"I think I'll stop here ", concluded Prof Andrew Wiles , his third and final lecture of the series at the Cambridge University, to the transfixed audiences which included some of the prominent mathematician of the world. Only a few understood those symbols and equations that was scribbled on the board. But most of them knew that they witnessed one of the biggest moment in the history of mathematics. The 300 year old, longest standing mathematical problem has now been solved. Fermat's Last Theorem, which puzzled, frustrated and challenged the brilliant minds across the globe for over three centuries is now been addressed. The Misery resolved, the mathematicians around the world is now rejoiced.
Some of the toughest problems of the world are simple, yet difficult to prove. We all have learned the Pythagoras' theorem of right angles triangles. a (2) + b(2) =c(2) where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
Mathematician who followed, Pythagoras , like Euclid , Dyophantus and the those of Hypatian school of Alexandria made early progress in the development of Mathematical Sciences. During renaissance period, the newly learnt wisdom of Arab's and the Eastern world, the European thinkers and mathematicians gave giant leap to the science of mathematics.
In the early 17th century Frenchman Pierre de Fermat, an amateur mathematician was famous for his riddles, usually aimed at those across the Channel at England. While exploring the Dyaphantus book 'Arithametica' he wrote his famous observation which later came to be known as 'Fermat's Last Theorem' , tormenting the mathematicians around the world for next three centuries. His theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. On the margins of the book he also noted that "I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. ( it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain)
The master riddler, now sent the world in a spin with rather innocent looking observation. The next few centuries saw mathematicians trying to find out the elusive proof, which Fermat supposed to have found but did not bother to write down for want of space in the margin ! The "number theory " branch of mathematics saw quantum progress, thanks to one riddler called Pierre de Fermat.
The world had been seeing many such conjectures and theorems over the years. Many were either proven or broken with the latest methods or the technological improvements, largely with the invention of computers. However, this one remained elusive. Some people even gave up trying, despite some large amount of money being promised by many institutes and individuals.
Simon Singh's book on the triumph over Fermat's theorem , takes us through this mathematical journey. From the early days of Mathematics, secretly hidden in the "Pythagoran sects" later flourished under the various rulers and thinkers in the early ages, to the days of Fermat and the later mathematicians, directly of indirectly influencing the progress in number theory. This book, resulted of a documentary he directed for BBC on Andrew Wiles ( see the you tube link here) , is a detailed exploration of the birth and its progress through the millennium, the centuries, decades and years of progress of mathematical wisdom.
There were tricksters as well. People who churn out 'theorems' and conjectures to the world, And there are many which at the first look appears to be true. Look at the below sequence.
31 ; 331 ; 3,331 ; 33,331 ; 333,331 ; 3,333,331 ; 33,333,331 ...all these are prime numbers. However, the next number in this pattern 333,333,331 is not a prime. 333,333,331 = 17 x 19,607,843.
Similarly, Euler , 18th century mathematician extrapolated the Fermat's Theorem with his on version. Euler's conjecture said , there are no whole number solution for x 4 + y4 + z4 =w4. For two hundred years nobody could prove Euler's conjecture.On the other hand , nobody could disprove it by finding a counter example. LAck of counter example is a strong evidence in favour of the conjecture. Then, in 1968, Naom Elkies of Harward University discovered the following solution
2682449(4) + 15365,639(4) + 18,796,760(4) = 20,615,673(4) , thus proving Euler's conjecture wrong..
Various mathematical wiz-kids attempted this theorem, moving towards the proof. In the 18th and 19th century, the proof for the power of 3,4,5 and upto 7 were established. But none of them is not good enough in the strict mathematical arena as the proof for the theorem. With the help of the computers and the progress over the centuries, mathematicians now are able to use the new techniques and tools that are available to them. Improvements in the other areas of mathematics, especially in the algebraic equation helped advancement in this regard. Then in 1959, two Japanese Mathematicians, made a great announcement comparing 'elliptical equations' to the 'modular forms', which later came to fame as Taniyama-Shimura Conjecture. There were attempts to prove this and a theorem comparing Taniyama-Shimura conjecture to the Fermat's Last Theorem was evidently established. Which in other words mean that if Taniyama-Shiumura Conjecture is proved, Fermat's Last Theorem is true. With the renewed vigour, mathematicians around the world got back to their scratch pads, trying to get the conjecture proved.
Andrew Wiles, by now a professor at Princeton , had his childhood dream on working and proving Fermat's theorem. The rest of the book is on his 7 years of efforts , his small achievements and many set backs on working on his dream project. to protect his efforts and achievements ( you do the 99% of the works and taking it from here someone else completing the job, thus taking all the claim is something he wanted to avoid) he worked in secrecy, even hiding from his own close associates and friends.
While Andrew Wiles was shut himself in a room working with single objective, a Japanese Mathematician Miyaoka in 1988 announced to the world that he has found a proof, shattering Wiles dreams. However, to the relief of Andrew, his claim was also failed to break the long pending wait for proof , triggering an interesting graffiti at the NY underground with the artist declaring " I have discovered a truly remarkable proof of this, but I can't write it now because my train is coming."
Then on the eventful day 23 June 1993, he announced to the world at his alma matar , in the presence of his mentor , that he had found the proof to the Fermat's Last Theorem. But the process of verification and authentication remain. The journal has to recruit some of the best judges to go through his manuscript and scrutinize every line to make sure there are no mistakes, any points overlooked or made any un-clarified assumptions. As feared, there was a glitch as one of the points did not work in all the possible hypothesis. To cut short, the next one and half years were heartburn for Andrew. As is in many cases, he was about to abandon his effort and accept the inevitable, the solution to the problem appeared to him. He was more relieved than elated.
Now that the long standing problem is solved, the mathematical world is suddenly has no new challenge (well, not to this level of complexity). That also means, the world need another motivation that can trigger the next wave of progress. As Wiles mentioned, the sense of purpose is now lost. The remaining of the days will not be as directed as it used to be. There is still hope for the enthusiasts. Most of the methods and tools by Andrew Wiles and his contemporaries were not available to Fermat while he worked on this. And if his claim is true, then there is a much simpler way to prove the theorem. The quest is now on to find out the proof Fermat had in his mind..Fermat's Enigma ( 1997)
NY Times, Leegruenfield,


Vladimir N. Strokov
Post-doctorate Researcher (UFES) & Senior Fellow Researcher (LPI RAS)
Current research interests: Gravity and cosmology, viz. singularity problem, quantum and physical cosmology, cosmological perturbation theory.
Postdoc at UFES (ES, Brazil) (2013-present)
Senior Fellow Researcher at LPI RAS (Moscow, Russia) (2011-present, on leave)
Postdoc at UFJF (MG, Brazil) (2012-2013)
PhD in astrophysics at LPI RAS (Moscow, Russia) (2007-2011)
B.S. and M.S. in applied physics and mathematics at PhysTech (MIPT, Dolgoprudny, Russia) (2001-2007)
List of papers: InspireHEP.
Curriculum: Lattes (Brazilian CV)
E-mail: @gmail, @asc.rssi.ru


One of Stanford mathematician Keith Devlin's pet peeves is the common “division is repeated addition” meme . He despises it so much he has something like a mantra, “Repeat after me. Division is NOT repeated addition.” Naturally, math teachers give him a lot of pushback because division is indeed repeated addition (except when it's not).
It seems there are actually two topics in play here. 1) Multiplication as repeated addition and 2) the skill of elementary math teachers. I completely understand his frustration with prospect of undoing the poor math instruction college students typically receive during their elementary school years. I also experience the same frustration as a secondary and college level instructor.
Multiplication as repeated addition is not a definition of multiplication, even though many elementary math teachers erroneously think the definition of multiplication is precisely repeated addition. Repeated addition is merely another name for the group model of multiplication. There are other models, such as the array model, the area model and the number line model, to name the ones most commonly presented to elementary students. Devlin rightly maintains that it is inaccurate to say that multiplication is repeated addition, period. As a misleading misstatement, it is right up there with “you cannot subtract a bigger number from a smaller number.”
However, as properly taught (a giant qualifier, I know), the group model is merely the first element in a teaching sequence which eventually progresses to the area model, then to the use of the area model to multiply fractions, and beyond. For example, you can definitely model a positive whole number times a negative rational number on the number line where it very much looks like repeated addition of the given negative number. Turn that number line vertically, and it makes even more sense to students because it reminds them of another number line they are very familiar with, the thermometer.
Devlin writes, “Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum.” Exactly, and this is precisely the way a good teacher presents the group model. Children sometimes ask questions like, “Instead of saying 8 + 8 + 8 + 8 + 8, and then saying the answer, can't we just say “8, five times” and then say the answer?” Of course we can, and that is what we do when we say 5 x 8 = 40. The group model is meant to express this particular connection between addition and multiplication. The group model is not meant to be a definition of multiplication. Nevertheless, I agree there are too many elementary math teachers who fail to make the distinction, or properly progress through the models.
An umbrella idea I like involves the word “of” as an English language expression of multiplication. We can say “5 groups of 3,” or “5 groups of -3,” or “1/2 of 3,” or “1/3 of 4/5,” or “16/100 of 40” or “75% of 200,” etc and neatly cover most examples of multiplication that children are likely to encounter before junior high. Devlin prefers scaling as the dominant meme and argues that children should readily understand scaling because examples of scaling surround them. The problem is most eight-year-olds have difficulty comprehending scaling as a model and effect of multiplication. Even though they can readily see that a scale model is a perfect replica of the original, they do not understand how it is possible that doubling the dimensions of a garden (to take a simple example) results in a garden four times larger. Most of the scaling children see is usually on maps where the scale is for them an unimaginably large (or small, depending on viewpoint) number.
Teachers are better off working their way up to the scalar model of multiplication. I have found this is best done by reminding younger students early and often with the idea that we have not yet exhausted the possible models and applications of multiplication. I have found it useful to show some examples of these applications, and say something like, “Later you will learn how you can use multiplication to produce an exact scale model, or use multiplication to produce a real-life-sized object from a scale model.”
Actually, most students get their first solid grip on scaling when they work with similar figures (typically triangles) during high school geometry. Personally, I have found success with older elementary students by giving them basic practice in scaling on the coordinate plane or increasing recipe yield and other types of problems. Students also enjoy the products of their work whether it be art or good eats. The number line model is also a good introduction to scaling because you are scaling only one dimension, as opposed to the two and three dimensions involved in scaling area and volume, respectively.
Devlin also laments the constant push to make math “real.”
No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in...many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.
His comments seem contradictory, but they are not. One of the most enjoyable aspects of teaching math is showing students the leap from concrete to abstract. For example, I love showing students a cube and showing them the edge e, the face e x e, then showing them cube e x e x e. I usually let the idea hang, and love when someone asks if I can show e x e x e x e on the cube. No, I answer, I have nothing to show them that. And therein lies the power of math. Math can help us express ideas we can understand, but for which we have no physical representation. I am thrilled when someone asks if e x e x e x e can be time. I ask what would e x e x e x e then express. The children are really products of this century, and one of them is likely to answer, “the coordinates of a time-traveling spaceship.” So much fun. One time, a child said, “Maybe someday we will have a real meaning for more es.”
To the extent that teachers present repeated addition as a property, which in some applications, connects the operations of addition and multiplication, no harm done. However, considering repeated addition as THE definition of multiplication is a serious problem, and Devlin is right to be concerned about it. As Denise Gaskins pointed out, “...if (a particular) model doesn’t work universally, then (the model) certainly cannot be used to define the operation.”
Devlin asks an interesting, and in today's pedagogical environment, nearly taboo question:
The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way?
I would say that technique over understanding has been the preferred approach for centuries, but by all accounts, many adults have never made it to the “understand later” stage. In my experience teaching concept concurrently with technique works the best. The problem I am seeing is that these days too many elementary teachers attempt to teach concepts they barely understand, and then give short shrift to technique because the kids have calculators for that. The result is legions of kids who not only have faulty understanding of the concept, but also lack the ability to perform the technique quickly and accurately.


Assuming that you will have an 7-8 litre reserve that is pretty impressive. By my calculations 77-80mpg.
Apologies was using us gallons
1 litre = us gallon = 0.264172
1 litre = uk gallon 0.219969
42-7=35, 35*0.219969=7.69892, 600/7.69892= 77.933 mpg
42-8=34, 34*0.219969= 7.47895
I normally manage to get mid 600s from a tank , which is working out to around 70-75 mpg , however , the car has the myp 6 speed gear box, plus a range of the 1.2 TDI drag reducing mods , the winglets , Exhaust cover , and air dam , along with a re-map and EGR delete
It’s normally just me in the car , with the cc set to 58 mpg hiding behind a large hgv .....


|Birth||(age 38–39) Bowral,Australia|
|Birth Place||(age 38–39) Bowral,Australia|
|Alma Mater||University of Sydney(BA) University of Freiburg(PhD)|
|Institution||University of Sydney
University of Oxford
Max Planck Institute for Mathematics)
|Thesis||Singular Soergel bimodules(2008)|
Events Occured in Scienctist Life
Geordie Williamson (born 1981 in Bowral, Australia) is an Australian mathematician at the University of Sydney.
He became the youngest living Fellow of the Royal Society when he was elected in 2018 at the age of 36.
Educated at Chevalier College, Williamson studied from 1999 at the University of Sydney and graduated with a Bachelor's degree in 2003 and then at the Albert-Ludwigs University of Freiburg, where he received his doctorate in 2008 under the supervision of Wolfgang Soergel.
After his PhD, Williamson was a post-doctoral researcher at the University of Oxford, based at St. Peter's College, Oxford and from 2011 until 2016 he was at the Max Planck Institute for Mathematics.
With Ben Elias, he gave a new proof and a simplification of the theory of the Kazhdan–Lusztig conjectures (previously proved in 1981 by both Beilinson–Bernstein and Brylinski–Kashiwara).
The conjecture was proved in 1994 by Henning Haahr Andersen, Jens Carsten Jantzen and Wolfgang Soergel for sufficiently large group-specific characteristics (without explicit bound) and later by Peter Fiebig for a very high explicitly stated bound.
He also found counterexamples to a 1990 conjecture of Gordon James on symmetric groups.
In 2016, he received the Chevalley Prize of the American Mathematical Society and the Clay Research Award.
In 2016 he was awarded the EMS Prize, for 2017 he was awarded the New Horizons in Mathematics Prize.
In 2018, he was plenary speaker at the International Congress of Mathematicians in Rio de Janeiro and was elected a Fellow of the Royal Society (FRS) and the Australian Academy of Science.
Williamson was awarded the 2018 Australian Mathematical Society Medal.


Prof. Dr.-Ing. Jens Deppler
Karlsruher Institut für Technologie (KIT)
Regularization of Nonholonomic Constraints in Multibody Systems
Rolling contacts are usual in various technical systems. Gear wheels in gearboxes, the motion of rolling elements in roller bearings or the Euler disk can be mentioned here as examples. Even the sliding of a clutch disk on an elastic support can become unstable transitioning into a rolling motion. In most cases rolling motion conditions yield non- holonomic constraint equations. Usually the non-holonomic constraints can be incorporated by the method of Lagrange multipliers. This formulation leads to index-2 differential algebraic problems. Different regularization approaches have been developed for higher index DAEs because of the numerical drift problems inherent for usual ODE methods. In the present paper we investigate a new regularization method that is motivated by physical considerations. Pure rolling is equal to “sticking” with a kinematically repositioned contact point. Usually sticking is modeled by introducing elasticity in the contact. Although constraints are mainly enforced by the elasticity in that case and the dissipative terms are necessary in order to avoid numerical oscillations in the contact. A suitable choice of the order of magnitude of the dissipative forces enhances the numerical performance of the method. The purpose of this kind of regularization is to get a consistent description for sliding, sticking and rolling contacts. Stamm applied this kind of regularization to a tangential contact law, extending the classical laws of friction to distributed contacts, where the problem of indeterminacy in the sticking state is circumvented by regularization. It this work we prove the convergency of the spring-damper regularization for the so called principal damping, which is motivated by the critical damping in the linear case, and the strong damping which is of the same magnitude order as the contact stiffness.
Contact: Prof. A. Fidlin, J. Deppler
|Auslegung eines Prüfstandskozepts zur Abbildung verbrennungsinduzierter Schwungraddynamik||Masterarbeit||
|Numerische Methoden für Mehrkörpersysteme||Diplomarbeit||
|Implementierung eines Kontaktmodells zur Simulation flächiger Reibkontakte||Masterarbeit||


#BetaPipol at 35, Abdon Atangana, is one of the best scientists in the world. He is a lecturer at the University of Free State (South Africa) and has been named among the top 1% of the best mathematicians whose work is the most cited in the world.
With an h index of 53 on Google Scholar, Abdon Atangana is on the prestigious global Clarivate Web of Science list. According to Clarivate, this distinction rewards true pioneers in their field over the past decade.
Mr. Atangana is placed on this list because of his cross-sectoral contribution, which includes mathematics and applications to real-world problems.
He has developed a new mathematical fractional operator used to model problems such as the passage of groundwater flow, the spread of infectious diseases, weather behaviour, and the type of cruise control in cars.
“Most maths formulas come from Europe, so when I introduced my operator, it was like a feeling of decolonization. It shows that Africans can also compete on the world stage.” – He said.
In 4 years, he has published in more than 20 leading journals of applied mathematics and mathematics; and in some, he serves as the first African selected as editor-in-chief.
His inspiring journey, includes the fact that he appears in the list of top 10 scientists in South Africa.
He recently received the TWAS Mohammad A. Hamdan Prize from the World Academy of Sciences for the advancement of science in developing countries.
Team 237 Oyeh 🇨🇲
Home » A Cameroonian Mathematician; Abdon Atangana, features among the Top 1% best mathematicians in the world!


2 edition of Bessel functions found in the catalog.
by Published for the Royal Society at the University Press in Cambridge (Cambridgeshire)
Written in English
Bibliography: p. xxi-xxiii.
|Other titles||Kelvin functions.|
|Statement||prepared on behalf of the Mathematical Tables Committee by Andrew Young and Alan Kirk. Part IV, Kelvin functions.|
|Series||Royal Society mathematical tables -- v. 10|
|Contributions||Kirk, Alan., Royal Society (Great Britain). Mathematical Tables Committee.|
|The Physical Object|
|Pagination||xxiii, 97 p. ;|
|Number of Pages||97|
Originally published in , this monumental treatise had two major objectives at that time--the development of applications of the fundamental processes of the theory of complex variables and the compiling of a collection of results of value for mathematicians and /5. Integrals of Bessel Functions concerns definite and indefinite integrals, the evaluation of which is necessary to numerous applied problems. A massive compendium of useful information, this volume represents a resource for applied mathematicians in many areas of academia and industry as.
Bessel functions. Cambridge, University Press, (OCoLC) Document Type: Book: All Authors / Contributors: W G Bickley; Frank W J Olver; Andrew Young; British Association for the Advancement of Science. Mathematical Tables Committee,; Royal Society (Great Britain). Bessel Functions Panel. As you can see from Fig. the Bessel function of first and second kind have a number of roots, i.e., values for x n at which J ν (x n) = 0 and Y ν (x n) = it turns out, these values are very important and frequently used when dealing with Bessel functions and in particular with Bessel differential equations.
A full, clear introduction to the properties and applications of Bessel functions, this self-contained text is equally useful for the classroom or for independent study. Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of /5(6). APPENDIX C Bessel Functions C.1 Series Solution of Bessel’s Equation In a certain class of differential or difference equations encountered in many branches of science and engineering, Bessel functions and - Selection from Digital Communication Systems [Book].
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Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order. More than problems throughout the book enable students to test and extend their understanding of the theory and applications of Bessel by: The zeroes of Bessel functions are of great importance in applications .
The zeroes, or roots, of the Bessel functions are the values of x where value of the Bessel function goes to zero (Jν(x) = 0). Frequently, the zeroes are found in tabulated formats, as they must the be numerically evaluated .
Bessel functions of the first FIG. Bessel functions are named for Friedrich Wilhelm Bessel ( - ), however, Daniel Bernoulli is generally credited with being the ﬁrst to introduce the concept of Bessels func-tions in He used the function of zero order as a solution to the problem of an oscillatingFile Size: KB.
Bessel function are an inexhaustible subject – there are always more useful properties than one knows. In mathematical physics one often uses specialist books. Back to top; Bessel Functions of General Order; Sturm-Liouville theory.
Bessel functions, being the solutions of an SL equation, are a set of orthogonal functions. Bessel functions Bessel functions book form (by the expansion theorem, section ) a complete set, just as with the other orthogonal functions developed in chapter 5 (complex exponentials) and in chapter 6 (Legendre polynomials and spherical harmonics).
The. G.N. Watson A Treatise on the Theory of Bessel Functions () Cambridge University Press The standard work on the subject.
Acrobat 7 Pdf Mb. Scanned by artmisa using Canon DRC + flatbed option. 1Introduction to Bessel Functions Bessel functions are the standard form of the solutions to Bessel’s di erential equation, x2 @2y @x2 + x @y @x + (x2 n2)y= 0; (1) where nis the order of the Bessel equation.
It Bessel functions book often obtained by the separation of the wave equation @2u @t2 = c2r2u (2) in cylindric or spherical coordinates. For this reason Author: Joella Rae Deal. Bessel functions are associated with a wide range of problems in important areas of mathematical physics.
Bessel function theory is applied to problems of acoustics, radio physics, hydrodynamics, and atomic and nuclear physics.
Bessel Functions and Their Applications consists of two parts. In Part One, the author presents a clear and rigorous intro3/5(2). 3 Bessel Function The Bessel function J s(z) is de ned by the series: J s(z) = z 2 sX1 k=0 (1)k k!(s+ k+ 1) z 2 2k (29) This series converges for all zon the complex plane, thus J s(z) is the entire function.
If z!0, then J s(z). z 2 s 1 (s+ 1) (30) If s2 is not an integer, then J s(z) is the second solution of the Bessel equation. Now: J s File Size: 2MB.
Bessel Functions For Engineers (Second Edition,Second Printing,Corrected) by Mclachlan, N. Seller Arroyo Seco Books Published A treatise on Bessel functions and their applications to physics by Andrew Gray, G.
Mathews and a great selection of related books, art and collectibles available now at Condition: New. Paperback. Integrals of Bessel functions concern definite and indefinite integrals, the evaluation of which is necessary to numerous applied problems.
A massive compendium of useful ng may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability. pages. Bessel’s equation Frobenius’ method Γ(x) Bessel functions Bessel’s equation Given p ≥ 0, the ordinary diﬀerential equation x2y′′ +xy′ +(x2 −p2)y = 0, x > 0 (1) is known as Bessel’s equation of order p.
Solutions to (1) are known as Bessel functions. Since (1) is a File Size: KB. geometric functions, see also ,or . Some well-known integrals are included for completeness. Here Z ν(x) denotes some Bessel function or modiﬁed Bessel function of the ﬁrst or second kind.
Partially the functions Y ν(x) [sometimes called Neumann’s functions or Weber’s functions and denoted by N ν(x)] and the Hankel File Size: 2MB. Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order.
More than problems throughout the book enable students to test and extend their understanding of the theory and applications of Bessel functions. Using original algorithms developed at Wolfram Research, the Wolfram Language has full coverage of all standard Bessel-related functions\[LongDash]evaluating every function to arbitrary precision with optimized algorithms for arbitrary complex values of its parameters, as well as supporting series and asymptotic expansions with full treatment of Stokes sectors, and an extensive web of symbolic.
Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order. More than problems throughout the book enable students to test and extend their understanding of the theory and applications of Brand: Dover Publications.
Bessel functions are a series of solutions to a second order di erential equation that arise in many diverse situations. This paper derives the Bessel functions through use of a series solution to a di erential equation, develops the di erent kinds of Bessel functions, and explores the topic.
G.N. Whittaker’s classic A Treatise on the Theory of Bessel Functions (Cambridge University Press, ) is a ponderous volume devoted entirely to the subject.
View chapter Purchase book. Book Description. Bessel functions are associated with a wide range of problems in important areas of mathematical physics.
Bessel function theory is applied to problems of acoustics, radio physics, hydrodynamics, and atomic and nuclear physics. Bessel Functions and.
Bessel Functions for Engineers. London: Oxford University Press, Hardcover. 9 1/2" X 6". pp. Blue cloth over boards, with spine lettered in gilt. Moderate edgewear and scattered rubbing and soiling to cloth, with corners and tail of spine bumped. Black marks to rear cover. Mild foxing to endpapers.
Pages clean and unmarked.Power Series Solutions to the Bessel Equation The Bessel equation The equation x2y00+ xy0+ (x2 2)y = 0; (1) where is a nonnegative constant, is called theBessel equation.
The point x 0 = 0 is a regular singular point. We shall use the method of Frobenius to solve this equation. Thus, we seek solutions of the form y(x) = X1 n=0 a nx n+r; x >0 File Size: KB.periodic. The Bessel functions are not quite periodic, though they do start to look more and more like scaled trigonometric functions for large x, roughly speaking:1 J n(x) ≈ r 2 πx cos(x−[n/2+1/4]π) Y n(x) ≈ r 2 πx sin(x−[n/2+1/4]π) For extensive detail on the theory of .


I am an Assistant Professor at Pontificia Universidad Católica de Chile. I have a Master's degree in Applied and Computational Mathematics and a Bachelor's degree in Statistics from Universidade Estadual Paulista Júlio de Mesquita Filho (UNESP). I completed my Ph.D. in Statistics at the University of São Paulo (USP) and Universidade Federal de São Carlos (UFSCar) through the Interinstitutional Program. I also participated in a sandwich Ph.D. program at the University of Connecticut (UCONN) in the United States. I hold two postdoctoral positions from the University of São Paulo (USP). Currently, I serve as a Ph.D. advisor in the Statistics program at the Faculty of Mathematics at Pontificia Universidad Católica de Chile and as an advisor in the Ph.D. program in Engineering and Sciences with Industry at the same university. I am an associate researcher at the Center for Research in Applied Mathematics to Industry, Cepid/CeMEAI, and a collaborating researcher in the Master's program in Applied and Computational Mathematics at FCT-UNESP, as well as an advisor in the professional Master's program in Applied Mathematics, Statistics, and Computation to Industry at the University of São Paulo. I have experience in the field of Probability and Statistics, with an emphasis on Bayesian and Classical Inference, primarily focusing on the following topics: model selection, reliability analysis, and quality control.
Ramos, P.; Jerez-Lillo, N.; Segovia, F.A.; Egbon, O.A.; Louzada, F. Power-law distribution in pieces: a semi-parametric approach with change point detection, Statistics and Computing, 32, 1-15, 2024.
Ramos, P.L, Rodrigues F. Ramos, E., Dey D. Louzada, F. Power laws distributions in objective priors. Statistica Sinica. v33(1), p. 1-26, 2023.
Ramos, P.L.; Almeida M.; Moala, F.A. Flores, E.; Louzada, F. Objective Bayesian inference for the Capability index of the Weibull distribution and its generalization, Computers & Industrial Engineering, 167, 108012, 2022
Instructor, São Paulo University (USP), Brazil, 2015.
Lecture, Trevisan Business School, Ribeirão Bonito.
Lecture, Paulista University, Bauru.
Collaborator Professor, São Paulo University (USP), Sãao Carlos, 2018.
Assistant Professor, Pontificia Universidad Católica de Chile, 2021-Currently.
HONORS, AWARDS AND GRANTS
First Place do Prêmio ANP de Inovação Tecnológica 2020, na Categoria I (joint work with USP, UFRGS, SENAI/SC), ANP.
Finalist of the 2019 ANP Technological Innovation Award, (joint work with other institutions) Brazilian National Oil Agency.
Best Poster Award on Statistics and Data Science, 2nd Conference on Statistics and Data Science, Bahia, Brazil , 2018 .
2 Place in the Best Doctoral Thesis Award from 2016-2018, Brazilian National Symposium on Probability and Statistic, 2018.
NSF/QPRC Award, University of Connecticut, 2017 .
Student Travel Grant - Visiting Scholar at University of Connecticut with scholarship granted by CAPES (Brazilian Federal Agency), 2017.
Invited Speaker - 16Th Brazilian Meeting Of Bayesian Statistics And Vi Latin American Conference On Statistical Computing, 2022.
Young doctor section - Regression Models School, Brazil, 2019.
Best thesis presentation - National Symposium on Probability and Statistics, Brazil, 2018.


Math Play has a large collection of free online math games for elementary and middle school students. Here you can find interactive games designed to make math drills fun and entertaining.
Unblocked Games 66 is home to over 2000+ games for you to play at school or at home. We update our website regularly and add new games nearly every day! Why not join the fun and play Unblocked Games here! Tron unblocked, Achilles Unblocked, Bad Eggs online and many many more. Jun 21, 2016 · Description: Slither.io MMO is the awesome game for you to play for free. The mission that you will accomplish is to help your snake grow longer and dominate the leaderboard with the best score as fast as possible. Download Hooda Math Games and enjoy it on your iPhone, iPad, and iPod touch. 12 Math Games for the Price of 1 Hooda Math Classics like PiMon, Prime Landing, Roots of Life, Carpenter's Cut, Number Eaters, Math Find, Multiplication Game, and Factor Feeder Two Math Games you won't find online: How Much Money? and How Much Change? 5th grade math games for free. Multiplication, division, fractions, and logic games that boost fifth grade math skills. Date added: 2017-08-03 Times played: 33 About DogfightX.com. Prepare yourself for a challenging Space Combat game called DogfightX.com! This IO-styled 2D Shooter game takes place in large space filled with tough opponents. Free Car Games. Now includes iPad Car also works on Kindle and Android - HTML5 Games at Hooda.net. Cool Mobile Games for Kids of All Ages. Pogo Swing Description. You Goal in this game is to get as far as you can from jumping off.
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One of the most popular thinking game is here on Friv-Games.com, the Hangman game from cool math. As you can see it, we have a cute boy who walks through the forest, when suddenly a big black monster appears in front of him who wants to hurt him. At this point, our child clings to the colorful balloons he has and floats above the monster, but you will need to help him by answering correctly ST Math starts by teaching the foundational concepts visually, then connects the ideas to the symbols and language. With visual learning, students are better equipped to tackle unfamiliar math problems, recognize patterns, and build conceptual understanding.
Play Papas Cupcakeria Game Prepare the cakes for customers as well as you can to earn tips. Try your best to become an expert cupcake maker! When youre done playing Papas
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Our preschool program features a multi-dimensional learning environment. Each child is afforded the opportunity to develop pre-reading, problem solving, oral and written language and hands-on art skills.
We focus on helping the child develop into a well-rounded student by focusing on the child’s personal and social development as well as starting their development in math and literacy. We rotate between discovery centers for all areas of learning; Math, Art, Reading, Dramatic Play and Circle Time.
The day is rounded out with time in our 5 different play yards, full of slides, swings, bikes and a sandbox.
- Self Concept
- Social Skills
- Emotional Development
- Critical Thinking
- Math Skills
- Literacy Development
- Motor Development
400 Stafford St. Folsom, CA 95630
Kenneth Ave, Orangevale Location
6822 Kenneth Ave., Orangevale, CA 95662
Elm Ave, Orangevale Location
9371 Elm St. Orangevale, CA 95662


Recently, Jie Li, a 2017 third-year phd student in our center, won the 2020 Hannan Graduate Student Travel Award from the Institute of Mathematical Statistics (IMS).
The International Association for Mathematical Statistics (IMS), headquartered in the United States, is one of the three authoritative statistical academic organizations in the world. It aims to develop and promote the theory and application of statistics and probability, and publish high-quality academic journals, such as Annals of Statistics and Annals of Probability. The honorary awards issued by IMS, such as IMS Fellow and various Travel Awards, are highly recognized by the international statistical community. The Hannan Graduate Student Travel Award, named after the late IMS Fellow and Emeritus Professor of Statistics and Probability at Michigan State University James Hannan, is awarded annually to graduate students and young scholars within two years of Ph.D. graduation all over the world，and rewards them for presenting outstanding papers at international statistical conferences sponsored or funded by IMS. Due to the epidemic this year, each winner can participate in any IMS international conference and receive a funding of US$500 before December 2022. This year, except for Li Jie, the other 23 winners are from universities in the United States, Britain, Canada, Australia and South Korea.
Jie Li’s award-winning paper "Statistical Inference for the Stationary Mean of Functional Time Series" proposes to use B-spline to estimate the mean function of functional time series, prove it "oracle efficient", and then construct the simultaneous confidence bands of the mean function. This method was applied to ElectroEncephalogram data and some meaningful findings were obtained.(Translated by Jie Li)


Now we have established that of the 16 lines going to the LCD panel there are four back-planes and twelve segment lines we can therefore imply that this combination can drive a panel with up to 4×12 i.e. 48 segments.
Each LCD Segment is electrically connected to one back plane and one segment line (although in reality these segments and back planes also interconnect, but we can ignore this for now as the clever multiplexing logic hides this from us), to derive if the segment will be ON (dark) or OFF (clear) we must determine the AC voltage applied to the segment. The easiest way to do this is to numerically difference the waveforms and to calculate the RMS (root mean square) voltage; If the voltage is above the LCD threshold the segment will be dark.
I arbitrarily assign a segment number to the combination of segment line and back plane and calculate the RMS for that combination. The RMS voltages seem to be either around 1v0 or 1v7 so we can imply the LCD threshold is somewhere in between (let’s assume 1v4). We then calculate if the segment should be ON or OFF.
When the scale is powered on but idle it displays 0.00kg which is roughly 6×3+2 i.e. 20 segments dark…. and we get 20 1’s… looks like we are on the right track.


Good Morning Monday!
Good Morning everyone!
So this morning I was wondering about the number 2.
It’s interesting to me in a mathematical sense because I had a mathematical thought today.
Here is how the thought happened or occurred or something….
The number 2 seems different than other numbers.
2 x 2 = 4 and
2 + 2 = 4.
1 + 1 = 2 and 1 x 1 = 1
3 x 3 = 9 but 3 + 3 = 6
4 + 4 = 8 but 4 x 4 =16
5 + 5 = 10 and 5 x 5 = 25
I did the calculations through 12 x 12 = 144 but 12 + 12 = 24 and oh man my mind went into overdrive for this thought. Why is this so? 🙂
Silly mind that I have. I mean there must be a mathematical reason for all of this but I didn’t get that far into the thought because I had to get up and start my day.
I think things that maybe no one else I know thinks. Maybe. At least I do when it’s early in the day.
What are thinking about today?
Whatever it is you are thinking, make it goodly thoughts that will lead to goodly action that will lead to a positive day for yourself.
p.s. you thought I was going to say goodly again didn’t you. 🙂
It’s still simmering there in the back of my mind though, waiting it’s turn to come up front.


Some of the basic probability and statistics concepts. joint probabilities, combinatories; conditional probabilities and independence and their examples; Bayes' rule.
This is the lecture note that I took on the lecture. This post serve as the overview of this series.
Microstrip transmission line is one of the basic type of transmission line in RF integrated circuit. Here is a basic/simple design example of it. This is a homework for course ECE810 RF Integrated Circuits in MSU.
Parameter estimation is one of the keystones in Adaptive control; the main idea of parameter estimation is to construct a parametric model and then use optimization methods to minimize the error between the true parameter and the estimation. Least-square algorithm is one of the common optimization methods.
MathJax is a commonly-used web-based math formula and expression render tool. Here is the test of it.
Blog migration again.


The purpose of this write up is to help the reader estimate the sExample before you hire the PE to do the official calculation.
All electrical systems are susceptible to short circuits . The short circuit currents can produce considerable thermal and mechanical stresses in electrical distribution equipment. Therefore, it’s important to protect personnel and equipment by calculating short-circuit currents during system upgrade and design. Because these calculations are life-safety related, they’re mandated by 110.9 of the NEC.
In the example show in Fig 1 our goal is to find the irrupt rating for the two breakers (1600A and 250A) and also find the withstand rating of the two panels (main and sub panel).
First will calculate the full load current for the 1MVA transformer
I F.L = P / (1.73 * V L-L ) ; where P is the transformer power rating in VA, V L-L is the line to line RMS voltage at the secondary side of the transformer
I F.L= 1,000,000/ 1.73*480= 1,202 A ; the I F.L is the full load current of the transformer
Now to find the short circuit rating of point 1 (i.e if we bolt all the three wires of the secondary side of the transformer how much current we will see at that point)
I S.C. for 3 Ph= I F.L /Z% ; where Isc is the short circuit current , Z% is the transformer impedance which is usually can be obtained from the local utility company
I S.C. 3 ph for point 1 = 1202 /0.05 = 24,506 A or 24.5 KA (this is the available short circuit current at point 1)
The short circuit current at point 2 will be lower than point 1 due to the line impedance ( line impedance for the cables will impose the current)
The calculation below will show the short circuit current at point 2
From the drawing ; cable length =L= 50 ft.
F = (1.73 * L* I S.C. 3 ph) / (C *n*V L-L) ; F for three phase fault ——–
F = (2 * L* I S.C. L-L) / (C *n*V L-L) ; F for line to line fault ——–
F = (2 * L* I S.C. L-N) / (C *n*V L-N) ; F for Line to Neutral fault ——–
In our case we will use for this example equation #1
C= the conductor factor , value can be obtained from table (1) by locating the conductors size in the proper configuration
n= the number of parallel runs
L= length in ft
These values are equal to one over the impedance per foot based upon resistance and reactance found in IEEE std. 241
F= (1.73 *50 *24,506)/(28,303*5*480)= 0.031206
M= 1/(1+F) =0.969738 ; where M is the multiplier
I S.C. 3 ph for point 2= 24506* M= 24506*0.969738= 23,764 A or 23.7 KA (this is the available short circuit current at point 2)
Its worth to mention is that the further the main panel from the transformer the lower is the short circuit current but you will have bigger voltage drop due to the larger line impedance.
The calculation below will show the short circuit current at point 3 (similar to point 2 calculation)
F = (1.73 * L* I S.C. 3 ph at point 2 ) / (C *n*V L-L)
F= (1.73 *100*23,764)/(16673*1*480) = 0.51370
M= 1/(1+F) = 1/1.51370 = 0.6606
I S.C 3 ph for point 3 = 23,764 *0.6606 = 15,699 A or 15.7 KA
I hope you find this example and the short circuit calculation useful. please let me know if you have any questions


We recommend a minimum of 12.7 nfa per foot.
Attic vents per square foot.
There are two important pieces of information required to calculate the adequate number of vents.
The general rule of thumb on the amount of total attic vent space needed is to have at least one square foot of vent space for every 150 square feet of attic area.
So, if your attic is 450 square feet, you will need 3 square feet of vent space, evened out throughout the entire space and divided equally between inlet and outlet vents.
Say the contractor is standing in front of a house that has an attic with 2,200 square feet.
Attic square footage ÷ 4 = square inches of exhaust and square inches of intake net free area needed.
Most local building codes require compliance with either the 1/150 method or the 1/300 method exception (refer to local.
This vent area should be divided equally between the soffit areas on both sides of the roof.
2,200 ÷ 2 = 1,100 square inches of exhaust net free area needed
According to the national roofing contractors association (nrca), a power vent with an airflow rate of 1.0 cubic foot per minute per square foot of attic space measured at the attic floor is generally considered to be equivalent to a 1:150 ventilation ratio.
Determine the net free ventilating area the attic vents you plan to.
To calculate the total minimum soffit vent intake area in square inches, divide the cfm of the pav by 300 and multiply the result by 144.
The general rule of thumb in these situations is of roughly one vent per every 300 square feet of attic area if the attic has a vapor barrier.
I am pretty sure you are wondering how i calculated this size.
The 10 square feet x 144 square inches per square foot =.
Federal housing authority recommends a minimum of 1 square foot of attic ventilation (evenly split between intake and exhaust) for every 300 square feet of attic floor space.
Example building 1,000 divided by 300 = 3.33 total square feet.
The net free area (nfa) rating indicates the amount of space each roof vent covers.
Attic square feet x 0.7 = cfm capacity for example, using the same dimensions as the previous example:


A post on comp.theory newsgroup referred to this paper, by Giorgi Japaridze. I read a bit of this paper, and believe the thoughts presented here are quite promising (at least, I found the goal of this study quite ambitious). I guess, people with interest in computer logic might find this interesting.
This paper introduces ptarithmetic (short for "polynomial time arithmetic") - a formal number theory similar to the well known Peano arithmetic, but based on the recent, computability logic instead of classical logic.
The arithmetic primitives defined in this theory, compute in polynomial time (as against classical logic, operations in which can compute in exponential or combinatorial time).


"1, 2, 3, 4… Can your child already count to five? The Learning Numbers set is a fun way to get acquainted with numbers!
This set includes numbers and arithmetic signs to help you learn how to recognize numbers and further develop other math skills, such as counting to ten and simple addition and subtraction. This set stimulates the cognitive development of your toddler!"
1, 2, 3, 4… can your little one count to five? The Learning numbers set is an educational set to help make learning the basics of mathematical thinking fun for your child! This set includes numbers and math symbols and signs to teach kids to recognize numbers to support the development of other number skills, such as counting to ten and basic math. In addition, this set stimulates the cognitive development of your toddler or pre-schooler!
The Numbers learning set is an educational set to help your child with numeracy. Recognition of numbers develops into more sophisticated abilities, for example counting and basic multiplication. The colourful blocks are bio-based and are made from the remains of sugar cane plants. Moreover, the educational blocks are eco-friendly, colourfast and safe. The toy blocks encourage children to play imaginatively and to build new creations to develop cognitive, motor and emotional skills! Kids practice lasting life skills through play and creativity. With this set you as a parent find out how toy blocks support your little one's developmental milestones, like learning colours, shapes and a lot more.
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Week Four Assignment. Following completion of your readings, complete exercise 4 in the “Projects” section on page 620 of Mathematics in Our World.
a. Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this.
b. After building your triples, verify each of them in the Pythagorean Theorem equation.
The assignment must include (a) all math work required to answer the problems as well as (b) introduction and conclusion paragraphs.
a. Your introduction should include three to five sentences of general information about the topic at hand.
b. The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
c. Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to include a statement as to what you learned and how you will apply the knowledge gained in this exercise to real-world situations.
Cite your resources in text and on the reference page.
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It's Certainly Man-Made But It's Interesting Regardless:
Another crop circle has appeared in the English countryside — and this one's clearly been made by someone, or something, that understands math.
At first glance, the strange ratcheted pattern, radiating out clockwise, doesn't look like much — modern art, perhaps.
But an electrical engineer from North Carolina — described by the London newspapers as a "retired astrophysicist" — figured out that it was an abstract representation of pi, the number at the center of Euclidean geometry.
The article says 'word quickly got around the crop-circle community'. I was unaware that such a community existed in any significant numbers but, hey, who am I to say what's significant. It's been proven time after time that these crop circles are man-made but that doesn't make them easy feats. I have to say, I'm impressed.


floating point types in C++ are :
double , and
long double .
They are fundamental types , as such they have a
mapping to the hardware of a computing machine ,
so they are as fast , as the machine is capable
of performing the computation .
floating point types in C++ , are
used to represent numbers
possibly have a fractional part . For example
C++ standard , does not specify , the
floating point algorithm
to be used , this is left
to the implementation , but what is typically
used , by all implementations , is the
IEEE floating point format.


Image size in millimeters is approximated by multiplying the sensor size in inches by 16. The circle of confusion is 1/500 of the image size for standard definition and 1/1000 for high definition video. The depth of field is then calculated using the equations given by KerryG.
Rather than implement a lookup table of common image sensor sizes, the computations used here estimate the diagonal measurement of the image in millimeters by multiplying the sensor size by 16. This gives exact results for 1", 1/2" sensors and results to within 2% for many consumer camcorders. Note, however, this approximation underestimates by 11% the image captured by a 1/3" sized sensor.
Another way to obtain the size of the captured image is by multiplying 43.3 by the actual focal length and then dividing by the equivalent 35mm focal length. The equivalent 35mm focal length may be reported by the manufacturer or measured experimentally. For more information see how to calculate 35mm equivalent focal length by George Kroonder.


WELCOME TO Mrs. Lokuta's SECOND GRADE!
St. Joseph Parish School Community is dedicated to Faith, Academics, Service, and Catholic Traditions. (We are F.A.S.T.)
Second Grade Pictures
Welcome to Second Grade!
Christopher Columbus sailed the ocean blue in 1492!
The Trenton Fire Fighters visited our school!
Ghosts! Ghosts! Ghosts!
For a Social Studies project we made "Where We Live."
Our fall trees!
Building a community!
Working on math facts with dice!
We played a math card game called "Trash!"
Four Signs of Baptism
We listened to the the story The Dot and then made our own dot designs!
Making our name out of little squares of paper!
Playing "Roll to 100"
Off to a great start!


At my school, we have interventions first thing in the morning. I LOVE starting the day with interventions! It is a great way to start our learning right away. For morning work, I created daily math and grammar practice for my students, since those are skills they always need to practice. I gave them about 15-20 math problems and about 20 sentences to correct/vocabulary practice. They received their packet on Monday, and it was due on Friday.
However, I noticed it was difficult giving my students this particular morning work for a few reasons:
1. A lot of my kids didn't finish their morning work because they were gone at interventions.
2. Students who didn't require interventions finished their morning work early, and then didn't use their remaining time as wisely as they should have. (The iPads really came into play here...)
3. I found myself grading a TON of morning work and having to interrupt my intervention group to help students, give them a direction on what they should be doing, etc.
For this upcoming school year, I came up with a new solution. I am still going to assign math and grammar/vocabulary practice daily, but I am going to cut back to just a few questions each. After my students finish their daily problems (which they should be able to complete no problem before interventions begin), they will choose one box from our "Brain Breakfast" sheet to complete.
At the bottom of the sheet there is a spot for students to check if they were at interventions. If they are in an intervention group, they are not required to complete a box a day. I love this idea because it holds kids accountable and gives them a focus for their morning. I can't wait to use it!


How much is 2/3 if you don’t have a measuring cup?
The equivalent of 2/3 cup is 5 1/3 fluid ounces, 1/3 pint and 10 tablespoons plus 2 teaspoons. Knowing these equivalent measurements allows the use of other types of measuring devices, like a liquid measuring cup or measuring spoons. The size of your fist or a baseball is about one cup.
Simply so, what can I use if I don’t have a 2 3 measuring cup?
Use a 1/3 of a cup and fill it twice if you don’t own or can‘t find your 2/3 measuring cup. You can also use 10 tablespoons plus 2 teaspoons in a pinch as a conversion for 2/3 of a cup.
One may also ask, how can I measure without a measuring cup? To measure liquids without a measuring cup, imagine an object of the same volume to help you visualize the correct amount. For example, a tablespoon is about the size of an ice cube and a cup is about the size of a baseball or apple. For 1/2 cup, imagine a tennis ball and for a 1/4 cup, picture a large egg.
People also ask, what is 2/3rds of a cup?
1 full 2/3 cup = 2/3 cup. 1/2 of a 2/3 cup = 1/2 x 2/3 =2/6 = 1/3 cup. 2/3 + 1/3 = 3/3 = 1 cup. The answer is 1 and 1/2 cups will make 1 cup when using a 2/3 cup.
How can I get 1/3 cup?
The information below shows measuring equivalents for teaspoons, tablespoons, cups, pints, fluid ounces, and more.
Enter your birthdate to continue:
|1 tablespoon (tbsp) =||3 teaspoons (tsp)|
|1/3 cup =||5 tablespoons + 1 teaspoon|
|3/8 cup =||6 tablespoons|
|1/2 cup =||8 tablespoons|
|2/3 cup =||10 tablespoons + 2 teaspoons|
What does 2/3 cup look like?
Measuring cups are used to measure liquids, or powders like flour or sugar. As we fill up the measuring cup, we need to look at which number it is filled to. Here we have 2/3 cup (two-thirds of a cup) of milk. This measuring cup holds up to 4 cups of liquid.
What 1/3 cup doubled?
Scale, Half and Double Quantity Amounts in a Recipe (Chart)
|Original Recipe Measure||Half Scaled Measure||Double Scaled Measure|
|1/3 cup||2 tbsp. + 2 tsp.||2/3 cup|
|1/2 cup (4 fl. oz.)||1/4 cup||1 cup|
|2/3 cup||1/3 cup||1 1/3 cups|
|3/4 cup||3 tbsp.||1 1/2 cups|
What is 2 3 of a cup of flour?
All-Purpose Flour and Confectioners’ Sugar
|1/2 cup||64 g||2.25 oz|
|2/3 cup||85 g||3 oz|
|3/4 cup||96 g||3.38 oz|
|1 cup||128 g||4.5 oz|
What do I use if I don’t have 3/4 cup?
For 3/4 cups you can just use the 1/4 cup three times. Or if you’re in a hurry and don’t mind washing extra you could use the 1/2 cup and 1/4 cup.
How many two thirds are in a cup?
It’ll take 3 pours of a 2/3 rd cup to make 2 cups. To prove this we’ll multiply 3 pours times 2/3 and get 2 cups.
How can I measure 3/4 cup of water?
Fill a cup with the thing you want to measure. Pour or take out half of it into another cup(this is 1/2 cup,). Now from either of the cups take out half of the thing you are measuring(it is 1/4 cup) . Add this one fourth quantity to half of the quantity now the final quantity in the cup is 3/4( 1/2 +1/4).
What is the size of a teaspoon?
A teaspoon is a unit of volume measure equal to 1/3 tablespoon. It is exactly equal to 5 mL. In the USA there are 16 teaspoons in 1/3 cup, and there are 6 teaspoons in 1 fluid ounce. “Teaspoon” may be abbreviated as t (note: lowercase letter t) or tsp.
How many ml is two thirds?
Imperial to Metric Volume Conversions
|2/3 cup||=||158 ml.|
|3/4 cup||=||177 ml.|
|1 cup||=||237 ml.|
|1 pint (U.S.)||=||473 ml.|
How many thirds make a cup?
We can do this by thinking through what 13 of a cup means: If the contents of one cup are split up into 3 equal parts, each will be one third of the cup. Each cup has three thirds, Therefore in 6 cups there will be six times as many thirds.
How many ounces is two thirds cup?
How Much Is Two–Thirds of a Cup? Since 1 cup holds 8 ounces, 2/3 of a cup is 5 1/3 ounces. As most measuring devices are not accurate to 1/3 of an ounce, an alternate method of measure may be easier to use.
What is half of 1/3 cup in cooking?
|Reducing the Size of Recipes|
|1/4 cup||1 tablespoon + 1 teaspoon|
|1/3 cup||1 tablespoon + 2-1/3 teaspoons (or round to 1 tablespoon + 2-1/4 teaspoons)|
|1/2 cup||2 tablespoons + 2 teaspoons|
|2/3 cup||3 tablespoons + 1-1/2 teaspoons|
How much of a solo cup is a cup?
Plastic Party Cup – 18 oz. The Red SOLO Cup is a necessity for your party supplies list. SOLO is well versed in good times and plastic cups are our specialty.
What can I use instead of measuring cup?
When you don’t have either of these basic baking sets, here’s what you can use as a substitute: measuring cup = standard coffee mug. measuring tablespoon = dinner spoon. measuring teaspoon = coffee spoon.
How can I measure ML at home?
Liquid Ingredients: Place the liquid measuring cup on a level surface. Bend down so your eye is level with the markings of the cup. Fill the cup to the right level. When measuring 1 Tablespoon (15 mL) or less, fill the appropriate measuring spoon to the top without letting it spill over.
How can I measure a cup of flour without a measuring cup?
How to Measure Without Measuring Cups
- 1/8 teaspoon is about one good pinch between your thumb and both your forefinger and middle finger.
- 1/4 teaspoon is about two good pinches between your thumb and both your forefinger and middle finger.
- A teaspoon is about the size of the tip of your finger (joint to tip).
What size is a solo cup?
Provide guests with your most popular refreshments with this Solo P16R red 16 oz. plastic cup. Boasting a 16 oz. capacity, the cup is just the right size for serving your guests ice-cold draft beer, signature punch, or soda.
How are cup sizes measured?
Place the tape at the fullest part of your bust. Make sure it’s parallel to your band again. Subtract the band size from your bust size to find the right cup size. Every inch equals one cup size up.


iD Tech Camps (Ages 7-17) and iD Tech Academies (Ages 13-18) run from June through August each year—but the learning doesn’t have to stop there. We offer a variety of online resources for year-round learning, including iD Tech 365 (our comprehensive online learning platform for all ages), a series of eBooks that demystify the coolest STEM (Science, Technology, Engineering, and Math) careers, and Free Monthly Online Workshops available to everyone. At iD, quality tech education knows no season.
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Just a quick update to let you know that things are coming along nicely in CIC at the moment. I’ve decided to put together my second tech demo (alpha 2) which wills how off movement and may have orbital mechanics involved. I’ve always been interested in orbits, both how they work, and the math that goes into entering and exiting them. I haven’t played a game to date that includes them so it seemed like a good mechanic for a tech demo. My hope is that I can hide all of the boring and complex 3D math from the player and let them concentrate on having fun sling shotting of planets, entering/exiting and changing orbits and other fun things.
Without accounting for the orbital parts, I’ve estimated about 55 hours of work to reach the tech demo. As I’m doing a lot of overtime at work at the moment I won’t be able to hit my usual 10 hours per week of work so I can’t set a date for the tech demo yet. Having said that, I’m making good time so far and will do as much as I can when I get the chance.
As always, any comments, ideas or feedback on the ‘fun’ parts of orbits or anything to do with moving in space is always welcome. I’m aiming for true* Newtonian physics which will allow for fun real 3D combat thanks to my as yet unannounced time algorithm.
*Many games claim to have Newtonian physics only to add in a ‘dampener drive’ (or equivalent) which magically takes away all the pains of REAL Newtonian physics. I hope to have a game that finally allows true physics without the player worrying about spending hours speeding up and slowing down.


One of the best ways to make math engaging is to use real life math projects for middle school. They will make the subject fun and relevant for older kids. Use them as part of your middle school math curriculum.
Tweens and teens don’t always connect how math connects to their actual life. They usually just think about it as a school subject that’s not very interesting.
However, real world math projects will show them the importance of mathematical concepts and their relevance outside of school.
Homeschooling moms often struggle with teaching practical math. But activities like these can help bridge the gap between academic math and real-world applications.
If your math curriculum doesn’t include interactive activities, use some of these to show your middle school students how to use math in everyday life.
Real Life Math Projects for Middle School
Have your tweens ever asked you “What does this have to do with real life?” or “When will I ever use this again?” during their math lessons? To be honest, they’re fair questions.
Of course, they need to learn some math concepts so they can move on to higher grade level math. But, it’s also important for kids to see how they will use math in their daily lives. But how can you make math topics relevant?
The best way is to incorporate activities to your homeschool lesson plans that focus on practical math.
How to Make Math Relevant and Fun
Here are some tips that will show middle schoolers the relevance of math and make it fun.
- Use real-life examples — Kids need to understand that math is everywhere, so give them everyday scenarios. Ask them how they would use a calculator, fractions, or basic geometry in a baking, budgeting, or a shopping project.
- Add a creative element — Make your math projects fun by adding an art or craft component. For example, draw and label shapes with the correct angles, draw pi, or use origami to create 3D geometry shapes.
- Incorporate games — Games are useful for helping kids develop and review essential skills. There are a lot of excellent math board games you can play.
Project-based learning is an excellent way to build student interest and enhance their learning.
This post contains affiliate links.
Benefits of Using Real World Projects for Teaching Math
There are many benefits of using practical activities to teach math. Here are three of the most important ones:
Build Problem-Solving Skills
Your older students will need to think critically to solve everyday problems and scenarios. This will help them in math class and develop their problem solving ability.
Also, they’ll probably have to get creative and use ingenuity to think of a solution. This will help them develop their logical and reasoning skills as well.
Grow Confidence in Math
When middle schoolers understand how much math is used in daily life and they get more comfortable with various scenarios, their confidence in math will soar.
And when students have a positive attitude toward math, they’re more likely to feel confident working through more complex concepts.
Develop Important Life Skills
Tying math to real-life applications will help them expand their math knowledge while also developing important life skills. Think about how much math you use as an adult. Budgeting, shopping, and cooking all involve math.
By teaching your children practical math with fun, interactive projects in middle school, you can set them up for success later on in life.
Life Skills Math Curriculum
As you’ve seen, it’s important to teach tweens practical math and the concepts in your math curriculum.
Luckily, there are plenty of fun math projects you can use in your homeschool that will teach functional math.
Teach Tweens How to Cook
Cooking is an essential life skill, so why not use it to beef up some math knowledge? This is an excellent way for younger teens to practice percentages, ratios, and fractions.
And when your tweens are done with their project, you’ll also have a delicious meal to enjoy.
Shopping and Budgeting
If you want to help older middle schoolers practice budgeting and the concept of spending money wisely, create a shopping project.
Give them a list of items and have them calculate the cost of each item and the total amount. They can also practice comparing products, sales tax and other money-related topics.
Combine the shopping and cooking projects and have tweens planning a week’s meals within a set budget and help prepare them.
If you have a teen who is not great with distance measurements, have them plan a road trip. They’ll learn how to calculate approximate mileage and time between destinations. They can also use a map to get an idea of the route to take.
Another fun project is planning a family vacation. Let them figure out the budget, considering factors such as fuel cost, meals, and accommodations.
Go Hiking Without a GPS
Hiking can be a fun way to teach angles. One project is to plan out a safe hiking route for your older kids using a compass. If you are hiking right away, you can also add some marks along the trail as reference points.
As you design the route, make an instruction list of the angles it takes.
Your instructions should say things such as “head East at a 45-degree angle until you reach a downed tree or red flag.”
Once you set the course up, give the list of angles to your kids, along with a compass, and see how well they navigate gauging the angles.
Instead of building a trail, you can also head out with your tweens and let them use math (and a compass) to traverse the area.
Students can use math to analyze and interpret data from their favorite sports teams. Ask them to look through team statistics, such as wins and losses, batting averages, or points scored in a season.
They can figure out the mean scores of each statistic they researched or create a graph to see trends. This is an excellent way for tweens to learn data analysis and statistics.
A room makeover can provide a fantastic opportunity for tweens to practice practical math while also giving them a chance to exercise their creativity.
They can start by measuring the room dimensions to understand the area and perimeter they have to work with. When choosing new furniture or decor, they can learn about geometry, scale, proportion, and budgeting.
They’ll use math to calculate whether a piece of furniture can fit in the room, or how much paint they’ll need to cover the walls. And also to choose and hang new drapes or wall hangings.
When they’re finished, they’ll have strengthened their math skills and have a beautiful new room!
Cost of Living Simulation
A great way to show teens how much math is needed in the real world is by teaching them personal finance through a budget simulation.
Give them a set amount of money as a salary and a list of bills they have to pay. Help them use this information to create a budget. They can also research the prices of items and services to compare costs.
Once they’re comfortable with this, you can take it a step further with topics like mortgages, interest rates, and loan percentages. This is a great activity for teaching financial literacy.
With these real-world math projects, they’ll learn valuable lessons while building their math skills.
As you can see, there are plenty of activities you can choose from that focus on practical math applications. And by teaching your tweens real-life math concepts, you’ll be preparing them for success in and out of the classroom.
So, this school year, include more real world math activities in your homeschool.
Additional Resources for Middle School Math
Need another math resource for your lesson plans? Check these out.
What is your favorite real life math project for middle school?


Smooth Operators is a complete solution for learning, practicing, and testing the order of operations. An interactive lesson teaches the math concepts thoroughly with explanations and example problems, including an optional section on exponents. Popular memory aids are introduced, but users also learn a logical approach to truly understand the order of operations.
In practice mode, students click on operators in the correct order, giving their complete attention to which operations come first. In test mode, the user performs all calculations and types in the answers. Score is always displayed.
Problems can include parentheses and exponents, and the learner or supervisor can select the number of operations per problem and the size of operands. In the course of solving problems, users have an excellent review of basic math skills as they add, subtract, multiply, and divide positive and negative numbers.
Smooth Operators helps students master crucial skills and prepare for algebra and higher math, while providing ample opportunity for arithmetic practice. This educational software is ideal for school or home, in class or independent study.
Smooth Operators is a free trial software application from the Mathematics subcategory, part of the Education category.
The app is currently available in English and it was last updated on 2006-09-30. The program can be installed on Win95, Win98, WinME, Windows2000, WinXP, Windows2003.
Smooth Operators (version 1.0) has a file size of 1.05 MB and is available for download from our website.
Just click the green Download button above to start. Until now the program was downloaded 273 times.
We already checked that the download link to be safe, however for your own protection we recommend that you scan the downloaded software with your antivirus.


The following is an email sent to my sister, who I have already called four times this morning, once at 4:30 am her time (sorry Becky!!) It's not my fault she lives in Colorado and happens to be a math tutor. She walked us through the following equations a couple of times, and she did such a good job that we thought we 'had' it! (When I say 'we' I mean Tawny and I). But we still didn't get the right answer, I think our answers are pretty good, but I guess they have to be all the way right...
By the way the funky Os are supposed to be square root symbols, I guess people don't blog enough about math to need those...
"Step by step:
Please give the standard form of a function with rational coefficients and the given zeros
Zeros: Ö3, 1-Ö5
What we’ve been doing-
(xÖ3)(x+Ö3) (x-(1-Ö5)) (x+(1-Ö5))
X2 +xÖ3-xÖ3-3 x2-3
we’re pretty much screwed up and desperately need your help!
we also don’t know what to do on the : 2, 2-i"
Now you may be wondering why I'm so into algebra all of a sudden. Tawny has finals this week and two algebra tests, one today. I had her skip school and told her we'd work on these problems until she felt confident enough to take the test. We did get all but 4 done, and that took us from 7:30 am to 5:30 pm, with a 1 1/2 hour break in there where I took a nap with Enoch and Danni.
So if anyone can help with these, or the Binomial Theorem, you would be my hero!


3Blue1Brown aka Grant Sanderson is an outstanding math animation artist, scholar, & educator. In this episode, I ask him about his workflow, the tools he uses, the nature of consciousness, machine learning, & so much more! Grant studied mathematics at Stanford as an undergraduate, then became a multivariable calculus fellow for KhanAcademy through their talent search program. He started making math animation videos on Youtube in 2015, and has since amassed over 2 million subscribers. It was a wonderful conversation, and I hope you enjoy it!


The polynomial model is an integral concept in mathematics, especially in determining relationships. It is a form of regression where the relationship between dependent and independent variables is determined (Fox & Burks, 2019). For example, the model can identify the relationship between Amazon’s net income/loss and sales. However, there are several steps required to create a polynomial model. One of the steps involves using the data collected to create a scatterplot. To achieve this, a Microsoft excel helps develop a trendline and regression equation. Therefore, creating a polynomial model is a process that starts with developing a scatterplot.
The Steps of Creating a Polynomial Model
The polynomial model determines the relationship between the independent variable X and the dependent variable Y. There are several steps followed to determine the polynomial model (Fox & Burks, 2019). The first step is using data collected on Amazon’s net income/loss and sales to create a scatterplot, as shown in the figure below. To achieve this, Microsoft excel is used to create a scatterplot automatically. In this case, the dependent variable Y is the company’s net income/loss, whereas the number of sales represents the independent variable X. As a result, the creation of a polynomial model starts with the development of a scatterplot.
The second step in developing the polynomial model is adding a trendline. A polynomial trendline is a curve that shows how data changes over time. The number of changes in the data can be used to figure out the polynomial order (Fox & Burks, 2019). With the created scatterplot, there is a need to add a trendline. In this trendline, polynomial orders two, three, and four can be used to create the model, as shown in the figures below. To do this, click on one of the points in the scatterplot. After that, right-click, select add a trendline, and a line connecting some points will appear automatically. As a result, excel allows the creation of different trendline based on the order chosen.
Second-order polynomial, also known as quadratic polynomials, is associated with a parabola. In this model, the x variable is squared and has a coefficient and a constant. It is usually used to determine factors in a curve (Fox & Burks, 2019). In addition, it plays an integral role in developing the second-order regression equation. This order distinguishes quadratic equations from the linear regression equation. Furthermore, a second-order polynomial has additional functions that can be used to determine more figures.
A third-order polynomial is different from the second and first order. It is among the high order polynomials due to the existence of X3. This order is of the form of y = ax3 + bx2+ cx + d where ‘a’ is not zero. It is also known as cubic polynomial because it has degree 3. Therefore, polynomial order three is more complex than the first and second order.
The third step in creating a polynomial model is the creation of polynomial regression equations for each of the orders chosen. On the trendline, it is possible to add various aspects to the scatterplot (Fox & Burks, 2019). One of the features that can be added is the regression equation. For order 2, the polynomial model is shown in the figure below. A second-order polynomial equation was created to predict Amazon’s net income/loss based on sales. The equation for the scatterplot was determined using Microsoft excel’s trendline function. For example, the equation for the model obtained is y = -2.2713x2 + 227.33x – 643.77. In this case, x represents the sales, whereas y is the net income/loss. Therefore, the last step in the process is using the trendline function in excel to generate a polynomial equation.
The order of the polynomial equation was kept as low as possible, and higher-order polynomials were avoided because randomly fitting higher-order polynomials often leads to serious misuse of regression analysis. In most cases, the second-order polynomial is used to determine which input drive responses (Fox & Burks, 2019). As the number of explanatory variables and, by extension, the number of cross-product terms goes up, it is evident that polynomial equations could get much more complicated. In addition, there are limited interactions with more than two factors. This helps keep the equations at a level that is easy to work with (Fox & Burks, 2019). Thus, second-order polynomial was considered in the creation of the model.
The development of a polynomial model begins with creating a scatterplot using data from Amazon’s net income/loss and sales. The polynomial model is an important concept used to determine relationships in figures and aspects. It is a type of regression that looks at the relationship between a dependent variable and an independent variable. For example, the model can determine how Amazon’s sales relate to its net income or loss. However, to make a polynomial model, there is a need to do several things. One of the steps is to make a scatterplot with the collected data. To achieve this, Microsoft Excel is used to make a trendline and regression equation.
Fox, W. P., & Burks, R. (2019). Applications of operations research and management science for military decision making. Springer, Cham.


CTRFL research activities result in a variety of databases that may benefit the research of others. Public, restriction free sharing of large datasets remains a data transfer challenge, so simple links to raw data are not feasible. However, if you are interested in using any of the datasets below, please contact us, and we discuss your needs and how we can help. Datasets currently available are described on this page, and additional databases will be added as they become available.
Multiscalar Mixing DNS Data
Direct Numerical Simulations of three-component mixing in forced isotropic turbulence at an average Taylor scale Reynolds number of 83 were performed to study joint PDFs of two mixture fractions. The simulations were performed on a triply periodic cubic domain with 256^3 grid points. As shown in the images below, eleven different initial scalar configurations were considered in this study, which were all mixed using identical turbulent velocity fields.
Three types of data from this study are available for all eleven configurations:
- Raw data: The values of the three velocity components and the two mixture fractions for each configuration at every grid point, stored for 102 temporal snapshots spanning the initial unmixed state to the final fully mixed state. These data are available as binary files (180 GB total), and a Python code can be provided for reading the data.
- Binned PDF data: The above raw data binned into joint and marginal PDFs of the two mixture fractions for each configuration, with 512 bins for each mixture fraction. These data are available as ASCII text files (184 MB total).
- Moment data: Mean, (co-)variance, and skewness of the two mixture fractions for each configuration, stored at every DNS time step. These data are available as ASCII text files (4 MB total).
- B.A. Perry, M.E. Mueller, Joint probability density function models for multiscalar turbulent mixing, Combustion and Flame 193 (2018) 344-362
Sydney Inhomogeneous Inlet Flame Boundary Conditions
The piloted jet burner with inhomogeneous inlets developed at the University of Sydney is a well characterized burner designed to probe partially premixed, multi-modal combustion. It has been a target configuration for combustion models at recent International Workshop on Measurement and Computation of Turbulent Flames (TNF). The key feature of this burner is the compositionally inhomogeneous mixture at the burner exit, generating by inhomogeneous partial premixing of fuel and air within the burner. To aid in the comparison of simulations from various groups, well-resolved LES calculations of the mixing within the nozzle have been conducted to provide a standard boundary condition for the mixture fraction and velocity profile at the nozzle exit.
The boundary condition datasets consist of unsteady cross-sections of the three components of the velocity, mixture fraction, and the square of the mixture fraction (to obtain the subfilter mixture fraction variance) at a location one diameter upstream of the burner exit. The data are provided in cylindrical coordinates (both the spatial components and the velocity components), corresponding to the axial (x/u), radial (y/v), and circumferential (z/w) directions. The cross-section has 161 points in the y-direction and 64 points in the z-direction. For each case, data are included for approximately 10,000 time steps spaced by 0.25 us. Datasets are available for the following cases:
- FJ200-5GP-Lr75-57 (fuel/air mixing and air/air mixing)
- FJ200-5GP-Lr75-80 (fuel/air mixing and air/air mixing)
- FJ200-5GP-Lr300-59 (fuel/air mixing)
These data are available as binary files (4 GB each), and a C code can be provided for reading the files.
- B.A. Perry, M.E. Mueller, Effect of multiscalar subfilter PDF models in LES of turbulent flames with inhomogeneous inlets, Proceedings of the Combustion Institute 37 (2019) 2287-2295
- B.A. Perry, M.E. Mueller, A.R. Masri, A two mixture fraction flamelet model for Large Eddy Simulation of turbulent flames with inhomogeneous inlets, Proceedings of the Combustion Institute 36 (2017) 1767-1775
Turbulent Planar Jet Flames
Information coming soon.


How to Get Ready for Math Test in College
It is hard to find ways on how to pass a math test without studying, then. If you need a good grade in math, work for it. Mathematics may seem hard, but if you study hard, it becomes easy to handle. If you find it hard to prepare for a mathematics test, here are some tips to help you prepare.
Begin by doing homework
The best strategy on how to do math test is doing your homework. Your tutor may give some assignments at the end of every lesson to help you understand the concept taught in class. If you need to learn how to pass a math test, begin by doing those assignments. You will learn new examinations skills, improve your calculation speed and internalize math concepts.
Homework also provides you with formulas and techniques you need to apply in your last test. Keep track of all the reading and try solving all the math questions supplied by your tutor for easy work.
Go an extra mile
You will not rely on the notes and examples from your tutor. You need to move ahead and work on the extra task. You can undertake an online math test that you can use for exercise and improve your working technique. The more you solve challenging questions, the more you sharpen your working skills. Some sites even have math test help where you can get assistance when facing a difficult question.
If you face trouble trying to solve math questions, find somewhere you can get math test help and learn step-by-step methods of solving challenging questions. The more you explore new questions, the more you earn new skills in working out difficult questions.
Set enough time for math
You won’t be rushing with questions and topics in math if you need to learn how to study for a math test. Math subject requires patience. You must dedicate ample time to the subject to go through the step-by-step procedures of solving math questions. Even if you don’t grasp the concept in a day, don’t give up. Spare some extra time and go through the process fresh until you internalize the challenging idea.
Prepare yourself both mentally and physically
You can’t walk into the math class exhausted and expect to make it in the math test. Before your big day, have time to relax and focus your mind on the test. Avoid spending the night cramming math formulas before walking up tired to the math test the following morning. When you are exhausted, your mind will be congested, and you won’t be productive in your math test.
Do online practice
You are looking for how to study for a math test? Do online practice or simply pay someone to take test for me online. Several sites are offering online math help to students facing a challenge in the subject. Such sites are full of online math tests that you can practice and test your speed and accuracy. As you expose yourself to more tasks, you learn and perfect your calculation speed.
Whether you need tips on passing a math test without studying or studying for math, you can now find what to do. Plan your time and do more practice. Start with the homework math questions as you keep improving on your skills.