International Schools provide students with linguistic skills, higher-order thinking skills, and a global perspective that will help them to contribute to, and succeed in, a 21st century world. International Schools help students engage in intellectually rigorous schoolwork and prepare them for college, career and life. International Schools help close both the academic achievement gap as well as the “global achievement gap”.
Common Core State Standards (CCSS)
The CCSS are academic standards that have been develop nationally and adopted by most states. They represent specific academic skills in reading, writing, communication and mathematics that students should learn at each grade level.
At Concord International we use multiple measures to assess student performance in school. We believe a single test should never be the sole judge of a student’s academic skills.Some measures we use are teacher observation, class work, portfolios, running records and standardized and classroom based assessments. The state and district also require testing and an explanation of those tests appear below.
Smarter Balanced (SBAC)
The SBAC is the state’s exam for students in grades 3-5. Its purpose is to assess a student’s progress in certain subject areas. Third, fourth, and fifth graders are tested in math and English Language Arts. Fifth graders are tested in science, still using the MSP. The assessment window is from Mid-March to early June. We will be sending out a schedule as it gets closer.
Measures of Academic Performance (MAP)
The MAP is an online test given to all students in grades K-5. Its purpose is to assess a student’s academic growth in reading and math. The test is given three times a year: fall, winter, late spring. Progress reports for each student are generated and are available on the Source. Scores from the MAP test are also included in the student’s report card and available at Parent-Teacher conferences.
WA Kids – Kindergarten
The WA kids assessment is started during Kindergarten Jumpstart and completed throughout September and October by the kindergarten teachers. Students are assessed in a variety of areas: social/emotional, physical, cognitive, language, literacy and mathematics. This information helps teachers tailor their instruction to meet the individual needs of the students in their classes.
Fountas and Pinnell Reading Assessment
This reading assessment is a one on one assessment given to students to assess their reading and comprehension level. It allows teachers to individualize instruction for students. This is done several times a year to monitor student growth.


delay differential equation - step size problem
I do have a problem with the following one-dimensional delay differential equation (dde):
(1) x'(t) = f(t,x(t-1)) := M * [ -x(t) - 2.35 * sin(x(t-1)) ] , t>0
with the initial condition
(I) x(t) = 2, -1 <= t <= 0.
If M is big, e.g. M = 100, this problem becomes stiff, as you can see in the plot of the attached *.jpg picture.
For this plot I solved (1) with a collocation method of order 4 with constant step size h = 0.001. As you can imagine, this is highly inefficient and I wish to apply a good step size control.
First, I like to explain briefly the mentioned collocation method:
Assume, that we have found an approximative solution s(t) for (1) with the initial condition (I) up to a number t(n)>0. Also, we have chosen a new step size h(n). Let
The collocation method of stage 3 (with Lobatto-points) consists in finding (the uniquely existing) polynomial
p(t) of degree less or equal 3 that satisfies:
(A) p(t(n)) = s(t(n)) [initial condition]
(B) p'(t(n)) = f(t(n),s(t(n)-1))
(C) p'(t(n)+0.5*h(n)) = f(t(n)+0.5*h(n),s(t(n)+0.5*h(n)-1))
(D) p'(t(n+1)) = f(t(n+1),s(t(n+1)-1))
(A)-(D) determines p(t) uniquely and you can find the coefficients of p(t) by simply doing a matrix multiplication.
This method is equivalent to the implicit Runge-Kutta (IRK) method with the coefficients
A = [ 0, 0, 0; 5/24, 1/3, -1/24; 1/6, 2/3, 1/6]
c = [ 0 ; 1/2 ; 1]
b = [1/6 , 2/3 , 1/6].
The big advantage of this collocation method is that you don't have to solve any implicit equations.
Finally, my problem:
I don't know how to find cheaply a method of order 3 (that works with stiff problems!!) to make some error estimation to choose the step size. As I used the collocation method, I do not know the values k(i), i=1,2,3 , of the mentioned IRK method.
So, we cannot just change the vector b to get a formula of order 3.
If anyone could give me any idea for an error estimation, I would be very, very grateful.
I thank you for your time!
^^ Your pilzgericht


Auguri is an integrated, general purpose data exploration, analysis and forecasting tool with emphasis on nonlinear methods. It provides tools for the manipulation and analysis of data throughout the process of predictive data mining.|
Details: Analysis and Operations
Common statistical analysis methods such as ANOVA, test of means and variances, histograms, IID tests and descriptive statistics are standard. Auguri also provides nonlinear methods such as generalized dimensions, Poincare sections, maximal Lyapunov exponent, false neighbors, space-time separation, mutual information, and phase portraits, as well as tools for the analysis of signals and series in the time and frequency domains, such as power spectrum estimation, Fourier transforms, auto- and cross- covariance and correlation, time-evolving statistics, solutions to linear equations and methods for the generation of random numbers in various distributions, as well as data scaling, normalization, differencing, summation and operations between series and includes a powerful functional expression parser that can be applied directly to data.
Model Specification and Solutions
In Auguri, models are specified straightforward. No need to expand data to specify complex, multivariate dependencies. All data expansion takes place automatically, following specifications, with concurrent solutions to models, which can be simulated, compared, or ran with external data.
Category: Software Applications


Tackling parent anxiety
Nathan Chisholm is the principal of Prahran High School, which is in its foundation year with only year 7 students.
“As a start-up school we were able to do things a little differently,” he said, starting with teachers designing their own curriculum in place of student textbooks.
Mr Chisholm said his school instructs parents not to tell their children that they didn’t enjoy or weren’t good at maths themselves.
“It’s a more common refrain in maths than perhaps in other discipline areas,” he said.
To demystify the subject, the Victorian education department and schools increasingly spruik resources such as the “numeracy at home” website, which guides parents on “11 ways to maths” like constantly asking if something is “more or less?”.
Engaging the community
On a weekend earlier this year, a (fake) meteorite landed at Boneo Primary School, on the Mornington Peninsula.
“We printed off signs onto a big vehicle that it was a HAZMAT recovery team, we had helmets,” said principal Mandy Whitworth.
On Monday, younger students measured parts of the meteorite and older ones were asked to explain why it couldn’t be a real meteor (clue: a meteor that size would have caused more collateral damage than it did).
“It just engages your whole community, parents get a real buzz out of it. It opened really great strong conversations at home and at school, it’s about really bringing maths alive,” Ms Whitworth said.
Family maths nights are hosted at schools such as Boneo and Lyndale Greens primary school in Dandenong.
“Parents come along and we show them what we do in maths, because it’s quite different to how they were taught,” said Amy Sommers, numeracy leader at Lyndale Greens.
One answer, many routes there
There is only ever one answer in maths, but Ms Sprakel said schools are encouraging students to embrace the different ways to the answer and adopt a “growth mindset”.
“That includes the power of the word ‘yet’. ‘I can’t do algebra’ versus ‘I can’t do algebra yet’ is a big difference,” she said.
Mr Chisholm said many students arrived at Prahran High with a “fixed idea about whether they’re good at maths or not”.
For primary school, Ms Sommers gives the example of students counting how many muffins are in a four by three formation.
“Some could use repeated addition, some could count 1 by 1, some could use multiplication,” she explained.
“Obviously we want them to be working towards multiplication, but they’re all equally acceptable. It lets kids work at their level,” which she said builds student confidence and adaptability.
Making it tangible
All schools should be using physical props – rocks, dice, dominos, drawings – from primary school to drill in the relevance of maths to everyday life, Ms Sprakel said.
High school students appreciate how maths is used in designing a mobile phone or how their bus timetable works.
Prahran High School this year asked students to design updates to the school playground.
“They collaborated, made their models, and pitched their ideas to myself and an architect for feedback. Now we’re enacting some of the ideas, we’ve laid some new turf,” Mr Chisholm said.
Prahran High will introduce an additional 75-minute weekly maths class from next year, aimed specifically at setting students problem-solving tasks.
Michael is a reporter for The Age.


As you know keyboard is one of the important parts of computer with the help of keyboard, we send instruction to computer to perform operation.
If you press the Shift button and hold the number then whatever symbol will be in that number key will be typed. The top row of the keyboard consists of letters such as " Q, W, E, R, T and Y ". That is why keypads in cell phones are called QWERTY Keypads.
the numeric keypad is on the right side of the keyboard. It has numbers from 0 to 9. In addition, there are also sign of addition, subtraction, division, multiplication and decimal sign.
We hope that this information about the different types of keyboards can help you to select the right keyboard as per your requirement which can make your job more convenient and your work task simple and effective.


Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.
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|Absolute Value Equations: How to solve absolute value equations||Understanding solutions For purposes of this course, we can consider an equation or inequality or compound statement with one variable solved if it is one of the following:|
|Absolute Value||Numbers you use for counting:|
|Inequalities Worksheets | Math Worksheets||Interval notation Video transcript - [Voiceover] What I hope to do in this video is get familiar with the notion of an interval, and also think about ways that we can show an interval, or interval notation.|
Chemistry addison-wesley answers, How to Convert 37 5 to a Common Fraction, algebra pizzazz worksheets, coordinate plane worksheets. Really hard math promblems that are solved, simplifying radicals, college math, square root equation calculator.Step and Piecewise Functions Part 1 – Piecewise Functions Day 1 Piecewise Functions – Domain – Range – Intervals that are Constant, 6.
Using the third graph, write the equations of the “pieces,” or segments, of the graph. Be sure to indicate the appropriate interval for each piece (for Interval Notation Please write the. Unit 5: Quadratic Equations Objective After completing this unit, students will be able to identify the parts of a parabola from a graph (ax2 + bx + c = 0), solve quadratic equations, write and evaluate quadratic equations using function notation, and find the domain and range of a quadratic graph.
Seventh grade absolute value, simultaneous equations fractions, holt biology tests, free primary one maths worksheets in singapore, decimal to mixed number calculator (free), to .
1 Absolute Value Equations and Inequalities Absolute Value Definition - The absolute value of x, is defined asâ€¦, â‰¥0 âˆ’, Absolute Value. Interval notation is a simplified form of writing the solution to an inequality or system of inequalities, using the bracket and parenthesis symbols in lieu of the inequality symbols.
Intervals with parentheses are called open intervals, meaning the variable cannot have the value of the endpoints. Learn algebra inequalities absolute value with free interactive flashcards. Choose from different sets of algebra inequalities absolute value flashcards on Quizlet.


Newton’s formula for speed of sound in a gas
v = √(P/ρ)
The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³
So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8
Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s
The velocity of sound in air comes as 280 m/s.
But the measured value of speed of sound in air is 332 m/s
Laplace suggested a correction. With Laplace’s correction the formula is
v = √( γ P/ρ)
where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.


Help with career options
By - ssilverrenown
If you like Maths, then look for opportunities at Indian Statistical Institute. Not only will it not cost you anything to study there, the Institute would pay you compensation every month.
And you can also go ahead and do research in maths ,statistics and computer science as well, which also comes with a compensation.
Oh thank you. I'll definitely try it
If you like maths and physics and would want to pursue a research based career in it, prepare for SAT, get into a good US University for bachelors and masters and get into a PhD later there. You needn't become a professor if you are not into teaching. There are labs and institutions which are just research focused which you can look at. Don't look at engineering because you won't really do maths and physics (especially) in engineering if it is not a tier 1 college.
The problem is that for joining a US institute, I need to take huge student loans which would be a problem in the future


My name is Magdaléna Tydrichová and I am a PhD candidate at LIP6, Sorbonne University, in Paris. I am working in the DECISION team, and have a chance to be supervised by Olivier SPANJAARD and Bruno ESCOFFIER.
I am working in the field called Computational Social Choice. The topic of my thesis tackles problems around structured preferences - at the first time, I mainly studied some structural relaxations of the single-peaked domain, then the nearly single-peaked domain, from algoritmical and axiomatic point of view.
When I was about a halfway through, I totally fell in love with the domain restriction known as Euclidean preferences. Basically, I am trying to understand and describe, with more or less pain, the underlying structure from mathematical point of view. The problems I am studying at the moment involve also a bit of geometry, graph theory or combinatorics.
Independently of my thesis, I am (and I have always been, as I can remember) passionated by fields like combinatorics (and especially Ramsey theory), finite geometry, graph theory (mostly geometric graph theory) or combinatorial game theory. I am also fascinated by topology or group theory, however, I am very limited by my poor mathematical background.
As you could notice, the cover photo has nothing to do with my thesis. Highly inspired by the poem Hora Říp by Jaroslav Seifert, it shows the place I come from.


UVA curriculum activity to do at home!
Materials: Clothespin (wooden), index cards with numbers 1-10 written on them
1. Have your child put the number of clothespins on the number card (example: card has a "4" on it, child puts four clothespins around the card).
2. Ask the child to do that for each card & count aloud as the clothespins are attached.
3. Remove the clothespins and start again.
Variation: write letters on the clothespin & match up the letters to the word.
What: Builds fine motor skills.
Lisa Howard, President & CEO


Tank Volume CalculatorA = π r 2 where r is the radius which is equal to d/2. Therefore: V (tank) = πr2h. The filled volume of a vertical cylinder tank is just a shorter cylinder with the same radius, r, and diameter, d, but height is now the fill height or f. Therefore: V (fill) = πr2f.
Mail: [email protected]
To calculate the fluid volume in a vertical or horizontal tank can be complicated depending of the fluid height and the caps. This article makes a synthesis of the calculations for the volumes of andorra vertical cylindrical tank building volume* Sloped Bottom Tank - arachnoid andorra vertical cylindrical tank building volumeThe easy part the cylindrical section above the slope, which has a volume of: (1) $ \displaystyle v = \pi r^2 h$ v = volume; r = tank radius; h = cylindrical section height; More difficult the tank's sloped section, which lies between the tank's bottom and the top of the slope where the tank
R increases the tank's size in the Y (vertical) and Z (depth) dimensions. A tank with a zero R entry will not have a volume, regardless of the other entries. The left and right end caps are fully (and separately) described: with a dimensional variable r that describes the reach of the end cap along the X Calculating Tank VolumeCalculating Tank Volume Saving time, increasing accuracy By Dan Jones, Ph.D., P.E. alculating fluid volume in a horizontal or vertical cylindrical or elliptical tank can be complicated, depending on fluid height and the shape of the heads (ends) of a horizontal tank or the bottom of a vertical tank.Online calculator: Tank Volume CalculatorsCylindrical Tank Volume Use this calculator to determine your cylindrical tank volume in cubic inches and gallons even if one or both ends are rounded. Especially useful if you've cut the tank in length. If you've got a flat end just leave it blank or enter 0. Tank Volume Calculator.
Meherwan P. Boyce, in Gas Turbine Engineering Handbook (Fourth Edition), 2012. Pressure Tanks. Vertical cylindrical tanks constructed with domed or coned roofs, which operate at pressures above 15 psia (1 Bar) but are still relatively close to atmospheric pressure, can be built according to API Standard 650. The pressure force acting against the roof is transmitted to the shell, which may have andorra vertical cylindrical tank building volumeCylindrical Tank CalculatorExample: Inputting liquid level = 3, diameter = 24, tank length = 30, then clicking "Inches" will display the total tank volume in cubic inches and US Gallons and will also show the volume at the 3 inch level. In addition, a dipstick chart is automatically generated. The default dipstick chart is in increments of one but you can change that by clicking on one of the increment buttons.Tank Volume Calculator - Inch CalculatorFor example, lets find the volume of a cylinder tank that is 36 in diameter and 72 long. radius = 36 ÷ 2 radius = 18 tank volume = × 18 2 × 72 tank volume = 73,287 cu in. Thus, the capacity of this tank is 73,287 cubic inches.
The tank size calculator on this page is designed for measuring the capacity of a variety of fuel tanks. Alternatively, you can use this tank volume calculator as a water volume calculator if you need to calculate some specific water volume. The functionality of Tank Shell - an overview | ScienceDirect TopicsTank shells shall be finished with galvanized steel jacketing applied directly over the insulation. The metal sheet may be corrugated or plain. Corrugations shall run vertically with vertical seams lapped a minimum of 2 corrugations and horizontal seams lapped a minimum of 75 mm and supported on S clips.Tank Storage | Glossary | OiltankingOil is usually stored in vertical cylindrical tanks made of steel. The appropriate type of construction and materials for storing these products is defined by DIN standards. Beyond this, the respective federal state building regulations based on the Construction Products Act, and all applicable fire protection regulations must also be observed andorra vertical cylindrical tank building volume
Jun 29, 2015 · Tank Vapor Volume For Horizontal Cylindrical Tanks - posted in Tank Blanketing and Venting: I am working in the calculation of the thermal in and out-breathing is a function of tank vapor volume. As per API 2000 for the tank vapor volume for vertical cylindrical tanks, use of volume based on the tank shell height is acceptable .Volume of a cylinder with calculator - Math Open ReferenceVolume of a partially filled cylinder. One practical application is where you have horizontal cylindrical tank partly filled with liquid. Using the formula above you can find the volume of the cylinder which gives it's maximum capacity, but you often need to know the volume of liquid in the tank Analysis of a draining tank - OlinA tank of water is drained with siphon. The cylindrical tank has a constant cross sectional area, A, and volume, V. The draining tube has diameter, D, and length L. The end of the tube and the top of the water are separate in vertical distance by a height, h(t). The water that leaves the tank
This phase of using of structures is associated with complex investigation and numerical analysis of thin-walled structures. In this paper the load bearing capacity of the steel wall of the existing over-ground vertical cylindrical tank in volume of 5,000 m3 with a single defect and with a few contiguous local defects of the shape is analyzed.Blank Worksheet to Calculate Secondary Containment This worksheet can be used to calculate the secondary containment volume of a rectangular or square dike or berm for a single vertical cylindrical tank. You may need a PDF reader to view some of the files on this page. See EPAs About PDF page to learn more. Blank Worksheet (PDF) (4 pp, 529 K)Calculate volume of cylindric tank - MATLAB Answers andorra vertical cylindrical tank building volumeI'm trying to build a script/function that will help me calculate the volume of a cylindrical fuel tank, and translate about 10 (every 10%) points of the tank to 4-20mA output. I'm using these tanks
A4 Above-ground, Vertical, Cylindrical Storage Tanks ----- 154 Appendix B Assessment of Seismic Designs for Under-ground Storage Tanks ----- 160 . Chapter2 1 1. General 1.1 Scope This Design Recommendation is applied to the structural design of water storage andorra vertical cylindrical tank building volumeDipcharts andorra vertical cylindrical tank building volumeDipping a tank is how one determines how much contents are in a tank. Unless the tank is perfectly rectangular there the reading from the measuring device and the contents of the tank are not a linear relationship. In other words the change in volume per unit measurement is different for every value.HOW TO CALCULATE THE VOLUMES OF PARTIALLY FULL 2.1. Cylindrical Tanks The majority of tanks used in the chemical industry are cylindrical tanks, either in horizontal or vertical configuration. Consider, for example, a cylindrical tank with length L and radius R, filling up to a height H. If you want to obtain the volume of the liquid that partially fills the tank
Please note that these charts are theoretical and are intended as a guide for estimating tank/vessel volumes. Required Choose Tank Style Horizontal Cylindrical Horizontal Cylindrical (Elliptical Heads) Horizontal Rectangular Vertical Flat Bottom Vertical Dished Bottom Vertical Coned BottomHorizontal Cylindrical Tank Volume Calculator - MetricTank Volume: Vertical Cylindrical Tank Volume: Rectangular Tank Volume: Directory: Inch: Inch: Inch: Inch: Horizontal Cylindrical Tank Volume Calculator, Dip Chart and Fill Times - Metric. Hemispherical Ends Ellipsoidal Ends - adj Ellipsoidal Ends - 2:1 Flat Ends: Straight LengthLiquid Volume of Horizontal Tank with Dished Ends andorra vertical cylindrical tank building volumeJan 18, 2013 · I am the Head of Department for quality control & Environment. we calculated our Horizontal Tank volume-Tori spherical Head, As per our knowledge it is giving spurious reading. Tank Dia is 2400 mm,LOS is 3000 mm, Torispherical head length from center are 472 mm. Internal lining material is natural rubber with thickness 4.3 mm (three layers).
Jan 18, 2013 · I am the Head of Department for quality control & Environment. we calculated our Horizontal Tank volume-Tori spherical Head, As per our knowledge it is giving spurious reading. Tank Dia is 2400 mm,LOS is 3000 mm, Torispherical head length from center are 472 mm. Internal lining material is natural rubber with thickness 4.3 mm (three layers).Mixing 101: Optimal Tank Design | Dynamix AgitatorsMar 10, 2015 · Vertical cylindrical tanks are the most common type of tank in use. A key consideration for cylindrical tanks is to ensure that they are either baffled or offset-mounted to prevent swirling from occurring. Refer to section 2 below (The Use of Baffling) for details. Generally baffles are not required for smaller tanks (<5,000 gallons in andorra vertical cylindrical tank building volumeOnline calculator: Cylindrical tank volumeCylindrical tank volume. Cylindrical tank volume. person_outlineTimurschedule 2011-07-30 10:41:00. andorra vertical cylindrical tank building volume To find it, we draw vertical line from center of circle. It divides triangle to two right-angle triangles with hypotenuse equals to R and top cathet R-m. Thus, and. The
containment type tanks are available in both vertical and horizontal configurations, with a maximum capacity of 30,000 gallons. Transport Tanks Process Fabricators, Inc. fabricates transport tanks in a wide variety of shapes and sizes. * Obround truck tanks * Cylindrical truck tanks * Rectangular truck tanks * Trailer tanks * Pallet tanksSpill Prevention Control and Countermeasure (SPCC) PlanV /V = ÷ = c is the secondary containment volume calculated in Step 1. d / e is the tank volume calculated in Step 2. c (ft 3) d or e (ft ) f % = x 100 = f g If percentage, g, is 100% or greater, the capacity of the secondary containment is sufficient to contain the shell capacity of the tank. If rain can collect in the dike or berm, continue to step 4.Storage Tank Calculator - Vertical & Horizontal Tank andorra vertical cylindrical tank building volumeFor vertical tanks, only the cylinder volume is used in calculations. The Top End Type and the Bottom End Slope of the tank are not factored into the estimated volume. In order to calculate the volume of the storage tank then, all we need is to calculate the main cylinder volume.
Jun 20, 2019 · Cylindrical Oil Tank. Lets say that I have a cylindrical oil tank which measures 7 yards in length and has a round face 5 feet in diameter (the distance across the circular end passing through the central point). I want to calculate the tank volume in cubic feet and work out how much oil will fit in the cylinder in US gallons.Tank Volume Calculator - Inch CalculatorFor example, lets find the volume of a cylinder tank that is 36 in diameter and 72 long. radius = 36 ÷ 2. radius = 18. tank volume = × 182 × 72. tank volume = 73,287 cu in. Thus, the capacity of this tank is 73,287 cubic inches.Tank Volume Calculator - Oil TanksThe tank size calculator on this page is designed for measuring the capacity of a variety of fuel tanks. Alternatively, you can use this tank volume calculator as a water volume calculator if you need to calculate some specific water volume. The functionality of
Vertical cylindrical tank volume calculator diagram Fraction Precision Set 1/8 1/16 1/32 1/64 Decimal All Inch inputs and dimensions are actual physical finished sizes (unless otherwise noted)Tank Volume Calculator for Ten Various Tank ShapesApr 17, 2019 · Cylindrical tank volume formula. To calculate the total volume of a cylindrical tank, all we need to know is the cylinder diameter (or radius) and the cylinder height (which may be called length, if it's lying horizontally).. Vertical cylinder tank; The total volume of a cylindrical tank may be found with the standard formula for volume - the area of the base multiplied by height.Tank Volume CalculatorA = r 2 where r is the radius which is equal to d/2. Therefore: V (tank) = r2h. The filled volume of a vertical cylinder tank is just a shorter cylinder with the same radius, r, and diameter, d, but height is now the fill height or f. Therefore: V (fill) = r2f.
If you want to do calculations for a horizontal cylinder, then go to this link: Horizontal Cylinder Calculator. Example: Inputting tank height = 12, liquid level = 3 and tank diameter = 6, then clicking "Inches" will display the total tank volume in cubic inches and US Gallons and will also show the volume at the 3 inch level. If the tank is filled with water, this calculator also displays the andorra vertical cylindrical tank building volumeVolume of a cylinder with calculator - Math Open ReferenceVolume of a partially filled cylinder. One practical application is where you have horizontal cylindrical tank partly filled with liquid. Using the formula above you can find the volume of the cylinder which gives it's maximum capacity, but you often need to know the volume of liquid in the tank how to caluclate vertical cylindrical water tank capacity andorra vertical cylindrical tank building volumeJun 09, 2007 · *Volume of a vertical cylindrical water tank =pi x r ^2 r xh. where, pi is 22/7[or 3.14] r= radius of cylinder. h= height of the cylinder. *If you want to calculate the volume of the tank at different level,then take the level (height)shown by the level gauge in to the formula.
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I’m a senior lecturer in statistics at the University of Bristol.
My current research is on the interface between Bayesian statistics and machine learning. I am particularly interested in developing approximate inference methods such as approximate Bayesian computation approaches and variational inference. One application is to likelihood-free inference, where simulation of data is possible but the likelihood function is unavailable. Another is to stochastic differential equations I’ve worked on applications to population genetics, physics, ecology and epidemiology. Another research interest is experimental design and how to quickly derive effective high dimensional designs.


What's the use? : how mathematics shapes everyday life / Ian Stewart.
- ISBN: 9781541699489
- ISBN: 1541699483
- Physical Description: 326 pages : illustrations ; 25 cm
- Edition: First US edition.
- Publisher: New York : Basic Books, 2021.
- Copyright: ©2021
|Bibliography, etc. Note:||
Includes bibliographical references (pages 299-307) and index.
|Formatted Contents Note:||
Unreasonable effectiveness -- How politicians pick their voters -- Let the pigeon drive the bus -- The kidneys of Königsberg -- Stay safe in cyberspace -- The number plane -- Papa, can you multiply triplets? -- Boing! -- Trust me, I'm a transform -- Smile, please! -- Are we nearly there yet? -- De-ising the arctic -- Call the topologist -- The fox and the hedgehog.
"An esteemed mathematician shows how math powers the world, from technology to health care and beyond"-- Provided by publisher
Almost all of us have sat in a math class, wondering if we would ever use it in the real world. Stewart argues that math isn't just about boring computations: it's a means to achieve great things, such as space exploration and organ donation. He shows that trigonometry is essential to keeping a satellite in orbit, and how the prime numbers power the world's most advanced security systems. Math isn't just relevant to our lives. It is the very fabric of our existence. -- adapted from jacket
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I’m a PhD student in Statistics at UC Berkeley, advised by Jacob Steinhardt. I work on making machine learning systems safer. Before moving to the Bay Area, I studied applied mathematics at École polytechnique.
Auditing Visualizations: Transparency Methods Struggle to Detect Anomalous Behavior
Jean-Stanislas Denain, Jacob Steinhardt
Grounding Representation Similarity with Statistical Testing
Frances Ding, Jean-Stanislas Denain, Jacob Steinhardt
[blog post], [code], [data]


2020 ArcGIS Online Competition
Esri is challenging students grades 4-12 to put their ArcGIS skills to work. Students are encouraged to participate with all states included and the chance to win prizes of $100 and the chance to earn a trip to the 2020 Esri Education Conference in San Diego, CA.
National STEM Honor Society
The National STEM Honor Society (NSTEM) inspires and recognizes the efforts and achievements of students to become academic, community, and business leaders in the fields of Science, Technology, Engineering, and Mathematics (STEM).
NSTEM inspires future leaders of our increasingly technology-driven world by nurturing STEM education and innovation through project-based learning, leadership development, and mutually beneficial collaboration among students, schools,
communities, and businesses.


“And he made the Sea of cast bronze, ten cubits from one brim to the other; it was completely round. Its height was five cubits, and a line of thirty cubits measured its circumference. Below its brim were ornamental buds encircling it all around, ten to a cubit, all the way around the Sea. The ornamental buds were cast in two rows when it was cast.”
I Kings 7:23-24 NKJV
“And this is how you shall make it: The length of the ark shall be three hundred cubits, its width fifty cubits, and its height thirty cubits. You shall make a window for the ark, and you shall finish it to a cubit from above, and set the door of the ark on its side. You shall make it with lower, second, and third decks.”
Genesis 6:15-16 NKJV
“So teach us to number our days, That we may gain a heart of wisdom.” Psalms 90:12 NKJV
Today, I was thinking about how mathematics has impacted our lives. Did you know the basic “code” of computers and devices is simply the numbers 0 and 1? Zero means numbers and one means on. Everything from gravity to how the Universe operates requires math. Time is mathematical. All structures around you require math. Digital currency is a series of digits and people are using mathematics to get rich. Oh, talking about rich, it is simply mathematical numbers and yet people look up to those who have more numbers to their name. Land ownership is mathematical too. Nation borders are based on numbers. People steal for numbers, wars are caused by numbers, money is measured by numbers and we work to earn numbers. Weather is number based.
Math has perfect rules too. You either get the right answer or a wrong answer. If your math isn’t right in your bank account, you could bounce a check. If the Universe isn’t mathematically perfect, we all die.
God is the best mathematician ever because He created it. His measurements and rules behind the numbers are perfect! They have to be for everything to run properly. We have grown technologically thanks to God allowing us to know some of the rules He created.
Now that much of what we value is purely numbers, let’s go deeper and discuss who created the number system in the first place? Without Him, the rules of how to use numbers would not exist, and neither would we.
Although numbers are important to build, create, and exist, they do not mean anything without a Creator. Thinking this way, we know where our true priority should lie and it’s not numbers.
by Kent Brosh a Kawaiaha’o Blog Contributor


θ(t) is the angular rotation as a function of time
Δθ is the change in angle (angular displacement)
w is the angular velocity
α is the angular acceleration
t is time
If we assume constant angular acceleration, we have the following formulas. Note that the subscripts 1 and 2 denote "initial" and "final".
These formulas apply to every particle in a rigid body that is experiencing rotation, such as the disk shown in the first figure. In other words, all the particles experience the same angular displacement, angular velocity, and angular acceleration regardless of their position relative to the center of rotation.


:Type:Area takes numerical values for denoting surface areas.
It supports the following units of measurement:
- 1 m², m2, sqm, square meter, square meters, square metre, square metres
- 0.000001 km², km2, sqkm, square kilometer, square kilometers, square kilometre, square kilometres
- 0.0001 ha, hectare, hectares
- 3.86102158542e-7 miles², sqml, sqmi, ml², mi², mi2, square miles
The main unit is m², i.e. values without a unit are interpreted in this way.
Feel free to add your own units and alternative representations. Note uses might also enter "squared" using 2 or a Unicode character, so you must enter separate "units" for each variation you wish to support.


Montessori Information Evenings
Preschool and Under 3 Playgroup families Welcome!
Each Term Principal Mary Batterham presents a free information session. We invite current families, Under 3 parents, waitlist families.
The topics vary between Language, Mathematics, Sensorial, Cultural, Practical Life and Montessori in the home.
Our second term topic is Mathematics.
In Montessori, we want each child to feel confident with maths. We introduce it in a way that makes sense to them, at a sensory level. We gradually lead them through a series of steps toward the abstract work; the symbols humans have created. By using the materials and following through the sequence of presentations that become more and more abstract, your child doesn’t just learn to mimic the steps of a maths process but really understands what it means and why it’s done a certain way with pencil and paper.
You are invited to come along and experience how easy it is to come to a greater understanding of mathematical concepts.
RSVP to Anna – [email protected]


How is the going rate determined anyway?
You can use regression analysis, but with what data and which variables?
When it comes to predictions using fuzzy logic a more-simple solution may be the Knowledge of the Crowds. Asking random people, experts and otherwise, to make predictions of the going rate and the value of different amenities. The more responses you get, the more accurate your prediction will be. This model may be more accurate than a regression equation because of the variation in valuation of amenities in different geographic regions. Recursively, knowing more information about individuals in your crowd will give you the opportunity to tease out the valuation of each amenity to a person in that market.


Math is a topic that many youngsters have bother with as they progress by way of faculty. It is important that math fundamental expertise be mastered early on, as these expertise construct the inspiration for understanding tougher ideas launched in larger grades. First grade math accommodates fairly just a few essential expertise, and utilizing on-line video games might be able to give college students the assistance they want so as to grow to be fluent in working with numbers.
Constructing on Primary Ideas
In kindergarten, youngsters be taught simple arithmetic expertise like quantity recognition and counting. They’re in a position to start to acknowledge that larger numbers are larger numbers and may perceive ideas like counting by twos and tens. Utilizing on-line video games with these kids after they attain first grade might help them transition between starting math expertise and extra complicated numerical ideas. Offering all kinds of various video games in an entertaining on-line setting will get youngsters enthusiastic about tackling new concepts and places math in a optimistic context. As soon as the inspiration has been laid for fundamental comprehension and computational expertise, kids can use these video games to progress even additional.
Mastering New Expertise
By the point they’re studying first grade math, youngsters must be able to sort out issues like the connection between addition and subtraction, the idea of including and subtracting two-digit numbers and studying to depend past 100. Having the ability to examine numbers as bigger, smaller or equal to one another can be essential, because it supplies the premise for recognizing whether or not or not the reply to a computation drawback is the proper one.
Kids must be allowed to grasp these and different important math expertise earlier than being requested to maneuver on to new concepts, however the fashionable classroom setting does not all the time permit for this. As deal with core curriculum begins to push complicated concepts into decrease grade ranges, youngsters are anticipated to be taught extra at a youthful age. First grade math nonetheless accommodates many basic ideas important for understanding larger math, and due to this fact should not be rushed by way of. By letting a toddler try to re-try every new factor because it comes, on-line math video games may give the additional time and observe that struggling college students want to attain success.
Taking it One Step at a Time
Each baby learns at a distinct fee, however it’s essential to keep in mind that all kids ought to have the ability to grasp comparable ideas as soon as they’ve accomplished first grade math. With out the power to depend, add, subtract and perceive fundamental relationships between all three, development to larger math turns into tough. Although on-line math video games cannot be used to show all the things a toddler must know, they are often a useful instrument in the case of augmenting what’s discovered within the classroom. Video games provide a distinct context and extra interactive setting than worksheets and are additionally extra dynamic, which supplies youngsters a brand new manner to consider the ideas they’re studying and may make clear issues they could be having bother with.
The interactive world of on-line video games 온라인바카라 makes first grade math gratifying for younger kids and permits them to observe new math ideas outdoors of faculty. Each mother and father and lecturers can use these instruments to boost conventional studying, thereby giving youngsters a novel and entertaining option to hone essential math expertise.


A bit hit!
I bought this puzzle because my 8 year old has been working on her times tables. I thought it might be a fun way to reinforce her learning (she's been using the Multiplication Learning Wrap-Ups, which she also highly recommends), but I never expected it to be such a hit! My kids LOVE this puzzle. It goes together pretty quickly, which I think is part of the appeal for them. We've put it together at least 4 times in the few weeks we've had it. Usborne puzzles are the best!
3/7/2022 12:56 PM


Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time.
Some other uses of differential equations include: In medicine for modelling cancer growth or the spread of disease In engineering for describing the movement of electricity In chemistry for modelling chemical reactions and to computer radioactive half life In economics to find optimum investment strategies In physics to describe the motion of waves, pendulums or chaotic systems. It is also used in physics with Newton's Second Law of Motion and the Law of Cooling. In Hooke's Law for modeling the motion of a spring or in representing models for population growth and money flow/circulation.


It seems as if many of those mathematicians are now saying that the equal sign, which has traditionally been used to show the exact relationships between objects may be on its way out.
Mathematicians don’t always agree on everything, but sometimes there are enough who agree on one thing to make it noteworthy. It seems as if many of those mathematicians are now saying that the equal sign, which has traditionally been used to show the exact relationships between objects may be on its way out. According to WIRED, they feel that it no longer holds up to newer mathematical models.
One important factor to understand when considering their arguments is set theory. This is a mathematical theory that has been around since the 1870s and perhaps even earlier. As an example, consider the 1+1=2 mathematical formula. To break it down even further, let’s say you have four items, a piece of celery, a tennis ball, and two oranges. If the celery and tennis ball are put on one side of the table and the two oranges are on the other side of the table, it’s an equation. Two different objects on one side of the table is equal to two of the same objects on the other side of the table. The collection of objects is the same, ergo they are equal.
Things start to get complicated when you put a piece of celery and an orange on one side of the table and the tennis ball and the other orange on the other side. It’s different than the first scenario but it is still written out using the same equation. You could continue to debate this in so many different ways, because the objects could continually be swapped out to an infinite number of scenarios.
“The problem is, there are many ways to pair up,” Joseph Campbell, a mathematics professor at Duke University, told Quanta Magazine. “We’ve forgotten them when we say ‘equals.’”
Some mathematicians are now saying that the concept of equivalence is a better idea. Equality is very strict but equivalence can come in different forms. For example, having two oranges on both sides of the table is a matter of strong equivalence because everything is the same. The scenario with a tennis ball and piece of celery on one side and two oranges on the other side is still a matter of equivalence but it is weaker.
Category theory is another concept that many mathematicians are considering. It tends to complicate matters by considering the relationships between different objects, but it may be, strictly speaking, accurate. Category theory is certainly more accurate than set theory when dealing with matters of equivalence. It can also be applied to many branches of mathematics.
Quanta also reported that switching to category theory is not something that will be done with a snap of a finger or a flip of a switch. Interpreting equations using equivalence and dropping equality altogether is complicated. It requires people to relearn things that they already know and everything that uses mathematics, down to basic arithmetic.
“This complicates matters enormously, in a way that makes it seem impossible to work with this new version of mathematics we’re imagining,” mathematician David Ayala told Quanta.
A number of mathematicians are taking the lead in research of category theory but the field is still in its infancy. We are not going to see the equal sign disappear overnight, but it may be on the horizon along with a new mathematical revolution.


SAXON CALCULUS DVD SET
Detailed explanation for every lesson by a gifted teacher
Recommended in Program(s): Challenge IV
Teaching Tape Technology videos are designed to meet the need of the homeschool teacher, student and others as a resource that establishes a solid foundation in math. These videos are not designed to replace Saxon books but rather to complement them. You must have Saxon books to use these videos.
Teaching Tape Technology is not affiliated with Saxon Publishers, Inc.
- 20 DVDs
- © 2004


Laboratoire LMPA, BP 699 , 62228 Calais CEDEX, FRANCE.
Téléphone : 33 3 21 46 55 92
Fax: 33 3 21 46 36 69
Abstracts of the main papers.I.Aspects of internal categories and 2-categories
I.Aspects of internal categories and 2-categories
1.-Natural anadeses and catadeses. Cahiers Top et Géom. Diff. XIV, 4 (1973) 371-480.
It is observed, in the appendix, that the three first cohomology groups of a group in the sense of Eilenberg-MacLane, can be successively calculated in terms of a limit for H0, a lax limit for H1, a 2-lax limit for H2. It is explicitely based on the fact that a given abelian group A has a canonical structure of n-category for each integer n, and it is the occasion of the general definition of n-category in the internal way (vs the " enriched category " way). The specific link between the abelian cohomology and this typical property of the abelian groups will be elucidated in .
2.-(with J. Penon).2-categories reductibles-Preprint Amiens (1978).
The 2-categories which are comonadic, via a left exact comonad, on a 2-category of the form Cat(E) (the 2-category of the internal categories in E) are characterized.
3.-Une autre propriété universelle pour le champ associé. Cahiers Top . et Géom. Diff.XXI,4 (1980) 403-409.
It is shown here that the stack completion C of an internal category C (in the sense of M. Bunge : Stack Completions and Morita equivalence for categories in a topos, Cahiers Top et Géom. Diff. XX,4, (1979)401-436) has another universal property : it classifies, the weakly representable profunctors with codomain C, ie the profunctors with codomain C representable up to weak equivalence.
The condition (BA) on a category E is considered : for every internal category C, the stack completion C is still internal, or equivalently the stack completion C is representable.
II Profunctors, shape theory and homotopylimits.
4.-(with J.M. Cordier) Distributeurs et théorie de la forme. Cahiers Top et Géom. Diff XXI,2(1980) 161-188.
The notion of shape category was introduced by Borsuk (1968) in order to study the homotopy properties of compact spaces. Axiomatic approaches of this notion were given by Holsztynski (1971) and Bacon (1975), as well as categorical ones by Deleanu and Hilton (1976). It is shown here that the axioms determined by these two ways of approach are actually universal properties (in terms of Kan extensions) with respect (not to ordinary functors but) to profunctors in the sense of Benabou . In this context, the category S is the shape category related to the functor K : Wà H if and only if it is the Kleisli category of the codensity promonad generated on H by the functor K. In the same way, further aspects of the Cech approach of the shape theory can be elucidated in terms of profunctor.
5.(with J.M. Cordier) A general formulation of homotopy limits Journal of Pure and Applied Algebra 29 (1983) 129-141.
Let å be the category of simplicial sets. The definition of the homotopy limit of a functor F :Ià å has been introduced by Bousfield and Kan (1972), taking into account the higher order homotopy coherences involved in certain constructions. This paper shows that this notion is a particular case of the general notion of indexed limits (Borceux, Kelly (1975)) in the context of the categories enriched in å , and that the Bousfield Kan replacement scheme holds in this extended context.
6.A canonical action on indexed limits. Category Theory, Springer LN 962 (1982) 23-32.
Given a V-category A and a profunctor j : A--->1,then this profunctor is an indexation for a certain type of V-limits (Borceux, Kelly (1975)). It is shown here that the jűindexed limit of the profunctor j itself is a V-monoïd and that this V-monoid has a canonical action on each j-indexed limit.
The particular situation which induced the idea of this result is the following classical one. Let D be the 2-category with a simple objet * and generated by a 1-morphism t :* à*, and two 2-morphism l :* =>t et m :t2 =>t such that m .l t=t=m .tl and m .m t=m .tm . It is clear that a monad on a category C is a 2-functor P from D to Cat, and well known that the lax limit of P is the category of algebras CT of this monad. Now the indexation P(D) :D ---> 1 determining this laxlimit is nothing but the laxlimit of the Yoneda embedding, and a simple but tedious computation shows that the Cat-monoid determined, following our result, by this indexation is Dco (the dual of D for the 2-cells). That this monoid acts on CT just emphasized that there is a comonad on CT. An application of the existence of such an action is then given in homotopy theory. Let us denote by H the sub 2-category of D obtained by cancelling the 2-cell l . As any 2-category, H is a simplicial category.Now let us call coherent homotopy idempotent a simplicial functor F :Hà Top. The action determined by H allows to show that the associated homotopy idempotent (i.e. the associated idempotent in the homotopy category) splits, although, in general, the homotopy idempotent do not split.
III Internal n-categories and cohomology .
7.-La tour de fibrations exactes des n-categories. Cahiers Top. Et Géom. Diff. XXV 4 (1984) 327-351.
A general setting is introduced here in order to control, in the case of the internal n-categories, the passage from n to n+1 in a way which allows to use proofs of inductive type (see 9 and 10 for applications).
Once defined the category CatE of internal categories in E, with the associated fibration ()0 : CatEàE assigning to each internal category its object of objects, the categorie (n+1)-CatE of internal (n+1) categories in E is defined by the following inductive process : suppose defined n-CatE and the fibration ()n-1 : n-CatEà (n-1)-CatE, then (n+1)-CatE is the full subcategory of Cat(n-CatE) whose objects are the categories internal to the fibres of ()n-1.
It is then shown that when E=Sets, this definition coincides with the enriched version of n-categories.
8.-The shift functor and the comprehensive factorization for internal groupoids Cahiers Top. Et Géom. Diff. XXVIII, 3 (1987), 197-226.
It is actually a preliminary paper to the following one. It contains the fact that the category GrdE of internal groupoids in E is monadic above the category SplE whose objects are the split pairs (an epimorphism with a given splitting) and morphisms the commutative squares between such pairs, as soon as E is left exact. This result allows to proof that if furthermore E is Barr exact then any internal functor between internal groupoids can be factorized, in a unique way up to isomorphism, into the composite of a discrete fibration with a final functor. It is called, according the result of Street and Walters (1973) when E=Sets , the comprehensive factorization.
9.-A right exactness property for internal n-categories Cahiers Top. Et Géom. Diff. XXIX, 2 (1988), 109-155.
When E is Barr exact, then the category SimplE of simplicial objects in E is still Barr exact. It is no more the case for the category n-CatE of internal categories in E. However it is possible to show that this category has the same kind of right exactness but only with respect to a certain class å n of regular epimorphisms. This class å n is defined by induction. Firstly it is easy to show that when E is Barr exact, the fibration ()n-1 :n-CatEà (n-1)-CatE is Barr exact : each fiber is Barr exact and each change of base functor is Barr exact.
The class å 1 is defined in the following way : the internal functor f1 is in å 1 if f0 is a regular epimorphism in E and its decomposition f1=g1h1 with respect to the fibration ()0 :Cat Eà E (h1 in the fibre, and g1 cartesian) is such that h1 is a regular epimorphism in the fibre.
The class å n is then defined inductively : the internal n-functor fn is in å n if fn-1 is in å n-1 and its decomposition fn=gn.hn with respect to ()n-1 : n-CatEà (n-1)-CatE is such that hn is a regular epimorphism in the fibre.
10.-Another denormalization theorem for abelian chain complexes. Journal of Pure and Applied Algebra 66 (190) 229-249.
Once the general setting of  introduced, it was very easy to show that for each additive category A and each integer n the category CnA of Complexes of length n in A is equivalent to the category n-GrdA of internal n-groupoids in A, what is an alternative of the classical Dold Kan equivalence between C¥A and SimplA.
Moreover these new denormalization equivalences exchange chain homotopies with what are expected to be n-lax natural transformations. Thus n-groupoids appear as possible substitutes for n-complexes in the non abelian context.
11.-The tower of n-groupoids and the long cohomology sequence Journal of Pure and Applied 62 (1989) 137-183.
Given a Barr exact category E, then the fibration ()0 :Grd EàE is Barr exact . Now if A is an abelian group object in E, the classical H0 (E,A) is nothing but the group E(1,A) of global elements of A. But A being abelian, it is also an abelian group, under the notation K1(A), in the fibre above 1 of the fibration ()0. Now given any object X in E with global support t :Yà 1, then the inverse image t *(K1(A)) determines an abelian group object in the fibre [X] above X. The starting point of this paper is the following remark : the abelian group H1(E,A) defined classically by the classes of A-torsors is nothing but the co-limit of the abelian groups H°([X], t *(K1(A))), each X having a global support.
This is clearly the beginning of an inductive process which defines Hn+1(E,A) as the co-limit of the Hn([X],t *(K1(A))). This process describes Hn+1 (E,A) as the set of connected components of the category whose objects are the aspherical internal n-groupoids in E equipped with a structure of Kn(A)-torsors with respect to the fibration ()n-1 : n-GrdEà (n-1)-GrdE, where Kn(A) is precisely the structure of n-groupoid in E associated with the abelian group A in E (as defined in the paper ) which is naturally an abelian group object in n-Grd(E).
These Hn has the property of the long cohomology sequence. Furthermore when E=A is abelian, this description of the Hn coincides with the classical YonedaĂs description of Extn, via the denormalization of . When E=Gp the category of abstract groups, this description is equivalent to those given by Holt (1979) and Huebschmann (1980) by means of crossed n-fold extensions.
IV Some applications of the n-categorical denormalization.
12.Pseudo functors and non abelian weak equivalences. Categorical algebra and its applications. Springer LN 1348 (1988), 55-70.
The usual Dold-Kan equivalence between the category C¥A of positive chain complexes in the additive category A and the category SimplA of simplicial objects in A did not strictly exchange chain homotopies with simplicial homotopies, while the new denormalization of  do strictly exchange chain homotopies with what is expected to be higher lax natural transformations. In that sense, the two notions of n-complexes and n-groupoids seem to be more closely connected than the two notions of n-complexes and n-truncated simplicial objects.
The aim of this paper is to make use of this strong relationship between CnA and n-Grd A as guide in exploring certain aspects of the categories n-GrdE, for any E, and especially in elucidating what could be the notion of weak n-equivalence between internal n-groupoids.
Indeed there are too many choices of possible definitions. The paper, here, is devoted to describe the notion of weak n-equivalence which, in the additive case, is in one to one correspondance, via the denormalization theorem, with the notion of weak equivalence between (n-truncated) chain complexes. We shall suppose, from now on, that the basic category E is Barr exact.
Of course, at level 1, the definition is clear and classical : a functor f1 :X1à Y1 is a weak equivalence when it is fully faithful (i.e. cartesian with respect to the fibration ()0 :GrdEà E) and essentielly surjective (i.e. P 0(f1) is an isomorphism). More generally a functor fn : Xnà Yn is a weak n-equivalence when it is cartesian with respect to ()n-1 : n-GrdEà (n-1)-GrdE (see ) and when P n-1(fn) is a weak (n-1)-equivalence, where P n-1 : n-GrdEà (n-1)-GrdE is the left adjoint to the discrete functor dis : (n-1)-GrdEà n-GrdE, left adjoint to ()n-1.
In the case E=Sets, a weak 2-equivalence f2 :X2à Y2 between the two 2-groupoids X2 and Y2 is a 2-functor such that :
1)f2 is essentially surjective, 2)f2 is " hom by hom " a weak equivalence.
Now, thanks to the axiom of choice, it is possible to characterize weak 2-equivalences in the category 2-Grd of 2-groupoids as those 2-functors which have (not an inverse equivalence as at level 1 but) a pseudo-inverse equivalence. Consequently every pseudo-functor : X2 --->Y2 can be represented by a pair of 2-functors : X2ß V2à Y2 where the left hand part is a weak 2-equivalence.
It is then possible to define abstract n-pseudofunctor in any n-Grd(E) as a pair of internal n-functors : Xnß Vnà Yn where the left hand part is a weak n-equivalence. This is all the more interesting than the Hn(E,A) of  appears then as the classes of n-pseudo-functors 1- - - >Kn(A).
13.-Produits tensoriels coherents de complexes de chaine. Bulletin de la Soc. Math. de Belgique (Série A), XLI,2 (1989), 219-248.
There are at least two different structures of closed monoidal category on the category 2-Grd of 2-groupoids : the cartesian closed structure and that which comes from the Gray tensor product and the lax natural transformations. It is clear that we may expect n different closed monoidal structures on the category n-Grd of n-groupoids, corresponding to the different level k, 1£ k£ n, where we stop the introduction of kĂ-cells, 1£ kĂ£ k, in the possible definitions of higher order lax natural transformations. But it seems very difficult to explicit them.
Now, in the additive situation, thanks to the new denormalization , n-Grd(Ab) is equivalent to CnAb. It is obviously much easier to work in CnAb, and the aim of this paper consists in expliciting the corresponding expected n different closed monoidal structures on CnAb.
One interesting point is that, for any additive category A, the category CnA is enriched in any of these n different closed monoidal structures on CnAb.
Thus the category C¥ Ab of positive chain complexes is endowed with infinitely many different closed monoidal structures and the category C¥A of positive chain complexes in A is endowed with infinitely many different enriched category structure.
14.-Normalization equivalence, kernel equivalence and affine categories category theory, Springer LN 1488 (1991) 43-62.
The starting point here is the caracterisation by Carboni (1989) of the categories of affine spaces by means of a " modularity " condition. Actually the denormalization theorem  leads naturally to an equivalent caracterisation. Let us denote by Pt(E) the category whose objects are the split pairs (an epimorphism with a given splitting) and morphisms the commutative squares between such pairs, and by p :Pt(E)à E the functor assigning to each pair the codomain of the epimorphism. This p is obviously a fibration as soon as E is left exact. It is called the fibration of the pointed objects in E.
Now a category E is modular if and only if
1)the fibration p is trivial (i.e. every change of base is an equivalence)
2)the functor -+1 (sum with the terminal object) satisfies a certain " modularity " condition.
The main result is that the condition 1 implies that the fibration p is additive, meaning that each fiber is additive and each change of base functor is additive. We shall call essentially affine a category satisfying the condition1, and consequently a category C is additive if and only if it is essentially affine and has a zero object.
Now a functor is an equivalence if and only if 1) it reflects the isomorphisms and 2) its left adjoint is a right inverse. So in presence of a zero object, the fact that C is essentially affine means that : 1)the short five lemma holds for split exact sequences and 2) the sequence 0à Aà A+Bà Bà 0 is exact. But the (split) short five lemma still holds for non abelian groups, and if we denote by Gp the category of groups then the fibration p :Pt(Gp)à Gp is such that every change of base functor reflects the isomorphisms.
Whence the definition : a category is protomodular when it is left exact and the fibration p :PtCà C is such that every change of base functor reflects the isomorphisms. Given any left exact category E, then the category Gp(E) of internal groups in E is protomodular as well as any fibre of the fibration
()o : GrdEà E (defined in ).
In any protomodular category C, as it is the case in Gp, the pullback functors reflect the monomorphism, any internal category is a groupoid. If furthermore C has a zero object and is Barr exact, any regular epi is the cokernel of its kernel.
V. Relationship between n-groupoids and n-truncated simplicial objects in the non abelian situation : the nerve functor for n-groupoids.
15.-Low dimensional geometry of the notion of choice. Category theory 1991 Canadian Math. Society Conference Proceedings, 13, (1992), 55-73.
In the same way as the category Grd E of internal groupoids in E is monadic above the category PtE of the split epimorphisms (see  and , the change of notation is due to the importance of the fibration p : PtEà E), the category 2-Grd E of internal 2-groupoids in E is shown to be monadic above the category N-GrdE of internal groupoids equipped with a normalization ; i.e. with the choice of an object in each connected component and a morphism between this object and any other object of this component. The underlying normalized groupoid of a 2-groupoid is the groupoid of the oriented triangles. This monadicity theorem yields a natural comparison functor N:2-GrdEà SimplE which is a synthetic version of the nerve functor for 2-groupoids and which, allowing to get free from the combinatorial geometric aspect of the pasting used in the case of 2-categories, has a meaning in the purely internal situation.
16.-Polyhedral monadicity of n-groupoids and standardized adjunction. Journal of Pure and Applied Algebra, 99 (1995) 135-181.
This paper is the natural continuation of the previous one, which, by the way, contained already inductive definition of normalized internal n-groupoids. Considering the fibration ()n-1 :n-GrdEà (n-1)-GrdE (see), a normalized n-groupoid is a n-groupoid Xn whose underlying (n-1)-groupoid Xn-1 is normalized and such that the terminal map t n :X n à G n (Xn-1), in the fibre above X n-1 , has a given splitting s n.
It is devoted to the proof that the category (n+1)-GrdE of internal (n+1)-groupoids is monadic above the category N-n-GrdE of normalized n-groupoids. The definition of the functor Un : (n+1)-GrdEà N-n-GrdE is inductively clear. On the contrary the proof of the monadicity is not easy and the geometric intuition is lost. More abstract and technical notions (such as those of standardized adjunctions and monads) are needed.
17.-The structural nature of the nerve functor for n-groupoids. To appear in Applied categorical structures
The notion of the nerve of a category is well known since Grothendieck. From that time, the question of the nerve for n-categories has been studied by various authors, mainly at the Australian (Street and all) and the Welsh (Brown and all) schools. It is always presented in a geometrical combinatorial way which is descriptively growing in a rather discouraging complexity as the level n grows up, and always defined only in the set-theoretical context.
The previous paper about the monadicity theorem for n-groupoids, using a kind of polyhedral presentation of them, led here to an intrinsic and synthetic definition of the nerve, in the case of n-groupoids. It has the advantage to use always the same inductive construction when going from the level n to the level n+1, and to make sense in any internal context.
The whole process in question is based upon the observation that the categories S-Simpl1E, S-Simpl2E,...,S-SimplnE of the split 1,2,..., n-truncated simplicial objects in E can be obtained successively, one from the previous one, by a unique iterative construction (E1,T1),(E2,T2),...,(En,Tn) from a category Ei endowed with a monad Ti to the category Ei+1 with the monad Ti+1, the starting point being E with the identity monad.
More precisely starting from a left exact fibration t :Eà B and a monad (T,l ,m ) on E, no relationship between t and (T,l ,m ) being required, the category TE of the T-elements of E is introduced, whose objects X1 are defined as triples (X0,d,t) of an object X0 of E, a map d :X1à TX0 in the fibre of TX0 and a map t :X0à X1 in E such that d.t=l X0. Such a T-element must be thought of as a map in the fibres of t which is split " modulo the monad T ". Moreover, there is on TE a monad (T1, l 1,m 1) with T1X1 = (X1,dĂ,tĂ) where the map dĂ is the pullback of d along m X0.Td, and the map tĂ=[1, l X1].
When the monad (T, l ,m ) is left exact, then the functor t o :TEà E defined by t o(X1)=Xo is again a left exact fibration and the monad (T1, l 1,m 1) is again left exact. It is therefore the beginning of an inductive process.
The relationship of this construction with the simplicial situation is the following one. The functor F :S-SimplEà SimplE from the split simplicial objects in E to the simplicial objects, forgetting the splittings, has a right adjoint U which determines a monad (T,l ,m ) on S-SimplE.
Actually this monad is stable on any subcategory S-SimplnE of n-truncated split simplicial objects, and we denote by (Tn, l n,m n) the monad at this level. Our main remark is that S-Simpln+1E=Tn (S-SimplnE), i.e. the category of Tn-elements of S-SimplnE, with respect to the fibration
t n-1 :S-SimplnEà S-Simpln-1E, which forgets the last level. The new fibration and the new monad on S-Simpln+1E determined by the inductive process are, of course, t n and (Tn+1, l n+1,m n+1).
Now this remark is the heart of the inductive definition of the nerve functor for the n-groupoids through the following result : given a category E with a left exact fibration t :Eà B and a left exact monad (T,l ,m ) on E there is always a comparison functor nT :GrdtAlgTà AlgT1 where GrdtAlgT is the full subcategory of the category GrdAlgT of those internal groupoids in AlgT which are actually groupoids in the fibres of t .
Thus, it is the fact that the category S-SimplnE possesses the structure of the iterated T-elements which determines entirely the existence of the nerve functor in the case of the n-groupoids.
18.-n-Groupoids from n-truncated simplicial objects as a solution ot a universal problem. Cahiers du Langal n°17 (1997) Preprint Univ-Littoral. To appear in Journal of Pure and Applied Algebra.
The previous paper  shew that the tower of fibrations :
Eß PtEß S-Simpl1Eß S-Simpl2E...S-SimplnEß S-Simpln+1EÓ was inductively built by the construction of the iterated categories of T-elements from the final fibration Eà 1 and the identity monad.
On the other hand, the paper  built inductively another tower of fibrations :
Eß PtEß N-GrdEß N-2-GrdE ... N-n-GrdEß N-(n+1)-GrdE..., also equipped with monads (Tn, l n,m n) on N-nGrdE whose category of algebras was the category (n+1)-GrdE of internal (n+1)-groupoids in E .
These two towers were related at the level of their algebras by the nerve functor . The aim of this paper is the study of the direct link which related them. This link is actually based on a very simple observation. Let t :Eà B be a left exact fibration, then any fibre has a terminal object. Now given a left exact monad (T,l ,m ) on E, it is always possible to associate with the fibration t , a left exact fibration t :Eà B in which the terminal objects in the fibres are also initial, with respect to the given monad (T,l ,m ).
If we denote by TE the combination of the two constructions : first, from t , the construction of the T-elements  t 0 :TEà E, then to :TEà E, we get a new inductive process, which, starting from the final fibration Eà 1 and the identity monad, produces the tower of the normalized n-groupoids.
Thus, and it is the claim of this paper, the construction which forces the terminal objects in a fibre to be also initial, determines the normalized n-groupoids from the n-truncated split simplicial objects. There is hidden, here, as intuitively felt in , a kind of 1,2,...,n-geometry of the notion of choice.
As a consequence, there is a characterization of the nerve of n-groupoids. This characterization simplifies these of Dakin and Ashley (1988) and has a meaning even in the internal context. Let us call q -complex in E a pair (X,T) where X is a simplicial object and T=(Ti),1£ i, a graded subobject of X, with TiÌ Xi. The elements of T are called thin and satisfy the following axioms, which can be simply expressed in terms of limits :
-All degenerate elements of X are thin
-Any initial horn is X has a unique thin filler
-If all but the last one of the faces of a thin element of X are themselves thin, then also is the last face thin.
Then (X,T) is the nerve of an internal groupoids as soon as (X,T) is of rank n (Tm=Xm, " m>n).
The method of this characterization, based upon an inductive algebraicization of the definition of the nerve leaves aside of course, but rather frustratingly, the geometric description for the benefit of a decisive algebraic property : the nerve functor n n+1 is cofibrant on the hypomorphisms.
VI Protomodularity and related topics.
19.-Mal'cev categories and fibration of pointed objects, Applied Categorical structures 4,(1996), 307-327.
The observation in  that when the fibration p :PtEà E of pointed objects in E is trivial (each change of base functors is an equivalence), then it is additive (each fibre is additive as well as any change of base functor), led here to some classifying properties for this fibration p.
-It is additive if and only if the category E is Naturally MalĂcev in the sense of Johnstone (1989)
-It is unital if and only if the category E is Mal'cev in the sense of Carboni, Lambek, Pedicchio (1990).
A fibration is unital when each fiber is unital and each change of base functor is left exact, a category C being unital when it is left exact, has a zero object and is such that for any pair of objects X,Y, the pair of maps ix :Xà XxY, iy :Yà XxY is jointly strongly epic, which means that any product XxY is generated by Xx1 and 1xY.
Actually other characterizations of Mal'cev and Naturally Mal'cev are given which show that :
20.-(with G. Janelidze) Protomodularity, descent and semi-direct products Theory and Applications of Categories, 4, n°2, (1998), 37-46.
An exact category with a zero object is protomodular  if and only if its satisfies the " split short five lemma ". This paper is devoted to two new observations about protomodular categories :
-using descent theory, one can generalize various form of the short five lemma to non-exact categories, which again gives conditions equivalent to protomodularity.
-the change of base functors of the fibration p :PtEà E, in many important cases as in the case of the category Gp of groups, is not only conservative but also monadic. This has an important application : a categorical definition of a semi-direct product which generalizes the algebraic notion of semi-direct product.


106 年 - 106 國立中山大學_碩士班招生考試_資工系(甲組)：離散數學#105791-阿摩線上測驗
106 年 - 106 國立中山大學_碩士班招生考試_資工系(甲組)：離散數學#105791
1. Let x1, x2, , be a sequence of n integers. A consecutive subsequence of x1,x2,... ., is a subsequence for some i, j, 1 ≤ i ≤ j ≤n. Show that for any k, I k n, there is a consecutive subsequence whose sum is divisible by k.
2. Assume that a sequence of numbers is deined by x0 = 0, x1 = 1, and = ＞ 1. Find generating function for the sequence, and then find an explicit expression for un.
There are 7 problems in this test. No calculators are allowed. Write down detailed steps for
the solution to each problem. Otherwise, no credits for that problem will be given.
4. A bipartite graph is a graph whose vertices can be partitioned into two subsets X and Y so that each edge has one end in X and the other end in Y: A cycle of G is a sequence of vertices u0, u1, ..., such that each vertex is distinct, except ,and ； are adjacent for each i = 1,2,. ,I. The length of the cycle is l. A k-cube is a graph whose vertices are binary strings of length k, for some integer k ＞ 0.Two vertices are adjacent if and only if they differ in exactly one bit.
6. Fibonacci numbers are deined as f0 = 0, f1= 1 and for n ＞.1. Show that is even, for every positive integer k. .
(a) Show that if'n is prime, then n divides for every i, 1 ≤i＜n.


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Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right. Graphs are one of the prime objects of study in discrete mathematics.


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Colloquium LML - conférence Y.D.Arthur (Ghana) [March 9, 2023]
Prof. Yarhands Dissou Arthur, Faculty of Applied Sciences and Mathematics Education - AMMUSTED - Ghana (visio)
« Modelling Student’s Interest in Mathematics : Role of History of Mathematics, Peer-assisted, and Student’s Perception. »
Abstract: The research was to ascertain the mediating effect of students’ perception in the relationship between the history of mathematics, peer-assisted learning, and students’ interest in mathematics. The study was based on 394 students from four selected secondary schools in the Ashanti region of Ghana. The study was a survey, adopting a structured questionnaire as the research instrument. Structural equation modelling was estimated using Amos (v.23) software, to test the hypothesized paths. History of mathematics had no direct effect on students’ perception towards mathematics, thereby, students’ perception acted as a full mediating variable. Peer-assisted learning had a direct effect on students’ perception towards mathematics, thereby, students’ perception acted as a partial mediating variable.
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Updated on:March 10, 2023, 10:31 a.m.
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The odds of winning at a casino are not very good at all. And for roulette, it's even worse. As financial author J.B. Maverick reports in Investopedia, the house has a 5.26 % edge on an American roulette wheel, a percentage that's much higher than other games like blackjack. What does this mean?
"A roulette wheel is numbered from one to 36, so you might think this puts the odds for winning a single number bet at 36 to one," Maverick writes. "However, roulette wheels also have a zero, and sometimes they have a double zero and even a triple zero. The actual odds of winning are thus 37 to one, 38 to one, or 39 to one, not 36 to one."


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I know we all expect mobile games to be rigged. Anything goes in the name of engagement and retention numbers, but it’s always hard to prove. Hidden odds, complicated algorithms and “power ups”, and incomprehensible stat sheets all obfuscate the actual pRNG (psuedo Random Number Generator). But recently, a mobile game I’d been playing held a special event that let you roll dice. This should be a clear look straight at the results of the RNG, but is it?
To find out, I rolled a pair of electronic dice 200 times, meticulously noting the values in a spreadsheet. The things we do for science. At first glance, intuitively, the data looks off:
That doesn’t look right at all. These should be even (aside from random noise), and instead we have a evenly sloped line – showing that higher numbers are encouraged on the dice (and in-game, higher values are rewarded). Now that we’ve got that down, lets look at doubles (also rewarded in-game):
We’d expect a 16% chance (first dice comes up X. Second dice has a 1:6 chance of matching X), but we’re at 23%. More damningly, Doubles occurred at a 30% rate in the first third of the data (not pictured, see spreadsheet at end). It seems the odds change as you play, starting strong to hook you before tapering towards (but never getting to) a natural rate.
The final aspect that I chose to examine was the rate of 7s. 7s are also rewarded, but rewarded so heavily it would make the game too easy if they came up too often.
But seven should be the most commonly rolled sum. If we put together a table we can calculate the exact chances:
There are 6 out of 36 possible results that are 7, more than any other number. However in our gameplay 7 came up less often than 3, which only has 2 possible creations! (1+2 or 2+1).
So here we have several indicators that the RNG is not truly random, nor even psuedorandom, but instead a weighted number generator that is watching the combined totals in order to manipulate the player into playing more, striving to get that 7 that should be just around the corner. But do we have proof? Perhaps I just got particularly unlucky several hundred times in a row. So how unlikely is it for this to occur?
For the two-dice example, we only had 6^2 options, so we could build a table and calculate the odds by hand. However, in order to build a table to match all 200 runs, we’d have (6^2)^200 options! Even google gives up at that point.
This is where we get to the hard math. Binomial distributions – essentially, how likely is a combined result of X, given N trials and a probability per-trial of p. This requires a lot of summations, factorials, and integrations… I don’t have time for that, I have fake dice to roll! Luckily excel will do it for us: BINOM.DIST([occurrences], [trials],[probability per trial], [cumulative or single])
So, for example, the odds of flipping 5 or more heads in 10 tosses is BINOM.DIST(5,10,.5, TRUE [meaning x or more, not exactly X]), and excel tells us this is likely 62% of the time. So lets look at our two biggest red-flags:
The chance that we got only 16 (or less) sevens, given 6/36 odds (6 of the 36 possible combinations result in 7), out of 200 rolls:
BINOM.DIST(16,200,6/36,TRUE) = .03%, about 1:3000 odds.
The chance that we got 46 (or more) pairs, in 199 trials, given 1:6 odds:
BINOM.DIST(46,199,1/6,TRUE) = .7%, about 1:120 odds.
And we can combine these and look for the chances that these both occur together, 1:(3000*120) = 1:360k. I didn’t adjust for the lack of 7s making doubles more likely, but these rough numbers are close enough to prove the point. I’m sure if we added the chances for the single dice rolls as well we’d be in 1:1M odds that these results are natural.
Potential mechanism of action: There may be a simple algorithm that would result in all of these red flags: If a 7 is rolled, flip a coin. If heads, return 7. If tails, increment one dice. This rule change would explain the slope of the singles, the lack of 7s, and the increase in doubles – but this increase would be located only on 4:4. So to test this hypothesis, lets create a graph showing the frequency of each pair:
We’d expect a flat chart, so this is ridiculously unnatural. However, it’s not unnatural in the way we hypothesized above, so the proposed algorithm is clearly not the one they are using. Fewer ones were rolled than any other number, and yet they are most likely to be in a pair. Clearly something else is going on.
So remember: When you’re not paying for the product, you may be the product.
More data more better, right? I gathered another 280 pairs and reran the analysis.
The rate of doubles is identical at 23%. The individual numbers were a bit more mixed, but still trended towards the most 6s, and not nearly the correct number of sevens.
Rerunning the binomial distribution on the combined data set leads to:
23% rate of doubles in 475 throws: .017% = 1:6000
(39)7s in 475: 1:17M
Postscript: All the raw numbers and calculations are available here: https://docs.google.com/spreadsheets/d/1h1KqF-IuFLbCTrlajyu_8774xoV1nsafQXCPMrw_5mU/edit?usp=sharing . If you’d like to do further analysis or correct any mistakes I may have made, please reach out, we’d love to get even deeper into these numbers.


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Additional resources for Algebra: Groups, rings, and fields
Nevertheless we must be careful not to overlook hidden isomorphisms. Proposition 17. If H and K are cyclic groups of respective orders m and n which are relatively prime, then H K (Zmn; +). Proof. In view of Theorem 2 we need only prove (Zm; +) (Zn; +) is cyclic of order mn. But ( 1]; 1]) obviously is an element of order mn, as desired. For example, Z3 Z2 Z6. The direct product of an arbitrary number of groups is de ned to be the Cartesian product with the operation de ned componentwise. This construction can provide groups that are generated only by large numbers of elements, cf.
G=N is a group homomorphism: (ab) = Nab = NaNb = (a) (b): (Ne) = fa 2 G : Na = Neg = N , by For the last assertion, ker = 0 Remark 2:9 . The group G=N is called the residue group, factor group, or quotient group. The homomorphism of Theorem 12 is called the canonical homomorphism. Let us now consider three instances when N / G and determine the structure of the group G=N . Example 13. (i) G = (Z; +) and N = nZ. Then G=N = fN + 0; N + 1; : : : ; N + (n 1)g; with the group operation given by (N + a) + (N + b) = N + (a + b).
7), and the second given in Exercise 13. 7. (Euler's theorem) Prove a'(m) 1 (mod m), for any relatively prime numbers a and m. 8. If m = p1 p2 for p1 ; p2 prime, show '(m) = m pm1 pm2 + 1. ) 9. For p prime, '(pt ) = pt pt 1 = pt (1 1p ). 10. Generalizing Exercise 8, show if m = pi1 pj2 for p1 ; p2 prime numbers, then '(m) = m(1 p11 )(1 p12 ). Can this be generalized to arbitrary numbers? 7) P 11. djn '(d) = n: (Hint: De ne f : f1; 2; : : :; ng ! ) 14 12. De ne the Mobius function 81 >< (n) = > 0 :( 13.
Algebra: Groups, rings, and fields by Louis Rowen


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The Seminars on Undergraduate Mathematics in Montreal (SUMM) is organized by undergraduate students from Montreal universities. The main objective is to create an environment facilitating exchange of ideas and interests as well as allowing students to network.
SUMM is for undergraduate students in mathematics or related domains. This year, the conference will be held on January 8 and 9, online.
The weekend consist of two days of presentations given by undergraduate students and invited professors. The presentations can cover a broad range of subject from mathematical physics to the applications of artificial intelligence as well as the history and philosophy of mathematics.
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