 4): [S:11/02:S] Postponed to 11/09.
Midterm 5 (Chapter 5): 11/21.
Final: 12/07.
Grade: 65% for Midterm Average (lowest score dropped) and 35% for the final.
Course Description and Information
Course Content
Math 307 is a basically a course on mathematical proofs. A proof is a series of logical steps based on predetermined assumptions to show that some statement is, beyond all doubt, true. Thus, there
are two main goals: to teach you how think in a logical and precise fashion, and to teach how to properly communicate your thoughts. Those are the "ingredients" of a proof.
Note that you also be graded on how well you write your proofs will affect your grade! A poorly written correct proof will not get full credit!
Thus, the topics of the course themselves play a somewhat secondary role in this course, and there are many difference possible choices. On the other hand, since these will be your first steps on
proofs, the topics should be basic enough so that your first proofs are as simple as possible. Therefore, you will be dealing at times with very basic mathematics, and will prove things you've
"known" to be true for a long time. But it is crucial that you do not lose sight of our real goal: do you know how to prove those basic facts? In fact, the truth is that you don't really know if
something is true until you see a proof of it! You might believe it to be true, based on someone else's word or empirical evidence, but only the proof brings certainty.
In any event, the topics to be covered in this course are: logic, set theory, relations and functions, induction and combinatorics. We will use also basic notions of real and integer numbers, but
these will be mostly assumed (without proofs).
Chapters and Topics
The goal would be to cover the following:
• Chapters 1 and 2: all sections, but these will be covered quickly and skipping some parts. These are sections in formal logic, which although crucial, I find better to be introduce in more
concrete settings as the need arises in the following chapters.
• Chapter 3: All sections, except 3.7.
• Chapter 4: All sections, except 4.5.
• Chapter 5: All sections, except 5.4.
• Chapter 6: All sections, except 6.5.
Other topics (and digressions) might also be squeezed in as time allows.
For the outcomes and problems (as well as videos) for each individual section, check Videos, Outcomes, Problems.
Homework Policy
Homework problems are posted below. As soon as we finish a section in class, you should start working on the problems from the section in the list. But, HW will not be collected or graded! (Also,
there are no quizzes.) The point of the HW is to learn and practice for the exams. In my opinion, doing the HW is one of the most important parts of the learning process, so even if it does not count
towards your grade, I recommend you take it very seriously.
Solutions to the HW will be posted Blackboard and you can bring your questions to class. In particular, I will try to set sometime to answer HW questions the day before each exam.
Also, you should make appointments for office hours having difficulties with the HW or the course in general! I will do my best to help you.
Piazza (Discussion Board)
We will use Piazza for online discussions. The advantage of Piazza (over other discussion boards) is that it allows us (or simply me) to use math symbols efficiently and with good looking results
(unlike Blackboard).
To enter math, you can use LaTeX code. (See the section on LaTeX below.) The only difference is that you must surround the math code with double dollar signs ($$) instead of single ones ($). Even if
you don't take advantage of this, I can use making it easier for you to read the answers.
You can access Piazza through the link on the left panel of Blackboard or directly here: https://piazza.com/utk/fall2016/math307/home. (There is also a link at the "Navigation" section on the top of
this page and on the Links section.)
To keep things organized, I've set up a few different folders/labels for our discussions:
• Chapters and Exams: Each chapter and exam has its own folder. Ask question related to each chapter or exam in the corresponding folder.
• Course Structure: Ask questions about the class, such as "how is the graded computed", "when is the final", etc. in this folder. (Please read the Syllabus first, though!)
• Computers: Ask questions about the usage of LaTeX, Piazza itself and Blackboard using this folder.
• Feedback: Give (possibly anonymous) feedback about the course using this folder.
• Other: In the unlikely event that your question/discussion doesn't fit in any of the above, please use this folder.
I urge you to use Piazza often for discussions! (This is specially true for Feedback!) If you are ever thinking of sending me an e-mail, think first if it could be posted there. That way my answer
might help others that have the same questions as you and will be always available to all. (Of course, if it is something personal (such as your grades), you should e-mail me instead.)
Note that you can post anonymously. (Just be careful to check the proper box!) But please don't post anonymously if you don't feel compelled to, as it would help me to know you, individually, much
Students can (and should!) reply to and comment on posts on Piazza. Discussion is encouraged here!
Also, please don't forget to choose the appropriate folder(s) (you can choose more than one, like a label) for your question. And make sure to choose between Question, Note or Poll.
When replying/commenting/contributing to a discussion, please do so in the appropriate place. If it is an answer to the question, use the Answer area. (Note: The answer area for students can be
edited by other students. The idea is to be a collaborative answer. Only one answer will be presented for students and one from the instructor. So, if you want to contribute to answer already posted,
just edited it.) You can also post a Follow Up discussion instead of (or besides) an answer. There can be multiple follow ups, but don't start a new one if it is the same discussion.
Also, you can send Private Messages in Piazza. So, if you have a math question not appropriate for the whole class, you can send me a private message instead of an e-mail. That way my reply can have
the math symbols nicely formatted.
Important: Make sure you set your "Notifications Settings" on Piazza to receive notifications for all posts: Click on the gear on the top right of the Piazza site, the choose "Account/Email Setting",
then "Edit Email Notifications" and then check "Automatically follow every question and note". Preferably, also set "Real Time" for both new and updates to questions and notes. I will consider a post
in Piazza official communication in this course, I will assume all have read every single post there!
You should receive an invitation to join our class in Piazza via your "@tennessee.edu" e-mail address before classes start. If you don't, you can sign up here: https://piazza.com/utk/fall2016/math307
. If you've register with a different e-mail (e.g., @vols.utk.edu) you do not need to register again, but you can consolidate your different e-mails (like @vols.utk.edu and @tennessee.edu) in Piazza,
so that it knows it is the same person. (Only if you want to! It is recommended but not required as long as you have access to our course there!) Just click on the gear icon on the top right of
Piazza, beside your name, and select "Account/Email Settings". Then, in "Other Emails" add the new ones.
I've recorded some videos for a course similar to this one taught online. The video has comments and solved problems from our textbook. Keep in mind that any comments about the course structure in
those videos should be disregarded, as the video were not made for this course!
You can access these videos here: Videos, Outcomes, Problems. In that page you can also see the expected outcomes and problems associated to each section of the book.
E-Mail Policy
I will assume you check your e-mail at least once a day, but preferably you should check your e-mail often. I will use your e-mail (given to me by the registrar's office) to make announcements. (If
that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in less than twenty four hours, and
it will be considered an official communication.
Moreover, you should receive e-mails when announcements are posted on Blackboard, or where there is a new post in Piazza. (Again, please subscribe to receive notifications in Piazza! Important
information my appear in those.)
Please, post all comments and suggestions regarding the course using Piazza. Usually these should be posted as Notes and put in the Feedback folder/label (and add other labels if relevant). These can
be posted anonymously (or not), just make sure to check the appropriate option. Others students and myself will be able to respond and comment. If you prefer to keep the conversation private (between
us), you can send me an e-mail, but then, of course, it won't be anonymous.
Legal Issues
All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:
Academic Integrity
The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing
unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has
adopted an Honor Statement.
All students should follow the Honor Statement: from Hilltopics, pg. 16:
Honor Statement
"An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I
will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity."
You should also be familiar with the Classroom Behavior Expectations.
We are in a honor system in this course!
Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.
Sexual Harassment and Discrimination
For Sexual Harassment, Sexual Assault and Discrimination information, please visit the Office of Equity and Diversity.
Campus Syllabus
Please, see also the Campus Syllabus.
Course Goals and Outcomes
Course Relevance
This course is clearly crucial to mathematicians, as our job is to prove things (and find things to be proved). But, this is a course also required for computer scientists, not only here at UT, but
virtually everywhere. The most obvious reason is that computer programs are written using formal logic. Another relevant connection is Artificial Intelligence, where you basically have to "teach" a
machine to come up with its own proofs.
Moreover, the skills taught in this course are universally important, and their benefits cannot be overstated! Everyone should be able to think clearly and logically to make proper choices in life,
and you should be able to communicate your thoughts clearly and concisely if you want to convince, teach, or explain your choices to someone else. In particular, Law Schools are often interested in
Math Majors, as the ability to think logically and clearly develop an argument is (or should be) the essence of a lawyer's job.
For teachers, it is important to help your students, from an early age, to be understand the importance of proofs! In my opinion, high school (at the latest!) students should be introduced to formal
proofs, even if in the most simple settings. This is important to foster analytic and critical thinking and to understand what mathematics is really about.
Course Value
The students will:
• develop analytic and critical thinking;
• broaden their problem solving techniques;
• learn how to concisely and precisely communicate arguments and ideas.
Student Learning Outcomes
At the end of the semester students should be able to:
• write coherent, concise and well-written proofs with proper language and terminology;
• use counting arguments for solving concrete numerical problems and as tools in abstract proofs;
• master standard proof techniques such as direct proofs, by contradiction or contrapositive, proofs by induction, proofs of and/or statements, proof of equivalencies, among others;
• master the terminology and notation of basic set theory (such as membership, containment, union, complement, partition, among others);
• master the terminology and notation of basic fucntion theory (such as injective/one-to-one, surjective/onto, bijective, invertible, etc.);
• understand and be familiar with examples of equivalency relations and its relation with partitions.
Study Guide
General Study Guide
Here are some comments on how to prepare for exams. To study, I recommend:
• Quickly review your notes and read the book. The most important is to review the definitions, Theorems, and how to compute things.
• I'd strongly recommend that you write these important facts in a different sheet of paper so that you don't have to browse your book every time you need to refresh your memory and so that you can
easily review them later.
• Do all HW problems! This is the most important thing! You can only learn by doing it.
• Don't be too quick to look for solutions (which are posted below). Sometimes it takes time to get a problem. Keep thinking about it, trying different approaches, look for similar problems/
examples in the book or notes (or videos!). This is an important part of the learning process. If you just look for solutions you might not learn enough. (See next item, though.)
• If you are stuck for a while on a problem, look at the solution. But, make sure to take notice of what you were missing. (Did you forget something? Is there an idea that you did not have before?)
Try to remember that. Moreover, redo the problem a few days later without looking at solutions, to make sure that you've assimilated the ideas.
• Look at all examples and solved problems (from book, class and videos!). If you have time, redo some of them (without looking at the solution).
• Do as many other problems as possible from the book.
• When you first start studying, you can look at the book or notes, but by the end, you should be able to do problems without looking.
• Look at problems from old exams, which will likely be posted in the individual sections below.
• If you have questions, bring them to class or post them on Piazza.
Midterm 1
I've just written (a preliminary version of) our Midterm 1. Here is some info about it (subject to change):
• It covers all sections of Chapter 1.
• It will be in class on Wednesday 09/07. (Usual room and time.)
• The exam has six questions, two worth 20 points (each has two parts worth 10 points) and four worth 15 points.
• The questions are very similar to HW problems, and in fact, two questions are HW problems.
• You can practice with the following problems (which might not cover all the topics you need to know):
Let me know if you have any questions. (Use the Piazza, if you can.)
Midterm 2
I've just written (a preliminary version of) our Midterm 2. Here is some info about it (subject to change):
• It covers all sections of Chapter 2.
• It will be in class on Wednesday 09/21. (Usual room and time.)
• The exam has six questions, two worth 20 points (each has two parts worth 10 points) and four worth 15 points.
• The questions are very similar to HW problems, and in fact, one question is a HW problem. (Although only one, the others are very similar.)
• You can expect the usual questions: analyzing logical forms (from English or set theory), intersection/union of families, indexed or not, logical equivalences (involving quantifiers), etc.
• Similar to the first midterm (e.g., Problem 4), there are a couple of question that should be very quick and easy, as long as you know the corresponding basic facts/definition, but since there is
so little to them, there is not much room for partial credit, so be careful.
• I would say this exam is a bit harder than our first, since the material is also. But there are, for those well prepared, at least 45 "easy" points. Also, in many of the other problems, would be
fairly easy to get some partial credit.
• You can practice with the following problems (which might not cover all the topics you need to know):
Let me know if you have any questions. (Use the Piazza, if you can.)
Midterm 3
I've just written (a preliminary version of) our Midterm 3. Here is some info about it (subject to change):
• It covers all sections of Chapter 3, except 3.7.
• It will be in class on Wednesday 10/12. (Usual room and time.)
• Feel free to bring questions to class on Monday (10/10). I will answer as many questions as I can that day.
• The exam has five questions, each worth 20 points, all proofs.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• You can expect the do all types of proofs we've worked on: contradiction/contrapositive (do you know how to negate statements?), proving statements with "and(s)" (possibly "if and only if" or
equality of sets), proving statements with "or(s)", using statements with "or(s)" (proof broken in cases), existence and uniqueness, etc.
• Partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you cannot finish the
proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• Some proofs have "parts", like cases or different statements that need to be proved. Sometimes some of these are hard, while others are easy. Make sure you try all parts (even if you cannot do
them all!), so that you can get the credit for the easier parts!
• Material from previous chapters, like families, indexed families and power sets do appear, as well as things like divisibility and odd/even (just like in the HW for this chapter).
• This might be the hardest of our exams, as writing proofs take some practice.
• You can practice with the following problems (which might not cover all the topics you need to know):
Let me know if you have any questions. (Use the Piazza, if you can.)
Make Up Midterm 3
I've just written (a preliminary version of) our Make Up Midterm 3. The description and study guide is virtually the same as for Midterm 3 above, except:
• It will be in class on Wednesday 10/19. (Usual room and time.)
• The exam has four questions, each worth 25 points, all proofs.
• The questions are very similar to HW problems, and in fact, one question is a HW problem. Two of the others come from examples from either book, class or video. The remaining one is similar to
previous HW problems.
• Besides all the studying suggestion above (for Midterm 3), I also suggest you try to redo Midterm 3. Look for ideas you've missed, and make sure you have good knowledge of the definitions and
proof techniques that were used.
Let me know if you have any questions. (Use the Piazza, if you can.)
Midterm 4
I've just written (a preliminary version of) our Midterm 4. Here is some info about it (subject to change):
• It covers all sections of Sections 4.1 to 4.4. (Section 4.6 is not in the exam.)
• It will be in class on Wednesday 11/09. (Usual room and time.)
• The exam has five questions, each worth 20 points, three of them are proofs, and two of those are HW problems.
• There is one question from 4.1, one from 4.2, one from 4.3 and two from 4.4.
• In one of the questions from 4.4 you are asked to find minimal/maximal elements, smallest/greatest elements, lower/upper bounds and least lower bound/greatest lower bound in a concrete example.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you
cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can practice with the following problems (which might not cover all the topics you need to know):
Let me know if you have any questions. (Use the Piazza, if you can.)
Midterm 5
I've just written (a preliminary version of) our Midterm 5. Here is some info about it (subject to change):
• It covers all sections of Sections 4.6 and 5.1 to 5.3.
• It will be in class on Monday 11/21. (Usual room and time.)
• The exam has four questions, each worth 25 points, all are proofs.
• The questions are very similar to HW problems, and in fact, two questions are HW problems. The others are very similar to HW problems.
• Like in the last exam, I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you
cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can practice with the following problems (which might not cover all the topics you need to know):
Let me know if you have any questions. (Use the Piazza, if you can.)
I've just written (a preliminary version of) our Final. Here is some info about it (subject to change):
• It will be on 12/07 (Wednesday), from 8am to 10am. (Usual room.)
• In principle it is comprehensive, covering all we've studied in this semester. On the other hand, emphasis will be given to Chapters 4, 5 and 6, as the previous sections are heavily used in these
• The exam has 8 questions, 4 worth 12 points, 4 worth 13 points. One from Chapter 3 (a problem from the book, involving families), two from Chapter 4 (one on ordering relations, one on equivalence
relations), two from Chapter 5 and three from Chapter 6.
• Note that there are three induction problems, worth 39 points in total.
• Note that it would be impossible to cover all topics from the course with only 8 questions, so many important things (on which you might work very hard) may not appear in the exam.
• I believe it covers all the "proof kinds" we've studied in Chapter 3.
• The questions are very similar to HW problems, and in fact, two questions are HW problems and other three are either examples done in class or in a video. The other three are very similar to HW
• Like in the last two exams, I will be generous with partial credit if you know your definitions, so make sure you do!
• Also, as before, partial credit (although sometimes small) will be given to choosing the appropriate proof technique (so make your approach clear!) and interpreting the notation. So, even if you
cannot finish the proof, write those down to make sure you get the partial credit for them. (This should also help you figure out the proof! That's why it grants you partial credit.)
• You will be graded on style! Please be neat, direct, clear and concise! State clearly your assumptions and, in parenthesis, your goals (as I've been doing in class).
• You can now do all problems from all the exam from past 307/504 courses. Of course you should focus on problems from Chapters 4 to 6. (Previous final exams had explicit material from Chapters 1
to 3.) You can practice with the following problems (which might not cover all the topics you need to know):
• We should have a review session before the exam. Please study before coming for the review, so that you can ask questions that would help you! Watch for an announcement (through Blackboard/
e-mail) soon.
Let me know if you have any questions. (Use the Piazza, if you can.)
This (LaTeX) is mostly irrelevant to our course! The only benefit for us is to help you post messages in Piazza with math symbols. So, feel free to skip the rest of this section.
LaTeX is the most used software to produce mathematics texts. It is quite powerful and the final result is, when properly used, outstanding! Virtually all professional math text you will ever see is
done with LaTeX, or one of its variants.
LaTeX is available for all platforms and freely available.
The problem is that it has a steep learning curve at first, but after the first difficulties are overcome, it is not bad at all.
One of the first difficulties one encounters is that it is not WYSIWYG ("what you see is what you get"). It resembles a programming language: you first type some code and then this code is processed
to produce a nice document (a non-editable PDF file, for example). Thus, one has to learn how to "code" in LaTeX, but this brings many benefits.
I recommend that anyone with any serious interest in producing math texts to learn it! On the other hand, I don't expect all of you to do so. But note that there are processors that can make it
"easier" to create LaTeX documents, by making it "point-and-click" and (somewhat) WYSIWYG.
Here are some that you can use online (no need to install anything and files are available online, but need to register):
If you want to install LaTeX in your computer (so that you don't need an Internet connection), check here.
A few resources:
• My web pages for Math 504 (a course very similar to this one) from Summer 2014 and Summer 2016. Includes all exams with solutions (in the section "Handouts").
• My web page for Math 307 - Introduction to Abstract Mathematics (Honors) - Spring 2013. Includes all midterms and final with solutions (in the section "Exams").
• My web pages for Math 300 - Fall 2009, Math 300 - Fall 2008. These courses had a different textbook, but still it might be useful to look at old exams.
• Carl Wagner has some interesting resources:
□ Logical Notation and Terminology: Here he discusses the symbols we use, as well as the notation of complement of a set. He also notes the difference between $\rightarrow$ and $\Rightarrow$
and $\leftrightarrow$ and $\Leftrightarrow$.
□ Logical Equivalences and Set Theoretical Identities: Here he lists the logical equivalences that we will learn and their set theoretical analogues. (Note I usually use $\sim$ instead of $\
Leftrightarrow$ for logical equivalence.)
□ Some Paradoxes in Propositional Logic: Here he shows some equivalences that often seem counter intuitive.
• Services for Current Students and MyUTK (registration, view your grades, etc.).
• Academic Calendars, including dates for add and drops, other deadlines, final exam dates, etc.
• Office of Equity and Diversity (includes sexual harassment and discrimination).
Solutions to Selected HW Problems
Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the "answer". If yours do not match mine, you can
try to figure out again. (Also, read the disclaimer below!) You can come to office hours or ask in class if you want explanations for the answers. Be careful that just because our "answers" were the
same, it doesn't mean that you solved the problem correctly (it might have been a "fortunate" coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat
detailed solutions to the harder problems, though.
Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when
preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.
Solutions to Selected HW Problems (Click on "Refresh" or "Reload" if you don't see the changes!)
• 11/29 -- 3:50pm Solutions for Chapter 6 posted.
• 11/16 -- 3:35pm Solutions for Sections 4.6 and Chapter 5 posted.
• 11/01 -- 9:20am Solutions for Sections 4.1 to 4.4 posted.
• 09/30 -- 3:30pm Solutions for Chapter 3 posted.
• 09/13 -- 5:15pm Solutions for Chapter 2 posted.
• 08/26 -- 3:10pm Solutions for Chapter 1 posted.
Homework Problems
Section 1.1: 1, 3, 6, 7.
Section 1.2: 2, 12.
Section 1.3: 2, 4, 6, 8.
Section 1.4: 2, 6, 7, 9, 11.
Section 1.5: 3, 4, 5, 9.
Section 2.1: 3, 5, 6.
Section 2.2: 2, 5, 7, 10.
Section 2.3: 2, 5, 6, 9, 12.
Section 3.1: 2, 3, 6, 10, 15, 16.
Section 3.2: 2, 4, 7, 9, 12.
Section 3.3: 2, 4, 10, 15, 18, 21.
Section 3.4: 3, 8, 10, 16, 24.
Section 3.5: 3, 8, 9, 13, 17, 21, 24.
Section 3.6: 2, 3, 7, 10.
Section 4.1: 3, 7, 9, 10.
Section 4.2: 2, 3, 5, 6(b), 8.
Section 4.3: 2, 4, 9, 12, 14, 16, 21.
Section 4.4: 2, 3, 6, 9, 15, 20, 22.
Section 4.6: 4, 8, 13, 20, 22.
Section 5.1: 9, 11, 13, 17.
Section 5.2: 3, 6, 11, 8, 9, 18.
Section 5.3: 4, 6, 10, 12.
Section 6.1: 2, 4, 9, 16.
Section 6.2: 3, 5, 6 (use the triangle inequality from Problem 12(c) of section 3.5; you don't need to do that exercise, just refer to it), 10.
Section 6.3: 2, 5, 9, 12, 16.
Section 6.4: 4, 6, 7, 19.
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Some of the most fundamental questions in science are at the cutting edge of modern mathematical physics. They address the very structure and origin of the universe, the nature of the constituents of
all matter and their interactions and the mathematical structures necessary for a quantitative formulation of the fundamental laws of nature. Two concrete goals of current focus in fundamental
physics are to find a quantum theory of gravity that avoids the inconsistencies that arise from trying to reconcile Einstein's general theory of relativity with quantum mechanics and to find a
unified theory which encompasses all of the forces of nature and describes all of the particles which are subject to those forces.
Superstring theory is a promising candidate for a physical theory that could simultaneously achieve both of these goals. String theory is a physical model which is postulated to describe fundamental
interactions at exceedingly small distances where quantum mechanical fluctuations of the geometry of spacetime would become important. As a physical dynamical system, it is still not completely
understood. It is clear that the full power of superstring theory will only be realized once significant progress has been made in understanding its mathematical and dynamical structure. This will
involve the development of new mathematics and will be an important frontier area for both mathematics and physics in the foreseeable future.
PIMS Distinguished Chair
Ashoke Sen (Harish-Chandra Research Institute), will give a series of lectures during July and August 2003 at UBC.
Ashoke Sen's significant contributions to string theory include his early work on string field theory, duality symmetries, and black hole solutions. This led him to the study of strong-weak coupling
duality of supersymmetric gauge theories. His proof of the existence of the conjectured bound states required by those dualities is now one of the classic works in the field. This work has had
enormous influence and is usually credited as the start of the "second superstring revolution" --- a shift of perspective which the field is still undergoing.
In 1998, Professor Sen initiated the study of non-supersymmetric states in string theory. Once again, this has proven to be a very fruitful study. Most recently, he has returned to the study of
string field theory in the context of tachyon condensation on D-branes. This has led to his current work on the cosmological consequences of tachyon condensation.
The CRG will have another Distinguished Chair in 2003 and two more in 2004. These chairs will visit the group for at least one month and give a minicourse of lectures
CRG Leaders:
U. Alberta:
• B. Campbell
• V. Frolov
• D. Page (Physics)
• T. Gannon
• E. Woolgar (Math)
• G. Semenoff
• M. Rozali
• M. Van Raamsdonk
• K. Schleich
• D. Witt
• M. Choptuik
• W. Unruh (Physics and Astronomy)
• J. Bryan
• K. Behrend (Math).
U. Lethbridge:
Perimeter Institute:
U. Toronto:
U. Washington:
Asia Pacific Center for Theoretical Physics, Korea:
PIMS Postdoctoral Fellows
• Jian-Jun Xu, PIMS PDF at SFU
• Jianying Zhang, PIMS/MITACS PDF at UBC
This CRG will include another PDF in 2003 and two more in 2004.
Graduate Students
K. Chu, P. de Boer, M. Laidlaw, J. Gardezi, B. Sussman, B. Ramadanovic, D. Young, K. Furuuchi, R. Fazio.
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The Best Number Line Graph Calculator with Steps 100% Free
Introduction to Number Line Graph Calculator:
Number line graph calculator is an online tool that helps you evaluate the linear equation to display on a number line system.
Our number line inequality calculator can be used to make an inequality relationalship between two algebraic expressions that are represented on a graph.
What is Graphing Inequalities on a Number Line?
Graphing inequality on the number line is a method that is used to solve the linear inequality equation (usually linear equation but has inequality in it) to get the point value, represented on
number line. It is same as a linear equation except for the inequality sign in it.
Rules Followed by Inequality Number Line Calculator:
To represent your point value on a number line graph, you have to follow the rules. The number line graph calculator uses the following rules to calculate the number line graph:
• If the y value is less than the x as y < x value, then shade the inside part of line graph.
• If the y value is greater than c values as y > x then the test fails and you shade the graph below the number line.
Evaluation Process of Number Line Inequality Calculator:
Inequality number line calculator uses an easy-to-understandable method to solve the linear differential equation.
Suppose you have a linear inequality equation with two variables. For its solution, you do not need to worry about its inequality sign, our graphing inequalities on a number line calculator first
gives the solution of a linear equation only irrespective of the inequality sign.
First, inequality line graph calculator checks the nature of the given linear equation ax + by + c = 0, and then it keeps one variable on one side of the equation and the second variable on the other
side as by = ax + c. Now it substitutes different values of x to get the cross-pounding value of y.
After getting some point values of x and y, it uses these point values and fixes these particular points on the number line graph and joins these points with the help of a line on the graph.
Let's observe an example of linear inequality equation whose solution comes from graph inequality on number line calculator. However, it helps you to understand the theory of its evaluation process
Example of Number Line Graph:
An example of graphing inequality linear equation is given below to let you know about the manual calculation of solving such problems and you can understand the results of our number line graph
Solve the following using the number line graph equation,
$$ 3x + 2y \le 6 $$
Suppose some point values of x and add them to in equation to get the y value.
$$ x \;=\; -5\; and\; get\; y \;=\; 5 $$
$$ x \;=\; -2\; and\; the\; solution\; gives\; y \;=\; -2 $$
$$ x \;=\; 2\; and\; y \;=\; 0\; and\; so\; on $$
$$ 3(-5) + 2(5) \le 6 $$
$$ \begin{matrix} (-5, 5) & -15 + 10 \le 6\\ \end{matrix} $$
$$ -5 \le 6 $$
$$ 3(-2) + 2(-2) \le 6 $$
$$ \begin{matrix} (-2, -2) & -6 + (-4) \le 6 \\ \end{matrix} $$
$$ -10 \le 6 $$
$$ 3(2) + 2(0) \le 6 $$
$$ \begin{matrix} (2, 0) & 6 + 0 \le 6 \\ \end{matrix} $$
$$ 6 \le 6 $$
Graphical representation of x and y values on a number line graph.
How to Use Number Line Graph Calculator?
Inequality number line calculator has a user friendly design that allows you to easily calculate the linear equation in less than a minute.
You should follow some of our guidelines for using the graph inequalities on a number line calculator for the evaluation of the inequality linear equation. These guidelines are given as:
1. Enter the linear equation (ax^2+ by + c < 0) in the input box of inequality line graph calculator.
2. Click the “Calculate” button to get the desired result of your given inequality of linear equation.
3. If you want to try our number line inequality calculator then you can use the load example option.
4. Check out your input equation again before clicking the calculate button so that you do not find any error in calculation.
5. Click on “Recalculate” button to get a new page for solving more inequality of linear equation problem.
Result of Graphing Inequalities on a Number Line Calculator:
Number line graph calculator provides you the solution of linear inequality equation problem when you give it an input. The result contain as:
• Result option gives you the solution for graphing inequalities on a number-line question.
• Possible step provides you all the steps of the graphing inequalities on a number line problem.
Advantages of Inequality Line Graph Calculator:
Number line inequality calculator will give you tons of advantages whenever you use it to calculate the linear inequality equation. The advantages are:
• Our inequality number line calculator saves your time from doing lengthy calculations of graphing inequalities on number line problem.
• Graphing inequalities on a number line calculator is a free-of-cost tool so you can use it to find some point values between algebraic expressions.
• It is a versatile tool that allows you to solve various types of inequality linear equations in no time.
• You can use this graph inequality on number line calculator for practicing so you get familiarity with the graphing inequalities on number line questions.
• It is a reliable tool that provides you accurate solutions every time when used to calculate the given inequality linear equation.
• Number line graph calculator provides a solution of linear equation in steps with no or minimal error so that you get a better understanding.
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The Year 11 Course – Why It’s Important and What You Need to Know
Afrina Islam
Why have I decided to dedicate an entire article to the year 11 course you ask? Well, it’s because as someone who sat the HSC last year, I saw first-hand just how examinable the year 11 topics are!
As you would know, ext.1 mathematics is one of the few subjects where the year 11 course content is explicitly tested in the HSC (not just assumed knowledge needed to answer some questions). And so
it’s of the utmost importance that you don’t overlook this section of the course in your revision.
Last year’s HSC exam consisted of over 50% of the marks going to year 11 topics??!
Now it can be argued that the year 11 course is in some ways more difficult than year 12 – it includes topics such as combinatorics, graphing and polynomials – all of which are some of the more
challenging concepts in the maths ext. 1 course. In this blog I’ll be giving a brief rundown of these topics and some tricks when answering questions.
• Polynomial division (essentially a form of long division)
• The factor theorem
• The remainder theorem
• Relationship between roots and coefficients
• Multiple roots
• The degree of the polynomial is determined by the power of the leading term (the highest power of the pronumeral)
• The degree of the polynomial tells you how many roots it can have (e.g. degree 5 means it has 5 roots), keeping in mind these could be multiple roots (e.g. a double root)
• The polynomial can be written as the divisor x the quotient + remainder (after doing polynomial division). This can be written as P(x) = divisor (what you were dividing by) times the quotient
(the answer to the division) plus the remainder
• Remainder theorem: if a polynomial P(x) is divided by (x – a) and the remainder R is a constant, then R = P(a). This means if I sub a into the polynomial, I will be left with the value of R.
(remember to rearrange the divisor into the form (x – a).
• Factor theorem: if a polynomial P(x) can be divided by (x – a) with a remainder of 0, then (x – a) is a factor of P(x) and P(a) = 0.
Topic breakdown and how I like to do it:
• Reciprocal functions (i.e. graphing 1/f(x), where f(x) is some function)
o Begin by drawing a vertical asymptote down any x-intercepts of the graph f(x) (this is because 1/f(x) cannot exist when f(x), the denominator, equals 0)
o Re-label the y-intercept as 1/y-intercept (e.g. 2 will become ½)
o Think about what the graph approaches on either side of the asymptote – you will essentially form a hyperbola shape with linear functions and a flat parabola shape between two vertical asymptotes
(as f(x) à ∞, 1/f(x) à 0 and vice versa)
o Label any known points when drawing the graph (e.g. 1/1 = 1, so the point 1 stays the same)
o Remember these graphs do not exist where f(x) is < 0 (you can ignore this part of the function/anything below the x-axis)
o Keep the x-intercept the same
o The square root of 0 is 0 and the square root of 1 is 1 (these points stay the same)
o Between 0 and 1, the graph will be slightly above the original graph
o Beyond 1 the graph will increase much less steeply
o If asked to sketch y2 = f(x), remember to reflect the graph in the x-axis as this means y = ±f(x)
o Note: remember to put a dot at 0 even if the graph becomes discontinuous as a square root graph exists at 0.
o These graphs have a sharp point and often have a V shape
o Remember absolute value signs will make any negative values positive
o Reflect the negative part of the graph in the x-axis, keeping the sharp point in the middle
o For y = |f(x)|, reflect everything below the x-axis above
o For y = f(|x|), reflect everything on the right of the y-axis to the left
o For y = |f(|x|)|, first reflect everything below the x-axis above, and then reflect anything on the right of the y-axis to the left
• The fundamental counting principle
• Pigeonhole principle
• Permutations
• Combinations
• Counting techniques in probability
• Pascal’s triangle
• Combinatorics is undoubtably one of the most difficult topics in extension 1 maths with questions ranging in difficulty from simple to complex
• The pigeonhole principle states that if n items are put into m container s, with n>m, then at least one container must contain more than one item.
• Permutations are used for ordered selections/arrangements (e.g. words, numbers, etc.); this can include arranging with restrictions or in a circle (remember to use (n – 1)! and to divide by the
no. of repeated items when some elements are identical)
• Combinations are used when order is not important/a selection (e.g. committees, selections, groups, etc.)
• Pascal’s triangle can be used to find the coefficient of terms in the expansion of a binomial and is as follows:
• Perms and combs can be used for probability problems involving real-world applications such as a deck of cards.
• If you are still struggling with this topic, below is a very useful link to the Khan Academy unit with video tutorials and practice questions I highly recommend for consolidating understanding
and some extra practice:
All in all, don’t forget to revise your year 11 content! I hope these tips have been helpful!
Commenting has been turned off.
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Average Calculator
* SD = standard deviation
Average calculation
The average (arithmetic mean) is equal to the sum of the n numbers divided by n:
The average of 1,2,5 is:
Weighted average calculator
Weighted average calculation
The weighted average (x) is equal to the sum of the product of the weight (w[i]) times the data number (x[i]) divided by the sum of the weights:
Find the weighted average of class grades (with equal weight) 70,70,80,80,80,90:
Since the weight of all grades are equal, we can calculate these grades with simple average or we can count how many times each grade appears and use weighted average.
= 78.33333
See also
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Comparison of 5th grade textbooks
As the author’s math-ignorant daughter and a full-fledged graduate of the
California education system is applying to PhD programs in comparative lit-
erature, her revengeful parent is compiling these notes in the aforementioned
The subjects for the comparison are Singapore and California, 5th grade
Primary Mathematics 5A, US Edition, Federal Publications, Ministry of
Education, Singapore , and
Mathematics, book 5, California Edition, Houghton Mifflin Company.
Figures 1 and 2 reproduce page 37 from Singapore and page 332 from California
respectively, representing the same topic chosen almost randomly.
In Figure 1, a boy from Singapore tells us a short story requiring the addition
of 1/3 and 1/2. The square prompts invite us to feed in the answer 5/6. This
was not hard, but the question remains: how did the boy guess to replace the
fractions with respectively 2/6 and 3/6? Is there a secret, a trick? Is he a
genius? No, the boy explains, it was a stroke of luck: as the picture shows,
the cake was precut into 6 equal parts, of which Ann took 2 and her brother 3.
Now the general idea is revealed: the problem of adding 1/3 and 1/2 looked
hard because the denominators were different, but using equivalent fractions
with the same denominator makes the problem easy.
In the next two pages, a girl and the boy will help us examine equivalent
fractions in 3 more addition and 3 subtraction examples with common de-
nominators climbing up to 30. These will be followed by 3 addition and 3
subtraction exercises to be solved on our own , and respectively — by two
pointers to homework sets from Workbook 5A. A page of Practice with 12
more exercises and several word problems will conclude unit 2. Addition and
Subtraction of Unlike Fractions .
│ │
│Addition and Subtraction of Unlike │
│Fractions │
│ │
│Ann ate 1/3 of a cake. │
│ │
│Her brother ate 1/2 of the same cake. │
│ │
│What fraction of the cake did they eat altogether?│
│ │
│They ate │
│ │
│They are called unlike fractions. │
│ │
│They are called like fractions . │
│ │
││We can change unlike fractions to like fractions││
││using equivalent fractions: ││
Figure 1
We note the precise and economical character of the text: not a sign is
On the contrary, the multicolor Figure 2 from California asks for an editor’s
red pen.
│Add Fractions With │
│Unlike Denominators │
│ │
│You will learn how to add fractions which have │
│different denominators. │
│ │
│┌────────────────────┐ │
││Review Vocabulary │ │
││equivalent fractions│ │
│└────────────────────┘ │
│ │
│Learn About It │
│ │
│Most of Earth’s surface is covered by water. │
│The Pacific Ocean covers about 1/3 of Earth’s │
│surface, and the Atlantic Ocean covers │
│about 1/5. What fractional part of Earth’s │
│surface is covered by these two oceans? │
│ │
│Add. │
│ │
││Find ││
││ ││
││Use the product of the denominators to write ││
││equivalent fractions with a common denominator. ││
││Step 1.Use number lines to │Step 2.Use the product of│Step 3.Rewrite ││
││model the fractions. Notice│the denominators to write│the problem ││
││that the fractions are │equivalent fractions with│using fractions││
││different unit lengths │like denominators. │Then add. ││
││To add fractions 1/3 and │┌──────────────────────┐ │ ││
││1/5, you need to first find││Think: Multiply by │ │ ││
││equivalent fractions with ││the denominator of the│ │ ││
││like denominators. ││other fraction. │ │ ││
││ │└──────────────────────┘ │ ││
Figure 2
The opening promise “You will learn how to add fractions which have dif-
ferent denominators” only reiterates the title (or does it? — we will come to
this later) and can be safely omitted. The satellite picture of the Earth is of
no use and better be dropped too. The scientifically true fact that “Most of
Earth’s surface is covered by water” does not really follow from 8/15 > 1/2
and, being presently irrelevant, should be removed as well .
“Find 1/3+1/5” is a perfect mathematical formalization of the problem and
stays. A cyborg’s thought process “Add. 1/3 + 1/5 = n” reads “a third and
a fifth add up to n” and goes, since it refers to an n which has not been
introduced (nor is going to show up later).
My limited English does not allow me to “Notice that the fractions are
different unit lengths”. Fortunately the entire Step 1 is redundant : drawing
the fractions on the number line does not facilitate the addition.
“Use the product of the denominators to write equivalent fractions with a
common denominator” explains the plan perfectly and leaves no reason to
repeat it in Step 2. Likewise, the instruction “Rewrite the problem using
fractions. Then add” in Step 3 adds nothing new after “Find 1/3+1/5”.
Removing it also helps one to realize that there is no need to chop the solution
into “steps”.
The result of our editing, shown in Figure 3, matches Figure 1 in clarity and
simplicity. Yet something still displeases the ear, doesn’t it? Who the heck
are these unlike denominators?
In Singapore (and most of the world), unlike fractions have different denom-
inators. Respectively, like fractions have equal denominators and are in this
sense similar, or “friendly” (as some teachers put it), “speaking the same lan-
guage” of sixths or fifteenths. Like fractions are not necessarily equal, so the
word comes handy. Embarrassingly, in California, the scholarly term unlike
denominators stands simply for different ones, so that like means nothing but
the same.
One can deepen the comparison by noting the variance in the methods of
addition of fractions in Singapore and California: the mental scan of equiva-
lent fractions until they become ”friendly” often yields smaller denominators
than the product routine. In fact the next Lesson in California introduces
Least Common Denominators and uses prime factorization, while the Singapore
math program postpones studying prime factorization until grade 7. One
may debate if this makes California ultimately more advanced, or argue that
in practice the method in Singapore is just as efficient, or probe educational
advantages of either approach.
One may further discuss how wise it is to fake scientific applications and
pretend doing algebra, or try to guess the consequences of replacing ideas with
algorithmic ”steps”. One may wonder what role is left to thinking when the
command think is used as an euphemism for do, or why Singapore students
don’t get a separate subtraction unit while California students need it.
│ │
│Add Fractions With │
│Unlike Denominators │
│ │
│The Pacific Ocean covers about 1/3 of Earth’s │
│surface, and the Atlantic Ocean covers │
│about 1/5. What fractional part of Earth’s │
│surface is covered by these two oceans? │
│ │
│Find │
│ │
│Use the product of the denominators to write │
│equivalent fractions with a common denominator.│
│ │
│┌───────────────────────┬┐ │
││Multiply by ││ │
││the denominator of the ││ │
││other fraction. ││ │
│└───────────────────────┴┘ │
Figure 3
All these subtleties are entirely beside the point, which is: California is
poorly written, period. The book is on the list of instructional materials
adopted by the California Department of Education in 2001 and features links
to California Math Standards pagewise, yet it is grossly redundant, full of
irrelevant details, misleading explanations, confusing comments, distracting
pictures, embarrassing mistakes.
Dear fellow mathematicians,
On those rare occasions when you are given the role of Content Reviewer of
a school textbook, please — may it even be the last sum of cash you receive
for such services — let common sense be your guide and the red pen your
I need your courage and sacrifice: my son has just entered the California
education system.
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OneClass: Identify the population and the sample. 1. 2. 3. Determine w
Identify the population and the sample.
Determine whether the sample is biased or unbiased. Explain.
4. A restaurant would like to evaluate the efficiency of its workers. So observe the workers during the second shift.
5. At the end of the year, a math teacher decides to survey his students to determine what they liked best about his class. He users his first-period class as his sample.
6. An airport randomly inspects every 8'' suitcase that goes through security.
7. Mark would like to determine their exercise habits of students in his school. He surveys people on his frack team.
8. To determine how many students plan to attend the school musical, Kaitlyn surveys 100 random people in the hallway.
9. To track the movement of sharks along the east coast, one shark is chosen at random and tagged.
10. To determine how many homes in his neighborhood plan to hand out candy on Halloween, Jack chooses one street at random and surveys each household on that street.
11. To check lightbulbs for defects, 500 random bulbs from a certain production line are checked.
12. Suppose you would like to determine how many students have a cell phone at your school.
Given an example of a biased sample and an unbiased sample.
Unlock all answers
Get 1 free homework help answer.
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What have Mathematicians Done for Us?
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• Downloads
• Extra Reading
Mathematics has played a vital role in the development of human civilisation, and is the foundation of much of modern technology and popular culture. However, the achievements of mathematics and
mathematicians are often unknown or misunderstood. The contribution of mathematicians over the centuries will be celebrated, showing how mathematical ideas have huge relevance today – varying between
Maxwell and the mobile phone, Florence Nightingale and modern statistics, Pythagoras and the development of music, Euclid and art, Euler and Facebook, and Cayley and Google. Even basic mathematics
can make a profound difference to our lives.
Download Text
11 October 2016
What have Mathematicians Done for Us?
Professor Chris Budd OBE
On the whole, mathematics and mathematicians has a rather bad image with the general public, the media, and (sadly) policy makers. It is often thought of as being remote, inaccessible, and of no
possible use to anyone. Similarly, mathematicians are regarded as cold, unemotional, mad, male nerds. People seem so frightened of mathematics that in a recent event, an aircraft was delayed because
someone on it saw a Professor working on a differential equation and alerted the cabin thinking that he must be a terrorist http://www.bbc.co.uk/news/world-us-canada-36240523.
This caricature of mathematics, and of mathematicians, is not only false: it is grotesquely untrue. In contrast to its perceived uselessness, mathematics has played an essential role in the
development of civilization, and lies at the heart of all modern technology. Things we now take for granted: the mobile phone, the internet, credit cards, satellite navigation, radio and TV, weather
forecasting, radar, train scheduling, and even microwave cooking. All these could not function at all without the application of clever and careful mathematical algorithms. It is these aspects of
mathematics that I want to explore not just in this lecture, but in all of the lectures in this series.
Part of the problem is that few people really know what mathematics is, or what it can do, beyond the use of simple arithmetic which terrified them at schools, and is now largely done by calculators
(which has led some people to claim that mathematics as a subject no longer exists). This confusion even extends to mathematicians themselves who tend to flounder around when trying to define their
own subject. (A glance at the Wikipedia page on mathematics does not help when trying to define it). One definition that I like is that mathematics is all about patterns which link different objects
together. Another is that maths is what mathematicians do. (Which I personally find very unsatisfactory). But my favourite definition (by far) is that maths is the subject in which we can write down,
and make sense of, statements like this:
Isn’t that wonderful! This beautiful and unexpected formula links , the ratio of the circumference of a circle to its diameter, to the odd numbers. It was derived by the Indian mathematician Madhava
of Sangamagrama in the 14th Century, and later rediscovered in the West by Leibniz and Gregory at the dawn of the invention of calculus. Arguably, the truth inherent in this formula is eternal.
Anyone not stunned by it has (in my opinion) no soul. As we shall see, formulae such as this play a vital role in modern technology, as does the number itself. However, it is not obvious how to go
from such seemingly abstract works of beauty to the technology of the modern world, nor how to use maths to unravel the secrets of the universe. This is the topic I want to explore in this lecture,
through a series of carefully chosen examples which I will develop in the rest of this lecture series.
However, it is not maths which has made the modern world: it is mathematicians. School maths text books usually display it as a dry, logical subject, with no human content. They frequently ignore the
people that created maths in the first place and the stories behind them. This led to one school child I was talking to saying “I don’t do maths, because I’m a people person!” It might possibly
surprise them to know that I’m a people person myself and that (with possibly one or two exceptions) mathematicians are people too. In fact mathematics is full of examples of exceptional people whose
achievements really have changed the world that we live in. Sadly most of these, except possibly Archimedes, Isaac Newton, and Albert Einstein, are unknown to the public or if they are known are not
recognized as being mathematicians. Leonhard Euler, Carl Friedrich Gauss, William Kelvin and Évariste Galois, are all giants of mathematics, but who in the general public knows about them?
One of the exercises I often do when talking to school children is to put up a picture of a mathematician who I can guarantee has changed all of their lives, and for them to tell me who it is. The
picture I put up is that of James Clerk Maxwell. It was Maxwell, by mathematically unifying the three subjects of electricity, magnetism and optics, discovered (purely by mathematical reasoning) the
existence of radio waves and thus ushered in the modern world. Without radio waves we wouldn't have mobile phones, WiFi, the Internet, computers, radar, Google, Sat Nav or any of the paraphernalia of
the modern world. As I said, Maxwell has utterly changed the world we live in today, but no one seems to know who he is. (Maxwell was in many ways a very interesting person, and did many other things
besides, including inventing colour photography, and deducing the structure of the rings of Saturn). Another exercise is to ask the same group to name a female mathematician who has changed the world
and who they all know. Usually I am treated with the name of a TV celebrity. Very rarely does someone come up with the name of Florence Nightingale! But it was Nightingale who basically invented the
subjects of medical statistics and the graphical representation of data. Indeed it can be argued that she was the forerunner of the Big Data Revolution that I will talk about in my next lecture.
I think that the real problem that (applied) mathematics, and the (applied) mathematician, faces is that like the air we breathe, it is vital for all that we do, but it is invisible and easy to
ignore. Hopefully this talk will make the role of mathematics a little more visible. Ultimately I want to address the issues raised in quote from Edward David, former head of Exxon Research and
Development, who said, in a report to the US Government:
“Too few people recognize that the high technology so celebrated today is essentially a mathematical technology”
I will do this by showing how mathematical technology really works.
The mathematical process, or how does applied mathematics work?
My job title is Professor of Applied Mathematics. But what does this mean? One view of mathematics (sadly believed by many of my colleagues) is that there is a factory called Pure Mathematics, which
creates maths, as an abstract concept unconnected to the real world. The job of the applied mathematician is to then take this and to find real world situations where it might be useful. I personally
believe this to be at best an oversimplification, and at worst wrong. Applying mathematics to the real world is hard, and nearly always you have to invent new maths to solve practical problems. This
mathematics, and the related mathematical tools, can then be abstracted leading to the discovery of new questions, patterns, structure and ultimately theorems. The wonderful thing is that these
theorems can then be used to help solve many other problems, which can look very different. Great mathematicians such are Carl Friedrich Gauss, Emmy Noether, Leonhard Euler, Bernhard Riemann, David
Hilbert and John von Neumann certainly all had this view of mathematics, seeing no distinction at all between what is ‘pure’ and ‘applied’. Thus (applied) mathematics can be seen as a way of finding
a bridge, which transfers ideas from one very different area to another, and creating new ideas along the way. A classical example of this is calculus, which had its roots in the need to solve
problems in celestial mechanics, and now in its refined and abstract form gives us insights into nearly every problem in the physical, biological and even social sciences. Another supreme example is
Euler’s fabulous theorem:
e^(i π)+1=0
Not only is this (as voted by the majority of the world’s mathematicians) the most beautiful formula in mathematics, it is also the central formula for any problem involving waves or rotational
motion. Whole books have been written just about this one formula [4].
Now let’s see how this all works by looking at a brief history of some of the ways in which maths has made the modern world. To do this I have chosen a series of mathematical achievements, which have
all impacted the world in a number of ways, starting from the ancients and moving up through the centuries. I hope that by doing this we will see both how maths made the modern world and how the
making of the modern world has stimulated much of modern mathematics.
Early maths, Quadratic equations and the tax man
We believe that early humans counted on their fingers. This led directly to the concept of the natural numbers 1,2,3,4,... From these we have built the rest of mathematics, as we now know it. Note
that I have used the word built. Maths is above all a highly creative subject!. The natural numbers when extended by the negative numbers and 0, give us the integers. When ratios are taken of these
such as 2/3, -3/4, or 7/11 we get the rationals, and when you take limits of sequences of rationals you get real numbers such as etc. Problems which are solved in terms of integers (or perhaps
rationals) are often called discrete, and those which involve real numbers are often called continuous.
The original problems in mathematics were all discrete such as ‘I have 6 cows and I sell 4, how many do I now have’. One of the first professions to look into the solution of such problems was,
inevitably the tax man. Whether or not this is the first example, it is certainly one of the first recorded examples, as we might expect from the tax profession. The earliest medium in which such
mathematics was recorded was cuneiform writing on clay tablets, and it was written down by Babylonian scribes. Many examples of such tablets can be found in the British Museum, and they give a
fascinating insight into the impact of mathematics on the ancient world, and in particular the growth of the early city state (and the large armies associated with it!). Discrete problems continue to
be very important in modern technology, and lie at the heart of modern digital technology, which uses natural numbers to represent music, pictures, and even films and exploits properties of the
integers to represent, store, manipulate and transmit these in an error free manner.
However, the Babylonians rapidly found that not all of the problems of interest to them were discrete. A number of these tablets concentrate, instead, on the continuous problem of the) calculation of
areas. This was again mainly for the benefit of the taxman, who would be assessing the size of fields to determine the level of tax for each, which was proportional to the area of the field. One
question they posed was what by proportion should the size of a field increase by if its area was to then double. To find the answer to this problem requires solving the quadratic equation .
Remarkably the Babylonians were able to fund a very good approximation to the solution (x = 1.41421356237…) of this equation. This and many other solutions of different quadratic equations (all of
value to the tax man) could be found in extensive tables, which were the forerunners of the navigational tables we will describe shortly.
Later on, continuous problems became more important as a way of describing the real world. Continuous problems still have this importance, indeed the calculus developed by Newton and Leibnitz in the
17th Century is based on them. Nature abounds with lovely examples of patterns and order, all of which much have inspired in our earliest ancestors the feeling that there was some structure in the
universe which could perhaps be explained. The ideas of calculus can be found in the order of the waves on the sea, the motion of the stars and planets, the regularity of the seasons, the shape of
clouds, the geometry of crystals, the arc of a rainbow, the hexagons in a beehive, the (fractal) shape of a lightning bolt, and much more. Calculus continues to be the best tool we have to understand
the modern world and to predict the future.
Maths for recreation, and why this can be serious
Maths isn’t always thought of as fun, and it certainly isn’t usually taught that way. However mathematics actually lies behind many areas of recreation and culture and it enriches our lives by doing
so, even if you are not a mathematician, or even realize that you are doing maths.
Mathematical puzzles as a form of recreation have been around almost as long as maths itself, see [5] for lots of great examples of this. Indeed some of the deepest problems in maths were originally
posed as puzzles. A good example of this being the problem of ‘doubling the cube’ (i.e. finding a cube which has twice the volume of an original one), which it is claimed was posed by the Oracle of
Delphi. Another example of an early recreational puzzle is the maze. The earliest maze (technically a labyrinth) recorded in literature can be found in the celebrated story of Theseus and the
Minotaur. In the 17th Century the puzzle maze became very popular as a recreational fashion item in gardens across the world. Now we see the mathematical process at work. The recreational stimulus of
solving the maze (i.e. finding the way to the centre and back again) occupied the minds of some of the leading mathematicians of the age, and was essentially solved by Euler, who devised the theory
of networks to do it. Network theory now has important applications in the study of communications (such as the Internet and Google), the spread of epidemics, the propagation of rumours, the spread
of Social Media, and even the voting patterns in the Eurovision song contest. One way to represent a large network, such as the World-Wide-Web, is to draw up a (very large) square table with each
side listing all of the web sites in the world (yes there are billions of them) with a 1 if one web-site points to the other, and a 0 otherwise. Such a table is an example of a matrix, which was a
mathematical object developed and studied by Cayley in the 19th Century. Matrices have applications in just about every branch of maths, physics and engineering. For our current purposes, without
Cayley’s invention we would not have Google. I return to this subject in more detail in the next lecture.
Another problem Euler was interested in was that of Latin Squares. These were square, made up of many smaller squares to form a grid, with numbers arranged in the grid such that each number appeared
once in each row and column of the grid. If that sounds familiar, then it is. It is the basis of all Sudoku puzzles, which engage a huge number of people, all of whom are using some form of
mathematical reasoning to solve them. Whilst purely recreational, the methods used to solve a Sudoku puzzle are closely related to techniques for scheduling (for example) aircraft, buses and trains,
and constructing a school timetable. Finding efficient algorithms to do this is both very important and very hard. Indeed the question of finding the best algorithm (and showing that it is the best)
is one of the hardest unsolved problems in modern mathematics.
Maths appears in many other games, ranging from Mornington Crescent [1] to game shows such as Deal or No Deal. Many of these games require an estimation of risk and a strategy for responding to your
opponents moves. The study of related situations involving two players led John von Neumann, and later John Nash, to develop the mathematical Game Theory which is now used to inform decision making
by economists, business leaders and even governments, all of which greatly impact on the modern world. Indeed it played a central role in the recent telecoms auctions where multiple millions of
pounds changed hands.
As a finale to this section on recreational mathematics, I can’t resist making a brief mention of the role that mathematics plays in music. Everyone has heard of the Greek mathematician Pythagoras
who is credited (falsely) with the discovery of the famous theorem named after him about the sides of a right angled triangle. Less well known is that Pythagoras made a study of music, and made the
momentous discovery that the frequencies of two notes which sounded pleasant when played together were in a rational proportion. For example the ratio of the frequencies of the notes of the Octave
(C:C) is exactly 2 and of the Perfect Fifth (C:G) is 3/2. From this observation he went on to construct a sequence of notes all of which sounded good when played together, and which had notes in
rational proportion. This sequence e.g. C:D:E:F:G:A:B:C we now know as the scale. In particular, the Just Scale is the sequence in the ratios:
1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8 : 2
This creation of the mathematical mind served musicians well until the 18th Century. Around this time the first keyboard instruments came into use. These were tuned once and for all, and the tuning
could not be changed by the player. It was rapidly found that a keyboard instrument tuned to the Just Scale in one key would sound out of tune in other keys. The solution to this problem was to
introduce a new scale in which the notes were all in equal proportion. In particular, as there are 12 semi-tones in the octave, the ratio of the frequencies of successive semitones was 1.0595… which
is the twelfth root of 2. Keyboard instruments tuned in this (mathematically based) equally tempered scale sounded in tune for every key, and the new scale was rapidly adopted. Bach was so pleased
with it that he wrote The Well-Tempered Clavier to celebrate a really nice application of mathematics to the art of music.
Maths tells you where you are.
Perhaps the leading scientific question of the 17th and 18th centuries was the question of finding Longitude at sea. It had almost a mythical status and appeared in Gulliver’s Travels as an example
of an impossible problem. This difficult question not only stimulated a lot of mathematics, but also led, directly, to the modern world in which mathematics and machines work together.
The first breakthrough in navigation came when it was realized that the position of the sun and the stars in the sky depended upon where you were on the (round) Earth. By seeing how the angle of the
sun changed the Greek mathematician Erastothsthenes was able to calculate the radius of the Earth to surprising precision. Having worked out a coordinate system for the Earth (Latitude and Longitude)
it was apparent that the latitude could be determined by measuring the angle of the noonday sun above the horizon. Doing this in itself required a good knowledge of angles and trigonometry. The angle
itself could be measured using a sextant, again using ideas from trigonometry. At that point the mathematics became much harder. In order to find the longitude it was necessary to determine the time
at the location with reference to some absolute standard. Mathematicians such as Newton struggled with this problem, and in a sense solved it, in that they found that the longitude could be
determined from very accurate measurements of the location of the moon, combined with a fearsome amount of calculation. Unfortunately, none of this was possible in the conditions at sea (and before
the invention of the pocket calculator). An accurate method of finding longitude had to wait until the development of the chronometer H4 by Harrison as a means of finding the time at Greenwich. See
Sobel’s book [7] for an excellent account of Harrison’s struggles to build this first chronometer. However, even then a large amount of calculation was needed and here mathematics came into its own.
In particular the development of spherical trigonometry, which was needed to solve the triangles on the surface of the Earth that were the results of the navigational measurements. (Further careful
mathematics was needed to make the necessary corrections for refraction, parallax and dip.) Tables were constructed which solved triangles with a vast range of different angles. These were used in
parallel with Ephemerides, which were tables of the location of the Sun, planets and many stars for frequent time intervals in every day throughout the year. Looking at nautical tables from the 18th
Century I am overwhelmingly impressed by the amount of calculation needed to produce them, most of which would have been done by human ‘computers’.
The result of all of this mathematics, combined with the mechanical brilliance of Harrison, was completely transformative. It utterly transformed navigation by sea, making it much safer and cheaper
to transport goods around the world. By doing do it revolutionized both the economy and also the process of exploration, leading to the modern world.
There was a nice side product of all of this effort. The process of producing the navigational tables was essentially routine and open for being mechanized. Driven by this idea, in the 19th Century,
Charles Babbage was inspired to design the difference engine arguably the ancestor of all modern computers. Sadly, Babbage did not live to see his machines actually working (although a working model
of the difference engine was built by the Science Museum), but his ideas have certainly led to the modern computer, and thus the modern world.
Fourier, his series, and where it led to
One of the nicest examples of a branch of maths devised to solve one problem, which then solves many other problems is that of Fourier series. Joseph Fourier was a 19th Century French mathematician
who was interested in how heat flowed through a body. His first contribution was in devising, what is now known as the heat equation to describe this. Fourier’s equation was an example of what is
called a partial differential equation and it described the way that the temperature T of a body depended both on time t and space x. In modern notation, Fourier’s equation (now usually called the
heat equation) is:
∂T/∂t=k (∂^2 T)/(∂x^2 )
Where k is the thermal conductivity. Fourier then set about solving this equation. His first extraordinary insight was that he could solve it by expressing T as a sum of simple functions and then
finding the solution in terms of these functions. A good way to think of this is that it is much easier to build a house brick by brick, than to build it in one go. His second extraordinary insight
was in his choice of functions to construct the temperature out of. His choice was to use the sine and cosine functions, which arise in trigonometry (the study of triangles) so that he wrote down the
expression for T as:
T(x,t)=1/2 a_0 (t)+∑_(n=1)^∞▒〖a_n (t) cosnx+b_n (t) sinnx 〗
(This expression is now called a Fourier series). At first sight this is an extraordinary choice of a way to represent T. After all, what was the possible link between triangles and the flow of heat?
However, it was exactly the right choice to solve the heat equation above. It allowed to problem to be broken down into a set of much simpler problems, each of which could then be solved and combined
to find the solution of the original problem. In fact it was found shortly after this original insight that the use of sines and cosines to build up a function could solve a huge number of other
problems as well, including: the motion of waves, the behaviour of gases, many problems in gravity, electrostatics, electromagnetism, and even the behaviour of the stock market. This exactly
illustrates my earlier point that mathematical ideas used to solve one problem (the shape of triangles) could be used in others which seem at first sight to have no connection with the original.
Following Fourier’s great discovery many mathematicians got to work on extending and generalizing his idea, finding many nice results on the way including a slick derivation of the wonderful formula
(originally discovered by Euler):
π^2/6=1+1/2^2 +1/3^2 +1/4^2 +1/5^2 +1/6^2 +1/7^2 +⋯
Fourier series, and their discrete generalisations on a computer, play a fundamental role in modern technology. In particular we use them to make up, and process, sounds, information and images, and
the music, TV and video industry would not exist without them. To learn more about the history and applications of Fourier series see Korner’s book [3]
Maths saves lives
My sister is a consultant doctor and I am very proud of her. Hopefully through her work she saves many lives. We think of doctors and nurses as occupations which save lives, and this is of course
true. However, mathematicians save lives too. We have already talked about the great statistician Florence Nightingale, who saved thousands of lives by combining statistics with nursing. Her work
continues today with the careful use of medical statistics to quantify the effectiveness (and potential risk) of different medical treatments through clinical trials, and ruling out the use of
possibly dangerous treatments. Mathematics is also used in my sister’s field of anaesthetics, where it plays a vital role both in determining the correct dose of the anaesthetic and also (through
statistics again) determining its effectiveness.
However my favourite application of mathematics to medicine (and also the one I am most closely involved with) is that of medical imaging. Until recently, if something was wrong with you, the only
way to find out the problem was to cut you open. Such a procedure was fraught with difficulties. It was usually hard to find the problem, it damaged the body and left it vulnerable to infection, it
could usually only be done once, and for parts of the body such as the brain it was quite impossible. The situation today has been completely transformed. Through medical imaging techniques ranging
from Ultrasound, Computerised Axial Tomography (CAT), Magnetic Resonance Imaging (MRI), and Positron Emission Tomography (PET) doctors can see very precisely what is going on inside your body
(including your brain) without ever having to cut you open. Indeed, with the recently developed Functional MRI (fMRI) methods it is even possible to monitor your brain in real time, and thus get an
insight into what you are thinking. Millions of scans of people are made every year, and they save countless lives.
The initiator of this medical revolution was the Austrian mathematician Johannes Radon. If you look at a picture of Radon he looks just like you would think a mathematician should look like. He is
old, white, balding, short-sighted and has a moustache. In pure mathematics he has a huge reputation for his work in the (difficult) field of measure theory. As part of his work in this, in the early
part of the 20th Century, he asked the seemingly abstract question of what information could you learn about an object by looking only at the shadows that it cast from a distant light source. He
published a paper in 1917 describing what is now called the Radon Transform linking an object to its shadows, and showed that although shadows are (only) two dimensional projections of a three
dimensional body, that if enough measurements were made it was possible to reconstruct the original three dimensional object. So far, so abstract. However, at around the same time that Radon was
doing his mathematical work, X-Rays were increasingly being used (recall that this was also during WWI) to diagnose medical conditions without having to cut open a body. However, the image given by
an X-Ray was effectively a shadow of the body and lacked most of the detail needed for diagnosis. It took fully fifty years to combine Radon’s ideas with the technology of X-Rays, and this was
finally achieved. The story of how this happened is remarkable in itself. Following Radon’s work other mathematicians developed faster ways of reconstructing the three dimensional object, most
notably Kaczmarz in the 1930s. Using these ideas, the first commercially viable X-ray (CAT) scanner was invented in 1967 by Hounsfield at the EMI Central Research Laboratories using X-rays. The first
EMI-Scanner was installed in England and produced the first brain-scan 1971. (It is rumoured that the funding for this came from the sales of the Beatles’ records). Almost independently, Cormack in
the USA invented a similar device based closely on Radon’s ideas (which Cormack said he had discovered independently). Cormack and Hounsfield were jointly awarded the Nobel Prize in 1979. Sadly there
is no Nobel Prize for mathematics. However, it is nice that you can still win a Nobel Prize for applying mathematics to the world.
Radon’s wonderful transform for a body of density f(x,y) is given by:
R(ρ,θ)=∫▒〖f(ρ cos(θ)-s sin(θ),ρ sin(θ)+s cos(θ) ) ds〗
Here R is the measured intensity of the X-ray at an angle to the body and a distance away from it at its closest point of approach. The integral expresses the way that the X-ray is absorbed, and the
function f(x,y) tells us what is going on inside the body. Basically, if you know R well enough then you can find f(x,y) and then work out what is going on inside the body. Anyone who has solved a
Griddler puzzle in a newspaper in order to find a hidden picture is basically doing this calculation themselves.
If I were to write Radon’s formula on an aircraft I might (possibly as we have already seen) get arrested for being a terrorist. But by doing so I might be helping to save many peoples’ lives.
We will return to the topic of medical imaging and how Maths Saves Lives, in much more detail in a later lecture, but you can read more in [?].
The world of communication
A defining aspect of human society throughout the ages is our need to communicate. This trend has accelerated a great deal in recent years, with our instant access to huge amounts of data and the
ability to communicate over vast distances via mobile phones. Sometimes we want our communications to be secret (such as when we send credit card details over the Internet), and sometimes we want
them to be as clear as possible (such as when we use a mobile phone to talk to someone in Australia). However, none of this would be possible without mathematics. The business of secret
communications started with the early ciphers used, for example, by Julius Caesar. Although Caesar was not a mathematician, his ciphers certainly had a mathematical basis, in this case addition
modulo 26. Since then mathematics has played a part in nearly all ciphers. A famous example was the machine based Enigma cipher developed by the Germans in the Second World War, and due to its
complexity it was thought to be unbreakable. As is now well known, a team of mathematicians at Bletchley Park, including Alan Turing and Bill Tutte, managed to exploit mathematical patterns in the
Enigma machine to crack the cipher. This changed the modern world in many ways. First the obvious, had they not cracked the ciphers the Allies may well have lost the war. Secondly, the method of
cracking the ciphers involved computers, and the design of the code breaking Colossus electronic computer by Tommy Flowers ushered in the modern computer age. Modern ciphers, such as the RSA cipher,
also rely very heavily upon mathematics, especially the ‘pure’ mathematics of prime numbers. For example it is easy to multiply two large prime numbers together, but very hard to find the original
two primes given the answer. This asymmetry is used in modern public key ciphers, which are used millions of times every day to keep communications secure. See [6] for more details.
Personally I find the whole business of security rather dark and I much prefer the subject of making communications as transparent as possible. One of the most important side products of the space
race (apart of course from non-stick frying pans) was the development of Error Correcting Codes. In order for a satellite to send back information through a noisy channel and across the vast
distances of space, it was essential to find a robust way of making sure that the information got through without error. The answer was again provided by mathematicians, in particular Claude Shannon
and Richard Hamming, working at Bell laboratories. Shannon’s seminal contribution was to develop the theory of communication, in which it was possible to predict how much information could be sent
through a noisy channel. Hamming’s equally important contribution was to work out how to send this information so that if an error occurred the computer receiving it could correct the error itself.
In a landmark paper in 1950, he created a family of error-correcting codes now called Hamming codes. These work by creating digital symbols for the different parts of a message which are so different
that even if one is corrupted by noise it can still be told apart from any other symbol. The original symbol can then be restored by the computer, and the message corrected. This wonderful idea
(which interestingly used mathematics created by the French mathematician Galois in the 19th Century) not only solved the particular problem of communicating a message over long distances, but also
opened up a whole new branch of mathematics, that of coding theory. Error correcting codes are now used everywhere, without them we would not be able to transmit pictures or documents over the
Internet, phone long distances on a mobile phone, or even listen to music on a CD. Truly they lie at the heart of the technology of the modern world.
I will return to many of these issues in more detail in my next lecture, which will be on the mathematics and relevance of Big Data.
And where next
This brief overview of what mathematicians have done for us is meant mainly to whet your appetite. Later on in this series we will see many more applications of mathematics to the modern world, and I
will explore both how the latest developments in maths are likely to lead to even newer technologies, and also what new maths I expect we might learn, stimulated by these technologies.
Some References
[1] C. Budd and J. Budd, How to win at Mornington Crescent, (2016), Plus Maths Magasine,
[2] C. Budd and C. Mitchell, Saving lives: the mathematics of tomography, (2008), Plus Maths Magasine, https://plus.maths.org/content/saving-lives-mathematics-tomography
[3] T. Korner, Fourier Series, (1984), CUP
[4] P.J. Nahin, Dr Euler’s Fabulous Formula. (2011), Princeton
[5] W. Rouse Ball and H. Coxeter, Mathematical; recreations puzzles and essays, (2016), University of Toronto Press.
[6] S. Singh, The code book, (2010), Harper-Collins
[7] D. Sobel , Longitude, (2011), Harper-Collins
© Professor Christopher Budd OBE, 2016
This event was on Tue, 11 Oct 2016
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Approximation algorithm for the cyclic swap problem
Given two n-bit (cyclic) binary strings, A and B. represented on a circle (necklace instances). Let each sequence have the same number k of l's. We are interested in computing the cyclic swap
distance between A and B. i.e., the minimum number of swaps needed to convert A into B. minimized over all rotations of B. We show that this distance may be approximated in 0(n + k' ^2) time.
Original language English
Title of host publication Proceedings of the Prague Stringology Conference '05
Pages 190-200
Number of pages 11
State Published - 2005
Event Prague Stringology Conference '05, PSC 2005 - Prague, Czech Republic
Duration: 29 Aug 2005 → 31 Aug 2005
Publication series
Name Proceedings of the Prague Stringology Conference '05
Conference Prague Stringology Conference '05, PSC 2005
Country/Territory Czech Republic
City Prague
Period 29/08/05 → 31/08/05
ASJC Scopus subject areas
Dive into the research topics of 'Approximation algorithm for the cyclic swap problem'. Together they form a unique fingerprint.
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Strings 2023
Edward Frenkel (UC Berkeley)
Richard Feynman's last blackboard at Caltech contains a number of tantalizing inscriptions about Bethe Ansatz in quantum integrable models, which fascinated him in the last years of his life. After
reviewing some basic examples, I will present a modern perspective on the subject, linking it to dualities in QFT and String Theory, as well as the Langlands duality in mathematics. I will then
discuss some recent developments that lend support to Feynman's intuition that these ideas could be useful in the study of 4d gauge theory, and formulate some open questions.
• a6ca0fa3-73b7-4e4d-af0e-1dd470483fea
• 7fd2cb45-08f9-4c25-83a3-3e330c771c5c
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Slant Height of a Right Circular Cone | Lexique de mathématique
Slant Height of a Right Circular Cone
Distance a
from vertex
of a cone (generally called its apex) to its
or to any point on the circle of the base.
• The directrix is the perimeter of the base of a cone.
• In the case of a right circular cone, the directrix is a circle.
• A right circular cone is also called a “cone of revolution”.
See also :
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Rotated Region3: Rotated Boxes in 3D Space
a guide written by EgoMoose
• CFrame and vector familiarity
Testing a point
To start off we’re going to try to find a fast way to tell if a point is within a region or not. There are many ways to do this, but one of the most straightforward is to use something called plane
In order to use plane culling it would make sense that we need to know a bit about planes. For those of you who read the “silhouettes and shadows” post most of what I’m about to explain will be very
familiar to you. For those of you who didn’t, no worries, planes are a pretty simple concept so let’s get into it.
A plane (not the vehicle) is a mathematical term we use to describe an infinite surface in space that has no thickness. Think about a sheet of wood extending forever but being infinitely thin. Sure
it’s not a thing you’d see in real life, but it’s a mathematical concept that has a lot of uses! When we draw planes we tend to give them edges. This isn’t because planes actually have finite
lengths, but rather because we don’t have infinite space for which to draw them on.
To define a plane we only need two simple pieces of information, a point on the plane, and the surface normal of the plane. If you aren’t familiar with a concept of a surface normal it’s just a fancy
way of saying a unit vector that defines the facing direction of a surface in space.
With these two pieces of information we have a way to theoretically define any point on the plane. If we take every possible vector that’s orthogonal to the normal and add it a point on the plane
whilst multiplying it by an infinite number of scalars we would be able to define every/any point on the plane.
Defining above and below
One of the cool things we can say about planes since they’re infinite is which side of the plane a point is on. This might seem like a really random and generally unusable thing to want to know, but
when used in the right circumstances it allows us to do some really neat stuff.
To start off we’re going to define the two sides of our plane. We’ll call points that lay on the side that the surface normal faces “above”, and we’ll call points that lay on the side that the normal
doesn’t face “below”. This rule maintains regardless of rotational orientation (as Ender says: “The enemy's gate is down!”).
The question then becomes how we tell if a point is above or below our plane? The answer is actually very simple. All we have do is get the point we want relative to a point on the plane and then get
said vector’s scalar projection onto the normal. Believe it or not that value is enough to tell us what side of the plane we’re on.
The scalar projection is very easy to calculate if we know the angle between the two vectors. Simply multiply the magnitude of the vector you want to project by the cosine of the angle between it the
vector you want to project onto.
That’s great and all, but we don’t have the angle between the two vectors so what’s to be done? Luckily our friend the dot product has a useful definition when it comes to getting projections.
That means if we want to get the projection of a onto b all we have to do is divide the dot product by the magnitude of b and if we wanted the projection of b onto a then we simply divide the dot
product by the magnitude of a. For our purposes things are even easier. Since we’re getting the projection of a vector onto the normal, which is a unit vector, division by 1 is redundant meaning in
this case our dot product value is the projection of the vector onto the normal!
Now that we’ve got that out of the way let’s go over why this projection value tells us which side of the plane we’re on. Simply put it has to do with Cos(theta). A vector’s magnitude is always going
to be positive so we know if our scalar projection is in fact negative it’s being caused by Cos(theta). Based on our knowledge of trig we know that when 0 <= theta <= 90° the value of Cos(theta) will
be between 0 and 1, and when the value of 90° < theta <=180° the value of Cos(theta) will be between 0 and -1. This ends up meaning that if our projection is negative we know there’s an angular
difference between the vectors greater than 90° meaning we’re below the plane. If on the other hand our projection is positive then we know that the angular difference is less than 90° and we’re
above the plane. Finally, if our projection is zero then we know the given point is actually on the plane.
local part = game.Workspace.Part;
function isAbove(point, planePoint, normal)
local relative = point - planePoint;
return relative:Dot(normal) > 0; -- if point is above plane
while true do
part.BrickColor = isAbove(part.Position, Vector3.new(0, 0, 0), Vector3.new(1, 0, 0)) and BrickColor.Green() or BrickColor.Red();
Defining a 3D region with planes
Now that we know how to check what side a point is on a plane our next step is to take that knowledge and apply it to our own rotated region. The concept is simple, we define a cube using six planes
with surface normals all facing away from the center of the cube. That way we can then check any given point against all 6 planes and know that if even one point is “above” any of the 6 planes we’re
outside the region and is inside if the point is “below” all six planes.
Hopefully now you can start to see how the pieces are coming together. All that’s left for us to do is find a way to define 6 planes to represent a cube. We want to be able to be able to do it
without having to manually define each plane for our region, so what’s the best way to do this? Use a cframe’s rotation matrix of course! For those of you who don’t know a cframe stores information
in vector form on what is left, right, and backwards relative to its rotation. With these three vectors we can get any relative direction but since we’re representing a cube we realistically only
want the enumerated normal ids (as they represent the surface normals). We could pull apart the matrix and get these vectors manually, but there’s actually an even easier way, vectorToWorldSpace!
local cf = CFrame.new(0, 0, 0) * CFrame.Angles(math.pi/8, math.pi, -math.pi/7);
local top = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Top));
local bottom = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Bottom));
local left = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Left));
local right = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Right));
local front = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Front));
local back = cf:vectorToWorldSpace(Vector3.FromNormalId(Enum.NormalId.Back));
That’s great we’ve got our normal covered via a simple CFrame. The next question is how do we get our point on the plane (which is the other thing we need)? The easiest way is to take the object
space enum vectors and multiply them by your standard size vector divided in half to get the magnitude you need to travel from the center by the worldspace normal to get onto the plane. All that said
and done we have our six planes!
local region = {};
function region.new(cf, size)
local self = setmetatable({}, {__index = region});
self.center = cf;
self.size = size;
self.planes = {};
for _, enum in next, Enum.NormalId:GetEnumItems() do
local lnormal = Vector3.FromNormalId(enum); -- object space normal
local wnormal = cf:vectorToWorldSpace(lnormal); -- world space normal
local distance = (lnormal * size/2).magnitude; -- distance to given surface
local point = cf.p + wnormal * distance; -- point on surface
table.insert(self.planes, {
normal = wnormal;
point = point;
return self;
Now that we have our planes defined we just need to run a check to see if our point is above or below the six planes. As mentioned before, if the point is below all six planes we know it’s within
region bounds and if it’s above any of them the point is outside the region.
function region:castPoint(point)
for _, plane in next, self.planes do
local relative = point - plane.point;
if relative:Dot(plane.normal) > 0 then
return false; -- above one of the planes
-- was above none of the planes. Point must be in region
return true;
With all that we now have a way to do a check to see if a point is within a given region and of course we can apply this to parts!
local part = game.Workspace.part;
local rpart = game.Workspace.rpart;
local region3 = require(script:WaitForChild("region")); -- the code from above
-- we're using a part to define our region for simplicity's sake
local region = region3.new(rpart.CFrame, rpart.Size);
-- if the region part changes we also update the region it represents
region = region3.new(rpart.CFrame, rpart.Size);
while true do
-- check if a given part is in bounds
local inBounds = region:castPoint(part.Position);
part.BrickColor = inBounds and BrickColor.Green() or BrickColor.Red();
A more accurate approach for parts
As we have it our system works, but it only detects points. This can lead to some difficulties for detecting if parts are in the region or not because as it stands be can only check their centers (or
corners) which leaves us with potential inaccuracies. This might be okay for some users, but others might want really accurate part detection. To do this we’re actually going to implement the concept
of “Separating axis theorem” (SAT for short) which is a collision detection algorithm that can be applied to 3D.
Separating axis theorem
SAT is an algorithm that’s all about projections. The idea is to take the points that make up two convex polygons, get their scalar projection onto an axis, then get the max and min scalar projection
of each shape and check for overlap. I’ve already written about SAT before in the wiki article on 2D collision detection which goes over applying the algorithm in 2D. It is recommended that you read
it as this post will mainly be covering the process of bringing this algorithm over to 3D, not how it actually works.
SAT in 3D is very similar to SAT in 2D. The main difference is that we are now testing eight corners for each shape (a total of 16!) on 15 axes. Some of you may be wondering why I say 15 axes.
Shouldn’t it only be six? After all the 2D article says to run the test against the unique surface normals of each shape and based on what we define as “unique”, each cube only has three. The
majority of the other 9 axes that we’re testing are actually the normals of each shape crossed with each other. This is to avoid false positives. For instance let’s see an example:
If we were to only test these six axes we’d get a positive return on collision, but as we can clearly see that’s not the case. If however we include the normals crossed with each other in our check
then we get an axis in which we can properly check and see collision in this case.
Note: The camera cannot look a perfect 180 degrees down, but there is in fact a gap on that axis
With all that in mind we can now run our SAT test on all 15 axis to check for collision almost the same as 2D!
A few extra things we have to wary of though. The first is what happens when we get a vector of zero magnitude or get a NAN vector? If that’s the case we can’t project onto these axis so we will
assume there is collision on these axis and continue on. The second thing we can take note of is the concept of the minimum translation vector (MTV). Since we know the minimum penetration depth and
on which axis that overlap exists, we can calculate the vector needed to move the two parts out of collision. This isn’t a necessity when it comes to rotated regions, but it’s pretty cool and you
might find uses for it in other projects.
<div class="blog-image-center" markdown="1"> <img src="http://i.imgur.com/iahiAzX.gif" width="600px"/> <img src="http://i.imgur.com/7sPeCVS.gif" width="600px"/> </div>
function getCorners(cf, size)
local size, corners = size / 2, {};
for x = -1, 1, 2 do
for y = -1, 1, 2 do
for z = -1, 1, 2 do
table.insert(corners, (cf * CFrame.new(size * Vector3.new(x, y, z))).p);
return corners;
function getAxis(c1, c2)
-- the 15 axes
local axis = {};
axis[1] = (c1[2] - c1[1]).unit;
axis[2] = (c1[3] - c1[1]).unit;
axis[3] = (c1[5] - c1[1]).unit;
axis[4] = (c2[2] - c2[1]).unit;
axis[5] = (c2[3] - c2[1]).unit;
axis[6] = (c2[5] - c2[1]).unit;
axis[7] = axis[1]:Cross(axis[4]).unit;
axis[8] = axis[1]:Cross(axis[5]).unit;
axis[9] = axis[1]:Cross(axis[6]).unit;
axis[10] = axis[2]:Cross(axis[4]).unit;
axis[11] = axis[2]:Cross(axis[5]).unit;
axis[12] = axis[2]:Cross(axis[6]).unit;
axis[13] = axis[3]:Cross(axis[4]).unit;
axis[14] = axis[3]:Cross(axis[5]).unit;
axis[15] = axis[3]:Cross(axis[6]).unit;
return axis;
function testAxis(corners1, corners2, axis)
if axis.Magnitude == 0 or tostring(axis) == "NAN, NAN, NAN" then
-- when a vector is crossed with itself or opposite
-- we already checked this axis anyway and know there was collision
return true;
local adists, bdists = {}, {};
for i = 1, 8 do
table.insert(adists, corners1[i]:Dot(axis));
table.insert(bdists, corners2[i]:Dot(axis));
-- get and check the max and mins
local amax, amin = math.max(unpack(adists)), math.min(unpack(adists));
local bmax, bmin = math.max(unpack(bdists)), math.min(unpack(bdists));
local longspan = math.max(amax, bmax) - math.min(amin, bmin);
local sumspan = amax - amin + bmax - bmin;
local pass, mtv = longspan < sumspan;
if pass then
-- calc mtv (cause y not?)
local overlap = amax > bmax and -(bmax - amin) or (amax - bmin);
mtv = axis * overlap;
return pass, mtv;
function collides(part1, part2)
local corners1 = getCorners(part1.CFrame, part1.Size);
local corners2 = getCorners(part2.CFrame, part2.Size);
local axis, mtvs = getAxis(corners1, corners2), {};
local include = true;
for i = 1, #axis do
local intersect, mtv = testAxis(corners1, corners2, axis[i]);
if not intersect then return false, Vector3.new(); end; -- no intersection
if mtv then table.insert(mtvs, mtv); end;
-- must be intersecting
table.sort(mtvs, function(a, b) return a.magnitude < b.magnitude; end);
return true, mtvs[1] or Vector3.new();
Casting every part into the region
So far we’ve learned how to check if individual base parts are colliding. We can very easily translate this over to our region class from above especially since the get corners function provided
above uses information that our region class already has. The question now becomes what can we do to find all the parts in our rotated region just like the standard axis-aligned region3?
Your first thought might be to check every part in the game, but that’s a huge resource hog not only in the fact that we have to recursively search our game every time we want to do a check, but also
because we’d then have to do a SAT test against every part in the game! Instead, we’re going to use the standard axis-aligned region 3 to get us an estimate and then check the parts that it returns
with SAT.
So, how do we get our estimate? The way that I chose to do so was by getting the world bounding box of the shape. To do this I take the three unique normals of a would-be axis-aligned box and project
all the corners of my rotated region onto them. I then get the points that have both the minimum and the maximum scalar projection for each normal. From there I take the max x, y, and z and the min
x, y, and z and give them their own vectors which define the two corners of my world bounding box for my non-rotated region 3. Once we have that region we just find the parts in it with built-in
methods and then do our SAT test for accuracy.
function shallowcopy(t)
local nt = {};
for k, v in next, t do
nt[k] = v;
return nt;
function region:cast(ignore, maxParts)
local ignore = type(ignore) == "table" and ignore or {ignore};
local maxParts = maxParts or 20; -- 20 is default for normal region3
-- world bounding box
local rmin, rmax = {}, {};
local copy = shallowcopy(self.corners); -- the order matters later so make a copy we can sort
for _, enum in next, {Enum.NormalId.Right, Enum.NormalId.Top, Enum.NormalId.Back} do
local lnormal = Vector3.FromNormalId(enum);
table.sort(copy, function(a, b) return a:Dot(lnormal) > b:Dot(lnormal); end);
table.insert(rmin, copy[#copy]);
table.insert(rmax, copy[1]);
rmin, rmax = Vector3.new(rmin[1].x, rmin[2].y, rmin[3].z), Vector3.new(rmax[1].x, rmax[2].y, rmax[3].z);
-- cast non-rotated region first as a probe
local realRegion3 = Region3.new(rmin, rmax);
local parts = game.Workspace:FindPartsInRegion3WithIgnoreList(realRegion3, ignore, maxParts);
-- now do real check!
local inRegion = {};
for _, part in next, parts do
if self:castPart(part) then -- :castPart(part) is the SAT test
table.insert(inRegion, part);
return inRegion;
Finding the intersection points
So we’ve got pretty much all the information we need to actually have rotated region3s now. This section is optional is it aims to show how we can get the points of intersection between a part and a
region. Normal region 3 doesn’t have this property, but we already have most of the information and knowledge we need to do it so, y’know… why not?
Last week we talked about ray plane intersections. We’ll be using that equation again so if you didn’t read last week’s post or need a refresher now’s a good time to check it out. The basis behind
finding the points of intersection between two shapes (or in this case region) is actually finding where and if the edges of a shape intersect with the surfaces of another. Luckily we already have
the surfaces in the form of planes and we know how to find the intersections with ray plane intersections. That means all we have to do is define the edges of our part as vectors and do the ray plane
intersection whilst making sure any intersection we calculuate is “underneath” all the other planes than the one it intersected with. We also have to remember that ray plane intersections provide a
scalar that acts as a percentage of the edge vector. If that percentage is greater than 0% or 100% then our intersection value isn’t actually included in the edge.
function planeIntersect(point, vector, origin, normal)
local rpoint = point - origin;
local t = -dot(rpoint, normal) / dot(vector, normal);
return point + t * vector, t;
function region:intersectionPoints(part)
local intersections = {};
-- check part against region
local corners = getCorners(part.CFrame, part.Size);
local attach = { -- edge vectors
[corners[1]] = {corners[3], corners[2], corners[5]};
[corners[4]] = {corners[3], corners[2], corners[8]};
[corners[6]] = {corners[5], corners[2], corners[8]};
[corners[7]] = {corners[3], corners[8], corners[5]};
-- do some plane ray intersections
for corner, set in next, attach do
for _, con in next, set do
local v = con - corner;
for i, plane in next, self.planes do
local p, t = planeIntersect(corner, v, plane.point, plane.normal)
if t >= 0 and t <= 1 then -- between 0 and 100 percent
local pass = true;
for i2, plane2 in next, self.planes do
if i2 ~= i then
-- underneath every other plane
local relative = p - plane2.point;
if relative:Dot(plane2.normal) >= 0 then
pass = false;
if pass then table.insert(intersections, p); end;
-- check region against part
local planes = {};
for _, enum in next, Enum.NormalId:GetEnumItems() do
local lnormal = Vector3.FromNormalId(enum);
local wnormal = part.CFrame:vectorToWorldSpace(lnormal);
local distance = (lnormal * part.Size/2).magnitude;
local point = part.CFrame.p + wnormal * distance;
table.insert(planes, {
normal = wnormal;
point = point;
local corners = self.corners;
local attach = { -- edge vectors
[corners[1]] = {corners[3], corners[2], corners[5]};
[corners[4]] = {corners[3], corners[2], corners[8]};
[corners[6]] = {corners[5], corners[2], corners[8]};
[corners[7]] = {corners[3], corners[8], corners[5]};
-- do some plane ray intersections
for corner, set in next, attach do
for _, con in next, set do
local v = con - corner;
for i, plane in next, planes do
local p, t = planeIntersect(corner, v, plane.point, plane.normal)
if t >= 0 and t <= 1 then -- between 0 and 100 percent
local pass = true;
for i2, plane2 in next, planes do
if i2 ~= i then
-- underneath every other plane
local relative = p - plane2.point;
if relative:Dot(plane2.normal) >= 0 then
pass = false;
if pass then table.insert(intersections, p); end;
return intersections;
So that concludes our journey through the math/code we’d need to make a rotated region3 module. You are of course encouraged to make your own, but you can always take the one that I made here too!
This was another long post, but I think it was worth writing (and hopefully reading) because there’s so much cool and useful stuff you can learn from it. As always I hope you enjoyed and learned
something new. Thanks for reading!
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Probabilistic Programming and Bayesian Inference for Time Series Analysis and Forecasting in Python
The original article can be found here at Towards Data Science
Probabilistic Programming and Bayesian Inference for Time Series Analysis and Forecasting in Python: A Bayesian Method for Time Series Data Analysis and Forecasting in Python – by Yuefeng Zhang, PhD
As described in [1][2], time series data includes many kinds of real experimental data taken from various domains such as finance, medicine, scientific research (e.g., global warming, speech
analysis, earthquakes), etc. Time series forecasting has many real applications in various areas such as forecasting of business (e.g., sales, stock), weather, decease, and others [2]. Statistical
modeling and inference (e.g., ARIMA model) [1][2] is one of the popular methods for time series analysis and forecasting.
There are two statistical inference methods:
The philosophy of Bayesian inference is to consider probability as a measure of believability in an event [3][4][5] and use Bayes’ theorem to update the probability as more evidence or information
becomes available, while the philosophy of frequentist inference considers probability as the long-run frequency of events [3].
Generally speaking, we can use the Frequentist inference only when a large number of data samples are available. In contrast, the Bayesian inference can be applied to both large and small datasets.
In this article, I use a small (only 36 data samples) Sales of Shampoo time series dataset from Kaggle [6] to demonstrate how to use probabilistic programming to implement Bayesian analysis and
inference for time series analysis and forecasting.
The rest of the article is arranged as follows:
• Bayes’ theorem
• Basics of MCMC (Markov chain Monte Carlo)
• Probabilistic programming
• Time series model and forecasting [3]
• Summary
1. Bayes’ Theorem
Let H be the hypothesis that an event will occur, D be new observed data (i.e., evidence), and p be the probability, the Bayes’ theorem can be described as follows [5]:
p(H | D) = p(H) x p(D | H) / p(D)
• p(H): the prior probability of the hypothesis before we see any data
• p(H | D): the posterior probability of the hypothesis after we observe new data
• p(D | H): likelihood, the probability of data under the hypothesis
• p(D): the probability of data under any hypothesis
2. Basics of MCMC
MCMC consists of a class of algorithms for sampling from a probability distribution. One of the widely used algorithm is the Metropolis–Hastings algorithm. The essential idea is to randomly generate
a large number of representative samples to approximate the continuous distribution over a multidimensional continuous parameter space [4].
A high-level description of the Metropolis algorithm can be expressed as follows [3]:
• Step 1: Start at the current position (i.e., a vector of n-parameter values) in a n-parameters space
• Step 2: Propose to move to a new position (a new vector of n-parameter values)
• Step 3: Accept or reject the proposed movement based on the prior probability at the previous position, the data, and the posterior probability calculated from the data and its prior
distributions according to Bayes’ theorem [3].
• Step 4: If the propose is accepted, then move to the new position. Otherwise, don’t move.
• Step 5: If a pre-specified number of steps has not yet been reached, go back to Step 1 to repeat the process. Otherwise, return all accepted positions.
The foundation of MCMC is Bayes’ theorem. Starting with a given prior distribution of the hypothesis and the data, each iteration of the Metropolis process above accumulates new data and uses it to
update hypothesis for selecting next step in a random walking manner [4]. The accepted steps are samples of the posterior distribution of the hypothesis.
3. Probabilistic Programming
There are multiple Python libraries that can be used to program Bayesian analysis and inference [3][5][7][8]. Such type of programming is called probabilistic programming [3][8] and the corresponding
library is called probabilistic programming language. PyMC [3][7] and Tensorflow probability [8] are two examples.
In this article, I use PyMC [3][7] as the probabilistic programming language to analyze and forecast the sales of Shampoo [6] for demonstration purpose.
4. Time Series Model and Forecasting
This section describes how to use PyMC [7] to program Bayesian analysis and inference for time series forecasting.
4.1 Data Loading
Once the dataset of three-year sales of shampoo in Kaggle [6] has been downloaded onto a local machine, the dataset csv file can be loaded into a Pandas DataFrame as follows:
df = pd.read_csv('./data/sales-of-shampoo-over-a-three-ye.csv')
The column of sales in the DataFrame can be extracted as a time series dataset:
sales = df['Sales of shampoo over a three year period']
The following plot shows the sales of shampoo in three years (36 months):
Figure 1: Sales of shampoo in three years (36 months).
4.2 Modeling
A good start to Bayesian modeling [3] is to think about how a given dataset might have been generated. Taking the sales of shampoo time series data in Figure 1 as an example, we can start by
• The dataset might have been generated by a linear function of time with random errors in sales since the dataset roughly forms a straight line from lower left corner to upper right corner.
• the random errors might follow a normal distribution with zero mean and an unknown standard deviation std.
We know that a linear function is determined by two parameters: slope beta and intercept alpha:
sales = beta x t + alpha + error
To estimate what the linear function of time might be, we can fit a linear regression machine learning model into the given dataset to find the slope and intercept:
import numpy as np
from sklearn.linear_model import LinearRegressionX1 = sales.index.values
Y1 = sales.values
X = X1.reshape(-1, 1)
Y = Y1.reshape(-1, 1)
reg = LinearRegression().fit(X, Y)reg.coef_, reg.intercept_
where reg.coef_ = 12.08 is the slope and reg.intercept_ = 101.22 is the intercept.
Y_reg = 12.08 * X1 + 101.22def plot_df(x, y, y_reg, title="", xlabel='Date', ylabel='Value', dpi=100):
plt.figure(figsize=(16,5), dpi=dpi)
plt.plot(x, y, color='tab:blue')
plt.plot(x, y_reg, color='tab:red')
plt.gca().set(title=title, xlabel=xlabel, ylabel=ylabel)
plt.show()plot_df(x=X1, y=Y1, y_reg=Y_reg, title='Sales')
The above code plots the regression line in red over the sales curves in blue:
Figure 2: Sales of shampoo in three years (36 months) with regression line in red.
The slope and intercept of the regression line are just estimations based on limited data. To take care of uncertainty, we can represent them as normal random variables with the identified slope and
intercept as mean values. This is achieved in PyMC [7] as below:
beta = pm.Normal("beta", mu=12, sd=10)
alpha = pm.Normal("alpha", mu=101, sd=10)
Similarly, to handle uncertainty, we can use PyMC to represent the standard deviation of errors as a uniform random variable in the range of [0, 100]:
std = pm.Uniform("std", 0, 100)
With the random variables of std, beta, and alpha, the regression line with uncertainty can be represented in PyMC [7]:
mean = pm.Deterministic("mean", alpha + beta * X)
With the regression line with uncertainty, the prior distribution of the sales of shampoo time series data can be programmed in PyMC:
obs = pm.Normal("obs", mu=mean, sd=std, observed=Y)
Using the above prior distribution as the start position in the parameters (i.e., alpha, beta, std) space, we can perform MCMC using the Metropolis–Hastings algorithm in PyMC:
import pymc3 as pmwith pm.Model() as model:
std = pm.Uniform("std", 0, 100)
beta = pm.Normal("beta", mu=12, sd=10)
alpha = pm.Normal("alpha", mu=101, sd=10)
mean = pm.Deterministic("mean", alpha + beta * X)
obs = pm.Normal("obs", mu=mean, sd=std, observed=Y)
trace = pm.sample(100000, step=pm.Metropolis())
burned_trace = trace[20000:]
There are 100,000 accepted steps, called trace, in total. We ignore the first 20,000 accepted steps to avoid the burn-in period before convergence [3][4]. In other words, we only use the accepted
steps after the burn-in period for Bayesian inference.
4.3 Posterior Analysis
The following code plots the traces of std, alpha, and beta after the burn-in period:
pm.plots.traceplot(burned_trace, varnames=["std", "beta", "alpha"])
Figure 3: Traces of std, alpha, and beta.
The posterior distributions of std, alpha, and beta can be plotted as follows:
pm.plot_posterior(burned_trace, varnames=["std", "beta", "alpha"])
Figure 4: Posterior distributions of std, alpha, and beta.
The individual traces of std, alpha, and beta can be extracted for analysis:
std_trace = burned_trace['std']
beta_trace = burned_trace['beta']
alpha_trace = burned_trace['alpha']
We can zoom into the details of the trace of std with any specified number of steps (e.g., 1,000):
Figure 5: Zoom into the trace of std.
Similarly we can zoom into the details of the traces of alpha and beta respectively:
Figure 6: Zoom into the trace of beta.
Figure 7: Zoom into the trace of alpha.
Figures 5, 6, and 7 show that the posterior distributions of std, alpha, and beta have well mixed frequencies and thus low autocorrelations, which indicate MCMC convergence [3].
4.4 Forecasting
The mean values of the posterior distributions of std, alpha, and beta can be calculated:
std_mean = std_trace.mean()
beta_mean = beta_trace.mean()
alpha_mean = alpha_trace.mean()
• std_mean = 79.41
• beta_mean = 12.09
• alpha_mean = 101.03
The forecasting of the sales of shampoo can be modeled as follows:
Sale(t) = beta_mean x t + alpha_mean + error
where error follows normal distribution with mean of 0 and standard deviation of std_mean.
Using the model above, given any number of time steps (e.g., 72), we can generate a time series of sales:
length = 72
x1 = np.arange(length)
mean_trace = alpha_mean + beta_mean * x1
normal_dist = pm.Normal.dist(0, sd=std_mean)
errors = normal_dist.random(size=length)
Y_gen = mean_trace + errors
Y_reg1 = mean_trace
Given 36 months of data, the code below plots the forecasting of sales in the next 36 months (from Month 37 to Moth 72).
plot_df(x=x1, y=Y_gen, y_reg=Y_reg1, title='Sales')
Figure 8: Forecasting sales in next 36 months (from Month 37 to Month 72).
5. Summary
In this article, I used the small Sales of Shampoo [6] time series dataset from Kaggle [6] to how to use PyMC [3][7] as a Python probabilistic programming language to implement Bayesian analysis and
inference for time series forecasting.
The other alternative of probabilistic programming language is the Tensorflow probability [8]. I chose PyMC in this article for two reasons. One is that PyMC is easier to understand compared with
Tensorflow probability. The other reason is that Tensorflow probability is in the process of migrating from Tensorflow 1.x to Tensorflow 2.x, and the documentation of Tensorflow probability for
Tensorflow 2.x is lacking.
1. R.H. Shumway and D.S. Stoffer, Time Series Analysis and Its Applications with R Examples, 4th Edition, Springer, 2017
2. C. Davidson-Pilon, Bayesian Methods for Hackers, Probabilistic Programming and Bayesian Inference, Addison-Wesley, 2016
3. J.K. Kruschke, Doing Bayesian Data Analysis, A Tutorial with R, JAGS, and Stan, Academic Press, 2015
4. A.B. Downey, Think Bayes, O’Reilly, 2013
5. Jupyter notebook in Github
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Carrying capacity by numbers
The application of mathematics to the prediction of population dynamics has challenged demographers for at least two hundred years. Various proponents have developed formulae for both the calculation
of population growth as well as the potential limits to such growth. While these formulae on their own have not always been able to accurately predict human carrying capacity limits, in many cases
they have contributed to the development of more complex carrying capacity models.
As such, they have often been theoretic in nature, rather than having direct applicability to a particular landscape.
One of the earliest known equations relating to population dynamics was Thomas Malthus’ exponential growth theory of 1798. According to Malthus,[ii] “[p]opulation, when unchecked, increases in a
geometric ratio,” while its means of subsistence, namely its food supply, increases only in a linear or arithmetic manner. The exponential growth formula is relatively simple and can be given as;
P(t) = P[o] e^rt,
where P(t) is the population at a point in time, P[o][ ]is the initial population, e is the base of natural logarithms (2.718...), r is the growth rate and t is time. This formula generates a
j-shaped curve with population reaching to infinity (figure 1a). However, according to Malthus, this infinite growth is inevitably halted by the inability of food production to keep up with the
population’s exponential expansion (figure 1b).
Figure 1a. (left): Malthusian exponential growth curve showing how the population increases infinitely. Figure 1b. (right): Malthus’ exponential population growth curve limited by the linearly
increasing food supply. The assumed carrying capacity is the point at which the population projection intersects with the food supply projection. The carrying capacity is assumed in this instance
because Malthus did not refer to it as carrying capacity.
Malthus’ theories have been largely derided for over 200 years, mainly due to the fact that his most dire predictions have yet come to pass. However, many authors such as physicist Albert Bartlett,
still warn of exponential population growth, stating, “[t]he greatest shortcoming of the human race is our inability to understand the exponential function.” The main objection that Malthus’
detractors level against his population theory involves the implausible nature of both linear agricultural growth and never-ending exponential population growth. Ginzburg & Colyvan
explain that the exponential formula, while theoretically correct, rarely reflects actual circumstances, describing the approach as the, “default situation for populations - how they behave in the
absence of any disturbing factors,” even though most environmental conditions are replete with disturbing factors. Despite ample evidence suggesting that human populations have sometimes experienced
periods of increasing exponential growth, the smooth curve that Malthus’ equation generates, rarely reflects the broader historic outcome. For instance, even though the 1,800 year period leading up
to Malthus’ era displays a strong correlation to his exponential growth curve, there is still a degree of fluctuation in the growth rate (figure 1a). In an analogy with financial accounting, Coutts
points out that the population curve generally follows a variable rate of compound growth rather than a smooth fixed compound rate. This variability of the growth rate, visible in an examination of
the last 200 year period (figure 1b), can be attributed not only to environmental irregularity but also, as Hussen
argues, to institutional and technological intervention in the population dynamic. He states, “There are social and economic factors that induce humans to check their own population growth under
adverse conditions”, which “make the Malthusian margin a moving target.”
Ultimately, criticism of Malthus on the impossibility of uninterrupted exponential population growth is most likely unfounded because Malthus
himself agrees that this formula is to be considered more as a theoretic foundation than a practical reality, stating that, “in no state that we have yet known has the power of population been left
to exert itself with perfect freedom.”
Figure 1a (left): Global population numbers from 1AD to 1800[ix] showing occasional falls, but overall growth. Figure 1b (right): Global annual population growth rate since 1800,[x] showing a
variable growth trend.
The second criticism of Malthus’ work is the suggestion that food production need not grow in a linear fashion. Lomborg,[xi] for example, points out that, “the quantity of food seldom grows linearly.
In actual fact, the world’s agricultural production has more than doubled since 1961.” Coutts[xii] supports such criticism, stating that Malthus’ position of first proposing a universal law of
exponential population growth but then effectively arguing against it by suggesting, “that food (which grows in populations!) grows arithmetically is logically contradictory.” Kendall and Pimentel,
[xiii] on the other hand, provide evidence to support Malthus’ assertions, stating that between 1950 and 1984, “[w]orld grain output expanded by a factor of 2.6… increasing linearly, within the
fluctuations.” Invoking a Malthusian disaster, Kendall and Pimentel continue,“[r]ising growth of population,… and a linearly increasing food production have persisted over the recent 40 years,” thus
potentially leading to “great human suffering.” Comparing these authors’ views, Coutts is at least clear about Malthus’ theorem, while it is clear that Lomborg is not. According to Malthus, the
difference between population and food production was not their potential for growth, but the rate at which that growth might potentially occur. He argued that the rate of growth may increase in
populations but remains relatively steady with food. Hence, Lomborg’s assertion that a doubling of food production since 1961 disproves linear growth is clearly unfounded. This statement merely
proves that food production grew, but does not explain whether the growth was constant (linear) or increasing (exponential). On the other hand, Kendall and Pimentel’s observations of one instance of
linear food growth and Coutts’ assertion that the growth rate of food production may be variable (sometimes constant, sometimes increasing dependant on timeframes and external circumstances) rather
than fixed doesn’t actually prove or disprove whether population growth is likely to ever outstrip food production.
Thus, it seems that Malthus’ theories for exponential population growth and linear food production growth are both highly conditional on the timeframe, societal influences and location to which they
are applied. Malthusian predictions of premature death visiting the human race[xiv] have opened his theories to much criticism but his warnings that infinite population growth will ultimately be
limited by the finite nature of its means of subsistence, or in other words, its carrying capacity, seem as pertinent today as they were in 1798.
Even though Malthus wrote of food constraints to population growth, his exponential population equation failed to actually incorporate it. However, four years after Malthus’ death, in 1838, Belgian
mathematician, Pierre-Francois Verhulst developed a theorem that began to incorporate these limits, in the form of the logistic curve, stated as;[xv]
dN/dt = rN (K - N)/K,
where N is the population size, r is the rate of population growth, K is the carrying capacity and dN/dt is the rate of population increase. This formula, when graphed (figure 2), takes on a
characteristic S-shaped sigmoid curve, beginning with exponential growth at low densities, but transitioning to a tapering off at higher densities, “as resources become insufficient to sustain
continued population growth.”[xvi]
Figure 2 (left): Verhulstian logistic growth curve illustrating eventual levelling-off of populations at a carrying capacity limit.
Even though the logistic equation may be instructive of theoretical population dynamics, in criticism analogous to that of Malthus, Fearnside[xvii] suggests that, applying it to human carrying
capacity assessment oversimplifies the complexity inherent in societal interactions. Price[xviii] also doubts the usefulness of the logistic equation in predictions of human population dynamics,
finding fault in the premise that environmental conditions might exert unchanging constraints on a population as well as the assumption that populations grow until automatically stabilising at a
carrying capacity limit. He points out that even in non-human populations, these aspects rarely hold true, stating, “seldom if ever does a natural population rise sharply and then stabilize in the
form of sigmoid curve.”[xix] There is some evidence however, to suggest that population growth over the last hundred years has followed a sigmoid curve pattern (figure 3), with the growth rate
initially rising gradually, accelerating after about 1950 and then starting to decline over the last twenty years. Whether this pattern will continue to follow the logistic curve remains to be seen,
but the United Nations[xx] does make predictions to this effect.
Figure 3 (left): The United Nations[xxi] record of global population growth over the last 100 years and their estimate (median fertility variant) for the next 90 years suggests a logistic growth
The aspect most clearly lacking from nineteenth century authors Malthus and Verhulst’s formulas is the effect of societal behaviour on population dynamics. Subsequent dramatic changes in science and
technology in the intervening 200 years has only served to exaggerate this omission.
A more recent population equation which attempts to address the societal influences of population dynamics was devised by Ehrlich and Holdren in the early 1970s. In 1971 they initially proposed the
I = PF,
where I is impact, P is population and F is a function measuring per capita impact.[xxii] In order to realign this formula to carrying capacity imperatives, it could also be given as a population
P = I / F,
where population is equal to its total environmental impacts divided by the impacts per person. Ehrlich and Holdren subsequently expanded on the F in this equation to also include affluence (A) and
technology (T) in order to highlight that environmental impacts are not only influenced by the population’s size but also by the consumption patterns (represented by A and T) of its participants
shown as;[xxiii]
I=PAT (or simply referred to as IPAT).
In this formula, affluence is defined by economic activity per person and technology as the environmental impact per unit of economic activity. It is perhaps not immediately obvious why technology is
an apt description of environmental impact per economic activity, but Dietz and Rosa[xxiv] suggest that it roughly represents the efficient utilisation of resources available to the population. In
other words, the T component is “determined by the technology used for the production of goods and services and by the social organization and culture that determine how the technology is mobilized.”
An alternate population-focused form of the IPAT formula would become;
P = I / (AT),
suggesting that population size has a direct correlation with impacts and an inverse relationship to affluence and technology. In this alternate equation, if P is considered to be the maximum
allowable population then it could also be thought of as the carrying capacity. As such, the equation shows that as acceptable impacts grow, so does the carrying capacity, but as personal consumption
grows, the carrying capacity falls. In other words, if a population wishes to grow, then it needs to accept either greater environmental impacts and/or a reduction in its per capita consumption.[xxv]
While this rearrangement of Ehrlich and Holdren’s equation serves to illustrate its potential in calculating carrying capacities, in reality, a method for assigning numeric values to the impacts,
affluence and technology components would have to first be derived. Schulze[xxvi] points out that, “[t]he equation is not intended as a formal mathematical model, but rather as a conceptual
framework.” In order to transform this conceptual equation into a comprehensive quantitative one, modes of impact such as habitat destruction, pollution levels, climate change and other measures of
environmental damage would need to be pursued; affluence would need to be further defined by elements such as economic performance and consumption of goods and services; and various facets of
technological usage would also need to be validated and quantified.
While a comprehensive approach to the IPAT equation is yet to be developed, there is evidence of some progress. For example, Dietz and Rosa[xxvii] developed a method of assessing societal carbon
dioxide impacts based on Ehrlich and Holdren’s work. They state, “[a]lthough there have been attempts to assess the validity of the [IPAT] model, they have typically relied on qualitative
assessments, field study demonstrations, or projections rather than on an assessment of the model’s overall fit to an appropriate data base. This was our main task.”[xxviii] Dietz and Rosa redefined
the components of the model to suit their own focus, with environmental impacts reflecting only industrial CO2 emissions, GDP representing affluence, and population data utilised on a national scale.
In a further alternate version of the IPAT formula, Dietz and Rosa rearranged it in order to derive the technology index. So their formula reads,
T = I / (PA).
The application of the IPAT formula proved useful for Dietz and Rosa in determining correlations between populations, economic growth and environmental impacts. They found that a population’s size is
roughly proportional to its impacts but that “when affluence approaches about $10,000 in GDP, CO2 emissions tend to fall below a strict proportionality.”[xxix] However, given that the authors didn’t
expect economic growth to rise to this level in most nations for two or three decades, they deduced that “[e]conomic growth in itself does not offer a solution to environmental problems.”[xxx]
While theoretically instructive, the exponential, logistic and IPAT formulae have not yet facilitated the accurate assessment of human carrying capacity. However, more quantitative approaches do
exist and according to Sayre,[xxxi] the earliest known carrying capacity assessment performed under that name, was conducted in Africa by William Allan[xxxii] in 1949. Although he didn’t pioneer the
particular food-based approach employed, he was amongst the first to clearly articulate the methodology. Firstly, Allan estimated the agricultural yield of regionally grown staple crops (Y) and this
was then divided by the average amount of food required per person (F). Then, drawing on existing ecological survey data of regional soil and vegetation types, he calculated the amount of land
available for growing staple crops (L) and divided this total land by the amount of land required per person. So, in summary, the formula reads, carrying capacity is equal to the area of land
available for food production; divided by the food required per person, divided by the area required per food, or:
K = L / (F / Y).
In its simplest form, the equation is merely the total area of land (L) divided by the area of land required per person (A):
K = L / A.
In this form, the carrying capacity equation mirrors Ehrlich and Holdren’s original formula (P = I / F)[xxxiii], except that land area is substituted for impacts. Consequently, Allen’s formula can be
seen as a resource-based approach focussing only on the constraint of food production and consumption, while the IPAT formula is an environmental impacts-based methodology.
While the formula developed by Ehrlich and Holdren predominantly serves to highlight societal trends, Allan’s resource-based approach actually generates a quantitative carrying capacity result.
Allen’s simple methodology only makes estimates of basic food production and consumption requirements for a small population. However, his methodology has subsequently been refined and developed by
other carrying capacity proponents[xxxiv] who added further detail to the equation relating to production techniques, resource demands beyond just food, land use variables and consumption choices.
This additional level of complexity allows more recent carrying capacity assessment approaches to be categorised as models rather than just formulae.[xxxv]
[i] This chapter only looks at population equations rather than carrying capacity models such as the Carrying Capacity Dashboard.
[ii] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[iii] BARTLETT, A. (2012) Al Bartlett, Professor Emeritus Physics. Boulder.
[iv] GINZBURG, L. R. & COLYVAN, M. (2004) Ecological orbits: how planets move and populations grow, New York, Oxford University Press.
[v] COUTTS, D. A. (2009) Reverend Thomas Robert Malthus - An Exponentialist View. Melbourne.
[vi] HUSSEN, A. M. (2004) Principles of environmental economics, London, Routledge.
[viii] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[ix] Graph derived from Cohen’s COHEN, J. (1995) How Many People Can the Earth Support?, New York, W. W. Norton. summary of global population estimates.
[x] WIKIPEDIA (2012) Malthusian catastrophe. Wikipedia.
[xi] LOMBORG, B. (2001) The Skeptical Environmentalist: Measuring the Real State of the World, Cambridge, Cambridge University Press.
[xii] COUTTS, D. A. (2012) Couttsian Growth Model. Academic Publishing Wiki.
[xiii] KENDALL, H. W. & PIMENTEL, D. (1994) Constraints on the Expansion of the Global Food Supply. Ambio, 23, 198-205.
[xiv] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[xv] PRICE, D. (1999) Carrying capacity reconsidered. Population and Environment. 21.
[xvi] FEARNSIDE, P. (1986) Human carrying capacity of the Brazilian rainforest, New York, Columbia University Press.
[xviii] PRICE, D. (1999) Carrying capacity reconsidered. Population and Environment. 21.
[xx] UNITED NATIONS (2011) World Population Prospects, the 2010 Revision. New York, United Nations.
[xxi] The United Nations (Ibid.) have used t
[xxii] EHRLICH, P. R. & HOLDREN, J. P. (1971) Impact of Population Growth. Science, 171, 1212-1217.
[xxiii] EHRLICH, P. R. & HOLDREN, J. P. (1974) Human Population and the Global Environment: Population growth, rising per capita material consumption, and disruptive technologies have made
civilization a global ecological force. American Scientist, 62, 282-292. and DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of
Sciences, 94, 175-179.
[xxiv] DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of Sciences, 94, 175-179.
[xxv] To further illustrate this relationship; if A is 10, I is 1 and T is 0.1, then the carrying capacity (P) equals 100; but if per capita consumption is decreased so that A is 10, I is 0.1 and T
is 0.01, then the carrying capacity would increase to 10,000; or if impacts are to decrease so that A is 1, I is 1 and T is 0.1, then the carrying capacity falls to only 10 people.
[xxvi] SCHULZE, P. C. (2002) I=PBAT. Ecological Economics, 40, 149-150.
[xxvii] DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of Sciences, 94, 175-179.
[xxxi] SAYRE, N. F. (2008) The Genesis, History, and Limits of Carrying Capacity. Annals of the Association of American Geographers, 98, 120-134.
[xxxii] ALLAN, W. (1965) The African husbandman, Munster, Lit Verlag.
[xxxiii] EHRLICH, P. R. & HOLDREN, J. P. (1971) Impact of Population Growth. Science, 171, 1212-1217.. Where P is population, I is impacts and F is the impact per person.
[xxxiv] More recent proponents include Fairlie FAIRLIE, S. (2007) Can Britain Feed Itself? The Land, 4, 18-26. and Peter, Wilkins and Fick PETERS, C. J., WILKINS, J. L. & FICK, G. W. (2007) Testing a
complete-diet model for estimating the land resource requirements of food consumption and agricultural carrying capacity: The New York State example. Renewable Agriculture & Food Systems, 22,
[xxxv] More on this later.
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Actual ATI TEAS 7 Test Questions Set 3
Question 1.
A baker is using a cookie recipe that calls for 2 whole and 1/4 cup of flour to yield 36 cookies. How much flour will the baker need to make 90 cookies using the same recipe?
Question 2:
In a local community college, there are 800 students enrolled in four allied programs as shown in the pie chart. What is the number of students enrolled in the respiratory care program?
Question 3:
A baker is using a cookie recipe that calls for 2 whole and 1/4 cup of flour to yield 36 cookies. How much flour will the baker need to make 90 cookies using the same recipe?
Question 4:
What is the equivalent in pounds of 50 kg? (1 kg = 2.2 lb)
Question 5:
A temporary gauge reads 95°F. Which of the following is the correct conversion to degrees Celsius ( Note: °C = 5/9 (°F = 32))
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generate list of random dates python
Let's first have a quick look over what is a list in Python and how can we find the maximum value or largest number in a list. A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure
you have the best browsing experience on our website. We can also get the index value of list using this function by multiplying the result and then typecasting it to integer so as to get the integer
index and then the corresponding list value. Since then, Python has gained two new formatting approaches: string.Template (added in This module provides an implementation of a heap queue algorithm
known as heapq. By using our site, you Not the answer you're looking for? WebBacktracking is a class of algorithms for finding solutions to some computational problems, notably constraint
satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a
valid solution.. Quite a few times this function will throw a KeyError because the last line. Python defines a set of functions that are used to generate or manipulate random numbers through the
random module. With you every step of your journey. This is valuable because there's some preparatory work that random.choices has to do every time it's called, prior to generating any samples; by
generating many samples at once, we only have to do that preparatory work once. Randomly select elements from list without repetition in Python, Select Drop-down list using select_by_index() in
Selenium - Python, Randomly select n elements from list in Python, Select row with maximum and minimum value in Pandas dataframe. A random seed specifies the start point when a computer generates a
random number sequence. In this, we perform a similar task as the above function, using a generator to perform the task of Date successions. How do I generate a random integer in C#? How to generate
random numbers from a log-normal distribution in Python ? Here is a clear, simple method that is guaranteed to work. Making statements based on opinion; back them up with references or personal
experience. The performance of this solution is improvable for sure, but I prefer readability. WebIn Python programming, you can generate a random integer, doubles, longs etc . 3. The order of the
(item, prob) pairs in the list matters in your implementation, right? To use TIO, simply click the arrow below, pick a programming language, and start typing. How to import CSV file in SQLite
database using Python . Lets discuss certain ways in which this can be done. The list uses comma-separated values within square brackets to store data. If you need actual python datetimes, as opposed
to Pandas timestamps: use instead of [] to get a date_generator. This single utility function performs the exact required as asked by the problem statement, it generated N no. Python Random module is
an in-built module of Python which is used to generate random numbers. random.random() method generates the floating-point numbers in the range of 0 to 1. the code is below import random import
datetime as dt import numpy as np import pandas as pd def The important property of the heap is that its smallest element is always the root element. Method 2: Generate random string using
secrets.choice() We can Generate Random Strings and Passwords in Python using secrets.choice(). Python - Generate k random dates between two other dates. We reduce the current item's weight from
cumulative beta, which is a cumulative value constructed uniformly at random, and increment current index in order to find the item, the weight of which matches the value of beta. Method #1 : Using
choices() + loop + timedelta() In this, we extract all dates in range using loop and timedelta() of 1-day difference. Example usage: let's set up a population and weights matching those in the OP's
question: Now choices(population, weights) generates a single sample, contained in a list of length 1: The optional keyword-only argument k allows one to request more than one sample at once. Does an
existing module that handles this exist? To generate random numbers we have used the random function along with the use of the random.randint function. The seed value needed to generate a random
number. This function is used to generate a floating point random number between the numbers mentioned in its arguments. This task is to perform in general using loop and appending the random numbers
one by one. @S.Lott: OK, about 10000*365 = 3650000 = 3.6 million elements. PDFsharp: MIT C# developer library to create, extract, edit PDF files. Creating a list of dates using Python Loop. (OK, I
know you are asking for shrink-wrap, but maybe those home-grown solutions just weren't succinct enough for your liking. based on other solutions, you generate accumulative distribution (as integer or
float whatever you like), then you can use bisect to make it fast, this is a simple example (I used integers here), the get_cdf function would convert it from 20, 60, 10, 10 into 20, 20+60, 20+60+10,
20+60+10+10, now we pick a random number up to 20+60+10+10 using random.randint then we use bisect to get the actual value in a fast way, you might want to have a look at NumPy Random sampling
distributions. But you can use numpy.random.choice(), passing the p parameter: I wrote a solution for drawing random samples from a custom continuous distribution. WebAbout Our Coalition. This is the
simplest and straightforward approach to find the largest element. Python Random module is an in-built module of Python which is used to generate random numbers. Selecting numerous items from a list
or a single item from a particular sequence is handy with the help of random.choices function. So a function doing something completely different is faster than the one I suggested. Here we have
grouped all elements in a pair of size k. Python Programming Foundation -Self Paced Course, Data Structures & Algorithms- Self Paced Course, Python program to select Random value form list of lists,
Random sampling in numpy | random() function. Python | Generate random numbers within a given range and store in a list, Generate five random numbers from the normal distribution using NumPy,
Generate Random Numbers From The Uniform Distribution using NumPy. Generating random number list in Python using random(). You could also look up the Python 3 source for random.choices and copy that,
if so inclined. For example, lets say you wanted to generate a random number in Excel (Note: Excel sets a limit of 9999 for the seed). But there is always a requirement to perform this in the most
concise manner. These intervals can be used to select from (and thus sample the provided distribution) by simply stepping through the list until the random number in interval 0.0 -> 1.0 (prepared
earlier) is less or equal to the current symbol's interval end-point. Since Python 3.6, there's a solution for this in Python's standard library, namely random.choices. Connect and share knowledge
within a single location that is structured and easy to search. In Python, we can generate a random integer, doubles, long, etc in various ranges by importing a "random" module. I need this because I
want to generate a list of birthdays (which do not follow any distribution in the standard random module). In this, we perform a similar task as the above function, using a generator to perform the
task of Date successions. If the items in the list are strings, first they are ordered alphabetically and then the largest string is returned. This method is not handy and sometimes programmers find
it useless. A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. The random module contains the random.sample() function. WebA
constructive and inclusive social network for software developers. Here we generate a million samples, and use collections.Counter to check that the distribution we get roughly matches the weights we
gave. 3. About random: For random we are taking .rand() numpy.random.rand (d0, d1, , dn) : scipy.stats.rv_discrete might be what you want. Is there a verb meaning depthify (getting more depth)? The
second solution shows an interesting construct using lambda function. Why is the federal judiciary of the United States divided into circuits? WebIf you wanted to generate a sequence of random
numbers, one way to achieve that would be with a Python list comprehension: >>> [ random . These particular type of functions is used in a lot of games, lotteries, or any application requiring a
random number generation. String Formatting in Python using % 8. Python | Sort list of dates given as strings. We add new tests every week. Browse other questions tagged, Where developers &
technologists share private knowledge with coworkers, Reach developers & technologists worldwide. By using our site, you If you have any feedback please go to the Site Feedback and FAQ page. Doesn't
need to be, but it will perform the fastest if it is sorted by probability biggest first. generis - Code generation tool providing generics, free-form macros, conditional compilation and HTML
templating. acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Full Stack Development with React & Node JS (Live), Fundamentals of Java Collection
Framework, Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, How
to randomly select rows from Pandas DataFrame, Python program to find number of days between two given dates, Python | Difference between two dates (in minutes) using datetime.timedelta() method,
Python | Convert string to DateTime and vice-versa, Convert the column type from string to datetime format in Pandas dataframe, Adding new column to existing DataFrame in Pandas, Create a new column
in Pandas DataFrame based on the existing columns, Python | Creating a Pandas dataframe column based on a given condition, Selecting rows in pandas DataFrame based on conditions, Get all rows in a
Pandas DataFrame containing given substring, Python | Find position of a character in given string, replace() in Python to replace a substring, How to get column names in Pandas dataframe. If the
element is greater than the root element, we assign the value of this item to the root element variable, and at last, after comparing we get the largest element. The list of pseudo-random numbers in
each list are unique because there is no user-supplied seed value for the initial invocation of the RAND function. Python Programming Foundation -Self Paced Course, Data Structures & Algorithms- Self
Paced Course, Python - Generate k random dates between two other dates, Python - Generate random number except K in list, Python | Generate random number except K in list. I found this to be 30%
faster than scipy.stats.rv_discrete. When would I give a checkpoint to my D&D party that they can return to if they die? Just a solution, the second which came to my mind (the first one was to search
for something like "weight probability python" :) ). How could my characters be tricked into thinking they are on Mars? This is because unlike pure functional languages, Python doesnt optimize for
tail recursion, so every call to max() is being held on the stack. Did the apostolic or early church fathers acknowledge Papal infallibility? By using our site, you Convert the timestamp back to date
string and you have a random time in that range. The below example uses an input list and passes the list to max function as an argument. A question, how do I return max(i , if 'i' is an object?
Output: 0.0023922878433915162. @christianbrodbeck: Thanks, fixed. The classic textbook example of the use of The last element in the list is list[-1]. Given a date, the task is to write a Python
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0), datetime.datetime(1997, 1, 5, 0, 0), datetime.datetime(1997, 1, 6, 0, 0), datetime.datetime(1997, 1, 7, 0, 0), datetime.datetime(1997, 1, 8, 0, 0)]. Using the given example, we use heapq.nlargest
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by an external signal and have to be reset by hand? Thanks! But it differs by the fact that it requires 2 mandatory arguments for range. While you need O(n) time and space for preprocessing, you can
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library of knowledge resources, and developer advocates ready to help. Here, the shuffling operation is in place. The Python max() function returns the largest item in an iterable. The list is
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generation for enums from code comments. WebOptional. to the original question. You can supply your probabilities via the values parameter. Are you aware, This looks impressive. If you want to sample
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datetime.datetime(1997, 1, 5), datetime.datetime(1997, 1, 6), datetime.datetime(1997, 1, 7), datetime.datetime(1997, 1, 8)]. Python | Output Formatting. You just need the funtion random_custDist and
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question should all work on Python 2.7. This module can be used to perform random actions such as generating random numbers, printing random a value for a list or string, etc. My recommendation would
still be to use the function that does what you want rather than a function that does something else, even if the function that does something else is faster. @S.Lott isn't that very memory intensive
for big differences in the distribution? 4. To learn more, see our tips on writing great answers. The numpy functions also seem to only support a limited number of distributions with no support for
specifying your own. random () for _ in range ( 5 )] [0.021655420657909374, 0.4031628347066195, 0.6609991871223335, 0.5854998250783767, 0.42886606317322706] In this method, we will use pandas
date_range to create a list of ranges of dates in Python. WebHow to generate a list of random integers bwtween 0 to 9 in Python Professional provider of PDF & Microsoft Word and Excel document
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selling dragon parts come from? If you are using Python 3.6 or above, you can use random.choices() from the standard library see the answer by Mark Dickinson. The first is the keyword max which finds
the maximum number and the second argument is iterable. Python example (output is almost in the format you specified, import datetime import radar # Generate random datetime (parsing dates from str
values) radar.random_datetime(start='2000-05-24', stop='2013-05-24T23:59:59') # Generate This decision was made to encourage developers to use loops, as they are more readable. Then, the sorted()
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which returns a random floating number between 0 and 1. This is the simplest implementation but a bit slower than the max() function because we use this algorithm in a loop. Creating a list that uses
a dictionary that has percentages. The numpy.random.randn() function creates an array of specified shape and fills it with random values as per standard normal distribution. How do you sample from a
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weights: An optimization may be to normalize amounts by the greatest common divisor, to make the target list smaller. This module can be used to perform random actions such as generating random
numbers, printing random a value for a list or string, etc. Just to put things in context, here are the results from 3 consecutive executions of the above code: ['Count of 1 with prob: 0.1 is: 113',
'Count of 2 with prob: 0.05 is: 55', 'Count of 3 with prob: 0.05 is: 50', 'Count of 4 with prob: 0.2 is: 201', 'Count of 5 with prob: 0.4 is: 388', 'Count of 6 with prob: 0.2 is: 193']..['Count of 1
with prob: 0.1 is: 77', 'Count of 2 with prob: 0.05 is: 60', 'Count of 3 with prob: 0.05 is: 51', 'Count of 4 with prob: 0.2 is: 193', 'Count of 5 with prob: 0.4 is: 438', 'Count of 6 with prob: 0.2
is: 181'] . and, ['Count of 1 with prob: 0.1 is: 84', 'Count of 2 with prob: 0.05 is: 52', 'Count of 3 with prob: 0.05 is: 53', 'Count of 4 with prob: 0.2 is: 210', 'Count of 5 with prob: 0.4 is:
405', 'Count of 6 with prob: 0.2 is: 196']. Functions in the random module rely on a pseudo-random number generator function random(), which generates a random float number between 0.0 and 1.0. Is
the EU Border Guard Agency able to tell Russian passports issued in Ukraine or Georgia from the legitimate ones? random.randrange() method is used to generate a random number in a given range, we can
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structures, to generate Lorem Ipsum which looks reasonable. With almost 10 years history of publishing the hottest porn videos online, Nuvid.com still rocks hard! If you see the "cross", you're on
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true generator on the Internet. It can also be used to find the maximum value between two or more parameters. generating random dates with a given probability distribution). OpenPDF: GNU LGPLv3 /
MPLv2.0: Open source library to create and manipulate PDF files in Java. Formatting containers using format() in Python. The rest of the code for selection and generating a arbitrarily long sample
from the distribution is below : Here is a more effective way of doing this: Just call the following function with your 'weights' array (assuming the indices as the corresponding items) and the no.
:-). Output: A random number from list is : 4 A random number from range is : 41 Method 3: Generating random number list in Python using seed(). Nice. Given a list and our task is to randomly select
elements from the list in Python using various functions. Connecting three parallel LED strips to the same power supply. WebRsidence officielle des rois de France, le chteau de Versailles et ses
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comment. go-linq - .NET LINQ-like query methods for Go. None of these answers is particularly clear or simple. Find centralized, trusted content and collaborate around the technologies you use most.
The registry also allows access to counters for profiling system performance. Lets us see the below example to use reduce() in two different ways in this case: The reduce() takes two parameters. I
pseudo-confirmed that this works by eyeballing the output of this expression: Maybe it is kind of late. The above example has defined a function large() to find maximum value and it takes the input
list as the argument. Default value is 2 6. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. In functional languages, reduce() is an important function and is very useful.
Both should Generating Random id's using UUID in Python, Generating random strings until a given string is generated, Generating Random Integers in Pandas Dataframe, Random sampling in numpy | random
() function, Reading and Generating QR codes in Python using QRtools, Generating hash id's using uuid3() and uuid5() in Python. rev2022.12.9.43105. In this article, we will look into the principal
difference between the Numpy.random.rand() method and the Numpy.random.normal() method in detail. Prepare for your next technical Interview. randint() is the method which return the random integer
between two specified values. Python - Generate k random dates between two other dates. It is also a very slow and memory-consuming program. Generate random string/characters in JavaScript,
Generating random whole numbers in JavaScript in a specific range, Random string generation with upper case letters and digits. acknowledge that you have read and understood our, Data Structure &
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Pandas dataframe. How to get 5 random numbers with a certain probability?
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My New Twin Prime Numbers
Re: My New Twin Prime Numbers
Re: My New Twin Prime Numbers
Hi phrontister;
The 2763 is the first answer for P = 7. Other numbers just mean there are more than one answer further down.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
Thanks for that, Bobby...but I don't understand it. I suppose that yours is a functional approach, and my procedural brain can't grasp that yet. My code's just LB under the M hood.
I'm trying to use the help files to work out what it all means, but it's slow going.
What does the answer 2763 represent? Also, entering higher upper limits give further answers after the initial 2763, but I don't know what they mean either...or even if they're supposed to be there.
Or maybe I should wait for your final version before going further with this.
Last edited by phrontister (2013-04-24 13:17:43)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
Hi Bobby,
Here is my M code. Got any suggestions to improve it? It finds P=13 in just under 7 seconds.
Is yours much different? Mine is very raw, just looking for the answer without any fancy output formatting.
I enjoyed writing the code, and learnt some new things (eg, 'Break', 'PrimeQ', 'NextPrime', the table of primes, and better understanding of 'If').
Last edited by phrontister (2013-04-23 23:20:30)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Good luck with the grant.
As far as this problem is concerned I think there will always be a solution it is just a matter of going far enough.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Registered: 2012-01-30
Posts: 266
Re: My New Twin Prime Numbers
Hi phrontister
Thanks for the solutions, 2^2+-1=3,5 it is part of the solutions, in which I overlooked it, it seems the equation can generate larger primes and it is becoming more interesting.
bobbym, I am drafting a research grant for my prime equations, that is why I put on hold for the grid computing. It would be kool to have a facility that could run very big calculations at a faster
rate. I have more prime equations, I am trying to get a research grant to study them and maybe trying to find the largest primes as alternative to Mersenne in the future. This is one of the prime
equations that could give big Prime at a smaller prime number input, the derivation of the equation is simple.
(10002^214+10003^214)/(10002^2+10003^2)=849 Digits Prime and (10002^562+10003^562)/(10002^2+10003^2)=2241 Digits Prime.
Last edited by Stangerzv (2013-04-23 11:24:42)
Re: My New Twin Prime Numbers
Hi Bobby,
I thought the pair meant the plus minus of n.
I was referring to this, from post #1:
Where all Pi are the consecutive primes
I'd still think that a single Pi is valid, though, even though it isn't a 'consecutive' series.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Here's my answer for the lowest 3-digit Prime-th Power.
For P=101
Last edited by phrontister (2013-04-23 11:15:01)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
I thought the pair meant the plus minus of n.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
For the short version of P=2 there's only one Pi, which means the answer doesn't contain the "consecutive primes" mentioned in post #1. But I'm pretty sure that I've misunderstood the intent and that
a single Pi is fine...which means I should change my code back to what it was originally.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
But can you tell from that if the answer should contain multiple primes?
I am not following you, can you explain?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
I'm off to bed now...but I've set the program to test a larger number that might take it a while.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Thanks...I remember now. But can you tell from that if the answer should contain multiple primes?
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
It is just a shorthand for summation.
For P = 31
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
Hi Bobby,
Thank you.
Yes, my original program found that small answer for P=2, but later when I saw the larger answer in Stranerzv's first post and reread the post I thought (correctly or incorrectly) that "consecutive
primes" meant there should be multiple primes in the answer...and so I changed my program to what it is now.
I don't really understand that sigma notation very well (even though stefy told me how it worked) and so I couldn't tell from the one in post #1 whether or not I made the right decision. I copied the
sigma thingy from your post and used it for mine.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Hi Stangerzv;
For P = 2, 2^2 ± 1 = {3,5} which is smaller.
For P = 5, phrontister is correct. The solution in post #1 is incorrect.
are not both prime.
Hi phrontister;
Very good work!
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
A couple more (hope I'm doing this right)!
For P=23
For P=29
Last edited by phrontister (2013-04-23 11:15:48)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
For P=13
Last edited by phrontister (2013-04-23 11:16:05)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
For P=19
Last edited by phrontister (2013-04-23 11:16:23)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Hi Stangerzv,
For P = 5 I get a different answer from the one in your first post:
Last edited by phrontister (2013-04-23 05:20:32)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Hi Bobby,
Yes...I worked that out soon after I asked.
Your code is excellent for finding multiple solutions!
When I tried P=13 the quickest time with your code was about 11 seconds, compared to about 7 seconds with mine. I don't know how to tweak yours to make it faster.
I suppose the main flaw is having to guess the upper limits, and the more accurate your guess the closer you are to the minimum processing time. 'Break' stopped my procedural program when the answer
was found, but I don't know how to exit early with your code.
I played around with your code for printing the results:
Last edited by phrontister (2013-04-24 16:29:07)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Yes, getting it to stop is something I am working on. Unfortunately connection problems have kept me on the
phone all day and I got nothing done.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers
Btw, the code in my previous post only works for t>=f, otherwise it prints an error (c.f. your original code, which just prints a blank if the upper limit is too low).
Edit: I just got an error message (not just a blank) with your original code for P=101, so what I said in the para above isn't quite right.
Last edited by phrontister (2013-04-24 17:07:10)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
Also, I used t for the two upper limits as I couldn't see any reason for them to differ. That cuts the time down a bit.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers
I am working on an idea to improve it but the problems with the connection just won't stop.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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Net Present Value Analysis
Champion Company is considering a contract that would require an
expansion of...
Net Present Value Analysis Champion Company is considering a contract that would require an expansion of...
Net Present Value Analysis
Champion Company is considering a contract that would require an expansion of its food processing capabilities. The contract covers five years. To provide the required products, Champion would have
to purchase additional equipment for $80,000. Champion estimates the contract will provide annual net cash inflows (before taxes) of $35,000. For tax purposes, the equipment will be depreciated as
Year 1 $10,000
Year 2 20,000
Year 3 20,000
Year 4 20,000
Year 5 10,000
Although salvage value is ignored in the tax depreciation calculations, Champion estimates the equipment will be sold for $10,000 after five years.
Assuming a 35% income tax rate and a 10% cutoff rate, compute the net present value of this contract proposal. Using net present value analysis, should Champion accept the contract?
Round answers to the nearest whole number. Use rounded answers for subsequent calculations.
Use a negative sign with net present value to indicate a negative amount. Otherwise do not use negative signs with your answers.
After-Tax Cash Flow Analysis
Amount Present Value
After-tax cash inflows for 5 years $Answer $Answer
Tax savings from depreciation
Year 1 Answer Answer
Year 2 Answer Answer
Year 3 Answer Answer
Year 4 Answer Answer
Year 5 Answer Answer
After-tax equipment sale proceeds Answer Answer
Total present value of future cash flows Answer
Investment required in equipment Answer
Net positive (negative) present value $Answer
Should Champion accept the contract?
Select the most appropriate answer below.
Champion should accept the contract because there is a negative net present value.
Champion should not accept the contract because there is a positive net present value.
Champion should accept the contract because there is a positive net present value.
Champion should not accept the contract because there is a negative net present value.
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Has David Howden Vindicated Richard von Mises’s Definition of Probability?
In my recent article on these pages entitled “On the Possibility of Assigning Probabilities to Singular Cases: Or, Probability is Subjective Too!” (Crovelli 2009) I argued that members of the
Austrian School of economics have adopted and defended a faulty definition of probability. I argued that the definition of probability necessarily depends upon the nature of the world in which we
live. I claimed that if the nature of the world is such that every event and phenomenon which occurs has a cause of some sort, then probability must be defined subjectively; that is, “as a measure of
our uncertainty about the likelihood of occurrence of some event or phenomenon, based upon evidence that need not derive solely from past frequencies of ‘collectives’ or ‘classes’” (Crovelli 2009, p.
3). I further claimed that the nature of the world is indeed such that all events and phenomena have prior causes, and that this fact compels us to adopt a subjective definition of probability.
David Howden has recently published what he claims is a refutation of my argument in his article “Single Trial Probability Applications: Can Subjectivity Evade Frequency Limitations” (Howden 2009).
Unfortunately, Mr. Howden appears to not have understood my argument, and his purported refutation of my subjective definition consequently amounts to nothing more than a concatenation of confused
ideas that are completely irrelevant to my argument.
My reply will focus on the first three sections of Mr. Howden’s paper in isolation, and a discussion of the final two sections together.
On the Supposed “Necessity” of the Frequency Definition of Probability
The first major section of Mr. Howden’s paper bears the subtitle “The necessity of a frequency interpretation for probability theory.” With a subtitle as ambitious as this, one would expect that Mr.
Howden would offer some sort of argument to the effect that the frequency “interpretation” is a necessity. Disappointingly, one searches in vain to find any argument whatsoever—let alone an argument
indicating the “necessity” of the frequency “interpretation.” It is thus unnecessary for me to reply to Mr. Howden’s argument, since there is no argument to which I might reply. I will, however,
reply to a few of the random ideas in the section.
First, I would point out that Mr. Howden’s claim that “a priori probability distributions cannot be ascertained without verification through empirical tests” (Howden 2009, p. 2) is baldly question
begging. To claim that all probabilities must be tested against experience begs the question: well, what is probability in the first place? The classical method for generating probabilities by
assuming equal likelihood of occurrence does not rely upon “testing” or empirical “verification” of any kind. Whether or not this method for generating numerical probabilities is legitimate depends
upon how we define probability. So, to assume from the outset that this method is illegitimate simply begs the question.
Another idea in this section of Mr. Howden’s paper that is apparently advanced as evidence of the “necessity” of the frequency “interpretation” revolves around my reclassification of Richard von
Mises’s definition as a “conceptual method,” rather than the definition of probability itself. In reply, I would point out that my claim was not that Richard von Mises created a “conceptual
definition of probability,” as Mr. Howden mistakenly claims (Howden 2009, p. 3 emphasis in original). On the contrary, I specifically claimed that Richard von Mises created what is best described as
a “conceptual method for generating numerical probabilities” (Crovelli 2009, p. 10). I made this claim based upon the observation that, since relative frequencies can never be calculated for infinite
or indefinite series of observations, the method people actually use must rely upon finite series of actual observations. And, this is precisely the point that Richard von Mises himself makes in the
quote Mr. Howden cites (Howden 2009, p. 3). Mr. Howden’s misunderstanding on this point ultimately leads him to attribute to my argument the idea that the relative frequency method requires infinite
repetitions, an argument that I never made.
Mr. Howden’s extremely important discussion of odds-makers in this section also deserves notice. In my paper I claimed that since bookies and casinos are able to consistently generate accurate odds
for singular events and phenomena like boxing matches this is strong prima fascia evidence that non-frequentist methods for generating numerical probabilities were not “meaningless,” as the brothers
von Mises had claimed. Mr. Howden will have none of this, and he denounces such odds as “an illusion fabricated by an odds-maker” (Howden 2009, p. 4). His reason for denying that such odds are
probabilities, however, is almost embarrassingly question begging. Indeed, his evidence that such odds are not probabilities amounts to nothing more than a mistaken restatement of how bookies go
about generating odds:
[T]he odds-maker only has to have an estimate of who will win and who will lose a fight. The odds established are used to entice individuals to bet against the expected winner, in the hopes of
pocketing more winnings. (Howden 2009, p. 4)
Setting aside the fact that this is not how bookies manage a sportsbook, (and setting aside the fact that Mr. Howden is here admitting that odds-makers can indeed accurately predict who will win a
fight!), it should be obvious that it is question begging to use claims such as this as evidence that probability must be defined as a frequency. Again, the question thus begged would be: well, what
is the definition of probability in the first place? To assume from the outset that the methods of odds-makers utilizing non-frequentist methods are not “exercise(s) in probability,” is to assume the
very thing one is attempting to prove!
In sum, despite the ambitious subtitle of this section of Mr. Howden’s paper, we are not offered any argument whatsoever for the supposed “necessity” of the frequency definition of probability.
What Does Risk Have to do with the Definition of Probability?
Turning now to the second section of Mr. Howden’s paper, we find errors in reasoning that rival those of the first section in their seriousness. In the first place, the section opens with a quote
drawn from my paper where I made the following claim:
[W]hen we deal with the subject of probability we must necessarily and concomitantly deal with the subject of uncertainty. The term ‘probable’ applies to statements and facts about which we are
uncertain—the word does not apply to statements and facts about which we are absolutely certain. (Crovelli 2009, p. 7)
Mr. Howden then makes the following claim:
That there are only two options available—complete certainty and complete uncertainty—seems to neglect the case of risk. (Howden 2009, p. 5)
At no point in my paper, however, did I ever claim that these are the “only two options available,” and it is thus completely invalid to ascribe this claim to me. In fact, my claim was that there
man’s uncertainty varies since probability acts as a numerical measure of man’s uncertainty about the world.
Mr. Howden then turns to what he apparently considers to be a critical concept when attempting to define probability: the concept of “risk.” I am chided for not citing Frank Knight, who made a
“distinction between risk and uncertainty in terms of the ability to quantify outcomes” (Howden 2009, p. 5). How Knight’s conception of risk is relevant to the problem of defining probability is
never explained, however. The supposed distinction between these two concepts is simply stated and then abandoned without any argument whatsoever about why it is critical for defining probability.
Mr. Howden’s lamentation that I did not analyze the concept of risk is thus difficult to fathom. For, if he cannot explain exactly why this distinction is important in the quest to define
probability, on what basis should I be condemned for not discussing it?
It is important to bear in mind that my paper aimed to define probability. My argument that probability must be defined subjectively cannot be refuted by simply restating what the frequentists have
said about the concepts of “risk” “uncertainty” “class probability” and “case probability.” But this is precisely what Mr. Howden’s argument, such as it is, amounts to, because he never even attempts
to explain how these concepts are relevant to my argument. There is also a very serious element of circular reasoning involved in citing the frequentists’ conception of “risk” and “uncertainty” as
the only evidence that the frequentists’ definition is the correct definition. The question at hand is whether the relative frequency conception of probability is correct one, and to cite their
conception as the only evidence that their conception is correct is circular.
In addition, as I have previously noted, the conception of “uncertainty” that I developed in my paper has absolutely nothing to do with the idea of “complete uncertainty” (Howden 2009, p. 5). Hence,
Mr. Howden’s repeated condemnation of applying probabilities to things about which we are completely uncertain is nothing more than a repetitious flaying of his own straw man. To argue that
subjectivists are seeking to apply numerical probabilities to things about which we are completely uncertain (e.g., does God have a beard?) is to completely misunderstand the debate.
In sum, the second section of Mr. Howden’s paper offers only irrelevant and distracting straw men and begged questions, and absolutely no discussion of my argument.
Richard von Mises Was an Indeterminist
The third section of Mr. Howden’s paper offers yet another restatement of the frequentists’ conception of “collectives” and “classes.” No reply on my part is necessary here, since a restatement of
the frequentists’ position is no argument against my position. The question at hand, after all, is whether the frequentists have a correct definition of probability, and a mere restatement of their
position is no evidence whatsoever that they are right and I am wrong.
Interestingly, however, Mr. Howden attempts to bring the idea of “randomness” into the discussion in this section. He cites Richard von Mises to the effect that randomness in the world is what
“necessitates the probabilistic approach” (Howden 2009, p. 8). As in previous portions of the paper, Mr. Howden does not explain how randomness is relevant to the definition of probability, but he
does not seem to be aware that my entire argument rests on the claim that there is no randomness whatsoever in the world. In other words, my claim was that if every event and phenomenon in the world
has a prior and certain cause, (and I argued that there are indeed causes for everything that occurs in the world), there is quite literally no such thing as randomness in the world. I claimed that
if every event and phenomenon has a prior and certain cause, then the reason why man is uncertain about those events and phenomena lies in man’s own mental limitations—not in some mysterious property
of randomness in the world itself. I concluded from this that probability, as a numerical measure of man’s uncertainty, must be defined as a measure man’s subjective uncertainty. Mr. Howden seems to
be unaware of all this when he cites Richard von Mises, an outspoken indeterminist, on the topic of randomness.
Has David Howden Vindicated Richard von Mises’s Definition of Probability?
The final two sections of Mr. Howden’s paper offer a summary of his purported refutation of the subjective definition of probability and a vindication of the frequentist definition. The penultimate
section of the paper is entitled “Probability—what is it?” In this section, one would expect to find a recap of the argument against the subjective definition, and this is indeed what one finds, in a
sense. The argument, in sum, is that subjective methods cannot be applied to “collectives.” That this type of argument is question begging when deployed as evidence that the subjective definition is
mistaken hardly needs to be mentioned, but it is worth noting that non-frequentist methods are in fact capable of being applied to problems involving “collectives.” To cite but one example, if we
generate a probability of throwing a six with a die by assuming that each side of the die has an equal likelihood of being thrown, (i.e., we utilize the classical method), this would give us a
probability of 1/6. Dogmatic frequentists such as Richard von Mises would no doubt object that this number is meaningless and absurd until the die is actually cast many, many times, but their
protestation would be based upon their assumption that probability must be defined as a frequency. In short, whether or not non-frequentist methods are capable of being applied to problems involving
“collectives” depends upon what definition of probability we adopt.
Mr. Howden also claims that the subjective definition of probability lacks the “generality of the frequentist approach, while offering no significant advantages in replacement” (Howden 2009, p. 9).
This claim is absolutely dumbfounding. In the first place, to think that the frequentists’ definition of probability is more general than the subjectivists is to state precisely the reverse of the
truth. It is the frequentists who condemn all attempts to quantify uncertainty that are not derived from past frequencies as absurd, and the subjectivists who allow for other methods. How the
frequentists’ dogmatic and virulent condemnation of other methods could be interpreted as more general is difficult to fathom. In addition, the fact that the subjective definition allows for other
methods for quantifying man’s uncertainty demonstrates the “significant advantages” of the subjective definition. For, the subjective definition allows for calculating numerical probabilities for
singular boxing matches, singular wars, singular elections…et cetera ad infinitum.
In his concluding remarks, Mr. Howden sums up very nicely his entire argument against the subjective definition of probability:
When we realize the distinction between two similar concepts—risk and uncertainty for Frank Knight, case and class probabilities for Ludwig von Mises, and collectives and unique events for Richard
von Mises—we understand that probability is not something which may be redefined as Crovelli assumes. (Howden 2009, p. 10)
There is not much more to Mr. Howden’s argument than that. The paper contains no argument showing that the definition of probability does not depend upon the nature of the world. There is no argument
claiming that the world is not governed by time-invariant causal laws. There is no argument or explanation whatsoever as to why we must reserve the term “probability” only for those situations where
we can construct a “collective.” And there is absolutely no argument as to why the concept of risk obliges us to adopt a frequentist definition.
The answer to the question of whether David Howden has vindicated Richard von Mises’s definition of probability must therefore be that he has failed.
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