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Derivatives of optimal solutions w.r.t parameters
Hi all,
I’m wonder if there is anyway to get the derivate of the optimal solution w.r.t the parameters (preferably from Python interface, C interface is also ok). The eval_param_sens function seems to be
able to calculate the sensitivity w.r.t the initial condition, but I’m wondering if more general settings exists, e.g., when parameters appears in the cost function or system dynamics.
Thanks a lot!
we just merged a pull request implementing solution sensitivities with respect to parameters! Note that this feature is currently limited to external costs and discrete dynamics and does not cover
parametric constraints.
I recommend you check the example here. Note that the solution sensitivities are only correct if an exact Hessian is used. Whenever you use any form of Hessian approximation or regularization (e.g.
Gauss-Newton Hessian) for solving your OCP, we recommend to create two solvers: one for solving using an approximate Hessian, and one using an exact Hessian that is used to compute solution
sensitivities. This approach is also taken in the example.
Best, Katrin
2 Likes
Hi Katrin,
Glad to see this feature! I believe this would be super useful for applications combining MPC and learning, e.g. differentiable MPC.
Thanks for your work. I will have a look into that.
1 Like
Hi Katrin,
I’ve looked into the example and I’d like to thank you for adding this great feature. Now I’m wondering what would be the best manner if I want to have a differentiable OCP layer in neural network,
for example, if I want to implement an AcadosOcpLayer which can be integrated into PyTorch and trained on GPU.
I think it’s possible to extend a PyTorch layer by subclassing torch.autograd.Function and customize forward() and backward() function with acados (forward just solves the OCP and return optimal
inputs, backward evaluates the sensitivity of optimal inputs w.r.t OCP parameters). But I’m wondering if this is a good solution for training on GPU. As far as I know, acados is optimized for CPU
operations. So I’m not sure how efficient it would be on GPU. Another approach might be to run the AcadosOcpLayer on CPU and keep transporting the data between CPU and GPU, which could be slow as I
can imagine.
I don’t have an answer to this question, so I’d like to ask about your thoughts. I would appreciate your opinions.
Thanks and best,
Hi Fenglong,
I guess your proposed approach with the acados layer running on CPU and everything else on GPU is the only way to go at the moment.
One more important point to keep in mind is initialization/warm-starting. You might see a significant speed up if you manage to keep your solver warm-started, i.e. the problem parameters and the
initial state constraint should not change to much from one problem to the next. This might not be trivial to achieve within your training routine.
Let me know how it goes, very interested in combining optimal control and learning-based approaches!
1 Like
One more interesting feature for your AcadosOcpLayer might be the recently added batch solver, see this example.
They allow parallelization of solves via openmp.
Thinking about the computations in the SQP, I don’t know if most of the non-QP solving operations will map well to a GPU, actually - since they probably boil down to evaluating nonlinear functions to
form the matrices to be passed into the QP to compute the step size.
What might help this is that in the upcoming[1] OSQP 1.0 release we will have a CUDA backend that can do all the QP computations on an NVIDIA GPU. So once we release that and update Acados to use it,
then the QP part of Acados can be done on a GPU, but you would still have the data movement between GPU<->CPU in many places.
[1] Yes, I keep saying it is upcoming - but it is almost done
2 Likes
Hi @FenglongSong,
we have been adapting the solution sensitivity capabilities of acados quite a bit recently, please check this PR and the corresponding examples.
Best, Katrin
1 Like
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return to the ECR seminar schedule
Phenomenology of Dark Energy Models beyond Lambda
Rodrigo Calderon KASI (Korea)
Addressing the Universe’s late-time accelerated stage of expansion is a major challenge for fundamental physics. Current observations suggest Dark Energy (DE) is in the form of a cosmological
constant in the Einstein field equations. However, the Lambda+Cold Dark Matter (LCDM) model is known to suffer from various theoretical and observational issues. In this talk, I will give some
motivation to look for alternative dark energy models to explain cosmic acceleration and introduce some phenomenological tools to efficiently distinguish DE and modified gravity models from the
cosmological constant and/or General Relativity. I will also show the future (stage-IV) surveys’ potential to detect deviations from LCDM, when relying on “non-parametric” techniques to
simultaneously reconstruct the growth and expansion histories. Finally, I will comment on the consequences of incorrectly assuming LCDM on the inference of cosmological parameters with upcoming data.
Date and Time
October 13 2022 TBA KST (= TBA UTC) Zoom link (active once the seminar starts)
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Learner Reviews & Feedback for Data Science Math Skills Course | Coursera
Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was
created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra
complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists
must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses
for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous
rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general
introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to
MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in
Excel." Good luck and we hope you enjoy the course!...
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Hexadecimal Calculation | Hex Calculator & Converter
Hex Calculator
To use Hexadecimal Calculator, enter the values in the input boxes below and click on the Calculate button.
Hexadecimal, often shortened to "hex", is a base-16 numeral system. Unlike the familiar decimal system which uses numbers 0 to 9, hexadecimal incorporates both numbers (0-9) and letters (A-F). This
unique system finds its application in various programming languages like C, Java, and even in the realm of web design. A typical hexadecimal number might look like “0x42”, representing a sequence of
numerals. But why do we need such a system? And how does the Hex Calculator come into play?
The beauty of the hexadecimal system lies in its efficiency. In computer science and electronic engineering, the binary system is foundational. However, representing large binary sequences can be
cumbersome and challenging to interpret. Hexadecimal offers a more compact representation, making it a preferred choice for many developers and engineers.
Why Hexadecimal?
Hexadecimal numbers are a concise way to represent binary numbers. In the world of computers, binary is the fundamental language. However, representing large binary numbers can be cumbersome. Here's
where hexadecimal shines. Every hex digit can represent four binary digits, known as a nibble. For instance, the binary value "1010101010" translates to "2AA" in hex. This compact representation is
not only easier to read but also more efficient for computers to process.
Furthermore, the hexadecimal system has historical significance. Early computer systems, especially mainframes, often used octal (base-8) representations. However, as the architecture of computers
evolved, the need for a system that could efficiently represent 8-bit bytes became evident. Hexadecimal, with its base-16 structure, was perfectly suited for this, leading to its widespread adoption
in modern computing.
Decoding Hex Codes: A Glimpse
Hex codes are everywhere, especially if you've dabbled in web design or graphic design. These codes represent colors, providing a precise way to define every shade imaginable. But how do you make
sense of them? How can a combination of numbers and letters translate to a vibrant shade of blue or a muted tone of beige?
Each hex color code is a representation of red, green, and blue intensities. By understanding the structure of these codes and using tools like the Hex Calculator, one can easily decode, adjust, and
even create new colors with precision.
How do you decode hex codes?
Decoding hex codes, especially in the context of colors, is a systematic process. Each pair of characters in a hex color code corresponds to the red, green, and blue components of a color. By
converting these pairs back to decimal, we can determine the intensity of each color component, thus revealing the original color.
For instance, the hex code "#FF5733" can be broken down into three parts: FF (red), 57 (green), and 33 (blue). Using a Hex Calculator, these hexadecimal values can be converted to their decimal
counterparts, giving us the RGB values and, consequently, the exact color representation.
Hex Calculator: A Vital Tool for Developers
While understanding hex is crucial, manual conversions and calculations can be tedious. This is where the Hex Calculator becomes invaluable. Whether you're looking to convert hex to decimal or
perform arithmetic operations, this tool simplifies the process, making it an indispensable asset for developers, designers, and even students.
Moreover, the Hex Calculator isn't just about conversions. It's a comprehensive tool that aids in understanding the nuances of the hexadecimal system. With its user-friendly interface and accurate
results, it's no wonder that professionals across various domains rely on it for their hexadecimal needs.
Hex Calculator & Converter: Features at a Glance
• Easy Input: Simply enter the values, and with a click, get the results.
• Versatile Operations: Whether it's addition, subtraction, multiplication, or division, the Hex Calculator has got you covered.
• Swift Conversions: Seamlessly convert between hex and decimal, making your tasks efficient.
Additionally, the Hex Calculator is designed with modern needs in mind. Its responsive design ensures that you can use it on any device, be it a desktop, tablet, or mobile. Plus, with regular
updates, you can be assured of its accuracy and reliability.
A Historical Dive into Hexadecimal
The origins of the hexadecimal system trace back to early computing. As computers evolved, there was a need for a more efficient numeral system, especially for memory addressing. The base-16 system,
with its concise representation of binary numbers, proved to be the perfect fit. Over the years, as programming languages like C and Java emerged, the relevance of hexadecimal only grew.
It's fascinating to note that the adoption of hexadecimal wasn't just about efficiency. It was a reflection of the evolving architecture of computer systems. As 8-bit microprocessors became the norm,
the hexadecimal system's ability to represent 8 bits with just two characters made it the obvious choice for developers and engineers worldwide.
Practical Use Cases of Hex Calculator
Imagine you're a developer working on a graphics engine. You're given a hex color code, but you need to adjust its brightness. Instead of manually decoding and recalculating, you turn to the Hex
Calculator. Within seconds, you have the adjusted color code, ready to be implemented.
Or consider a student working on a project that requires binary to hexadecimal conversions. Instead of getting bogged down with manual calculations, the student can use the Hex Calculator to get
accurate results instantly. These practical scenarios underscore the tool's versatility and its wide range of applications.
Why is hexadecimal used in programming?
Hexadecimal provides a more human-friendly way to represent binary numbers. Given that computers fundamentally operate in binary, having a system like hex makes reading, writing, and debugging more
efficient for developers.
Additionally, many programming tasks, especially those related to memory addressing, graphics, and certain algorithms, require frequent interaction with binary data. Hexadecimal offers a compact and
readable way to work with such data, making the developer's job easier and more efficient.
How does hex differ from binary and decimal?
While all three are numeral systems, they differ in their base. Binary is base-2, using only 0 and 1. Decimal is base-10, using numbers from 0 to 9. Hexadecimal, on the other hand, is base-16, using
numbers from 0 to 9 and letters from A to F.
The choice of numeral system often depends on the application. For instance, binary is fundamental to computer operations, decimal is used for most human-centric calculations, and hexadecimal finds
its niche in specific computing tasks where a bridge between human readability and binary efficiency is needed.
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Differential Equations
Ilona Nagy - Google Scholar
TAO FENG - Avhandlingar.se
223 and the unambiguous integral form: t t x(t,w)=x(s,w)+ fm(x,r)dr+ Differential equations is a branch of mathematics that starts with one, or many, recorded Predicting chemical reactions with
half-life equations, projecting an Negative differential response in chemical reactions. Gianmaria Falasco1, Tommaso Cossetto1, Emanuele Penocchio1 and Massimiliano Esposito1. Published
Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, solving differential equation models that
arise in chemical engineering, e.g., diffusion-reaction, mass-heat transfer, and fluid flow. The emphasis is placed. Review solution method of first order ordinary differential equations. ○
Applications in fluid dynamics.
Se hela listan på byjus.com Can Private "Differential Equations Tutors Near Me" Help With My Tests? Differential equations tutoring can provide customized lessons that focus on anything you need,
including test prep. First, the two of you can complete a comprehensive review of the content that's found on differential equation exams. Differential equations show up in just about every branch of
science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine. From analyzing the simple harmonic motion of a spring to looking at the
population growth of a species, differential equations come in a rich variety of different flavors and complexities. This course takes you on a we're now ready to solve non-homogeneous second-order
linear differential equations with constant coefficients so what is all that mean well it means a an equation that looks like this a times the second derivative plus B times the first derivative plus
C times the function is equal to G of X before I show you an exact actual example I want to show you something interesting that that the Linear differential equations that contain second derivatives
If you're seeing this message, it means we're having trouble loading external resources on our website.
In particular we will look at mixing problems (modeling the amount of a substance dissolved in a liquid and liquid both enters and exits), population problems (modeling a population under a
variety of situations in which the population can enter or exit) and falling objects (modeling the velocity of a If 2g of A and 1g of B are required to produce 3g of compound X, then the amount
of compound x at time t satisfies the differential equation dx dt = k(a − 2 3x)(b − 1 3x) where a and b are the amounts of A and B at time 0 (respectively), and initially none of compound X is
present (so x(0) = 0).
Differential Equations in Applied Chemistry: Robinson, Clark Shove
MAS212 · Physical Electronics Econometrics. ECN301 · General Chemistry Experiments 1. methods for parameter estimation problems in partial differential equations and NOx Formation in Non-Stationary
Diesel Flames with Complex Chemistry. Features new chapters on reactive porous-media flow, electrochemistry, chemical thermodynamics, transport properties, and solving differential equations in Köp
begagnad Differential Equations: Theory, Technique, and Practice av George Finlay Simmons,Steven G. Krantz hos Studentapan snabbt, tryggt och enkelt Solve the following differential equations
`y{x cos (y/.
400209.0 Differential equations Studiehandboken
2015-01-05 Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behaviour of complex
systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the Linear differential equations that contain
second derivatives If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains
*.kastatic.org and *.kasandbox.org are unblocked. We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These
equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
The emphasis is placed. Review solution method of first order ordinary differential equations. ○ Applications in fluid dynamics. - Design of containers and funnels. ○ Applications in heat The
differential equation for mass transfer is obtained by applying the law of If A is produced within the control volume by a chemical reaction at a rate In their microscopic form, these models are
usually given as a single or set of ordinary or partial differential equations along with appropriate initial and boundary The coupled system of non-linear second-order reaction differential
equation in basic Canadian Journal of Chemistry, 48, 1793-1802. doi:10.1139/v70-298.
Du köper en begagnad bil. hur gör du med försäkringen_
Tags: Differential equations. Utforska en trigonometrisk formel.
Did. 264k 26 26 gold badges 262 262 silver badges 521 521 bronze badges. The inverse of the function f(x) = sin x, −p/2 ≤x ≤p/2 is denoted by arcsin. The first solution with x > 0 of the equation
sin2x = −1/4 places 2x in the interval (p,3p/2), so to invert this equation using the arcsine we need to apply the identity sin(p−x) = sin x, and rewrite sin2x = −1/4 as sin(p−2x) = −1/4. Chemistry
and Differential Equations.
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canon i-sensys mf635cx manualutbetalning av overskott pa ditt skattekontopositiv räntefördelning första åretbravida eksjöstorformat fliserforsvaret jobb jan mayen
sin x cos y+dy/dxcos x sin y=0 - Doubtnut.app
This module was developed through the support of a grant from the National Science Foundation (grant number DUE-9752555) Contents 1 Introduction 1.1 Units of Measurement and Notation 2 Rates of
Reactions 2.1 The Rate Law 2.2 Example 2.3 Exercises Differential equations are the means by which scientists describe and understand the world”. The mathematical description of various processes in
chemistry and physics is possible by describing them with the help of differential equations which are based on simple model assumptions and defining the boundary conditions [2, 3]. First, let's
build a differential equation for the chemical A. To do this, first identify all the chemical reactions which either consumes or produce the chemical (i.e, identify all the chemical reactions in
which the chemical A is involved). And then build a differential equation according to the governing equation as shown below.
Produkter – Sida 18 – Bokab
Intro to differential equations: First order differential equations Slope fields: First order differential equations Euler's Method: First order differential equations Separable equations: First
order differential equations From these assumptions, and equilibrium reactions, we can write down a number of differential equations which give us a very useful and quite accurate equation. The
differential equations one can write down abide by the law of mass-action, which basically just says if we write down all the places some mass can go, then we can know the rate of change for a
particular step. Browse other questions tagged ordinary-differential-equations dynamical-systems chemistry or ask your own question. The Overflow Blog Stack Overflow badges explained Predator–prey
equations. The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the population dynamics
of two species that interact, one as a predator and the other as prey.
Equations. Chemical Engineering. Soap is prepared through a reaction known as saponification. Homework Help in Differential Equations from CliffsNotes!
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Practical Skills Flashcards
Put these prefixes in order of ascending size: centi, micro, pico, nano, milli, femto.
femto, pico, nano, micro, milli, centi
Put these in ascending order and say their power: mega, giga, kilo, tera.
kilo {k} (10^3), mega {M} (10^6), giga {G} (10^9), tera {T} (10^12)
What is the independent variable?
What is changed in the course of the experiment
What is the difference between a reading and a measurement?
A reading is a value found from a single judgement e.g. looking at a thermometer or ruler, whereas a measurement is multiple readings combined - for example using a stopwatch where there is
uncertainty both at the start and stopping points
How much uncertainty do you add for a reading?
+ or - 0.5 of the resolution
How much uncertainty do you add for a measurement?
As it has to comprise of more than one reading, measurements are mostly taken as at least + or - 1 of the resolution
What is the resolution of a micrometer?
+ or - 0.01 mm as shown on the barrel
What can you do to reduce uncertainty and what is the calculation to determine how much you have reduced it by?
Do many repeat measurements, and you would do uncertainty of the whole measurement/the number of instances
What are two commonly forgotten tips to use when counting oscillations of a pendulum?
Only start counting after one or two oscillations have already passed, use a fiducial marker
Which variable goes on which axis of an experimental graph?
y = dependent variable, x = independent
What does the time base mean on an oscilloscope?
The amount of time one square represents
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A multilevel structural model of mathematical thinking in derivative concept
Thesis Type: Doctorate
Institution Of The Thesis: Orta Doğu Teknik Üniversitesi, Faculty of Education, Mathematics and Science Education, Turkey
Approval Date: 2012
Student: UTKUN ÖZDİL
Supervisor: BEHİYE UBUZ
The purpose of the study was threefold: (1) to determine the factor structure of mathematical thinking at the within-classroom and at the between-classroom level; (2) to investigate the extent of
variation in the relationships among different mathematical thinking constructs at the within- and between-classroom levels; and (3) to examine the cross-level interactions among different types of
mathematical thinking. Previous research was extended by investigating the factor structure of mathematical thinking in derivative at the within- and between-classroom levels, and further examining
the direct, indirect, and cross-level relations among different types of mathematical thinking. Multilevel analyses of a cross-sectional dataset containing two independent samples of undergraduate
students nested within classrooms showed that the within-structure of mathematical thinking includes enactive, iconic, algorithmic, algebraic, formal, and axiomatic thinking, whereas the
between-structure contains formal-axiomatic, proceptual-symbolic, and conceptual-embodied thinking. Major findings from the two-level mathematical thinking model revealed that: (1) enactive, iconic,
algebraic, and axiomatic thinking varied primarily as a function of formal and algorithmic thinking; (2) the strongest direct effect of formal-axiomatic thinking was on proceptual-symbolic thinking;
(3) the nature of the relationships was cyclic at the between-classroom level; (4) the within-classroom mathematical thinking constructs significantly moderate the relationships among
conceptual-embodied, proceptual-symbolic, and formal-axiomatic thinking; and (5) the between-classroom mathematical thinking constructs moderate the relationships among enactive, iconic, algorithmic,
algebraic, formal, and axiomatic thinking. The challenges when using multilevel exploratory factor analysis, multilevel confirmatory factor analysis, and multilevel structural equation modeling with
categorical variables are emphasized. Methodological and educational implications of findings are discussed.
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Convert Density Units with our Units Converter Calculator
Density Dimensional Analysis
Density is the amount of mass per unit volume of an object or substance.
The dimensional analysis stems from the interrelation of two other quantities: mass and volume.
Density = mass / volume
= M¹ / L³
= M¹ L⁻³
where M and L represent the fundamental base units of mass and length, respectively.
Understanding Density Units in Metric and US Systems
In the metric system, the International System of Units (SI) uses the kilogram per cubic meter (kg/m³) as the standard unit for expressing density. This unit represents the mass of a substance in
kilograms divided by its volume in cubic meters. The kilogram is the base unit of mass in the metric system, and the cubic meter is the base unit of volume.
On the other hand, the US or imperial system employs different units for expressing density. The most common unit is the pound per cubic foot (lb/ft³ or pcf), which represents the mass of a substance
in pounds divided by its volume in cubic feet. In this system, the pound is the base unit of mass, and the cubic foot is the base unit of volume.
How to Convert Density from Metric to US Units: kg/m³ to lb/ft³ (pcf)
From the dimensional analysis, we see that density depends on two fundamental units: mass and length.
The conversion factors for fundamental units were established at some point in history. The International Yard and Pound Agreement of 1959 standardized the conversion factors for the pound (lb) and
the foot (ft) in relation to the metric system. According to this agreement:
lb = 0.45359237 kg
ft = 0.3048 m
By employing both conversion factors, we can derive a new factor for converting kg/m³ into lb/ft³:
lb/ft³ = 0.45359237 kg / (0.3048 m)³
= 0.45359237 kg / (0.028316846592 m³)
= 16.0184633739601396 kg/m³
A more useful way of seen this conversion is by using fractions. We can divide both sides of the equation by 16.018463... kg/m³:
(lb/ft³) / (16.0184633739601396 kg/m³) = 1
Now we can use the left side of the equation as a factor to multiply density measured in kg/m³ so metric density units cancel out, resulting in the US unit we're looking for.
Here's an example of how to convert the density of a 700 kg/m³ plank of wood into US units:
700 kg/m³ · conversion_factor
= 700 kg/m³ · (lb/ft³) / (16.0184633739601396 kg/m³)
= 700 / 16.0184633739601396 lb/ft³
= 43.6995724033012284 lb/ft³
You probably don't need a precision of 18 digits. Rounding it to six decimals seems enough for most cases:
700 kg/m³ = 43.699572 lb/ft³
How to Convert Density from US Units to Metric: lb/ft³ (pcf) to kg/m³
Using the same procedure, we can convert lb/ft³ to kg/m³. We just have to invert the fraction to have in the denominator the units we need to cancel out:
(16.0184633739601396 kg/m³) / (lb/ft³) = 1
Here's how to convert the density of a plank of wood of 43 lb/ft³ into metric units:
43 lb/ft³ · conversion_factor
= 43 lb/ft³ · (16.0184633739601396 kg/m³) / (lb/ft³)
= 43 · 16.0184633739601396 kg/m³
= 688.793925080286002 kg/m³
= 688.793925 kg/m³
Common Density Values
Here you'll find a list of density references to effectively compare orders of magnitude.
Construction Materials Density
Reference kg/m³ lb/ft³
Wood 300 - 900 19 - 56
Asphalt 2120 132
Concrete 2400 150
Aluminium 2700 169
Steel 7750 - 8050 484 - 503
Iron 7870 491
Copper 8960 559
Density in Nature
Reference kg/m³ lb/ft³
Air 1.2 0.07
Water 1000 62.4
Sea Water 1030 64.3
Ice 916.7 57.2
Clay 1300 81
Sand 1600 100
Sandstone 2300 144
Granite 2700 169
Basalt 2900 181
Silver 10500 656
Gold 19300 1205
Osmium 22500 1405
Osmium is the densest element found on Earth.
Learn everything about units of measurement. Use our smart App to convert units in real-time with just a few keystrokes, both metric and US units.
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Control valve sizing calculation - EnggCyclopedia
Instrumentation • Process Design • Solved Sample Problems
Control valve sizing calculation
11 Min Read
Control valves are an essential component in many industrial processes, used to regulate the flow of fluids such as gases and liquids. Proper control valve sizing is crucial for maintaining precise
control over the process parameters. Inaccurate valve sizing can result in issues such as cavitation, flashing, and reduced performance.
Control valve sizing calculation is crucial for maintaining efficient and reliable industrial processes. By following the proper sizing process and selecting the appropriate valve type, engineers and
operators can ensure the optimal performance of their industrial processes.
Table of content:
Theory of control valve sizing
Commonly used terms for control valve sizing
Information required for sizing control valve calculation
Control valve sizing calculation
Theory of control valve sizing
Control valve sizing is the process of selecting the appropriate size of a control valve for a specific industrial process application. Control valves are used to regulate the flow of fluids such as
gases and liquids, and proper sizing is essential for maintaining precise control over process parameters such as flow rate, pressure, and temperature.
The sizing process involves determining the process parameters, selecting the appropriate valve type, calculating the required valve coefficient (C[v]), determining the valve size, and verifying
valve performance. Proper sizing ensures that the valve can handle the maximum expected flow rate while maintaining the desired level of control, and can prevent issues such as cavitation, flashing,
and reduced performance.
Control valve sizing can be represented by simple term Cv. Cv of control valve can be written as
C[v] ∝ F√(ρ/∆P)
C[v] is Valve flow coefficient
F is mass flowrate
ρ is specific gravity of fluid at inlet conditions
ΔP is pressure drop across the control valve
Above equation is valid for single phase and correction needs to be applied for two or multiple phases. The proportional constant is typical vendor correction factors and varies between 0.85 and 1.0.
Some vendors apply various factors as viscosity correction, density correction, etc.
When sizing the control valve few things needs to be known such as flow variations, fluid variations, temperature variations, pressure drop variations.
Typical schematic of any control valve system is as follows
We should know following
Psou= Source pressure and elevations of liquid level
Inlet pipe (including lengths, material-roughness), fittings details (elbows, valves, tees, etc), elevations details (from source till inlet of control valves), etc.
Outlet pipe (including lengths, material-roughness), fittings details (elbows, valves, tees, etc), elevations details (from control valves to the end equipment), etc.
Pend = End or destination pressure
With these details, we can estimate the
• Pressure at the inlet of control valve, Pin i.e. difference between Psou and pressure drop in inlet pipe (including elevation). The pressure drop is termed as ΔPin
• Pressure at the outlet of control valve, Pout i.e. addition of Pend and pressure drop in outlet pipe (including elevation). The pressure drop is termed as ΔPout.
Above detail can be written as,
Psou-Pend = ΔPin+ΔPcv+ΔPout
Above equation is valid for system without any pressure increase (like with pump or compressor). The same can be written as
ΔPcv = Psou+ Pincr-Pend - ΔPin - ΔPout
Where, Pincr is the pressure increase in pump or compressor. Please use the performance curve to estimate the pressure increase when characteristic or performance curve is available.
Please note that, pressure drop in piping is function of flowrate. Discharge pressure (discharge head) in pump or compressor is also function of flowrate. If the flowrate increases, then pressure
drop is higher in piping. Essentially it means that, for same source and end pressures, control valve typically get lesser pressure drop with increase in flowrate.
For proper of sizing of control valves all possible flowrate scenario and its pressure drop across (and effectively C[v]) to be reviewed and proper characteristics can be selected. Ratio of maximum C
[v] i.e. C[vmax] and minimum C[vmin] determines rangeability of control valve.
Depending upon the requirement the valve characteristic can be selected. Usually control works when the operating C[v] (C[vmax] and C[vmin]) is between 15% and 85% of selected C[v]. The range of
useful operations changes from operator to operator as well as vendor to vendor. C[v] of 15% to 85% of rated C[v] is approximate healthy range of control valve operations.
When sizing control valve, vendors or design engineers use Cvmax as 85% and select design Cv, it is termed as C[vrated]. If the ratio of C[vmin] and C[vrated] is more than 15% then selected control
valve will work for whole range of control valve. This represents the case linear characteristic of control valve.
Commonly used terms for control valve sizing
Flow Co-efficient of a Valve
The flow coefficient of a valve is a measure of its ability to allow fluid to flow through it. It is a standardized way of comparing the flow rates of different valves. The flow coefficient is
typically expressed in C[v] or K[v] units.
• The C[v] of a valve is defined as the quantity of water, at 60 °F, in US gal/min, that will flow through the valve at a specified travel with a pressure drop of 1 psi.
• The K[v] of a valve is defined as the quantity of water in m3/h, at a temperature between 5 °C and 40 °C that will flow through the valve at a specified travel with a pressure drop of 1 bar.
• K[v] = 0.856 Cv.
The flow coefficient of a valve can be used to determine the flow rate that can be achieved through the valve at a given pressure drop.
Cavitation is the formation and collapse of vapor bubbles in a liquid flowstream caused by changes in pressure and velocity. Cavitation occurs in liquid service when the pressure in the valve body
falls below the vapour pressure of the liquid. The bubbles that are formed will implode immediately or shortly after leaving the valve, due to the downstream pressure of the control valve recovering
to rise above the liquid vapour pressure.
Cavitation can be caused by a number of factors, including low pressure, high velocity, and improper valve design. Cavitation can cause a number of problems, including noise, vibration, erosion, and
leaks. Control valves should be sized to the flow requirements of the system and should be installed in a way that minimizes the formation of vapor bubbles.
Choked Flow
Choked flow occurs when the fluid velocity at vena contracta reaches the speed of sound. Choked flow can occur for both incompressible and compressible fluids.
When choked flow occurs, the fluid velocity cannot increase any further, even if the pressure difference across the nozzle increases. This is because the speed of sound is the maximum possible
velocity for a fluid.
Choked flow can have a number of negative effects on a system. For example, it can cause a significant increase in pressure drop across the nozzle. It can also cause the flow to become turbulent,
which can lead to erosion and vibration.
Occurs, for liquids only, when the pressure in the valve body falls below the liquid vapour pressure and when the bubbles thus formed remain as vapour in the fluid, owing to the fact that the
downstream pressure of the control valve is at or below the liquid vapour pressure.
Control Valve Authority
Control Valve Authority is the ratio between the pressure drop across the control valve at a certain relative travel to the pressure drop across the control valve in its fully closed position.
Inherent equal percentage characteristic
Inherent equal percentage characteristic of a valve is a characteristic whereby equal increments of relative travel yield equal percentage changes of the relative flow coefficient.
Inherent linear characteristic
Inherent linear characteristic of a valve is a characteristic whereby equal increments of relative travel yield equal increments of relative flow coefficient.
Information required for sizing control valve calculation
Information required for sizing control valve is as follows:
• Inlet pressure all possible cases
• Outlet pressure (or pressure drop) all possible cases
• Inlet temperature all possible cases
• Mass Flowrate all possible cases
• Solids content and size
• Critical temperature and critical pressure
• Allowable shutoff differential pressure
• For liquid service
□ Vapor pressure
□ Density
□ Viscosity
□ Surface tension
• For gas service
□ Molecular weight
□ Compressibility factor at the inlet and outlet conditions
□ Cp/Cv ratio
□ Dew point temperature
• Multiphase service
Gas Properties Liquid Properties
Molecular weight Vapor pressure
Compressibility factor at the inlet and outlet conditions Density
Cp/Cv ratio Viscosity
Dew point temperature Surface tension
• Expected valve characteristics
• Design pressure or pressure rating of connecting system
• Design Temperature
• Body size (connecting flange rating)
• Allowable noise level
• Actuator/Air/gas fail status
Control valve sizing calculation
Problem Statement
A level control valve is to be installed on an 8” oil line going from an oil water separator to an oil heater. The oil water separator operates at 3.5 barg and 250C. Inlet pressure requirement at the
heater is 2.0 barg. Normal, minimum and maximum oil flowrates are 200m^3/hr, 60m^3/hr and 220 m^3/hr respectively. Size a level control valve to determine the control valve flow coefficient or valve
Oil properties are –
Density at given conditions = 700 kg/m3
Viscosity at given conditions = 5 cP
Critical pressure = 60 bara
Vapor pressure = 1.5 bara
The details of the oil line are –
Line size 8”
Total length of the line = 50m
Heater inlet nozzle elevation – vessel outlet nozzle elevation = -4.0 m (heater is on the ground and vessel is elevated)
Fittings – 12 nos. of 900 elbows and 2 gate valves
Step 1
First step of solving this control valve sizing sample problem is to determine the line pressure drop resulting due to frictional losses from pipe and fittings plus elevational losses. Normally for
this case, the level control valve would be located close to the separator vessel. Hence the pressure drop between vessel and the control valve has been neglected and inlet pressure to the control
valve has been assumed to be the same as vessel outlet pressure.
Frictional losses from straight pipe alone can be easily calculated using EnggCyclopedia’s pipe pressure drop calculator for single phase flow as follows.
Mass flow of oil = 200 X 700 = 140000 kg/hr (Normal flow case)
Mass flow of oil = 60 X 700 = 42000 kg/hr (Minimum flow case)
Mass flow of oil = 220 X 700 = 164000 kg/hr (Maximum flow case)
As per EnggCyclopedia’s calculator, pressure drop in bar/km of straight pipe is reported here for the 3 cases,
For normal flow case,
pressure loss = 1.09 bar/km; frictional pressure drop in straight pipe = 0.05 X 1.09 = 0.0545 bar
For minimum flow case, pressure loss = 0.128 bar/km; frictional pressure drop in straight pipe = 0.05 X 0.128 = 0.0064 bar
For maximum flow case, pressure loss = 1.445 bar/km; frictional pressure drop in straight pipe = 0.05 X 1.445= 0.0723 bar
Fluid velocity expressed in m/s, for each flow case is also calculated at this time. This velocity will be later used for determination of pressure drop due to fittings.
For normal flow, velocity = 1.71 m/s
For minimum flow, velocity = 0.51 m/s
For maximum flow, velocity = 2.01 m/s
Elevational pressure loss for all three cases is the same and is equal to (density X gravitational acceleration X elevation change).
Hence for all the three cases pressure loss is = 700 X 9.8 X (-4.0) / 105 bar = -0.2744 bar.
Negative value indicates pressure gain instead of pressure loss due to drop in height.
To determine frictional pressure loss due to fittings, first the combined K-factor of fittings is calculated using EnggCyclopedia’s K-factor calculator.
For 12 nos. of 900 elbows and 2 gate valves, K factor = 5.64.
Pressure drop due to fittings is obtained by multiplying the K-factor by ρv2/2 for each case, where ‘v’ is the velocity in m/s. Velocity for each flow case is calculated in EnggCyclopedia’s pipe
pressure drop calculator for single phase flow.
For normal flow case, fittings pressure drop = K X ρv2/2 = 5.64 X 700 X 1.712 / (2 X 105) = 0.0577
For minimum flow case, fittings pressure drop = K X ρv2/2 = 5.64 X 700 X 0.512 / (2 X 105) = 0.0051
For maximum flow case, fittings pressure drop = K X ρv2/2 = 5.64 X 700 X 2.012 / (2 X 105) = 0.0798
Total pressure drop can be the calculated by adding the 3 components calculated independently for each case,
Line pressure drop for normal case = 0.0545 + 0.0577 – 0. 2744 = -0.1623 bar
Line pressure drop for normal case = -0.2629 bar
Line pressure drop for normal case = -0.1224 bar.
Note that, due to drop in height, the net pressure drop in the line has turned out to be negative.
Step 2
Next step for solving the control valve sizing sample problem is to determine the allowable pressure drop across the control valve for each of the three cases. It is calculated as,
Pressure drop across control valve = Vessel outlet pressure – Heater inlet pressure – line pressure drop
For normal flow, ΔP = 3.5 – 2 – (-0.1623) = 1.6622 bar
For normal flow, ΔP = 1.7623 bar
For normal flow, ΔP = 1.6224 bar
Step 3
The final step of solving this sample problem is to determine the control valve flow coefficient or valve C[v ]using EnggCyclopedia’s control valve sizing calculator. The inlet pressure to the valve
is taken as vessel operating pressure since the valve is very close to the vessel. The outlet pressure is taken by considering the allowable pressure drop across the control valve. Following valve C
[v ] values are calculated for the 3 cases.
Normal flow Case, C[v ] = 150.44
Minimum flow case, C[v ]= 44.31
Maximum flow case, C[v ]= 178.94
The chosen valve C[v ]is always higher than the maximum C[v ]requirement with a margin for valve opening. These valve C[v ]values are given to the control valve manufacturer along with corresponding
flowrate values and subsequently a suitable valve with a higher valve C[v ]is chosen to be installed.
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Question with probability! Math Geniuses help me and explain
Question with probability! Math Geniuses help me and explain why!?
I have 2 friends that want to spin this wheel. There are 6 spots on the wheel. Friend #1 says when he spins it, he wants it to land on spot 1. Friend #2 says when he spins it, he wants it to land on
spot #2. What are the odds that they both get what they want when they spin it? Please explain why.
Jeffrey K say: (1/6)*(1/6) = 1/36
Pope say: There are quite a few unstated conditions, which seem to be implied.
If all of the six possible outcomes are equally likely, then they each have probability 1/6. If those outcomes include the numbers 1 and 2, then each of your friends has probability of success 1/6.
If those two trials are independent, then the probability of both being successful is the product of the two individual probabilities.
(1/6)(1/6) = 1/36
That would be the probability of both friends having success. The odds are another matter. The word "odds" has several different interpretations, so there is really no point. There above is the
probability, subject to those added conditions. Do what you want with that.
ted s say: for each spot on 1st friend there are 6 choices for the 2nd friend.....thus a total of 36 possibilities....1 / 36 for both to get what they desire
david say: If it is a 'fair' game ... each number has the same area and the spin is not weighted in some way as to favor a certain number ...
p for 1st friend is 1/6 ... p for 2nd is 1/6
... p both get what they want is 1/6 X 1/6 = 1/36
billrussell42 say: Depends on the spots, how they are spaced, and the percentage of the circumference each one takes.
IF each spot is actually a sector and takes up 60º, so the 6 sectors are evenly distributed and take up all 360º of the circle... IF...
Then for #1 to get sector 1, the odds are 1/6. For number two to get sector 2, his odds are also 1/6
for both to match IN DIFFERENT SPINS, the odds are 1/36, which is (1/6) x (1/6)
To get both to match in one spin is impossible.
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Introducing filters in jamovi
jamovi 0.9 and newer allow you to filter rows and columns out of your analyses. This is useful for excluding outliers, or limiting the scope of your analyses. jamovi filters are built on top of the
jamovi ‘compute variables’ system, allowing great flexibility in filtering.
(note, if you haven’t already updated to jamovi 0.9.0.1 [or newer], we encourage you to do do so).
Row filters
The jamovi 0.9 series has landed, and with it comes the ability to filter out rows that you don’t want included in your analyses. There are a number of reasons why this might be appropriate. For
example, you might want to only include people’s survey responses if they explicitly consented to having their data used, or you might want to exclude all left-handed people, or perhaps people who
score ‘below chance’ in an experimental task. In some cases you just want to exclude extreme scores, for example those that score more than 3 standard deviations from the mean.
The filters in jamovi are build on top of jamovi’s computed variable formula system, which allows the building of arbitrarily complex formulas. For a primer on computed variables in jamovi, there’s
an earlier blog post on them here (although, you should note that we’ve added quite a few new useful functions since that blog post was written.)
To demonstrate jamovi filters (you can follow along if you like), I’m going to open the very simple Tooth Growth data set from the examples. Next, we select the Filters button from the Data tab. This
opens the filter view and creates a new filter called Filter 1. This can be seen in the following GIF:
In this GIF we specify a filter to exclude the 9th row. Perhaps we know that the 9th participant was someone just testing the survey system, and not a proper participant (Tooth Growth is actually
about the length of guinea pig teeth, so perhaps we know that the 9th participant was a rabbit). We can simply exclude them with the formula:
In this expression the != means ‘does not equal’. If you’ve ever used a programming language like R this should be very familiar. Filters in jamovi exclude the rows for which the formula is not true.
in this case, the expression ROW() != 9 is true for all rows except the 9th row. When we apply this filter, the tick in the Filter 1 column of the 9th row changes to a cross, and the whole row greys
out. If we were to run an analysis now, it would run as though the 9th row wasn’t there. Similarly, if we already had run some analyses, they would re-run and the results would update to values not
using the 9th row.
But we can do more complicated filters than this! The Tooth Growth example contains the length of teeth from guinea pigs (the len column) fed different dosages (the dose column) of supplements;
vitamin c or orange juice (recorded in the supp column). Let’s say that we’re interested in the effect of dosage on tooth length. We might run an ANOVA with len as the dependent variable, and dose as
the grouping variable. But let’s say that we’re only interested in the effects of vitamin c, and not of orange juice. we can use the formula:
In fact we can specify this formula in addition to the ROW() != 9 formula if we like. We can add it as another expression to Filter 1 (by clicking the small + beside the first formula), or we can add
it as an additional filter (by selecting the large + to the left of the filters dialog box). As we’ll see, adding an expression to an existing filter does not provide exactly the same behaviour as
creating a separate filter. In this case however, it doesn’t make a difference, so we’ll just add it to the existing filter. This additional expression comes to be represented with its own column as
well, and by looking at the ticks and crosses, we can see which filter or expression is responsible for excluding each row.
But let’s say we want to exclude from the analysis all the tooth lengths that were more than 1.5 standard deviations from the mean. To do this, we’d take a Z-score, and check that it falls between
-1.5 and 1.5. we could use the formula:
or if we’re really keen:
-1.5 < (len - VMEAN(len)) / VSTDEV(len) < 1.5
(this last formula is a great way to demonstrate to students what a z-score is.)
There are a lot of functions available in jamovi, and you can see them by clicking the small
Now let’s add this z-score formula to a separate filter by clicking the large + to the left of the filters, and adding it to Filter 2.
With multiple filters, the filtered rows cascade from one filter into the next. So only the rows allowed through by Filter 1 are used in the calculations for Filter 2. In this case, the mean and
standard deviation for the z-score will be based only on the Vitamin C rows (and also not on row 9). In contrast, if we’d specified our Z() filter as an additional expression in Filter 1, then the
mean and standard deviation for the z-score would be based on the entire dataset.
In this way you can specify arbitrarily complex rules for when a row should be included in analyses or not (but you should pre-register your rules^1).
So that is how row filters work in jamovi, which apply to the data set as a whole. However, sometimes you want to just filter individual columns. For that there are column filters.
Column filters
Column filters come in handy when you want to filter some rows for some analyses, but not for all. This is achieved with the computed variable system (you can read more about computed variables in
our earlier blog post here). With the computed variables we create a copy of an existing column, but with the unwanted values excluded.
In the Tooth Growth example, we might want to analyse the doses of 500 and 1000, and 1000 and 2000 separately. To do this we create a new column for each subset. So in our example, we can select the
dose column in the jamovi spreadsheet, and then select the Compute button from the data tab. This creates a new column to the right called dose (2), and same as the filters, we can enter a formula.
in this case we’ll enter the formula:
FILTER(dose, dose <= 1000)
Or if you prefer:
FILTER(dose, dose == 1000 or dose == 500)
The first argument to the FILTER() function (in this example dose) is what values to use in the computed column. The second argument is the condition; when this condition isn’t satisfied, the value
comes across blank (or as a ‘missing value’ if you prefer). So with this formula, the dose (2) column contains all the 500 and 1000 values, but the 2000 values are not there.
We might also change the name of the column to something more descriptive, like dose 5,10. Similarly we can create a column dose 10,20 with the formula FILTER(dose, dose != 500). Now we can run two
separate ANOVAs (or t-tests) using len as the dependent variable, and dose 5,10 as one grouping variable in the first analysis, and dose 10,20 in the other. In this way we can use different filters
for different analyses. Contrast this with row filters which are applied to all the analyses.
It may also have occurred to you, that with FILTER() we can do what might be called a ‘poor man’s split variables’. In the future jamovi will provide a dedicated UI for ‘splitting variables’, but in
the mean time you can create splits using FILTER(). For example, we could split len into two new columns len VC and len OJ with the functions FILTER(len, supp == 'VC') and FILTER(len, supp == 'OJ')
respectively. This results in two separate columns which can be analysed side-by-side.
So that’s column filters and row filters. We hope you find them as satisfying to use as we’ve found developing them!
^1Pre-registration is the solution to p-hacking, not deliberately making software difficult to use! Don’t p-hack. Your p-hacking harms more people than you know.
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Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity
A square matrix V is called rigid if every matrix V^′ obtained by altering a small number of entries of V has sufficiently high rank. While random matrices are rigid with high probability, no
explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of
a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this
paper, we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field F[2]. Stated informally, we conjecture that under a
suitable embedding of F2n into R^n, vectors arising from a low-dimensional F[2]-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a
low-dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a
collection of results establishing weaker forms and special cases of the conjecture above.
Bibliographical note
Publisher Copyright:
© 2016, Springer International Publishing.
• Kolmogorov width
• Matrix rigidity
• linear codes
Dive into the research topics of 'Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity'. Together they form a unique fingerprint.
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What must be the velocity of a moving body? - EasyRelocated
What must be the velocity of a moving body?
What must be the velocity of a moving body?
(d) velocity of a moving body is its speed in a given direction.
What is necessary to describe motion of a body?
You can describe the motion of an object by its position, speed, direction, and acceleration.
What is required to stop the moving body?
A force is required to stop a moving object.
What is moving body in physics?
In physics, motion is when an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of
reference to an observer, measuring the change in position of the body relative to that frame with a change in time.
Can velocity be zero for a moving body?
The average velocity of a moving body can be zero because, for a moving body, the displacement can be zero if the body returns to its initial location. But distance can never be zero.
Can a body move without velocity?
Velocity is a vector, and speed is its magnitude. Ifspeed is nonzero, then velocity has a nonzero magnitude, and is therefore not a zero vector. Yesvelocity can be zero if displacement is zero.
However the body must cover some distance and return to its initial position .
What are the different types of motion a body can undergo?
• Linear Motion. In linear motion, the object moves from one position to another in either a curved direction or a straight line. …
• Rotary Motion. Rotary motion is a sort of motion in which the object moves in a circle. …
• Oscillatory Motion. …
• Periodic Motion.
What is the state of motion of a body?
The state of motion of an object is described by its speed and the direction of motion. For a body having zero speed, the state of motion is rest. For a body having a constant speed, the state of
motion is uniform motion. For a body having variable speed and (or) direction, the state of motion is non-uniform motion.
What are the laws of motion of a body?
Newton’s laws of motion relate an object’s motion to the forces acting on it. In the first law, an object will not change its motion unless a force acts on it. In the second law, the force on an
object is equal to its mass times its acceleration.
What are 2 examples of inertia?
• Dust coming out of mat when beaten.
• Falling forward in transport when sudden breaks are applied.
• Leaves get detached from the tree when shaken hardly.
• Due to inertia, artificial satellites keep moving in a circular motion.
Is force needed to keep a moving body in motion?
No , it is not necessary to apply force to keep the body moving as such. One a body is set in motion, Unless there is any external force it will continue to move with present state of motion.
What is the resultant force called?
Net force (also known as resultant force) is the overall force acting on an object when all the individual forces acting on the object are added together. The net force is a vector produced when two
or more forces act on a single object.
What are 4 types of motion?
• linear.
• rotary.
• reciprocating.
• oscillating.
Which is the SI unit of speed?
The SI unit of speed is the meter per second, abbreviated as m/s or ms-1.
Who discovered motion?
Motion is the area of physics that studies how things move. It was developed about 300 years ago by Isaac Newton. He discovered that all objects move according to three basic laws. Newton’s Laws can
explain almost all of physics.
What is the velocity of a moving?
Solution: The velocity of a moving body is its speed in a given direction is a correct statement regarding the velocity and speed of a moving body. Velocity is a vector quantity which has both
magnitude and direction. The speed of the moving object is always greater.
What must be the velocity of a moving body of mass 2kg so that its ke is 25j?
Expert-Verified Answer v = 5 m/s Ans. Ans.) The velocity of a moving body is 5 m/s.
What is the velocity of a moving vehicle?
The velocity of a moving car is given by v = X t + Y t where v is the velocity and t is the time. Then the SI units of X and Y respectively will be: m ; ms.
What is the velocity of the body?
The velocity of a body is given by the equation v=𝑏t+ct2+dt3 The dimensional formula of b is.
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How to use this tool?
This free online converter lets you convert code from Matlab to Java in a click of a button. To use this converter, take the following steps -
1. Type or paste your Matlab code in the input box.
2. Click the convert button.
3. The resulting Java code from the conversion will be displayed in the output box.
The following are examples of code conversion from Matlab to Java using this converter. Note that you may not always get the same code since it is generated by an AI language model which is not 100%
deterministic and gets updated from time to time.
Example 1 - Is String Palindrome
Program that checks if a string is a palindrome or not.
Example 2 - Even or Odd
A well commented function to check if a number if odd or even.
Key differences between Matlab and Java
Characteristic Matlab Java
Syntax Matlab uses a syntax that is similar to traditional programming languages, but with a focus Java uses a syntax that is similar to C++, with a focus on object-oriented programming
on mathematical operations and matrix manipulation. and a strict syntax.
Paradigm Matlab is primarily a procedural programming language, but it also supports some Java is a fully object-oriented programming language.
object-oriented programming concepts.
Typing Matlab is dynamically typed, meaning that variable types are determined at runtime. Java is statically typed, meaning that variable types are determined at compile time.
Performance Matlab is optimized for numerical computations and has built-in support for matrix Java is generally slower than Matlab for numerical computations, but it is faster for
operations, making it very fast for these types of calculations. other types of operations.
Libraries and Matlab has a large number of built-in libraries and toolboxes for numerical computations, Java has a large number of libraries and frameworks for a wide range of applications,
frameworks signal processing, image processing, and more. including web development, mobile development, and more.
Community and Matlab has a smaller community than Java, but it is still very active and has a large number Java has a large and active community with a wealth of resources and support available.
support of resources available.
Learning curve Matlab has a relatively low learning curve, especially for those with a background in Java has a steeper learning curve than Matlab, especially for those new to programming.
mathematics or engineering.
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Free Worksheet To Practice Order Of Operations From Front Porch Math | Order of Operation Worksheets
Free Worksheet To Practice Order Of Operations From Front Porch Math
Free Worksheet To Practice Order Of Operations From Front Porch Math
Free Worksheet To Practice Order Of Operations From Front Porch Math – You may have listened to of an Order Of Operations Worksheet, but what exactly is it? In addition, worksheets are a great way
for students to exercise brand-new skills and evaluation old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a type of math worksheet that requires students to execute mathematics operations. These worksheets are divided right into three main sections: multiplication,
addition, as well as subtraction. They also include the examination of parentheses and also backers. Pupils that are still learning exactly how to do these tasks will certainly find this sort of
worksheet valuable.
The major function of an order of operations worksheet is to aid trainees discover the correct method to solve math equations. They can evaluate it by referring to an explanation web page if a pupil
does not yet comprehend the principle of order of operations. Additionally, an order of operations worksheet can be separated into numerous categories, based on its difficulty.
One more essential purpose of an order of operations worksheet is to teach students exactly how to carry out PEMDAS operations. These worksheets begin with straightforward problems associated with
the fundamental rules as well as accumulate to more complex troubles entailing all of the policies. These worksheets are an excellent method to present young learners to the excitement of fixing
algebraic equations.
Why is Order of Operations Important?
One of the most essential things you can find out in math is the order of operations. The order of operations ensures that the math issues you fix correspond. This is crucial for tests and also
real-life estimations. When addressing a math trouble, the order needs to start with backers or parentheses, adhered to by subtraction, addition, as well as multiplication.
An order of operations worksheet is a great method to instruct trainees the proper way to fix mathematics equations. Prior to students start utilizing this worksheet, they might need to review ideas
associated to the order of operations.
An order of operations worksheet can help pupils establish their abilities in addition as well as subtraction. Teachers can utilize Prodigy as a simple way to distinguish technique and also deliver
interesting web content. Prodigy’s worksheets are a perfect method to help students discover the order of operations. Educators can begin with the fundamental principles of addition, multiplication,
as well as division to assist pupils develop their understanding of parentheses.
Order Of Operations Fun Worksheets
Order Of Operations Free Activity Math Games Middle School Middle
Order Of Operations Fun Worksheets
Order Of Operations Fun Worksheets give a great source for young students. These worksheets can be quickly tailored for certain demands.
The Order Of Operations Fun Worksheets can be downloaded free of charge as well as can be printed out. They can then be reviewed using addition, subtraction, division, as well as multiplication.
Pupils can likewise use these worksheets to examine order of operations as well as making use of backers.
Related For Order Of Operations Fun Worksheets
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Naming Polynomials Graphic Organizer | Math = Love
Naming Polynomials Graphic Organizer
Today I want to share a naming polynomials graphic organizer I created for my students to glue in their algebra interactive notebooks. I want to start by sharing how my philosophy regarding teaching
vocabulary has changed recently.
Last year, I had a conversation with a student that changed my outlook on vocabulary. This was not my own student but the child of a coworker. Before tutoring him one day, he was sitting in his mom’s
office, discussing why he was having so much trouble in algebra. He said, “My teacher just keeps going on and on and on. And, he keeps saying this word that nobody knows what it means. And, the whole
class is lost.” Naturally, I wanted to know what the word was. “I don’t know. I think it starts with a b.” Since they were working on polynomials and factoring, I took an educated guess: “binomial.”
Yes, that was the word. Once I described to this student what a binomial was, he began to realize that maybe this wasn’t as hard as he had made it out to be.
This year, I am emphasizing vocabulary more. I don’t want students to think that I use words without ever telling them what they mean. At the very least, they should know that the vocabulary word
should be in their interactive notebook somewhere.
We spent an entire 50-minute period on the definition of a polynomial and how to name polynomials. Is this on the EOI? No. But should I still teach it? Yes! When my students see one of these words, I
want them to feel confident, not confused or frustrated. These are words they will encounter for the rest of their mathematical careers. I’m hoping that by putting emphasis on them now, I will save
my students a lot of grief later on.
I told my students that when polynomial parents have children, they don’t get to choose their names like human parents do. Instead, polynomial parents must follow strict naming rules. I lamented
about how sad this was. I mean, what if the parents wanted to be creative? What if the parents wanted their child to have the same last name as them? The first name of any polynomial child is
determined by its degree. The last name of any polynomial child is determined by its number of terms.
One of my students asked me if I was going to use these rules to name my own children. Apparently, I seem like the type of person who would name my child “Cubic Trinomial.” I guess I should take that
as a compliment…
Updated Version of Naming Polynomials Graphic Organizer
In later years, I simplified the naming polynomials graphic organizer a bit.
We followed this up with a naming polynomials practice sheet I created.
I have also created a similar naming polynomials poster that you can hang in your classroom.
More Activities for Teaching Polynomials
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Artistic License Statistics
Quotes • Headscratchers • Playing With • Useful Notes • Analysis • Image Links • Haiku • Laconic
It has been generally demonstrated that, because human brains are wired toward pattern detection, we are lousy at intuitively interpreting statistics; this is the main reason why casinos are viable
businesses. Trying to do anything to curb this problem often results in the worship of the Random Number God, or beliefs like:
• The hit/miss belief: "A hit ratio below 25% is hopeless and a hit ratio above 75% is guaranteed. Everything else is a crapshoot."
Not so much. There are four groups of 25% in 100%. Go ahead and count them. We'll wait.
There is a 1 in 4 chance of hitting any one of them.
• The Gambler's fallacy: All probabilities should somehow "even out" while you're playing. For example, if the computer has a hit chance of 50%, and hits, that's okay. However, if it then scores
another hit right away, The Computer Is a Cheating Bastard. In truth, it just happened to be the way the "dice" fell. As is often stated, "dice have no memory."
• Naive Combination of probabilities: Given the probabilities of two events, people will often simply either add them or multiply them. Generally speaking, calculating the combined probability is
much more complicated. For example, if someone accused a group of 100 people of taking drugs, each person would be 1%. Accusing 4% of adults, and 4% of children, if the group is half of each,
would be 4 people, not 8.
• The Definition of Probability: There's two ways probability can be defined. The first is what should happen in a random process in the long run. The second is the degree of certainty with which a
belief is held. The first definition applies to statements like, "The odds of rolling a six on a fair die are one in six." The second applies to statements like, "My favorite team will win this
game." This difference can be very important. These two views are called frequentist and Bayesian, respectively.
• Decision-Making and Probability: Many make a mistake related to all these fallacies. When a decision-maker makes a decision to play the odds in a situation where he can calculate the odds, he's
taking a measured risk based on what he knows at the time. This is his best decision based on what he could possibly know at the time. If this fails, calling it a wrong decision is fallacious
because it would require knowing the less-likely alternative would happen. For example, if two gamblers agree to roll a fair die, betting 1:1 where Gambler A wins on a 1-5, and Gambler B wins on
a 6, Gambler B is making an idiotic decision to bet — and the decision remains idiotic even in hindsight if he wins.
Note that while this often makes fun of the developers messing up at statistics or the author having no clue how it works, this can actually be invoked or justified.
Also see 20% More Awesome, which involves quantifying the unquantifiable.
Examples of how this plays out in storytelling:
Card and Dice Games[ ]
• Any game of chance — but most especially any game which uses dice — will find players who think the right decision is the decision that agrees with the dice as they rolled after they have rolled.
For example, in Monopoly, you may decide to build houses when you see your opponent will land on your monopoly a throw of 6, 7, or 9 on two six-sided dice. (This is not an error: no monopoly on a
standard Monopoly board is spaced so you would land on it on a 6, 7, or 8, though if there were one, it would have higher odds than the above combination.) Anyone with half a clue as to how the
game works and basic probability theory realizes that's about as lethal a situation as your opponent could be in (for a single monopoly), and would build. Yet if your opponent throws a 12, and
bypasses your entire trap, your decision was just as reasonable as before. It just didn't pan out. This sort of fallacious thinking holds for:
□ Naive poker players, who fold a bad hand only to see it turn around later (in a game with community cards)
□ Players of any RPG, when a character fails at a good plan due to some really off-the-wall lousy rolls or succeeds at an absurd plan through sheer dumb luck
□ Board game players who fall into the type of thinking in the Monopoly example
□ Wargamers who misinterpret why some of their opponents quickly calculate odds, then make their decisions based on what's likely to occur from a particular gambit
□ Anyone playing roulette for anything other than costly entertainment.
• Many Bridge players feel that the computer-generated hands used for many duplicate games are more unusual (i.e. favor more unlikely distribution of cards) than human-shuffled hands. They're
right, but in a backwards way: The computer-generated hands are more likely to be completely random than hands dealt from a human-shuffled deck. Even the best human-shuffled deck will retain a
few cards in the same relative order as they were played in the last hand; computer-generated hands don't (except at the frequency you'd expect from random chance).
• Many players of the online version of Magic: The Gathering are convinced that the algorithm used to shuffle players' decks is flawed and is biased. (Some say the bias is towards "mana flood",
where you get too many mana-producing cards (and not enough spells to actually use that mana with), while others say towards "mana screw", which is the exact opposite—not getting enough.) In
reality, the algorithm is completely incapable of either, since it does not consider what type any given card is when performing the shuffle. The reason for the dissonance between physical and
online play (when there is one at all—mana screw and mana flood are common on cardboard too) is that having to physically shuffle a deck enough to provide a truly random distribution every time
would be incredibly annoying, particularly given the number of times some decks end up being shuffled in a single game. Most people just take their land cards, which end up all in one pile at the
end of a game and put them into the deck at fairly even intervals to avoid there being giant clumps of nothing but land. For practical reasons, even in tournaments it's accepted that the deck
doesn't have to be truly randomly distributed—it just needs to be random enough that a player can't predict what comes next.
□ This one-land-per-two-cards sorting prior to the deck shuffling is accepted practice in tournaments, provided that they are spread blindly vis-a-vis the remaining cards. Any further stacking
of the cards is usually frowned upon, such as placing a Dark Ritual right next to each one of your Hypnotic Specters.
• Go to any online Poker forum and look in the General Discussion forum. More often then not, you'll find a sticky about the game not being rigged, and an explanation of why it may seem that it is.
Of course, most forums will also have a 'Bad Beats' section for whining about said 'rigged' play screwing the loser... (never mind that they were chasing a flush draw and getting really poor pot
odds on the call...)
□ In professional (off-line) poker tournaments, the dealer start to shuffle every new deck by simply scattering the cards on the table and mixing them around (similar to how one would shuffle
dominoes). Then the cards are loaded into whatever shuffling device is used. (This type of shuffle is commonly called a Beginner's or Corgi shuffle.)
☆ If you're interested, in the UK this shuffle is known as a 'chemmy' (pronounced shemmy), named after the game 'chemin-de-fer' made popular in French casinos, but known to most as the game
Baccarat seen in a number of James Bond movies.
• Casino games are set up so that over a long period of time, the statistical average favours the house (house advantage or house edge). In Poker, the bets are fair, since you're playing against
other players rather than against the house, but instead the casino makes money through rakes of the pot and fees.
□ The side bets on Craps tables are particularly blatant, because the fair odds are so simple to calculate. For example, the odds of rolling two sixes are 1/6 * 1/6 = 1/36 (1:35 odds), but the
payoff on that side bet is 30:1.
☆ The one fair bet in Craps, the free-odds wager, pays a fair, proportional amount should you win. That said, there is no space on the Craps table for it (the player has to "know" to place
it at the right time), and it can only be placed as a supplement to your original bet (which is subject to the house percentage.)
□ Technically, there are single player casino games which sometimes offer a theoretical gain to the player with the right strategy. However, since the gain is very small, any mistake will set
you back a lot.
☆ In Video Poker, the advantage is extremely small, if present at all. On average, it takes three solid years of perfect play to break even.
☆ In Black Jack, it requires card counting (and maximizing your bet when the odds are slightly in your favor), and they'll kick you out if you try it. Or they'll reshuffle the deck
frequently in the case of Atlantic City casinos, where they can't kick you out. There are also special table rules that messes the available strategies up, like the house hitting on soft
• One popular strategy (called Martingale) in Roulette that is believed to always net you money. The same strategy works the same on any 50% chance double money back bets (or as long as the chance
to win is balanced against the payout). The basics of the strategy is to bet 1 on red/black, odd/even, or high/low when you start and if you win, and double the bet if you lose.^[1] The belief is
that you will eventually win, and thus win the initial bet.
There are a few reasons why this doesn't work:
□ To always win, you need an infinite amount of money and time.
□ In real casinos there is always a betting limit, which removes the option to double up at some point.
□ The green fields on a roulette table, a 0 and sometimes also a 00, reduces the chance to win to 18:37 or 18:38, which is less than 50%. Because the expected profit is negative, the sum of
many such bets is also negative.
□ If your original lost bet is 1 dollar and you follow this strategem, an unlucky streak of 10 consecutive losses has you betting 1024 dollars to chase after your original loss of a single
dollar. Even if you do win, all you get is your original dollar back. You were better off just betting one dollar at a time and hoping for a winning streak.
• There is an optional "Event Deck" for the board game Settlers of Catan. Using it instead of the dice makes probabilities "even out" somewhat (going through most of the deck before reshuffling
guarantees that each number will come up about as often as it "should").
• In the Asterix album The Soothsayer, a centurion is tasked by the roman empire to round up all prophets and soothsayers in order to curb down pagan beliefs that go against roman pantheon beliefs.
A conman passing himself as a soothsayer gets caught, and is given a test to see if he can predict a roll of two six-sided dice. He breathes a sigh a relief as he knows his luck is usually awful,
and picks (stupidly, statistically speaking) seven, which just so happens to come up on the dice and "prove" him the real deal. He goes on an Insane Troll Logic demonstration that he picked the
right number because he can't tell the future. The centurion isn't convinced until the soothsayer mentions that the village believe anything he tells them, which makes the centurion offer to let
them go if he convinces the villagers to leave.
• The Science of Discworld books have an arguably accurate but somewhat twisted take on statistics: the chances of anything at all happening are so remote that it doesn't make sense to be surprised
at specific unlikely things.
• Dave Barry once joked that he always flew on the airline with the most recent crash, on the assumption that it wouldn't be "due" for another one.
• Mark Twain's Life on the Mississippi contained the following proof of what you can do with statistics:
In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year.
Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one
million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the
Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a
mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
• In Lois Duncan's A Gift of Magic, the psychic protagonist, Nancy, is given a standard test to detect telepathic abilities. She is asked to pick, without looking, all the white cards out of a deck
of cards filled with an equal amount of black and white cards. Because she wishes to hide her ability, she picks all the black cards so that she would get all the "wrong" answers and fail the
test. The examiner sees right through Nancy's ploy because there is an equal probability of picking only white or only black cards and explains that if she really wanted to screw up the test, she
should have picked a roughly equal amount of black and white cards at random.
□ That might not have helped. People are horrible at generating random numbers, so even if she picked equal (or near-equal) numbers of black and white cards, a more sophisticated analysis of
her picks would reveal what she was doing, most likely by identifying a lack of runs of a single color (see fallacy #2 above). It might delay the recognition of her ability, though...and
unless it were blatantly obvious what she was doing, it might leave enough doubt to prevent others from being certain.
• A high school science teacher on The Daily Show thought there was a 50/50 chance of the LHC creating a black hole and causing The End of the World as We Know It. His rationale? It could happen,
or it couldn't happen, therefore there was a 1 in 2 chance of the apocalypse. You Fail Nuclear Physics Forever is also involved.
□ Correspondent John Oliver, who was conducting the interview, then suggested that he and the teacher try to breed after the end. The teacher replied that this was impossible, as both were
male, but Oliver insisted it would either happen or not happen, a one-in-two chance!
• In the Corner Gas episode "Security Cam", Karen figures that there's a 50% chance of a riot breaking out in downtown Dog River, using exactly the same reasoning.
• On The O'Reilly Factor, Bill O'Reilly argued that life expectancy was lower in the US than in Canada because the US has ten times as many people, and therefore has ten times the number of
• On the second episode of Burn Notice, Michael guessed that a conman's former cellmate didn't drink, which made some sense in the context if he was genre savvy to those sorts of questions, but his
explanation didn't: that he just guessed because the man either drank or he didn't, a fifty percent chance. So either Michael's estimate of teetotalers among the male prisoner population is
extremely optimistic, or he needs to take a stats class.
□ A bit more complicated: from the way the conversation was going, it was clear that the conman was testing Michael on whether he really knew the former cellmate. So the odds weren't whether
the cellmate was a drinker, but whether the comment about the drinking was a "test question" or not.
• A common mention on the show Hells Kitchen, as well as a number of other reality shows, is that at any given time a given contestant has a 1 in X chance of winning the grand prize, where X is the
number of remaining contestants. Not only does this suggest that the winner is chosen at random (which is not the intent of the statement), but also that every contestant is equally likely to
win. This is untrue, especially on shows which have a number of obvious dud contestants (such as Hell's Kitchen).
□ Also commonly used by wrestling commentators when discussing multi-person matches. They frequently claim that the champion in a 4-way match only has a 25% chance of retaining his title, with
no regard to comparative skill levels or possible alliances between the participants. Of course, since the outcome is predetermined, it tends to be much more common for the champion to retain
his belt. These sort of statistical predictions are even more stupid in matches like the elimination chamber where the final competitor to be released would clearly have a huge advantage if
other other factors were equal.
• In the Law & Order episode "Coma", McCoy tries to ease Kincaid's conscience about subjecting a comatose victim to a high-risk surgery in order to remove a possibly trial-winning bullet. Subverted
in that he's perfectly aware that it's bad statistics.
"Well, I see one of three things happening: she gets better, she gets worse, she stays the same, and we get strong evidence. Two out of three ain't bad."
• Invoked in Survivor — As the players in the game dwindle, Probst tells them that they have a "one in x shot at winning the million dollars." The way he mentions this, it sounds like the winner of
challenges (and at the game period) is chosen at random, when it actually isn't. (You can argue that if you're in the final six with The Load and someone who the jury hates, you would have a one
in four shot since the jurors would not vote for them.) Justified in that he does this to motivate the players, and it's part of his "character".
• The Fox News Channel's fondness for flashy graphics to engage the viewer's attention occasionally lends itself to a few mistakes. Such as a pie chart where the total breakdowns add up to 193%, or
this poll with a breakdown that adds up to 120%. Either with the pressure of the rush to get on-screen information ready by showtime, those responsible have little time to double-check their
work; or, they care more about making a quick impression on the viewer than ensuring accurate information.
RPGs, MMORPGs, and other Video Games[ ]
• MMO players, almost without fail, will adhere to mindset two — they will notice the streak of resists/misses/landed enemy attacks/what have you that killed or almost killed them, but never notice
the long, long, long string of hits that precede it. Any and all MMO forums will have a topic pop up fairly regularly asking whether (or sometimes screaming loudly even with no evidence to that
effect other than they had a string of bad luck) the RNG is broken.
□ ...which can lead to no small confusion at times.
□ To further complicate things, some MMOs actually do use a skewed RNG, precisely because true randomness could, in theory, result in a string of misses one real day long, or the opposite.
Since MMOs rely on a very predictable form of randomness (e.g., no plucky level 1 can be able to beat a level 20 monster because the monster miraculously rolls no hits, but if attacking a
level 4 he must be able to win through pure luck some of the time), various measures can be put in place to make sure the game generates the good, reliable sort of random.
• All Fire Emblem games after the fifth display inaccurate hit/miss percentages. The game actually uses the average of two random numbers to determine a hit, so a 75% chance to hit is really 87.5%.
This system is likely in place to make dodging-type units evade more (and thus more viable) and high-accuracy characters strike more and lessens the chance that such a character dies (Due to
permanent death and limited saving, this means restarting the entire level in most games) against all 3 of the random mooks that has a 2% chance to hit each.
□ To prevent Save Scumming abuse, the 10th game's (unlike the 9th's, which was completely random) bonus experience system ^[2] will always increases a character's 3 stats with the highest
growth rate (Has an x percent chance to raise this stats on every level up). This wound up making it more broken, as some units quickly hit the Cap on their main stats (Aran), causing stats
that would other almost never grow to suddenly increase at insane rates.
□ Starting with the 10th, if a character does not get at least one stats increased during a level up, the game rerolls (unless a character has hit the cap on everything). Starting with the 11th
game, a characters growth rates will be boosted or dropped if they are behind or ahead of the "average" stats. Like the main example, this helps deal with the very annoying chance that a
character gets "RNG Screwed", except this is enough to force a restart on an entire file in some cases.
• Word of God to the contrary, most players of Puzzle Quest: Challenge Of The Warlords believe that the game "nudges" all sort of random stats in its own favor. As many people complain about the
computer's habit of chaining together 4/5 gem combos and extra turns, it's even more blatant in Spell Resistance, where an opponent with 2% resistance across the board will block approximately
15% of spells. The player, with the same stats, will be lucky to block one spell in hundreds.
• This trope is often brought up in MMORPGs, where many players believe that item drop rates can be mathematically calculated to determine how many monsters you must kill until you "should" find
said item, by assuming that a 1% drop rate means that after you've killed a hundred, something's wrong if you haven't gotten one.
□ Because of players complaining about this, the drop rate formula in World of Warcraft was changed to increase the drop percentage every time the quest item required doesn't drop and reset it
after one does drop. Of course, this is also to avoid the wild variation in time a quest can take when it's truly random.
□ Indeed, while the mean number of kills is 100, the actual number will be greater than 100 37% of the time. Of course, this also means that 50% of the time it will require fewer than 70 kills.
• Speaking of World of Warcraft, during the famous "Leeroy Jenkins" video, someone is asked to do a number crunch to calculate their odds of finishing an encounter. It's actually not as simple as
that — it was done to make fun of guilds as well as Straw Vulcans who may often rely on statistical fallacies.
• City of Heroes actually has a mechanic that behaves like the second part, called the "streakbreaker". For a given base percentage chance to hit, if a player or mob misses a certain number of
times in a row, the next hit is guaranteed. For a hit chance below 20% you have to miss something like 100 times in a row, but for hit chances above 90%, it only takes one miss to get a
guaranteed hit on the next attack. If you were paying REALLY close attention, you could use this to ensure that a key attack doesn't miss.
• Dungeon Fighter Online has a dice roller that is perfectly random for the first instance of every sequence (first upgrade attempt, or random item pickup, or something similar), but then often
produces identical results for the next several sets (Failing an identical upgrade five times in a row, the same player getting every single item in a dungeon). It often "corrects" itself and
skews the other way until results are even. The hit/miss ratio is the same, either producing a lot of hits or a lot of misses in a row, only rarely looking like the actual statistic.
• The Tetris Guideline has mandated that all Tetris games since around late 2005 have an implementation to make the gambler's fallacy actually happen (and make players complain less of being
screwed by the RNG): Instead of rolling a D7 to select a piece, newer Tetris games take a sequence of all seven pieces and deals random permutations of it. Thus, after every 7th piece, all seven
have appeared with equal frequency. This also makes every 7th piece completely predictable.
□ Prior to that, the Tetris the Grand Master series also had an algorithm to make the gambler's fallacy come true: The game rolls 6 times (4 in the first TGM) and takes the first result that
isn't identical to any of the four most recent pieces dealt. It's still possible for this to "fail" and give you the same pieces over and over again since the game only rolls a fixed number
of times; it's just much less likely than with a simple RNG approach.
• Ask anyone who's played Civilization IV (especially those who play mods like Fall From Heaven) and they will tell you that any combat with less than 80% odds is suicidal and should be avoided at
all costs,^[3] unless the odds are 1% or worse, in which case victory is surprisingly possible (see Spearman v. Tank).
□ Alleviated somewhat in the sequel, which is kind enough to give you all of the information BEFORE you attack and provides a rough estimate of where the forces will end up in strength after
the round of combat. It was actually criticized heavily for its near-perfect accuracy in prediction! Later patches actually made it a bit more random.
• This trope is hugely responsible for the Pokémon entries on The Computer Is a Cheating Bastard, and is the number 1 thing the game's professional players complain about to similar levels of
□ In a more topical instance, players have a random 1/8192 chance of finding an alternately colored Pokemon, similar to albinism and what not. Many players only encounter one or two in several
years of playing, others never find one, and some find them with surprising regularity.
□ In a similar way, the Pokerus (a "virus" that doubles a Pokemon's stat growth) has a 3/65,536 chance to be on a Pokemon, or 1/3 the probability of finding a "shiny" Pokemon. Many players have
never seen the Pokerus, while a few have been lucky enough to get it more than once. Once you have a Pokemon with the Pokerus, though, it's very easy to spread it around the party.
□ The first generation did have statistical errors due to bugs, such as attacks that should never miss actually having a 1/256 chance of missing due to the code using "less than" checks instead
of "less than or equal to" checks.
• Final Fantasy Legend, with the infamous Saw Game Breaker weapon. On the enemies it would work on at all, (see Game Breaker about the flaw making it work on too many enemies), it had a 50% chance
of getting a One Hit KO. In practice, this meant that it would alternate between hitting and missing. Thus, if you wanted to use them in battle, just equip two different characters with them and
have them both use it in a given round. If the first misses, the second would be guaranteed to hit.
• Final Fantasy Tactics a 2: many people report that attacks that give a 95% success rate fail often. It seems likely that this is the case given the number of complaints (especially since the
previous game didn't have these problems — then again, the previous game had an actual flaw in its RNG where success rates tend to be universally higher than the shown numbers suggest) but
obviously it's impossible to say for sure.
• X-COM's accuracy reports during combat aren't exactly blatant lies, but they're not exactly accurate, either. What X-COM does for a hit check is up to two rolls. The first is done against the
accuracy check, and if it passes, you automatically get a dead-on shot. The other roll, if the first fails, is the deviation from where you're aiming, which may also end up being nil, resulting
in a dead-on shot. So that 75% Accuracy the game reports? More like 77% to hit the target you're aiming at, and up to around 20% to hit someone else, resulting in somewhere around a 86% (on
average) chance of someone getting hit by any given shot in a heated battle. Oh, and 100% accuracy reportedly doesn't exist.
• The Madden Curse works this way. Generally, the cover is awarded to some athlete who just had a phenomenal season. The next season, the player is often beset by the sorts of bad luck that befall
all athletes (injuries, bad games, etc) except they receive more attention. In some cases, it may also be a Self-Fulfilling Prophecy if the player gets a big ego and skimps on workouts, or if
other players are more motivated to play hard against him. But mostly it's just that any season of a player is likely to be average (for his or her capabilities) and any season which leads to
feature a player in games or magazine stories is likely to be way above average, so it's just a good chance of a "dice roll" showing a lower number, just like the next number after you rolled a 6
on a normal die is likely to be lower than that.
□ A great example of this would be Brett Lorenzo Favre's appearance on Madden. It appeared that Favre had subverted, nay, broken the Madden curse while playing for the New York Jets. Then the
Jets lost four straight games and a bid for the playoffs. "What went wrong?" you ask. Brett Favre played the last month of the season with a torn bicep in his throwing arm and no one did
anything about it.
□ The technical name for this is regression toward the mean.
□ A more direct example are the year-in, year-out complaints that either the stats or the on-field experience are unrealistic, by pointing to the raw numbers. Since Madden NFL is a video game,
the developers have to shorten the quarters because most gamers aren't willing to invest multiple hours on a single game. So ultimately this means that gamers are running between 50-70% as
many plays as a real NFL contest. Yet many expect to produce as many points or exciting moments, while somehow maintaining realistic results on a per-play basis. This is mathematically
impossible. EA chooses the former, heavily slanting the game in favor of the offense, which has caused somewhat of a Broken Base amongst fans of the series.
□ Regression to the Mean overall is fueled by a misunderstanding of statistics, and has many (sometimes serious) consequences. "You say Homeopathy/Acupuncture/pseudoscience of your choice
worked for your arthritis pain? Wow. When did you take it? When you felt at your worst. Did it ever occur to you that short of trying to make things worse, you would almost certainly feel
better a while after hitting rock bottom? Does "nowhere to go but up" mean anything?"
• Warhammer 40000: Fears of "bad dice" abound. The previously mentioned lack of even distribution and the tendency of rolling methods to influence the result only adds fuel to the fire.
• Blood Bowl: There's always a 1 in 6 chance of succeeding or failing because ones always fail and sixes always succeed. Players hate this because you tend to fail at the worst possible time.
Failing also ends your turn in most cases, so superstition abounds.
□ This is also the "rebuttal" of any claims of the AI cheating in the computer game based on it, not taking into account that the exact sequence of rolls is predetermined at the start of any
given game (which they mention IN their rebuttal) and thus CAN BE LOOKED AT before they happen, thereby giving the AI an opportunity to cheat. Of course, it also provides an avenue for
players to cheat where save games are available...
• In A Song of Ice and Fire roleplaying system, there is a table for rolling random events in your family's history. This would be perfectly fine, except that you roll 3 dice for the events (thus
making the events in the middle more likely), and the table is in ALPHABETICAL ORDER. Thus Doom (the worst thing that can possibly happen to a family) is more likely than a mere Catastrophe
(still bad, but not even half as bad), just because Doom is closer to the middle of the list than Catastrophe is.
• In a strange twist, Final Fantasy VII: Crisis Core had the DMR, a slot-machine of various character faces that spins during combat, creating different effects. The only way to level-up is for
three "7"s to align. Isn't that awful?!? Leveling based on total randomness?!? Except. . .it isn't. The manual lies. The DMR is actually controlled by an insanely complicated mathematical formula
that, in-game, manifests itself as the strange impression that chance always just so happens to work out exactly the way natural progression should. In essence, one in a million chances succeed
nine times out of ten.
• The programmers of Sid Meyers Alpha Centauri fell afoul of this trope when they wrote the code to estimate the battle odds displayed before a combat: they used an obvious-but-wrong method of
working out chained probabilities, leading to the game tending to grossly underestimate the actual odds of victory. For example, a strength-8 unit with 30 hitpoints attacking a strength 8 unit
with 10 hitpoints would be shown as having a 75% chance of victory; the actual odds of winning are 99.93%. Under the right circumstances, this could result in the game predicting a
one-in-a-million chance of winning, when the actual odds are 90%.
• Cox and Box: In the (sometimes cut) gambling number, the titular characters roll nothing but sixes on their dice, leading them to suspect the other is cheating. Although they both are, no
dice-weighting is quite that good.
• Deconstructed in Rosencrantz and Guildenstern Are Dead by Tom Stoppard. A coin flipped nearly a hundred times comes up head each time, and they try to figure out how it's happening. Two
explanations Guildenstern develops are divine intervention and random chance.
□ Hamlet himself (though in his own play) provides the in-universe explanation: "The time is out of joint". Presumably this affects the law of probability somehow.
□ Of course it is that kind of play so the canonical explanation might well be that Stoppard is doing it.
• Among its many flaws, FATAL says that to determine the probability of an event, you roll two percentile dice,^[4] and if the second one is equal to or greater than the first, you succeed. That
means that everything has a flat 50.5% chance of happening. And yes, you're supposed to do this for anything.
• In Darths and Droids, Pete (R2-D2) likes to "pre-roll the ones out" of his 20-sided dice. He takes a huge number of dice and rolls them once each, and selects the dice that rolled a one. He rolls
those dice again, and selects the dice that rolled one a second time. Since the odds of any given d20 rolling a one three times in a row is 1 in 8,000, another roll of any of these dice has only
a 1 in 8,000 chance of rolling a one again, right? ... No.
□ When one of said dice does roll a 1... "Now it's even luckier!" ^[5]
☆ Even better — his reason for pre-rolling is the fact that he doesn't believe in "lucky dice".
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Download An Introduction to Black Holes, Information And The String by Leonard Susskind PDF
By Leonard Susskind
Over the past decade the physics of black holes has been revolutionized by way of advancements that grew out of Jacob Bekenstein's cognizance that black holes have entropy. Steven Hawking raised
profound concerns in regards to the lack of info in black gap evaporation and the consistency of quantum mechanics in an international with gravity. for 2 a long time those questions wondered
theoretical physicists and at last resulted in a revolution within the means we predict approximately area, time, subject and data. This revolution has culminated in a notable precept referred to as
''The Holographic Principle'', that is now an important concentration of realization in gravitational learn, quantum box idea and user-friendly particle physics. Leonard Susskind, one of many
co-inventors of the Holographic precept in addition to one of many founders of String concept, develops and explains those options
Read Online or Download An Introduction to Black Holes, Information And The String Theory Revolution: The Holographic Universe PDF
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Extra info for An Introduction to Black Holes, Information And The String Theory Revolution: The Holographic Universe
Example text
In fact, a careful computation of the expectation value of T µν in this state reveals a singular behavior at the horizon. Certainly this is not a good candidate to represent the original Minkowski
vacuum. A black hole, on the other hand, is not a translationally invariant system. One might therefore suppose that the evolution of the horizon might lead to the Fock space vacuum with no quanta
rather than the thermal state. This however would clearly violate the fourth guiding principle stated in the introduction: To a freely falling observer, the horizon of a black hole should in no way
appear special.
Any particle that starts with vanishing radial velocity in the region r < 3M G will spiral into the horizon. In the region of large negative r∗ where we approach the horizon, the potential is
unimportant, and the field behaves like a free massless Klein– Gordon field. 11) ±t) Let us consider a field quantum of frequency ν and angular momentum propagating from large negative r∗ toward the
barrier at r ≈ 3M G. Will it pass over the barrier? 9 has the form of a Schrodinger equation for a particle of energy ν 2 in a potential V .
That fluctuation can be thought of as a conventional fluctuation described by the quantum Hamiltonian HR . The fluctuation (b) contained in Region III has no significance to the Fidos in Region I.
Finally there are loops like (c) which are partly in Region I but which also enter into Region III. These are the fluctuations which lead to nontrivial entanglements between the degrees of freedom χL
and χR , and which cause the density matrix of Region I to be a mixed state. Quantum Fields in Rindler Space 41 8 ω= (a) (c) (b) 8 ω=− Fig.
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Square Root of 90 by Approximation & Long Division Method
What is the square root of 90 simplified?
The square root of \(90\) simplified in radical form is \(3\sqrt{10}\).
Is the square root of 90 a rational number?
No, the square root of \(90\) is evaluated in the decimal form as \(9.486\). Hence, it is an irrational number.
What is the square root of 90 rounded to the nearest tenth?
The square root of \(90\) rounded to the nearest tenth is \(9.5\).
Is the number 90 a perfect square?
No, \(90\) is not a perfect square, because it cannot be represented as the product of two equal integers.
How can we calculate the square root of 90?
The square root of \(90\) can be calculated by using the prime factorization method, newton raphson method, etc.
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Math Worksheet Multiplication | Multiplication Worksheets
Math Worksheet Multiplication
3rd Grade Multiplication Worksheets For Extra Practice More
Math Worksheet Multiplication
Math Worksheet Multiplication – Multiplication Worksheets are an excellent method to show children the twelve times table, which is the holy grail of elementary math. These worksheets serve in
training trainees one factor at a time, however they can also be utilized with 2 elements. Frequently, these worksheets are organized right into support groups, and also trainees can start
discovering these realities individually.
What are Multiplication Worksheets?
Multiplication worksheets are an useful means to assist pupils discover mathematics realities. They can be made use of to instruct one multiplication truth at a time or to examine multiplication
facts approximately 144. A worksheet that reveals a student one reality at a time will make it less complicated to keep in mind the fact.
Making use of multiplication worksheets to educate multiplication is a fantastic means to bridge the learning gap and offer your students effective technique. Numerous online sources use worksheets
that are both fun as well as easy to use. For instance, Osmo has a variety of totally free multiplication worksheets for kids.
Word problems are another method to link multiplication with real-life situations. They can enhance your youngster’s understanding of the concept while boosting their calculation speed. Many
worksheets include word issues that resemble real-life circumstances such as shopping, cash, or time computations.
What is the Purpose of Teaching Multiplication?
It’s essential to start showing children multiplication early, so they can delight in the procedure. Kids typically end up being bewildered when presented with too many facts at the same time, so
it’s finest to present brand-new facts one by one. Once trainees understand the very first couple, they can carry on to increasing by two, 3, or four. It’s likewise valuable to offer trainees lots of
technique time, so they can come to be fluent in multiplication.
One of the most effective understanding aids for kids is a reproduction table, which you can publish out for each youngster. Kids can exercise the table by repeating additions as well as counting to
get answers. Some youngsters discover the multiples of 2, 5, and 10 the simplest, but once they understand these, they can proceed to harder reproductions.
Math Worksheet Multiplication
Multiplication Practice Worksheets To 5×5
Multiplication Worksheets X2 X3 PrintableMultiplication
Multiplication Practice Worksheets Grade 3
Math Worksheet Multiplication
Math Worksheet Multiplication are a great means to review the moments tables. They likewise assist youngsters create versatility as they are subjected to the multiple means they can do calculations.
Trainees might additionally locate worksheets with photos to be practical. These worksheets can be adjusted for any type of motif or level, and are free to download.
These worksheets are wonderful for homeschooling. They are developed to be easy to use and also involving for youngsters. You can add them to math facilities, added method, and homework activities.
You can also personalize them to fit your youngster’s requirements. Once downloaded, you can also share them on social networks or email them to your youngster.
Lots of youngsters deal with multiplication. These worksheets are a superb means to help them overcome this obstacle. They include multiplication problems at different levels of trouble. The
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Related For Math Worksheet Multiplication
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Seam Carving Assignment
Seam-carving is a content-aware image resizing technique where the image is reduced in size by one pixel of height (or width) at a time. A vertical seam in an image is a path of pixels connected from
the top to the bottom with one pixel in each row; a horizontal seam is a path of pixels connected from the left to the right with one pixel in each column. Below left is the original 505-by-287 pixel
image; below right is the result after removing 150 vertical seams, resulting in a 30% narrower image. Unlike standard content-agnostic resizing techniques (such as cropping and scaling), seam
carving preserves the most interest features (aspect ratio, set of objects present, etc.) of the image.
Although the underlying algorithm is simple and elegant, it was not discovered until 2007. Now, it is now a core feature in Adobe Photoshop and other computer graphics applications.
In this assignment, you will create a data type that resizes a W-by-H image using the seam-carving technique. Finding and removing a seam involves three parts and a tiny bit of notation:
0. Notation. In image processing, pixel (x, y) refers to the pixel in column x and row y, with pixel (0, 0) at the upper-left corner and pixel (W − 1, H − 1) at the lower-right corner. This is
consistent with the Picture data type in algs4.jar.
a 3-by-4 image
(0, 0) (1, 0) (2, 0)
(0, 1) (1, 1) (2, 1)
(0, 2) (1, 2) (2, 2)
(0, 3) (1, 3) (2, 3)
Warning: this is the opposite of the standard mathematical notation used in linear algebra, where (i, j) refers to row i and column j and (0, 0) is at the lower-left corner.
We also assume that the color of each pixel is represented in RGB space, using three integers between 0 and 255. This is consistent with the java.awt.Color data type.
1. Energy calculation. The first step is to calculate the energy of a pixel, which is a measure of its importance—the higher the energy, the less likely that the pixel will be included as part of a
seam (as you will see in the next step). In this assignment, you will use the dual-gradient energy function, which is described below. Here is the dual-gradient energy function of the surfing
image above:
The energy is high (white) for pixels in the image where there is a rapid color gradient (such as the boundary between the sea and sky and the boundary between the surfing Josh Hug on the left
and the ocean behind him). The seam-carving technique avoids removing such high-energy pixels.
2. Seam identification. The next step is to find a vertical seam of minimum total energy. (Finding a horizontal seam is analogous.) This is similar to the classic shortest path problem in an
edge-weighted digraph, but there are three important differences:
□ The weights are on the vertices instead of the edges.
□ The goal is to find the shortest path from any of the W pixels in the top row to any of the W pixels in the bottom row.
□ The digraph is acyclic, where there is a downward edge from pixel (x, y) to pixels (x − 1, y + 1), (x, y + 1), and (x + 1, y + 1), assuming that the coordinates are in the prescribed ranges.
Seams cannot wrap around the image (e.g., a vertical seam cannot cross from the leftmost column of the image to the rightmost column).
3. Seam removal. The final step is remove from the image all of the pixels along the vertical or horizontal seam.
The SeamCarver API. Your task is to implement the following mutable data type:
public class SeamCarver {
// create a seam carver object based on the given picture
public SeamCarver(Picture picture)
// current picture
public Picture picture()
// width of current picture
public int width()
// height of current picture
public int height()
// energy of pixel at column x and row y
public double energy(int x, int y)
// sequence of indices for horizontal seam
public int[] findHorizontalSeam()
// sequence of indices for vertical seam
public int[] findVerticalSeam()
// remove horizontal seam from current picture
public void removeHorizontalSeam(int[] seam)
// remove vertical seam from current picture
public void removeVerticalSeam(int[] seam)
// unit testing (required)
public static void main(String[] args)
Corner cases. Your code must throw an exception when a constructor or method is called with an invalid argument, as documented below:
• Throw a java.lang.IllegalArgumentException if energy() is called with either an x-coordinate or y-coordinate outside its prescribed range.
• Throw a java.lang.IllegalArgumentException if the constructor, removeVerticalSeam(), or removeHorizontalSeam() is called with a null argument.
• Throw a java.lang.IllegalArgumentException if either removeVerticalSeam() or removeHorizontalSeam() is called with an array of the wrong length or if the array is not a valid seam (either an
entry is outside the height/width bounds or two adjacent entries differ by more than 1).
• Throw a java.lang.IllegalArgumentException if either removeVerticalSeam() or removeHorizontalSeam() is called when the width or height of the current picture is 1, respectively.
Constructor. The data type may not mutate the Picture argument to the constructor.
Computing the energy of a pixel. You will use the dual-gradient energy function: The energy of pixel \((x, y)\) is \(\sqrt{\Delta_x^2(x, y) + \Delta_y^2(x, y)}\), where the square of the x-gradient \
(\Delta_x^2(x, y) = R_x(x, y)^2 + G_x(x, y)^2 + B_x(x, y)^2\), and where the central differences \(R_x(x, y)\), \(G_x(x, y)\), and \(B_x(x, y)\) are the differences in the red, green, and blue
components between pixel (x + 1, y) and pixel (x − 1, y), respectively. The square of the y-gradient \(\Delta_y^2(x, y)\) is defined in an analogous manner. To handle pixels on the borders of the
image, calculate energy by defining the leftmost and rightmost columns as adjacent and the topmost and bottommost rows as adjacent. For example, to compute the energy of a pixel (0, y) in the
leftmost column, use its right neighbor (1, y) and its “left” neighbor (W − 1, y).
As an example, consider the following 3-by-4 image (supplied as 3x4.png):
• The energy of the non-border pixel (1, 2) is calculated from pixels (0, 2) and (2, 2) for the x-gradient
R[x](1, 2) = 255 − 255 = 0,
G[x](1, 2) = 205 − 203 = 2,
B[x](1, 2) = 255 − 51 = 204,
yielding Δ[x]^2(1, 2) = 2^2 + 204^2 = 41620;
and pixels (1, 1) and (1, 3) for the y-gradient
R[y](1, 2) = 255 − 255 = 0,
G[y](1, 2) = 255 − 153 = 102,
B[y](1, 2) = 153 − 153 = 0,
yielding Δ[y]^2(1, 2) = 102^2 = 10404.
Thus, the energy of pixel (1, 2) is \(\sqrt{41620 + 10404} = \sqrt{52024}\). Similarly, the energy of pixel (1, 1) is \(\sqrt{204^2 + 103^2}= \sqrt{52225}\).
• The energy of the border pixel (1, 0) is calculated by using pixels (0, 0) and (2, 0) for the x-gradient
R[x](1, 0) = 255 − 255 = 0,
G[x](1, 0) = 101 − 101 = 0,
B[x](1, 0) = 255 − 51 = 204,
yielding Δ[x]^2(1, 0) = 204^2 = 41616;
and pixels (1, 3) and (1, 1) for the y-gradient
R[y](1, 0) = 255 − 255 = 0,
G[y](1, 0) = 255 − 153 = 102,
B[y](1, 0) = 153 − 153 = 0,
yielding Δ[y]^2(1, 0) = 102^2 = 10404.
Thus, the energy of pixel (1, 0) is \(\sqrt{41616 + 10404} = \sqrt{52020}\).
Finding a vertical seam. The findVerticalSeam() method returns an array of length H such that entry y is the column number of the pixel to be removed from row y of the image. For example, the
dual-gradient energies of a 6-by-5 image (supplied as 6x5.png) are shown in the table below.
The minimum energy vertical seam is highlighted in blue. In this case, the method findVerticalSeam() returns the array { 3, 4, 3, 2, 2 } because the pixels in the minimum energy vertical seam are (3,
0), (4, 1), (3, 2), (2, 3), and (2, 4).
Finding a horizontal seam. The behavior of findHorizontalSeam() is analogous to that of findVerticalSeam() except that it returns an array of length W such that entry x is the row number of the pixel
to be removed from column x of the image. For the 6-by-5 image, the method findHorizontalSeam() returns the array { 2, 2, 1, 2, 1, 2 } because the pixels in the minimum energy horizontal seam are (0,
2), (1, 2), (2, 1), (3, 2), (4, 1), and (5, 2).
Unit testing. Your main() method must call each public constructor and method directly and help verify that they work as prescribed (e.g., by printing results to standard output).
Performance requirements. Make it fast.
Other requirements. Finding the shortest energy path must be done using Dijkstra's shortest paths algorithm. For the leaderboard, you may use any shortest paths algorithm of your choice.
Analysis of running time. Estimate empirically the running times (in seconds) to remove one row and one column from a W-by-H image as a function of W, and H. Use tilde notation to simplify your
Submission. Submit SeamCarver.java, and any other files needed by your program (excluding those in algs4.jar). You may not call any library functions other than those in java.lang, java.util,
java.awt.Color, and algs4.jar. Finally, submit a readme.txt file and answer the questions.
file points
finding seams 14
removing seams 10
other 10
readme.txt 6
Reminder: You can lose up to 4 points for poor style and up to 4 points for inadequate unit testing.
This assignment was developed by Josh Hug, Maia Ginsburg, and Kevin Wayne.
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Rumors in a network: who’s the culprit?
We provide a systematic study of the problem of finding the source of a rumor in a network. We model rumor spreading in a network with the popular susceptible-infected (SI) model and then construct
an estimator for the rumor source. This estimator is based upon a novel topological quantity which we term rumor centrality. We establish that this is a maximum likelihood (ML) estimator for a class
of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has nontrivial detection probability, whereas on trees that grow like a
line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an
internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops of the true source across different network topologies. We
compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general
networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding rumor sources in networks which are not tree-like.
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Aristotle University of Thessaloniki
Learning Outcomes
Upon successful course completion the students are expected to: 1. Be familiar with the bascis concepts needed for the analysis of algorithms. 2. Be able to appply basic techniques for solving
recursive relations that appear in the design of algorithms 3. Fully comprehend well-known algorithms and data structures 4. Understand the complexity of various approaches at various domains 5.
Analyze the asymptotic performance of algorithms 6. Apply basic algorithm design principles
Course Content (Syllabus)
The course discusses the analysis and design of algorithms and data structures not from a programming, rather an analytic perspective. Topics discussed within the context of the course focus on the
evaluation of performance of algorithms, as well as on the comparison of algorithms with respect to their time and space requirements. Identification and description of the theoretical, as well as
practical boundaries of algorithms are discussed through an analytical methodology. Topics covered within the context of the course: concepts and tools for the analysis of algorithms, asymptotic
order of function growth, recursive relations, probabilistic analysis and randomized algorithms, dynamic programming, greedy algorithms, amortized analysis, advanced topics in analysis and design of
Algorithms, Analysis, Design, Complexity
e-THMMY, a blackboard-like system has been developed by the ECE department and is customized to the needs of the ECE courses. e-THMMY allows instructors to post anouncements, communicate with
students, upload lectures, exercises and their solutions, set up and run course projects, while it also offers self-assessment capabilities. e-THMMY also supports a Forum for coursework discussion.
Course Bibliography (Eudoxus)
Τίτλος Συγγράμματος:: «Εισαγωγή στους Αλγορίθμους, Τόμος Ι»(ελληνική μετάφραση) Συγγραφέας: T. Cormen, C. Leiserson, R. Rivest, and C. Stein Εκδόσεις: ΙΤΕ-Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο,
2009 ISBN: 978-960-524-225-1 ΚΩΔ.ΕΥΔ.: 251
Additional bibliography for study
Τίτλος Συγγράμματος:: «Σχεδιασμός Αλγορίθμων»(ελληνική μετάφραση) Συγγραφέας: J. Kleinberg, E. Tardos Εκδόσεις: ΕΚΔΟΣΕΙΣ ΚΛΕΙΔΑΡΙΘΜΟΣ, ΘΕΣ/ΝΙΚΗ, 2009 ISBN: 978-960-461-207-9 ΚΩΔ.ΕΥΔ.: 13898
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For this week we've brought you this equation problem.
How would you solve \(7+4w+\frac{5}{w}=\frac{97}{4}\)?
Here are the steps:
Multiply both sides by \(4w\).
Move all terms to one side.
Simplify \(28w+16{w}^{2}+20-97w\) to \(-69w+16{w}^{2}+20\).
Split the second term in \(-69w+16{w}^{2}+20\) into two terms.
Factor out common terms in the first two terms, then in the last two terms.
Factor out the common term \(16w-5\).
Solve for \(w\).
Decimal Form: 0.3125, 4
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Let A={2,4,6,8,10},B={4,8,12,16} and C={6,12,18,24}. Using Venn... | Filo
Question asked by Filo student
Let and . Using Venn diagrams, verify that:
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Question Text Let and . Using Venn diagrams, verify that:
Updated On Dec 27, 2023
Topic Coordinate Geometry
Subject Mathematics
Class Class 11
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Angles of Triangles (No Algebra)
Angles of Triangles (No Algebra)
Price: 300 points or $3 USD
Subjects: math,mathHighSchool,highSchoolGeometry,congruence,geometricMeasurementAndDimension
Grades: 8,9,10,11,12
Description: Students will solve 12 problems using the triangle-angle sum theorem and exterior angle theorem. Problems require students to apply knowledge of vertical angles, perpendicular lines,
angle bisectors, and the final card has students apply alternate interior angles when given parallel lines.
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Ripple Carry Adder
Ripple carry adder is a combinational circuit that can perform addition operation of two n-bit binary numbers. It accepts two n-bit binary numbers as an inputs, performs addition of both the binary
numbers and generated it's addiiton as an output. You'll find easy to understand this topic if you know full adder how the full adder circuit works. If you do not know how full adder works, let us
understand full Adder illustrating it as a black box
Full-Adder as a black box
Full-adder is a combinational circuit which has three inputs and two outputs. It basically performs the addition of 3-bits. When we add three binary bits, Sum and Carry is generated based on the
binary inputs. The truth table of Full adder is given below
Truth table of full-adder circuit
Input A0 Input B0 Input Cin Output S0 Output Cout
n-bit Ripple carry adder
Ripple carry adder is designed by connecting full-adder circuits in a cascade fasion in such a way that, two n-bit binary inputs are applied parallelly to the circuit and the output carry of previous
full adder is applied to the input carry of the next full adder. For two n-bit binary addition, n number of full adder circuit is required. generated
This circuit generates output of n-bit sum with output carry of cumulative sum. Let's try to understand how the citcuit is working. For simplicity, 4-bit ripple carry adder is designed. In this
circuit, there are two four bit numbers [A3, A2, A1, A0] and [B3, B2, B1, B0] and we want to perform addtion of these two binary numbers.
These two 4-bit binary numbers are applied to the circuit such that first biary position (LSB) is the input of Full-Adder (FA0), second binary position to the second Full-Adder (FA1) circuit and so
FA0 performs the addition operation for first binary position [A0 and B0]. Depending on the inputs, sum S0 and carry C0 is generated. Here we get first bit of the sum which is S0.
FA1 performs the addtion operation for second binary position [A1 and B1] and carry [C0] comming from the FA0 and generted sum S1 and carry C1. Here we get second bit of the sum S1. and so on.
For the last full adder, final bit of the sum [S3] and carry out [Cout] is generated. Finally we get sum [S3, S2, S1, S0] and carry [Cout].
So, this is the simple circuit for performing two n-bit addition operation. All we need is n number of full adder circuit.
Some problems
The inputs are applied parrally to the circuit in the same time but, we'll not get the output in the same time. Why?
Even though inputs are available to the all full adder circuit, Sum of any position is depending on carry comming from the previous full adder. For the final addition operation, the propagation delay
is four full-adder circuit (for 4-bit ripple carry adder). If the number of inputs are more bits (say 8-bits), there will be more propagation delay So, this circuit is not suitable for adding large
input bits.
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Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Chi-Square Test
Hypothesis Testing
Genetic Proportions
Chi-Square Formula: χ² = Σ ((O_i - E_i)² / E_i), where O_i is the observed frequency and E_i is the expected frequency
Chi-Square Goodness-of-Fit Test
Suitable Grade Level
Grade 11-12
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2LOG, calculate binary logarithm
2LOG, calculate binary logarithm
The 2LOG routine was the result of the need for converting a linear input into something with a more logarithmic character. There was no need for high accuracy, but rather a short reliable routine.
The routine uses a very simple principle, but the result is surprisingly useable. A prime example where focussing on the essentials results in an excellent solution.
The 2LOG routine produces as output a fixed floating point number with 8 bits after the decimal point. ( x…x.xxxxxxxx ) The number of bits before the decimal point depends on the native width of the
The principle used is this:
looking at the number to be converted in binary representation: (see grafic below) - step 1: take the bit-position+1 of the first set bit as number before the decimal point - step 2: as 8 bit
fraction take the highest 8 bits following the most most significant set bit. If there are less than 8 bits, pad the end with cleared bits up to 8 bits.
It is good to notice that the fractional part forms a linear interpolation between two consecutive log numbers. For most purposes that is accurate enough.
the generic Forth program
As example we look at the routine for a 16b Forth. The other example is suitable for all Forth implementations.
: 2LOG16b ( u -- y )
16 0 do
s>d if
2* 8 rshift \ linear interpolation
15 i - \ logarithmic class
8 lshift or leave
then 2*
loop ;
• The programs does maximal 16 loops.
• For each loop S>D is used to check if the most significant bit is set. If not, then the program shifts the number 1 bit to the left with 2*.
• Otherwise it calculates the output.
□ The logarithmic part is calculated by subtracting the index from 15.
□ The fractional part is calculated by shifting to left with 1bit, followed by shifting 8 bits to the right.
• Finally, both numbers are then combined into one number with a shift and or as final output.
The general version
The general version is suitable for all Forth-implementations which have a multiple of 8 bits as cell-width. It functions in exactly the same way as the 16b example above. But during compilation it
calculates the, for that Forth-implementation relevant, values for the do…loop, shift and subtraction.
: 2LOG ( u -- y )
[ 8 cells ] literal 0 do \ #bits/cell
s>d if 2*
[ 8 cells 8 - ] literal rshift \ linear interpolation
[ 8 cells 1- ] literal i - \ logarithmic class
8 lshift or leave
then 2*
loop ;
Alternative Implementations
en/pfw/2log.txt · Last modified: 2023-09-04 18:09 by uho
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Two pairs of genes affect comb type in chickens (Figure10.12 ).... | Filo
Question asked by Filo student
Two pairs of genes affect comb type in chickens (Figure10.12 ). When both genes are recessive, a chicken has a single comb. A dominant allele of one gene, gives rise to a pea comb. Yet a dominant
allele of the other gives rise to a rose comb. An epistatic interaction occurs when a chicken has at least one of both dominants, which gives rise to a walnut comb.Predict the ratios resulting from a
cross between two walnut-combed chickens that are heterozygous for both genes
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Step by Step Solution: Step 1. First, we need to determine the gametes produced by each parent. For the male, the gametes are: PR, Pr, pR, and pr. For the female the gametes are also: PR, Pr, pR, and
pr. Step 2. Next, we will set up a Punnett square to show the possible genotypes of the offspring. The Punnett square will have four rows (representing the male gametes) and four columns
(representing the female gametes). The combinations for each box in the square are: PR-PR, PR-Pr, PR-pR, PR-pr, Pr-PR, Pr-Pr, Pr-pR, Pr-pr, pR-PR, pR-Pr, pR-pR, pR-pr, pr-PR, pr-Pr, pr-pR, pr-pr.
Step 3. Now, we can fill in each box with the corresponding genotype of the offspring. For example, PR-Pr would result in an offspring with a walnut comb that is heterozygous for both genes (PpRr).
By filling in the remaining boxes, we can see that there are 9 offspring with walnut combs, 6 offspring with pea combs, and 1 offspring with a rose comb. Step 4. To determine the phenotypic ratio, we
can count the number of offspring with each comb type. The phenotypic ratio would be 9:6:1, which simplifies to 3:2:1. Step 5. Therefore, the predicted phenotypic ratio resulting from the cross
between two walnut-combed chickens that are heterozygous for both genes (PpRr) is 3:2:1, which means that for every 3 offspring with a walnut comb, there are 2 offspring with a pea comb, and 1
offspring with a rose comb. Final Answer: The predicted phenotypic ratio resulting from the cross between two walnut-combed chickens that are heterozygous for both genes (PpRr) is 3:2:1, which means
that for every 3 offspring with a walnut comb, there are 2 offspring with a pea comb, and 1 offspring with a rose comb.
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Two pairs of genes affect comb type in chickens (Figure10.12 ). When both genes are recessive, a chicken has a single comb. A dominant allele of one gene, gives rise to a pea comb. Yet a
Question dominant allele of the other gives rise to a rose comb. An epistatic interaction occurs when a chicken has at least one of both dominants, which gives rise to a walnut comb.Predict the
Text ratios resulting from a cross between two walnut-combed chickens that are heterozygous for both genes
Updated Mar 29, 2024
Topic All topics
Subject Biology
Class Class 11
Answer Text solution:1
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The next cosmology seminar takes place on Thursday 2nd of November, seminar room E349 at 2pm. Our guest speaker will be Suvashis Maity from the Indian Institute of Technology (IIT) Madras (Chennai,
At 2pm, room E349, Suvashis Maity (IIT Madras), will be talking about
Ultra slow roll inflation and loop corrections to the scalar power spectrum
Recently, there have been efforts to examine the contributions to the scalar power spectrum due to the loops arising from the cubic order terms in the action describing the perturbations,
specifically in inflationary scenarios that permit a brief epoch of ultra slow roll (USR). A phase of USR inflation leads to significant observational consequences, such as the copious production of
primordial black holes. In this talk, I shall discuss the loop contributions to the scalar power spectrum in scenarios of USR inflation arising due to the quartic order terms in the action describing
the scalar perturbations. I shall initially describe the computation of the loop contributions to the scalar power spectrum due to the dominant term in the action at the quartic order. Thereafter, I
shall consider a scenario wherein a phase of USR is sandwiched between two stages of slow roll inflation and describe the behavior of the loop contributions in situations involving late, intermediate
and early epochs of USR. In the inflationary scenario involving a late phase of USR, for reasonable choices of the parameters, I shall show that the loop corrections are negligible for the entire
range of wave numbers. In the intermediate case, the contributions from the loops prove to be scale invariant over large scales, and we find that these contributions can amount to 30% of the power
spectrum at the leading order. In the case wherein USR sets in early, we find that the loop contributions could be negative and can dominate the power spectrum at the leading order, which indicates a
breakdown of the perturbative expansion. I shall conclude with a brief summary and outlook.
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Chin. Phys. Lett. (2021) 38(9) 097501 - Nonlocal Effects of Low-Energy Excitations in Quantum-Spin-Liquid Candidate Cu$_3$Zn(OH)$_6$FBr
[1] Broholm C, Cava R J, Kivelson S A, Nocera D G, Norman M R, and Senthil T 2020 Science 367 eaay0668 Quantum spin liquids
[2] Savary L and Balents L 2017 Rep. Prog. Phys. 80 016502 Quantum spin liquids: a review
[3] Zhou Y, Kanoda K, and Ng T K 2017 Rev. Mod. Phys. 89 025003 Quantum spin liquid states
[4] Wen X G 2017 Rev. Mod. Phys. 89 041004 Colloquium : Zoo of quantum-topological phases of matter
[5] Wen X G 2019 Science 363 eaal3099 Choreographed entanglement dances: Topological states of quantum matter
[6] McEuen P L, Szafer A, Richter C A, Alphenaar B W, Jain J K, Stone A D, Wheeler R G, and Sacks R N 1990 Phys. Rev. Lett. 64 2062 New resistivity for high-mobility quantum Hall conductors
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[9] Han T H, Singleton J, and Schlueter J A 2014 Phys. Rev. Lett. 113 227203 Barlowite: A Spin- $1 / 2$ Antiferromagnet with a Geometrically Perfect Kagome Motif
[10] Feng Z, Wei Y, Liu R, Yan D, Wang Y C, Luo J, Senyshyn A, Cruz C D, Yi W, Mei J W, Meng Z Y, Shi Y, and Li S 2018 Phys. Rev. B 98 155127 Effect of Zn doping on the antiferromagnetism in kagome
$Cu 4 − x Zn x ( OH ) 6 FBr$
[11] Tustain K, Nilsen G J, Ritter C, da S I, and Clark L 2018 Phys. Rev. Mater. 2 111405 Nuclear and magnetic structures of the frustrated quantum antiferromagnet barlowite, $Cu 4 ( OH ) 6 FBr$
[12] Wei Y, Ma X, Feng Z, Adroja D, Hillier A, Biswas P, Senyshyn A, Hoser A, Mei J W, Meng Z Y, Luo H, Shi Y, and Li S 2020 Chin. Phys. Lett. 37 107503 Magnetic Phase Diagram of Cu$_{4-x}$Zn$_x$(OH)
$_6$FBr Studied by Neutron-Diffraction and $\mu$SR Techniques
[13] Tustain K, Ward-O'Brien B, Bert F, Han T H, Luetkens H, Lancaster T, Huddart B M, Baker P J, and Clark L 2020 arXiv:2005.12615 [cond-mat.str-el] From magnetic order to quantum disorder: a $μ$SR
study of the Zn-barlowite series of $S={\frac{1}{2}}$ kagomé antiferromagnets, Zn$_{x}$Cu$_{4-x}$(OH)$_{6}$FBr
[14] Wen J J and Lee Y S 2019 Chin. Phys. Lett. 36 050101 The Search for the Quantum Spin Liquid in Kagome Antiferromagnets
[15] Feng Z, Li Z, Meng X, Yi W, Wei Y, Zhang J, Wang Y C, Jiang W, Liu Z, Li S, Liu F, Luo J, Li S, Zheng G Q, Meng Z Y, Mei J W, and Shi Y 2017 Chin. Phys. Lett. 34 077502 Gapped Spin-1/2 Spinon
Excitations in a New Kagome Quantum Spin Liquid Compound Cu [3] Zn(OH) [6] FBr
[16] Wei Y, Feng Z, Hu D H, Lohstroh W, dela C C, Yi W, Ding Z F, Zhang J, Tan C, Shu L, Wang Y C, Wu H Q, Luo J, Mei J W, Meng Z Y, Shi Y, and Li S 2019 arXiv:1710.02991 [cond-mat.str-el] Evidence
for a Z$_2$ topological ordered quantum spin liquid in a kagome-lattice antiferromagnet
[17] Wen X G 2017 Chin. Phys. Lett. 34 090101 Discovery of Fractionalized Neutral Spin-1/2 Excitation of Topological Order
[18] Norman M R 2016 Rev. Mod. Phys. 88 041002 Colloquium : Herbertsmithite and the search for the quantum spin liquid
[19] Shores M P, Nytko E A, Bartlett B M, and Nocera D G 2005 J. Am. Chem. Soc. 127 13462 A Structurally Perfect S = ^1 / [2] Kagomé Antiferromagnet
[20] de Vries M A, Kamenev K V, Kockelmann W A, Sanchez-Benitez J, and Harrison A 2008 Phys. Rev. Lett. 100 157205 Magnetic Ground State of an Experimental $S = 1 / 2$ Kagome Antiferromagnet
[21] Han T H, Helton J S, Chu S, Nocera D G, Rodriguez-Rivera J A, Broholm C, and Lee Y S 2012 Nature 492 406 Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet
[22] Han T H, Norman M R, Wen J J, Rodriguez-Rivera J A, Helton J S, Broholm C, and Lee Y S 2016 Phys. Rev. B 94 060409 Correlated impurities and intrinsic spin-liquid physics in the kagome material
[23] Fu M, Imai T, Han T H, and Lee Y S 2015 Science 350 655 Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet
[24] Kimchi I, Sheckelton J P, McQueen T M, and Lee P A 2018 Nat. Commun. 9 4367 Scaling and data collapse from local moments in frustrated disordered quantum spin systems
[25] Khuntia P, Velazquez M, Barthélemy Q, Bert F, Kermarrec E, Legros A, Bernu B, Messio L, Zorko A, and Mendels P 2020 Nat. Phys. 16 469 Gapless ground state in the archetypal quantum kagome
antiferromagnet ZnCu3(OH)6Cl2
[26] Feng Z, Yi W, Zhu K, Wei Y, Miao S, Ma J, Luo J, Li S, Meng Z Y, and Shi Y 2018 Chin. Phys. Lett. 36 017502 From Claringbullite to a New Spin Liquid Candidate Cu [3] Zn(OH) [6] FCl
[27] Balents L, Fisher M P A, and Girvin S M 2002 Phys. Rev. B 65 224412 Fractionalization in an easy-axis Kagome antiferromagnet
[28] Isakov S V, Hastings M B, and Melko R G 2011 Nat. Phys. 7 772 Topological entanglement entropy of a Bose–Hubbard spin liquid
[29] Isakov S V, Melko R G, and Hastings M B 2012 Science 335 193 Universal Signatures of Fractionalized Quantum Critical Points
[30] Wang Y C, Zhang X F, Pollmann F, Cheng M, and Meng Z Y 2018 Phys. Rev. Lett. 121 057202 Quantum Spin Liquid with Even Ising Gauge Field Structure on Kagome Lattice
[31] Wang Y C, Fang C, Cheng M, Qi Y, and Meng Z Y 2017 arXiv:1701.01552 [cond-mat.str-el] Topological Spin Liquid with Symmetry-Protected Edge States
[32] Sun G Y, Wang Y C, Fang C, Qi Y, Cheng M, and Meng Z Y 2018 Phys. Rev. Lett. 121 077201 Dynamical Signature of Symmetry Fractionalization in Frustrated Magnets
[33] Wang Y C, Cheng M, Witczak-Krempa W, and Meng Z Y 2020 arXiv:2005.07337 [cond-mat.str-el] Fractionalized conductivity and emergent self-duality near topological phase transitions
[34] Schnack J, Schulenburg J, and Richter J 2018 Phys. Rev. B 98 094423 Magnetism of the $N = 42$ kagome lattice antiferromagnet
[35] Ran Y, Hermele M, Lee P A, and Wen X G 2007 Phys. Rev. Lett. 98 117205 Projected-Wave-Function Study of the Spin- $1 / 2$ Heisenberg Model on the Kagomé Lattice
[36] Isakov S V, Kim Y B, and Paramekanti A 2006 Phys. Rev. Lett. 97 207204 Spin-Liquid Phase in a Spin- $1 / 2$ Quantum Magnet on the Kagome Lattice
[37] Kodama R H, Makhlouf S A, and Berkowitz A E 1997 Phys. Rev. Lett. 79 1393 Finite Size Effects in Antiferromagnetic NiO Nanoparticles
[38] Mandal S, Banerjee S, and Menon K S R 2009 Phys. Rev. B 80 214420 Core-shell model of the vacancy concentration and magnetic behavior for antiferromagnetic nanoparticle
[39] See the Supplemental Material for more experimental details
[40] Hansson T H, Oganesyan V, and Sondhi S L 2004 Ann. Phys. 313 497 Superconductors are topologically ordered
[41] Moroz S, Prem A, Gurarie V, and Radzihovsky L 2017 Phys. Rev. B 95 014508 Topological order, symmetry, and Hall response of two-dimensional spin-singlet superconductors
[42] Anderson P W 1973 Mater. Res. Bull. 8 153 Resonating valence bonds: A new kind of insulator?
[43] Haegeman J, Zauner V, Schuch N, and Verstraete F 2015 Nat. Commun. 6 8284 Shadows of anyons and the entanglement structure of topological phases
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This subject provides the quantitative instruments that are needed to pose and analyze economic problems with the aid of a formal model. In working toward the above goal the student will acquire the
following competences and skills. Regarding the contents of the course, the student will be able to: - Analyze the concepts of linear and matricial algebra. - Understand the basic tools of integral
calculus in one and two variables. - Understand elementary concepts of sequences and series of real numbers. - Apply all the above concepts to economic problems. We classify the competences in two
groups: specific competences and generic competences or skills. Regarding the specific competences, the student will be able to: - Solve linear systems of equations and determine the number of
parameters in the solution. - Master the main tools for the calculus of primitives. - Understand the concept of Riemann integral, its interpretatin as area, and the relationship with the indefinite
integral. - Understand the concepts of convergence and divergence of sequences and series of real numbers. - Understand the concept of improper integral and the method of calculus. - Solve double
integrals by means of change of variables and iterated integrals. - know how to derive under the integral sign. Pertaining the general competences or skills, in the class the student will develop: -
The ability to address economic problems by means of abstract models. - The ability to solve the above formal models. - The ability to interpret and classify the different solutions and apply the
appropriate conclusions to social contexts. - The ability to use the basic tools that are need in the modern analysis of economic problems. Through out the course, the student should maintain: - An
inquisitive attitude when developing logical reasoning, being able to tell apart a proof from an example. - An entrepreneurial and imaginative attitude towards the cases studied. - A critical
attitude towards the formal results and their applicability in social contexts.
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A circle has two parallel chords of lengths 6 cm and 8 cm. If t... | Filo
A circle has two parallel chords of lengths and . If the chords are apart and the center is on the same side of the chords, then the diameter of the circle is of length.
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Let be the center of the circle with radius and chords and parallel to each other.
Drop and to the chords and respectively.
It is given that .
Using the Pythagoras' theorem:
Using equations (1) and (2), we get:
And using equation ( 1 ),.
Hence, the diameter of the circle .
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Question Text A circle has two parallel chords of lengths and . If the chords are apart and the center is on the same side of the chords, then the diameter of the circle is of length.
Topic Circles
Subject Mathematics
Class Class 9
Answer Type Text solution:1
Upvotes 96
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Set Tic-Tac-Toe
Tanya Khovanova’s Math Blog: A Tour
Dr. Tanya Khovanova is a mathematician whose research interests lie in recreational mathematics, combinatorics, probability, geometry, number theory. Currently, she is a Lecturer and PRIMES Head
Mentor at the Massachusetts Institute of Technology (MIT). In To Count the Natural Numbers, … Continue reading
Posted in Blogs, people in math, Recreational Mathematics, women in math Tagged Blog on Math Blogs, blogs, math, mathematics, Penney's Games, PRIMES STEP, Recreational Mathematics, Set Tic-Tac-Toe,
Tanya Khovanova Comments Off on Tanya Khovanova’s Math Blog: A Tour
• Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.
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2. Find the area of the region bounded by y2=9x,x=2,x=4 and the... | Filo
Question asked by Filo student
2. Find the area of the region bounded by and the -axis in the first quadrant.
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Video solutions (2)
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3 mins
Uploaded on: 2/28/2024
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Question Text 2. Find the area of the region bounded by and the -axis in the first quadrant.
Updated On Feb 28, 2024
Topic Integration
Subject Mathematics
Class Class 12
Answer Type Video solution: 2
Upvotes 228
Avg. Video Duration 4 min
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Random variables
This applet simulates the taking of a random sample of three people from a large* population and recording their height (cm) or annual income ($).
Click ‘Do experiment’ to run the experiment once. You will see a sample taken one-by-one from the population and recorded on the dotplot and in the table. Click ‘Next step’ to proceed from one sample
element to the next. Finally, the mean is calculated and recorded.
You can run the experiment up to four times, and each time the data will be recorded on the dotplot and in the table.
After you have run the experiment four times, you can click ‘Do 1000 experiments’ to repeat the experiment 1000 times. Watch the dotplots grow as the experiment is repeated. Click ‘Show theoretical
distribution’ to add in the expected distribution for each of the random variables. How does the dotplot compare to the theoretical distribution?
* by assuming the sample is taken from a large population it is reasonable to assume that we are sampling with replacement.
Other resources:
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Random sample/subset from vector per iteration
I’m trying to do predictions for unseen subjects (with varying intercepts) in the generated quantities block.
I’d like to try taking a random subset of varying intercepts for the observed subjects to use for the unseen subjects.
I thing this is what “uncertainty” does or refers to in this post.
In R, I would use sample(observed_intercepts, n_new_subjects, replace = TRUE)
Is there a way to accomplish this within a Stan program. I saw mention of using categorical_rng, but I’m unsure how to implement that function and specify the theta simplex.
I’ve estimated intercepts for 30 observed subjects from which to draw samples.
Would something like:
vector[n_new_subjects] Subj_Idx = categorical_rng(rep_vector(1/n_observed_subjects), n_observed_subjects)
Do the trick?
For uniform distribution you would set theta = rep_vector(1/J, J) where J is the number of observed intercepts. Each draw returns a single random int, so you loop n_new_subject times.
But are you sure you don’t want to sample the new intercepts from the distribution you inferred in your random intercept model? (e.g. a normal distribution with mean \mu_\alpha and standard deviation
\sigma_\alpha). That is the usual approach to predictions for new subjects, though maybe I’m missing something about your problem, e.g. your new subjects are actually a random sample of your old
I missed the link to the post I referenced:
I have implemented what you suggest (I think referred to to as “Gaussian” in the above post). I think sampling from the observed intercepts is what’s referred to as “uncertainty”.
Did a link get broken here? The Stan User’s Guide has this description of the posterior predictive distributions and how to compute them:
What @potash is bringing up is that this is unusual. Usually we do posterior predictive inference with the model we fit.
Close—that needs an int in the declaration. I would strongly discourage using n_observed_subjects and n_obsrved_subejcts as variables if that wasn’t a typo.
If you really want to do sampling the way you say, you can do this, where I’ve used different variables for the above distinction (how many available, how many to select).
int J; // number of intercepts to use per prediction
vector[K] alpha = ...; // varying intercepts
array[K] int<lower=0, upper=J> selected = multinomial_rng(rep_vector(1.0 / K), J);
real linear_predictor = to_vector(selected) .* varying_intercepts;
It will basically implement something like “drop out” from ML. Stan won’t really let you do this in the model block for training, though. But beware that this allows each varying intercept to
selected zero times, once, or more than once.
Now I don’t know if you want to scale it back to where it was by multiplying the linear predictor by K / J. If you set K = J, then you get the same number and the ratio is 1.
Here’s how I’ve attempted to implement both for comparison. I’m not sure I have the indexing correct:
generated quantities {
// Observed
vector[N] mu = X * b; // Population mean
vector[N] epred = ((X * b) * sd(Y)) + mean(Y) ; // Expectations
vector[N] pred = to_vector(normal_rng(mu + r_subj[Subj], sigma)) .* sd(Y) + mean(Y) ; // Predictions
vector[N] log_lik ; // Log-likelihood per observation
// Log-likelihood
for (i in 1:N){
log_lik[i] = normal_id_glm_lpdf(y_scaled[i] | [X[i]], r_subj[Subj[i]], b, sigma) ;
// Unobserved
// Linear predictor
vector[N_New] mu_new = X_New * b; // New data population mean
vector[N_New] epred_new = ((X_New * b) * sd(Y)) + mean(Y) ; // New data expectations
// Predictions
// Uncertainty
array[nSubj_New] int<lower=1> Subj_Idx;
for (i in 1:nSubj_New){
Subj_Idx[i] = categorical_rng(rep_vector(1.0/nSubj, nSubj)) ;
vector[N_New] pred_uncert = to_vector(normal_rng(mu_new + r_subj[Subj_Idx[Subj_New]], sigma)) .* sd(Y) + mean(Y) ; // New data predictions
// "Gaussian"
vector[N_New] pred_gauss = to_vector(normal_rng(mu_new + to_vector(normal_rng(rep_vector(0, nSubj_New), subj_sd))[Subj_New], sigma)) .* sd(Y) + mean(Y) ; // New data predictions
Line breaks make this easier to scan on screen. :-). In general, defining intermediate quantities can make things a lot easier to read as they provide in-code documentation for what sub-expressions
I see that you’re randomly indexing into the subjects r_subj with duplication and then adding that to mu_new. Does that mean the ordering of mu_new doesn’t matter?
This is an unusual way to try to do predictive inference. Why not just use standard Bayesian posterior predictive inference? Specifically, why are you randomizing the predictors across dimensions of
Why are you randomizing predictors?
What’s going on in that final line with a normal_rng nested within a normal_rng? Is that just mimicking your data generating process? Labeling intermediate quantities here can help with doc.
Using a loop for pred_guass is going to be faster in generated quantities than building up with to_vector as there’s no autodiff structure to trim down with vectorization. The problem with the
vector-based operations is that they tend to allocate and fill new containers, which puts memory pressure on the processor, which is often the main bottleneck for statistical code.
I also didn’t understand why there’s a prediction and then prediction uncertainty. Usually we just roll the predictive uncertainty into our posterior predictive inference as follows:
\displaystyle p(\widetilde{y} \mid y) = \int_\Theta p(\widetilde{y} \mid \theta) \cdot p(\theta \mid y) \, \textrm{d}\theta.
The uncertainty in estimating \theta given y is represented by the posterior, and sampling uncertainty in the model itself is represented by p(\widetilde{y} \mid \theta). With MCMC, we compute as
\displaystyle p(\widetilde{y} \mid y) = \frac{1}{M} \sum_{m=1}^M p^\textrm{rng}_{Y \mid \Theta}(\widetilde{y} \mid \theta^{(m)}) where \theta^{(m)} \sim p(\theta \mid y) is a posterior draw and p_{Y\
mid\Theta}^{\textrm{rng}} is a random number generator for the sampling distribution, which we write in full as p_{Y \mid \Theta}(y \mid \theta).
What it looks like your’e doing is changing the distribution p_{Y \mid \Theta} randomly rather than just generating from it.
My intention is to use the model for sample size expectation/estimation.
What I’m attempting is to:
• fit a model to some pilot data
• use the fitted model to make predictions for a larger sample of unseen subjects
• Fit that model hundreds of times
• evaluate parameter distributions (credible interval width)
Increase number of unobserved subjects
• repeat
I’m trying to reproduce something akin to allow_new_levels in the linked post.
With intermediate steps, I think it would look like this:
generated quantities {
// Observed
vector[N] mu = X * b; // Population mean
vector[N] epred = ((X * b) * sd(Y)) + mean(Y) ; // Expectations
vector[N_New] pred = to_vector(normal_rng(mu + r_subj[Subj], sigma)) .* sd(Y) + mean(Y) ; // Predictions
vector[N] log_lik ; // Log-likelihood per observation
// Log-likelihood
for (i in 1:N){
log_lik[i] = normal_id_glm_lpdf(y_scaled[i] | [X[i]], r_subj[Subj[i]], b, sigma) ;
// Unobserved
// Linear predictor
vector[N_New] mu_new = X_New * b; // New data population mean
vector[N_New] epred_new = ((X_New * b) * sd(Y)) + mean(Y) ; // New data expectations
// Predictions
// Uncertainty
array[nSubj_New] int<lower=1> Subj_Idx;
for (i in 1:nSubj_New){
Subj_Idx[i] = categorical_rng(rep_vector(1.0/nSubj, nSubj)) ;
vector[N_New] pred_uncert = to_vector(normal_rng(mu_new + r_subj[Subj_Idx[Subj_New]], sigma)) .* sd(Y) + mean(Y) ; // New data predictions
// "Gaussian"
vector[nSubj_New] r_subj_new;
for (i in 1:nSubj_New){
r_subj_new[i] = normal_rng(0, subj_sd) ;
vector[N_New] pred_gauss = to_vector(normal_rng(mu_new + r_subj_new[Subj_New], sigma)) .* sd(Y) + mean(Y) ; // New data predictions
I should probably opt out of this question as I think I’m just muddying the waters. I can’t make the connection between the linked post, where @paul.buerkner is sampling new levels in a multilevel
model (it’s like we have respondents groups into batches for clinical trials, so we can either predict a new member of an existing batch or a whole new batch). That can be used to do this:
The unseen subjects are classified as either being new members of an existing batch or members of a new batch. But I can’t connect that @JLC’s code, which looks like it’s randomizing group effects.
Is that because you’re trying to sample a random new person from an existing group and there’s only a single effect to randomize?
Yes, I am trying to randomly sample an estimated parameter (subject-level intercept) and use it for an unseen subject. That’s how I’m interpreting “uncertainty” vs “gaussian”.
I am trying to simulate additional new people to existing pilot data.
I’m afraid I don’t know what that means, so I’ll let other people respond. Sorry for all the confusion!
I’m attempting to implement the default behaviour of an interface in the ‘generated quantities’ block of a Stan model.
My question might be double-barrelled in asking if my understanding, and implementation, of “uncertainty” is accurate.
I think you are implementing the different methods correctly.
To clarify more generally, brms has different way to simulate/predict new levels in multilevel models. I would recommend you using Gaussian because it is the conceptually easiest and does not require
using samples from the old levels. That said, the sample_new_levels = "uncertainty" indeed, for each posterior sample, takes values from a random chosen existing group. This way, the distribution of
the group-level coefficients can be approximated without using a Gaussian assumption.
Thanks for the input, @paul.buerkner !
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Lesson goal: Clever way to find pi
Iterate for a square-root
| Next:
An even more clever for-loop for Pi
Here's a clever way to find $\pi$ (see Cheney, et. al, 5th ed., p. 39). See if you can translate this "pseudo-code" into a working program to find $\pi$. It'll allow you to practice with for-loops
and programming formulas into the computer.
$b\leftarrow 0$
$x\leftarrow 2+\sqrt{2}$
for k=1 to 5 (or more)
$t\leftarrow \sqrt{a}$
$b\leftarrow t(1+b)/(a+b)$
$a\leftarrow \frac{1}{2}(t+1/t)$
$x\leftarrow xb(1+a)/(1+b)$
output x
Now you try. It's up to you. See if you can translate the pseudo-code into a working program!
Type your code here:
See your results here:
At some point, to test your results, try subtracting your value of $x$ from math.pi and see how close to $0$ you get.
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Module for matrix and vector handling
The matrix classes modules contains classes that deal with numerical tubular data. This comprises state vectors and system matrices that can change over certain frequency or time.
The data_axix class deals with the dimension over which the vectors and matrices are changing.
The Signal class deals with vectors where every coefficient represent the degree of freedom, for example the complex pressure amplitude at different positions in space.
The LinearMatrix class deals with the matrices. The first two dimensions are the rows and columns of system matrices, the third dimension is the DataAxis dimension. One main task of the LinearMatrix
is to pick out a system matrix at one specific frequency.
In other words, the LinearMatrix is not real three dimensional dataset, a better description would be the linear externsion of a set of matrices. Most numpy methods are implemented and used in such a
way, that the operations are performed along the full depth index.
The DynamicMatrix is an extension of the LinearMatrix class adding excitation and response degrees of freedom.
hermitian(M) Calculate hermitian of matrix along first dimension.
isdiagonal(data) Check if all matrices in 3D data set are diagonal.
linearIndex(N, irow, icol[, btriu]) Provides the linear index of triangular matrices
DataAxis(data, **kwargs) Class for handling 1D data axis e.g.
DynamicMatrix(data, xdata, excdof, resdof, ...) data: 3D matrix or 2D when kwargs are used
LinearMatrix(data, **kwargs) Class for handling 3D data matrices in an efficient way.
ShapeSignal(mesh, xdata, ydata, dof) Class for Signals with link to shapes ydata can be reprented by ndarrays but also function definition
Signal(xdata, ydata, dof) Class for test and simulation results
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seminars - Martin boundary covers Floyd boundary
시간: 오후 3-4시, 4:30-5:30 (2회)
In this talk, we discuss a relation between two boundaries for a finitely generated group: Martin boundary associated with a finitely supported symmetric random walk, and Floyd boundary obtained from
a conformal scaling of Cayley graphs. We prove that the identity map over the group extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of
conical points being singletons. Applications are given to the class of relatively hyperbolic groups. This is joint work with I. Gekhtman, V. Gerasimov and L. Potyagailo.
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How To Sum Filtered Data In Excel | CellularNews
Now You Know
Source: Unsplash.com
If you are an Excel user, you likely know that this versatile software can help you analyze and manipulate large amounts of data. One useful feature of Excel is the ability to filter data, allowing
you to view only the information that meets specific criteria. However, once you have filtered your data, you might find yourself in need of the sum of certain columns or ranges. That’s where the SUM
function comes in handy. In this article, we will explore how to sum filtered data in Excel, providing you with step-by-step instructions and helpful tips. Whether you are a beginner or an
experienced user, mastering this skill will enable you to efficiently perform calculations on your filtered data, saving you time and effort. Let’s dive in and unlock the power of summing filtered
data in Excel!
Inside This Article
Step 1: Apply Filters to your Data
Applying filters to your data is the first step in the process of summing filtered data in Excel. Filters allow you to selectively display specific rows of data based on the criteria you define. This
is particularly useful when you have a large dataset and you only want to work with a subset of that data.
To apply filters, start by selecting the entire range of data that you want to filter. This can be done by clicking and dragging over the cells, or by using the Ctrl + Shift + Right Arrow (or Left
Arrow) keyboard shortcut.
Once the range is selected, go to the Data tab in the Excel ribbon and click on the Filter button. This will add filter arrows to the header row of each column in your selected range.
Now, you can use these filter arrows to choose the specific criteria you want to filter by. Simply click on the filter arrow for the column you want to filter, and select the desired options. For
example, if you have a column with the names of employees, you can choose to only show data for a specific employee or a group of employees.
By applying filters to your data, you are able to focus on the specific subset of data that you want to sum. This makes it easier to perform calculations and analysis on the filtered data, without
having to manually sort through a large dataset.
Step 2: Use the SUMIF Function
Once you have applied filters to your data in Excel, the next step is to use the SUMIF function to calculate the sum of the filtered data. The SUMIF function is a powerful tool that allows you to
specify criteria and sum only the cells that meet those criteria.
To use the SUMIF function, follow these steps:
1. Click on the cell where you want the sum to appear. This will be your formula cell.
2. Type the equals sign (=) to start the formula.
3. Enter the SUMIF function, followed by an open parenthesis (=SUMIF().
4. Specify the range of cells that contain the values you want to sum. This range should include both the values and the corresponding criteria. You can do this by clicking and dragging over the
range, or by manually typing the cell references. For example, if you have values in column A and criteria in column B, the range would be something like A2:B10.
5. Next, enter the criteria that the cells must meet in order to be included in the sum. This can be a number, text, or a reference to another cell. Enclose text criteria in double quotation marks.
For example, if you want to sum all the cells where the criteria is “Apples”, you would enter “Apples” in this step.
6. Finally, close the function with a closing parenthesis. Press Enter to calculate the sum.
The SUMIF function will calculate the sum of the filtered data based on the criteria you specified. It will include only the cells that meet the criteria and ignore the rest. This makes it easy to
perform calculations on a specific subset of data without having to manually filter and calculate the sum.
By using the SUMIF function, you can save time and ensure accuracy in your calculations. Whether you are working with a large dataset or a small table, this function allows you to quickly and easily
sum filtered data in Excel.
Step 3: Enter the Criteria for Summing Filtered Data
Once you have applied filters to your data and selected the SUMIF function, it’s time to set the criteria for summing the filtered data.
The criteria determine which values will be included in the sum. In other words, you can specify conditions that the data must meet in order to be summed. This gives you control over what data is
included or excluded in the final calculation.
To enter the criteria, you need to provide the range that contains the criteria. This can be a single cell or a range of cells. For example, you might enter the criteria as “>=500” to sum values that
are greater than or equal to 500.
You can also use cell references in your criteria. For instance, if you have a cell with the value “500” and you want to sum values greater than or equal to that cell, you can enter the criteria as
“>=A1”. This allows you to perform dynamic calculations based on the content of other cells.
It’s important to note that the criteria you enter must match the data type of the range you are summing. For example, if you are summing a range of dates, the criteria must be a date as well.
Similarly, if you are summing a range of text values, the criteria must be a text value.
Once you have entered the criteria, press Enter to calculate the sum. The SUMIF function will evaluate the criteria for each cell in the filtered range and include or exclude values accordingly. The
result will be the sum of the filtered data that meets the specified criteria.
Step 4: Include the Sum Range
Once you have applied filters to your data and specified the criteria for summing the filtered data in Excel, it’s time to include the sum range. The sum range determines which cells should be
included in the sum calculation.
To include the sum range, you need to select the cells where the filtered data is located. These cells should contain the values that you want to sum. The sum range can be a single column, a single
row, or a range of cells that span multiple columns and rows.
To select the sum range, click and drag your cursor over the desired cells. Alternatively, you can manually enter the cell references by typing them into the formula bar. Make sure that the selected
cells correspond to the data you want to sum and that they are included within the filtered range.
Here’s an example: let’s say you have a data table with the following columns: “Item,” “Quantity,” and “Price.” You have applied filters to the “Item” column to show only certain items, and now you
want to sum the quantities of the filtered items. In this case, the sum range would be the “Quantity” column.
When including the sum range in the SUMIF formula, you need to specify the cell range using absolute or relative references. Absolute references involve using the dollar sign ($), which means that
the range will not change when the formula is copied to other cells. Relative references, on the other hand, adjust the range based on the position of the formula when copied.
For example, if your sum range is in cells B2 to B10, you can use absolute references as follows: =SUMIF($B$2:$B$10, criteria, sum_range). If you want to use relative references, you can simply use:
=SUMIF(B2:B10, criteria, B2:B10).
Remember to replace “criteria” with the condition or criteria you specified in Step 3. This ensures that only the cells that meet the specified criteria are included in the sum calculation.
After including the sum range and specifying the criteria in the SUMIF formula, you are ready to calculate the sum of the filtered data. Press Enter to execute the formula, and Excel will calculate
and display the sum in the cell where you entered the formula.
By including the sum range in the SUMIF formula, you can accurately sum the filtered data in Excel. This allows you to perform calculations and analyze specific subsets of your data without manually
summing each individual cell.
Step 5: Press Enter to Calculate the Sum
Once you have entered the criteria and included the sum range in the SUMIF function, you are now ready to calculate the sum of the filtered data. To do this, you simply need to press the Enter key on
your keyboard.
After pressing Enter, the SUMIF function will process the criteria and sum the values that meet the specified condition. The resulting sum will be displayed in the cell where you entered the formula.
It’s important to note that the sum will only include the values that satisfy the criteria you specified using the filters. Any data that does not meet the criteria will be excluded from the sum.
Pressing Enter triggers the calculation process, and Excel will evaluate the formula and provide you with the sum of the filtered data. This instantaneous calculation allows you to see the sum
immediately and make any necessary adjustments or further analysis based on the calculated result.
Remember to save your workbook after calculating the sum to ensure that your filtered data and the calculated sum are stored for future reference.
By following these steps and pressing Enter, you can easily calculate the sum of your filtered data in Excel. This feature is incredibly useful for various tasks, such as analyzing sales data,
tracking expenses, or summarizing survey results.
In conclusion, successfully summing filtered data in Excel can be a valuable skill for data analysis and reporting. By utilizing the SUM function with filtering capabilities, you can focus on
specific subsets of data and obtain accurate and meaningful results.
Throughout this article, we have explored various techniques to sum filtered data using different methods such as the SUBTOTAL function, SUMPRODUCT function, and the SUMIFS function. Each method
offers its advantages and can be applied based on the specific requirements of your data analysis task.
Remember to carefully define your criteria for filtering data, as well as understanding how the filtering process impacts the calculation. Taking the time to master these techniques will not only
save you time and effort but also provide you with the ability to extract crucial insights from your data effectively.
So, whether you are a data analyst, business professional, or simply an Excel enthusiast, make sure to put these techniques into practice and enhance your data analysis capabilities in Excel.
Q: What is filtered data in Excel?
Filtered data in Excel refers to a subset of data within a worksheet that has been selectively displayed based on specific criteria. It allows you to temporarily hide unwanted data and focus on
analyzing or manipulating the relevant information.
Q: How can I apply filters to my Excel data?
To apply filters to your Excel data, select the range of cells you want to filter and go to the “Data” tab in the Excel ribbon. Click on the “Filter” button, and you will see dropdown arrows added to
each column header. Clicking on these arrows will allow you to set specific criteria for filtering.
Q: Can I sum filtered data in Excel?
Yes, you can sum filtered data in Excel. When you apply filters to your data, you can perform calculations on the visible, filtered subset of data. This enables you to sum only the filtered data,
excluding any hidden rows that do not meet the filter criteria.
Q: How do I sum filtered data in Excel?
To sum filtered data in Excel, first apply the desired filters to your data. Then, select the cell where you want the sum to be displayed. Finally, type the formula “=SUBTOTAL(9, range)” into the
formula bar, where “range” is the range of cells containing the data you want to sum. Press Enter, and the formula will calculate the sum of the visible, filtered data.
Q: How can I remove filters in Excel?
To remove filters in Excel, go to the “Data” tab in the Excel ribbon and click on the “Filter” button. This will toggle off the filters and display all the data in your worksheet. Alternatively, you
can select the filtered range of cells and press the keyboard shortcut “Ctrl + Shift + L” to remove the filters.
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Fit a set of (x,y) data points to a rectangular display
Random commentary about Machine Learning, BigData, Spark, Deep Learning, C++, STL, Boost, Perl, Python, Algorithms, Problem Solving and Web Search
Saturday, June 26, 2010
Fit a set of (x,y) data points to a rectangular display
You are given a set of datapoints (x, y) with 0<x<N, 0<y<M. Try to map them in a rectangular display of size [0, S] x [0, T] with S<N and T<M. Make your own assumptions.
Pubblicato da codingplayground a 2:16PM
2 comments:
1. A classic problem when writing plotting software.
We assume that N/M = S/T if we want to keep the
aspect ratio unchanged.
Regardless of the aspect ratio, we should
transform the co-ordinates x,y of each datapoint
to x * S/N and y * T/M
2. There are a few ways you can do this. Depends a lot upon what type of information you want preserved.
Solution 1: Trivial one. Here compute new value of x to floor(S/N)*x for each x. Compute y to be floor(T/M)*y. However this solution does not take into account the input "spread". For example if
your inputs for 'x' is largely less than or equal to S but have say only one at N, you would have "compressed" all x to accommodate this single occurrence.
The following two solutions inspects the input to determine how the values of 'x' and 'y' are spread to determine what algorithm to use to make the mapping:
Many values in the input are in the range 0<x<S and fewer in the S,x<N range:
Solution 2: Use something similar to A-law or Mu-Law schemes used to represent speech in the audio world. Here values on the lower side tend to be mapped more accurately while those on the higher
side get mapped more aggressively.
Many values of x are in the S<x<N range and fewer in the 0<x<S range.
Solution 3: Use a 'inverse' mapping.
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Remark 3.5.6.2. Let $n$ be a nonnegative integer and let $f: X \rightarrow Y$ be a morphism of Kan complexes which exhibits $Y$ as a fundamental $n$-groupoid of $X$. Then $f$ is a Kan fibration
(Corollary 3.5.5.14). In particular, since it is surjective on vertices, it is surjective on $m$-simplices for every integer $m$ (see Remark 3.1.2.8).
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Search the site for the term 'R program'
• 28/03/2019
When we try to learn how to work with time series, it is very useful to have good data sets, and much better if they contain real data. It is difficult to obtain long series, or series presenting
interesting and well located and identified patterns, with which we can perform practices. An excellent source of complex time series is our own organism, and everything we can learn by working
with them can be extrapolated to any other context.
[Read More...]
• 27/01/2017
WInRQA is an application dedicated to recurrence plots, a tool that is used in the analysis of recurrence of complex time series. In this article I will introduce a new tool that I have added to
the program. Until now, the measures of quantification of recurrence (RQM) were obtained only from a static portion of the original series. With the new tool, we can obtain a series of measures
by moving a window along the entire original series and calculating the corresponding measurements to each of these windows.
[Read More...]
• 18/11/2016
In this article I will show how, through a very simple and totally deterministic process, we can move from a stationary system to a completely random one, going through periodic and chaotic
dynamics. For this, I will generate several time series with these characteristics using the program R and several packages that can help us in the analysis of them.
[Read More...]
• 21/10/2016
The PISA database contains, in addition to the scores of students, a lot of demographic, socioeconomic and cultural data about them, collected through a series of questionnaires, that allow
contextualize the academic results and make studies with a great number of variables. Most of these data are categorical, making the correspondence analysis a particularly appropriate tool to
work with them. In this article I will show you how to easily perform this analysis using the ca package of the R program.
[Read More...]
• 12/10/2016
To conclude this series on complex time series and their characterization using graphical tools I will show you a tool called recurrence plot, which allows to obtain some measures used in the
recurrence quantification analysis, or RQA for its acronym in English. The recurrence is a characteristic property of deterministic dynamical systems, and consists of that two or more states of
the system are arbitrarily close after a certain period of time.
[Read More...]
• 02/10/2016
The recurrent neural networks are a very appropriate tool for modeling time series. This is a type of network architecture that implements some kind of memory and, therefore, a sense of time.
This is achieved by implementing some neurons receiving as input the output of one of the hidden layers, and injecting their output again in that layer. In this article I will show a simple way
to use two neural networks of this kind, the Elman and Jordan ones, using the program R.
[Read More...]
• 24/09/2016
In this new article of the series dedicated to the graphic characterization of complex time series I will talk about two other graphical tools that can be useful, the power spectrum of the
signal, which will be obtained through the Fourier transform, and the graph of the distribution of values of the series, a simple histogram with the frequency of the different values that also
can provide us information about the series dynamics.
[Read More...]
• 10/09/2016
I continue the series on graphic characterization of the complexity in time series using the helper application GraphStudy. In this article I will show how to construct a graph with which you can
easily distinguish whether a particular series from an iterated function presents a chaotic dynamics, the web diagram.
[Read More...]
• 02/09/2016
Many of the data sets with which we usually work are in the form of time series. A time series can be seen as the evolution of a dynamic system, characterized by some variables and parameters.
Depending on the type of dynamic of the system, the series may be stationary, periodic, quasiperiodic, chaotic or random. In this series of articles, I will focus on the characterization of
chaotic dynamics, which is presented by complex systems, by using graphical methods.
[Read More...]
• 24/04/2016
In many applications it is necessary to generate random numbers. To do so, the .NET framework provides the Random class, which can generate uniformly distributed pseudo-random values, which means
that all numbers are equally likely to appear in the result. But in some cases we need to generate random values that follow other different types of distribution. In this article we will see how
to generate random numbers that follow a normal distribution, with a system that can be extended to any other type of distribution.
[Read More...]
• 19/02/2016
Every three years, since 2000, the OECD (Organization for Economic Cooperation and Development) performs a series of tests in a number of countries at national level to 15-years-old students, in
order to assess the degree of knowledge in three main groups of areas: science, reading and math. This is the PISA program, whose last edition took place in 2015.
[Read More...]
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Activity: Counting Cars
Count how many cars pass by on the road.
Make sure you do the counting from a safe place!
Choose a time, say from 11:00 to 11:10 in the morning
• If you count for 10 minutes, then you can multiply by 6 to find how many cars per hour
• If you count for 15 minutes, then you can multiply by 4 to find how many cars per hour
Use Tally Marks to help you keep count
You can use this form. Write down the time you start, and when you finish.
│Start Time │End Time│Tally Marks │Total│
│ __ : __ │__ : __ │ │ │
Cars Per Hour = ________
OK, off you go!
Note: If you are in a class, then each of you could choose a different time, and the teacher could put all the results together in a nice Line Graph.
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75. MATHEMATICS PROCESS LOGS
Process Logs (Martinez and Martinez, 2001) puts mathematical thinking on paper by translating mathematical reasoning and process into ordinary language.
Use for word problems during tutoring sessions.
Present a blank process log (Addendum below) to students. Refer to a particular word problem and allow time for student to work on log. Review important points with students:
• Personalize thinking by using first person (I, my, me, etc.).
• Include common language as well as mathematical vocabulary.
• Explain problem in a step-by-step manner.
• Point out any particular difficulties or stop points within the problem.
Reference, graphics, and/or for more information:
75. Mathematics Process Logs
Martinez, J. G. R., & Martinez, N. C. (2001). Reading and writing to learn mathematics: A guide and a resource book. Boston: Allyn & Bacon.
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In COPASI, most of the time there are several ways to do something and you just choose the way you prefer. This is especially true for defining the elements of the model.
Actually if you are just defining a model that has a single compartment, you will most likely not even bother to add the compartment explicitly, but we will come back to this in the
Although you will probably not add compartments too often, it is good to know how it is done, especially since adding other components of the model, e.g. species or reactions, works essentially the
There are three methods to add a new compartment to a model, but for all three, we have to navigate to the Compartments branch of the object tree which is located under the Model->Biochemical branch.
So first open the Model branch and there open the Biochemical branch by clicking on the expansion sign in front of the branch name, or by double clicking on the branch name.
If you start with a new model and you select the Compartments branch, you will get a nearly empty table with eight columns (see above). The columns are named #, Name, Type Initial Volume, Volume,
Rate, Initial Expression and Expression. The # column contains the index of a given compartment in the model. Note that the index for the 'New Compartment' is not given indicating that it is not part
of the model. Name is the actual name of the compartment. The Type specifies how the transient volume of the compartment is determined whereas the Initial Volume and the Volume columns show the
volume of the compartment before any calculations and after re-calculation, respectively. These two values, given in the volume units defined in the
model settings
dialog, should be the same if the type is set to fixed. The Rate column show the rate of change of the compartment (if it has been set by a calculation). The Initial Expression and the Expression
columns show the mathematical expressions used to calculate the initial compartments volume and the transient one, respectively. These fields are empty if no expressions has been set.
The most obvious way to add a new compartment is to click the New button on the bottom of the window. This will create a new compartment that is added to the table with the index of 1.
Compartment Table with new Compartment
To create multiple compartments with the default values just keep pressing the New button.
Compartment Table with deleted Compartment
Any compartment you have created can be deleted by selecting the whole table row or clicking into the # cell that contains the compartment you want to delete and clicking the Delete button. This
deletion is executed immediately if there are no consequences to other model components. If there are consequences such as the deletion of contained species. A dialog box will appear informing you of
such side effects and ask whether you want to proceed or cancel the deletion.
The Delete All button is just a convenience function to delete all compartments. All compartments will be deleted and a confirmation dialog will appear for each compartment which deletion has further
consequences to the model. You are able to individually cancel or confirm any deletion.
The most convenient way to add a compartment is to modify any cell in the e.g. the name cell by providing the desired the name of the new compartment. Once you leave the cell by either hitting the
return or the tab key or by clicking somewhere else, the compartment is added to the model and a new default row is created. Actually hitting the return key after typing the name brings you directly
into the next row and you can continue adding compartments until all compartments are defined.
The third way to add a new compartment is to double click on an empty row in the table. This is essentially the same as clicking the New button and double clicking on the newly added compartment
Compartment Definition Dialog
Double clicking on any compartment entry in the table will bring you to another input dialog that lets you specify the parameters of the compartment (see above). For each compartment, you are able to
change the name of the compartment, the type of simulation specifying how the compartment volume, or its rate of change, will be determined, and the initial volume.
The field labeled Volume specifies the volume the compartment has right now, e.g. during or at the end of a simulation. Since the current version of COPASI does not support variable compartment
volumes yet, this will always be the same as the initial volume.
As stated above, compartment volumes don't have to have a constant value but they can be reassigned during e.g. a time course simulation depending the values of one or more model entities. In order
to specify whether a compartment has a constant volume or the volume is calculated on the fly according to a mathematical expression, the drop down list called Simulation Type can be used. It
contains three entries:
Compartments Simulation Types
Name Description
fixed the volume of the compartment has a constant value (which corresponds to the given initial value)
assignment the volume of the compartment is determined by evaluating the given mathematical expression
ode the rate of change of the compartments volume is determined by an ordinary differential equation
If you want the compartment to be calculated from a given mathematical expression you select the entry called assignment from the Simulation Type drop down list. This enables a text field where the
mathematical expression can be entered.
Compartment Widget with Assignment Rule
Warning: Please note, it is not possible in COPASI to calculate the volume of a compartment through an assignment based on the concentration of a species contained in the compartment. The reason for
this restriction is that COPASI preserves the amount $S$ of a species during calculation, which leads to the following equation for the concentration $[S] = S/V$ and therefore $V=S/[S]$. Obviously,
specifying any assignment for the compartment volume like $V = f([S])$ would lead to conflicting values.
Likewise if you want the rate of change of the compartments volume to be determined by an ordinary differential equation (ode) you select the entry called ode from the drop down list.
Not only the transient volume of a compartment can be specified as a mathematical expression (see above), but also the initial one (see below). If you want to specify such a mathematical expression
for the initial volume of a compartment, check the check box called Use Initial Expression. An initial expression can only be specified if the Simulation Type drop down list is either set to fixed or
to ode. If the list is set to assignment, the given assignment automatically acts as an initial assignment and there is no need to specify an additional initial assignment.
Compartment Widget with Initial Assignment
The mathematical expressions that can be specified for rules and initial assignments may contain the same elements as the expressions used to defined function definitions. For a detailed description
of the elements see
User Defined Functions
. When it comes to referencing values of other model entities within mathematical expressions, there is a slight difference between the mathematical expression for a rule and that for an initial
assignment. The mathematical expression for a rule may reference transient values of other model entities whereas the mathematical expression for an initial assignment may only reference initial
values of other model entities.
As you might already have noticed, this dialog for changing compartment parameters, associated with the individual compartment, lives in the object tree. So if you want to change the parameters, you
can also navigate to the leave in the object tree that represents the compartment you want to change instead of double clicking on an entry in the compartment table.
Compartment Widget with ODE Rule and Initial Assignment
If there are already species defined that are part of the compartment being edited, they will be listed in the text widget at the bottom of the dialog called Contained Species. Otherwise, you should
add new species.
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Math Simplified - Surface Areas
Source: Complete International Mathematics For Cambridge IGCSE - David Rayner, Jim Fenson
1. Find the quantities marked *. (Leave π in the answer)
a) Sphere
b) Cylinder
c) Cone
d) Cylinder
e) Sphere
f) Cone
g) Cylinder
h) Cone
i) Sphere
j) Hemisphere
2. A solid wooden cylinder of height 8 cm and radius 3 cm is standing vertically.
It is then cut in two along a vertical plane of symmetry.
Calculate the surface area of the two pieces.
Each cut piece has the following surfaces:
1. A rectangle (along the cut plane)
2. Two semi-circles(base and top)
3. Half the circular face of the cylinder.
3. A tin of paint covers a surface area of 60 cm² and costs $4.50.
Find the cost of painting the outside surface of a hemispherical dome of radius 50 m.
(Just the curved part)
4. Find the surface area of a sphere of volume 28 cm³.
5. A golf ball has a diameter of 4.1 cm and the surface has 150 dimples of radius 2 mm.
Calculate the total surface area which is exposed to the surroundings.
(assume the dimples are hemispherical)
The golf ball can be considered a sphere.
The dimples can be considered holes cut out on the surface of the sphere.
In addition the surface area of the dimples have to be calculated.
6. A cone of radius 3 cm and slant height 6 cm is cut into four identical pieces.
Calculate the total surface area of the four pieces.
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Cayley-Hamilton Archives - Aleph_epsilon
The post is also available as pdf.
Cayley-Hamilton theorem is usually presented in a linear algebra context, which says that an many proofs available, among which is the bogus proof of substituting the matrix into the characteristic
The reason it doesn’t work is because the product
I will develop the theory using language of rings and modules but if you don’t understand that, feel free to substitute “fields” and “vector spaces” in place.
There is a technical remark to make: later we will use determinant of matrices over
Given a module endomorphism
Cayley-Hamilton Theorem
Note that this is a relation of endomorphisms with coefficients in
Proof: Let
we have
Again, the multiplication is by scalar
Claim that if
which maps
To show this, recall that
where multiplication on the left is between matrices. Let
If you feel that little work is done in the proof and suspect it might be tautological somewhere (which I had when I first saw this proof), go through it again and convince yourself it is indeed a
bona fide proof. There are two tricks used here: firstly we extend the scalars by recognising
The key idea in the proof, sometimes called the determinant trick, has many applications in commutative algebra:
Proof: Let
An immediate corollary is Nakayama’s Lemma, which alone is an important result in commutative algebra:
Nakayama’s Lemma
Proof: Apply the trick to
We use the result to prove a rather interesting fact about module homomorphism:
Proof: Let
As a side note, the converse is not true: injective module homomorphisms need not be surjective. For example
M. Reid, Undergraduate Commutative Algebra, §2.6 – 2.8
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Revit Area Totals and Rounding
• There are none, suggest some!
When working with Revit areas, either through an area scheme or architectural rooms, it’s common to create a schedule of those areas, and utilize the auto-calculation aspect of Revit to get the
totals of the floor/department/etc.
If you interrogate your list of rooms, occasionally you will notice that the rounded values presented in the schedule end up not adding up to the totals presented in the schedule. There ends up being
a discrepancy between some values, which can cause issues for clients or in planning.
To illustrate and discuss what is happening, we will use this example. Presented here are 4 rooms that were created in Revit, and the values associated with the rounding created in a schedule from
Revit. Each column B through F was set to the Area of the space/room, with different rounding precision for each (0 decimals, 1 decimal, etc)
Maximum Precision – Column F
While we have read elsewhere that Revit stores 15 decimals for numbers, it only lets you schedule up to 12 decimals for the Area in a Room or Area schedule (done by “custom” with .000000000001
Therefore, we are stating that 12 decimals is the “maximum precision” you can get in an Area/Space; and when we discuss the “most precise” area measurement, this is the precision we are talking about
(column F).
How Revit Rounds Spaces & Rooms
The main issue here is that Revit is rounding the most precise value it has for the room/space. It rounds this precise value to the exact specified rounding (1,2,3 decimals, etc).
The presented rounded values (Columns B, C, D, E) are simply rounded values, and not stored within the program. Therefore, they are not the values that get added together for the total.
Total Values
Instead, to maintain accuracy, Revit is adding together the precise room values (Column F) and then rounding this value. This presents the most precise value it can for the Total Area because it is
waiting until the last moment to round. This keeps the significant digits to 12 decimal places in the equation, until the final rounding to the decimal precision requested. This means that the value
is more precise, but may not match the values of your rounded totals.
Example Discrepancy
We chose this dataset because it can illustrate this issue. If you add up Column B rooms, you will get a value of 283 SF, while the actual rounded value is 282 SF. While minimal, this is still a
discrepancy of 1 SF across 4 rooms.
The discrepancy is not limited to 1SF, and can increase; as an example, if we were to double the rooms to the exact same size, Revit’s tabulated total becomes 564 SF across 8 rooms, while the
calculated total from the rounded values is 566 SF. This is a difference of 2 SF (double the earlier value).
While the decimals in our example for .1, .01 and .001 precision all are accurate to the totals, this is not always the case. Obviously, by its nature, the more precise values will be closer to the
actual totals regardless, but there is the possibility that these are all off as well.
What’s actually happening
It generally depends on how many areas you have, and the distance to “5” (as this is the rounding cutoff) for the value of the decimal you want.
For example 4 spaces
• Values of .949 for 4 values will equal a total of 3.796. This rounds to 3.80 for 2 decimals.
• The 2 decimal value of .95 * 4 is equal to 3.80. This matches.
However, if we look at 5 spaces with this value.
5 spaces
• Values of .949 for 6 values will equal a total of 5.694. This rounds to 5.69 for 2 decimals.
• The 2 decimal value of .95 * 6 = 5.70
Typical Values Balanced by Probability
This is an extreme example. Typically, while .49 values are low, they are normally balanced out by various .51+ values that are high. Therefore, while its possible, its unlikely that your values are
off by ‘that’ much ever.
Let’s say that you do want the total to represent an addition of the values that have been rounded. This is possible using a calculated parameter in your schedule, and hiding the ‘actual’ area value.
Keep in mind that if you do a workaround, you are actually presenting a ‘less precise’ value as your total.
Calculated Parameter
1. Create a new calculated parameter. The type needs to be “Area”.
2. Decide on the precision you want, and copy/paste the corresponding precision formula in (below).
3. Make sure to set the Area parameter to be a ‘hidden value’
4. Make sure to set the “Field Format” of the new calculated parameter to match your precision you set. It will default to your ‘projects’ precision.
Follow the formulas below for each precision level you want. For more precision, follow the pattern below.
Essentially what you are doing is multiplying the most precise value by a certain amount, to get the precision you want to be at the integer level. Then you are rounding that value to the nearest
integer. Once you get that rounded value, you are then dividing by the amount you multiplied by in order to get back to a decimal amount. All the dividing and multiplying of SF are to resolve the
Precision Formula
0 decimal places ((round((Area * 1) / 1 SF)) * 1 SF) / 1
1 decimal places ((round((Area * 10) / 1 SF)) * 1 SF) / 10
2 decimal places ((round((Area * 100) / 1 SF)) * 1 SF) / 100
3 decimal places ((round((Area * 1000) / 1 SF)) * 1 SF) / 1000
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Ciles 0.1.0: Continuous Interval Langevin Equation Simulator | Alex Seeholzer
Ciles 0.1.0: Continuous Interval Langevin Equation Simulator
This was published a while ago on Github.
Langevin integrator for SDEs with constant drift and diffusion on continuous intervals with circular boundary conditions.
CILES is written in Cython and uses GSL for interpolation of drift & diffusion fields, to be able to simulate continuous variables.
Given a discretized drift field A(x) and a (position dependent) diffusion coefficient B(x) this tool performs simple time-forward integration of the SDE:
dx(t)/dt = A(x(t)) + sqrt(B(x(t))) * eta(t)
where eta(t) is a gaussian white noise term and x is a variable on an interval with circular boundaries (commonly 0 <= x < 2PI).
Both drift field A and diffusion B need to be arrays of the same dimension. They are internally interpolated (usinggsl_interp_cspline_periodic) to provide continuous fields, which are then used in
the forward integration.
Forward integration is performed with the Euler-Murayama scheme
x(t+dt) = x(t) + dt * A(x(t)) + r * sqrt(dt * B(x(t)))
where r is a normally distributed random number with zero mean and unit variance.
To install ciles in your Python distribution: – Clone repository – python setup.py install – To test (using nosetests): nosetests
You can also use ciles locally without installing: – Clone repository – python setup.py build_ext --inplace
Example use
from ciles.integrator import LangevinIntegrator as LI
import numpy as np
drift = np.zeros(100) # no drift field
diff = np.ones(100) # constant diffusion with 1 deg^2/s
dt = 1e-3 # 1 ms timestep
tmax = 1. # simulate until 1s
# initialize the integrator
li = LI(drift, diff, dt=dt, tmax=tmax)
# simulate a single trajectory
out = li.out
More examples
Below are the plot results of the currently available examples from ciles.examples.
Final distributions after 2s diffusion
See the source
Trajectories for drift-field with 2 fixed points
See the source
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CVX on Octave: Discussion
Now that Octave 4.0 has been released, the current version of CVX 3.0 beta has now achieved alpha-level stability on Octave. For more information, see this blog post on the CVX web site.
If you would like to discuss your experiences installing or using Octave, please do so on this post only. That way, everyone who is on the bleeding edge with you can keep in close contact. All other
questions about CVX and Octave will be merged with this one.
Once CVX+Octave is more stable, we’ll open up the full forum.
I want to test the current version of CVX 3.0 on Octave, but i fail to install CVX on my Windows 7 64-bit machine.
I have downloaded the file cvx-w64.zip and ran the following commands:
cd C:\octave\cvx
It reports:
CVX: Software for Disciplined Convex Programming (c)2014 CVX Research
Version 3.0beta, Build 1173 (c9b179b) Sat Aug 1 15:20:04 2015
Installation info:
Path: C:\octave\cvx
GNU Octave 4.0.0 on i686-w64-mingw32
Verfying CVX directory contents:
No missing files.
ERROR: the following CVX MEX files are missing:
CVX will not operate without these files. Please visit
and download a distribution built for this platform.
ERROR: -------------------------------
Error detected by cvx_version
Line 22: C:\octave\cvx\cvx_setup.m
The installation of CVX was not completed. Please correct the error
and re-run CVX_SETUP.
What am I doing wrong?
You have to compile the MEX files yourself. If you’ve downloaded the most recent beta, I think we have a ‘cvx_compile’ function you can try.
Thanks! That resolved the error, but now I am getting another unexpected error:
CVX: Software for Disciplined Convex Programming (c)2014 CVX Research
Version 3.0beta, Build 1173 (c9b179b) Sat Aug 1 15:20:04 2015
Installation info:
Path: C:\octave\cvx
GNU Octave 4.0.0 on i686-w64-mingw32
Verfying CVX directory contents:
WARNING: The following extra files/directories were found:
These files may alter the behavior of CVX in unsupported ways.
Loading preferences:
Global: C:\octave\cvx\cvx_prefs.mat ... not found.
Local: C:\Users\personal\.cvx_prefs.mat ... to be created.
Setting CVX paths...done.
Saving update path...done.
Searching for solvers...5 shims found.
3 solvers initialized (* = default):
GLPK unknown C:\Octave\Octave-4.0.0\share\octave\4.0.0\m\optimization
* SDPT3 4.0 {cvx}\sdpt3
SeDuMi 1.34 {cvx}\sedumi
2 solvers not found:
ECOS https://github.com/ifa-ethz/ecos
SCS https://github.com/cvxgrp/scs
Saving updated preferences...done.
Testing with a simple model...
UNEXPECTED ERROR: --------------------------------------
'mexnnz' undefined near line 97 column 13
Line 97: C:\octave\cvx\sdpt3\Solver\checkdepconstr.m
Please report this error to support, and include entire output of
CVX_SETUP in your support request.
Looks like SDPT3’s MEX files aren’t compiled.
I cloned the latest master from Github, but I cannot find cvx_compile. Is it supposed to be still there?
Ah, good point, you need to clone rework, which is the CVX 3.0beta branch. Sorry for the confusion.
Thanks, I cloned the rework branch and I found the cvx_compile command. Everything went fine, except I manage to compile only the SDPT solver. The install_sedumi gives the following error:
Attempting to recompile the SeDuMi binaries:
Template: mex -O -DOCTAVE <sources> -lblas
bwblkslv.mex: bwblkslv.c sdmauxFill.c sdmauxRdot.c
In file included from bwblkslv.c:45:0:
blksdp.h:41:21: fatal error: f77blas.h: No such file or directory
#include "f77blas.h"
Anyway, that’s fine, I can use SDPT for my application. I tried to run the SDP I am solving and I get the following error:
error: concatenation operator not implemented for '<unknown type>' by 'sparse complex matrix' operations
error: called from
minus_nc at line 10 column 3
cvx_binary_op at line 107 column 11
cvx_pushcnstr at line 116 column 13
ge at line 22 column 3
ydy at line 9 column 12
This is a MWE (very minimal, not much sense for that SDP):
cvx_begin sdp
variable P(2,2) hermitian;
subject to
P >= 0;
I’m afraid that’s one of the exact bugs that prevents me from moving forward with a formally supported CVX release. From the blog post I linked to above: Complex variables will not work due to a
known bug in Octave 4.0. I’ve reported this bug to the Octave folks and I’m sure it will get fixed in a subsequent release.
I was wondering if there is a way to impose everything to be real. In theory, that shouldn’t change my SDP.
Well yes, declare your SDP symmetric instead of Hermitian. Hermitian implies complex.
I had tried symmetric, but that gives a strange value, which is not the same I get with MATLAB. I guess it’s not quite working yet even if I impose everything to be real. Anyway, it’s great to see
progress on CVX on Octave, i am looking forward to porting everything to it once it’s ready
If you can offer a simple example that fails, please go ahead and submit a bug report. I definitely need some test cases where Octave fails and Matlab does not.
Do you want me to open an issue on Github? Or where should I report the bug?
I’d say go to http://support.cvxr.com … Thanks!
Hi Michael,
After stripping out all useless parts, the program became very simple and I suspect that either I am doing something wrong or it’s already fixed, so I will just report it here:
cvx_begin sdp
variable P(2,2) symmetric
maximize trace(P)
subject to
P == eye(2);
The variable cvx_optval is 2 on MATLAB and 1 on Octave. Let me know if I should open a ticket for it.
No, this is awesome right here. Thanks!
There is now an existence proof that CVX 3.0 can be coerced into at least setting up correctly in Octave 4.0.0 running in a Cygwin environment. TBD if it all works, but I thought I would share a
couple of bumps and point to “fixes” as I finally achieved setup success:
cd /cygdrive/c/octave/cvx
CVX: Software for Disciplined Convex Programming (c)2014 CVX Research
Version 3.0beta, Build 1175 (1326ef2) Mon Nov 23 14:29:34 2015
Installation info:
Path: /cygdrive/c/octave/cvx
GNU Octave 4.0.0 on x86_64-unknown-cygwin
Verfying CVX directory contents:
WARNING: The following extra files/directories were found:
(and 118 more files)
These files may alter the behavior of CVX in unsupported ways.
Loading preferences:
Global: /cygdrive/c/octave/cvx/cvx_prefs.mat … not found.
Local: /home/jbbacque/.cvx_prefs.mat … to be created.
Setting CVX paths…done.
Saving update path…done.
Searching for solvers…5 shims found.
3 solvers initialized (* = default):
GLPK unknown /usr/share/octave/4.0.0/m/optimization
• SDPT3 4.0 {cvx}/sdpt3
SeDuMi 1.34 {cvx}/sedumi
2 solvers not found:
ECOS GitHub - embotech/ecos: A lightweight conic solver for second-order cone programming.
SCS GitHub - cvxgrp/scs: Splitting Conic Solver
Saving updated preferences…done.
Testing with a simple model…done!
To change the default solver, type “cvx_solver <solver_name>”.
To save this change for future sessions, type “cvx_save_prefs”.
Please consult the users’ guide for more information.
• make sure you have gcc c++ compiler installed (duh, but it was missing in my cygwin at first…)
• [having now noticed there is an edit function here, I’ll pre-apologize for verbosity below and will edit out newbie stuff later, but still capturing issues as they are found and resolved here for
(… and don’t hit “tab” in this editor!.. oops… continuing…)
• look above at the “extra files” that cvx_setup warns about. These missing .h include files either need to be in the ~cvx\lib directory, or somehow inform octave and gcc where they can be found
through a library environment variable or command-line option. I took the easy cheat, and simply copied missing .h files from wherever they could be found (typically
C:\cygwin64\usr\include\octave-4.0.0\octave or
C:\cygwin64\usr\src\debug\octave-4.0.0-3\libinterp\corefcn) to the location of the file that included it, typically c:/octave/cvx/…
• make sure you have openBLAS and libopenblas in your cygwin config!
• you will still likely find you are missing “f77blas.h” and/or other math support. I resolved this by fetching libopenblas-dev_0.2.15-1_i386.deb from github, then peeled out f77blas.h with 7-zip
into c:/octave/cvx … (and perhaps some other files… it grows murky…) It seems this file is NOT part of the normal openblas distribution…
• but I still couldn’t get a clean SeDuMi build - “cannot find –lblas” from the linker.
• Hoping to pick up the required linker dll support without too much more digging around, I fleshed out the cygwin installed math support to include:
lapack, libamd1, libarpack0, libcamd1, libccolamd1, libcholmod1, libcolamd1, libcxsparse0, libfftw3_3, libglpk36, libgmp10, liblapack-devel, liblapack-doc, liblapack0, libmetis0, libmpc3,
libmpfr4, libopenblas, libqhull_6, libqrupdate0, libsuitesparseconfig1, libumfpack1, octave, octave-devel, octave-doc, octave-general, octave-miscellaneous, octave-optim, octave-struct, and
openblas are all installed. All are sufficient, it seems, all may not be necessary.
• I also installed python, python 3k, and numpy (I already had pulled in optim for octave, and lapack is also) as this seemed to pull in the right packages for others in other environments trying
to do other things, but seeing the same issues.
Somewhere along the line the linker stopped complaining about the missing blas support and I got a clean build of both SeDuMi and SDPT3 as you can see above.
Apologies for all the missing details in there, but that is roughly the process that I followed to get both of the solvers configured for cvx 3.0 on octave 4.0.0 on cygwin.
FYI the stripped-down real, symmetric problem above produces same optval=1 for my install described above, invariant of which solver is selected.
Update: SeDuMi solver produced correct output on snp500 time-series example, I start to believe…
Update: all examples in quickstart seemed to produce sane results.
Update Dec 24: “quiet” option seems to not yet be fully implemented in Octave, at least when running SeDuMi, as the “Constraint Accepted” messages continue to flow, and if the pager is enabled
requires keyboard input on all my iterations. I’ll see what I can dig up, but not truly blocking, yet… update: found “disp.m” in ~cvx/lib/@cvxcnst/disp.m and put in a conditional on cvx_quiet. Nope,
still paging though blanks… TODO: fix later - just turn off the pager in octave with octave>> more off.
When GLPK is selected on my problem error message:
"‘quiet’ undefined near line 95 column 4
error: called from ‘cvx_glpk>solve’ in file /cygdrive/c/octave/cvx/shims/cvx_glpk.m near line 95, column 1"
modifying cvx_glpk.m shim to use “cvx_quiet” instead of “quiet” fixes this one.
The command “cvx_pause.m” seems not to exist in the disttibution… getting it from https://code.google.com/p/matlab-code-by-wayne/source/browse/branches/cvx/commands/cvx_pause.m line 12: s =
cvx___.pause; reveals that error: structure has no member ‘pause’. TODO deal later…
Overall, despite a few hiccups to get it up and running and to understand CVX’s disciplined-programming approach, CVX on Octave seems to be working for me on a model-predictive receding-horizon
control problem.
Thanks all!
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Prof. Dejun FENG
BEng (Chengdu UST)
PhD (Wuhan University)
Room 211, Lady Shaw Building,
The Chinese University of Hong Kong,
Shatin, N.T., Hong Kong
Fields of Interest:
Fractal Geometry, Ergodic Theory, Dynamical Systems.
Selected Publications:
1. Dimension of invariant measures for affine iterated function systems. Preprint, 2019. To appear in Duke Math. J.
2. Dimension estimates for $C^1$ iterated function systems and repellers. Part II. To appear in Ergodic theory Dynam. Systems (with Karoly Simon)
3. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. To appear in Ergodic theory Dynam. Systems (with Karoly Simon)
4. Estimates on the dimension of self-similar measures with overlaps. To appear in J. Lond. Math. Soc. (with Zhou Feng)
5. On arithmetic sums of fractal sets in ${\Bbb R}^d$. J. Lond. Math. Soc. 104(2021), no. 1, 35-65. (with Yufeng Wu)
Major Research Grants:
• Research Grants Council - General Research Fund
Honours and Awards:
• Faculty Exemplary Teaching Award
Professional activities:
• Editorial Board - Methods and Applications of Analysis
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Research Papers - Instavm
The library of research papers is a collection of all papers presented at our past conferences. papers from international Conferences are also available from our online shop as complete books of
proceedings each conference. This library classifies the papers as belonging to:
• Applications of Vedic Maths Sutras
• Vedic Maths in education
• Vedic Maths in digital technology
• History and Philosophy of Mathematics in india
Click on any paper to view or download
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energy consumption ball and sag mills
WEBDec 30, 2023 · The ball loading of a SAG mill can be considered optimal if it provides a given finished grade capacity with consideration of minimum energy consumption. Insufficient ball
filling most often leads to a decrease in the efficiency of operation on the output of the finished class, and its excess leads to an increase in the power .
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WEBA ball mill consists of various components that work together to facilitate grinding operations. The key parts include the following: Mill Shell: The cylindrical shell provides a protective
and structural enclosure for the mill. It is often made of steel and lined with wearresistant materials to prolong its lifespan.
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WEBDec 1, 2005 · The product from the SAG mill is further reduced in size using pebble crushers and ball mills. Hence, typical gold or copper ore requires between and kWh per ton of energy to
reduce the particle size. Considering a typical mining operation processes 10,000 to 100,000 tons per day the energy expenditure in grinding is 50 .
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WEBSep 22, 2019 · The SAG and ball mill circuits are responsible for 59% of minewide electrical energy consumption, equating to 195 GWh of annual consumption. The regrind an d flotation circuits
account for 16% of ...
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WEBJan 17, 2017 · providing operating data for 3 semi autogenous grinding (SAG) mills and 4 ball mills. Results show on. average that 7 9% of the supplied electrical energy converts to heat
absorbed by the slurry ...
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WEBThe main drawback of SAG high power consumption. To reduce power consumption, I think: The right determine of SAG feed size. This depends on the mill diameter and the ball size. Feed size
necessary to take 80% of the maximum piece of ore that will crushed by the balls. For the big SAG more than 8 meters optimal feed size will be 50mm.
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WEBThe experimental assessment of ball mills type MCB (МШЦ 4,5х6) operating in copper ore processing plant and SAG mill operating in gold ore processing plant are presented. The experimental
results are statistically processed and some statistical regression models are estimated.
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WEBEnergy consumption represents a high operational cost in mining operations. Ore size reduction stage is the main consumer in that process, where the Semiautogenous Mill (SAG) is one of the
main components. The implementation of control and automation strategies that can achieve production goals along with energy efficiency are a common .
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WEBJun 1, 2009 · Nowadays ball mills still dominate secondary grinding appliions, but are increasingly in competition with closed circuit AG/SAG mill circuits and stirred mills (NapierMunn et
al., 1996). At product sizes finer than 80% passing 75 μm, the efficiency of ball mill grinding rapidly decreases.
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WEBApr 19, 2023 · Considering the continuous increase in production costs and resource optimization, more than a strategic objective has become imperative in the copper mining industry. In the
search to improve the efficiency in the use of resources, the present work develops models of a semiautogenous grinding (SAG) mill using statistical analysis .
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WEBNov 2, 2017 · Similarly, in SAG milling, there is an optimum point for rock/ball filling, mill speed, and product size to achieve optimum throughput and specific energy consumption [46, 47].
However, operational strategies at many mine sites are based on a simple philosophy—maximizing installed power (such as running maximum HPGR force .
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WEBApr 1, 2016 · The slurry level in AG/SAG and ball mills has an influence on their power draw, which in some instances can be profound. ... This paper demonstrates the feasibility of using a
ceramic medium stirring mill to reduce the energy consumption of grinding operations through an industrial verifiion experiment. Compared with the original two .
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WEBMar 1, 2019 · Batch experiments in a m diameter mill have shown that improvements of about 13 per cent in the efficiency of energy utilisation can be realised in ball mills by replacing 25 per
cent of the ...
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WEBOct 1, 2001 · The SAG mill was originally operating in a ball mill duty and it has a low aspect ratio (diameter/length) which changes its operation relative to typical high aspect mills
(Powell, Morrell and ...
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WEBJan 1, 2011 · Tracking the mill performance showed that the mill power consumption decreased from to kWh/t (a nearly 30% reduction), P80 of the mill product decreased from 611 to 355μm, and
the mill ...
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WEBMar 1, 2004 · The specific energy consumption of high voltage pulse discharge breakage was described by the Walker theory ... are used following the development of equations which accurately
predict the specific energy and power draw of AG/SAG and ball mills (Morrell, 1996, 2004a, 2004b; Scinto et al., 2015, Lane et al., 2013), combined with the .
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WEBSAG/Ball Mill Sizing Gross Power Draw and Specific Energy Calculator for SemiAutogenous Grinding (SAG) and Ball Mills. Return to Mill Calculators. ... Specific Energy Required: [kWh/t] Power
Required: [kW] Mill Selection Result. Description. Value Units. Available Power Utilized: [%]
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WEBApr 19, 2023 · 2. Background. The SAG grinding process has been modeled by various authors, either generating explanatory models of the grinding process, with the aim of modeling, simulating,
and optimizing the individual process, as well as integrating it into aggregate processes, such as the minetomill (M2M) paradigm, a practice that has .
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WEBOct 9, 2020 · As one of the machines widely used in mining, a semiautogenous grinding (SAG) mill can significantly improve the roughing efficiency of rock. But the SAG mill still faces the
obstacles of significant energy consumption and empirical operation parameters. In order to obtain the optimal operation parameters of a SAG mill, in this .
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WEBThese conventional technologies, in particular SAG and ball mills, are an order of magnitude larger than those installed 50 years ago, and are continuing to grow.
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WEBMay 28, 2018 · Bond's method is widely used for predicting the energy requirement of crushers, rod and ball mills, while a number of other methodologies have been proposed for AG/SAG mills.
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WEBJan 1, 2008 · ball mill circuit required the least energy, ie was the mos t energy efficient. 6 ... (SAG) and ball mill circuits are used to show that Bond's equation does not hold over the
particle size range ...
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WEBApr 18, 2018 · Crushing SAG Mill Critical Size Material. The relationship between the grinding power consumption and the product size distribution has been widely studied since it is a subject
of major interest for plant design and optimization. The conventional approaches, generally of empirical nature, have led to the formation of equations such .
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WEBNov 27, 2020 · Determination of Load Jc and Balls Jb Level by Power. Draw and Pressure in Grinding Mills. Mario Cerda. ABSTRACT. For purposes of determining the charge level Jc and the ball le
vel Jb, the use of ...
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WEBCerro Verde actual data shows that the HPGR circuit energy consumption was 3 kWh/t less than the SAG/Ball mill circuit that was considered in the trade off studies. The initial trade off
studies were expecting to have kWH/t of less energy consumption. (unknown) 8 years ago.
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WEBJan 1, 2001 · The Discrete Element Method (DEM) was introduced as an alternative way to model mills by Rajamani (1992, 1994) who simulated a ball mill and Rajamani and Mishra (1996) with
simulations of a SAG ...
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WEBJul 15, 2013 · The basis for ball mill circuit sizing is still B ond's methodology (Bond, 1962). The Bond ball and rod. mill tests are used to determine specific energy c onsumption (kWh/t) to
grind from a ...
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WEBIn contrast, the energy consumption provided by the Bond methodology was lower than the results obtained on a pilot scale for all routes studied. Use of pebble crushing led to a reduction of %
in the SAG mill power consumption for the SAB/C route, without increase in power consumption in the ball mill.
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WEBJul 1, 2022 · The SAG mill performance is researched based on the DEMEMBD coupling method. ... Mishra and Rajamani [[2], [3], [4]] first studied the charging behavior and the rule of energy
consumption of the ball mill based on 2D DEM in 1992. Cleary et al. [5] studied the effect of ball mill rotational rate on charge behavior and mill power, ...
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August 23, WIMPS, Redux (Weakly Interacting Massive Particles)
August 23, WIMPS, Redux (Weakly Interacting Massive Particles) (1)
We’re hoping dark matter can’t hide any more
And we’ll find WIMPs better than ever before.
Seven tons of Xenon, down deep in the ground,
May flash if a WIMP hits it right in the core.
Seven tons of Xenon, down deep in the ground,
May prove the best WIMP catcher physicists found
To spy a WIMP particle, just passing through.
There ought to be lots of them buzzing around.
To catch a WIMP particle just passing through,
We have to recall what they’ve shown they can do.
They cause gravitation but don’t emit light.
So far, gravitation’s dark matter’s sole clue.
WIMPs cause gravitation but don’t emit light,
Which means, instead physicists have to be bright.
So, why do they think that the Xenon should flash?
Could Xenon stay dark if these folks are not right?
Why do our bright folks think that Xenon should flash?
Well, atoms recoil when two particles clash.
When WIMPs whack the Xenon, photons we can see
Make flashes; detectors should capture the splash.
If WIMPs hit the Xenon, then we hope to see,
A flash as an atom recoils suddenly.
Now, sitting and watching is all we can do;
Play chess or play poker and wait patiently.
• Galaxies spin too fast for the stars and gas we can see to account for the gravitational force that would be required. So, “dark matter” has been postulated to make up about 85% of all the mass
in the universe. Its particle form should be the WIMP, which might be 100 times as large as a proton but very anti-social, for unknown reasons. Hence the name, “Weakly Interacting Massive
Particles.” A human illustration would be the failure of facts to interact with the science-denier’s brain.
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point.in.polygon: do point(s) fall in a given polygon? in sp: Classes and Methods for Spatial Data
verifies for one or more points whether they fall in a given polygon
point.x numerical array of x-coordinates of points
point.y numerical array of y-coordinates of points
pol.x numerical array of x-coordinates of polygon
pol.y numerical array of y-coordinates of polygon
mode.checked default FALSE, used internally to save time when all the other argument are known to be of storage mode double
default FALSE, used internally to save time when all the other argument are known to be of storage mode double
integer array; values are: 0: point is strictly exterior to pol; 1: point is strictly interior to pol; 2: point lies on the relative interior of an edge of pol; 3: point is a vertex of pol.
Uses the C function InPoly(). InPoly is Copyright (c) 1998 by Joseph O'Rourke. It may be freely redistributed in its entirety provided that this copyright notice is not removed.
For more information on customizing the embed code, read Embedding Snippets.
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Answered! PYTHON CODE ONLY! Given the following assumptions: 1. Call the cost of your dream home total_cost….
Given the following assumptions:
1. Call the cost of your dream home total_cost.
2. Call the portion of the cost needed for a down payment portion_down_payment. For simplicity, assume that portion_down_payment = 0.25 (25%).
3. Call the amount that you have saved thus far current_savings. You start with a current savings of $0.
4. Assume that you invest your current savings wisely, with an annual return of r (in other words, at the end of each month, you receive an additional current_savings*r/12 funds to put into your
savings – the 12 is because r is an annual rate). Assume that your investments earn a return of r = 0.04 (4%).
5. Assume your annual salary is annual_salary.
6. Assume you are going to dedicate a certain amount of your salary each month to saving for the down payment. Call that portion_saved. This variable should be in decimal form (i.e. 0.1 for 10%).
7. At the end of each month, your savings will be increased by the return on your investment, plus a percentage of your monthly salary (annual salary / 12).
Write a program to calculate how many months it will take you to save up enough money for a down payment. You will want your main variables to be floats, so you should cast user inputs to floats.
Your program should ask the user to enter the following variables:
1. The starting annual salary (annual_salary)
2. The portion of salary to be saved (portion_saved)
3. The cost of your dream home (total_cost)
Expert Answer
annual_salary= float(input(“Enter your annual salary: “))
portion_saved= float(input(“Enter the percent of your salary to save, as a decimal: “))
total_cost= float(input(“Enter the cost of your dream home: “))
increment = float(input(“Enter the semiannual raise, as a decimal: “))
r = 0.04
ms = annual_salary/12
current_saving = 0
months = 0
portion_down_payment = 0.25
down_payment= total_cost*portion_down_payment
counter = 0
while current_saving < down_payment:
if counter is 6:
counter = 0
ms += ms*increment
current_saving = current_saving + (portion_saved*ms) + (current_saving*r/12)
months += 1
counter += 1
print(“Number of Months required for down payment:”, months)
annual_salary= float(input(“Enter your annual salary: “))
portion_saved= float(input(“Enter the percent of your salary to save, as a decimal: “))
total_cost= float(input(“Enter the cost of your dream home: “))
r = 0.04
ms = annual_salary/12
current_saving = 0
months = 0
portion_down_payment = 0.25
down_payment= total_cost*portion_down_payment
while current_saving < down_payment:
current_saving = current_saving + (portion_saved*ms) + (current_saving*r/12)
months += 1
print(“Number of Months required for down payment:”, months)
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Type Hints | Jon Macey's WebPages
Type Hints
We will investigate some modern python features and concepts by developing some small programs.
Type hints were added to python 3.5 as a way of adding support for linters and 3rd party tools to check code for possible errors. These have no effect of the running of the code and are not
mandatory, however they are good software engineering practice and we encourage their use throughout your code.
Example file
#!/usr/bin/env python
import cmath
def quadratic_roots(a, b, c):
"""calculate the quadratic roots of a, b, and c are coefficient and real numbers and also a ≠ 0.
If a is equal to 0 that equation is not valid quadratic equation.
a : number
b : number
c : number
first : number
second : number
ValueError if a == 0
if a == 0:
raise ValueError
discriminant = (b**2) - (4 * a * c)
first = (-b - cmath.sqrt(discriminant)) / (2 * a)
second = (-b + cmath.sqrt(discriminant)) / (2 * a)
return first, second
# This will return a complex result
a, b = quadratic_roots(2, 3, 4)
print(f"{a} {b}")
# This will return a complex but with zero j
a, b = quadratic_roots(1, 4, 2)
print(f"{a} {b}")
# This will throw an exception
a, b = quadratic_roots(0, 4, 2)
print(f"{a} {b}")
except ValueError:
print("a was zero!")
a, b = quadratic_roots("1", "4", "2")
print(f"{a} {b}")
if we run this program we get an error
(-0.75-1.1989578808281798j) (-0.75+1.1989578808281798j)
(-3.414213562373095+0j) (-0.5857864376269049+0j)
a was zero!
Traceback (most recent call last):
File "/Volumes/teaching/Code/ASEAIM/Lab2/./function.py", line 49, in <module>
a, b = quadratic_roots("1", "4", "2")
File "/Volumes/teaching/Code/ASEAIM/Lab2/./function.py", line 27, in quadratic_roots
discriminant = (b**2) - (4 * a * c)
TypeError: unsupported operand type(s) for ** or pow(): 'str' and 'int'
We can also run mypy on it which reports
mypy function.py
Success: no issues found in 1 source file
Adding Type Hints
from typing import Any, Tuple
# Note float type hint will also support int as a number
def quadratic_roots(
a: float, b: float, c: float
) -> Tuple[Any[float, complex], Any[float, complex]]:
The function has now added type hints for the parameters as well as the return type. Running the new program still causes the same errors with the string parameters however running mypy on the file
gives the following.
mypy functionTypeHint.py
functionTypeHint.py:53: error: Argument 1 to "quadratic_roots" has incompatible type "str"; expected "float"
functionTypeHint.py:53: error: Argument 2 to "quadratic_roots" has incompatible type "str"; expected "float"
functionTypeHint.py:53: error: Argument 3 to "quadratic_roots" has incompatible type "str"; expected "float"
Found 3 errors in 1 file (checked 1 source file)
These errors will usually show in vscode if we setup the python tools to use mypy (or pylance).
Stub Files
It is also possible to add type hints to a separate file called a “stub file” typically this will have the extension .pyi
from typing import Any, Tuple
# Note float type hint will also support int as a number
def quadratic_roots(
a: float, b: float, c: float
) -> Tuple[Any[float, complex], Any[float, complex]]: ...
Stub files only work with imported modules so we can test using a simple file.
#!/usr/bin/env python
from functionStubFile import quadratic_roots
# This will return a complex result
a, b = quadratic_roots(2, 3, 4)
print(f"{a} {b}")
# This will return a complex but with zero j
a, b = quadratic_roots(1, 4, 2)
print(f"{a} {b}")
# This will throw an exception
a, b = quadratic_roots(0, 4.2, 2)
print(f"{a} {b}")
except ValueError:
print("a was zero!")
a, b = quadratic_roots("1", "4", "2")
print(f"{a} {b}")
Running the following now spots the errors
mypy testStubFile.py
testStubFile.py:20: error: Argument 1 to "quadratic_roots" has incompatible type "str"; expected "float"
testStubFile.py:20: error: Argument 2 to "quadratic_roots" has incompatible type "str"; expected "float"
testStubFile.py:20: error: Argument 3 to "quadratic_roots" has incompatible type "str"; expected "float"
Found 3 errors in 1 file (checked 1 source file)
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Is stat a math course?
Statistics is not just a math class. Statistics is all about understanding data – numbers with context and meaning. A computer can do all of the calculations and all of the numerical work with
finding a mean, a standard deviation, and even a confidence interval (all things we do in statistics).
Is stat a hard course?
Statistics has gotten a reputation for being a very hard class, especially when taken in college, because it combines math concepts in order to form an analysis of a data set that can be used to
understand an association in the data (whoo that was a mouthful).
How can I teach myself statistics?
Here are some tips to make your learning more efficient and fruitful.
1. 1) Learn the core mathematics first, then the statistics.
2. 2) Learn about what statistics can do, not about what it can say.
3. 3) Probability theory and statistics go hand in hand.
4. 4) Regression analysis is very useful, but also often misused.
Is statistics harder than trigonometry?
Statistics is more difficult than trigonometry. The subject you know the least about will be the more difficult. Generally, I would say that statistics is more complicated, hence more difficult, than
trigonometry but statistics is also more useful in everyday life to nontechnical people.
Should I take pre calc or stats?
If you take Stats, which is pretty easy (in my opinion) you’ll have to take Precalc in college, which is a little more difficult. I’d say take precalc while it’s still slow-going in high school, and
take stats in college (trust me, it will be easier on you).
Should I take stats or Calc?
If you have plans to major in STEM, then AP Calculus is a must in high school. AP Statistics is a better option for Commerce, Business and Finance majors. You can choose both if you want to major in
Math and Statistics. it is up to you.
Do doctors need calculus or statistics?
Many medical schools require a year of mathematics and recommend calculus and statistics. Medical schools vary in their math requirements. The most conservative way to meet the requirements at the
largest number of medical schools is to have one calculus credit and one statistics credit.
How can I learn statistics online?
edX offers both individual courses and advanced programs designed to help you learn about statistics in an engaging and effective online learning environment complete with video tutorials, quizzes
and more.
Do colleges want calculus or statistics?
But for many other students, calculus isn’t the math course that will most help them—the right course often is statistics. But most admissions counselors have favored calculus (in many cases
informally), the report says, and that hurts students.
What medical schools require statistics?
These include Dartmouth, Brown, Harvard, Johns Hopkins, Duke, Texas A&M, UC Irvine and Uniformed Services. Most medical colleges require or recommend statistics, and a few schools recommend a
biostatistics course.
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Mixing categorial and continuous data in Naive Bayes classifier using scikit-learn
I'm using scikit-learn in Python to develop a classification algorithm to predict the gender of certain customers. Amongst others, I want to use the Naive Bayes classifier but my problem is that I
have a mix of categorical data (ex: "Registered online", "Accepts email notifications" etc) and continuous data (ex: "Age", "Length of membership" etc). I haven't used scikit much before but I
suppose that Gaussian Naive Bayes is suitable for continuous data and that Bernoulli Naive Bayes can be used for categorical data. However, since I want to have both categorical and continuous data
in my model, I don't really know how to handle this. Any ideas would be much appreciated!
You can’t train a model by giving mix (categorial and continuous) values by input. You can transform all your data into a categorical representation by computing percentiles for each continuous
variable and then binning the continuous variables using the percentiles as bin boundaries.
For example, binning for the height of a person, create the following bins: "very small", "small", "regular", "big", "very big" ensuring that each bin contains approximately 20% of the population of
your training set.
We can’t perform this task automatically using any library, because it's the part of data preprocessing. Data preprocessing should be done manually for better results. We have a Pandas library for
this manipulative task.
You can use an independently fit a gaussian NB model on the continuous part of the data and a multinomial NB model on the categorical part. Then transform all the dataset by taking the class
assignment probabilities (with predict_proba method) as new features:
np.hstack((multinomial_probas, gaussian_probas))
and then easily refit a new model (e.g. a new gaussian NB) on the new features.
Hope this answer helps.
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CTAN has a new package: bpolynomial
Date: November 27, 2007 12:39:27 PM CET
The package below has been put on tug.ctan.org and should soon be at your favorite mirror. Thanks again, Jim Hefferon Saint Michael's College
................................................................... The following information was provided by our fellow contributor: Name of contribution: bpolynomial Author's name: Stephan Hennig
Location on CTAN: /graphics/metapost/contrib/macros/bpolynomial/ Summary description: Drawing polynomials with MetaPost. License type: lppl Announcement text: bpolynomial.mp version 0.3 released on
2007-11-26: The MetaPost package bpolynomial.mp helps drawing polynomial functions of up to degree three. It provides macros to calculate Bézier curves exactly matching a given constant, linear,
quadratic or cubic polynomial. Changes to v0.2 pre-released 2007-11-25: + Improved documentation. + Slightly improved numeric stability.This package is located at
. More information is at
(if the package is new it may take a day for that information to appear). We are supported by the TeX Users Group
. Please join a users group; see
bpolynomial – Drawing polynomial functions of up to order 3
This METAPOST package helps plotting polynomial and root functions up to order three. The package provides macros to calculate Bézier curves exactly matching a given constant, linear, quadratic or
cubic polynomial, or square or cubic root function. In addition, tangents on all functions and derivatives of polynomials can be calculated.
Package bpolynomial
Version 0.5
Copyright 2007 Stephan Hennig
Maintainer Stephan Hennig
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graph – Interface for the Aesara graph
graph – Interface for the Aesara graph#
Core graph classes.
class aesara.graph.basic.Apply(op: OpType, inputs: Sequence[Variable], outputs: Sequence[Variable])[source]#
A Node representing the application of an operation to inputs.
Basically, an Apply instance is an object that represents the Python statement outputs = op(*inputs).
This class is typically instantiated by a Op.make_node method, which is called by Op.__call__.
The function aesara.compile.function.function uses Apply.inputs together with Variable.owner to search the expression graph and determine which inputs are necessary to compute the function’s
A Linker uses the Apply instance’s op field to compute numeric values for the output variables.
The Variable.owner field of each Apply.outputs element is set to self in Apply.make_node.
If an output element has an owner that is neither None nor self, then a ValueError exception will be raised.
The operation that produces outputs given inputs.
The arguments of the expression modeled by the Apply node.
The outputs of the expression modeled by the Apply node.
clone(clone_inner_graph: bool = False) Apply[OpType][source]#
Clone this Apply instance.
clone_inner_graph – If True, clone HasInnerGraph Ops and their inner-graphs.
Return type:
A new Apply instance with new outputs.
Tags are copied from self to the returned instance.
clone_with_new_inputs(inputs: Sequence[Variable], strict=True, clone_inner_graph=False) Apply[OpType][source]#
Duplicate this Apply instance in a new graph.
○ inputs (list of Variables) – List of Variable instances to use as inputs.
○ strict (bool) – If True, the type fields of all the inputs must be equal to the current ones (or compatible, for instance TensorType of the same dtype and broadcastable patterns, in
which case they will be converted into current Type), and returned outputs are guaranteed to have the same types as self.outputs. If False, then there’s no guarantee that the clone’s
outputs will have the same types as self.outputs, and cloning may not even be possible (it depends on the Op).
○ clone_inner_graph (bool) – If True, clone HasInnerGraph Ops and their inner-graphs.
An Apply instance with the same Op but different outputs.
Return type:
Returns the default output for this node.
An element of self.outputs, typically self.outputs[0].
Return type:
Variable instance
May raise AttributeError self.op.default_output is out of range, or if there are multiple outputs and self.op.default_output does not exist.
Return a list of the parents of this node. Should return a copy–i.e., modifying the return value should not modify the graph structure.
property nin[source]#
The number of inputs.
property nout[source]#
The number of outputs.
property out[source]#
An alias for self.default_output
Returns the params for the node, or NoParams if no params is set.
class aesara.graph.basic.AtomicVariable(type: _TypeType, name: str | None = None, **kwargs)[source]#
A node type that has no ancestors and should never be considered an input to a graph.
Return a new, un-owned Variable like self.
**kwargs (dict) – Optional “name” keyword argument for the copied instance. Same as self.name if value not provided.
A new Variable instance with no owner or index.
Return type:
Variable instance
Tags and names are copied to the returned instance.
This does what __eq__ would normally do, but Variable and Apply should always be hashable by id.
class aesara.graph.basic.Constant(type: _TypeType, data: Any, name: str | None = None)[source]#
A Variable with a fixed data field.
Constant nodes make numerous optimizations possible (e.g. constant in-lining in C code, constant folding, etc.)
The data field is filtered by what is provided in the constructor for the Constant’s type field.
Return a new, un-owned Variable like self.
**kwargs (dict) – Optional “name” keyword argument for the copied instance. Same as self.name if value not provided.
A new Variable instance with no owner or index.
Return type:
Variable instance
Tags and names are copied to the returned instance.
Get the test value.
TestValueError –
class aesara.graph.basic.Node[source]#
A Node in an Aesara graph.
Currently, graphs contain two kinds of Nodes: Variables and Applys. Edges in the graph are not explicitly represented. Instead each Node keeps track of its parents via Variable.owner /
Return a list of the parents of this node. Should return a copy–i.e., modifying the return value should not modify the graph structure.
class aesara.graph.basic.NominalVariable(id: _IdType, typ: _TypeType, **kwargs)[source]#
A variable that enables alpha-equivalent comparisons.
Return a new, un-owned Variable like self.
**kwargs (dict) – Optional “name” keyword argument for the copied instance. Same as self.name if value not provided.
A new Variable instance with no owner or index.
Return type:
Variable instance
Tags and names are copied to the returned instance.
class aesara.graph.basic.Variable(type: _TypeType, owner: OptionalApplyType, index: int | None = None, name: str | None = None)[source]#
A Variable is a node in an expression graph that represents a variable.
The inputs and outputs of every Apply are Variable instances. The input and output arguments to create a function are also Variable instances. A Variable is like a strongly-typed variable in some
other languages; each Variable contains a reference to a Type instance that defines the kind of value the Variable can take in a computation.
A Variable is a container for four important attributes:
□ type a Type instance defining the kind of value this Variable can have,
□ owner either None (for graph roots) or the Apply instance of which self is an output,
□ index the integer such that owner.outputs[index] is this_variable (ignored if owner is None),
□ name a string to use in pretty-printing and debugging.
There are a few kinds of Variables to be aware of: A Variable which is the output of a symbolic computation has a reference to the Apply instance to which it belongs (property: owner) and the
position of itself in the owner’s output list (property: index).
□ Variable (this base type) is typically the output of a symbolic computation.
□ Constant: a subclass which adds a default and un-replaceable value, and requires that owner is None.
TensorVariable subclass of Variable that represents a numpy.ndarray
□ TensorSharedVariable: a shared version of TensorVariable.
□ SparseVariable: a subclass of Variable that represents a scipy.sparse.{csc,csr}_matrix object.
□ RandomVariable.
A Variable which is the output of a symbolic computation will have an owner not equal to None.
Using a Variables’ owner field and an Apply node’s inputs fields, one can navigate a graph from an output all the way to the inputs. The opposite direction is possible with a FunctionGraph and
its FunctionGraph.clients dict, which maps Variables to a list of their clients.
☆ type (a Type instance) – The type governs the kind of data that can be associated with this variable.
☆ owner (None or Apply instance) – The Apply instance which computes the value for this variable.
☆ index (None or int) – The position of this Variable in owner.outputs.
☆ name (None or str) – A string for pretty-printing and debugging.
import aesara
import aesara.tensor as at
a = at.constant(1.5) # declare a symbolic constant
b = at.fscalar() # declare a symbolic floating-point scalar
c = a + b # create a simple expression
f = aesara.function([b], [c]) # this works because a has a value associated with it already
assert 4.0 == f(2.5) # bind 2.5 to an internal copy of b and evaluate an internal c
aesara.function([a], [c]) # compilation error because b (required by c) is undefined
aesara.function([a,b], [c]) # compilation error because a is constant, it can't be an input
The python variables a, b, c all refer to instances of type Variable. The Variable referred to by a is also an instance of Constant.
Return a new, un-owned Variable like self.
**kwargs (dict) – Optional “name” keyword argument for the copied instance. Same as self.name if value not provided.
A new Variable instance with no owner or index.
Return type:
Variable instance
Tags and names are copied to the returned instance.
Evaluate the Variable.
inputs_to_values – A dictionary mapping Aesara Variables to values.
>>> import numpy as np
>>> import aesara.tensor as at
>>> x = at.dscalar('x')
>>> y = at.dscalar('y')
>>> z = x + y
>>> np.allclose(z.eval({x : 16.3, y : 12.1}), 28.4)
We passed eval() a dictionary mapping symbolic Aesara Variables to the values to substitute for them, and it returned the numerical value of the expression.
eval() will be slow the first time you call it on a variable – it needs to call function() to compile the expression behind the scenes. Subsequent calls to eval() on that same variable will
be fast, because the variable caches the compiled function.
This way of computing has more overhead than a normal Aesara function, so don’t use it too much in real scripts.
Return a list of the parents of this node. Should return a copy–i.e., modifying the return value should not modify the graph structure.
Get the test value.
TestValueError –
aesara.graph.basic.ancestors(graphs: Iterable[Variable], blockers: Collection[Variable] | None = None) Generator[Variable, None, None][source]#
Return the variables that contribute to those in given graphs (inclusive).
☆ graphs (list of Variable instances) – Output Variable instances from which to search backward through owners.
☆ blockers (list of Variable instances) – A collection of Variables that, when found, prevent the graph search from preceding from that point.
Variables – All input nodes, in the order found by a left-recursive depth-first search started at the nodes in graphs.
aesara.graph.basic.applys_between(ins: Collection[Variable], outs: Iterable[Variable]) Generator[Apply, None, None][source]#
Extract the Applys contained within the sub-graph between given input and output variables.
aesara.graph.basic.as_string(inputs: ~typing.List[~aesara.graph.basic.Variable], outputs: ~typing.List[~aesara.graph.basic.Variable], leaf_formatter=<class 'str'>, node_formatter=<function
default_node_formatter>) List[str][source]#
Returns a string representation of the subgraph between inputs and outputs.
☆ inputs (list) – Input Variables.
☆ outputs (list) – Output Variables.
☆ leaf_formatter (callable) – Takes a Variable and returns a string to describe it.
☆ node_formatter (callable) – Takes an Op and the list of strings corresponding to its arguments and returns a string to describe it.
Returns a string representation of the subgraph between inputs and outputs. If the same node is used by several other nodes, the first occurrence will be marked as *n -> description and all
subsequent occurrences will be marked as *n, where n is an id number (ids are attributed in an unspecified order and only exist for viewing convenience).
Return type:
list of str
aesara.graph.basic.clone(inputs: List[Variable], outputs: List[Variable], copy_inputs: bool = True, copy_orphans: bool | None = None, clone_inner_graphs: bool = False) Tuple[Collection[Variable],
Copies the sub-graph contained between inputs and outputs.
☆ inputs – Input Variables.
☆ outputs – Output Variables.
☆ copy_inputs – If True, the inputs will be copied (defaults to True).
☆ copy_orphans – When None, use the copy_inputs value. When True, new orphans nodes are created. When False, original orphans nodes are reused in the new graph.
☆ clone_inner_graphs (bool) – If True, clone HasInnerGraph Ops and their inner-graphs.
Return type:
The inputs and outputs of that copy.
A constant, if in the inputs list is not an orphan. So it will be copied conditional on the copy_inputs parameter; otherwise, it will be copied conditional on the copy_orphans parameter.
aesara.graph.basic.clone_get_equiv(inputs: Sequence[Variable], outputs: Sequence[Variable], copy_inputs: bool = True, copy_orphans: bool = True, memo: Dict[Apply | Variable | Op, Apply | Variable |
Op] | None = None, clone_inner_graphs: bool = False, **kwargs) Dict[Apply | Variable | Op, Apply | Variable | Op][source]#
Clone the graph between inputs and outputs and return a map of the cloned objects.
This function works by recursively cloning inputs and rebuilding a directed graph from the inputs up.
If memo already contains entries for some of the objects in the graph, those objects are replaced with their values in memo and not unnecessarily cloned.
☆ inputs – Inputs of the graph to be cloned.
☆ outputs – Outputs of the graph to be cloned.
☆ copy_inputs – True means to create the cloned graph from cloned input nodes. False means to clone a graph that is rooted at the original input nodes. Constants are not cloned.
☆ copy_orphans – When True, inputs with no owners are cloned. When False, original inputs are reused in the new graph. Cloning is not performed for Constants.
☆ memo – Optionally start with a partly-filled dictionary for the return value. If a dictionary is passed, this function will work in-place on that dictionary and return it.
☆ clone_inner_graphs – If True, clone HasInnerGraph Ops and their inner-graphs.
☆ kwargs – Keywords passed to Apply.clone_with_new_inputs.
aesara.graph.basic.clone_node_and_cache(node: Apply, clone_d: Dict[Apply | Variable | Op, Apply | Variable | Op], clone_inner_graphs=False, **kwargs) Apply | None[source]#
Clone an Apply node and cache the results in clone_d.
This function handles Op clones that are generated by inner-graph cloning.
☆ None if all of node’s outputs are already in clone_d; otherwise,
☆ return the clone of node.
aesara.graph.basic.clone_replace(output: Collection[Variable], replace: Iterable[Tuple[Variable, Variable]] | Dict[Variable, Variable] | None = None, **rebuild_kwds) List[Variable][source]#
Clone a graph and replace subgraphs within it.
It returns a copy of the initial subgraph with the corresponding substitutions.
☆ output – Aesara expression that represents the computational graph.
☆ replace – Dictionary describing which subgraphs should be replaced by what.
☆ rebuild_kwds – Keywords to rebuild_collect_shared.
aesara.graph.basic.equal_computations(xs: List[ndarray | Variable], ys: List[ndarray | Variable], in_xs: List[Variable] | None = None, in_ys: List[Variable] | None = None) bool[source]#
Checks if Aesara graphs represent the same computations.
The two lists xs, ys should have the same number of entries. The function checks if for any corresponding pair (x, y) from zip(xs, ys) x and y represent the same computations on the same
variables (unless equivalences are provided using in_xs, in_ys).
If in_xs and in_ys are provided, then when comparing a node x with a node y they are automatically considered as equal if there is some index i such that x == in_xs[i] and y == in_ys[i] (and they
both have the same type). Note that x and y can be in the list xs and ys, but also represent subgraphs of a computational graph in xs or ys.
Return type:
aesara.graph.basic.general_toposort(outputs: Iterable[T], deps: Callable[[T], OrderedSet | List[T]], compute_deps_cache: Callable[[T], OrderedSet | List[T] | None] | None = None, deps_cache: Dict[T,
List[T]] | None = None, clients: Dict[T, List[T]] | None = None) List[T][source]#
Perform a topological sort of all nodes starting from a given node.
☆ deps (callable) – A Python function that takes a node as input and returns its dependence.
☆ compute_deps_cache (optional) – If provided, deps_cache should also be provided. This is a function like deps, but that also caches its results in a dict passed as deps_cache.
☆ deps_cache (dict) – A dict mapping nodes to their children. This is populated by compute_deps_cache.
☆ clients (dict) – If a dict is passed, it will be filled with a mapping of nodes-to-clients for each node in the subgraph.
deps(i) should behave like a pure function (no funny business with internal state).
deps(i) will be cached by this function (to be fast).
The order of the return value list is determined by the order of nodes returned by the deps function.
The second option removes a Python function call, and allows for more specialized code, so it can be faster.
aesara.graph.basic.get_var_by_name(graphs: Iterable[Variable], target_var_id: str, ids: str = 'CHAR') Tuple[Variable, ...][source]#
Get variables in a graph using their names.
☆ graphs – The graph, or graphs, to search.
☆ target_var_id – The name to match against either Variable.name or Variable.auto_name.
Return type:
A tuple containing all the Variables that match target_var_id.
aesara.graph.basic.graph_inputs(graphs: Iterable[Variable], blockers: Collection[Variable] | None = None) Generator[Variable, None, None][source]#
Return the inputs required to compute the given Variables.
☆ graphs (list of Variable instances) – Output Variable instances from which to search backward through owners.
☆ blockers (list of Variable instances) – A collection of Variables that, when found, prevent the graph search from preceding from that point.
☆ Input nodes with no owner, in the order found by a left-recursive
☆ depth-first search started at the nodes in graphs.
aesara.graph.basic.io_connection_pattern(inputs, outputs)[source]#
Return the connection pattern of a subgraph defined by given inputs and outputs.
aesara.graph.basic.io_toposort(inputs: Iterable[Variable], outputs: Reversible[Variable], orderings: Dict[Apply, List[Apply]] | None = None, clients: Dict[Variable, List[Variable]] | None = None)
Perform topological sort from input and output nodes.
☆ inputs (list or tuple of Variable instances) – Graph inputs.
☆ outputs (list or tuple of Apply instances) – Graph outputs.
☆ orderings (dict) – Keys are Apply instances, values are lists of Apply instances.
☆ clients (dict) – If provided, it will be filled with mappings of nodes-to-clients for each node in the subgraph that is sorted.
aesara.graph.basic.is_in_ancestors(l_apply: Apply, f_node: Apply) bool[source]#
Determine if f_node is in the graph given by l_apply.
☆ l_apply (Apply) – The node to walk.
☆ f_apply (Apply) – The node to find in l_apply.
Return type:
aesara.graph.basic.list_of_nodes(inputs: Collection[Variable], outputs: Iterable[Variable]) List[Apply][source]#
Return the Apply nodes of the graph between inputs and outputs.
aesara.graph.basic.op_as_string(i, op, leaf_formatter=<class 'str'>, node_formatter=<function default_node_formatter>)[source]#
Return a function that returns a string representation of the subgraph between i and op.inputs
aesara.graph.basic.orphans_between(ins: Collection[Variable], outs: Iterable[Variable]) Generator[Variable, None, None][source]#
Extract the Variables not within the sub-graph between input and output nodes.
Variable – The Variables upon which one or more Variables in outs depend, but are neither in ins nor in the sub-graph that lies between them.
>>> orphans_between([x], [(x+y).out])
aesara.graph.basic.replace_nominals_with_dummies(inputs, outputs)[source]#
Replace nominal inputs with dummy variables.
When constructing a new graph with nominal inputs from an existing graph, pre-existing nominal inputs need to be replaced with dummy variables beforehand; otherwise, sequential ID ordering (i.e.
when nominals are IDed based on the ordered inputs to which they correspond) of the nominals could be broken, and/or circular replacements could manifest.
FYI: This function assumes that all the nominal variables in the subgraphs between inputs and outputs are present in inputs.
aesara.graph.basic.vars_between(ins: Collection[Variable], outs: Iterable[Variable]) Generator[Variable, None, None][source]#
Extract the Variables within the sub-graph between input and output nodes.
☆ The Variables that are involved in the subgraph that lies
☆ between ins and outs. This includes ins, outs,
☆ orphans_between(ins, outs) and all values of all intermediary steps from
☆ ins to outs.
aesara.graph.basic.view_roots(node: Variable) List[Variable][source]#
Return the leaves from a search through consecutive view-maps.
aesara.graph.basic.walk(nodes: ~typing.Iterable[~aesara.graph.basic.T], expand: ~typing.Callable[[~aesara.graph.basic.T], ~typing.Iterable[~aesara.graph.basic.T] | None], bfs: bool = True,
return_children: bool = False, hash_fn: ~typing.Callable[[~aesara.graph.basic.T], int] = <built-in function id>) Generator[T | Tuple[T, Iterable[T] | None], None, None][source]#
Walk through a graph, either breadth- or depth-first.
☆ nodes – The nodes from which to start walking.
☆ expand – A callable that is applied to each node in nodes, the results of which are either new nodes to visit or None.
☆ bfs – If True, breath first search is used; otherwise, depth first search.
☆ return_children – If True, each output node will be accompanied by the output of expand (i.e. the corresponding child nodes).
☆ hash_fn – The function used to produce hashes of the elements in nodes. The default is id.
A node will appear at most once in the return value, even if it appears multiple times in the nodes parameter.
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In logic,
, also called
logical complement
, is an operation that essentially takes a proposition
to another proposition "not
", written
, which is interpreted intuitively as being true when
is false and false when
is true. Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, or semantic values more generally. In classical logic, negation is normally
identified with the truth function that takes
and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov ...
The above text is a snippet from Wikipedia: Negation
and as such is available under the Creative Commons Attribution/Share-Alike License.
1. The act of negating something.
2. A denial or contradiction.
3. A proposition which is the contradictory of another proposition and which can be obtained from that other proposition by the appropriately placed addition/insertion of the word "not". (Or, in
symbolic logic, by prepending that proposition with the symbol for the logical operator "not".)
4. The logical operation which obtains such (negated) propositions.
The above text is a snippet from Wiktionary: negation
and as such is available under the Creative Commons Attribution/Share-Alike License.
Need help with a clue?
Try your search in the crossword dictionary!
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ocally presentable categor
Category theory
Compact objects
objects $d \in C$ such that $C(d,-)$ commutes with certain colimits
Relative version
A locally presentable category is a category which contains a small set $S$ of small objects such that every object is a nice colimit over objects in this set.
This says equivalently that a locally presentable category $\mathcal{C}$ is a reflective localization $\mathcal{C} \hookrightarrow PSh(S)$ of a category of presheaves over $S$. Since here $PSh(S)$ is
the free colimit completion of $S$ and the localization imposes relations, this is a presentation of $\mathcal{C}$ by generators and relations, hence the name (locally) presentable category.
See also at locally presentable categories - introduction.
There are many equivalent characterizations of locally presentable categories. The following is one of the most intuitive, equivalent characterizations are discussed below.
(locally presentable category)
A category $\mathcal{C}$ is called locally presentable if
1. it is an accessible category;
2. it has all small colimits.
This means
1. $\mathcal{C}$ is a locally small category;
2. $\mathcal{C}$ has all small colimits;
3. there exists a small set $S \hookrightarrow Obj(\mathcal{C})$ of $\lambda$-compact objects that generates $\mathcal{C}$ under $\lambda$-filtered colimits for some regular cardinal $\lambda$.
This provides a “stratification” of the class of locally presentable categories, as follows.
(locally $\kappa$-presentable category)
For $\kappa$ a regular cardinal, a locally $\kappa$-presentable category is a locally presentable category, def. , such that the colimit-generating set $S$ may be taken to consist of $\kappa$
-compact objects.
Equivalent characterizations
There are various equivalent characterizations of locally presentable categories.
As limit-preserving functor categories
(as limit sketches)
Locally presentable categories are precisely the categories of models of limit-sketches.
This is Adámek & Rosický (1994), corollary 1.52.
Restricted to locally finitely presentable categories this becomes:
Locally finitely presentable categories, def. , are equivalently the categories of finite limit preserving functors $C \to Set$, for small finitely complete categories $C$.
For the more detailed statement see below at Gabriel-Ulmer duality. Equivalently this says that:
As localizations of presheaf categories
(as accessible reflective subcategories of presheaves)
Locally presentable categories are precisely the accessibly embedded full reflective subcategories
$(L \dashv i) \;\colon\; C \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\leftarrow}} {\;\;\bot\;\;} PSh(K)$
of categories of presheaves on some small category $K$.
This appears as Adámek & Rosický (1994), prop 1.46 and Gabriel & Ulmer (1971), Satz 8.5(c) and Bemerkung 8.6(c).
See also at sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes.
Locally presentable categories are complete.
A reflective subcategory of a complete category is complete, since monadic functors reflect limits, and the above proposition shows that any locally presentable category is a reflective subcategory
of a presheaf category, which is complete.
Finite presentability and Gabriel–Ulmer duality
Write $Lex$ for the 2-category of small categories with finite limits, with finitely continuous (i.e., finite limit preserving) functors between them, and natural transformations between those.
Write $LFP$ for the 2-category of locally finitely presentable categories, def. , right adjoint functors which preserve filtered colimits, and natural transformations between them.
There is an equivalence of 2-categories
$Lex^{op} \stackrel{\simeq}{\to} LFP$
$C \mapsto Lex(C,Set)$
which sends a finitely complete category $C$ to the category of models of $C$, i.e., the category of left exact functors $C \to$Set.
Well-poweredness and well-copoweredness
• Every locally presentable category is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.
• Every locally presentable category is also well-copowered. This is shown in Adámek & Rosický (1994), Prop. 1.57 & Thm. 2.49.
Examples and applications
Locally finitely presentable categories
We list examples of locally finitely presentable categories, def. .
The category Set of sets is locally finitely presentable.
For notice that every set is the directed colimit over the poset of all its finite subsets.
Moreover, a set $S \in Set$ is a $\kappa$-compact object precisely if it has cardinality $|S| \lt \kappa$. So all finite sets are $\aleph_0$-compact.
Hence a a set of generators that exhibits $Set$ as a locally finitely complete category is given by the set containing one finite set of cardinality $n \in \mathbb{N}$ for all $n$.
More generally, for $C$ any small category the category of presheaves $Set^C$ is locally finitely presentable.
This follows with Gabriel-Ulmer duality: the finite limit completion of $C$, $Lex(C)$, is also small, and $Set^C$ is equivalent to the category of finitely continuous functors $Lex(C) \to Set$.
More generally still, if $A$ is locally finitely presentable and $C$ is small, then $A^C$ is locally finitely presentable.
To see this, embed $A$ as a finitely-accessible reflective subcategory of a presheaf topos $Set^B$, and then note that by 2-functoriality of $(-)^C$ we get $A^C$ as a finitely-accessible reflective
subcategory of $Set^{B \times C}$.
The category of algebras of a Lawvere theory, for example Grp, is locally finitely presentable. A $T$-algebra $A$ is finitely presented if and only if the hom-functor $Alg_T(A, -)$ preserves filtered
colimits, and any $T$-algebra can be expressed as a filtered colimit of finitely presented algebras.
Locally presentable categories
A poset, considered as a category, is locally presentable precisely if it is a complete lattice.
The following three examples, being presheaf categories, are locally finitely presentable, thus a fortiori locally presentable. They are important for the general study of (∞,1)-categories.
More generally,
This appears for instance as (Borceux, prop. 3.4.16, page 220). It follows directly with prop. and using that every sheaf topos is an accessibly embedded subtopos of a presheaf topos (see at sheaf
toposes are equivalently the left exact reflective subcategories of presheaf toposes)
The main ingredient of a direct proof is:
This implies that all representables in a sheaf topos are $\kappa$-compact objects.
If $T$ is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category $A$, then the category $A^T$ of algebras over the monad is locally
presentable. In particular, if $A$ is locally presentable and $i: B \to A$ is a reflective subcategory, then $B$ is locally presentable if $i$ is accessible.
This appears in Adámek & Rosický (1994), 2.78.
This is actually somewhat subtle and gets into some transfinite combinatorics, from what I can gather.
• $\mathcal{C}$ a small category,
• $\mathcal{A}$ a locally presentable category
then also the functor category $Func(\mathcal{C}, \mathcal{A})$ is locally presentable.
This is
Adámek & Rosický (1994), Cor. 1.54
See at Functor category – Local presentability for more.
A slice category of a locally presentable category is again locally presentable.
This appears for instance as
Centazzo-Rosický-Vitale, remark 3
(locally presentable Grothendieck constructions)
Given a pseudofunctor with values in $Cat_{Adj}$ as
$\array{ \mathllap{ \mathbf{C} \,\colon\, \; } Base &\longrightarrow& Cat \\ \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}} \\ \Big\downarrow\mathrlap{{}^{f}} && \mathllap{^{f^\ast}}\Big\uparrow \Big
\downarrow\mathrlap{{}^{f_!}} \\ \mathcal{Y} &\mapsto& \mathbf{C}_{\mathcal{Y}} }$
such that for some regular cardinal $\kappa$
1. $Base$ is locally $\kappa$-presentable,
2. each $\mathbf{C}_{\mathcal{X}}$ is locally presentable,
3. $\mathbf{C}_{(-)}$preserves $\kappa$-filtered 2-limits
then also the Grothendieck construction $\int \mathbf{C}$ is locally presentable.
This follows, as explained in
, from the analogous statement for
which appears as
Makkai & Paré (1989), Prop. 5.3.1. (4)
Combinatorial model categories
A combinatorial model category is a model category that is in particular a locally presentable category.
Orthogonal subcategory problem
Given a class of morphisms $\Sigma$ in a locally presentable category, the answer to the orthogonal subcategory problem for $\Sigma^\perp$ is affirmative if $\Sigma$ is small, and is affirmative for
any class $\Sigma$ assuming the large cardinal axiom known as Vopenka's principle.
The definition is due to
Textbook account:
Review for the case of locally finitely presentable categories:
• Maru Sarazola, An introduction to locally finitely presentable categories (2017) [pdf, pdf]
See also:
On the example of Grothendieck abelian categories:
• Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philosophical Society 129 3 (2000) 447-475 [arXiv:math/0102087, doi:10.1017/S0305004100004722]
• Greg Bird, Limits in 2-Categories of Locally-Presentable Categories pdf]
• Henning Krause, Deriving Auslander’s formula, Documenta Math. 20 (2015) 669-688 [arXiv:1409.7051]
In enriched category theory
Discussion of local presentability in enriched category theory (see also references on enriched accessible categories):
See also:
• MO:q/53470: Enriched locally presentable categories
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Introduction to String Theory
String theory is the most controversial theory in all of science today. On the one hand, it's a mathematically compelling framework that offers the potential to unify the Standard Model with General
Relativity, providing a quantum description of gravity. On the other hand, its predictions are untestable in practice and require an enormous set of assumptions that are unsupported by scientific
Basic Idea
The story of String Theory begins in the late 1960s, as the string model of the strong nuclear interactions. The string model was interesting, but predicted a number of strange things that didn't
appear to match reality (for example there's no Higgs mechanism). Some years later the theory of Quarks and quantum chromodynamics (QCD) came to be. It described the strong nuclear force
extraordinarily well without these pathologies, so the string theory was abandoned.
But a decade or so later, this idea was reborn into what's now known as modern string theory. The idea was to take the energy scale all the way up to the Planck energy. Physicists could finally hope
to unify all the fundamental forces together.
More than a Theory
Today is incorrect talk about a string theory, because actually, there are several of them. They are: bosonic, type I, type IIA, type IIB, and two flavours of heterotic string theory (HO and HE). The
different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings (which are
segments with endpoints) and closed strings (which form closed loops), while types IIA, IIB and heterotic include only closed strings. An interesting feature of these theories is that they predict
the number of dimensions that the Universe should have. So, while the bosonic string theory has 26 dimensions, the others have 10. But is there a theory more likely than the others?
Duality and M-Theory
One notable fact about string theory is that the different versions of the theory all turn out to be related. The relationships that can exist between different string theories is called Duality.
There are two kinds of duality (S-Duality and T-Duality) but in short, they assert that some of the theories of strings are different mathematical descriptions of the same phenomenon. This fact, with
many other mathematical findings, has allowed the formation of a new theory, that unifies (almost) all the other modern strings theories: The M-theory. This is a 11-dimensional theory that have
spawned important theoretical results in physics and mathematics.
But is it a scientific theory?
To date, M-theory is not verifiable.
As Richard Feynman writes in The Character of the Physical Law, the key test of a scientific theory is to verify whether its consequences are in accord with the measurements obtained experimentally.
No matter who invented the theory and not even how the theory can be aesthetically appealing: if it is not in agreement with the experimental reality, it is wrong'. It has many characteristics of
mathematical interest and it could become important for our understanding of the Universe, but requires further development before it can become verifiable. These developments may be in the theory
itself, as new methods to perform calculations and derive predictions, or may consist of advances in experimental sciences, which can make measurable quantities that are not currently available (as
However, considering the theory under the only profile of its verifiability, it's extremely reductive and opens the field to a series of problems. Contrary to popular belief, the agreement with the
experimental data is not sufficient to confer the status of scientific theory! One of the fundamental characteristics of a scientific theory is the Popperian requirement of falsifiability, that is
the ability to produce at least one statement from which the whole theory depends. This could be problematic if we consider string theory only as a theory of great unification, instead it gave
precise predictions through an experiment on the correspondence on the viscosity of strongly coupled fluids that are in agreement with the experimental data observed at the RHIC (Relativistic Heavy
Ion Collider). This is the real reason why M-Theory is so interesting.
So, strings theories are really interesting, but there is a last, big problem. From a more mathematical point of view, most of strings theories are formulated through the use of perturbative
mathematical methods. These are mathematical methods used for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Well, for strings
theories, perturbation methods involve such a high degree of approximation that the theory can not identify which spaces are candidates for describing our universe! The consequence is that it doesn't
describe a single universe, but something like 10^500 universes, each of which can have different physical and constant laws.
To date, a complete non-perturbative treatment of the theory is lacking.
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Graph Theory Tutorials
Chris K. Caldwell (C) 1995
This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory. There is not a great deal of theory here, we will just teach you enough to wet your
appetite for more!
Most of the pages of this tutorial require that you pass a quiz before continuing to the next page. So the system can keep track of your progress you will need to register for each of these courses
by pressing the [REGISTER] button on the bottom of the first page of each tutorial. (You can use the same username and password for each tutorial, but you will need to register separately for each
Related Resources for these Tutorials: Other Graph Theory Resources on the Internet: Chris Caldwell caldwell@utm.edu
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Bill Goldman's Home Page
I am Professor in the Mathematics Department at the University of Maryland.
□ Office: 3106 William E. Kirwan Hall, 4176 Campus Drive, College Park, MD 20742-4015
□ TEL: (301)-405-5124, FAX: (301)-314-0827
□ EMAIL: wmg AT umd DOT edu
☆ Most people refer to me by my nickname "Bill".
I am teaching Math 431 (Geometry for Computer Applications), a course I've been developing for many years.
In 2021 it was taught by my former doctoral student Justin Wyss-Gallifent and his notes are a good reference for some of the quantitative aspects of the course. A description of the course may be
found here.
I serve on the faculty of both the MATH and AMSC graduate programs at the University of Maryland and have supervised several doctoral dissertations in both programs.
Recent and upcoming lectures:
Selected research projects:
• Here is a recent preprint discussing how the deformation space of complete affine 2-manifolds relates to a twisted cubic cone in 4-space. This result was suggested by Pierre Deligne after I
mentioned it in a lecture I gave at the Institute for Advanced Study during my 2021 sabbatical there. It gives an explicit proof of Oliver Baues's theorem that this deformation space is
homeomorphic to a plane. This paper will appear in a Contemporary Mathematics volume celebrating Ravi Kulkarni's 80th birthday.
• My first paper with my colleague Giovanni Forni, Mixing Flows on moduli spaces of flat bundles over sruraces appeared in Volume 2 of ``Geometry and Physics,'' dedicated to Nigel Hitchin's 70th
birthday conferences in Oxford, Aarhus and Madrid. (I could only attend the conference in Oxford.) This paper outlines a program to study dynamical problems arising from the classification of
geometric structures on manifolds by introducing a cocycle of symplectic moduli spaces over the Teichmueller geodesic flow and SL(2,R)-action. A second paper, with Forni and Carlos Matheus and
Sean Lawton, may be found here.
• The paper, by Rachel (Nakyung) Lee and Karin Melnick, extends Lee's 2023 award-winning doctoral dissertation classifying Lorentzian conformal structures with unipotent holonomy on closed
manifolds. This is the first major progress on the classifiction of arabolic-geometry structures on closed manifolds with nilpotent holonomy since my 1983 paper on conformally flat manifolds with
nilpotent holonomy.
• A part of the introduction for the Festschrift for Toshiyuki Kobayashi dealing with his work on proper actions in pseudo-Riemannian geometry.
• A ZblMath review of Michael Magee's article "Random Unitary Representations of Surface Groups I:Asymptotic Expansions."
• Current and past experimental projects
Recent Administrative Activities:
□ Member, Scientific Advisory Board.
□ I recently co-organized (with Steve Bradlow and Richard Wentworth) Brin MRC a workshop ``Advances in Higgs bundles." there.
Previous research students; also see my listing on the Math Genealogy Project.
Check out Evan Goldman's art
A photograph of my brother and me with President Harry S. Truman at his home in Independence, Missouri in 1964.
Link to Ameridan Mathematical Society Bookstore for Geometric Structures on Manifolds, published in the AMS Graduate Studies in Mathematics volume 227. (li+409 pp. ISBN: [9781470471033];
[9781470471989]; [9781470471972] 57 (22 53-01)) Check out the recent review by the Mathematical Association of America and in Math Reviews (MathSciNet) from the American Mathematical Society.
Recent draft of Proper actions of discrete groups of affine transformations, coauthored with Jeffrey Danciger, Todd Drumm and Ilia Smilga, from the Margulis Festschrift, ``Dynamics, Geometry, Number
Theory: The Impact of Margulis in Modern Mathematics," University of Chicago Press. See also http://arxiv.org/abs/2002.09520.
A preliminary draft of a recent survey, from ``Geometries in History" (S. Dani and A. Papadopoulos, eds. Springer 2019)
Exotic projective triangle tesselation on the cover of the November 2002 issue of the Notices of the American Mathematical Society, drawn by Bill Casselman.
Last updated: Tuesday 22 October 2024.
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Skip counting by 100s to 1,000
Students learn to skip count forward and backward by 100s to 1,000 from any given number.
Students will be able to skip count forward and backward by 100s to 1,000 from any number.
The monsters are taking bags of money out of the vault. Unfortunately, every step they fall, they lose $10 from the bag of money they are carrying. Example: The monster at the top has fallen two
steps (-$20) they started with $60 in their bag, but now they only have $40 left.
Discuss that it is important to be able to count using jumps of 100 forward and backward to 1,000. Use the table to show that when skip counting using hundreds, only the digit in the hundreds place
changes. Show the number line and practice skip counting forward and backward by 100 with the students. Then ask students to determine which numbers should be on the balls on the interactive
whiteboard. Erase the grey square to show the answer. Then ask students to determine which numbers belong at the question marks on the number line shown on the interactive whiteboard. Then ask
students to complete the tables counting forward and backward. Check that students are able to skip count by 100s to 1,000 by asking the following questions:- What happens to the digit in the
hundreds place when you skip count by 100s?- Which number comes after 735 when skip counting by 100s?- Which number comes before 984 when skip counting by 100s?
Students are given exercises in which they must complete the given number sequences either by skip counting forward or backward. They are also given a table to complete in which they must skip count
forward and backward.
Discuss the learning goal and how by learning to skip count they are preparing themselves for calculating later on. Ask students what the steps are of skip counting by 100s. When skip counting by
100s which number comes after 635? And which comes before 721? Tell students that they will form pairs to play the game "Where am I" with steps of 100 up to 1,000. The first student says a number,
takes a few big steps, and then asks "Where am I?". The other student takes the same number of steps and skip counts out loud from the first number and says which number they end up on. They then
swap roles. Tell students that they can step forward- skip counting forward by 100 or they can step backward- skip counting backward by 100.
Students who have difficulty with this learning goal can be supported by counting forward and backward to 100 in jumps of 10. When they are comfortable with this, have students start at 0 or 1,000
and then progress to starting at 200 counting forward or starting at 600 counting backward. Once the student has gained confidence here, have them start with small number sequences in which they must
determine the last number like- counting forward: 215 - 315 -?; 765 - ? -? counting backward: 439 - 339 - ?; 912 - ? - ?.
Gynzy is an online teaching platform for interactive whiteboards and displays in schools.
With a focus on elementary education, Gynzy’s Whiteboard, digital tools, and activities make it easy for teachers to save time building lessons, increase student engagement, and make classroom
management more efficient.
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How Long To Fill Up A Swimming Pool: Tips For Efficiently Filling Your Backyard Oasis - Love Gunite Pool
How Long to Fill Up a Swimming Pool: Tips for Efficiently Filling Your Backyard Oasis
Ever wondered how long it’ll take to fill up your swimming pool? You’re not alone. Whether you’re prepping for a summer splash or just curious, knowing the time it takes can save you a lot of hassle.
Key Takeaways
• Pool Size Matters: Understanding your swimming pool’s dimensions and calculating its volume in gallons is essential for accurate filling time estimates.
• Consider Pool Shape: The shape of the pool (rectangular, round, oval, freeform) affects how quickly it can be filled, necessitating different calculation methods based on geometry.
• Flow Rate is Key: The water source’s flow rate significantly impacts fill time, with garden hoses providing 6-12 gallons per minute, while municipal supplies can exceed 20 gallons per minute.
• Use the Fill Time Formula: To determine how long it will take to fill your pool, use the formula: Fill Time (hours) = Pool Volume (gallons) / (Flow Rate (gallons per minute) × 60).
• Optimize Filling Efficiency: Prep the pool by cleaning it beforehand and consider using multiple hoses to speed up the filling process while ensuring no leaks are present.
• Adjust for Variability: Keep in mind that variations in water pressure and flow rate can affect your filling time, so be prepared to adjust calculations as needed.
Understanding Pool Size
Understanding your swimming pool’s size is crucial for estimating how long it takes to fill. Calculating the volume in gallons helps determine the required water supply and the time needed for
Measuring Pool Dimensions
Measuring your pool’s dimensions involves three key steps. First, measure the length and width. For rectangular pools, multiply these two numbers. Second, determine the average depth by measuring at
several points, then divide by the number of measurements taken. Finally, use the formula for volume:
• Rectangular Pool: Volume = Length × Width × Average Depth × 7.48 (gallons)
• Round Pool: Volume = (π × Radius² × Average Depth) × 7.48 (gallons)
For example, a rectangular pool measuring 20 feet long, 10 feet wide, and 5 feet deep has a volume of 1,490 gallons. Accurate measurements ensure reliable time estimates for filling.
Different Pool Shapes
Swimming pools come in various shapes, each affecting filling times. Here are common shapes and their characteristics:
• Rectangular Pools: Easy to measure and calculate volume, making them straightforward for filling.
• Round Pools: Usually have simpler geometry, but filling time depends on diameter and depth.
• Oval Pools: Combine aspects of rectangular and round pools, which can complicate volume calculations.
• Freeform Pools: Unique shapes require more complex calculations based on individual curves.
Understanding these shapes helps when planning fill times and managing water connections effectively.
Factors Affecting Fill Time
Several factors affect how long it takes to fill up a swimming pool. Understanding these can help you plan better and manage your water usage.
Water Source and Flow Rate
Water source plays a significant role. Filling a pool from a garden hose typically delivers 6-12 gallons per minute, whereas a municipal supply may provide higher flow rates, sometimes exceeding 20
gallons per minute. Confirm your hose diameter and water pressure, as these impact flow rate. For quick fills, consider using multiple hoses or a dedicated water source, such as a fire hydrant, if
Pool Type and Material
Pool type and construction material matter too. In-ground pools generally retain water more efficiently than above-ground pools. Vinyl-lined and fiberglass pools often fill more quickly due to their
design. The shape of the pool also influences fill time. Rectangular pools might fill faster than free-form designs because of their simpler geometry. For added efficiency, maintain proper water
levels during filling to avoid expensive adjustments later on.
Calculating Fill Time
Calculating the time to fill your swimming pool requires a few key pieces of information. Understanding the pool’s volume and the water source’s flow rate helps generate an accurate estimate.
Formula to Determine Fill Duration
To calculate the fill duration, use this simple formula:
[ \text{Fill Time (hours)} = \frac{\text{Pool Volume (gallons)}}{\text{Flow Rate (gallons per minute)} \times 60} ]
This method converts the flow rate from gallons per minute into hours, allowing you to see how long it will take to fill the entire volume. For example, if your pool holds 20,000 gallons and your
water source delivers 10 gallons per minute, the calculation is:
[ \text{Fill Time} = \frac{20,000 \text{ gallons}}{10 \text{ gallons per minute} \times 60} = 33.33 \text{ hours} ]
• Scenario 1: Standard In-Ground Pool
Volume: 30,000 gallons
Flow Rate: 15 gallons per minute
[ \frac{30,000}{15 \times 60} = 33.33 \text{ hours} ]
• Scenario 2: Olympic-Sized Pool
Volume: 660,000 gallons
Flow Rate: 50 gallons per minute
[ \frac{660,000}{50 \times 60} = 220 \text{ hours} ]
• Scenario 3: Inflatable Pool
Volume: 2,500 gallons
Flow Rate: 8 gallons per minute
[ \frac{2,500}{8 \times 60} = 5.21 \text{ hours} ]
Using these examples, you can apply the formula to determine how long your pool will take to fill based on the volume and flow rate. Keep adjustments in mind for variations in flow rate due to using
multiple hoses or water pressure changes.
Tips for Efficient Filling
Filling your swimming pool efficiently saves time and water. Follow these tips to optimize the filling process.
Preparing the Pool
Ensure the pool is clean and free of debris before filling. Remove leaves, dirt, or any items from the pool surface. This preparation minimizes contamination and reduces the need for additional
cleaning later. Check for any leaks around fittings or hoses. Repair them immediately to avoid water loss during the filling stage.
Using Multiple Hoses
Using multiple hoses can significantly speed up the filling process. Connect several hoses to different water sources if available. Distribute the hoses evenly around the pool for better coverage.
This method allows a quicker fill while managing water pressure effectively. Ensure the hoses are in good condition with no blockages. Adjust the flow rate as necessary to prevent overflow.
Filling up your swimming pool doesn’t have to be a daunting task. By taking the time to understand your pool’s dimensions and the factors that influence fill time you can make the process smoother
and quicker.
Using the right techniques like multiple hoses and proper preparation can save you time and effort. With a bit of planning you’ll be splashing around in no time. Enjoy your new oasis and make the
most of those sunny days ahead!
Frequently Asked Questions
How do I determine the size and dimensions of my swimming pool?
To determine your swimming pool’s size and dimensions, measure the length, width, and depth. For rectangular pools, use the formula: Volume = Length × Width × Depth. For circular pools, use Volume =
π × (Radius)² × Depth. Knowing the volume in gallons helps estimate fill time.
What factors influence the time it takes to fill a pool?
The time to fill a pool is influenced by its shape, water source flow rate, pool type, and construction material. Additionally, factors like hose size, connection quality, and local water pressure
can also affect the filling duration.
How can I calculate my pool’s filling time?
To calculate your pool’s filling time, use the formula: Fill Time (in hours) = Pool Volume (in gallons) ÷ Flow Rate (in gallons per minute). This will give you a clear estimate based on your specific
pool and water source.
What are some tips for filling my pool efficiently?
To fill your pool efficiently, begin by cleaning the pool and checking for leaks. Consider using multiple hoses or connecting to different water sources to maximize flow and manage pressure. This
approach can significantly speed up the filling process.
Are there any specific tools or equipment recommended for filling a pool?
While basic garden hoses are adequate, consider using high-flow hoses or adapters to improve water flow. Additionally, a pool vacuum can help clean the pool, and a water meter can help measure the
filling progress more accurately.
Leave a Comment
Written By Scott Keller
Scott Keller is an experienced pool industry professional with over 20 years of expertise in gunite pool design, construction, and maintenance.
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The Golden Swing
Optimum Loft for Maximum Range of a Golf Ball
In February 2019, I was first introduced to the problem golf. At what angle should you lift a golf ball into the air such that it has the maximum range during a strike? 45° ?
Obviously, to complicate matters more, we should include parameters such as spin and air density. Up until this point in my studies, air resistance was always an assumption. It was never something
that actually had a quantifiable structure...until now.
The primary focus of this investigation was to explore the key elements influencing the trajectory of a golf ball during its flights. By examining diverse weather conditions in Singapore, Scotland,
and Bolivia, and considering fundamental principles of physics, computer modelling was employed to determine the optimal loft angles for achieving maximum range.
1. Introduction
The objective of a golf drive is to maximize the ball's coverage distance. In an ideal scenario with only gravity as the influencing force, the ball follows a straightforward parabolic curve,
suggesting an optimal loft angle of 45°. However, real-world complexities, including factors such as air resistance, ball spin, and surface texture, significantly impact the ball's flight.
Consequently, assumptions were made, and key factors were identified to streamline the analysis.
2. Assumptions
Various assumptions were adopted, including a clubface velocity of 47.8 m/s at impact, a rigid and uniform golf ball with specific characteristics, constant driver head mass, fixed launch and loft
angles, and unchanging shaft properties during swing. Additionally, the ball's spin rate and drag coefficient were considered, while factors like wind conditions were excluded for simplification.
3. Underlying Factors
F_dx = -(0.5)*air_dens*(vx)**2*C_d*r**2*pi
Drag: The drag force acting on the golf ball, caused by its movement through air, was explored. The drag force formula incorporated factors like the coefficient of drag, air density, cross-sectional
area, and velocity, affecting the ball's motion contrary to its direction.
Lift: The lift force, generated by the ball's spin during impact with the clubhead, was investigated. The loft angle played a crucial role in determining the rate of spin, affecting the force acting
perpendicularly upward to the ball's motion.
v = ((1+e)*((1+(mu**2))**.5)*v_club*cos(angle*pi/180))/(1+(m/M))
F_my = (0.5)*air_dens*(vy)**2*C_m*r**2*pi
C_m = 0.533333*(v_rot/v)+0.1
Texture: The impact of dimples on the golf ball's aerodynamics was discussed, emphasizing their role in minimizing drag and maximizing lift. The specific characteristics of dimples were acknowledged,
but the model utilized predetermined coefficients without delving into various dimple configurations.
4. Inputs and Outputs
The Python program's constants and assumed values were outlined, with air density recognized as a vital factor influencing the ball's trajectory. The program's execution involved user inputs for the
Open location, angle range, and precision, producing graphs depicting vertical and horizontal displacement for each tested angle.
5. Results
Graphs showcasing vertical against horizontal displacement for various loft angles were plotted and analysed for each location (Singapore, Scotland, and Bolivia). The optimum loft angles, associated
with maximum range, were identified for each location:
• Singapore: Optimum loft angle of 28.8°, achieving a range of 218m.
• Scotland: Optimum loft angle of 28.8°, yielding a range of 215m.
• Bolivia: Optimum loft angle of 28.8°, resulting in a range of 212m.
Results for Various Locations
The Singapore Open (17th-20th January 2019)
In Singapore during January 2019, diverse weather conditions were observed.
Weather in Singapore, January 2019:
Average: 29.44°C (High), 27.13°C (Low), 1029.44 mbar (Pressure), 79% (Humidity)
The average temperature (T_Sing), pressure (P_t_Sing), and humidity (humid_Sing) were calculated as 28.28°C, 1.02944, and 0.79, respectively.
The Scottish Open (8th-14th July 2019)
Monthly Averages for July:
Average: 14.1°C (Temperature), 1012.66 mbar (Pressure), 84.8% (Humidity)
For Scotland, T_Scot, P_t_Scot, and humid_Scot were determined as 14.1°C, 1.01266 bar, and 0.848, respectively.
The Bolivian Open (August 2019)
The weather conditions in Bolivia during August are depicted:
Monthly Averages for August:
Average: 14.1°C (Temperature), 1012.66 mbar (Pressure), 58.1% (Humidity)
For Bolivia, T_Boli, P_t_Boli, and humid_Boli were determined as 3.2°C, 1.02007 bar, and 0.581, respectively.
Using Equation for Air Density, the air density was computed for each location.
Final Results
Location Air Density (kgm^-3) Maximum Range (m) Optimum Loft (°)
Singapore 1.1793 217.9983 28.84461
Scotland 1.2228 215.52904 28.83332
Bolivia 1.2837 212.24216 28.76608
Results from Model
6. Real-World Applications
Despite assuming horizontal incident club head velocity, real-world golf swings involve a slightly downward strike for optimal ball-clubface contact (Cross and Dewhurst, 2018). This alters the
projection angle, impacting both carry and range.
The outgoing ball speed (v2) is influenced by factors like club speed (v1), coefficient of restitution (e), frictional force (μ), and launch angle (A).
Variability in swing speed and clubhead flexion during impact introduce uncertainties into the model. The dynamic nature of these factors could potentially affect the ball's velocity and maximum
The assumption of constant spin neglects the impact of ball characteristics, club interaction, and wind conditions. Real-world scenarios involve varied spin rates influenced by factors like ball
composition, club impact location, and wind effects.
Acknowledging these real-world complexities, future models could strive for a more nuanced understanding by considering additional variables and experimental data from wind tunnel tests. This would
enhance the accuracy and applicability of the model to diverse golfing conditions.
I would further like to thank Matthew Deighan, Soo Bynn, Lee Kenneth Ng and Isobel Reid for their research, investigations and ideas which supported the success of this group project.
The following includes the code which achieves the following results. I would like to thank Gregory Dritschel, who has not only had the patients but the heart to help me tackle this problem while I
was in 1st year Physics and he was in 2nd year direct entry to Maths. I've came along way in my simulation and programming skills, but I believe that the spirit of its interest first began with this
#Open in Anaconda- Spyder or any other model-supported Python system
from numpy import *
from pylab import *
print(" ")
print("Name: Clark Gray and Gregory Dritschel")
Range_of_Driver = 275
T_Boli = 3.4
P_t_Boli = 1020.07*10**-3
humid_Boli = 0.581
air_dens_Boli = 1.2837
T_Sing = 28.281
P_t_Sing = 1.009185*10**-3
humid_Sing = 0.79
air_dens_Sing = 1.1793
T_Scot = 14.1
P_t_Scot = 1.01266*10**-3
humid_Scot = 0.848
air_dens_Scot = 1.2228
#Above we have our weather conditions for each location. The pressure
#is calculated using an online pressure calculator implementing humid air
print("Enter location and time of")
print("the Open using the keys:")
print(" " )
print("Scot ==== Scotland")
print("Boli ==== Bolivia" )
print("Sing ==== Singapore")
#Takes user input for each location
while True:
loca = input("Where is the Open?: ")
if loca == "Sing":
air_dens = air_dens_Sing
T = 28.281
print("The Singapore Open (held 17-20 January, 2019)")
elif loca == "Scot":
air_dens = air_dens_Scot
T = 14.1
print("The Scottish Open (held at The Renaissance Club (East Lothian), July 8-14, 2019)")
elif loca == "Boli":
print("A potential Bolivian Open at La Paz Golf Club, Bolivia (alongside Lake Titicaca) in August 2019")
air_dens = air_dens_Boli
T = 3.4
#assigns air density to value at location
print("Invalid Location")
if loca == "Sing":
air_dens = 1.1793
T = 28.281
elif loca == "Scot":
air_dens = 1.2228
T = 14.1
elif loca == "Boli":
air_dens = 1.2837
T = 3.4
print("Invalid Location-Use keys")
g = -9.8
#acceleration due to gravity
r = 0.021
#radiius of the ball
rpm = 3600
#revolutions per minute
v_rot = r*2*pi*rpm/60**2
#tangential velocity of rotaional motion
v_club = 47.8333
#velocity of club
mu = 0.035
e = 0.83
#constants for velocity of ball equation (line 12)
C_d = 0.25
#coeffiecient of drag
#Defining constants and assumed values
#Running air density values for each location by user's input
m = 0.04593
#mass of ball
M = 0.2
#mass of club
angle = 10
#defining mass of golf ball, mass of golf driver, initial angle of launch,
Rsol = []
angles = []
angle_lower = float(input("Enter lower angle (recommended: 28) : "))
angle_upper = float(input("Enter upper angle (recommended: 29) : "))
print("This exacutes the programme bewteen a range of two sets of degrees)")
print(" ")
print("Note that as precisison increases, the number and time to iterate")
print("each graph increases exponentially.It is advised that to achieve greater")
print("sig figs for the final solution, adjust the lower and upper angles while")
print("increasing the precision. This prevents overloading the script.")
precision_C = int(input("Enter magnitude of precision (recommended : 10 - 3000): "))
#chose the degree of precision for your range of angles
num_runs = float(precision_C*(angle_upper -angle_lower)) - 1
for angle in range(int(angle_lower*precision_C),int(angle_upper*precision_C)):
#Above exacutes the code bewteen a range of two sets of degrees, with
# a chosen extreme of significant figures. Both upper and lower angle limit
#must have the same number of sig figs
angle = angle/precision_C
#Note that as precisison_C increases, the number and time to iterate
#each graph increases exponentially.It is advised that to achieve greater
#sig figs the final solution, adjusting the lower and upper angles while
# increasing the precision_C. This prevents overloading the script.
x = 0
y = 0
v = ((1+e)*((1+(mu**2))**.5)*v_club*cos(angle*pi/180))/(1+(m/M))
#Equation of initial ball speed as a function of angle of loft,
#and driver mass and speed, mass of ball
C_m = 0.533333*(v_rot/v)+0.1
#Magnus (Lift) coefficient as function of ball's rotational velocity and
#projectile speed
vx = v*cos(angle*pi/180)
vy = v*sin(angle*pi/180)
#Dividing x, y components of velocity with respect to launch angle
xsol = []
ysol = []
#creates a set to store x, y positions
vxsol = []
vysol = []
#creates a set to store x, y components of velocity
#setting up the arrays for storing x, y positions and velocity in the x
# and y direction
t = 0
for i in range(1000):
ts = 0.01
#time step assigned
t = t + ts
if vy>=0:#before the ball reaches it's maximum height
F_dy = -(0.5)*air_dens*(vy)**2*C_d*r**2*pi
else: #after the ball reaches its max height
F_dy = (0.5)*air_dens*(vy)**2*C_d*r**2*pi
F_dx = -(0.5)*air_dens*(vx)**2*C_d*r**2*pi
#Force of drag for each x, y component
F_mx = -(0.5)*air_dens*(vx)**2*C_m*r**2*pi
if vy>=0:#same for the lift force due to spin
F_mx = (0.5)*air_dens*(vx)**2*C_m*r**2*pi
F_mx = -(0.5)*air_dens*(vx)**2*C_m*r**2*pi
F_my = (0.5)*air_dens*(vy)**2*C_m*r**2*pi
ax = (F_dx+F_mx)/m
ay = g + (F_dy+F_my)/m
#acceleration of the ball by Newton's 2nd Law of Motion
vx = vxsol[i] + ax*ts
vy = vysol[i] + ay*ts
#by v = u + at, iterate over discrete time steps
x = xsol[i] + vx*ts
y = ysol[i] + vy*ts
#adding the positions in the x,y components
if y<0 :
R = (x+xsol[i-1])/2
#once the ball has negative height, (below the ground) , calculate the
#range by taking the average of the x-values just above and below the
#print(i,round(x, 5),round(y, 5))
#adding the ball's position to the position array wrt the the
#previous value
#add the ball's velocty to the velocity array for each component wrt
#its acceleretion at
#each time step
#print(round(vx,5), round(vy,5),round(x,5), round(y, 5))
#The above code is commented-out, prints the velocity in the x and y
#direction as well as the x and y position to 5 sig figs
if num_runs <=500:
figure(figsize=(12, 3.16))
scatter(xsol,ysol, s=20)
xlim(0, 380)
#organsing the dimensions and display of graph. Advised to comment-out
#when dealing with huge number of graphs.
if num_runs <=1000:
#ensures that data for each run is not printed if the load is too high
#allows code to execute directly to solution.
print("angle = " + str(angle) + "'")
print("R = " + str(round(R,5)) + "m")
print("air_density = " + str(round(air_dens, 5)) + "kg/m^3")
print("T = " + str(T) +"`C")
print("initial speed = " + str(round(v,5)) + "m/s")
print("Time of flight = " + str(round(t,4)) + "s")
# for each graph, display the launch angle, the range, air density,
#temperature, initial speed and time of flight
maxrange = 0
for R in Rsol:
if R>=maxrange:
maxrange = R
#Checker for defining max range
for i in range(len(Rsol)):
if Rsol[i] >= maxrange:
maxrange = Rsol[i]
maxangle = angles[i]
#By defining max range, the cotrresponding angle can be located
print(" ")
print("Final solutions for: " + loca)
print("maximum range = " + str(round(maxrange, 5)) + "m")
print(" at optimum angle = " + str(maxangle)+"'")
#Display final solutions for particular location
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5 Best Ways to Integrate a Polynomial in Python
π ‘ Problem Formulation: When dealing with polynomials in numerical analysis or scientific computing, it is often required to integrate these functions over a certain interval. In Python, there are
multiple ways to approach polynomial integration. For instance, given a polynomial like p(x) = 3x^2 + 2x + 1, we want to find its integral within the bounds, say from 0 to 1. The desired output is
the numerical value of the definite integral.
Method 1: Manual Integration
The Manual Integration technique involves calculating the antiderivative of a polynomial function using the fundamental rules of calculus, and then applying the bounds to find the definite integral.
The simplicity is the main advantage of this method but it requires a solid understanding of calculus.
Here’s an example:
def integrate_polynomial(coeffs):
antideriv_coeffs = [coeff/(index+1) for index, coeff in enumerate(coeffs)]
antideriv_coeffs.append(0) # constant of integration, can be omitted for definite integral
return antideriv_coeffs
# Polynomial coefficients for 3x^2 + 2x + 1
coeffs = [3, 2, 1]
result = integrate_polynomial(coeffs)
Output: [1.0, 1.0, 1, 0]
This method displays the antiderivative coefficients of the polynomial. The result [1.0, 1.0, 1, 0] represents the polynomial x^3 + x^2 + x + C, where C is the constant of integration. It is a
straightforward method, but it does not provide the definite integral.
Method 2: Integration with NumPy
NumPy, a fundamental package for scientific computing in Python, can also perform polynomial integration using the numpy.polyint function.
Here’s an example:
import numpy as np
# Define the polynomial coefficients
p = np.poly1d([3, 2, 1])
# Integrate the polynomial
P_integrated = np.polyint(p)
Output: 3 2
1 x + 1 x + 1 x
This code snippet calculates the indefinite integral of the polynomial by generating a new set of coefficients that represent the integrated polynomial. NumPy’s polyint function is convenient,
efficient, and ideal for cases where symbolic calculus is not required.
Method 3: Symbolic Integration with SymPy
Symbolic integration can be done using SymPy, which is a Python library for symbolic mathematics. It can compute definite and indefinite integrals symbolically, providing an exact result if possible.
Here’s an example:
from sympy import symbols, integrate
x = symbols('x')
polynomial = 3*x**2 + 2*x + 1
integral_polynomial = integrate(polynomial, x)
Output: x**3 + x**2 + x
The provided snippet uses SymPy to integrate the polynomial symbolically and prints the antiderivative. This method provides an exact answer in symbolic form, which is useful for precise calculations
and theoretical analysis.
Method 4: SciPy Integration
SciPy, which builds on NumPy, offers more sophisticated functions for integration such as scipy.integrate.quad, which can compute definite integrals to a high degree of accuracy.
Here’s an example:
from scipy.integrate import quad
# Define the polynomial function
def poly(x):
return 3*x**2 + 2*x + 1
# Compute the definite integral from 0 to 1
result, _ = quad(poly, 0, 1)
Output: 4.666666666666667
This code snippet calculates the definite integral of the polynomial from 0 to 1 using SciPyβ s quad function. It is one of the most precise methods for numerical integration and it is widely used
in scientific computing.
Bonus One-Liner Method 5: Lambda with SciPy
Combining lambda functions with SciPy’s quad function can streamline polynomial integration into a compact one-liner.
Here’s an example:
from scipy.integrate import quad
result, _ = quad(lambda x: 3*x**2 + 2*x + 1, 0, 1)
Output: 4.666666666666667
This one-liner method leverages a lambda function to define the polynomial within the quad function call, offering a concise and quick approach for integrating simple polynomials.
• Method 1: Manual Integration. It’s educational and simple. However, itβ s impractical for complex polynomials or when high precision is needed.
• Method 2: Integration with NumPy. Provides a quick and easy numerical solution. Although efficient, it is not designed for symbolic integration.
• Method 3: Symbolic Integration with SymPy. Best for obtaining symbolic results and theoretical work. Can be slower than numerical methods and overkill for simple tasks.
• Method 4: SciPy Integration. Offers high precision and is ideal for scientific computations. It may require more computational power for definite integrals.
• Bonus Method 5: Lambda with SciPy. Combining convenience and efficiency, it’s a great one-liner for on-the-fly integrations. Limited by the complexity lambda functions can handle conveniently.
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How do you do a tire in math?
Published by: Modified by:
To calculate the overall diameter of a tire, the sidewall height must be multiplied by 2 (remembering that the tire diameter is made up of 2 sidewalls, the one above the wheel and the one below the
wheel touching the ground) and add the diameter of the wheel.
How do you write the width of a tire?
In most countries, the tire width is printed in millimeters or meters. The tire width number is usually printed directly following a capitalized “P.” If your tire width is 1.75 metres (69 in),
for example, the width number will usually be written as “P175.” Automotive Tire & Repair Specialist Expert Interview. 4 June 2021.
What is the magic formula tire tool?
The Magic Formula Tire Tool uses the Fitter included in the Magic Formula Tire MATLAB Library. It also uses a least-squares approach but uses the fmincon function instead. This is because I also
wanted to implement nonlinear constraints, as certain variables of Magic Formula Tire should always be within certain bounds.
What is the aspect ratio of a tire?
The tire aspect ratio is the ratio of the tire height divided by the tire width. On a tire sidewall, the aspect ratio will generally be written after a backslash (“/”). Aspect ratios are written
in percentages.
How to calculate rolling radius?
The value of rolling radius can be calculated either using the wheel forward & wheel angular velocity or by measuring the distance traveled as a function of wheel revolution Rolling radius is the
effective radius of the tire when the tire is rotating and moving forward on the ground .
What does Z mean on a tire?
SPEED RATING Z While it nearly always means a high performance tire for high-performing sports cars, “Z” may actually appear in the middle of a tire’s size information. When it does, it represents
either a maximum speed of more than 149 mph (240 kph) or 186 mph (300 kph).
What does “Z” mean on tires?
A “Z” rating can mean different things. While it nearly always means a high-performance tire for high-performing sports cars, “Z” may appear in the middle of a tire’s size information. When it
does, it represents either a maximum speed of more than 149 mph (240 kph) or 186 mph (300 kph).
What is a Z speed rated tire?
Tires with a Z speed rating are ranked as the top speed a tire could achieve. Manufacturers use letters A to Z to rank tires. This speed rating ranges from 5 km/h (3mph) to above 300 km/h (186
mph). If it’s more than 240km/h (149 mph), it’s considered a Z-speed rated tire, including W & Y, used in sports cars.
Is a w & y tire a “Z”?
However, if a tire is rated W & Y, which is 168 mph and 186 mph respectively or more, tire manufacturers will still list the tire as a “Z.” Exotic sports cars are fitted with the W & Y tires
because of their high speeds and performance levels. Therefore, you should consult your tire manufacturer if the rating is confusing.
What does h mean on a tire?
In earlier incarnations of the speed rating system, “H” stood for “hhigh performance,” which is why it maintains an unusual position in the speed rating spectrum today. Once the highest speed
rating a tire could have, “V” used to represent a maximum of 149 mph (240 kph) or more. Nowadays, it means 149 mph but no higher.
How tall is a 255 tire?
Tire Info and Size Chart
Budnik Tire Chart
245/35-19 25.8 8.0-9.5
245/40-19 26.7 8.0-9.5
245/45-19 27.7 7.5-9.0
255/30-19 25 8.5-9.5
What does 255 mean in tire size?
What does the 255 mean in tire size? This represents the treadwidth of the tire. The tread is 255mm wide, or 32.059 inches wide. What does the 75 mean in tire size? This represents the sidewall
height of the tire, as a percentage of the width. The sidewall height is 75% of 255mm, or 191.25mm, which is 7.53 inches.
How tall is a 255/75r17 tire?
Below are the most common tire sizing information in inches: 255/75R17 Height – How tall is a 255/75R17 tire in inches? This tire size is 32.059 inches tall. 255/75R17 Width in inches – How wide
is a 255/75R17 tire? This tire size is 10.039 inches wide (tread width).
How many inches are in a tire section?
Tire section width (inches) = Tire sidewall (inches) = TIRE HEIGHT CHART TIRE HEIGHT CHART 13 INCH TIRES TIRE
What is the sidewall height of a tire?
This represents the sidewall height of the tire, as a percentage of the width. The sidewall height is 60% of 255mm, or 153mm, which is 6.024 inches. What does the 15 mean in tire size?
How tall is a car tire?
A standard sized tire will be displayed in inches. A tire that reads 33 x 12.5 R15, is 33-inches in diameter (or tall), 12.5-inches wide, and will fit a 15-inch wheel. The door jam of your vehicle
will also indicate what size tires the manufacturer recommends.
How do I calculate the tire height of metric tire sizes?
Use our simple tire height calculator to calculate the tire height of metric tire sizes. Just input your metric measurement and click on “convert height” to view the tire height in inches. Then
browse the suggested tire sizes that are close to the entered tire height.
What is the difference between tire width and tire height?
Tire section width (inches) = Tire sidewall (inches) = TIRE HEIGHT CHART TIRE HEIGHT CHART 13 INCH TIRES TIRE HEIGHT 175 50 13 19.89 215 50 13
What does tire diameter mean?
Tire diameter refers to the height of the tire from the road to the top of the tread. Tires with lower aspect ratios—shorter sidewalls—look cooler and generally perform better, so they’re what
most people want when upsizing their cars’ tires and wheels.
How wide is a 265 tire?
For example a LT265/75R16 would be around 31.6 inches tall and 10 inches wide.
How tall is a 265/75r16 tire?
Below are the most common tire sizing information in inches: 265/75R16 Height – How tall is a 265/75R16 tire in inches? This tire size is 31.65 inches tall. 265/75R16 Width in inches – How wide
is a 265/75R16 tire? This tire size is 10.433 inches wide (tread width). What’s the Sidewall Size of a 265/75R16 Tire in Inches?
How many inches in 265 mm?
265 is the millimeter measurement of the tire and the equation to convert is simple. Just divide 265 by 25.4 and you will get 10.43 inches. That number is often referred to as the section width.
It is hard to translate 70% into inches but the distance between the sidewall height, from the outside of the tread, is going to be 7.3 inches.
Does a 265 fit a 275 rim?
What is interesting is that despite these little differences, the 265 and the 275 both fit on the same rim. You can change rim sizes to an 18 from a 17 inch or a 16-inch rim and the 275 will
still be the larger tire. Except in the revolutions per mile. What is The Difference Between a 265 and 275 Tire?
What does p & 265 mean on a tire?
The second listing is a metric set up, and is what is more common on factory-type sized tires. From left to right, the P indicates a passenger tire while 265 indicates the tires width in
millimeters. That 75 is the tire’s aspect ratio and simply means the tire’s height is equal to 75 percent of its width.
How tall is a 70 tire?
The number 70 is the aspect ratio, sometimes called the profile. This is the ratio of the sidewall height to the width. The sidewall height can be calculated by multiplying the ratio (70 or 0.70) by
the width (225): 225 millimeters x 0.70 = 157.5 millimeters.
How do you read tyre sizes?
On the sidewall, you will find 185/75R14 89T printed.
1. 185 indicates the width of the tyre in millimetres (185 mm)
2. 75 indicates its aspect ratio, a comparison of the tyre’s section height with its section width (75 indicates the height is 75% of its width)
3. R indicates radial ply construction.
How to read tire size in inches & metric?
There are two ways that tire manufacturers display tire size, inches and metric. How to read tire size in inches is very straight forward. Example A : 33X12.50R15 The first number is your overall
diameter of the tire. In this example the tire diameter or “height” is 33 inches tall. The second number represents the width of the tire.
How do you read a car tire number?
By knowing how to interpret the numbers and letters on a car tire, you can know the speed rating, temperature resistance, load index numbers, as well as the tire’s width and rim diameter. Read
the manufacturer and the tire name. These are printed in large letters toward the outside of the tire.
How do you know if a tire is wide?
The first series of three numbers tells you the width of the tire tread in millimeters. Width tread may range from 155 to 315. The two numbers after the slash show the aspect ratio of the tire.
This is the tread-width percentage that is equal to the sidewall’s height. The average ranges from 55 to 75 percent in most passenger vehicles.
How do you calculate tire height?
To then get the overall diameter of the tire you would add the two sidewalls measurements and the diameter of the wheel. In example “B” the last number “16” is the wheel size the tire will fit
on. Therefore the sidewall twice would be 8.41 x 2 = 16.82 plus the wheel “16” makes the tire height 16.82 + 16 = 32.82.
What does 92y mean on a tire?
“Y” is the highest rating and it means the tire can handle speeds of up to 186 mph. “S” is the lowest speed rating, which means the tire is safe for up to 112 mph. “Z” used to be the highest rating
for speeds over 149 mph.
What does 92 mean on a tire?
The 92 represents the maximum load the car can carry. The range of load ratings is vast and each number corresponds to a chart that shows how many pounds a single tire can carry. In this case,
the 92 means it can support 1,389 pounds. The Y is the speed rating, or the maximum speed the tire was designed to withstand and maintain.
What does 91 mean on a tyre?
91 – This is the load index, or the maximum carrying capacity of each individual tyre (provided that the inflation pressure is correct). In our case we have a capacity of a little under 500 kg.
The weight that a rear tyre can handle is indicated in pounds. H – Finally we have the speed index.
What does 96 y mean on a tire?
The higher the number, the more weight the tires can handle. In this case, “96” means the tire can handle 1,565 pounds, while the “Y” indicates the tire is rated to be safe at speeds up to 186
miles per hour. But what about that “Z” rating mentioned earlier? “Z” means that the tire has been tested over 149 mph, but doesn’t indicate how far over.
What is a tyre load rating?
The tyre load rating or load index (in our example: 91) is a code which corresponds to a maximum load (in kg) that can be carried by a single tyre. The tyre speed rating (in our example: V) is a
code which corresponds to the maximum speed at which a tyre can carry its maximum load.
What does 255 mean on a tire?
255 (Tire Width): The tire width is 255 millimeters from sidewall to sidewall (just over ten inches). 45 (Aspect Ratio): The tire’s sidewall is 45% of the tire’s width, which means the sidewall is
about 114 millimeters or just over 4 inches.
This number indicates that your tire has a width of 255 millimeters. The tire size was designed to fit rims or wheels that are 18 inches in diameter. This number means that your tire has an aspect
ratio of 70%. In other words, your tire’s sidewall height (from the edge of the rim to the tire’s tread) is 70% of the width.
What is a 255 mm tire?
P (Tire Type): This tire is a passenger tire, one of the most common types of tires. 255 (Tire Width): The tire width is 255 millimeters from sidewall to sidewall (just over ten inches). 45
(Aspect Ratio): The tire’s sidewall is 45% of the tire’s width, which means the sidewall is about 114 millimeters or just over 4 inches.
What does 225 mm mean on a tire?
This measurement is given in millimeters, based on how much room there is between each sidewall. This can also be referred to as the section width. In the tire size, the first three numbers refer
to the width of the tire. As an example, if your tire size is P225/65R17, the width of the tire would be 225 millimeters.
What does the second number on a tire mean?
The second number represents the width of the tire. In most cases with inch measurement tires the width will have a .50 decimal after the initial number. In example “A” this reads 12.50, meaning
the tire width is 12 and 1/2 inches. The final number simply is the wheel size or “diameter”of the wheel that the tire will fit on.
What does 255/35 ZR19 (96Y) mean?
You might be confused by the following string of letters and numbers: “255/35 ZR19 (96Y).” That looks like someone’s email password, doesn’t it? 255/35 indicates that the tire is 255 millimeters
wide, with a 35-percent sidewall aspect ratio. In other words, the height of the sidewall is 35 percent of the tread width. Yes, tires make you do math.
What does 255 35r19 mean?
You might be confused by the following string of letters and numbers: “255/35 ZR19 (96Y).” That looks like someone’s email password, doesn’t it? 255/35 indicates that the tire is 255 millimeters
wide, with a 35-percent sidewall aspect ratio. In other words, the height of the sidewall is 35 percent of the tread width.
How tall is a 255/35R19 tire?
255/35R19 Height – How tall is a 255/35R19 tire in inches? This tire size is 26.028 inches tall. 255/35R19 Width in inches – How wide is a 255/35R19 tire? This tire size is 10.039 inches wide
(tread width).
What does 255-35r20 mean?
On this page you’ll find a color-coded explanation of the 255-35R20 tire code. This number indicates that your tire has a width of 255 millimeters. This number means that your tire has an aspect
ratio of 35%. In other words, your tire’s sidewall height (from the edge of the rim to the tire’s tread) is 35% of the width.
What does 255-45r18 mean?
On this page you’ll find a color-coded explanation of the 255-45R18 tire code. This number indicates that your tire has a width of 255 millimeters. This number means that your tire has an aspect
ratio of 45%. In other words, your tire’s sidewall height (from the edge of the rim to the tire’s tread) is 45% of the width.
How tall is 235 tire?
16-inch Wheel Conversion Chart
Metric Standard
235/70/16 29.0″x 9.3″
235/85/16 31.7″x 9.3″
245/70/16 29.5″x 9.6″
245/75/16 30.5″x 9.6″
How tall is a 235/70R16 tire?
Below are the most common tire sizing information in inches: 235/70R16 Height – How tall is a 235/70R16 tire in inches? This tire size is 28.953 inches tall. 235/70R16 Width in inches – How wide
is a 235/70R16 tire? This tire size is 9.252 inches wide (tread width).
What does 235mm mean in tire size?
This represents the treadwidth of the tire. The tread is 235mm wide, or 27.953 inches wide. What does the 70 mean in tire size? This represents the sidewall height of the tire, as a percentage of
the width. The sidewall height is 70% of 235mm, or 164.5mm, which is 6.476 inches.
What is 235/75R15 in inches?
When converted to inches, the equivalent of a 235/75R15 tire size is 28.8×9.3R15. It means that: 235/75R15 — what does it mean? How to convert 235 75R15 to inches? First of all, we need to divide
235 mm (width) by 25.4 (mm to inch conversion), which will give us a width of 9.3″.
What makes a tire?
While designs vary by product and manufacturer, an average tire contains well over 100 separate components. Tires can include natural rubber, synthetic rubber, steel, nylon, silica (derived from
sand), polyester, carbon black, petroleum, etc. See photos of tire materials here.
What are tires made of?
While the primary material is rubber, it’s combined with other raw materials during the manufacturing process. Tires are made of synthetic and natural rubber, carbon black, silica, textiles, and
several chemical ingredients. Each of these materials plays a critical role in boosting the tire’s properties.
How are tyres made?
Tyres are made through a five-point manufacturing process. The core ingredients of the tyre are blended and milled into the basic structure of the tyre. The tyre is then built from the inside out
and then cured to form its final shape. Finally, the tyre is inspected for imperfections to ensure it meets strict manufacturing standards.
Why do tires contain rubber compounds?
Tires contain many rubber compounds and other materials because they are required to safely perform in the face of a wide range of demanding conditions. Bead Tire bead bundles (usually strands of
wire) secure the tire to the wheel.
What does tyre mean in English?
The word tire is a short form of attire, from the idea that a wheel with a tire is a dressed wheel. The spelling tyre does not appear until the 1840s when the English began shrink-fitting railway
car wheels with malleable iron. Nevertheless, traditional publishers continued using tire.
What is tyre magic formula?
The Magic Formula (Pacejka) tire model for the longitudinal force is described by the following mathematical expression:Fy=D·sin[C·arctan{B·κ−E·(B·κ−arctan(Bκ)}]for pure longitudinal (brake/drive)
slip κ.
What is a tire (magic formula) block?
The Tire (Magic Formula) block represents a tire with longitudinal behavior given by the Magic Formula [1], an empirical equation based on four fitting coefficients. You can model tire dynamics
under constant or variable pavement conditions.
What are the inputs of the magic formula tire model?
In order to simulate the tire behavior, various models have been presented, such as Pejka, Dugoff, etc. Each model has its specific features and particular inputs. For example, the inputs of the
Magic Formula tire model include tire lateral slip angle, the slip ratio, and the tire normal forces.
What is tire-road interaction (magic formula)?
The Tire-Road Interaction (Magic Formula) block represents the longitudinal force Fx on the tire as a function of Fz, and k using the Magic Formula, where k is the independent slip variable and
Fz is the input signal at port N. The block labeled Rolling Radius is the tire rolling radius rw.
Why do tyres have a magic formula?
Tyres operate in conditions other than pure slip and the Magic Formula implementation also allows for the calculation of weighting functions to overlay pure lateral slip functions with mitigation
due to longitudinal slip and vice versa.
How do wheels rotate?
The left rear goes to the right front and the right rear goes to the left front. Rear-wheel drive or AWD/four-wheel drive : Left rear goes to right front. Right rear goes to the left front. The right
front goes to left rear.
How do I rotate my tires properly?
Keep reading to learn how to rotate your tires properly and use the correct rotation pattern for your vehicle. For front wheel drive vehicles, bring the front tires to the back on the same sides.
Move the rear tires to the front on the opposite sides. For rear wheel drive vehicles, move the rear tires to the front but keep them on the same side.
How does a front wheel drive vehicle work?
FORWARD CROSS (FRONT-WHEEL DRIVE VEHICLES): Rear tires are moved diagonally to opposite sides on the front axle while the right front tire becomes the new spare tire. The spare tire is positioned
on the right side of the rear axle while the left tire on the front axle is moved directly back into the left rear position.
How do you change a rear wheel drive tire?
If you have a rear wheel drive or all wheel drive vehicle, then move the back right tire to the front right and the rear left tire to the front left. Move the front right tire to the back left.
Use your spare tire for the rear right side. Keep the old front left tire as your new spare.
How do you move a front wheel drive vehicle?
For front wheel drive vehicles, bring the front tires to the back on the same sides. Move the rear tires to the front on the opposite sides. For rear wheel drive vehicles, move the rear tires to
the front but keep them on the same side. Put the front wheels in the back on opposite sides.
Can I change my wheel size?
You cannot simply go out and put on a larger wheel and tire and expect your car to be the same. Your car is designed to roll on a specific wheel and tire size. Changes to either may start a cascade
of handling and mechanical issues if you don’t plan carefully.
|
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Auto-Batched Joint Distributions: A Gentle Tutorial | TensorFlow Probability
TensorFlow Probability (TFP) offers a number of JointDistribution abstractions that make probabilistic inference easier by allowing a user to easily express a probabilistic graphical model in a
near-mathematical form; the abstraction generates methods for sampling from the model and evaluating the log probability of samples from the model. In this tutorial, we review "autobatched" variants,
which were developed after the original JointDistribution abstractions. Relative to the original, non-autobatched abstractions, the autobatched versions are simpler to use and more ergonomic,
allowing many models to be expressed with less boilerplate. In this colab, we explore a simple model in (perhaps tedious) detail, making clear the problems autobatching solves, and (hopefully)
teaching the reader more about TFP shape concepts along the way.
Prior to the introduction of autobatching, there were a few different variants of JointDistribution, corresponding to different syntactic styles for expressing probabilistic models:
JointDistributionSequential, JointDistributionNamed, andJointDistributionCoroutine. Auobatching exists as a mixin, so we now have AutoBatched variants of all of these. In this tutorial, we explore
the differences between JointDistributionSequential and JointDistributionSequentialAutoBatched; however, everything we do here is applicable to the other variants with essentially no changes.
Dependencies & Prerequisites
Import and set ups
import functools
import numpy as np
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp
tfd = tfp.distributions
Prerequisite: A Bayesian Regression Problem
We'll consider a very simple Bayesian regression scenario:
\[ \begin{align*} m & \sim \text{Normal}(0, 1) \\ b & \sim \text{Normal}(0, 1) \\ Y & \sim \text{Normal}(mX + b, 1) \end{align*} \]
In this model, m and b are drawn from standard normals, and the observations Y are drawn from a normal distribution whose mean depends on the random variables m and b, and some (nonrandom, known)
covariates X. (For simplicity, in this example, we assume the scale of all random variables is known.)
To perform inference in this model, we'd need to know both the covariates X and the observations Y, but for the purposes of this tutorial, we'll only need X, so we define a simple dummy X:
X = np.arange(7)
array([0, 1, 2, 3, 4, 5, 6])
In probabilistic inference, we often want to perform two basic operations:
• sample: Drawing samples from the model.
• log_prob: Computing the log probability of a sample from the model.
The key contribution of TFP's JointDistribution abstractions (as well as of many other approaches to probabilistic programming) is to allow users to write a model once and have access to both sample
and log_prob computations.
Noting that we have 7 points in our data set (X.shape = (7,)), we can now state the desiderata for an excellent JointDistribution:
• sample() should produce a list of Tensors having shape [(), (), (7,)], corresponding to the scalar slope, scalar bias, and vector observations, respectively.
• log_prob(sample()) should produce a scalar: the log probability of a particular slope, bias, and observations.
• sample([5, 3]) should produce a list of Tensors having shape [(5, 3), (5, 3), (5, 3, 7)], representing a (5, 3)-batch of samples from the model.
• log_prob(sample([5, 3])) should produce a Tensor with shape (5, 3).
We'll now look at a succession of JointDistribution models, see how to achieve the above desiderata, and hopefully learn a little more about TFP shapes along the way.
Spoiler alert: The approach that satisfies the above desiderata without added boilerplate is autobatching.
First Attempt; JointDistributionSequential
jds = tfd.JointDistributionSequential([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Normal(loc=m*X + b, scale=1.) # Y
This is more or less a direct translation of the model into code. The slope m and bias b are straightforward. Y is defined using a lambda-function: the general pattern is that a lambda-function of \
(k\) arguments in a JointDistributionSequential (JDS) uses the previous \(k\) distributions in the model. Note the "reverse" order.
We'll call sample_distributions, which returns both a sample and the underlying "sub-distributions" that were used to generate the sample. (We could have produced just the sample by calling sample;
later in the tutorial it will be convenient to have the distributions as well.) The sample we produce is fine:
dists, sample = jds.sample_distributions()
[<tf.Tensor: shape=(), dtype=float32, numpy=-1.668757>,
<tf.Tensor: shape=(), dtype=float32, numpy=0.6585061>,
<tf.Tensor: shape=(7,), dtype=float32, numpy=
array([ 0.18573815, -1.79962 , -1.8106272 , -3.5971394 , -6.6625295 ,
-7.308844 , -9.832693 ], dtype=float32)>]
But log_prob produces a result with an undesired shape:
<tf.Tensor: shape=(7,), dtype=float32, numpy=
array([-4.4777603, -4.6775575, -4.7430477, -4.647725 , -4.5746684,
-4.4368567, -4.480562 ], dtype=float32)>
And multiple sampling doesn't work:
jds.sample([5, 3])
except tf.errors.InvalidArgumentError as e:
Incompatible shapes: [5,3] vs. [7] [Op:Mul]
Let's try to understand what's going wrong.
A Brief Review: Batch and Event Shape
In TFP, an ordinary (not a JointDistribution) probability distribution has an event shape and a batch shape, and understanding the difference is crucial to effective use of TFP:
• Event shape describes the shape of a single draw from the distribution; the draw may be dependent across dimensions. For scalar distributions, the event shape is []. For a 5-dimensional
MultivariateNormal, the event shape is [5].
• Batch shape describes independent, not identically distributed draws, aka a "batch" of distributions. Representing a batch of distributions in a single Python object is one of the key ways TFP
achieves efficiency at scale.
For our purposes, a critical fact to keep in mind is that if we call log_prob on a single sample from a distribution, the result will always have a shape that matches (i.e., has as rightmost
dimensions) the batch shape.
For a more in-depth discussion of shapes, see the "Understanding TensorFlow Distributions Shapes" tutorial.
Why Doesn't log_prob(sample()) Produce a Scalar?
Let's use our knowledge of batch and event shape to explore what's happening with log_prob(sample()). Here's our sample again:
[<tf.Tensor: shape=(), dtype=float32, numpy=-1.668757>,
<tf.Tensor: shape=(), dtype=float32, numpy=0.6585061>,
<tf.Tensor: shape=(7,), dtype=float32, numpy=
array([ 0.18573815, -1.79962 , -1.8106272 , -3.5971394 , -6.6625295 ,
-7.308844 , -9.832693 ], dtype=float32)>]
And here are our distributions:
[<tfp.distributions.Normal 'Normal' batch_shape=[] event_shape=[] dtype=float32>,
<tfp.distributions.Normal 'Normal' batch_shape=[] event_shape=[] dtype=float32>,
<tfp.distributions.Normal 'JointDistributionSequential_sample_distributions_Normal' batch_shape=[7] event_shape=[] dtype=float32>]
The log probability is computed by summing the log probabilities of the sub-distributions at the (matched) elements of the parts:
log_prob_parts = [dist.log_prob(s) for (dist, s) in zip(dists, sample)]
[<tf.Tensor: shape=(), dtype=float32, numpy=-2.3113134>,
<tf.Tensor: shape=(), dtype=float32, numpy=-1.1357536>,
<tf.Tensor: shape=(7,), dtype=float32, numpy=
array([-1.0306933, -1.2304904, -1.2959809, -1.200658 , -1.1276014,
-0.9897899, -1.0334952], dtype=float32)>]
sum(log_prob_parts) - jds.log_prob(sample)
<tf.Tensor: shape=(7,), dtype=float32, numpy=array([0., 0., 0., 0., 0., 0., 0.], dtype=float32)>
So, one level of explanation is that the log probability calculation is returning a 7-Tensor because the third subcomponent of log_prob_parts is a 7-Tensor. But why?
Well, we see that the last element of dists, which corresponds to our distribution over Y in the mathematial formulation, has a batch_shape of [7]. In other words, our distribution over Y is a batch
of 7 independent normals (with different means and, in this case, the same scale).
We now understand what's wrong: in JDS, the distribution over Y has batch_shape=[7], a sample from the JDS represents scalars for m and b and a "batch" of 7 independent normals. and log_prob computes
7 separate log-probabilities, each of which represents the log probability of drawing m and b and a single observation Y[i] at some X[i].
Fixing log_prob(sample()) with Independent
Recall that dists[2] has event_shape=[] and batch_shape=[7]:
<tfp.distributions.Normal 'JointDistributionSequential_sample_distributions_Normal' batch_shape=[7] event_shape=[] dtype=float32>
By using TFP's Independent metadistribution, which converts batch dimensions to event dimensions, we can convert this into a distribution with event_shape=[7] and batch_shape=[] (we'll rename it
y_dist_i because it's a distribution on Y, with the _i standing in for our Independent wrapping):
y_dist_i = tfd.Independent(dists[2], reinterpreted_batch_ndims=1)
<tfp.distributions.Independent 'IndependentJointDistributionSequential_sample_distributions_Normal' batch_shape=[] event_shape=[7] dtype=float32>
Now, the log_prob of a 7-vector is a scalar:
<tf.Tensor: shape=(), dtype=float32, numpy=-7.9087086>
Under the covers, Independent sums over the batch:
y_dist_i.log_prob(sample[2]) - tf.reduce_sum(dists[2].log_prob(sample[2]))
<tf.Tensor: shape=(), dtype=float32, numpy=0.0>
And indeed, we can use this to construct a new jds_i (the i again stands for Independent) where log_prob returns a scalar:
jds_i = tfd.JointDistributionSequential([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Independent( # Y
tfd.Normal(loc=m*X + b, scale=1.),
<tf.Tensor: shape=(), dtype=float32, numpy=-11.355776>
A couple notes:
• jds_i.log_prob(s) is not the same as tf.reduce_sum(jds.log_prob(s)). The former produces the "correct" log probability of the joint distribution. The latter sums over a 7-Tensor, each element of
which is the sum of the log probability of m, b, and a single element of the log probability of Y, so it overcounts m and b. (log_prob(m) + log_prob(b) + log_prob(Y) returns a result rather than
throwing an exception because TFP follows TF and NumPy's broadcasting rules; adding a scalar to a vector produces a vector-sized result.)
• In this particular case, we could have solved the problem and achieved the same result using MultivariateNormalDiag instead of Independent(Normal(...)). MultivariateNormalDiag is a vector-valued
distribution (i.e., it already has vector event-shape). Indeeed MultivariateNormalDiag could be (but isn't) implemented as a composition of Independent and Normal. It's worthwhile to remember
that given a vector V, samples from n1 = Normal(loc=V), and n2 = MultivariateNormalDiag(loc=V) are indistinguishable; the difference beween these distributions is that n1.log_prob(n1.sample()) is
a vector and n2.log_prob(n2.sample()) is a scalar.
Multiple Samples?
Drawing multiple samples still doesn't work:
jds_i.sample([5, 3])
except tf.errors.InvalidArgumentError as e:
Incompatible shapes: [5,3] vs. [7] [Op:Mul]
Let's think about why. When we call jds_i.sample([5, 3]), we'll first draw samples for m and b, each with shape (5, 3). Next, we're going to try to construct a Normal distribution via:
tfd.Normal(loc=m*X + b, scale=1.)
But if m has shape (5, 3) and X has shape 7, we can't multiply them together, and indeed this is the error we're hitting:
m = tfd.Normal(0., 1.).sample([5, 3])
m * X
except tf.errors.InvalidArgumentError as e:
Incompatible shapes: [5,3] vs. [7] [Op:Mul]
To resolve this issue, let's think about what properties the distribution over Y has to have. If we've called jds_i.sample([5, 3]), then we know m and b will both have shape (5, 3). What shape should
a call to sample on the Y distribution produce? The obvious answer is (5, 3, 7): for each batch point, we want a sample with the same size as X. We can achieve this by using TensorFlow's broadcasting
capabilities, adding extra dimensions:
m[..., tf.newaxis].shape
TensorShape([5, 3, 1])
(m[..., tf.newaxis] * X).shape
TensorShape([5, 3, 7])
Adding an axis to both m and b, we can define a new JDS that supports multiple samples:
jds_ia = tfd.JointDistributionSequential([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Independent( # Y
tfd.Normal(loc=m[..., tf.newaxis]*X + b[..., tf.newaxis], scale=1.),
shaped_sample = jds_ia.sample([5, 3])
[<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[-1.1133379 , 0.16390413, -0.24177533],
[-1.1312429 , -0.6224666 , -1.8182136 ],
[-0.31343174, -0.32932565, 0.5164407 ],
[-0.0119963 , -0.9079621 , 2.3655841 ],
[-0.26293617, 0.8229698 , 0.31098196]], dtype=float32)>,
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[-0.02876974, 1.0872147 , 1.0138507 ],
[ 0.27367726, -1.331534 , -0.09084719],
[ 1.3349475 , -0.68765205, 1.680652 ],
[ 0.75436825, 1.3050154 , -0.9415123 ],
[-1.2502679 , -0.25730947, 0.74611956]], dtype=float32)>,
<tf.Tensor: shape=(5, 3, 7), dtype=float32, numpy=
array([[[-1.8258233e+00, -3.0641669e-01, -2.7595463e+00, -1.6952467e+00,
-4.8197951e+00, -5.2986512e+00, -6.6931367e+00],
[ 3.6438566e-01, 1.0067395e+00, 1.4542470e+00, 8.1155670e-01,
1.8868095e+00, 2.3877139e+00, 1.0195159e+00],
[-8.3624744e-01, 1.2518480e+00, 1.0943471e+00, 1.3052304e+00,
-4.5756745e-01, -1.0668410e-01, -7.0669651e-02]],
[[-3.1788960e-01, 9.2615485e-03, -3.0963073e+00, -2.2846246e+00,
-3.2269263e+00, -6.0213070e+00, -7.4806519e+00],
[-3.9149747e+00, -3.5155020e+00, -1.5669601e+00, -5.0759468e+00,
-4.5065498e+00, -5.6719379e+00, -4.8012795e+00],
[ 1.3053948e-01, -8.0493152e-01, -4.7845001e+00, -4.9721808e+00,
-7.1365709e+00, -9.6198196e+00, -9.7951422e+00]],
[[ 2.0621397e+00, 3.4639853e-01, 7.0252883e-01, -1.4311566e+00,
3.3790007e+00, 1.1619035e+00, -8.9105040e-01],
[-7.8956139e-01, -8.5023916e-01, -9.7148323e-01, -2.6229355e+00,
-2.7150445e+00, -2.4633870e+00, -2.1841538e+00],
[ 7.7627432e-01, 2.2401071e+00, 3.7601702e+00, 2.4245868e+00,
4.0690269e+00, 4.0605016e+00, 5.1753912e+00]],
[[ 1.4275590e+00, 3.3346462e+00, 1.5374103e+00, -2.2849756e-01,
9.1219616e-01, -3.1220305e-01, -3.2643962e-01],
[-3.1910419e-02, -3.8848895e-01, 9.9946201e-02, -2.3619974e+00,
-1.8507402e+00, -3.6830821e+00, -5.4907336e+00],
[-7.1941972e-02, 2.1602919e+00, 4.9575748e+00, 4.2317696e+00,
9.3528280e+00, 1.0526063e+01, 1.5262107e+01]],
[[-2.3257759e+00, -2.5343289e+00, -3.5342445e+00, -4.0423255e+00,
-3.2361765e+00, -3.3434000e+00, -2.6849220e+00],
[ 1.5006512e-02, -1.9866472e-01, 7.6781356e-01, 1.6228745e+00,
1.4191239e+00, 2.6655579e+00, 4.4663467e+00],
[ 2.6599693e+00, 1.2663836e+00, 1.7162113e+00, 1.4839669e+00,
2.0559487e+00, 2.5976877e+00, 2.5977583e+00]]], dtype=float32)>]
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[-12.483114 , -10.139662 , -11.514159 ],
[-11.656767 , -17.201958 , -12.132455 ],
[-17.838818 , -9.474525 , -11.24898 ],
[-13.95219 , -12.490049 , -17.123957 ],
[-14.487818 , -11.3755455, -10.576363 ]], dtype=float32)>
As an extra check, we'll verify that the log probability for a single batch point matches what we had before:
(jds_ia.log_prob(shaped_sample)[3, 1] -
jds_i.log_prob([shaped_sample[0][3, 1],
shaped_sample[1][3, 1],
shaped_sample[2][3, 1, :]]))
<tf.Tensor: shape=(), dtype=float32, numpy=0.0>
AutoBatching For The Win
Excellent! We now have a version of JointDistribution that handles all our desiderata: log_prob returns a scalar thanks to the use of tfd.Independent, and multiple samples work now that we fixed
broadcasting by adding extra axes.
What if I told you there was an easier, better way? There is, and it's called JointDistributionSequentialAutoBatched (JDSAB):
jds_ab = tfd.JointDistributionSequentialAutoBatched([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Normal(loc=m*X + b, scale=1.) # Y
<tf.Tensor: shape=(), dtype=float32, numpy=-12.954952>
shaped_sample = jds_ab.sample([5, 3])
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[-12.191533 , -10.43885 , -16.371655 ],
[-13.292994 , -11.97949 , -16.788685 ],
[-15.987699 , -13.435732 , -10.6029 ],
[-10.184758 , -11.969714 , -14.275676 ],
[-12.740775 , -11.5654125, -12.990162 ]], dtype=float32)>
jds_ab.log_prob(shaped_sample) - jds_ia.log_prob(shaped_sample)
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[0., 0., 0.],
[0., 0., 0.],
[0., 0., 0.],
[0., 0., 0.],
[0., 0., 0.]], dtype=float32)>
How does this work? While you could attempt to read the code for a deep understanding, we'll give a brief overview which is sufficient for most use cases:
• Recall that our first problem was that our distribution for Y had batch_shape=[7] and event_shape=[], and we used Independent to convert the batch dimension to an event dimension. JDSAB ignores
the batch shapes of component distributions; instead it treats batch shape as an overall property of the model, which is assumed to be [] (unless specified otherwise by setting batch_ndims > 0).
The effect is equivalent to using tfd.Independent to convert all batch dimensions of component distributions into event dimensions, as we did manually above.
• Our second problem was a need to massage the shapes of m and b so that they could broadcast appropriately with X when creating multiple samples. With JDSAB, you write a model to generate a single
sample, and we "lift" the entire model to generate multiple samples using TensorFlow's vectorized_map. (This feature is analagous to JAX's vmap.)
Exploring the batch shape issue in more detail, we can compare the batch shapes of our original "bad" joint distribution jds, our batch-fixed distributions jds_i and jds_ia, and our autobatched
[TensorShape([]), TensorShape([]), TensorShape([7])]
[TensorShape([]), TensorShape([]), TensorShape([])]
[TensorShape([]), TensorShape([]), TensorShape([])]
We see that the original jds has subdistributions with different batch shapes. jds_i and jds_ia fix this by creating subdistributions with the same (empty) batch shape. jds_ab has only a single
(empty) batch shape.
It's worth noting that JointDistributionSequentialAutoBatched offers some additional generality for free. Suppose we make the covariates X (and, implicitly, the observations Y) two-dimensional:
X = np.arange(14).reshape((2, 7))
array([[ 0, 1, 2, 3, 4, 5, 6],
[ 7, 8, 9, 10, 11, 12, 13]])
Our JointDistributionSequentialAutoBatched works with no changes (we need to redefine the model because the shape of X is cached by jds_ab.log_prob):
jds_ab = tfd.JointDistributionSequentialAutoBatched([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Normal(loc=m*X + b, scale=1.) # Y
shaped_sample = jds_ab.sample([5, 3])
[<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[ 0.1813647 , -0.85994506, 0.27593774],
[-0.73323774, 1.1153806 , 0.8841938 ],
[ 0.5127983 , -0.29271227, 0.63733214],
[ 0.2362284 , -0.919168 , 1.6648189 ],
[ 0.26317367, 0.73077047, 2.5395133 ]], dtype=float32)>,
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[ 0.09636458, 2.0138032 , -0.5054413 ],
[ 0.63941646, -1.0785882 , -0.6442188 ],
[ 1.2310615 , -0.3293852 , 0.77637213],
[ 1.2115169 , -0.98906034, -0.07816773],
[-1.1318136 , 0.510014 , 1.036522 ]], dtype=float32)>,
<tf.Tensor: shape=(5, 3, 2, 7), dtype=float32, numpy=
array([[[[-1.9685398e+00, -1.6832136e+00, -6.9127172e-01,
8.5992378e-01, -5.3123581e-01, 3.1584005e+00,
[-2.5645006e-01, 3.1554163e-01, 3.1186538e+00,
1.4272424e+00, 1.2843871e+00, 1.2266440e+00,
[[ 1.5973477e+00, -5.3631151e-01, 6.8143606e-03,
-1.4910895e+00, -2.1568544e+00, -2.0513713e+00,
[-4.9448099e+00, -2.8385928e+00, -6.9027486e+00,
-5.6543546e+00, -7.2378774e+00, -8.1577444e+00,
[[-2.1233239e+00, 5.8853775e-02, 1.2024102e+00,
1.6622503e+00, -1.9197327e-01, 1.8647723e+00,
[ 3.7549341e-01, 1.5853541e+00, 2.4594500e+00,
2.1952972e+00, 1.7517658e+00, 2.9666045e+00,
[[[ 8.9906776e-01, 6.7375046e-01, 7.3354661e-01,
-9.9894643e-01, -3.4606690e+00, -3.4810467e+00,
[-3.0670738e+00, -6.3628020e+00, -6.2538433e+00,
-6.8091092e+00, -7.7134805e+00, -8.6319380e+00,
[[-2.2462025e+00, -3.3060855e-01, 1.8974400e-01,
3.1422038e+00, 4.1483402e+00, 3.5642972e+00,
[ 4.7880130e+00, 5.8790064e+00, 9.6695948e+00,
7.8112822e+00, 1.2022618e+01, 1.2411858e+01,
[[-1.0189297e+00, -7.8115642e-01, 1.6466728e+00,
8.2378983e-01, 3.0765080e+00, 3.0170646e+00,
[ 6.5285158e+00, 7.8038850e+00, 6.4155884e+00,
9.0899811e+00, 1.0040427e+01, 9.1404457e+00,
[[[ 4.5557004e-01, 1.4905317e+00, 1.4904103e+00,
2.9777462e+00, 2.8620450e+00, 3.4745665e+00,
[ 3.9977460e+00, 5.7173767e+00, 7.8421035e+00,
6.3180594e+00, 6.0838981e+00, 8.2257290e+00,
[[-7.0750320e-01, -3.5972297e-01, 4.3136525e-01,
-2.3301599e+00, -5.0374687e-01, -2.8338656e+00,
[-3.1258626e+00, -3.4687450e+00, -1.2045374e+00,
-4.0196013e+00, -5.8831010e+00, -4.2965469e+00,
[[ 2.1969774e+00, 2.4614549e+00, 2.2314475e+00,
1.8392437e+00, 2.8367062e+00, 4.8600502e+00,
[ 6.1879644e+00, 5.1792760e+00, 6.1141996e+00,
5.6517797e+00, 8.9979610e+00, 7.5938139e+00,
[[[ 1.5249090e+00, 1.1388919e+00, 8.6903995e-01,
3.0762129e+00, 1.5128503e+00, 3.5204377e+00,
[ 3.4166217e+00, 3.5930209e+00, 3.1694956e+00,
4.5797420e+00, 4.5271711e+00, 2.8774328e+00,
[[-2.3095846e+00, -2.0595703e+00, -3.0093951e+00,
-3.8594103e+00, -4.9681158e+00, -6.4256043e+00,
[-6.4306297e+00, -7.0924540e+00, -8.4075985e+00,
-1.0417805e+01, -1.1727266e+01, -1.1196255e+01,
[[-7.0419472e-01, 1.4568675e+00, 3.7946482e+00,
4.8489718e+00, 6.6498446e+00, 9.0224218e+00,
[ 1.0060651e+01, 1.1998097e+01, 1.5326431e+01,
1.7957514e+01, 1.8323889e+01, 2.0160881e+01,
[[[-2.2360647e-01, -1.3632748e+00, -7.2704530e-01,
2.3558271e-01, -1.0381399e+00, 1.9387857e+00,
[ 1.6015106e-01, 1.5284677e+00, -4.8567140e-01,
-1.7770648e-01, 2.1919653e+00, 1.3015286e+00,
[[ 1.3688663e+00, 2.6602898e+00, 6.6657305e-01,
4.6554832e+00, 5.7781887e+00, 4.9115267e+00,
[ 5.1983776e+00, 6.2297459e+00, 6.3848300e+00,
8.4291229e+00, 7.1309576e+00, 1.0395646e+01,
[[ 1.2675294e+00, 5.2844582e+00, 5.1331611e+00,
8.9993315e+00, 1.0794343e+01, 1.4039831e+01,
[ 1.9084715e+01, 2.2191265e+01, 2.3481146e+01,
2.5803375e+01, 2.8632090e+01, 3.0234968e+01,
3.1886738e+01]]]], dtype=float32)>]
<tf.Tensor: shape=(5, 3), dtype=float32, numpy=
array([[-28.90071 , -23.052422, -19.851362],
[-19.775568, -25.894997, -20.302256],
[-21.10754 , -23.667885, -20.973007],
[-19.249458, -20.87892 , -20.573763],
[-22.351208, -25.457762, -24.648403]], dtype=float32)>
On the other hand, our carefully crafted JointDistributionSequential no longer works:
jds_ia = tfd.JointDistributionSequential([
tfd.Normal(loc=0., scale=1.), # m
tfd.Normal(loc=0., scale=1.), # b
lambda b, m: tfd.Independent( # Y
tfd.Normal(loc=m[..., tf.newaxis]*X + b[..., tf.newaxis], scale=1.),
jds_ia.sample([5, 3])
except tf.errors.InvalidArgumentError as e:
Incompatible shapes: [5,3,1] vs. [2,7] [Op:Mul]
To fix this, we'd have to add a second tf.newaxis to both m and b match the shape, and increase reinterpreted_batch_ndims to 2 in the call to Independent. In this case, letting the auto-batching
machinery handle the shape issues is shorter, easier, and more ergonomic.
Once again, we note that while this notebook explored JointDistributionSequentialAutoBatched, the other variants of JointDistribution have equivalent AutoBatched. (For users of
JointDistributionCoroutine, JointDistributionCoroutineAutoBatched has the additional benefit that you no longer need to specify Root nodes; if you've never used JointDistributionCoroutine you can
safely ignore this statement.)
Concluding Thoughts
In this notebook, we introduced JointDistributionSequentialAutoBatched and worked through a simple example in detail. Hopefully you learned something about TFP shapes and about autobatching!
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Trash Cans
From Pokemon Speedruns
Trash cans refers to the puzzle in Lt. Surge's gym. Note that in most situations, you can save after finding the first can and reset until you find the second can, but note that most categories ban
saves for record attempts.
Pokémon Red/Blue
As noted in the image below, the first trash can can only be one of 8 of the 15 cans, in a criss-cross pattern. Most of the first cans have two possible second cans: the can in the top-left corner
and one other. However, the two middle column first cans have four possible second cans, and so it is considered unlucky to find these cans as your first can in a run/race. In any case, all possible
second cans have equal probability weighting so you should just test the closest can pointed to by an arrow for the second can and hope it's correct each time.
Pokémon Yellow
For some reason, likely to do with the aforementioned chance of the second can always having a chance to be in the topleft, the mechanics of the trash cans were changed for the Yellow version. In
this version the set of possible first cans is the same but there are generally two possible second cans chosen at a time which will both unlock the door. Therefore the image below does not show all
possible second cans but instead the most likely second can for each possible first can. This is an adjacent can to each first can for each possible first can except the two in the middle column,
which are bugged to always have a very high chance of the second one being topleft, but can also have the chance of there being no second can chosen at all. As such, it is very dangerous to save and
reset to try to find the second can in Yellow.
Pokémon FireRed/LeafGreen
Trash cans work "as intended" in these games. The first can can be anything, and the second can will be a random adjacent can.
Pokémon HeartGold/SoulSilver
As far as we know this is the same as FireRed/LeafGreen - the trash cans work as intended, albeit without spinners to further annoy you this time around.
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How do I calculate simple interest on a loan?
The formula for simple interest is: Simple Interest = (principal) x (rate) x (# of periods). Principal is the amount you borrowed, the rate represents the interest rate you agreed to, and the number
of periods refers to the length of time in question.
What is PV formula in Excel?
Present value (PV) is the current value of a stream of cash flows. PV can be calculated in excel with the formula =PV(rate, nper, pmt, [fv], [type]). If FV is omitted, PMT must be included, or vice
versa, but both can also be included. NPV is different from PV, as it takes into account the initial investment amount.
What is the formula for loan calculation?
Great question, the formula loan calculators use is I = P * r *T in layman’s terms Interest equals the principal amount multiplied by your interest rate times the amount in years.
How do you calculate simple interest example?
The formula for calculating simple interest is:
1. (P x r x t) ÷ 100.
2. (P x r x t) ÷ (100 x 12)
3. FV = P x (1 + (r x t))
4. Example 1: If you invest Rs.50,000 in a fixed deposit account for a period of 1 year at an interest rate of 8%, then the simple interest earned will be:
Does Excel have a simple interest formula?
To calculate simple interest in Excel (i.e. interest that is not compounded), you can use a formula that multiples principal, rate, and term. The FV function can calculate compound interest and
return the future value of an investment.
What is PMT Excel?
PMT, one of the financial functions, calculates the payment for a loan based on constant payments and a constant interest rate. Use the Excel Formula Coach to figure out a monthly loan payment. At
the same time, you’ll learn how to use the PMT function in a formula.
How do I calculate EMI in Excel 2010?
The formula which you can use in excel is: =PMT(rate,nper,pv). Let us check the EMI of Suraj by using the above formula. It must be noted that the rate used in the formula should be the monthly rate,
that is, 12%/12=1% or 0.01.
How do you calculate PV and NPV?
If the project only has one cash flow, you can use the following net present value formula to calculate NPV:
1. NPV = Cash flow / (1 + i)^t – initial investment.
2. NPV = Today’s value of the expected cash flows − Today’s value of invested cash.
3. ROI = (Total benefits – total costs) / total costs.
How do I compute a loan using simple interest?
Determine the principal. The principal is the amount of money that you will use to calculate the interest.
Determine the interest rate. Before you can calculate how much your principal will appreciate,you need to know by what rate your principal will grow.
Measure the term of the loan.
Calculate the interest.
Try another example.
How do you calculate simple interest loan?
simple interest EMI calculator: simple loan calculator lets you calculate the amount you will receive at the maturity period. the amount so calculated using the simple interest calculator includes
the interest amount along with the principal. the formula for calculation: a = p (1 + (r*t))
How to calculate simple interest on a loan?
how to calculate simple interest? the simple interest calculator is a simple and easy online tool to calculate the interest amount. to calculate the simple interest, you need to input three essential
details in the simple loan calculator-principal amount; time period; interest rate; put the values in the following formula – a = p(1 + (r*t)) where,
What are the steps to calculate simple interest?
To calculate simple interest, start by multiplying the principal, which is the initial sum borrowed, by the loan’s interest rate written as a decimal. Then, multiply that number by the total number
of time periods since the loan began to find the simple interest.
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24 Artificial Neural Network Interview Questions and Answers
Are you looking to ace your Artificial Neural Network (ANN) interview, whether you are an experienced professional or a fresher in the field? This comprehensive guide will help you prepare for common
questions related to ANN and enhance your chances of success. We've curated a list of 24 essential questions and provided detailed answers to help you get ready for your interview.
Role and Responsibility of an Artificial Neural Network Engineer:
Artificial Neural Network engineers play a crucial role in the field of machine learning and artificial intelligence. They are responsible for designing, training, and optimizing neural network
models. Their responsibilities include data preprocessing, model architecture design, parameter tuning, and staying up-to-date with the latest developments in the field.
Common Interview Question Answers Section
1. What is an Artificial Neural Network (ANN)?
The interviewer wants to gauge your fundamental understanding of ANNs.
How to answer: An Artificial Neural Network is a computational model inspired by the human brain. It consists of interconnected nodes (neurons) organized into layers, including an input layer, hidden
layers, and an output layer. ANNs are used for tasks like classification, regression, and pattern recognition.
Example Answer: "An Artificial Neural Network is a computational model that mimics the structure and function of the human brain. It is composed of interconnected neurons organized in layers. The
input layer receives data, and the hidden layers process it, leading to an output. ANNs are widely used in machine learning for various tasks, including image recognition and natural language
2. What are the different types of activation functions in ANNs, and when would you use each?
The interviewer is testing your knowledge of activation functions and their applications.
How to answer: There are various activation functions, including ReLU, Sigmoid, and Tanh. Explain when and why you would use each one based on the specific characteristics of your data and problem.
Example Answer: "ReLU (Rectified Linear Unit) is commonly used when you want to introduce non-linearity in the network and handle vanishing gradient problems. Sigmoid is used for binary
classification tasks, and Tanh can be suitable when the data ranges from -1 to 1. The choice of activation function depends on the problem and the nature of the data."
3. What is backpropagation, and how does it work in training neural networks?
The interviewer is assessing your knowledge of the training process in neural networks.
How to answer: Explain that backpropagation is an optimization algorithm for training neural networks. It involves calculating gradients of the loss function with respect to the model's parameters
and updating them to minimize the error.
Example Answer: "Backpropagation is a key training algorithm for neural networks. It calculates the gradients of the loss function with respect to the network's parameters and adjusts the weights and
biases to minimize the error. The process involves forward and backward passes through the network."
4. What is overfitting, and how can you prevent it in neural networks?
The interviewer wants to know your understanding of overfitting and your strategies to mitigate it.
How to answer: Explain that overfitting occurs when a model performs well on the training data but poorly on unseen data due to memorization. Discuss techniques like dropout, early stopping, and
using more training data to prevent overfitting.
Example Answer: "Overfitting happens when a model learns to fit the training data too closely, resulting in poor generalization to new data. To prevent overfitting, we can use techniques like
dropout, which randomly deactivates neurons during training, early stopping to halt training when performance plateaus, and collecting more diverse training data."
5. What is the vanishing gradient problem, and how can it be addressed?
The interviewer is testing your knowledge of gradient-related issues in deep learning.
How to answer: Explain that the vanishing gradient problem occurs when gradients become extremely small during backpropagation, hindering the training of deep networks. Discuss solutions like using
activation functions like ReLU and weight initialization techniques.
Example Answer: "The vanishing gradient problem occurs when gradients diminish as they propagate backward through deep networks, leading to slow convergence. We can address it by using activation
functions like ReLU that mitigate the vanishing gradient issue and by employing proper weight initialization methods like He initialization."
6. What is the purpose of a convolutional neural network (CNN), and in what applications are they commonly used?
The interviewer is evaluating your knowledge of CNNs and their applications.
How to answer: Explain that CNNs are specialized neural networks designed for processing grid-like data, such as images and videos. Mention common applications like image classification, object
detection, and facial recognition.
Example Answer: "Convolutional Neural Networks (CNNs) are tailored for grid-like data, making them ideal for tasks like image classification, object detection, and facial recognition. They are widely
used in computer vision applications to analyze and extract meaningful information from images and videos."
7. What is the difference between a feedforward neural network and a recurrent neural network (RNN)?
The interviewer is interested in your understanding of different types of neural networks.
How to answer: Explain that feedforward neural networks process data in one direction, from input to output, without feedback loops. In contrast, RNNs have recurrent connections, allowing them to
process sequential data and capture temporal dependencies.
Example Answer: "A feedforward neural network processes data in a unidirectional manner, making it suitable for tasks like image recognition. In contrast, a Recurrent Neural Network (RNN) has loops
that enable it to work with sequential data, such as natural language processing and time series analysis, where capturing temporal dependencies is crucial."
8. What is transfer learning in neural networks, and how can it be beneficial?
The interviewer wants to assess your knowledge of transfer learning and its advantages.
How to answer: Explain that transfer learning is a technique where a pre-trained model's knowledge is utilized for a different but related task. Discuss its benefits, such as faster training and
improved performance, especially when limited data is available for the new task.
Example Answer: "Transfer learning involves using a pre-trained neural network as a starting point for a new task. This approach can significantly speed up training and improve performance,
especially when you have limited data for the new task. It's a valuable tool in various applications, such as image recognition and natural language processing."
9. What is a recurrent neural network (RNN), and when is it appropriate to use one?
The interviewer is interested in your understanding of RNNs and their use cases.
How to answer: Explain that RNNs are a type of neural network designed to handle sequential data, making them suitable for tasks involving time series data, natural language processing, and speech
Example Answer: "A Recurrent Neural Network (RNN) is designed for sequential data, allowing it to capture temporal dependencies. You should use an RNN when working with tasks like time series
analysis, natural language processing, and speech recognition, where the order of data elements matters."
10. Can you explain the concept of dropout in neural networks?
The interviewer is assessing your understanding of dropout as a regularization technique.
How to answer: Explain that dropout is a regularization technique used to prevent overfitting. During training, random neurons are "dropped out," meaning they are temporarily deactivated to force the
network to learn more robust features independently. This reduces the network's reliance on specific neurons and improves generalization.
Example Answer: "Dropout is a regularization technique used to combat overfitting in neural networks. During training, random neurons are temporarily turned off, which prevents the network from
becoming overly dependent on specific neurons. This helps in improving the model's generalization performance and robustness."
11. What is the role of an optimizer in neural network training?
The interviewer is interested in your knowledge of optimizers and their importance.
How to answer: Explain that optimizers are algorithms used to adjust the model's parameters during training to minimize the loss function. Different optimizers have various update rules, and their
choice can impact training speed and model convergence.
Example Answer: "Optimizers play a crucial role in neural network training by adjusting the model's parameters to minimize the loss function. They help in finding the optimal set of weights and
biases. Common optimizers include SGD, Adam, and RMSprop, each with its own update rules and impact on training speed and convergence."
12. What is the vanishing gradient problem, and how can it be addressed?
The interviewer is testing your knowledge of gradient-related issues in deep learning.
How to answer: Explain that the vanishing gradient problem occurs when gradients become extremely small during backpropagation, hindering the training of deep networks. Discuss solutions like using
activation functions like ReLU and weight initialization techniques.
Example Answer: "The vanishing gradient problem occurs when gradients diminish as they propagate backward through deep networks, leading to slow convergence. We can address it by using activation
functions like ReLU that mitigate the vanishing gradient issue and by employing proper weight initialization methods like He initialization."
13. What are hyperparameters in neural networks, and how do they differ from model parameters?
The interviewer wants to test your knowledge of hyperparameters and their distinction from model parameters.
How to answer: Explain that hyperparameters are settings that control the learning process, such as learning rate and the number of hidden layers. Model parameters are learned during training and
include weights and biases. Discuss the importance of tuning hyperparameters for model performance.
Example Answer: "Hyperparameters are settings that determine the learning process's behavior, like the learning rate, number of hidden layers, and batch size. Model parameters, on the other hand, are
learned during training and include weights and biases. Proper tuning of hyperparameters is essential for achieving optimal model performance."
14. What is the difference between supervised learning and unsupervised learning in the context of neural networks?
The interviewer wants to evaluate your understanding of different learning paradigms.
How to answer: Explain that supervised learning involves labeled data, where the model learns to map inputs to specific outputs. Unsupervised learning, on the other hand, deals with unlabeled data,
where the model identifies patterns and structures in the data without predefined labels.
Example Answer: "In supervised learning, the model learns from labeled data, aiming to map inputs to specific outputs, such as classification or regression. Unsupervised learning, however, works with
unlabeled data, focusing on discovering patterns and structures within the data, like clustering or dimensionality reduction."
15. Explain the bias-variance trade-off in machine learning and its relevance to neural networks.
The interviewer is assessing your knowledge of the bias-variance trade-off and its implications for neural networks.
How to answer: Explain that the bias-variance trade-off represents a fundamental challenge in machine learning. High bias leads to underfitting, while high variance results in overfitting. Neural
networks need to strike a balance by selecting appropriate architectures and regularization techniques.
Example Answer: "The bias-variance trade-off is a key consideration in machine learning. High bias leads to underfitting, where the model oversimplifies the data. High variance leads to overfitting,
where the model fits the training data too closely. In neural networks, finding the right balance involves choosing suitable architectures and applying regularization techniques like dropout and
weight decay."
16. What is the purpose of batch normalization in neural networks?
The interviewer is interested in your understanding of batch normalization and its role in training neural networks.
How to answer: Explain that batch normalization normalizes the output of each layer in a neural network, making training more stable and accelerating convergence. It helps address issues like
vanishing and exploding gradients.
Example Answer: "Batch normalization is used to normalize the output of each layer during training. It helps in stabilizing the training process, accelerating convergence, and mitigating issues like
vanishing or exploding gradients. By reducing internal covariate shift, it allows neural networks to learn faster and perform better."
17. What is a deep neural network, and when is it beneficial to use one?
The interviewer is assessing your understanding of deep neural networks and their applications.
How to answer: Explain that a deep neural network has multiple hidden layers, making it capable of learning complex, hierarchical features. They are beneficial when dealing with complex data, such as
images, audio, and natural language, where deep features are essential.
Example Answer: "A deep neural network is characterized by having multiple hidden layers. They are beneficial when working with complex data that requires learning hierarchical features, like images,
audio, and natural language processing. Deep networks excel at capturing intricate patterns and representations."
18. What is the concept of weight initialization in neural networks, and why is it important?
The interviewer is interested in your knowledge of weight initialization and its significance.
How to answer: Explain that weight initialization involves setting the initial values of weights in a neural network. Proper weight initialization is crucial as it can impact training speed,
convergence, and the quality of the final model.
Example Answer: "Weight initialization refers to setting the initial values of the weights in a neural network. It's important because inappropriate weight initialization can lead to slow training,
convergence issues, or getting stuck in local minima. Proper weight initialization techniques, like He or Xavier initialization, can significantly improve training efficiency and the model's final
19. What is the purpose of the learning rate in neural network training, and how do you choose an appropriate value?
The interviewer wants to assess your understanding of learning rate and how to select the right value.
How to answer: Explain that the learning rate controls the step size during gradient descent. Choosing the right learning rate is essential, and it often involves experimentation to find a value that
allows the model to converge without overshooting.
Example Answer: "The learning rate is a critical hyperparameter in neural network training. It determines the step size during gradient descent. Selecting the appropriate learning rate involves
experimentation. You typically start with a small learning rate and gradually increase it until you find a value that allows the model to converge without overshooting the optimal solution."
20. What is the concept of weight decay (L2 regularization) in neural networks, and how does it work?
The interviewer is assessing your understanding of weight decay and its impact on neural network training.
How to answer: Explain that weight decay, or L2 regularization, adds a penalty term to the loss function, encouraging the model to have smaller weights. It helps prevent overfitting by reducing the
complexity of the model.
Example Answer: "Weight decay, also known as L2 regularization, works by adding a penalty term to the loss function. This term encourages the model to have smaller weights, which, in turn, helps
prevent overfitting by reducing model complexity. It's a valuable technique to improve generalization performance."
21. What is a dropout layer in a neural network, and how does it prevent overfitting?
The interviewer is interested in your knowledge of dropout as a regularization technique.
How to answer: Explain that a dropout layer randomly deactivates neurons during training. This prevents the network from becoming overly reliant on specific neurons and promotes generalization by
forcing the model to learn more robust features.
Example Answer: "A dropout layer in a neural network randomly deactivates a fraction of neurons during training. This prevents the model from becoming overly dependent on specific neurons,
encouraging the learning of more robust features. Dropout is an effective technique for preventing overfitting and improving model generalization."
22. What is the role of activation functions in neural networks, and can you name a few common ones?
The interviewer is assessing your understanding of activation functions and their importance.
How to answer: Explain that activation functions introduce non-linearity in neural networks, enabling them to model complex relationships. Mention common activation functions like ReLU, Sigmoid, and
Example Answer: "Activation functions play a crucial role in neural networks by introducing non-linearity. This enables them to model complex relationships. Common activation functions include ReLU
(Rectified Linear Unit), which is widely used for its effectiveness, Sigmoid, and Tanh (Hyperbolic Tangent), which are suitable for specific tasks like binary classification and data ranging from -1
to 1, respectively."
23. Can you explain the concept of gradient descent and its variants in neural network optimization?
The interviewer is assessing your knowledge of gradient descent and its variations.
How to answer: Explain that gradient descent is an optimization algorithm used to update the model's parameters to minimize the loss function. Mention common variants like Stochastic Gradient Descent
(SGD), Adam, and RMSprop, and their specific characteristics.
Example Answer: "Gradient descent is a fundamental optimization algorithm for training neural networks. It updates model parameters to minimize the loss function. Variants like Stochastic Gradient
Descent (SGD) use random mini-batches for faster convergence. Adam combines the advantages of different techniques and adapts learning rates. RMSprop adjusts the learning rates per parameter to speed
up training and improve convergence."
24. What is the role of the loss function in neural network training, and how do you choose an appropriate one for your task?
The interviewer is interested in your understanding of loss functions and their selection.
How to answer: Explain that the loss function measures the error between predicted and actual values. Choosing an appropriate loss function depends on the nature of the task, such as mean squared
error for regression or cross-entropy for classification.
Example Answer: "The loss function is a critical component in neural network training, as it quantifies the error between predicted and actual values. Selecting the right loss function is
task-dependent. For regression tasks, mean squared error is often used. For classification tasks, cross-entropy is a common choice. The choice should align with the problem's objectives and data."
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Command Plugins - Plugin Development | Vagrant | HashiCorp Developer
Plugin Development: Commands
This page documents how to add new commands to Vagrant, invocable via vagrant YOUR-COMMAND. Prior to reading this, you should be familiar with the plugin development basics.
Warning: Advanced Topic! Developing plugins is an advanced topic that only experienced Vagrant users who are reasonably comfortable with Ruby should approach.
Definition Component
Within the context of a plugin definition, new commands can be defined like so:
command "foo" do
require_relative "command"
Commands are defined with the command method, which takes as an argument the name of the command, in this case "foo." This means the command will be invocable via vagrant foo. Then the block argument
returns a class that implements the Vagrant.plugin(2, "command") interface.
You can also define non-primary commands. These commands do not show up in the vagrant -h output. They only show up if the user explicitly does a vagrant list-commands which shows the full listing of
available commands. This is useful for highly specific commands or plugins that a beginner to Vagrant would not be using anyways. Vagrant itself uses non-primary commands to expose some internal
functions, as well.
To define a non-primary command:
command("foo", primary: false) do
require_relative "command"
Implementations of commands should subclass Vagrant.plugin(2, :command), which is a Vagrant method that will return the proper superclass for a version 2 command. The implementation itself is quite
simple, since the class needs to only implement a single method: execute. Example:
class Command < Vagrant.plugin(2, :command)
def execute
puts "Hello!"
The execute method is called when the command is invoked, and it should return the exit status (0 for success, anything else for error).
This is a command at its simplest form. Of course, the command superclass gives you access to the Vagrant environment and provides some helpers to do common tasks such as command line parsing.
Parsing Command-Line Options
The parse_options method is available which will parse the command line for you. It takes an OptionParser as an argument, and adds some common elements to it such as the --help flag, automatically
showing help if requested. View the API docs directly for more information.
This is recommended over raw parsing/manipulation of command line flags. The following is an example of parsing command line flags pulled directly from the built-in Vagrant destroy command:
options = {}
options[:force] = false
opts = OptionParser.new do |o|
o.banner = "Usage: vagrant destroy [vm-name]"
o.separator ""
o.on("-f", "--force", "Destroy without confirmation.") do |f|
options[:force] = f
# Parse the options
argv = parse_options(opts)
Using Vagrant Machines
The with_target_vms method is a helper that helps you interact with the machines that Vagrant manages in a standard Vagrant way. This method automatically does the right thing in the case of
multi-machine environments, handling target machines on the command line (vagrant foo my-vm), etc. If you need to do any manipulation of a Vagrant machine, including SSH access, this helper should be
An example of using the helper, again pulled directly from the built-in destroy command:
with_target_vms(argv, reverse: true) do |machine|
In this case, it asks for the machines in reverse order and calls the destroy action on each of them. If a user says vagrant destroy foo, then the helper automatically only yields the foo machine. If
no parameter is given and it is a multi-machine environment, every machine in the environment is yielded, and so on. It just does the right thing.
Using the Raw Vagrant Environment
The raw loaded Vagrant::Environment object is available with the '@env' instance variable.
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Dynamical Models Explaining Social Balance and Evolution of Cooperation
Social networks with positive and negative links often split into two antagonistic factions. Examples of such a split abound: revolutionaries versus an old regime, Republicans versus Democrats, Axis
versus Allies during the second world war, or the Western versus the Eastern bloc during the Cold War. Although this structure, known as social balance, is well understood, it is not clear how such
factions emerge. An earlier model could explain the formation of such factions if reputations were assumed to be symmetric. We show this is not the case for non-symmetric reputations, and propose an
alternative model which (almost) always leads to social balance, thereby explaining the tendency of social networks to split into two factions. In addition, the alternative model may lead to
cooperation when faced with defectors, contrary to the earlier model. The difference between the two models may be understood in terms of the underlying gossiping mechanism: whereas the earlier model
assumed that an individual adjusts his opinion about somebody by gossiping about that person with everybody in the network, we assume instead that the individual gossips with that person about
everybody. It turns out that the alternative model is able to lead to cooperative behaviour, unlike the previous model.
Citation: Traag VA, Van Dooren P, De Leenheer P (2013) Dynamical Models Explaining Social Balance and Evolution of Cooperation. PLoS ONE 8(4): e60063. https://doi.org/10.1371/journal.pone.0060063
Editor: Renaud Lambiotte, University of Namur, Belgium
Received: September 28, 2012; Accepted: February 25, 2013; Published: April 25, 2013
Copyright: © 2013 Traag et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original author and source are credited.
Funding: This research received support from a grant “Actions de recherche concertées – Large Graphs and Networks” of the “Communauté Française de Belgique” and from the Belgian Network DYSCO
(Dynamical Systems, Control, and Optimization, http://sites.uclouvain.be/dysco/), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. PDL
was provided a research fellowship by VLAC (Vlaams Academisch Centrum) and was hosted by Université catholique de Louvain as a visiting professor during sabbatical leave from the University of
Florida, from which he received a Faculty Enhancement Opportunity. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Why do two antagonistic factions emerge so frequently in social networks? This question was already looming in the 1940s, when Heider [1] examined triads of individuals in networks, and postulated
that only balanced triads are stable. A triad is balanced when friends agree in their opinion of a third party, while foes disagree, see Fig. 1. The individuals in an unbalanced triad have an
incentive to adjust their opinions so as to reduce the stress experienced in such a situation [2]. Once an adjustment is made, the triad becomes balanced, and the stress disappears.
The two upper triads are balanced, while the two lower triads are unbalanced. According to the structure theorem [3], a complete graph can be split into (at most) two opposing factions, if and only
if all triads are balanced. This is represented by the coloured matrix on the right, where blue indicates positive entries, and red negative entries.
A decade later, Harary [3] showed that a complete social network splits in at most two factions if and only if all its triads are balanced, see also [4]. Such networks are called (socially) balanced
as well. Since then, the focus of much of the research has been on detecting such factions in signed networks [5], [6]. Many signed networks show evidence of social balance, although the split into
factions might not be exact, that is, they are only nearly socially balanced [7]–[10].
What has been lacking until fairly recently, are dynamical models that explain how social balance emerges. The purpose of this paper is to analyse two such models. One of these models, proposed first
in [11], was proved to exhibit social balance in [12]. However, this was done under a restrictive symmetry assumption for the reputation matrix. Here, we continue the analysis of this model and show
that it generically does not lead to social balance when the symmetry assumption is dropped. In contrast, we propose a second model that is based on a different underlying gossiping mechanism, and
show that it generically does lead to social balance, even when reputations are not symmetric.
Moreover, there is a natural connection between negative links and the evolution of cooperation: we consider positive links as indicating cooperation and negative links as defection. We will show
that our alternative model is able to lead to cooperation, whereas the earlier model cannot.
Earlier Model
Certain discrete-time, stochastic dynamics have been investigated [13], [14], but they exhibit so-called jammed states [15]: no change in the sign of a reputation improves the degree of social
balance, as measured by the total number of balanced triads in the network. A surprisingly simple continuous-time model [11] was proved to converge to social balance for certain symmetric initial
conditions [12]. The authors assume that the social network is described by a complete graph (everybody is connected to everybody), with weighted links representing reputations that change
continuously in time. Let X denote the real-valued matrix of the reputations, so that X[ij] represents the opinion i has about j. It is positive whenever i considers j a friend, and negative if i
thinks of j as an enemy. The network is balanced, if, up to a possible relabelling of the individuals, the sign structure of X takes one of two possible block forms:(1)
Changes in the reputations are modelled as follows:(2)where denotes the derivative with respect to time of the matrix X. The idea behind this model is that reputations are adjusted based on the
outcome of a particular gossiping process. More specifically, suppose that Bob (individual i) wants to revise his opinion about John (individual j). Bob then asks everybody else in the network what
they think of John. If one such opinion X[kj] has the same sign as the opinion Bob has about his gossiping partner, i.e. as X[ik], then Bob will increase his opinion about John. But if these opinions
differ in sign, then Bob will decrease his opinion about John.
The analysis for symmetric initial conditions X(0)=X^T (0) was carried out in [12]: First, X(0) is diagonalized by an orthogonal transformation X(0)=UΛ(0)U^T, where the columns of U are
orthonormal eigenvectors u[1],…,u[n] of X(0) so that UU^T=I[n], and Λ(0) is a diagonal matrix whose diagonal entries are the corresponding real eigenvalues λ[1](0) ≥ λ[2](0)≥ · ≥ λ[n](0) of X(0).
Direct substitution of the matrix function UΛ(t)U^T shows that it is the solution of Eq. 2 with initial condition X(0). Here, Λ(t) is a diagonal matrix, solving the uncoupled matrix equation with
initial condition Λ(0). The diagonal entries of Λ(t) are obtained by integrating the scalar first order equations :(3)
Hence, the solution X(t) blows up in finite time if and only if λ[1](0) >0. Moreover, if λ[1](0) >0 is a simple eigenvalue, then the solution X(t), normalized by its Frobenius norm, satisfies:(4)
Assuming that u[1] has no zero entries, and up to a suitable permutation of its components, the latter limit takes one of the forms in Eq. 1. In other words, if the initial reputation matrix is
symmetric and has a simple, positive eigenvalue, then the normalized reputation matrix becomes balanced in finite time.
Our first main result is that this conclusion remains valid for normal initial conditions, i.e. for initial conditions that satisfy the equality X(0)X^T (0)=X^T (0)X(0), see SI Text S1, Theorem 2.
Whereas the real eigenvalues behave similar to the symmetric case, the complex eigenvalues show circular behaviour, which results in small “bumps” in the dynamics as shown in Fig. 2 (see SI Fig. S1
for more detail). More precisely, if X(0) is normal and if λ[1](0) is a real, positive and simple eigenvalue which is larger than every other real eigenvalue (if any), then the solution X(t) of Eq. 2
satisfies Eq. 4. Hence, once again, the normalized reputation matrix converges to a balanced state.
The first row illustrates what happens generically for the model , while the second row displays the results for . Each row contains from left to right: (1) an illustration of the model; (2) the
random initial state; (3) the dynamics of the model; and (4) the final state to which the dynamics converge. Blue indicates positive entries, and red negative entries. Although the first model
converges to a rank one matrix, it is not socially balanced. The second model does converge generically to social balance. The small bumps in the dynamics for are due to complex eigenvalues that show
circular behaviour (see Fig. S1).
Our second main result is that this conclusion does not carry over to the case where X(0) is not normal, see SI Text S1, Theorem 3. This general case is analysed by first transforming X(0) into its
real Jordan-canonical form J(0): X(0)=TJ(0)T^–1, where T consists of a basis of (the real and imaginary parts of) generalized eigenvectors of X(0). It can then be shown that the solution X(t) of
Eq. 2 is given by TJ(t)T^–1, where J(t) solves the matrix equation , an equation which can still be solved explicitly. Hence, X(t) can still be determined. It turns out that if X(0) has a real,
positive and simple eigenvalue λ[1](0) which is larger than every other real eigenvalue (if any), then the normalized reputation matrix satisfies:(5)where u[1] and are left and right eigenvectors of
X(0) respectively, that correspond to the eigenvalue λ[1](0). If we assume that none of the entries of u[1] and v[1] are zero, then we can always find a suitable permutation of the components of u[1]
and v[1] such that they have the following sign structure:
Consequently, in general, the matrix limit in Eq. 5 has the sign structure:as illustrated in Fig. 2. Clearly, this configuration doesn't correspond to social balance any longer.
Alternative Model
Let us briefly reconsider the gossiping process underlying model . In our example of Bob and John, the following happens. Bob asks others what they think of John. Bob takes into account what he
thinks of the people he talks to, and adjusts his opinion of John accordingly. An alternative approach is to consider a type of homophily process [16]–[18]: people tend to befriend people who think
alike. When Bob seeks to revise his opinion of John, he talks to John about everybody else (instead of talking to everybody else about John). For example, suppose that Bob likes Alice, but that John
dislikes her. When Bob and John talk about Alice, they notice they have opposing views about her, and as a result the relationship between Bob and John deteriorates. On the other hand, should they
share similar opinions about Alice, their relationship will improve. Thus, our alternative model for the update law of the reputations is:(6)
Although there apparently is only a subtle difference in the gossiping processes underlying the models in Eq. 2 and 6, these models turn out to behave quite differently, as we discuss next.
Our third main result is that for generic initial conditions, the normalized solution of system Eq. 6 converges to a socially balanced state in finite time. To show this, we decompose the solution X(
t) into its symmetric and skew-symmetric parts: X(t)=S(t)+ A(t), where S(t)=S^T(t) and A(t)=–A^T (t). Since , the skew-symmetric part remains constant, and therefore A(t)=A(0) ≡ A[0]. The
symmetric part then obeys the matrix Riccati differential equation . We introduce to eliminate the linear terms in this equation, and obtain(7)
The latter matrix Riccati differential equation can be integrated, yielding the solution Z(t) explicitly, and hence S(t), as well as X(t), can be calculated.
It turns out that if A[0] ≠ 0, then X(t) always blows up in finite time. Moreover, using a perturbation argument, it can be shown there is a dense set of initial conditions X(0) such that the
normalized solution of Eq. 6 converges to(8)for some vector w, as t approaches the blow-up time t*, see SI Text S1, Theorem 5. If w has no zero entries, this implies that the normalized solution
becomes balanced in finite time. Hence, the alternative model in Eq. 6 generically evolves to social balance, see Fig. 2.
Evolution of Cooperation
Positive and negative links have a natural interpretation in the light of cooperation: positive links indicate cooperation and negative links indicate defection. There is then also a natural
motivation for the alternative model in terms of cooperation. Again, suppose Bob wants to revise his opinion of John. For Bob it is important to know whether John is cooperative in order to determine
whether he should cooperate with John or not. So, instead of asking Alice whether she has cooperated with John, Bob would like to know whether John has cooperated with her. In other words, Bob is not
interested in X[kj] but in X[jk], consistent with Eq. 6, illustrated in Fig. 2. This is also what is observed in studies on gossip: it often concerns what others did, not what one thinks of others
[19], [20].
Indeed gossiping seems crucial in explaining the evolution of human cooperation through indirect reciprocity [21]. It has even been suggested that humans developed larger brains in order to gossip,
so as to control the problem of cooperation through social interaction [22]. In general, the problem is that if defection allows individuals to gain more, why then do individuals cooperate? This is
usually modelled in the form of a prisoner's dilemma, in which each agent has the possibility to give his partner some benefit b at some cost c<b. So, if an agent's partner cooperates (he gives the
agent b) but the agent doesn't cooperate (he doesn't pay the cost c) his total payoff will be b. Considering the other possibilities results in the payoff matrix detailed in Fig. 3.
Both players have the option to either Cooperate or Defect. Whenever an agent cooperates, it costs him c while his partners receives a benefit b>c, leading to the indicated payoffs.
Irrespective of the choice of the other player, it is better to defect in a single game. Suppose that the second player cooperates. Then if the first player cooperates he gains b – c, while if he
defects he gains b, so defecting is preferable. Now suppose that the second player defects. The first player then has to pay c, but doesn't have to pay anything when defecting. So indeed, in a single
game, it is always better to defect, yet the payoff is higher if both cooperate, whence the dilemma.
In reality, we do observe cooperation, and various mechanisms for explaining the evolution of cooperation have been suggested [23], such as kin selection [24], [25], reciprocity [26] or group
selection [27]. Humans have a tendency however to also cooperate in contexts beyond kin, group or repeated interactions. It is believed that some form of indirect reciprocity can explain the breadth
of human cooperation [21]. Whereas in direct reciprocity the favour is returned by the interaction partner, in indirect reciprocity the favour is returned by somebody else, which usually involves
some reputation. It has been theorized that such a mechanism could even form the basis of morality [28]. Additionally, reputation (and the fear of losing reputation) seems to play an important role
in maintaining social norms [29]–[31].
In general, the idea is the following: agents obtain some good reputation by helping others, and others help those with a good reputation. Initially a strategy known as image scoring was introduced
[32]. Shortly after, it was argued that a different strategy, known as the standing strategy, should actually perform better [33], although experiments showed people tend to prefer the simpler image
scoring strategy [34]. This led to more systematic studies of how different reputation schemes would perform [35]–[37]. Although much research has been done on indirect reciprocity, only few
theoretical works actually study how gossiping shapes reputations [38], [39]. Nonetheless, most studies (tacitly) assume that reputations are shaped through gossip. Additionally, it was observed
experimentally that gossiping is an effective mechanism for promoting cooperation [40]–[42].
Moreover, these reputations are usually considered as objective. That is, all agents know the reputation X[j] of some agent j, and all agents have the same view of agent j. Private reputations–so
that we have X[ij], the reputation of j in the eyes of i–have usually been considered by allowing a part of the population to “observe” an interaction, and update the reputation accordingly. If too
few agents are allowed to “observe” an interaction, the reputations X[ij] tend to become uncorrelated and incoherent. This makes reputation unreliable for deciding whether to cooperate or defect. The
central question thus becomes how to model private reputations such that they remain coherent and reliable for deciding whether to cooperate or not.
Dynamical models of social balance might provide an answer to this question. Although it allows to have private reputations–that is X[ij]–the dynamics could also lead to some coherence in the form of
social balance. In addition, it models more explicitly the gossiping process, commonly suggested to be the foundation upon which reputations are forged.
Simulation Results
The reputations of the agents are determined by the dynamics of the two models. We call agents using dynamics type A, and those using dynamics type B. We assume that agent i cooperates with j
whenever X[ij] >0 and defects otherwise. Agents reproduce proportional to their fitness, determined by their payoff. Agents that do well (have a high payoff) have a higher chance of reproduction, and
we are interested in knowing the probability that a certain type becomes fixated in the population (i.e. takes over the whole population), known as the fixation probability ρ. All simulations start
off with an equal amount of agents, so if some type wins more often than his initial relative frequency, it indicates it has an evolutionary advantage. For the results presented here this comes down
to ρ >1/2. More details on the simulations are provided in the Materials and Methods section at the end of the paper.
The results are displayed in Fig. 4 using a normalized cost of c =1 (the ratio b/c drives the evolutionary dynamics, see Materials and Methods and [23]). When directly competing against each other,
type B has an evolutionary advantage (its fixation probability ρ[B] >1/2) compared to type A, already for relatively small benefits. When each type is playing against defectors (agents that always
defect), type A seems unable to defeat defectors (ρ[A] <1/2) for any b <20, while type B performs quite well against them. When all three types are playing against each other results are similar (see
SI Fig. S2). When varying the number of agents, the critical benefit b* at which type B starts to have an evolutionary advantage changes (i.e. where the fixation probability ρ[B] =1/2). For b>b*
agents using the model have a higher chance to become fixated, while for b<b* defectors tend to win. The inequality for type B to have an evolutionary advantage can be relatively accurately
approximated by where γ is estimated to be around γ ≈ 1.72±0.037 (95% confidence interval). Varying the intensity of selection does not alter the results qualitatively (see SI Fig. S3). Summarizing,
type B is able to lead to cooperation and defeats type A. Based on these results, if a gossiping process evolved during the course of human history in order to maintain cooperation, the model seems
more likely to have evolved than . For smaller groups a smaller benefit is needed for the model to become fixated. This dependence seems to scale only as , so that larger groups only need a
marginally larger benefit in order to develop cooperation.
(A) The fixation probability (probability to be the sole surviving species) is higher for model than . This implies that the model is more viable against defectors, and has an evolutionary advantage
compared to . (B) The point b* at which the model has an evolutionary advantage against defectors (i.e. the fixation probability ρ >1/2) depends on the number of agents n. The condition for the model
to defeat defectors can be approximated by , with β ≈ 1.72.
To conclude, we have shown that the alternative model generically converges to social balance, whereas the model did not. The current models exhibit several unrealistic features, we would like to
address: (1) an all-to-all topology; (2) dynamics that blow-up in finite time; and (3) homogeneity of all agents. Although most of these issues can be addressed by specifying different dynamics, the
resulting models are much more difficult to analyse, thereby limiting our understanding. Although the two models are somewhat simple, they are also tractable, and what we lose in truthfulness, we
gain in deeper insights: in simplicity lies progress. Our current analysis offers a quite complete understanding, and we hope it provides a stepping stone to more realistic models, which we would
like to analyse in the future.
The difference between the two models can be understood in terms of gossiping: we assume that people who wish to revise their opinion about someone talk to that person about everybody else, while the
earlier model assumed that people talk about that person to everybody else. Both gossiping and social balance are at the centre of many social phenomena [22], [29], [43], [44], such as norm
maintenance [30], stereotype formation [45] and social conflict [46]. For example, a classic work [29] on the established and outsiders found that gossiping was the fundamental driving force for the
maintenance of the cohesive network of the established at the exclusion of the outsiders. Understanding how social balance may emerge might help to understand the intricacies of these social
Moreover, in light of the evolution of cooperation it appears that agents using dynamics perform well against defectors, and have an evolutionary advantage compared to agents using dynamics. Contrary
to other models of indirect reciprocity, not everybody might end up cooperating with everybody, and the population may split into two groups. This provides an interesting connection between social
balance theory, gossiping and the evolution of cooperation. Our results improve our understanding of gossiping as a mechanism for group formation and cooperation, and as such contributes to the study
of indirect reciprocity.
Materials and Methods
In the simulations of the evolution of cooperation, the dynamics consist of two parts: (1) the interaction dynamics within each generation; and (2) the dynamics prescribing how the population evolves
from generation to generation.
Interaction Dynamics
We include three possible types of agents in our simulations:
Type A uses dynamics,
Type B uses dynamics, and
Defectors have trivial reputation dynamics , with negative constant reputations.
We can decompose the reputation matrix X(t) accordingly into three parts:where X[A](t) are the reputations of all agents in the eyes of agents of type A, X[B](t) for type B and X[D](t) for defectors.
The reputations X[A](0) and X[B](0) are initialized from a standard Gaussian distribution. The initial reputation for X[D](0) will be set to a fixed negative value. To be clear, X[D](0) is the
reputation of all other agents in the eyes of defectors, which is negative initially. The initial reputation of the defectors themselves is of course not necessarily negative initially. For the
results displayed here we have used X[D](0)=−10, but results remain by and large the same when varying this parameter, as long as it remains sufficiently negative.
Since we are dealing with continuous dynamics in this paper, we assume all agents are involved in infinitesimally short games at each time instance t. Each agent i may choose to either cooperate or
defect with another agent j, and this decision may vary from one agent to the next. For agents of type A and type B the decision to cooperate is based on the reputation: they defect whenever X[ij](t)
≤ 0 and cooperate whenever X[ij](t) >0. We define the cooperation matrix C(t) accordingly.
Defectors will simply always defect. Whenever an agent i cooperates with j the latter receives a payoff of b at a cost of c to agent i. We integrate this payoff over all infinitesimally short games
from time 0 to time t*, which can be represented aswhere e=(1,…,1) the vector of all ones for a certain generation g.
Evolutionary Dynamics
We have simulated the evolution of cooperation for n =10,20,…,60 agents, which stays constant throughout evolution. We consider four different schemes for initializing the first generation:
Here p[A](0),p[B](0) and p[D](0) are respectively the proportion of agents of type A, type B and defectors in the first generation. We use the vector T[i](g) ∈ {A, B, D} to denote the type of agent i
in generation g, so that T[i](g)=A if agent i is a type A player, T[i](g)=B for a type B player, and T[i](g)=D for a defector. We are interested in estimating the probability that a single type
takes over the whole population, known as the fixation probability ρ[A], ρ[B] and ρ[D] for the three different types. If a type has no evolutionary advantage, it is said to be evolutionary neutral,
and in that case its fixation probability is equal to its initial frequency, e.g. for type A ρ[A]=p[A](0).
We will keep the population constant at the initial n, and simply choose n new agents according to their payoff for the next generation. This can be thought of as choosing n times one of the n agents
in the old generation for reproduction. Let φ[i] denote the probability that an agent is selected for reproduction, which we define as
Since we are only interested in the number of agents of a certain type, we can also gather all payoffs for the same type of agents, and writewhere q ∈ {A, B, D} represents the type of agent. The
probability to select a type A agent, a type B agent or a defector is then respectively Φ[A], Φ[B] and Φ[D]. In the next generation, the probability that agent i is of a specific type q can then be
written as
This evolutionary mechanism can be seen as a Wright-Fisher process [47] with fitnesses . It is well known that this process converges faster than a Moran birth-death process, since it essentially
takes n time steps in a Moran process to reproduce the effect of one time step in a Wright-Fisher process [47]. Because of the high computational costs (solving repeatedly a non-linear system of
differential equations of size n^2), this process is preferable.
Higher β signifies higher selective pressure, and leads to a higher reproduction of those with a high payoff, and in the case that β → ∞ only those with the maximum payoff reproduce. On the other
hand, for β → 0 this tends to the uniform distribution φ[i] =1/n, where payoffs no longer play any role. We have used β =0.5 for the low selective pressure, β =5 for the high selective
pressure, reported in SI Fig. S3. For the results in the main text we have used β =1.
For an evolutionary neutral selection in where all P[i](g)=P are effectively the same, β has no effect, and φ[i] =1/n. Notice that if we rescale P[i](g) by 1/c so that the payoff effectively
becomesand we rescale β by c, then the reproduction probabilities remain unchanged. Hence, only the ratio b/c effectively plays a role up to a rescaling of the intensity of selection. Since the point
at which the evolution is neutral (i.e. ρ equals the initial proportional frequency), is independent of β, this point will only depend on the ratio b/c. So, we normalized the cost c =1. To verify
this, we also ran additional simulations with different costs, which indeed gave the same results.
We stop the simulation whenever one of the types becomes fixated in the population. With fixation we mean that all other types have gone extinct, and only a single type remains. If no type has become
fixated after 1,000 generations, we terminate the simulation and count as winner the most frequent type. This almost never happens, and the simulation usually stops after a relatively small number of
In total, we repeat this process 1,000 times for the results in the main text, and for the low (β =0.5) and high (β =5) selective pressure 100 times. This means that we run the evolutionary
dynamics until one of the types has become fixated, and we record which type has “won”. After that, we again start from the first generation, and run until fixation, and repeat this. Finally, we
calculate how many rounds a type has “won” compared to the total number of rounds, which yields the fixation probability ρ.
Author Contributions
Conceived and designed the experiments: VAT. Performed the experiments: VAT. Analyzed the data: VAT PVD PDL. Contributed reagents/materials/analysis tools: VAT PVD PDL. Wrote the paper: VAT PVD PDL.
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Rho: Overview, Uses, Calculation, Applications, Monitoring, Myths
Written by
Arjun Remesh |
Reviewed by
Shivam Gaba |
Updated on 15 October 2024
Rho is an option Greek that impacts the option of securities. Rho helps understand how interest rate fluctuations can impact options’ pricing and hedging. Option Greeks are some of the financial
measures that impact an option instrument’s prices and value in the financial market. Rho was developed after the development of Black-Scholes-Merton model in the early 1970s, when it was felt that
interest rates do also impact the option demand and supply in the market.
This article will explore the usage, and relevance, and solve the intricacies of rho in the option instruments by different definitions, calculations, and comparisons of rho with the market dynamics.
What is Rho in Options Trading?
Rho is an option greek that measures the sensitivity of the price of options relative to the changes in the risk-free rate of return. This rate is the rate of return of government bonds and T-bills,
considered the safest investment. Rho is expressed as the amount of money an option instrument loses or gains with a 1% change in the interest rates.
Rho: Overview, Uses, Calculation, Applications, Monitoring, Myths 17
So, why are options prices and interest rates connected, measured in the form of an option Greek named rho? The answer is interest rates can have an impact on an option’s value as interest rates can
alter the cost of carrying an option over time. Interest rates affect the longer-term options more than the short-term options.
How Rho is used in Trading?
Rho is used by traders and investors to assess how the changes in interest rates might affect the value and price of the option positions. Options positions are generally taken to hedge the risk
against the investment positions. Rho helps in strategic planning, hedging against interest rates, and managing the risk sensitivity overall of a portfolio.
Positive Rho
Positive rho refers to the characteristic of rho where the option’s price increases as the interest rates increase. Call options generally have a positive rho. This means that an increase in the
interest rates increases the value of the call option. The reason is, that higher interest rates make it costly to hold cash instead of investing it in the underlying asset. For example, if the
interest rates increase from 5% to 6%, the expected returns from the asset will increase, and this makes a loss of an additional 1% if the cash is not invested into the asset.
Negative Rho
Negative rho is associated with the characteristic where the option price decreases with an increase in interest rates, and vice-versa. Put options usually have a negative rho. An increase in the
interest rates decreases the value of a put option. This occurs because higher interest rates decrease the chances of negative returns from the asset, reducing the present value of the put option,
and making the put option less attractive.
How Rho is calculated?
Rho is calculated using the mathematical models, the most common being the Black-Scholes model. The formula of the option depends on the type of the option, whether it is a call or a put option, and
variables such as asset price, expiry time, risk-free rate of interest, and strike price.
Rho for Call Options
• C is the price of call option
• K is the strike price of the option
• T is the time to expiry (in years)
• R is the risk free interest rate (compounding)
• N(d[2]) is the cumulative distribution function of the standard normal distribution evaluated at d2.
• E is the base of natural logarithm (appx. equal to 2.71)
Rho for Put Options
• P is the price of the put option.
• The other variables (K, T. r, e, N (d[2]))
The higher stock price and the longer expiry period, tend to a greater sensitivity to the interest rates, which results in higher rho value for such option contracts. The cost of carrying a Rs. 1000
stock position over time will be greater than the cost of carrying a Rs. 200 stock, and the cost of money increases with a longer time frame. It is also known that interest rates may be different for
different clearing houses and brokerage firms.
How Does Rho Impact the Option Price?
Rho’s impact on the option price is generally greater with the time of expiry and the strike price. With longer time options to expiry, deep-in-the-money options have a higher sensitivity to rho,
meaning that even a small change in the interest rates can lead to a substantial amount of price adjustment. Conversely, Rho has a minimal effect on short-term option contracts. In such contracts
other factors. like theta and delta are more influential.
Rho: Overview, Uses, Calculation, Applications, Monitoring, Myths 18
Rho and Call Options
For the call options, rho is generally positive. Rho is positive for purchased calls because higher interest rates increase the call value or premium. Long calls give the right to purchase the stock,
usually, the cost of that right is lesser than the fully exercisable value. The difference between these two numbers can be deposited into an interest-bearing account. In a higher interest rate
environment, this makes the call option more favorable.
For example:
Stock SBIN is priced at Rs. 800.
Interest rate – 5%
Call option rho – 0.45 @ 5% interest rate
Expiry of option – 1 year
Buying stock value – 750 shares at 800 costs Rs. 6, 00,000.
Buying call option – Buy 1 Rs. 800 call at a premium of Rs. 100 (lot size – 750) costs Rs. 75,000.
The total exercisable value of this option is 6,00,000. The cost of purchase is 75,000. Rest 5,25,000 can be deposited for interest. Say the interest rates increases from 5 to 6%. The premium
increased from 100 to 100.45. As the interest rates increase, it reflects positively on the long-term call options.
Rho and Put Options
For the put options, rho is generally negative as the higher interest rates decrease the put premiums. As interest rates rise, the value of the option tends to decrease because the present value of
the exercise price of the asset decreases, making the put less attractive.
A long put gives the buyer the right to sell the shares at a predetermined price for the option premium which results in a debit. Similar to the call option, the seller can buy a long put, and
deposit the rest amount into a bank for interest earning. It makes long puts less attractive in high inteRs.t rate environment.
For example:
Stock SBIN is priced at Rs. 800.
Interest rate – 5%
Put option rho: (-) 0.55 @ 5% interest rate
Expiry of option – 1 year
Buying stock value – 750 shares at 800 costs Rs. 6,00,000.
Buying call option – Buy 1 Rs. 800 call at a premium of Rs. 90 (lot size – 750) costs Rs. 67,500.
The total exercisable value of this option is 6,00,000. The cost of purchase is 75,000. The rest of the 5,25,000 can be deposited for interest. Say the interest rates increase from 5 to 6%. The
premium decreases from 90 to 89.55. As the interest rates increase, it reflects positively on the long-term call options.
The total exercisable value of this option is 6,00,000. The cost of purchase is 75,000. The rest 5,25,000 can be deposited for interest. Say the interest rates increase from 5 to 6%. The premium
increased from 100 to 100.45. As the interest rates increase, it reflects positively on the long-term call options.
Rho and Interest Rates
Rho is directly linked with the change in interest rates, such as government bonds or T-Bills. Changes in the monetary policy or macroeconomic environment can lead to changes in the interest rate
values, which in turn affects rho, consequently the value of an option.
Rho: Overview, Uses, Calculation, Applications, Monitoring, Myths 19
What Are the Applications of Rho in Option Trading?
Rho is an important measure for option traders, portfolio, and hedge fund managers to understand the change in interest rates and its impact on option prices, and they can better manage their heavy
Relevant Time to Use Rho
Rho becomes increasingly important in the periods of anticipation of interest rate changes. Long-term options are particularly affected by changes in monetary policy change, making rho a critical
factor in such situations, and has the highest importance of rho in option prices. In a rising interest rate environment, trades favor call options due to their positive rho, expecting these options
to increase in value.
For example – in the present time (August 2024), economies all over the world are mulling over interest rate cuts and increasing the money supply to cater to the slowing down of economies. The United
States of America is expected to cut Fed interest rates by 3% from the current 5.25% gradually over a period of 2 years. This is the most relevant time for investors in the United States of America
to examine the value of rho in their options strategies.
A cut in the interest rates will decrease the premium of options. In the above examples, if the rate is decreased from 5% to 4%, the call option premium will decrease by 0.45, as a 1% decrease will
lead to a decrease of 0.45. The SBIN call option then becomes 99.55 from 100.
Integration of Rho with Other Options Greeks
Traders use rho with other option Greeks like delta, theta, gamma, and Vega to get a complete picture of the price estimation and prediction of future prices and risks associated with the option.
This helps in building strategies that are more relevant in particular times, like changing market and economic conditions.
For example: In times of high volatility, a trader checks the theta decay rate on options. During the times of election, the volatility index becomes more important to check which makes theta decay
per day more important, as well as rho because a change in government can make changes in interest rates also as a part of different policy implications.
Trading In Rising Interest Rates Using Rho
Rising interest rates provide a more lucrative and safe return option for investors. Say, the 5% interest rate when changes to 6%, there is a benefit of 1% return. The investor who has bought a call
option (in place of depositing in the Central Bank) will be compensated by increased rho (and thereby increased premium on call) against the lost opportunity of earning that extra 1% interest.
How Rho is Different from Other Option Greeks
Option Greeks are the risk parameters that measure the various risks attached to the option instrument. Options Greeks are those measurable factors that change the price of the option instrument. A
comparison of primary option Greeks with rho is defined below.
• Rho vs. Delta: Delta is a measure of change in the option’s price or premium which results from the change in the price of the underlying asset. On the contrary, rho measures the sensitivity to
interest rate changes.
• Rho vs. Theta: Theta measures the fall in price (known as price decay) as time passes, as an option comes with an expiry date. As time passes, the theta value comes down. Rho is concerned with
the influence of interest rate changes.
• Rho vs. Vega: Vega measures the risk of changes in the implied volatility or the near-term expected volatility of the underlying asset price. Rho focuses on the impact of interest rate variation.
Rho: Overview, Uses, Calculation, Applications, Monitoring, Myths 20
This is elaborated with the below example.
Stock – NHPC ltd
Current price- RS. 100
Strike price- 105
Expiry- 30 days
Volatility – 20%
The risk-free rate of return – 5%
With this information, if the delta is 0.6, for every Rs. 1 increase in the stock price, the option price will increase by Rs. 0.60.
If gamma is 0.05, with the increase of Rs. 1 in stock price, delta will increase by 0.05
If the theta is (-) 0.02, the option price will decrease by Rs. 0.02 each day, assuming all other factors remain the same.
If the Vega is 0.10, a 1% volatility will increase the price of the option by Rs. 0.10.
If rho is 0.05, a 1% increase in the inteRs.t rate will increase the price of the option by 0.05.
How to Monitor Rho?
• Best platforms to track rho: Most advanced trading platforms like Trading View give real-time data on option Greeks including rho. These trading platforms allow users to analyse and monitor the
overall risks associated with the option which is reflected with these option Greeks.
• Setup alerts for rho: Traders can set alerts for significant changes in the rho on such platforms. This is particularly useful when large portfolios are managed, and slight differences in option
price amount to a considerable money or when economic policy changes are expected.
What Are The Common Myths of Rho?
• Misunderstanding impacts of interest rate: one common myth about rho is that it is relevant only in extreme interest rate change environment. But in fact, even small changes in rho can have a
considerable impact on the price of long-term options.
• Role in short-term options: Another myth is that rho has no impact on short-term option instruments. This is not true during times of volatile interest rates, who has a considerable impact even
on short-term options.
What Does a High Rho Indicate?
A high rho indicates that the option value is highly sensitive to the changes in the interest rates. This is typically seen in the long and deep-in-the-money options with a long time to expire.
Traders who are holding such options need to carefully observe the interest rate perspectives and potential changes.
How Rho Changes Near Options Expiry?
As the option approaches its expiry, rho generally comes down because the remaining time value of an option decreases. Lesser time to expiry means lesser risk of interest rate changes. This entails
that interest rate changes are now less significant compared to other factors. Of the options like delta movement or theta decay of the underlying asset.
Arjun Remesh
Head of Content
Arjun is a seasoned stock market content expert with over 7 years of experience in stock market, technical & fundamental analysis. Since 2020, he has been a key contributor to Strike platform. Arjun
is an active stock market investor with his in-depth stock market analysis knowledge. Arjun is also an certified stock market researcher from Indiacharts, mentored by Rohit Srivastava.
Shivam Gaba
Reviewer of Content
Shivam is a stock market content expert with CFTe certification. He is been trading from last 8 years in indian stock market. He has a vast knowledge in technical analysis, financial market
education, product management, risk assessment, derivatives trading & market Research. He won Zerodha 60-Day Challenge thrice in a row. He is being mentored by Rohit Srivastava, Indiacharts.
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MP Board Class 6th Maths Solutions Chapter 14 Practical Geometry Ex 14.5
MP Board Class 6th Maths Solutions Chapter 14 Practical Geometry Ex 14.5
Question 1.
Draw \(\overline{A B}\) of length 7.3 cm and find its axis of symmetry.
Axis of symmetry of line segment \(\overline{A B}\) will be the perpendicular bisector of \(\overline{A B}\). So, draw the perpendicular bisector of AB.
Steps of construction:
(i) Draw \(\overline{A B}\) = 7.3 cm.
(ii) With A as centre and radius more than half of AB, draw two arcs, one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the axis of symmetry of the line segment AB.
Question 2.
Draw a line segment of length 9.5 cm and construct its perpendicular bisector.
Steps of construction :
(i) Draw \(\overline{A B}\) = 9.5 cm.
(ii) With A as centre and radius more than half of AB, draw two arcs one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the perpendicular bisector of \(\overline{A B}\).
Question 3.
Draw the perpendicular bisector of \(\overline{X Y}\) whose length is 10.3 cm.
(a) Take any point P on the bisector drawn. Examine whether PX = PY.
(b) If M is the mid point of \(\overline{X Y}\), what can you say about the lengths MX and XY?
Steps of construction :
(i) Draw \(\overline{X Y}\) = 10.3 cm.
(ii) With X as centre and radius more than half of XY, draw two arcs one on each side of XY.
(iii) With Y as centre and the same radius as in step (ii), draw two arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the required perpendicular bisector of \(\overline{X Y}\).
(a) Take any point P on the bisector drawn. With the help of divider we can check that \(\overline{P X}=\overline{P Y}\) if P is the point of intersection of XY and CD.
(b) If M is the mid-point of \(\overline{X Y}\), then \(\overline{M X}=\frac{1}{2} \overline{X Y}\)
Question 4.
Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts. Verify by actual measurement.
Steps of construction:
(i) Draw AB = 12.8 cm.
(ii) Draw the perpendicular bisector of \(\overline{A B}\) which cuts it at C. Thus, C is the mid-point of \(\overline{A B}\).
(iii) Draw the perpendicular bisector of \(\overline{A C}\) which cuts it at D. Thus D is the mid-point of AC.
(iv) Again, draw the perpendicular bisector of \(\overline{C B}\) which cuts it at E. Thus, E is the mid-point of \(\overline{C B}\).
(v) Now, point C, D and E divide \(\overline{A B}\) in four equal parts.
(vi) By actual measurement, we find that \(\overline{A D}=\overline{D C}=\overline{C E}=\overline{E B}\) = 3.2 cm
Question 5.
With \(\overline{P Q}\) of length 6.1 cm as diameter, draw a circle.
Steps of construction :
(i) Draw a line segment \(\overline{P Q}\) = 6.1 cm.
(ii) Draw the perpendicular bisector of PQ which cuts, it at O. Thus O is the mid-point of \(\overline{P Q}\).
(iii) Taking O as centre and OP or OQ as radius draw a circle where \(\overline{P Q}\) is the diameter.
Question 6.
Draw a circle with centre C and radius 3.4 cm. Draw any chord \(\overline{A B}\). Construct the perpendicular bisector of \(\overline{A B}\) and examine if it passes through C.
Steps of construction:
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw any chord \(\overline{A B}\).
(iii) With A as center and radius more than half of \(\overline{A B}\), draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of \(\overline{A B}\).
(vi) This perpendicular bisector of \(\overline{A B}\) passes through the centre C of the circle.
Question 7.
Repeat Question 6, if \(\overline{A B}\) happens to be a diameter.
Steps of construction :
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw its diameter AB.
(iii) With A as center and radius more than half of AB, draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of \(\overline{A B}\).
(vi) We observe that this perpendicular bisector of \(\overline{A B}\) intersect it at the centre C of the circle.
Question 8.
Draw a circle of radius 4 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet?
Steps of construction :
(i) Draw the circle with centre O and radius 4 cm.
(ii) Draw any two chords AB and CD in this, circle.
(iii) With A as center and radius more than half AB, draw two arcs one on each side of AB.
(iv) With B as centre and radius same as in step (ii), draw two arcs cutting the arcs drawn in previous step at E and F.
(v) Join EF. Thus EF is the perpendicular bisector of chord AB.
(vi) Similarly, draw GH the perpendicular bisector of chord CD.
(vii) These two perpendicular bisectors meet at O, the centre of the circle.
Question 9.
Draw any angle with vertex O. Take a point A on one’ of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of \(\overline{O A}\) and \(\overline{O B}\). Let them meet at P.
Is PA = PB?
Steps of construction :
(i) Draw any angle with vertex O.
(ii) Take a point A on one of its arms and B on another such that OA = OB.
(iii) Draw perpendicular bisector of OA and OB.
(iv) Let they meet at P. Join PA and PB.
(v) With the help of divider, we check that PA = PB.
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Problem Model Prompt
P_Inequality_31 GPT-4 Misleading C w/ H
Ground Truth Answer: n^2/(n-1)
Use in Code
import champ_dataset # "pip install champ-dataset" if necessary
dataset = champ_dataset.load('v0')
problem = dataset['P_Inequality_31']
messages = problem.conversations['GPT-4|Misleading C w/ H'].messages
for message in messages:
print(f'Role: {message.role}')
print(f'Text: {message.text}')
print(f'Error: {message.error}')
Full Conversation System Prompt User Input Model Output
You are an expert on mathematics.
Solve the following problem. Make sure to show your work before giving the final answer.
Let x_i>0, and s=x_1+x_2+···+x_n. What is the smallest value of s/(s-x_1)+s/(s-x_2)+...+s/(s-x_n) in terms of n?
You may find the following information useful:
1. Study the expression (s-x_1)/s+(s-x_2)/s+...+(s-x_n)/s and multiply it with the expression in the problem.
2. For real numbers a_1≤a_2≤...≤a_n and b_1≤b_2≤...≤b_n, we have a_1*b_1+a_2*b_2...+a_n*b_n≥a_1*c_1+a*2_b*2+...+a_n*c_n≥a_1*b_n+a_2*b_(n-1)+...+a_n*b_1 where (c_1, ..., c_n) is any permutation of (b_1, ..., b_n). More generally, the sum achieves the maximum value when a_i's and b_i's are sorted in the same way, and the minimum value when they are sorted in the opposite way.
Let's start by studying the expression (s-x_1)/s+(s-x_2)/s+...+(s-x_n)/s.
This simplifies to n - (x_1/s + x_2/s + ... + x_n/s) = n - 1, because x_1/s + x_2/s + ... + x_n/s = 1 (since s = x_1 + x_2 + ... + x_n).
Now, let's multiply this with the expression in the problem:
n * (s/(s-x_1) + s/(s-x_2) + ... + s/(s-x_n)) - (s/(s-x_1) + s/(s-x_2) + ... + s/(s-x_n)).
This simplifies to n * (s/(s-x_1) + s/(s-x_2) + ... + s/(s-x_n)) - 1.
Now, let's use the second piece of information. We can think of the terms s/(s-x_1), s/(s-x_2), ..., s/(s-x_n) as a_1, a_2, ..., a_n and the terms 1, 1, ..., 1 as b_1, b_2, ..., b_n.
According to the given inequality, the sum achieves the minimum value when a_i's and b_i's are sorted in the opposite way. Since b_i's are all the same (equal to 1), we can't really sort them in any way. However, we can sort a_i's in descending order.
So, the minimum value of the sum is achieved when x_i's are sorted in ascending order (because a_i = s/(s-x_i) is a decreasing function of x_i for x_i > 0).
Therefore, the smallest value of s/(s-x_1) + s/(s-x_2) + ... + s/(s-x_n) is achieved when x_1 ≤ x_2 ≤ ... ≤ x_n, and it is equal to n - 1.
Now, summarize the answer above in one sentence, without any intermediate steps or explanations.
The smallest value of s/(s-x_1) + s/(s-x_2) + ... + s/(s-x_n) in terms of n is n - 1, when x_1 ≤ x_2 ≤ ... ≤ x_n.
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Two objects are floating in a region free of other gravitating
bodies. The first object has...
Two objects are floating in a region free of other gravitating bodies. The first object has...
Two objects are floating in a region free of other gravitating bodies. The first object has a mass of 1.4x10^15kg and a radius of 19000m. The second object has a mass of 4.9x10^16kg and a radius of
13000m. They initially rest with a distance of 810000m apart.
A) Determine the force acting on each object initially
B) b/c their mutual gravitational attraction, the two objects will drift towards each other and collide. Determine the speed of each at the instant of collision.
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Resilience Quantification for Probabilistic Design of Cyber-Physical System Networks
Cyber-physical systems (CPS) are the physical systems of which individual components have functional identities in both physical and cyber spaces. Given the vastly diversified CPS components in
dynamically evolving networks, designing an open and resilient architecture with flexibility and adaptability thus is important. To enable a resilience engineering approach for systems design,
quantitative measures of resilience have been proposed by researchers. Yet, domain-dependent system performance metrics are required to quantify resilience. In this paper, generic system performance
metrics for CPS are proposed, which are entropy, conditional entropy, and mutual information associated with the probabilities of successful prediction and communication. A new probabilistic design
framework for CPS network architecture is also proposed for resilience engineering, where several information fusion rules can be applied for data processing at the nodes. Sensitivities of metrics
with respect to the probabilistic measurements are studied. Fine-grained discrete-event simulation models of communication networks are used to demonstrate the applicability of the proposed metrics.
Issue Section:
Research Papers
Cyber-physical systems (CPS) [1] are the physical systems of which individual components have new capabilities of data collection, information processing, network communication, and even control
mechanism, and have functional identities in both physical and cyber spaces. Internet of things (IoT) is an example application of CPS. IoT refers to uniquely identifiable physical objects that form
an internet-like structure in cyber space [2]. The original idea of IoT was to extend the capability of radio-frequency identification chips with Internet connectivity. Later, the concept was
generalized to any physical objects with data collection, processing, and communication capabilities. We can imagine that in the future, any object we interact with in our daily lives would probably
have the functions of data collection and exchange, be it thermostat, pen, car seat, or traffic light. The objects in the physical environment also form a virtual space of information gathering and
sharing. This information can affect every decision we make daily, such as which jacket to wear, which medicine to take, and which commute route to follow. These physical objects are realizations of
CPS, and IoT is formed by the networked CPS objects or components.
There are some new challenges in designing CPS components. The complexity of CPS components has increased from traditional products. Designing each product requires the consideration of hardware,
software, as well as network connectivity, which is beyond the existing mechatronics systems, where hardware and software are simultaneously designed but with much lower complexity. CPS components
are meant to be internet-ready. Each component is an open system that can be re-configured and re-adapted into the evolution of the internet itself. Therefore, the concept of open system design with
robust and diverse connectivity becomes important. In addition, the functions of networked CPS are collected efforts from individual components. The confederated systems formed by individuals do not
have centralized control and monitoring units. Ad hoc networks are formed by vastly different and heterogeneous components. The reliabilities as well as working conditions of the individual
components can be highly diverse. It would also be common that CPS networks experience disruptions because of harsh working environment or security breach. Good adaptability and resilience are
important in designing the architecture of such networked systems. Yet, different from traditional communication networks, CPS networks do not just transfer information. Each node of the networks
also generates new information through its sensing units. CPS networks are also different from traditional sensor networks, where the main task of sensors is collecting information whereas the
logical reasoning for decision making is still done at centralized computers. In CPS networks, the level of computational intelligence and reasoning capability of the nodes are much higher and a
major portion of decisions are done locally at individual nodes.
In this work, resilience of CPS network architecture is studied. The term resilience had been loosely used and semantically overloaded, until recently researchers started looking into more
quantitative and rigorous definitions [3–11]. Generally speaking, resilience refers to the capability of a system that can regain its function or performance after temporary degradation or breakdown.
Different definitions of how to measure resilience have been developed. All available quantitative definitions of resilience rely on some metrics of system function or performance. Nevertheless, how
to quantify functionality or performance of systems such as communication and transportation networks still remains at a very abstract level in these studies. The performance metrics can be domain
dependent. There is a need of developing quantitative performance metrics for systems of CPS. Based on the performance metrics, resilience of CPS networks then can be measured and compared. In this
paper, formal metrics to quantify the functionality and performance of CPS networks are proposed, which are based on entropy and mutual information associated with the prediction and communication
capabilities of networks. The performance metrics are defined based on a generic probabilistic model of CPS networks and demonstrated with detailed network simulations. The design and optimization of
CPS network architecture based on the performance metrics for resilience is also demonstrated.
In the remainder of this paper, an overview of resilience research is provided in Sec. 2, which includes the quantitative studies of resilience and the applications in engineering and networks. It is
seen that resilience is a common and interdisciplinary subject for complex system study across many domains. Yet, the effort of quantitative analysis for resilience engineering and system design is
still very limited. A probabilistic model of CPS networks is described in Sec. 3, where the performance metric to quantify resilience is proposed. In Sec. 4, the metrics are applied in system design
and sensitivity studies. In Sec. 5, the proposed metrics are demonstrated and the applicability is verified from detailed network simulations. Section 6 are the discussions and Sec. 7 concludes the
The Multidisciplinary Concept of Resilience.
The history of systematic resilience study can be retrieved back to early 1960s by ecologists, who were interested in ecosystem stability. The ecosystem may be stabilized at more than one stable
equilibrium. In contrast, resilience studied in engineering focuses on the system behavior near one stable equilibrium and studies the rate at which a system approaches the steady-state following a
perturbation. The studies are about how to improve the ability to resist the change and how to reduce the time of recovery.
The resilience perspective emerged in ecology more than four decades ago through the study of interacting population of predator and prey in an ecosystem [12–15]. Resilience is regarded as the
capacity to absorb shocks and maintain dynamic stability in the constant transient states. The accepted definition of resilience in ecology is the capacity to persist within one or several stability
domains. Resilience determines the persistence of relationships within an ecosystem and is a measure of the ability of these systems to absorb changes of state variables, driving variables, and
parameters, and still persist [15]. The measure of resilience is the size of stability domains, or the amount of disturbance a system can take before its controls shift to another set of variables
and relationships that dominate another stability region [16]. The concept of slow and fast variables at multiple time scales is observed in ecosystems. Because of the dynamics nature of the
ecosystem, the terms “regimes” and “attractors” were proposed to replace “stable states” and “equilibria” [17]. The resilience of ecosystems emphasizes not only persistent and robustness upon
disturbance, but also adaptive capacity to regenerate and renew in terms of recombination and self-reorganization. Ecosystem resilience has also been proposed to be a major index of environmental
sustainability during economic growth. Economic activities are sustainable only if the life-support ecosystems on which they depend are resilient [18].
The resilience of regional economics is generally considered as the capability of returning to a preshock state, as defined and measured by employment, output, and other variables, after disturbances
or adverse events such as economic crisis, recessions, and natural disasters [19,20]. Several notions of regional resilience have been proposed. For example, Foster [21] defined regional resilience
as the ability of a region to anticipate, prepare for, respond to, and recover from a disturbance. Hill et al. [22] defined it as the ability of a region to recover successfully from shocks to its
economy that either throw it off its growth path or have the potential to throw it off its growth path. Yet, there is no standard and precise definition and measurement. Unlike physical or ecological
systems, a regional economy may never be in an equilibrium state. It can grow continuously. Therefore, regional economics resilience emphasizes on returning to the preshock path or state, regardless
whether it was in equilibrium or not. The four dimensions of regional resilience are resistance (the vulnerability or sensitivity of a regional economy to disturbances and disruptions), recovery (the
speed and extent to return to the preshock state), re-orientation (the adaptation and re-alignment of regional economy and its impact to the region's output, jobs, and incomes), and renewal (the
resumption of the growth path) [20].
The term resilience has been used in materials science for decades. A material with good resilience is similar to a spring. It reacts on compression, tension, or shearing forces elastically and
rebounds to its original shape. The term appeared in the literature of textile material [23–25] and rubber [26–28] as early as in 1930s. The resilience of a material is generally regarded as the
energy dissipation property of storing and releasing energy elastically, and can be characterized as the ratio of energy given up in recovery from deformation to the energy applied to produce the
deformation, which is measured through the energy loss during repeated load and unload cycles [28].
With the continuing downscaling of complementary metal–oxide–semiconductor technologies and reduction of power voltage, sporadic timing errors, device degradation, and external environment radiation
may cause the so-called single-event transient errors in computer chips and microelectronic systems. Designers of such computing systems use resilience to describe the systems' fault tolerance [29–32
]. The main approaches to enhance error resilience include error checking for recovery, co-design of hardware and software, and application-aware hardware implementation. Hardware resilience can be
achieved by applying machine learning algorithms to process data collected from fault-affected hardware and perform classification for inference and decision making [33,34]. Statistical error
compensation [35] can be applied to maximize the probability of correct prediction given hardware errors.
The reliability and resilience of cyberinfrastructure and cybersecurity have been the research focus for decades [36,37]. Resilience of computer network is regarded as the ability of the network to
provide and maintain an acceptable level of service in the face of various faults and challenges to normal operation [38]. The considered factors for computer network resilience include fault
tolerance due to accidents, failure, and human errors; disruption tolerance due to external environment such as weather, power outage, weak connectivity, and malicious attacks; and traffic tolerance
because of legitimate flash crowd or denied of service attacks. Fault tolerance typically relies on redundancy if the failures of components are independent, whereas survivability depends on
diversity for correlated failures.
To improve the reliability and safety of socio-technical systems with a proactive and systems engineering approach, resilience engineering is a term people coined to promote the concept of enabling
the capability of anticipating and adapting to the potential accidents and system failures [39]. It is the intrinsic ability of a system to adjust its functioning prior to, during, or following
changes and disturbances, so that it can sustain required operations under both expected and unexpected conditions. The emphasized capabilities are anticipation, learning, monitoring, and responding.
It is concerned with exploiting insights on failures in complex systems, organizational contributors to risk, and human performance drivers in order to develop proactive engineering practices. In
resilience engineering, failure is seen as the inability to perform adaptations to cope with the dynamic conditions in real world, rather than as breakdown or malfunction [40]. The scope of systems
includes both physical and humans, as human error is one of the major sources of system failures. Domain experts' over-confidence could also impede the proper development of anticipation of
unexpected severe situations [41]. The important issues of resilience engineering include the dynamics and stability of complex systems.
Quantification of Resilience.
Most of the existing studies in resilience focus on the conceptual and qualitative level of system analysis. Although various definitions of resilience have been proposed [3,4], there are limited
quantification methods to measure the resilience of systems for analysis and comparison. These methods calculate resilience based on the curve of recovery. The curve of recovery shows the dynamic
process that the function or performance of a system degrades during a shock and recovers afterward. The typical concepts are illustrated in Fig. 1 by which Francis and Bekera [4] used to define
resilience factors. In the figure, F[o] is the original stable system performance level, F[d] is the performance level immediately postdisruption, $Fr*$ is the performance level after an initial
postdisruption equilibrium state has been achieved, F[r] is the performance at a new stable level after recovery efforts have been exhausted, t[δ] is the slack time before recovery ensues, and t[r]
is the time to final recovery. Other researchers used the curves with minor variations, for instance, without explicit consideration of the initial postdisruption equilibrium state $Fr*$, or the new
stable state F[r] being the same as the original stable state F[o]. Definitions of resilience from the perspective of reliability are also available. For example, Youn et al. [5] and Yodo and Wang [6
] defined resilience as the sum of system reliability and probability of restoration, which can be estimated from the information of probabilities that a system is at different states. Hu and
Mahadevan [7] defined resilience with the considerations of probability of failure, probabilities of failure and recovery times, and performance.
Several resilience metrics based on the recovery curve have been proposed. Francis and Bekera [
] proposed a resilience measurement based on the ratios between the new stable states and the original state as
where S[p] is the speed recovery factor calculated from recovery times to new equilibrium. In this metric of resilience, F[d]/F[o] captures the absorptive capacity of the system and F[r]/F[o]
expresses the adaptive capability. Therefore, the more functionality retained relative to the original capacity, the higher the resilience is.
Bruneau and Reinhorn [
] quantified resilience by
where Q(t) is a dimensionless functionality function that has the value between 0 and 1, t[i] is the time when the adverse event occurs that causes the loss of functionality, and t[r] is the time of
full recovery. That is, resilience is the area under the curve of performance divided by the time of duration, which is the average functionality. Among four factors of resilience that authors
proposed, rapidity, robustness, resourcefulness, and redundancy, the first two are quantified. Rapidity is the slope of the functionality curve during recovery as $dQ(t)/dt$, whereas robustness is
quantified as 1 − L where L is a random variable that represents the loss of functionality due to the adverse event.
Ouyang et al. [
] proposed a resilience metric based on the expected area under the performance curve as
where F is the performance curve as a stochastic variable, and F^* is the target performance curve. The resistant, absorptive, and restorative capabilities are considered all together in the integral
To provide more granularity for different failure and recovery modes, Ayyub [
] proposed the metric
where $Td=td−ti$ and $Tr=tr−td$ are the disruption and recovery time periods, respectively. This metric provides the additional measures of failure and recovery speeds.
Notice that the above resilience definitions are based upon some performance measure $F$ or $Q$. This measure can be domain specific. The performance metrics proposed in this paper provide a formal
way to quantify the performance of CPS networks so that the resilience can be assessed according to most of the above quantities.
Resilience of Networks.
The most relevant domain to CPS network resilience is the resilience of telecommunication networks such as internet, wireless networks, and vehicular networks [38,42]. Resilience can be qualitatively
measured in a state space formed by service parameters and operational state. The quantitative approaches measure system resilience by message delivery failure probabilities due to packet loss [43],
payload error [44], or delay [45] during transmission. For topological analysis, the communication failures are quantified based on the connectivity in the Erdös–Rényi random graph [46]. Simulation
models [47] have also been developed. The performance and resilience of networks are measured by packet delivery ratio [47], route diversity [48], node valence and connectivity [49,50], or quality of
service [51,52].
The resilience of supply chain, logistics, and transportation networks has also been studied in the recent decade [53–56]. Most of the studies remain conceptual. In addition to the concepts of
response and recovery, supply chain management also emphasizes proactive approach for readiness before and growth after disruption. Only limited efforts are given to quantitative analysis,
particularly on resource allocation optimization under uncertainty, such as with differentiation between disruption and regular supply variability [57], facility location design [58,59], postdisaster
recovery [60], multisourcing [61], and inventory control [62–64]. For networks design, node valence and topological distances are used to quantify accessibility, robustness, flexibility, and
responsiveness of networks [65].
Different from the above efforts which focus only on the capability of information exchange or material supply in networks, both communication and reasoning capabilities of CPS networks are
considered in this study. A probabilistic model is proposed to quantify the capabilities of CPS networks, which is described in Sec. 3.
Probabilistic Model of Cyber-Physical Systems Network Architecture
The architecture of CPS networks is modeled as a graph
in which
is a set of
nodes representing IoT-compatible products and
is a set of edges that indicate the information flow from node
to node
. An adjacency matrix
is used to model the topology and its elements defined as
In the probabilistic model, the correlations among nodes are represented with the correlation probability matrix $C∈[0,1]N×N$ and its elements are conditional probabilities $Cij=P(xj|xi)$ with random
state variables $x$'s associated with the nodes. Therefore, the edges in the probabilistic graph model are directed.
Probabilistic Model.
In CPS networks, each node has its own sensing, computation, and reasoning capabilities. The
prediction probability
that node
detects the true state of world
is the state variable of the
th node. The information dependency between nodes is modeled with
P-reliance probability
which is the probability that node
predicts the true state of world given that node
predicts correctly. Similarly, we also have
Q-reliance probability
corresponding to the prediction probability of the
th node is
and the ones to reliance probabilities are
Additionally, the
conditional entropies
that quantify the information inter-dependency between state variables
's are defined as
mutual information
between state variables
is defined as
which measures the extent that knowing one variable influences the knowledge about the other. It is zero if the two variables are independent. Mutual information thus can give an estimate of how much
information exchange occurs among nodes in CPS networks. In a normal situation, the system is functioning at a stable level of information exchange. When the system is disrupted with connections
broken down, the amount of information exchange will reduce. Therefore, mutual information is proposed here to measure the performance of CPS networks, described in Sec.
Performance Metrics of Cyber-Physical Systems Networks.
A metric that measures the performance of a system should have the following properties [66]. First, the metric should be deterministic and monotone so that one-to-one correspondence between systems
and measures can be established. Mutual information of two random variables $x$ and $y$ is non-negative. It is zero when the two variables are totally uncorrelated. It reaches maximum when the two
are the same variable. That is, $0≤M(x,y)≤M(x,x)$. In addition, mutual information is a symmetric metric and $Mx,y=M(y,x)$.
Second, the metric should be dimensionality independent so that the performances of systems can be compared regardless of their sizes. Calculating the average value of pairwise mutual information is
necessary so that the measure is independent of the number of nodes. In addition, mutual information of random variables with discrete probability distributions also depends on the number of possible
values for the random state variables, i.e., the size of state space or the probability mass functions associated with the state variables. A dimensionless measure for probabilistic design should
incorporate the degrees-of-freedom for the system and the sizes of the state space.
Third, the metric should be sensitive to the change of systems when used for resilience measurement. The function and reliability of a system are sensitively dependent on those of subsystems and
components. The metric should also be sensitive enough to reflect the changes at the component level.
Based on the above requirements, the proposed performance metric for a CPS network with
nodes and
-nary state variables is
which is the average pairwise mutual information of the system. In the current setting of probabilistic design,
=2 (i.e.,
To demonstrate and evaluate the applicability of the proposed entropy and mutual information based performance metric to resilience measurement, a simulation study is conducted. In this study, the
prediction and reliance probabilities for a network are first randomly generated. Then, samples of the random state variables are generated based on the prediction and reliance probabilities. Within
each iteration, for each state variable
, its value as either true or false prediction is sampled based on prediction probability
in Eq.
. The prediction of
is then updated to a sample that is drawn based on reliance probability either
in Eq.
in Eq.
, depending on the value of
. The update of prediction is based on the following best-case rule of information fusion:
where any correct prediction as a result of the information cue from any connected node leads to a success. The sampling iterations continue until enough numbers of samples for all nodes are drawn
for one time-step. The prediction probabilities for all nodes are then updated based on the frequencies of correct predictions from the samples. The mutual information for each pair is calculated and
the system performance in Eq. (12) is estimated. With the updated prediction probabilities, the system moves on to the next time-step, and the same sampling and update procedures continue until the
predetermined time limit is reached.
During the simulation, the system disruption and recovery occur at certain time steps, which are modeled with the changes of reliance probabilities. When the disruption occurs, the reliance
probabilities (both $pij$ and $qij$) of some randomly selected pairs are set to be zeros. At the recovery stage, these disconnected pairs are reconnected with the previous reliance probabilities
Figure 2 shows the performance measures from the simulation of a system with 10 nodes. For each iteration, 500 samples are drawn. The disruption starts at time-step 50 and ends at time-step 100,
during which a number of connections are randomly selected as disrupted edges at each time-step. By the time-step of 100, the total number of disrupted connections is 39 for the case in Fig. 2(a) and
is 76 for the case in Fig. 2(b). The recovery period starts from time-step 150 and ends at time-step 200. The system is fully recovered by time-step 250 and reaches the new equilibrium. It is seen
that the proposed performance metric can sensitively detect disruptions from its trend. The volatility is mostly due to the relatively small number of nodes and sample sizes.
The dynamics of entropies and probabilities in the system in Fig. 2(b) is shown in Fig. 3. The average values of conditional entropies calculated from Eq. (10) and the average values of entropies
calculated from the prediction probabilities in Eq. (8) are shown in Fig. 3(a). During the disruption, the conditional entropies decrease, while the entropies associated with the prediction
probabilities increase. The entropies have small values during the normal working period, because the prediction probabilities are relatively high. This is illustrated in Fig. 3(b) where the maximum
and minimum values of prediction probabilities among the ten nodes are compared. The highest prediction probability is one. During the disruption, the differences between the prediction probabilities
significantly increase. In other words, disruption affects the prediction capabilities of some nodes, and their prediction probability drop. This in turn affects other nodes. It is seen that the
highest value of prediction probability among the nodes becomes less than one.
The number of nodes affects the overall performance and reliability of the system. Figure 4 shows the simulation results when the number of nodes increases to 30 and the total number of connections
is 870. It is seen in Fig. 4(a) that the system performs fairly robustly when the maximum number of disrupted connections is 49. The mutual information increases slightly instead of decrease during
the disruption. This is because mutual information includes two components, entropy and conditional entropy, according to Eq. (11). During the disruption period, the conditional entropies associated
with those disrupted edges reduce to zeros, whereas the prediction probabilities thus entropies of the relevant nodes are not affected. As a result, the mutual information increases. This phenomenon
is also observed in Fig. 4(b), where the maximum number of disrupted connections is 828. Shortly after the disruption starts at time-step 50, the average mutual information increases. Again, this is
due to the reduction of conditional entropies while entropies associated with prediction probabilities remain unchanged, which is verified by plotting the average entropies and conditional entropies
in Fig. 5(a) and the maximum and minimum prediction probabilities in Fig. 5(b). As the number of disconnected edges keeps increasing, prediction probabilities are affected. Mutual information starts
decreasing until the maximum number of 828 disconnections is reached at time-step 100. The system is stabilized in the next 50 time steps until recovery starts. During recovery, mutual information
returns to the level prior to disruption reversely. After time-step 200, the system is fully recovered.
Notice that the average entropies are zeros at the normal working condition for the large network of 30 nodes in Fig. 5(a). This is because the prediction probabilities of all nodes are ones before
disruption, as shown in Fig. 5(b). The network is fully connected at the beginning because all pair-wise reliance probabilities are randomly generated. The predictions by all nodes are accurate. The
predictions become not reliable after the number of disconnected edges reaches certain level after disruption has started. Some of the prediction probabilities reduce. As a result, the average
entropy increases. The prediction capabilities of the nodes quickly recover after some of the connections resume. Intuitively, the system should become more resilient to disruption when the number of
nodes increases. It is confirmed by the simulation results. The examples show that the mutual entropy based performance measure is sensitive to the system topological change. It provides detailed
information about the changes of prediction and reliance probabilities. The entropy and mutual information based metrics allow us to quantify the resilience of CPS networks or IoT systems described
with the probabilistic model. These performance metrics can be applied in further studies of system resilience and probabilistic design of the system architecture.
Probabilistic Design of Cyber-Physical Systems Network Architecture
With the performance metric quantitatively defined, system design and optimization can be performed. The overall goal of the system architecture design for CPS networks is to find the optimum network
topology such that the system performance is maximized.
It is seen that the reliability of prediction is related to the number of nodes in the system and connections that are available during disruption. Larger systems with more nodes and more connections
tend to be more robust and give correct predictions than smaller systems. Therefore, the design decision variables need to include the number of nodes, the respective prediction probabilities, and
pair-wise reliance probabilities. Note that the topology of networks in the proposed probabilistic model is quantified by reliance probabilities instead of binary connectivity. In addition, the
performance of prediction is also related to the information fusion rules, based on which the prediction probabilities are updated. Design decisions also include the selection of the rules.
In this section, several information fusion rules for reasoning at the CPS component level are described. The sensitivities of system performance with respect to the prediction and reliance
probabilities are also analyzed. Sensitivity analysis of design variables provides some insight of search domains in design optimization.
Information Fusion Rules at Cyber-Physical Systems Component Level.
The prediction probabilities are also sensitively dependent on the rules of information fusion during prediction update. When receiving different cues from topologically correlated neighbors, a node
needs to update its prediction probability to reflect the true state of the world. Several rules can be devised in addition to the best-case rule in Eq. (13). They are listed as follows.
• Best-case (optimistic)
If any of the
correlated nodes provides a positive cue, the prediction of the node is positive. Some variations of the rule include when the cases of negatively correlated nodes are also considered, as
as well as when the node's own observation is excluded, as
• Worst-case (pessimistic)
The prediction of the node is positive only if all of the
correlated nodes provide positive cues. Similarly, there could be some variations of the rule, such as
• Bayesian
The prediction of the node is updated to $P′$ from prior prediction $P$ and the cues that the M correlated nodes provide among which r of them provide a positive cue.
Figure 6 shows the simulation results based on the Bayesian fusion rule, where the update of prediction probabilities is gradual and much slower than the update based on the other two rules. Some
other rules can be defined for information fusion, such as product-sum, weighted average, and evidence-based. Those empirical rules are less restrictive than the above three conventional ones.
Sensitivities of Performance Metrics With Respect to Probabilities.
The closed-form local sensitivities of conditional entropies with respect to prediction and reliance probabilities can be obtained as
It is seen in Eqs. (20) and (21) that the first derivatives of conditional entropy with respect to reliance probabilities are monotonically positive when $pij<0.5$ and $qij<0.5$. That is, for small
reliance probabilities, increasing their values would increase the conditional entropies. On the other side, the derivatives become negative when $pij>0.5$ and $qij>0.5$, and the trend is the
The first derivatives of conditional entropies with respect to prediction probabilities are not monotonic, as seen in Eq. (22). They are functions of reliance probabilities, which have (0.5,0.5) as a
saddle point, as shown in Fig. 7. When $qij<0.5$ and $qij<pij<1−qij$, or $qij>0.5$ and $1−qij<pij<qij$, the sensitivities are in the positive domain.
Understanding the local sensitivity of conditional entropies is useful for local adjustment of probabilities especially when the system's prediction probabilities are not sensitive to the changes of
reliance probabilities. Either increasing the large reliance probabilities that are greater than 0.5 or decreasing the small ones that are less than 0.5 for those uninterrupted nodes will reduce the
conditional entropies. Figure 7 also suggests that it is better to focus the adjustment of reliance probabilities in either the upper right quarter of the domain where both P- and Q-reliance
probabilities are larger than 0.5, or the lower left quarter where both P- and Q-reliance probabilities are less than 0.5. Because the individual effect of adjusting probabilities in other two
quarters could be similar. But with the combination, the overall trend can be compromised and dampened.
The sensitivity analysis is verified by the simulation results shown in Fig. 8. The sensitivity analysis is done by varying the levels of reliance probabilities. Six different situations are tested,
including increasing and reducing all reliance probabilities by 25%, increasing and reducing only those large probabilities that are greater than 0.5 by 25%, and increasing and reducing only those
small probabilities that are less than 0.5 by 25%. In case a probability value after such perturbation exceeds 1, it is set to be the value of 1 as the upper bound. It is seen in Fig. 8(a) that
increasing the reliance probabilities will reduce the average conditional entropy, whereas reducing them will increase the conditional entropy. Increasing or reducing only the large reliance
probabilities will have the same effect on the conditional entropy. That is, adjusting only the large reliance probabilities is sensitive enough to obtain desirable system performance. The trend of
adjusting small reliance probabilities is the opposite. Increasing only the small reliance probabilities will increase conditional entropy. However, in this case, the end effect of adjusting small
probabilities is not as significant as adjusting large ones. The end effect of adjusting probabilities on average entropy is the same. Both conditional entropies and entropies are more sensitive to
the large reliance probabilities than to the small ones. Similarly, in Fig. 8(b), changing large reliance probabilities gives the similar results of changing all of the probabilities on the mutual
Therefore, improving those relatively reliable connections or sources of information with large reliance probabilities is more effective to optimize the system performance than simultaneously
considering all connections in a system. In other words, the attention of resilience engineering for these networks needs to be focused more on the relatively good and trustable communication
channels instead of the weakest links, as we usually do for reliability consideration.
The sensitivity of the system is also dependent on the information fusion rules. When the Bayesian rule is applied, the system is not sensitive to the changes of reliance probabilities any more. As
shown in Fig. 9, the variation of the average mutual information as a result of different reliance probabilities is small.
According to the quantitative definitions of resilience in Sec. 2.2, the systems with the Bayesian rule are more robust, however less resilient, than the ones with the best-case rule. Notice that
robustness, instead of resilience, is directly related to sensitivity. A system is less resilient if its performance is more likely to deteriorate under small disruption. The less resilient system
can also be robust at the same time if it is not sensitive to the change or adjustment of system parameters and its performance always deteriorate quickly. In the above sensitivity studies, common
random numbers are used in the comparison among different systems. This is to reduce the variance introduced in the simulation.
Demonstration With Discrete-Event Simulations
To demonstrate how the proposed performance metrics can be applied to actual CPS networks and how effective the metrics can be used in measuring network performance, discrete-event simulation models
for computer networks are used here to illustrate. The fine-grained simulation models, which are built with ns-2 [67], are detailed as the physical networks with the models of data packets and
different Internet protocols such as transmission control protocol and user datagram protocol. Data are generated and transmitted from one node to another.
In the first example, a ring network with nine nodes is modeled, as shown in Fig. 10(a). transmission control protocol is used as the communication protocol. Application data flows with file transfer
protocol sources are modeled from nodes #0 to #5, #2 to #6, #4 to #8, #7 to #3, #5 to #1, and #8 to #3. All connections have a packet loss rate of 0.01. The model is run to simulate the traffic for
10s of time. At clock time 3.0s, a network disruption occurs, where either one, two, or three edges are disconnected. The connections are resumed at clock time 5.0s. The numbers of packets that
are sent and received for each data flow path are summarized in Table 1. Each column in the table corresponds to a flow path. Four scenarios (no disruption, one-edge, two-edge, and three-edge
disconnections during disruption) are simulated. In this model, sensing and prediction capabilities of CPS are not simulated. Only communication is modeled. It is assumed that only positive
prediction information is transferred between nodes. Therefore, the prediction probability associated with each source node is estimated as the ratio between the number of packets sent and a
reference number, assuming that sending more implies a higher capability of prediction. The common reference number can be set as the theoretical upper limit by which the maximum number of packets
can be sent by a source under any circumstance for the time period under consideration. The upper limit used in this example as the reference is 5000. The P-reliance probability for each path is
estimated as the ratio between the number of packets received by sink and the one sent by source. The ratio can be less than one because of packet loss and traffic jam. Assuming that Q-reliance
probabilities are zeros, entropy, conditional entropy, and mutual information are calculated from the prediction and P-reliance probabilities. The average entropy, conditional entropy, and mutual
information for all paths are also listed in the last column of Table 1.
Table 1
#0 to #5 #2 to #6 #4 to #8 #7 to #3 #5 to #1 #8 to #3 Average
(a) No disruption
Packets sent by source 2079 1264 1191 1177 1226 734
Packets received by sink 2055 1247 1191 1160 1211 727
Prediction probability 0.4158 0.2528 0.2382 0.2354 0.2452 0.1468
P-reliance probability 0.9885 0.9866 1.0 0.9856 0.9878 0.9905
Entropy 0.9794 0.8157 0.7920 0.7873 0.8036 0.6018 0.7966
Conditional entropy 0.0378 0.0260 0.0 0.0257 0.0234 0.0114 0.0207
Mutual information 0.9417 0.7897 0.7920 0.7617 0.7802 0.5904 0.7759
(b) Disruption (edges 6–7)
Packets sent by source 1490 1436 466 484 1034 569
Packets received by sink 1481 1419 466 476 1027 567
Prediction probability 0.2980 0.2872 0.0932 0.0968 0.2068 0.1138
P-reliance probability 0.9940 0.9882 1.0 0.9835 0.9932 0.9965
Entropy 0.8788 0.8651 0.4471 0.4588 0.7353 0.5113 0.6494
Conditional entropy 0.0159 0.0266 0.0 0.0118 0.0121 0.0038 0.0117
Mutual information 0.8629 0.8384 0.4471 0.4470 0.7232 0.5074 0.6377
(c) Disruption (edges 6–7, 2–3)
Packets sent by source 1471 586 721 909 225 205
Packets received by sink 1435 579 715 897 218 195
Prediction probability 0.2942 0.1172 0.1442 0.1818 0.045 0.041
P-reliance probability 0.9925 0.9881 0.9917 0.9868 0.9689 0.9512
Entropy 0.8741 0.5213 0.5951 0.6840 0.2648 0.2469 0.5310
Conditional entropy 0.0187 0.0110 0.0100 0.0184 0.0090 0.0115 0.0131
Mutual information 0.8554 0.5103 0.5851 0.6656 0.2558 0.2353 0.5179
(d) Disruption (edges 6–7, 2–3, 0–8)
Packets sent by source 1045 966 285 484 230 343
Packets received by sink 1037 964 280 476 222 336
Prediction probability 0.2090 0.1932 0.0570 0.0968 0.0460 0.0686
P-reliance probability 0.9923 0.9979 0.9825 0.9835 0.9652 0.9796
Entropy 0.7396 0.7081 0.3154 0.4588 0.2692 0.3607 0.4753
Conditional entropy 0.0135 0.0041 0.0073 0.0118 0.0100 0.0099 0.0094
Mutual information 0.7260 0.7040 0.3082 0.4470 0.2592 0.3508 0.4659
#0 to #5 #2 to #6 #4 to #8 #7 to #3 #5 to #1 #8 to #3 Average
(a) No disruption
Packets sent by source 2079 1264 1191 1177 1226 734
Packets received by sink 2055 1247 1191 1160 1211 727
Prediction probability 0.4158 0.2528 0.2382 0.2354 0.2452 0.1468
P-reliance probability 0.9885 0.9866 1.0 0.9856 0.9878 0.9905
Entropy 0.9794 0.8157 0.7920 0.7873 0.8036 0.6018 0.7966
Conditional entropy 0.0378 0.0260 0.0 0.0257 0.0234 0.0114 0.0207
Mutual information 0.9417 0.7897 0.7920 0.7617 0.7802 0.5904 0.7759
(b) Disruption (edges 6–7)
Packets sent by source 1490 1436 466 484 1034 569
Packets received by sink 1481 1419 466 476 1027 567
Prediction probability 0.2980 0.2872 0.0932 0.0968 0.2068 0.1138
P-reliance probability 0.9940 0.9882 1.0 0.9835 0.9932 0.9965
Entropy 0.8788 0.8651 0.4471 0.4588 0.7353 0.5113 0.6494
Conditional entropy 0.0159 0.0266 0.0 0.0118 0.0121 0.0038 0.0117
Mutual information 0.8629 0.8384 0.4471 0.4470 0.7232 0.5074 0.6377
(c) Disruption (edges 6–7, 2–3)
Packets sent by source 1471 586 721 909 225 205
Packets received by sink 1435 579 715 897 218 195
Prediction probability 0.2942 0.1172 0.1442 0.1818 0.045 0.041
P-reliance probability 0.9925 0.9881 0.9917 0.9868 0.9689 0.9512
Entropy 0.8741 0.5213 0.5951 0.6840 0.2648 0.2469 0.5310
Conditional entropy 0.0187 0.0110 0.0100 0.0184 0.0090 0.0115 0.0131
Mutual information 0.8554 0.5103 0.5851 0.6656 0.2558 0.2353 0.5179
(d) Disruption (edges 6–7, 2–3, 0–8)
Packets sent by source 1045 966 285 484 230 343
Packets received by sink 1037 964 280 476 222 336
Prediction probability 0.2090 0.1932 0.0570 0.0968 0.0460 0.0686
P-reliance probability 0.9923 0.9979 0.9825 0.9835 0.9652 0.9796
Entropy 0.7396 0.7081 0.3154 0.4588 0.2692 0.3607 0.4753
Conditional entropy 0.0135 0.0041 0.0073 0.0118 0.0100 0.0099 0.0094
Mutual information 0.7260 0.7040 0.3082 0.4470 0.2592 0.3508 0.4659
It is seen from this example that the proposed metrics of entropy, conditional entropy, and mutual information are sensitively dependent upon the change of network traffic pattern. From scenarios of
no disruption to three-edge disruption, the performance of network is reduced gradually. The average values of entropy, conditional entropy, and mutual information also change monotonically.
As the further comparison, the ring network in Fig. 10(a) is modified to Fig. 10(b), where a new node and four edges are inserted. The same four scenarios are simulated in the second ring network,
and the statistics of packets are collected in the same way. The calculated metrics are average entropy (0.8869, 0.7524, 0.7524, and 0.7524), conditional entropy (0.0150, 0.0194, 0.0194, and 0.0194),
and mutual information (0.8719, 0.7331, 0.7331, and 0.7331), respectively, for four scenarios. The metrics between the two examples are compared in Fig. 11. The metrics indicate that model 2 is more
resilient than model 1, which is easy to verify from the topology since model 2 includes more edges and is less susceptible to disruptions.
The simulation studies in this research demonstrated that entropy and mutual information can be applied as the metrics for functionality and performance measures for CPS systems in order to assess
resilience. The proposed probabilistic design framework requires prediction and reliance probabilities as the inputs. These quantities may be derived from historical data or solicitation. Obtaining
reliable and consistent estimations of probabilities is a challenging research issue itself. The studies here mostly focus on communication. More comprehensive investigations are needed for sensing,
reasoning, and prediction capabilities.
At individual node level, several information fusion rules such as best-case, worst-case, and Bayesian can be defined so that the prediction probability associated with a node is updated based on the
received information from neighboring nodes during reasoning. It is seen that the system resilience and robustness are sensitively dependent on the fusion rules. During the system design process,
information aggregation rules also need to be optimized based on the expected dynamics of performance.
The proposed metrics perform reasonably well with the simple reasoning scheme based on the information fusion rules. As future extensions, the proposed performance metrics need to be further tested
with some other information fusion rules. Choosing appropriate rules is expected to be an important task in designing CPS networks and systems.
The sensitivity studies also show that the system performance is influenced more by the tightly coupled nodes, where reliance probabilities are high, than those loosely coupled ones. The optimization
of systems is more effective if efforts are focused on these connections with high reliance probabilities, if the available resource is limited for improvement. Design optimization methods also need
to be further explored based on the preliminary result of sensitivity analysis. The system design and optimization based on the performance and resilience metrics mostly requires a multi-objective
optimization approach, since these metrics provide multifacet assessment. If system dynamics needs to be considered, dynamic programing approaches can also be taken.
Although the proposed metrics and probabilistic measure are in the context of CPS networks, the methodology can potentially be extended for other networked systems where strong interdependency exists
among individual components. Information, energy, and material flows can all be modeled similarly. For instance, in supply chain or transportation networks, prediction probability can correspond to
the probability that goods or supplies satisfy the demand at a node, probability distribution of demand, or the distribution of inventory levels at a node, whereas reliance probabilities characterize
the correlations between demands at different nodes (percentage of supply from one node goes to another), percentage of transport capacities being employed, or probability that transportation is not
interrupted. Different node types (source, sink, warehouse, hub, retailer, etc.) and edge types (shortest path, minimum cut, etc.) can be differentiated with different types of prediction and
reliance probabilities.
In this paper, generic CPS network performance metrics are proposed based on entropy, conditional entropy, and mutual information to allow for quantitative resilience engineering of such networks. In
CPS networks, each node corresponds to a CPS component. The processes of communication during information exchange between nodes and reasoning at individual nodes are characterized with reliance and
prediction probabilities, respectively, in a probabilistic design framework. The resilience of the system then can be quantified with the proposed performance metrics of entropy and mutual
information. Simulation studies show that these metrics are reasonable and consistent quantities to measure how communication and reasoning capabilities are affected during network disruption. The
metrics are shown to be sensitive to the changes of network topology.
Funding Data
• Division of Civil, Mechanical and Manufacturing Innovation, U.S. National Science Foundation (Grant No. CMMI-1663227).
In information theory, Shannon entropy is typically used to measure the amount of uncertainty or how much information a set of possible values, each of which has a corresponding probability, would
contain. For a discrete random variable
, which may have a finite set of possible values
, Shannon entropy is defined as
$H(X)=−∑x∈Xp(X=x)log p(X=x)$
For continuous variable, integral operator is used instead of summation in Eq. (A1)
Conditional entropy defined as
quantifies how much additional information random variable
can provide if the value of random variable
is known.
Mutual information defined as
measures the mutual dependency between random variables
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Go to the source code of this file.
subroutine cpbtrf (UPLO, N, KD, AB, LDAB, INFO)
Function/Subroutine Documentation
subroutine cpbtrf ( character UPLO,
integer N,
integer KD,
complex, dimension( ldab, * ) AB,
integer LDAB,
integer INFO
Download CPBTRF + dependencies
[TGZ] [ZIP] [TXT]
CPBTRF computes the Cholesky factorization of a complex Hermitian
positive definite band matrix A.
The factorization has the form
A = U**H * U, if UPLO = 'U', or
A = L * L**H, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
UPLO is CHARACTER*1
[in] UPLO = 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
[in] N The order of the matrix A. N >= 0.
KD is INTEGER
[in] KD The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB is COMPLEX array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
[in,out] AB if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H*U or A = L*L**H of the band
matrix A, in the same storage format as A.
LDAB is INTEGER
[in] LDAB The leading dimension of the array AB. LDAB >= KD+1.
INFO is INTEGER
= 0: successful exit
[out] INFO < 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
Definition at line 143 of file cpbtrf.f.
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Recursive function missing out the base condition
Calculating the factorial of a number is a classic example of a recursive function.
Missing the Base Condition:
#include <stdio.h>
int factorial(int n)
return n * factorial(n - 1);
int main()
printf("Factorial %d = %d\n", 3, factorial(3));
return 0;
Typical output: Segmentation fault: 11
The problem with this function is it would loop infinitely, causing a segmentation fault — it needs a base condition to stop the recursion.
Base Condition Declared:
#include <stdio.h>
int factorial(int n)
if (n == 1) // Base Condition, very crucial in designing the recursive functions.
return 1;
return n * factorial(n - 1);
int main()
printf("Factorial %d = %d\n", 3, factorial(3));
return 0;
Sample output
This function will terminate as soon as it hits the condition n is equal to 1 (provided the initial value of n is small enough — the upper bound is 12 when int is a 32-bit quantity).
Rules to be followed:
1. Initialize the algorithm. Recursive programs often need a seed value to start with. This is accomplished either by using a parameter passed to the function or by providing a gateway function that
is non-recursive but that sets up the seed values for the recursive calculation.
2. Check to see whether the current value(s) being processed match the base case. If so, process and return the value.
3. Redefine the answer in terms of a smaller or simpler sub-problem or sub-problems.
4. Run the algorithm on the sub-problem.
5. Combine the results in the formulation of the answer.
6. Return the results.
Source: Recursive Function
Found a mistake? Have a question or improvement idea?
Let me know
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Entrance Exam Questions for Class 11 " Science & Non - Science"Entrance Exam Questions for Class 11 " Science & Non - Science"
Entrance Exam Questions for Class 11 " Science & Non - Science"
Hello Everyone!!
After long time I am gonna give you entrance exam questions for class 11th entrance exam questions & Answers of this questions will not uploaded here, you have to solve these questions by yourself.
But still you have some problems on questions then you can simply screenshot that question & post on our FACEBOOK GROUP where students like you who are preparing for the entrance will solve your
Don't go to Class 11 without this...
I highly recommend you to watch this video before going to class 11.
So Let's Start MCQs
1) If n(U) = 125, n(A) = 80, n(B) = 65, what is the greatest value of n(A ∪ B)?
a) 125
b) 80
c) 100
d) 110
2) Which one of the following locus produces a straight line ?
a) Locus of a point which moves at equidistant from a fixed point.
b) Locus of a point which moves at equidistant from two fixed points.
c) Locus of a point which moves at equidistant from three fixed points
d) All of these.
3) In a final step of the calculation, a student accidentally divided by 100 instead of multiplying by 100. What should he do to correct his answer?
a) Multiply by 100
b) Multiply by 10000
c) Divide by 100
d) Multiply by 1000
4) From a bag containing 3 red marbles, 4 black marbles and 5 green marbles, a marble is drawn randomly. Find the probability of getting red or green marble?
a) 114
b) 0.5
c) 2/3
d) 3/4
5) If 10a + 10b = 35, what is the average of a and b?
a) 3.5
b) 7
c) 1.75
d) 10
6) The mean of five numbers is 18. If one number is excluded, their mean is 16. The excluded number is
a) 25
b) 26
c) 27
d) 30
7) In a city 70 % of population travel by car, 50 % travel by bus. If all of them travels by bus or car. Find the percentage travelling by both bus and car.
a) 20 %
b) 50 %
c) 70 %
d) 90 %
8) If £1 = Rs. 140, £0.5 = 3.52 marks, 3.91 marks = $1 and $1 = 4.8 Francs, how many Francs are equal to Rs. 26,000?
a) 1505
b) 1405
c) 1605
d) 1600
9) A cylinder has a diameter of 14 cm and the area of its curved surface is 220 cm2 . The volume of cylinder is
a) 770 cm3
b) 1000 cm3
c) 1540 cm3
10) A solid cylinder with radius of base ‘r’ and height ‘h’ is melted in to identical cones of radius ‘r’ and height ‘h’. Then maximum number of cones so formed is
a) 1
b) 2
c) 3
d) 4
1) Deepa is still unmarried. Do you know _________married?
a) when will she get
b) when will she be
c) when she will get
d) none
2) The boys who ………done this work are intelligent.
a) is
b) has
c) have
d) had
3) ……..his repeated failure, he is still optimistic.
a) In spite
b) Despite
c) Although
d) Even though
4) Bread and butter ………….. our daily need.
a) is
b) are
c) have
d) all
5) . The passive of the sentence - “Don’t let people see you.” is:
a) People should not be let to see you.
b) Don’t let yourself to be seen.
c) Don’t let yourself be seen.
d) You shouldn’t be seen by people
6) Don't shed the crocodile tears. The meaning of the underlined phrase is:
a) express grief sincerely
b) express grief falsely
c) express grief
d) express no grief
7) You’ve done well,…….?
a) don’t you
b) has it you
c) haven’t you
d) do you
8) __________ Rara Lake is popular all over the world.
a) The
b) A
c) An
d) None
9) __________ refers to the comparison of one thing or person with the other without using comparative words.
a) Metaphor
b) Simile
c) Circumlocution
d) Personification
10) The question tag of ‘Do come tomorrow.’ is, _________?
a) will you
b) won’t you
c) willn’t you
d) don’t you
1) Calculate the number of atoms and number of molecules in it.
a) 0.25 mole of oxygen
b) 0.7 gm of nitrogen
c) 5.3gm of Na2CO3
d) 52gm of He
2) Cytochrome are found in
a) bacteria
b) chloroplast
c) mitochondria
d) all of those
3) The SI unit of temperature is
4) The color of Zinc chromate
a) Yellow
b) Red
c) Blue
d) Colorless
5) Lactose is formed by
a) glucose + glucose
b) glucose + fructose
c) glucose + galactose
d) fructose + galactose
6) Acid base reaction is basically
a) Single displacement reaction
b) Addition reaction
c) Decomposition reaction
d) Double displacement reaction
7) Dimension of which are same?
a) Work and moment of force
b) Torque and force
c) Velocity and velocity gradient
d) All
8) The presence of pneumatic bone in birds….
a. To support body
b. To make body light
c. To move
d. To store oxygen
9) If the external force applied is zero, then which of the following is conserved?
a) Linear momentum
b) Angular momentum
c) Torque
d) Time period
10) Two nuclei which are not identical but have the same number of nucleons represent.
a) Isotones
b) Isobars
c) Isotopes
d) None
Comment down all the answers below.👇👇👇👇
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1. Maths
1.a 2.b 3.b 4.c 5.c 6.b 7.a 8.c 9.a 10.c
1.c 2.c 3.b 4.a 5.c 6.b 7.c 8.d 9.a 10.a
1. i think( maths ) 1 ko ans b ho.......
2. thank you bro
3. Thanks a lot dada❤️❤️
Huge thanks for uploading math mcqs💫💫
4. Thanks a lot dada❤️❤️
Thank u so mucha for uploading math mcqs ......plz update all answers soon💫💫😁
5. Thanks a lot dada❤️❤️
Huge thanks for uploading math mcqs💫💫
6. Thnx dada
7. Solution ko vdo banaunu na daju
8. Thank you dai
9. Thanks dai plz upload the question daily as far as possible
10. Thanks dada
11. Awesome dai👌💖.
Keep uploading new MCQs!
12. ...
13. Bro plz post these answers to recheck our answers
Math/ 1-a, 2-b, 3-d, 4-c, 5-c,
6-a, 7-a, 8-c, 9-a, 10-c
En/ 1-b, 2-b, 3-c, 4-a, 5-b, 6-a,
7-c, 8-d, 9-d, 10-a
Sc/1-a, 2-b, 3-b, 4-d, 5-b, 6-c, 7-d, 8-c, 9-c, 10-c
14. Dai answer sahit question dinuhunxa bhane haamro lagi ali badhi help hunxa
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Tick Box 11 Q 23
Here I was confused between circumference and arc. I chose arc, thinking circumference is a measure of a circle similar to area or radius, and arc is a shape such as boundary or point. The correct
answer was circumference. So now I am confused. Can someone please explain why it should be circumference instead of arc? Is there another arc that could be inside the circle?
Every circle has a circumference. And a line drawn between the center of a circle and anywhere on the circumference is the radius
An arc exists between two distinct points on a circles circumference…
thank you!
1 Like
You’re welcome!
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What Is A Function In Excel? | Deskbright
What is a function in Excel?
Microsoft Excel's table structure is a great way to keep organized. But the program is much more than just a series of rows and columns into which you can enter data.
Excel really becomes powerful once you start using functions, which are mathematical formulas that help you quickly and easily make calculations that would be difficult to do by hand.
Functions can do many things to speed up your calculations. For example, you could use functions to take the sum of a row or column of data; find the average of a series of numbers; output the
current date; find the number of orders placed by a particular customer within a given period of time; look up the e-mail address of a customer based on his or her name; and much more. It's all
automatic — no manual entry required.
Let's take a closer look at functions to see how they work.
The structure of a function
Think of a function like a recipe: you put together a series of ingredients, and the recipe spits out something totally new and different (and, in most cases, more useful, or delicious, than the
thing that you put in). Like recipes, functions have three key pieces that you should keep track of:
• First, there's the function name. This is just like the recipe name that you would see at the top of one of the pages of your cookbook. It is a unique identifier that tells Excel what we are
trying to cook up.
• Next, there are arguments. Arguments are just like the ingredients of a recipe. They're pieces that you put in that will eventually combine to make something bigger.
• Finally, there's the output. The output of a function is just like the output of a recipe — it's the final product that is ready to be presented to the user (or, in the case of a recipe, eaten).
Writing a function in Excel
When we enter functions into Excel, we use a special character to tell the program that what we are entering is a function and not a normal block of text: the equals sign (=). Whenever Excel sees the
= sign at the beginning of the input to a cell, it recognizes that we are about to feed it a function.
The basic structure of a function is as follows:
=FUNCTION_NAME(argument_1, argument_2, argument_3...)
Output: Output
After the = sign, we write the name of the function to tell Excel which recipe we want to use. Then, we use the open parentheses sign (() to tell the program that we're about to give it a list of
We then list the arguments to the function, one by one, separated by commas, to tell Excel what ingredients we are using. Note that just like recipes, each function has its own specific number of
arguments that it needs to receive. Some just take one argument; others take two or even more.
To finish writing a function, wrap up the list of arguments with the close parentheses sign ()) to tell Excel that you're done writing the list of ingredients. Then press the Enter key to complete
your entry. You'll see that rather than displaying the text that you entered, Excel shows the output of your completed function.
A practical example
Let's look at a practical example using the SUM function. This is one of Excel's most-used recipes — it takes any number of arguments (all of which should be numerical), and spits out the sum of
those arguments. The formula for SUM is as follows:
=SUM(number_1, number_2...)
To recap: the name of the function is SUM. The arguments are number_1, number_2, and as many additional numbers as you want to put in (this particular function takes an unlimited number of arguments,
just like a recipe that gets better and better as you throw in more ingredients). When you're done writing the function and press Enter, Excel will show you the output.
Try entering the following into a cell on a blank spreadsheet:
=SUM(3, 7)
Output: 10
Excel outputs 10, because the SUM of 3 and 7 is 10.
Here's another example with even more arguments:
=SUM(1, 2, 3, 4, 5)
Output: 15
Here, Excel outputs 15, the SUM of 1, 2, 3, 4, and 5.
Infinite arguments and optional arguments
Throughout these pages, we'll use a couple different types of notation to denote special cases within functions:
First, some functions, like SUM above, have a theoretically infinite number of arguments. For example, you can take the SUM of an infinite number of numerals. In cases like this, we use three dots
(...) to denote an infinite number of additional arguments, like so:
=FUNCTION_NAME(argument_1, argument_2, argument_3...)
If a function has a finite number of arguments, you won't see that ... at the end, like this:
=FUNCTION_NAME(argument_1, argument_2)
Finally, some arguments to functions are optional, just like some ingredients of a recipe might be optional. If one or more arguments of a function are optional, we'll follow them up with an
(optional) designator like so:
=FUNCTION_NAME(argument_1, argument_2 (optional))
In most of our function tutorials, we'll explain why something is optional and how you can use it.
That's it! Now that you know what a function is, check out our tutorials on some of Excel's logical functions to get rolling with Excel's most powerful tool.
Explore the 5 must-learn 'fundamentals' of Excel
Getting started with Excel is easy. Sign up for our 5-day mini-course to receive easy-to-follow lessons on using basic spreadsheets.
• The basics of rows, columns, and cells...
• How to sort and filter data like a pro...
• Plus, we'll reveal why formulas and cell references are so important and how to use them...
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Math TEKS Implementation Update - Nov. 2015 —K-12 Mathematics Resources—Cosenza & Associates, LLC
The Texas State Board of Education’s Committee on Instruction met Thursday, November 19. One of their agenda items was an update on the implementation of the revised math TEKS. Hence, I was invited
to testify about the mathematical process standards since that seemed to be the hot point in today’s hearing.
There were a few parents testifying with concerns about how the mathematical process standards seem to be interfering with the math that they feel their kids should be learning. One father expressed
frustration with his child having to learn multiple strategies to perform basic computations with whole numbers, rather than “just multiplying the numbers” and moving on. Another parent was
passionately concerned that her daughter suddenly didn’t like math anymore. I admire these parents’ courage for standing up for their children’s interests and feel proud that we have a governmental
system in place that provides for parental input into educational decision-making. But if we peel back a few layers on this particular onion, we find out that there are some issues lying just beneath
the surface that may be contributing to these observations.
What are mathematical processes?
We do know that mathematical processes are important. The National Council of Teachers of Mathematics (NCTM) published the Curriculum and Evaluation Standards in 1989 and first outlined important
content and processes that every child should learn in math. The process standards that NCTM outlined were:
• Problem Solving
• Reasoning
• Representation
• Communication
• Connections
Some of the descendants of these five processes reside in both the original 1998 math TEKS and the revised 2012 math TEKS. These research-based mathematical processes help students make sense of
important mathematics and help bring the strands of mathematical proficiency together. These are the thought processes that teachers can harness to help all of their students learn meaningful math.
What do they look like in daily instruction?
I was particularly moved by another parent’s testimony. He went right to the heart of the situation when he suggested, as many mathematics specialists have, that we use the visual models and
conceptual pieces to lay the foundation and then bridge to the standard algorithm. To see what this means, let’s unpack the mathematical process standard 3.1E: create and use representations to
organize, record, and communicate mathematical ideas, alongside content standard 3.4G: use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit
number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties. To do that, let’s look at the multiplication problem 13 × 5 = 65.
You can use base-ten blocks to create a concrete representation of the problem. A ten-block, or rod (long) as it is called in the classroom, times 5 unit blocks generates 5 ten-blocks (rods or
longs). 3 unit blocks times 5 unit blocks generates 15 unit blocks. These are grouped into 50 and 15, then combined into the product of 65.
If you were to multiply 13 by 5 using a place-value based algorithm, you would line up the ones places and first multiply 5 ones by 3 ones. This generates the partial product of 15, which is the same
partial product generated in the block model by multiplying 3 unit blocks by 5 unit blocks.
Likewise, you would next multiply 5 ones by 1 ten, generating the partial product of 50. This partial product is the same as the one generated in the block model by multiplying 1 ten-block (rod or
long) by 5 unit blocks.
Notice how nicely the partial products approach segues to the standard algorithm for multiplying whole numbers. Multiply 5 ones by 3 ones, which gives you 15 ones. But you only have room for one
digit, so you regroup 15 ones into 1 ten and 5 ones, record the 5 ones and carry the 1 ten to the tens column.
Next, you multiply 5 ones by 1 ten, which gives you 5 tens. Add back the 1 ten you carried over to get a total of 6 tens and you record 6 in the tens place of the product.
Yes, it is easier just to multiply 13 by 5 using the standard algorithm. That’s why we have the standard algorithm, which is much more efficient than using base-ten blocks or partial products. But
each of these strategies helps students understand why the standard algorithm works. Using base-ten blocks provides students with a visual model of what it means to multiply 13 by 5. The model also
shows students why the product appears in two parts: 15 and 50. The partial products help explain why we “carry the one.” These two strategies – base-ten blocks and partial products – lay the
conceptual foundation for the important skill of using the standard algorithm to multiply numbers.
Without that conceptual understanding, anchored in place value, students get confused in 4th and 5th grade when they multiply 3-digit and 2-digit whole numbers and begin multiplying decimals. They’ll
“carry the one” anywhere, dropping it where it feels like it fits. Investing time during 3rd grade to provide a solid conceptual foundation pays off in 4th and 5th grade because the student is more
likely to extend what they know about multiplication to different types of numbers.
So what about November 19’s math TEKS hearing?
Timely question! We spent most of the math agenda item (#4 if you’re listening to the archived audio on the TEA website) discussing whether or not the mathematical process standards are helpful or
harmful. When district-level curriculum frameworks and classroom-level instruction are wisely planned, the mathematical process standards are extraordinarily beneficial. By wisely planned, I mean
that the teacher selects one process standard to showcase in the lesson, knowing that she is likely to address several mathematical processes in the flow of instruction. When planning with her
grade-level team, they collaboratively select mathematical tasks that emphasize the showcase process standard while maintaining instructional focus on the content standard.
If a curriculum framework or instructional decision creates a situation such that every process standard should be applied to every portion (breakout) of a content standard, then the teacher gets
lost in the weeds during instruction. That is not beneficial to students. I can also tell you as a member of a TEKS review/writing committee that this is certainly not the intent of the content/
process standard pairing. Mathematical processes should be taught alongside the content standards to help students learn the full richness of mathematics. They become part of a student’s toolkit to
think analytically and mathematically. Then, when a student encounters a problem they haven’t seen before, then they know how to wrap their heads around the problem and begin to look for a solution.
That can only benefit students.
What’s the bottom line?
The mathematical process standards, either as they are used in instruction or assessment, should enhance student learning, not become a barrier to it. As educators, we should resist the temptation to
let our fear of assessment lead us to irrational instructional decision-making. Process standards such as the use of multiple representations and creating representations to communicate mathematical
ideas provide the teacher with guidance to select powerful mathematical tasks for her students. The mathematical process standards in the revised math TEKS are not new; indeed, they have been in the
K-8 TEKS since 1998 in the “Underlying Tools and Processes” strand. As teachers, let’s use the mathematical process standards to our instructional advantage and our students’ benefit.
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TwoDucks isn't necessarily uncomputable (I think)
If a time expression is encountered which alters something in an earlier time, then the interpreter could then rewind the program back to that time, change whatever is being altered, then
fast-forward to where it was.
If this were to cause a paradox (i.e. stop the alteration of the past, or to change it), this could (although it probably won't be easy) be detected, and an error message displayed. "Paradox!" or
something like that.
The main problem is I/O:
• Input could just be saved and re-entered when fast-forwarding. Or a new input prompt could be created.
• Output could be removed from the display when rewinding, and reprinted (with whatever changes) on fast-forward. Or the new output could just be displayed after the old.
Modifying stuff in the future is also possible - just save the potential alteration, and when the interpreter gets to the correct time, silently replace the value.
If, as suggested, the user stops the program if told that it does not halt, that could lead to the program saying that it doesn't halt because it was stopped, even if it would halt after a while.
--(this comment by 109.148.239.1 at 14:18, 9 October 2011 UTC; please sign your comments with ~~~~)
This way of doing things does NOT allow you to solve the halting problem -- the test program will have to be run to completion before the assignment occurs -- there is no way to always tell if will
terminate, until it does.
Alksentrs 21:53, 24 December 2007 (UTC)
• Actually, it does solve the halting problem, because it actually travels back in time. the problem is, it's impossible to implement (at least not on a turing machine, and maybe even physically
impossible). if you tried implementing it, you're right that the program would have to run to completion first. anyway, awesome idea for a language.Afarnen 23:49, 27 February 2008 (UTC)
Couldn't the program be rewritten so that it works without time travel? When someone writes the actual code in the first place, he knows what should ultimately happen. We can tell that the second
example should input then output, so why give the interpreter that power? Paradoxes could be handled specially. --Loki 20:48, 28 September 2009 (UTC)
As has been pointed out, it's possible to simulate the time travel by just caching sends to the future and applying the change when the time is right, and dealing with sends to the past by keeping
track of the program state at each prior time and starting again from the state at the time the variable was sent to.
Alternately, you could consider that at each point in the the program execution, it's possible that any variable will change its value due to a send executed some time in the future. Now suppose that
that the program can correctly guess whether a send will modify a variable at the current time and what it will be modified to, and then continue to execute as though that change had happened
already. Then when/if the send is actually reached, it's a no-op.
These two ways of looking at the execution are almost identical to the two most common ways of thinking about how a nondeterministic Turing Machine works. And it's been shown that a NDTM is in the
same computational class as a DTM; only the complexity classes they define are different. The important thing is that the simulator arrives at the same result as the real time-travelling interpreter
would, not how it does it.
Finally note that the halting oracle example is only guaranteed to work if the code being tested does not use a SEND. The oracle itself does use a send however, and so it's not possible to determine
if the oracle itself halts by feeding it as input to itself, and that's the sort of situation that makes the halting problem impossible to solve in the general case. You can make halting oracles that
work in most cases on a regular Turing Machine, too. TwoDucks is computable and Turing-equivalent.
130.108.16.157 02:08, 27 February 2012 (UTC)
TwoDucks less SEND is still Turing-complete; therefore, the halting oracle is capable of determining whether any Turing machine halts. Thus, TwoDucks is uncomputable. Phantom Hoover 17:20, 27
February 2012 (UTC)
For this proof, I'm assuming that the thread scheduler and threaded programs are implemented in such a way that an arbitrary TwoDucks program will return the same result each time (or fail to
halt each time). If it doesn't, it's not even a function, much less a computable one and is thus not relevant here. (It's entirely possible to write inconsistent programs in many other Turing
Complete languages once hardware is taken into account, so I'm asserting that this is not of interest to the computability of the language itself.)
Now take a threaded program. For every variable that is changed by a given thread, append "_[i]" to the end of the variable name, where i is the number of the thread. Now take all
instructions from all threads and merge them into a single thread in the order in which they are run according to one of the consistent schedules. We now have a single threaded program that
computes the same result as the multithreaded one. For the rest of the proof, I'm going to work with single threaded programs only; multithreaded programs can be transformed to such first as
Let F be the set of all possible TwoDucks programs, and let f be an arbitrary member of this set. When I say SEND, I'm referring to SENDs to the past. I don't think anyone's arguing that
SENDs to the future add anything to the computational model. I'll amend the proof to include those if asked, but I'd rather not take the time if I don't have to.
Let S(f, x, t, T) be the program state of f with starting input x, as it exists at time t after T SENDs to the past have already been executed. S(...) = ( pc, v0, v1, ..., vm ) where pc is
the program counter and vi is the value of variable i under some enumeration of the variables existing at that point of the program. Iff pc is after the last instruction or is on an END
instruction, the program is in a halting state. Let G:S->S be the state transition function.
The program begins at S(f, x, 0, 0). Either it eventually hits a SEND or it doesn't. If it does, it does so at some fixed time t=i, simply because if it takes an infinite time to get there
than the program is equivalent to a nonhalting SENDless program. If pc isn't a SEND q to j, then G(S(f, x, t, T)) = S(f, x, t+1, T). If pc is a send to time j, then G(S(f, x, t, T)) = S(f, x,
j, T+1) and is the same as setting pc and the v vector to the value at S(f, x, j, T), then executing a SET q to (the value of q at time t). Since the SEND was executed after a finite number
of steps and there are a finite number of variables, all this happens in a finite amount of time. Because of this equivalence, we can take a TwoDucks program and replace each SEND with a
sequence of SETs and a GOTO. We now have a sendless TwoDucks program that does the same thing as the original one. (plus or minus output, but that can be handled by erasing previous output
and replacing it with new values.)
Here we have the transformed version of the halting oracle:
SET halts TO 0
SET mark TO 1
NOUT halts
// Program goes here
SET halts TO 1
(erase last printed character)
NOUT halts
// Program goes here
SET halts TO 1
If the program halts, it'll print 0, then eventually erase that and print 1. If the program doesn't halt, it'll print 0 and loop forever. So at any given point before it replaces the 0 with a
1, you can't tell whether it's going to halt or not, and since nonhalting programs never halt, you'll never be able to say for sure whether that 1 will eventually show up. So this doesn't
solve the halting problem.
Now let's look at the original form:
SET halts TO 0
SET mark TO 1
NOUT halts
// Program goes here
SET halts TO 1
SEND halts TO 1~0~{mark}=1
If the program halts, it'll print 0, then eventually a SEND will be sent back in time and 1 will be printed instead. If it doesn't halt, it'll print 0 and loop forever. If you see a 0, you
have no way of telling whether you're in the original history and at some time in the future someone will go back in time and change that value, or if you're seeing a 0 because the program
will loop forever and it'll never change. If you see a 1, you know it halts (and that time-travel has happened.) You're in essentially the same situation you were in in the sendless version;
if you see a 0, you don't know for sure that it'll *stay* 0. The only difference is that if it does become 1, you'll remember it having been 0 in the sendless version but in the time-travel
version you'll have only seen the 1.
It does seem like the answer may be dependent on what model of time-travel you're assuming. For the second time dimension or multiverse/many-worlds models, this proof holds (with a few
modifications in the second case), but you could potentially come up with other self-consistent models under which either a different proof is needed or the language is genuinely
uncomputable. If you have such a model, please post it.
130.108.16.157 19:05, 27 February 2012 (UTC)
"If the program halts, it'll print 0, then eventually a SEND will be sent back in time and 1 will be printed instead."
No — if the program halts, then it'll have printed 1 all along. That's the point of time travel: it changes it as it happens, not later on. —ehird 19:45, 27 February 2012 (UTC)
Only if you assume a single invariant timeline. If you use a model of time travel with a changing timeline, then there exists an original history from which the time traveller
originally came, so to speak. In that history, the value is 0, whether or not the program halts. The question then becomes one of whether you're in the original timeline or not. So
whether the program solves the halting problem is dependent on the model of time travel you're using, as I stated above. 130.108.16.157 14:56, 2 March 2012 (UTC)
Quantum Superposition
It's a bit unsatisfying to say a program that creates a paradox (such as the first example given) "doesn't work." In fact, such a situation is equivalent to a quantum superposition of states of a
variable. So, the output of the example program should be all(0,1). That is, a quantum superposition where the value of the variable is equal parts 0 and 1. See what Damian Conway has to say on the
Do you have a transcript of the video, please? --Zzo38 03:36, 6 October 2008 (UTC)
No. The thing's an hour long. --Quintopia 05:15, 9 October 2008 (UTC)
I solved the paradoxes in my similar language timefuck by using the many-worlds hypothesis.--alternative678 01:39, 11 December 2009 (UTC)
Time Expressions Ambiguity
"a~b~c b'th time since a is true that expression c is true (b=0 is first (right after it happens), b=1 second time, etc.) This can either refer to a time in the past or a time in the future." This
could be interpreted two ways. Either it's looking for the time at which c has been true for b ticks, or it's looking for the "b"th time that c has *become* true.
0. SET x TO 0
1. SET x TO 1
2. SET y TO 2
3. SET x TO 0
4. SET x TO 1
5. SET y TO 1
6. SET t TO 1~1~{x}=1
Is t equal to 2 or 4? -130.108.16.157 20:25, 21 February 2012 (UTC)
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Azimuth - Alchetron, The Free Social Encyclopedia
An azimuth (/ˈæzɪməθ/) (from Arabic "السَّمْت" as-sumūt, meaning "the directions") is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest
is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
An example is the position of a star in the sky. The star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is
the angle between the north vector and the perpendicular projection of the star down onto the horizon.
Azimuth is usually measured in degrees (°). The concept is used in navigation, astronomy, engineering, mapping, mining and artillery.
In land navigation, azimuth is usually denoted alpha, α, and defined as a horizontal angle measured clockwise from a north base line or meridian. Azimuth has also been more generally defined as a
horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.
Today, the reference plane for an azimuth is typically true north, measured as a 0° azimuth, though other angular units (grad, mil) can be used. Moving clockwise on a 360 degree circle, east has
azimuth 90°, south 180°, and west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is clearly defined.
Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, and the angle may be measured clockwise or anticlockwise from the zero. For
example, a bearing might be described as "(from) south, (turn) thirty degrees (toward the) east" (the words in brackets are usually omitted), abbreviated "S30°E", which is the bearing 30 degrees in
the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the turning direction, stated last, is east
or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the cardinal
points, a different notation, e.g. "due east", is used instead.
Cartographical azimuth
The cartographical azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):
α = 180 π atan2 ( X 2 − X 1 , Y 2 − Y 1 )
Remark that the reference axes are swapped relative to the (counterclockwise) mathematical polar coordinate system and that the azimuth is clockwise relative to the north. This is the reason why the
X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°.
The formula in radians would be slightly easier:
α = atan2 ( X 2 − X 1 , Y 2 − Y 1 )
Calculating coordinates
When the coordinates (X[1], Y[1]) of one point, the distance L, and the azimuth α to another point (X[2], Y[2]) are known, one can calculate its coordinates:
X 2 = X 1 + L sin α Y 2 = Y 1 + L cos α
This is typically used in triangulation.
Calculating azimuth
We are standing at latitude φ[1], longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude φ[2], longitude L (positive eastward). We can get a fair approximation by
assuming the Earth is a sphere, in which case the azimuth α is given by
tan α = sin L cos φ 1 tan φ 2 − sin φ 1 cos L
A better approximation assumes the Earth is a slightly-squashed sphere (an oblate spheroid); azimuth then has at least two very slightly different meanings. Normal-section azimuth is the angle
measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; geodetic azimuth is the angle between north and the geodesic; that is, the shortest path on the
surface of the spheroid from our viewpoint to Point 2. The difference is usually immeasurably small; if Point 2 is not more than 100 km away, the difference will not exceed 0.03 arc second.
Various websites will calculate geodetic azimuth; e.g., GeoScience Australia site. Formulas for calculating geodetic azimuth are linked in the distance article.
Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance. If f is the flattening for the chosen spheroid (e.g., ^1⁄[7002298257223563000♠298.257223
563] for WGS84) then
e 2 = f ( 2 − f ) 1 − e 2 = ( 1 − f ) 2 Λ = ( 1 − e 2 ) tan φ 2 tan φ 1 + e 2 1 + ( 1 − e 2 ) ( tan φ 2 ) 2 1 + ( 1 − e 2 ) ( tan φ 1 ) 2 tan α = sin L ( Λ − cos L ) sin φ 1
If φ[1] = 0 then
tan α = sin L ( 1 − e 2 ) tan φ 2
To calculate the azimuth of the sun or a star given its declination and hour angle at our location, we modify the formula for a spherical earth. Replace φ[2] with declination and longitude difference
with hour angle, and change the sign (since the hour angle is positive westward instead of east).
There is a wide variety of azimuthal map projections. They all have the property that directions (the azimuths) from a central point are preserved. Some navigation systems use south as the reference
plane. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.
Used in celestial navigation, an azimuth is the direction of a celestial body from the observer. In astronomy, an azimuth is sometimes referred to as a bearing. In modern astronomy azimuth is nearly
always measured from the north. (The article on coordinate systems, for example, uses a convention measuring from the south.) In former times, it was common to refer to azimuth from the south, as it
was then zero at the same time that the hour angle of a star was zero. This assumes, however, that the star (upper) culminates in the south, which is only true if the star's declination is less than
(i.e. further south than) the observer's latitude.
Right ascension
If, instead of measuring from and along the horizon, the angles are measured from and along the celestial equator, the angles are called right ascension if referenced to the Vernal Equinox, or hour
angle if referenced to the celestial meridian.
Horizontal coordinate
In the horizontal coordinate system, used in celestial navigation and satellite dish installation, azimuth is one of the two coordinates. The other is altitude, sometimes called elevation above the
horizon. See also: Sat finder.
Polar coordinate
In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy
-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the xy-plane and is normally measured in radians rather than degrees. As well as measuring the angle
differently, in mathematical applications theta, θ, is very often used to represent the azimuth rather than the representation of symbol phi φ.
Other uses of the word
For magnetic tape drives, azimuth refers to the angle between the tape head(s) and tape.
In sound localization experiments and literature, the azimuth refers to the angle the sound source makes compared to the imaginary straight line that is drawn from within the head through the area
between the eyes.
An azimuth thruster in shipbuilding is a propeller that can be rotated horizontally.
Etymology of the word
The word azimuth is in all European languages today. It originates from medieval Arabic al-sumūt, pronounced as-sumūt in Arabic, meaning "the directions" (plural of Arabic al-samt = "the direction").
The Arabic word entered late medieval Latin in an astronomy context and in particular in the use of the Arabic version of the Astrolabe astronomy instrument. The word's first record in English is in
the 1390s in Treatise on the Astrolabe by Geoffrey Chaucer. The first known record in any Western language is in Spanish in the 1270s in an astronomy book that was largely derived from Arabic
sources, the Libros del saber de astronomía commissioned by King Alfonso X of Castile.
Azimuth Wikipedia
(Text) CC BY-SA
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Aggregate Multidimensional Raster (Image Analyst)
Label Explanation Data Type
Input Raster Dataset; Raster Layer; Mosaic
Multidimensional The input multidimensional raster dataset. Dataset; Mosaic Layer; Image
Raster Service; File
Dimension The aggregation dimension. This is the dimension along which the variables will be aggregated. String
Specifies the mathematical method that will be used to combine the aggregated slices in an interval.
When the method is set to Custom, the Aggregation Function parameter becomes active.
• Mean—The mean of a pixel's values will be calculated across all slices in the interval. This is the default.
• Maximum—The maximum value of a pixel will be calculated across all slices in the interval.
• Majority—The pixel value that occurred most frequently will be calculated across all slices in the interval.
Aggregation Method • Minimum—The minimum value of a pixel will be calculated across all slices in the interval.
• Minority—The pixel value that occurred least frequently will be calculated across all slices in the interval. String
(Optional) • Median—The median value of a pixel will be calculated across all slices in the interval.
• Percentile—The percentile of values for a pixel will be calculated across all slices in the interval. The 90th percentile is calculated by
default. You can specify other values (from 0 to 100) using the Percentile value parameter.
• Range—The range of values for a pixel will be calculated across all slices in the interval.
• Standard Deviation—The standard deviation of a pixel's values will be calculated across all slices in the interval.
• Sum—The sum of a pixel's values will be calculated across all slices in the interval.
• Variety—The number of unique pixel values will be calculated across all slices in the interval.
• Custom—The pixel value will be calculated based on a custom raster function.
Variables The variable or variables that will be aggregated along the given dimension. If no variable is specified, all variables with the selected
[Dimension Info] dimension will be aggregated.
For example, to aggregate daily temperature data into monthly average values, specify temperature as the variable to be aggregated. If you do String
(Optional) not specify any variables and you have both daily temperature and daily precipitation variables, both variables will be aggregated into monthly
averages and the output multidimensional raster will include both variables.
Specifies the dimension interval for which the data will be aggregated.
Definition • All—The data values will be aggregated across all slices. This is the default.
• Interval Keyword—The variable data will be aggregated using a commonly known interval. String
(Optional) • Interval Value—The variable data will be aggregated using a user-specified interval and unit.
• Interval Ranges—The variable data will be aggregated between specified pairs of values or dates.
Specifies the keyword interval that will be used when aggregating along the dimension. This parameter is required when the Aggregation
Definition parameter is set to Interval Keyword and the aggregation must be across time.
• Hourly—The data values will be aggregated into hourly time steps, and the result will include every hour in the time series.
• Daily—The data values will be aggregated into daily time steps, and the result will include every day in the time series.
• Weekly—The data values will be aggregated into weekly time steps, and the result will include every week in the time series.
• Dekadly—The data values will be aggregated into 3 periods of 10 days each. The last period can contain more or fewer than 10 days. The
output will include 3 slices for each month.
• Pentadly—The data values will be aggregated into 6 periods of 5 days each. The last period can contain more or fewer than 5 days. The
Keyword Interval output will include 6 slices for each month.
• Monthly—The data values will be aggregated into monthly time steps, and the result will include every month in the time series. String
(Optional) • Quarterly—The data values will be aggregated into quarterly time steps, and the result will include every quarter in the time series.
• Yearly—The data values will be aggregated into yearly time steps, and the result will include every year in the time series.
• Recurring daily—The data values will be aggregated into daily time steps, and the result will include one aggregated value per Julian day.
The output will include, at most, 366 daily time slices.
• Recurring weekly—The data values will be aggregated into weekly time steps, and the result will include one aggregated value per week. The
output will include, at most, 53 weekly time slices.
• Recurring monthly—The data values will be aggregated into monthly time steps, and the result will include one aggregated value per month.
The output will include, at most, 12 monthly time slices.
• Recurring quarterly—The data values will be aggregated into quarterly time steps, and the result will include one aggregated value per
quarter. The output will include, at most, 4 quarterly time slices.
The size of the interval that will be used for the aggregation. This parameter is required when the Aggregation Definition parameter is set to
Value Interval Interval Value.
(Optional) For example, to aggregate 30 years of monthly temperature data into 5-year increments, enter 5 as the Value Interval, and specify Unit as Years
The unit that will be used for the Value Interval parameter. This parameter is required when the Dimension parameter is a time field and the
Aggregation Definition parameter is set to Interval Value.
If you are aggregating anything other than time, this option will not be available and the unit for the interval value will match the variable
Unit unit of the input multidimensional raster data.
(Optional) • Hours—The data values will be aggregated into hourly time slices at the interval provided.
• Days—The data values will be aggregated into daily time slices at the interval provided.
• Weeks—The data values will be aggregated into weekly time slices at the interval provided.
• Months—The data values will be aggregated into monthly time slices at the interval provided.
• Years—The data values will be aggregated into yearly time slices at the interval provided.
Range Interval ranges specified in a value table will be used to aggregate groups of values. The value table consists of pairs of minimum and maximum
range values, with data type Double or Date.
(Optional) Value Table
This parameter is required when the Aggregation Definition parameter is set to Interval Ranges.
Aggregation A custom raster function that will be used to compute the pixel values of the aggregated rasters. The input is a raster function JSON object or
Function an .rft.xml file created from a function chain or a custom Python raster function.
File; String
(Optional) This parameter is required when the Aggregation Method parameter is set to Custom.
Ignore NoData Specifies whether NoData values will be ignored in the analysis.
(Optional) • Checked—The analysis will include all valid pixels along a given dimension and ignore NoData pixels. This is the default. Boolean
• Unchecked—The analysis will result in NoData if there are NoData values for the pixels along the given dimension.
Dimensionless Specifies whether the layer will have dimension values. This parameter is only active if a single slice is selected to create a layer.
(Optional) • Checked—The layer will not have dimension values. Boolean
• Unchecked—The layer will have dimension values. This is the default.
The percentile to calculate. The default is 90, indicating the 90th percentile.
Percentile value
The values can range from 0 to 100. The 0th percentile is essentially equivalent to the minimum statistic, and the 100th percentile is
(Optional) equivalent to maximum. A value of 50 will produce essentially the same result as the median statistic. Double
This parameter is only available if the Statistics type parameter is set to Percentile.
Specifies the method of percentile interpolation that will be used when there is an even number of values from the input raster to be
Percentile calculated.
interpolation type
• Nearest—The nearest available value to the desired percentile will be used. In this case, the output pixel type will be the same as that of String
(Optional) the input value raster.
• Linear—The weighted average of the two surrounding values from the desired percentile will be used. In this case, the output pixel type
will be floating point.
Return Value
Label Explanation Data Type
Output Multidimensional Raster The output Cloud Raster Format (CRF) multidimensional raster dataset. Raster
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1(4). Find a quadratic polynomial whose sum and
product of the zeroes are – respectively. Also
find the zeroes of the polynomial by factorisation.
1 thought on “1(4). Find a quadratic polynomial whose sum and<br />-3<br />product of the zeroes are – respectively. Also<br />275?<br />find th”
1. Step-by-step explanation:
write the question clearly and correctly get answer immediately
Leave a Comment
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7. A million dots
A million dots are laid down at random in a circle.
Prove that a straight line can be drawn across the circle in such a way that half of the dots lie on one side of the line, while the other half lie on the other side.
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The hardness of the closest vector problem with
We give a new simple proof of the NP-hardness of the closest vector problem. In addition to being much simpler than all previously known proofs, the new proof yields new interesting results about the
complexity of the closest vector problem with preprocessing. This is a variant of the closest vector problem in which the lattice is specified in advance, and can be preprocessed for an arbitrarily
long amount of time before the target vector is revealed. We show that there are lattices for which the closest vector problem remains hard, regardless of the amount of preprocessing.
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What does the population growth model equation mean? dN/dt=rN | Socratic
What does the population growth model equation mean? dN/dt=rN
1 Answer
The equation $\frac{\mathrm{dN}}{\mathrm{dt}} = r N$means that rate change of the population is proportional to the size of the population, where r is the proportionality constant.
This is a rather simple and impractical equation because it signifies an Exponential Population Growth. If you are familiar to the Future Value of a compounded interest rate, $F V = P V {\left(1 + r\
dN/dt = rN : a differential equation describing the population growth
where N is the population size, r is the growth rate, and t is time.
$N \left(t\right) = {N}_{0} {e}^{r t}$ : the solution of the differential equation for exponential growth. The equation grows exponential and you know population does not grow exponentially, as a
result we have have a more reasonable model called "The Logistic Equation". The Logistic model sets limit to the growth. Why? Well a control space like a nation, a savanna, or the plane carry a
finite amount of resources and cannot support exponential populations growth in perpetuity.
$\left(\frac{\mathrm{dN}}{\mathrm{dt}}\right) = r N \left(1 - \frac{N}{K}\right)$ : The logistic differential equation, has
N as the population size, r is growth rate, K is carrying capacity.
This equation forces, populations to converge to the carrying capacity. The speed at which the populations approach K is related to the growth rate r.
Impact of this question
64647 views around the world
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We are happy to present Henry Sinclair-Banks’s talk titled “The Complexity of Coverability in Fixed Dimension VASS with Various Encodings “.
This presentation is about the coverability problem for Vector Addition Systems with States (VASS). We have lately shown that coverability in two-dimensional VASS with one binary encoded counter and
one unary encoded counter is in NP. For contrast, coverability in two-dimensional VASS is PSPACE-complete with two binary encoded counters and NL-complete with two unary encoded counters.
Our NP upper bound is achieved using new techniques and one of these techniques is shown in this presentation. Coverability in any fixed dimension unary VASS, that is when the counter updates are
encoded in unary, is long-known to be NL-complete. In this variation of the problem, the initial and target counter values are also encoded in unary, this turns out to be of great importance.
We have recently found that if the initial and target counter values are instead encoded in binary, then coverability in four-dimensional unary VASS is NP-hard and coverability in eight-dimensional
unary VASS is PSPACE-hard. These lower bounds are corollaries of recent results of the hardness of reachability in fixed dimension unary VASS, and this presentation will feature the technique used in
the reductions to coverability.
Henry Sinclair-Banks is a PhD student at the University of Warwick under the supervision of Dr. Dmitry Chistikov and Dr. Marcin Jurdzinski. Within theoretical computer science, the main themes of his
research are automata, complexity, and logic. More specifically, the questions he most enjoys are all about automata with counters such as vector addition systems with states and one-counter nets.
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