Composition operators on weighted Bergman spaces.
Jones, Matthew (1999) Composition operators on weighted Bergman spaces. PhD thesis, University College London.
- Published version (with publisher's formatting)
In the late 1960’s, E.A. Nordgren and J.V. Ryff studied composition operators on the Hardy space H2. They provided upper and lower bounds on the norms of general composition operators and gave the exact norm in the case where the symbol map is an inner function. Composition operators themselves, on various other spaces, have been studied by many authors since and much deep work has been done concerning them. Recently, however B.D. MacCluer and T. Kriete have developed the study of composition operators on very general weighted Bergman spaces of the unit disk in the complex plane. My starting point is this work.
Composition operators serve well to link the two areas of analysis, operator theory and complex function theory. The products of this link lie deep in complex analysis and are diverse indeed. These include a thorough study
of the Schr¨oeder functional equation:
and its solutions, see  and the references therein, in fact some of the well known conjectures can be linked to composition operators. Nordgren, ,has shown that the Invariant Subspace Problem can be solved by classifying
the minimal invariant subspaces of a certain composition operator on H2, and de Branges used composition operators to prove the Bieberbach conjecture. In this thesis, I use various methods from complex function theory to prove results concerning composition operators on weighted Bergman spaces of the unit disk, the main result is the confirmation of two conjectures of T. Kriete, which appeared in . I also construct, in the final chapter, inner functions which map one arbitrary weighted Bergman space into another.
|Item Type:||Thesis (PhD)|
|Research Areas:||B. > Theses
A. > School of Science and Technology > Design Engineering and Mathematics
|Depositing User:||Dr Matthew M Jones|
|Date Deposited:||25 Nov 2009 07:52|
|Last Modified:||13 Oct 2016 14:16|
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...I have been teaching and researching high school, college and graduate school math, science, and engineering for more than 20 years. Currently I am teaching high school Calculus II. I enjoy teaching all level mathematics. (read more)
...At Florida State University, I completed a 3000 level genetics class and a 4000 level molecular biology class with grades of B and A, respectively. I also completed the first year MGC (molecules, genes, and cells) class at SUNY Stony Brook School of Medicine. I received an HP (high pass).
Some... (read more)


Computerized voice: Click the triangle. Triangle.
Interviewer: Advancing that knowledge also means bringing technology into the equation. Building Blocks incorporates the use of technology in the classroom. Why?
Dr. Sarama: It offers the advantage of giving one more environment in order to work with the concepts we're talking about. So, yes, we want children counting stairs, we want them counting blocks, we want them counting each other, we want them counting everything. Being on the computer and counting the things on the screen is just one more environment, and we can structure the technology-- and this is very important--to support children's counting wherever they are developmentally.Cerrar
Este video es parte del módulo Uso de los medios interactivos para el aprendizaje temprano, uno de los que conforman la serie de Módulos de aprendizaje de la Alianza EarlyEdU (video en inglés).


The distribution of generalized exponential was invented by Rameshwar D. Gupta and Debasis Kundu in 2007. The distribution was the result of a generalized transformation of the exponential distribution. This paper explained the generalized exponential Marshall-Olkin distribution which is the result of the expansion of the generalized exponential distribution using the Marshall-Olkin method. The generalized exponential Marshall-Olkin distribution has a more flexible form than the previous distribution, especially in its hazard function which has various forms so that it can represent survival data better. The flexibility characteristic is due to the addition of new parameters to the generalized exponential Marshall-Olkin distribution. We explained some characteristics of the Marshall-Olkin generalized exponential distribution such as the probability density function (PDF), cumulative distribution function (CDF), survival function, hazard function, mean, and moments. Parameter estimation was conducted using the maximum likelihood method. In the application, it was shown data with generalized exponential Marshall-Olkin (GEMO) distribution. The GEMO distribution was modelled to the waiting time data until the damage to a lamp. The data was taken from Aarset data (1987). The results of modelling the waiting time data until the damage to a lamp on the distribution of GEMO and was compared to the distribution of alpha power Weibull. A comparison of models using Akaike information criteria (AIC) and Bayesian information criteria (BIC) shows that the distribution of GEMO is more suitable in modelling the lamp damage waiting time data than the distribution of alpha power Weibull.
|Journal||Journal of Physics: Conference Series|
|Publication status||Published - 12 Jan 2021|
|Event||2nd Basic and Applied Sciences Interdisciplinary Conference 2018, BASIC 2018 - Depok, Indonesia|
Duration: 3 Aug 2018 → 4 Aug 2018
- Generalized transformation
- Hazard function
- Maximum likelihood estimation


30 Aug Low tech but great tool-Draftsman
Even though I appreciate technology, it is important to remember there are low-tech tools that help our learning process immensely.
Click Draftsman Tactile Drawing Board by APH.org to see what students can use to practice their print letters and handwriting. These boards can also be used in math class. Students can draw what the math teacher is drawing and follow along with the class. The boards are also used to draw pictures and diagrams, hey even a game of tic-tac-toe!


I am not sure how to calculate my LSTM weights based on this link and my Keras programing as below:
model = Sequential() model.add(LSTM1(A1, input_shape=(2,B1), return_sequences=True, stateful=False)) model.add(LSTM2(A2, return_sequences=True, stateful=False)) model.add(LSTM3(A3, return_sequences=False, stateful=False)) model.add(Dense(B2, activation='softmax')) model.summary()
Is that correct?
Input = A1 Output = B2 Hidden Layers = 2 Cells in layer number one = A2 Cells in layer number two = A3
4*ni*(ni−1+ni) for each layer is calculated as below: X1=4*A1*(A1-1+A1) X2=4*A2*(A2-1+A2) X3=4*A3*(A3-1+A3) X4=4*B2*(B2-1+B2) Total number of weights = X1+X2+X3+X4
All my problem is about understanding the concept of cell in Keras and not in LSTM itself.


Friday Lunch Seminar
September 11, 2020
12:15 〜 13:00
(On-line for CiNet members only)
“Data-driven modeling of nonlinear dynamics: From mode decomposition to interaction analysis”
by Yoshinobu Kawahara
Institute of Mathematics for Industry, Kyushu University and Advanced Intelligence Project
Host PI: Okito Yamashita
Data-driven modeling of complex systems has received much attention over the recent years, largely due to the availability of large datasets. In particular, the analysis of nonlinear dynamical system with Koopman operator has been actively discussed in applied mathematics and various scientific fields for this purpose. This is because it can provide physical interpretations of the dynamics based on deep theoretical backgrounds and is endowed with prominent estimation methods such as dynamic mode decomposition (DMD). DMD is a numerical method for estimating spectra of Koopman operator, and has been attracting attention as a way of obtaining global modal descriptions of nonlinear dynamics from data without requiring explicit prior knowledge. In this talk, I first overview the recent advances on this research topic, focusing on spectral analysis of dynamical systems with Koopman operator and DMD. Then, I describe several recently-proposed related algorithms using machine learning principles. One limitation of DMD is that it assumes inherent-scale spatial or temporal dynamics independently contribute to its whole dynamics, which is often not satisfied in dynamic phenomena investigated in scientific studies. Thus, I describe a neural network model to estimate interactions among different scale dynamics to address this issue. In the talk, I occasionally show some applications of these method to several real-world data.
About CiNet’s Friday Lunch Seminars:
The Friday Lunch Seminar is CiNet’s main regular meeting series, held every week at 12:15 in the beautiful main lecture theatre on the ground floor at CiNet. The talks are typically 40mins long and orientated towards an inter-disciplinary audience. They are informal, social, and most people bring their own lunch to eat during the talk. They are open to anyone who is feeling curious and wants to come, regardless of where you work.


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What is a tree diagram in probability?
Tree diagrams are a way of showing combinations of two or more events. Each branch is labelled at the end with its outcome and the probability is written alongside the line. To work out the probabilities of each combination, multiply the probabilities together.


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Projects per year
We study a class of symmetric critical points in a variational 2D Landau-de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k/2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when |k|≥2.
|Number of pages||22|
|Journal||Annales de l'Institut Henri Poincaré (C) Non Linear Analysis|
|Publication status||Published - 16 Apr 2015|
- Liquid crystal defects
- Nonlinear elliptic PDE system
- Singular ODE system
FingerprintDive into the research topics of 'Instability of point defects in a two-dimensional nematic liquid crystal model'. Together they form a unique fingerprint.
- 1 Finished
16/09/13 → 15/03/17


Wedges and the Half-Space
Structures that can be idealized to wedges are found at every scale from mas-sive gravity dams to gear teeth. Moreover, for wedge angles greater than 90_, no strength-of-materials type analysis is possible and only for very small angles should we expect such simplified analysis to give even a decent approximation. Therefore, as for thick rings and plates with small holes, two-dimensional solutions from elastic field theory are mandatory for the design of such structures.
Concentrated Loadings at the Apex
Consider first a wedge of angle 2_ in plane stress such as shown in Figure 8.1 with an axial load, P, at the apex. If the wedge angle were very small, then we might expect . Let us assume, then, a stress func-tion of type III:
But we know the stresses must be symmetric with θ and the first term gives nothing so, through this physical reasoning process, we can deduce that:
A comparison of σx from the elasticity solution to the average value σx = P/A is shown in Figure 8.1b. We see that only for small wedge angles is the strength-of-materials approximation adequate.
This solution when applied to the half-space is particularly important as it corresponds to the punch or bearing capacity situation. It also serves as one of the fundamental building blocks for 2D boundary-element modeling. The result for ϕ2 = π, often called the Flamant Solution,
is shown in Figure 8.1c. The stresses die off as 1/r and the isopachics and iso-chroniatics (contour lines of equal σ1 + σ2 and σ1 - σ2) are identical circles:
The load, P, must actually be distributed over a small “circular” area near the apex in which plastic flow takes place redistributing the stress to its yield
value at r=ryield. Outside this plasticized zone the strains can be computed from the stresses using the elastic relationships and integrated to determine displacements.*
For a vertical transverse load Q at the apex, the reasoning is similar. For a very small wedge angle, we would expect that σr == σx == My/I where, as shown in Figure 8.2, M = Qr, y = rθ, and I = (2rφ)3/12. Therefore σ r =3Qθ/2rφ3 , so again the radial stress dies off as 1/r and we should assume m = 1 in the stress function (a). In this case, however, since there is bending and no axial force, only the asymmetric terms apply. The term Crsinθ gives no stress and therefore:
The boundary conditions like the overall moment and axial-force equi-librium requirements are already satisfied in Region R but to satisfy vertical equilibrium:
For any significant wedge angle, a strength-of-material approximation is unsatisfactory. For the half-space, Q becomes a concentrated shear producing radial stress bulbs,
as shown in Figure 8.2c. They are identical to those for P but 90o out of phase.
Clearly the solutions for concentrated line loads P and Q at the apex of a wedge can be superimposed to give the field for a line load at any inclination. Moreover, a line load within an “infinite” elastic body (ϕ=π), sometimes called the Kelvin Problem, can also be solved.*
Finally let us solve for the stresses in a wedge with a concentrated line moment at the apex (Figure 8.3). Again, considering a narrow wedge with ϕ
so that the strength-of-materials approximation breaks down quickly for significant wedge angles. The compar-ison of normal stress is similar to that for a transverse load in Figure 8.2b, but the shears are very different. More significant, perhaps, is that the field is highly localized near the point of application since the stresses induced by the moment die off as 1/r2 compared to 1/r for a concentrated load.
For larger wedge angles the solution is suspect. At 2ϕ=257.4o the denominator sin2ϕ = 2ϕcos2ϕ goes to zero and the solution blows up. Sternberg and Koiter term this ambiguity in the two-dimensional solution for a line moment on a wedge for 2ϕ>πa paradox, which they do not entirely resolve. We shall see that such ambiguity is present in a number of wedge solutions and that all questions in “simple” two-dimensional elasticity theory are not yet answered.
A tapered Epoxy beam (wedge) in plane stress is loaded by shear trac-tions of s on the upper face and -s on the lower face as shown. The stress function F = Ax2 - By 2 is proposed. Evaluate A and B and show that this is the solution. Are the support conditions appropriate? For v = 1/3, E = 200,000 psi determine the deformed shape of a 60o wedge (2ϕ=60o) if l=10 ft. for s=10,000 lb/ft per inch thickness. Sketch the deformed shape.


Math Tutor DVD - Calculus Help: The Calculus 3 Tutor: Volume 2
English | XviD 761 Kbps, 640x386, 29.97 fps | MP3 96 Kbps | 3.6G
Calculus 3 is considered by most to be a very difficult course to master in the realm of Calculus. This is because you will learn about many different topics, and each topic builds on the previous. If you don't understand something early on, the chances of "catching up" are drastically reduced as time goes on.


British Columbia should look to the East for lessons in how to teach math and science because its current approach is not preparing students for the modern world, says a group of Chinese-educated tutors who are lobbying for curriculum change.
The tutors, who have formed the Educational Quest Society, say they have valuable insight because they were educated in China, earned graduate degrees in the West and have worked for many years in Metro Vancouver as private tutors. In their dealings with struggling students, they say they learned about the inadequacies of the math and science curriculum in K-12 schools.
tap here to see other videos from our team.
“It’s weak and getting weaker,” said Sharon Shen, who owned and operated a tutoring school called Elite Education Centre in Burnaby and Richmond for more than a dozen years before her recent retirement. “We feel there is a crisis.”
Although B.C. teachers do not agree, John Yuan, also a society member, said he began noticing a drop in math standards in particular 10 years ago and now sees many students who lack a basic foundation. They make simple mistakes, such as thinking that 7.52 is larger than 7.9, he said.
For proof of a performance decline, the society points to the latest results from the Pan-Canadian Assessment Program (PCAP), which tested the math skills of 32,000 Grade 8 students from across the country in 2010. While those in Quebec, Ontario and Alberta had results equal to or above the Canadian average, B.C. participants fell below.
Rankings are not as important as the message they send, the society says. International assessments focus on the skills students need for life in the modern world and thus the scores “are actually indicators of our children’s future and (the) competency of our economy in the world.”
The society says there are many important differences between B.C. and China when it comes to teaching math and science, and those have led to runaway success for certain Chinese students in international assessments. For example, Chinese schools place greater emphasis on basic concepts, assign more homework and employ teachers with special math training (which is not the case in B.C. elementary schools).
But its main beef is with a recent education ministry decision to drop all Grade 12 provincial exams except English 12 or its equivalent. That, according to the society, has had a negative impact on educational standards, student motivation, teacher enthusiasm and public accountability.
Jerry Mussio, an education consultant who was Canada’s representative in the development of the Programme for International Student Assessment (PISA) between 2000 and 2004, has also issued a warning about B.C.’s declining standards and international assessment results and shares the society’s dismay over the loss of many provincial exams.
But he said China, which emphasizes rote learning from textbooks, could also learn from B.C.’s concentration on problem solving and creativity. During a meeting with society members earlier this month, he noted the ministry is reviewing the Grade 12 graduation program as part of the B.C. Education Plan and urged them to join the discussions.
“We’re always trying to improve the curriculum, but we’re not going to satisfy every stakeholder group.”
A new math curriculum has been implemented over the past two years and introduced in Grade 12 only this year. Therefore, he said any complaints about the competencies of senior students would pertain to the old curriculum, not the new one.
While the new curriculum isn’t perfect, it is better than the old one, he said, noting that it emphasizes understanding concepts, not memorizing formulas.
Becker acknowledged that China places more weight on math education than Canada does, but said that isn’t necessarily positive, especially if it means young children are over-burdened with homework. “It might be a little bit better for their understanding of raw facts and mathematical (concepts), but we’re also trying to create students who are productive in whatever field they choose.”


Modern problems of mechanics, Collected papers
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, 351 pp.
The volume presents studies on various issues in mechanics and dynamical systems theory, including the self-similar piston problem in a Prandtl–Reuss elastoplastic medium with special properties, homogenization of acoustic equations for a heterogeneous layered medium consisting of creep materials, spectral stability of shock waves in singular limits of smooth heteroclinic solutions to an extended system of equations, and stability of periodic orbits of a planar Birkhoff billiard. The problem of Arnold diffusion, dynamics of nonholonomic systems, integrable systems in analytical mechanics, and problems of the KAM theory in infinite-dimensional Hamiltonian systems are also discussed.
The volume is of interest to researchers, postgraduates, and students specializing in analytical mechanics and continuum mechanics.
Kozlov V. V., Sergeev A. G., Modern problems of mechanics, Collected papers, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, 351 pp.


Hi guys in this problem we have X one X two and up to X. M be a sample from an exponential distribution with parameter lambda. So it can be shown that to Lemaitre times some nation over I from one to end for X. I has a tai chi squared distribution with two N degrees of freedom. So we use this fact to drive a test statistics and critical region for testing such that each note the null hypothesis lambda equals lambda. Note against the alternative hypothesis such as that llamada is less than La Madonna. Okay, so it's given that statistic statistics is uh to limit the times is a submission of X. I. Which has a chi squared distribution with two and degrees of freedom. So if this is true, F H note is true. Obviously we are going to reject each note for extreme values for statistics. Okay, for the one sided alternative, we must think carefully about which deal of the distribution to use. Should we be rejecting? It's not for every large or very small values of the test. So under the alternative hypothesis, um we have two llamada times as a submission for X. I from R equals one to end has a chi square distribution. So this sequins lama donald over llamada times two lambda. Times submission of X I from R equals one to end. And lambda over llamada note over a kilometer. There's more than one. Okay, so we expect to get large values of the test statistics under each one. Okay, so we therefore reject each note in favor of each one. Such that llama to less than la Madonna. Statistics statistics we see it's it's appropriate other percentage point of the chi square distribution. So the critical value is chosen to obtain the specialized type one error probability. So we'll let X square of alpha over to an M denote the upper 100 alpha percent a percentage point of excess square distribution. Okay so that assuming that each note is true. Therefore our test is to reject each note. Okay so the critical region for the test such that X note of alpha and to end and infinity. Okay then we consider the right tail. Test that test it's note lambda equals lambda and all against each one. Such a debt llamada moors and llama donut. Okay again statistics statistics is um to la Mcdonald times assassination over X I from equals one to end. Right. Okay. Which has a chi squared distribution with two n degrees of freedom of each note is true. So applying the same process as in the first part we conclude that appropriate to reject its nose for some very small values of statistics statistics then the critical value is now that lower down 100 alpha percentage point. So it's X squared of one minus alpha over two. Yeah. Yeah it's one over alpha and to em okay it's not all for over two of the chi square distribution was two and degrees of freedom. So that probability of to lambda and the submission from equals one to end for X. I. Um This is less than X squared for one minus alpha and two and it's alpha. Okay, assuming that it's not, it is true. Okay. So we reject it's not uh this value and the critical region for the test from zero two X squared of one minus alpha and to end. Okay, okay. Them for the last part of the problem we consider the two tail test. That's the test. Each note lambda equals lambda node against each one. Such as that llama. That does not equal la Madonna. Again, that is the statistics to lambda note times the submission for X Y. Which has a chi square distribution with two and degrees of freedom. Okay, so if each one is true in this case is a critical values are ah x squared of alpha over two and um and next one squared X. Described of one minus alpha over two and two. And so the upper and lower limits are 100 alpha over two percentage points. Of the chi square distribution was two and degrees of freedom. Okay, so let's find the probability of X squared of one minus alpha over two and two and it's less than or equal to land an old times submission from I equals one to N for X I and less than or equal X squared off also over two and two. And so it's one minus. Okay, now we reject it, know what if the test, the statistics falls outside this interview, that is. We reject each not in favor of each one if, either to lambda times as a submission ah for X. I less than X squared of one minus alpha over two and two and over the second part to limit the note times and submission for X I from I equals one to enter more than X squared of also over to and two. And okay, that's it.


Good morning. Today will be another big one in London — we'll have a preview of the action in a jiffy. For now, here are some stories that caught our eye:
- The London Olympics are a ratings hit, as NBC's coverage has broken records. "Through the weekend NBC averaged 35.8 million viewers in London, five million more than Beijing, and over a million more than the previous record-holder, Atlanta," says the TVNewser blog.
Morning Editioncatches up today with one New Jersey mom's way of teaching math to her children: bedtime problems "that soon became a beloved routine."
Laura Overdeck, as it says on her Bedtime Math website, "along with her husband, John, started giving math problems to their two older kids. ... [And] when their 2-year-old started hollering for his own math problem, they knew they were onto something."


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174. Hence we conclude that we may transpose a term from one member of an inequality to the other, provided we change its sign. Thus, suppose
a? +62>362—2a. Adding 2aż to each member of the inequality, it becomes
a? +62+2a2>362. Subtracting b2 from each member, we have
a2 + 2a > 312-22,
175. 2d. If we add together the corresponding members of two of more inequalities which subsist in the same sense, the resulting in. equality will always subsist in the same sense. Thus,
7> 3 Adding, we obtain 16>9.
176. 3d. If one inequality be subtracted from another which sudsists in the same sense, the result will not always be an inequality subsisting in the same sense.
Take the two inequalities 4<7 and
2 <3 Subtracting, we have 4-2<7–3, or 2<4, where the result is an inequality subsisting in the same sense. But take
6< 8 Subtracting, we have 9-6>10–8, or 3>2, where the result is an inequality subsisting in the contrary sense.
We should therefore avoid as much as possible the use of this transformation, or, when we employ it, determine in what sense the resulting inequality subsists.
177. 4th. If we multiply or divide each member of an inequality by ihe same positive quantity, the resulting inequality will subsist in the same sense.
pose we have
Hence an inequality may be cleared of fractions. Thus, sup
2d 3а Multiplying each member by bad, it becomes
178. 5th. If we multiply or divide each member of an inequality by the same negative number, the resulting inequality will subsist in the contrary sense.
Take, for example, 8>7.
Multiplying each member by —3, we have the opposite in. equality
-24 <-21. So, also,
15>12. Dividing each member by -3, we have
-5<-4. Therefore, if we multiply or divide the two members of an inequality by an algebraic quantity, it is necessary to ascertain whether the multiplier or divisor is negative, for in this case the resulting inequality subsists in a contrary sense.
179. 6th. If the signs of all the terms of an inequality be changed, the sign of inequality must be reversed.
For to change all the signs is equivalent to multiplying each member of the inequality by –1.
- 180. Reduction of Inequalities. The principles now established enable us to reduce an inequality so that the unknown quantity may stand alone as one member of the inequality. The
other member will then denote one limit of the unknown quantity.
EXAMPLES. 1. Find a limit of x in the inequality
7 53 95
6 412 Multiplying each member by 12, we have
x <9. 2. 2x+ -8<6.
5x 4 3. 3x-2>
12 Ans. «>
2 6. Given
to find the limits of x. 50-15
1 cc>2. Ans.
<3. 7. A man, being asked how many dollars he gave for his watch, replied, If you multiply the price by 4, and to the product add 60, the sum will exceed 256; but if you multiply the price by 3, and from the product subtract 40, the remainder will be less than 113. Required the price of the watch.
8. What number is that whose half and third part added together are less than 105; but its half diminished by its fifth part is greater than 33?
9. The double of a number diminished by 6 is greater than 22, and triple the number diminished by 6 is less than double the number increased by 10. Required the number.
INVOLUTION. 181. A power of a quantity is the product obtained by taking that quantity any number of times as a factor. Thus the first power of 3 is 3; the second power of 3 is 3 x 3, or 9;
3x3 the fourth power of 3 is 3x3x3 x3, or 81, etc. Involution is the process of raising a quantity to any power.
182. A power is indicated by means of an exponent. The exponent is a number or letter written a little above a quantity to the right, and shows how many times that quantity is taken as a factor.
Thus the first power of a is a', where the exponent is 1, which, however, is commonly omitted.
The second power of a is a xa, or a?, where the exponent 2 denotes that a is taken twice as a factor to produce the power
The third power of a is axaxa, or a', where the exponent 3 denotes that a is taken three times as a factor to produce the power aaa. The fourth power of a is axaxaxa, or a*.
Also the nth power of a is a xaxaxa, etc., or a repeated as a factor n times, and is written an.
The second power is commonly called the square, and the third power the cube.
183. Exponents may be applied to polynomials as well as to monomials. Thus (a +b+c)3 is the same as
(a+b+c)(a+b+c) (a+b+c), or the third power of the entire expression a+b+c.
Powers of Monomials. 184. Let it be required to find the third power or cube of 2a352.
According to the rule for multiplication, we have (2a’62)3=20°32 x 2a3b2 x 2a3b2=2 x 2 x 2aʼaʼa?l?6262=8aolo.
In a similar manner any monomial may be raised to any power.
Hence, to raise a monomial to any power, we have the fol. lowing
Raise the numerical coefficient to the required power, and multiply the exponent of each of the letters by the exponent of the required power.
185. Sign of the Power.-— With respect to the signs, it is obvious from the rules for multiplication that if the given monomial be positive, all of its powers are positive; but if the monomial be negative, its square is positive, its cube negative, it, fourth power positive, and so on. Thus
-ax-a=ta, -ax-ax-ar-a, -ax-ax-ax-a=ta*, -ax-ax-ax -ax
=-a, etc. In general, any even power of a negative quantity is positive, and every odd power negative; but all powers of a positive quan. tity are positive.
EXAMPLES. 1. Find the square of 11a2bcd2.
Ans. 121a%b2c-24. 2. Find the square of -18xyz. 3. Find the cube of 7ab2c2. 4. Find the cube of - 8xy2z3. 5. Find the fourth power of 4ab2c3. 6. Find the fourth power of -5a3b2x. 7. Find the fifth power of 2ab3x2. 8. Find the fifth power of — 3ab2x*. 9. Find the sixth power of 3ab2x'.


A locally one-dimensional finite difference time domain method for the analysis of a periodic structure at oblique incidence
Article first published online: 9 SEP 2011
Copyright 2011 by the American Geophysical Union.
Volume 46, Issue 5, October 2011
How to Cite
2011), A locally one-dimensional finite difference time domain method for the analysis of a periodic structure at oblique incidence, Radio Sci., 46, RS0F03, doi:10.1029/2011RS004730., , , and (
- Issue published online: 9 SEP 2011
- Article first published online: 9 SEP 2011
- Manuscript Accepted: 7 JUN 2011
- Manuscript Revised: 25 MAY 2011
- Manuscript Received: 31 MAR 2011
- finite difference time domain;
- locally one-dimensional;
- oblique incidence;
- periodic boundary condition
 An implicit finite difference time domain (FDTD) method is developed to analyze periodic structures at oblique incidence. The split-field technique is applied to the locally one-dimensional (LOD) FDTD method with the periodic boundary condition. In addition, the dispersion control parameters are introduced to reduce the numerical dispersion error. The effectiveness of the present method is investigated through the analysis of a photonic band gap structure. It is shown that the computational time is reduced to ≅16% of that of the explicit FDTD method with acceptable results being maintained. As an application, a broadband mirror consisting of a subwavelength grating is analyzed and discussed.


User Group Conference 2019
- User Group Conference 2019
This year the teacher participation will happen at the end of each break-away in the format of sharing questions, answers and discussion among participants.
The NumberSense Mathematics Programme User Group Conference is being held on 10 and 11 May 2019, at the Avenue Conference Centre, V&A Waterfront in Cape Town.
A quick preview of the programme:
1. Pre-Conference session: Visits to the Western Cape Shikaya NumberSense schools to observe the implementation of the NumberSense mathematics routines in Grades 1-4 (limited to 80 participants on a first come, first served basis).
2. Three break-away workshop sessions. This year the teacher participation will happen at the end of each break-away in the format of sharing questions, answers and discussion among participants.
Plenary sessions to include:
- The NumberSense Mathematics Programme – what it is and what it is not! Highlighting achievements and addressing misconceptions.
- How the NumberSense Mathematics Programme begins preparing students for high school mathematics, from Grade R.
- Developing mathematical reasoning, understanding and application.
3. Networking activity: Using the GeoGenius Construction Kit to set a world record in geometry!
And evening entertainment.
Please mail conference@NumberSense.co.za for further information.


To look at data for a class, choose CLASSES from the blue ribbon.
Choose a class from the left column.
Click on Mixed Review in the middle column.
To understand and interpret this data--
The pie chart on the left shows how many points the students earned in assignments compared to points earned in mixed review. For long-term retention, GMM recommends that students earn more points in Mixed Review than assignments.
In the circle graph, we recommend the green portion (Mixed Review points) be larger than the purple portion (assignment points).
This number is the number of squares the class has when working in Mixed Review.
The median number of days students take to spiral through the active skills.
GMM recommends that this number is below 30 days.
If the number is too high, look at the Points Distribution and the number of Active Skills. Perhaps, students are not spending enough time in Mixed Review, or students have too many skills to complete. Consider deactivating some skills.
Check out this article on deactivating skills. https://intercom.help/get-more-math/en/articles/4523713-how-can-i-remove-skills-from-mixed-review-i-deleted-the-assignment-why-is-the-skill-still-in-my-student-s-mixed-review
Accuracy (in the upper right corner of the screen)
The accuracy for the class is the average of the accuracy for each active skill.
In the report above, the 94% accuracy rate pertains to the 47 active skills. Accuracy is the percentage of time students answer correctly on the first attempt.
The proficiency for each skill is the percentage of students who are green, silver, or gold on a skill. This metric provides valuable information about the long-term retention of skills for a class.
Remember that students must have a skill for at least three days and continue to answer problems correctly to change the color of a square to green; at least five days for silver; and for a longer period of time for gold.
When viewing Proficiency, if students have had the skill for several days and have NOT changed the color of the square, consider re-teaching the skill with guided practice or provide more practice time in Mixed Review.
View other columns by adding, or removing, using the manage columns button.
Accuracy (for each skill) in a row
The accuracy of a skill is the average of the accuracy for each student who has tried that skill at least once.
The number of students who have not attempted the skill.
The number of students whose square is red for the skill.
The number of students whose square is yellow for the skill.
The number of students whose square is green for the skill.
The number of students whose square is silver star for the skill.
The number of students whose square is gold star for the skill.
For more information, contact us at email@example.com or click the green chat bubble.


KMV-Merton model Probability of Default represented by Jin-Chuan Duan, Genevi`eve Gauthier and Jean-Guy Simonato (2005).
This code calculates the probability of default based on Moody’s KMV where firms equity follows a geometric Brownian motion presented by Merton and the probability of default is calculated bas on European call option of the firms market value. Newton-Raphson method is used to calculate the equity value provided the volatility of the equity.
No updates, just added few comments to explain lines in the code.
Download apps, toolboxes, and other File Exchange content using Add-On Explorer in MATLAB.


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unpack "L= 3 yds less than triple the W"
Length = 3 yards less than triple the Width
"triple with width W" would be written as \(3\times W\) or \(3W\)
3 yards less than triple the width: \(3W-3\)
so far so good?
for example, if the width was say \(100\) (which it is not, just an example) then the length would be 3 less than triple 100 or \(300-3=297\)
on how to figure out the Length
ok lets try this: we are forgetting everything but the line "L= 3 yds less than triple the W"
only focusing on that one suppose i say the width is 50 what is the length?
if it is not clear, let me know and we can walk through it
L = 147??
and suppose the width is 25 what is the length?
L = 72
k good now we know what we are doing how did you get your answers?
oopss forgetting ft
i don't see the mention of feet in the problem so lets assume the units are yards, and ignore feet
3(25) - 3 = 3*25 = 75 - 3 = 72
k good so if you know the width, you know the length how do you find it? multiply by 3, then subtract 3 if we replace \(25\) by the variable \(W\) then the equation tells you \[L=3W-3\] which is what "L= 3 yds less than triple the W" says in math
ok so that part is done now how about this part " P= 450"
we need an expression for the perimeter in terms of the width \(W\) and length \(L\) do you know it? ("no" is a fine answer, just asking)
actually . no
wouldn't it be p =l * w
so since we have the Perimeter wouldn't we divide the L and W
to get the Dimensions
that is an expression for the Area and as you see by your multiplication it comes in square units (2 dimension) if the width is 4 and the length is 3 yards, the area is \(3\times 4=12\) square yards perimeter is a length, one dimension, a number of yards in this case
oooppps wouldn't we divide to get them?
so no, it is not \(P=LW\) but rather \(A=LW\) we need an expression for the perimeter
measure of the perimeter is the total length in this case it would be \(10+10+15+15=50\) or i could write (suggestively) \[2\times 10+2\times 15=50\]
now with this picture |dw:1377786057826:dw| what is the perimeter?
you have \(P=W+W+L+L\) which we would write as \[P=2L+2W\]
good so far?
ok now we know that \(P=450\) i.e. we know \[2L+2W+450\]
and we also know that \(L=3W-3\) so we can replace the \(L\) in \(2L+2W=450\) by \(2W-3\) with judicious use of parenthese and get the equation \[2(2W-3)+2W=450\] which we can then solve for \(L\) because now there is only one variable used \(L\)
typo there sorry equation is \[2(3W-3)+2W=450\]
final job, which will take a few steps, is to solve this for \(W\)
should we work through the steps?
w = 8
is that correct??
that is not what i get lets work through the steps
\[2(3W-3)+2W=450\] first step is to multiply on the left using the distributive property \[6W-6+2W=450\] so far so good?
ok, that is what I got
second step is to combine like terms on the left, i.e. add \(6w+2W\) to get \[8W-6=450\]
ok, that is what I got
add \(6\) to both sides and get \[8W=456\]
then I added 6 to 450
k good last step?
ooppss ans. should be 57
ok good, that is the width, right now we also need the length
I miss read my calculations. I switched the numbers around
ok did you get the length?
so, we would put 3(57) -3 = L
L = 168
that is what i get as well
ok, so the actual set up for the problem would be: 3w-3 = L; 2(3w -3) + 2w = 450; 6w - 6
6w - 6 + 2w = 450; 8w = 456; 456/8 = 57 = W = 57; 3(57) -3 = L --- L= 168. Is that all my daughter needs to do to show her work for this type of problem??
wow ok i am not sure what your sentence "Is that all my daughter needs to do to show her work" means lets go back and see what all we need first we need to know how to write "L is 3 yards less than triple the width" in math then we need to know the formula for a perimeter also we need to know how to substitute \(3W-3\) for \(L\) in \[2L+2W=450\] and finally we need to know how to solve \(2(3W-3)+2W=450\) for \(W\) the last part, finding \(3\times 57-3\) is not much
my daughter has to show her work, and how the equation should be set up.
show her work to whom?
i guess so i have no way of knowing what her teacher wants
is sitting her with me and she did all the calculations except for the last one where I came up with 8 instead of 57
i suppose she could ask her teacher what kind of work she wants to see
on line class?
algebra is hard to learn on line you have to do it all yourself, and it requires a lot of maturity good luck
thanks. so far she is doing good. just this one question through her
Thank you so much for you help.


Here's the "Troll Pi" or "Pi equals 4" image.
Video Lectures, Video Courses, Science Animations, Lecture Notes, Online Test, Lecture Presentations. .
A collection of free, web browser-based apps that aim to make mathematics intuitive for learners of all ages.
TED is a small nonprofit devoted to Ideas Worth Spreading -- through TED.com, our annual conferences, the annual TED Prize and local TEDx events.
The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day
You don't have permission to access /home/wdj/art/hypotrochoid_R_equals_7,_r_equals_2,_d=3.gif on this server.
(click to fire) | | | | |
Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h. Upon reaching the other train, the fly turn...
Neil Bickford calculated the first 458 million terms for the continued fraction of pi, breaking the previous record of 180 million. He used to develop his code and verify his results—which he posted shortly after he turned 13.
The #1 tool for creating Demonstrations and anything technical.


1.3 Evaluating Limits Algebraically part 2
This is the third lesson in the Limits Unit. The lesson contains Guided Notes, homework, SmartBoard lesson, and all solutions. Don’t forget the added bonus of a math joke embedded into the Smartboard lesson ☺! You may have also opt to purchase the video lesson to the notes.
In this lesson students will learn 3 more methods on how to solve limits algebraically. They will learn basic substitution, how to evaluate infinite limits, and trigonometric limits. By the end of the lesson, students should understand how to recognize an infinite limit and determine whether the graph will head toward positive or negative infinity. They will also learn how to use the special sine and cosine trigonometric limit properties.
I hope you find the notes easy to follow. Please feel free to offer any suggestions on how to make them better as well! Through a community of collaboration, we can better aid our students and their learning process.
All the best,


Dear Matthew Alice: My pa sez, if ya read it in the Reader, it gots ta be true. So I have enclosed the following example of mathematics [list of winners to Reader puzzle #869], and I’m confused by the results as stated. Is this some of the new math (e.g., 2+2 = 5 for all large values of 2), or am I missing links? It says 47% of the entries got the answer right, 53% got it wrong; “of the 87 entrants, 73 were correct,” but then it lists 23 incorrect answers received. But 87 minus 73 is 14; and 73 is 84% of 87, making 14 only 13% of 87. So if the “47% correct” is correct, and “53% wrong” is correct, and 87 is the total “sample,” making A = 41 and B = 46, what did I miss, and how did I miss it? — Carl Ferguson, San Diego
Well, Carl, I’m sure everybody else is digging around for a pencil and paper, trying to figure out what’s going on, so in the meantime, I’ll say congratulations for submitting the only winning entry to Reader Puzzle #870, titled “What’s Wrong with Reader Puzzle #869?” Your prize is a year’s free subscription, if you don’t mind picking it up every week at the nearest convenience store. We tried getting it in the mail, but just couldn’t manage the nine-digit ZIP code. I think we were subtracting the last four digits from the first five, and the paper kept ending up in Virginia. We do like Pa Ferguson’s attitude and wonder if he’d be interested in filling a vacancy in the Reader puzzle department.


More Maths, or Better Maths?
Interesting view from a Biosciences perspective about the recent recommendations to increase the number of students taking Mathematics at A-level.
I’ve always had a problem with the way Statistics is taught at A-level, which is largely as a collection of recipes without much understanding of the underlying principles; would more emphasis on probability theory be a better way to go?
The introduction of post-16 maths is in the news again with a report from the House of Lords committee on Higher Education in STEM and many of the headlines from the Guardian, Independent and Times Higher have picked up on the recommendations regarding maths study post-16.
I have written a few thoughts here on my first impressions but would very much welcome comments.
Though I was pleased to see that some of my work showing that only GCSE maths is required for undergraduate biosciences was cited, the conclusion from this was that more students should take maths A level and this is a little worrying.
The lack, or low level, of maths requirements for admission to HEIs, particularly for programmes in STEM subjects, acts as a disincentive for students to take maths and high level maths at A level. We urge HEIs to introduce more demanding maths requirements…
View original post 634 more words


Aveline Giulia October 1, 2020 Worksheets
Between the ages of 3 and 6 years, your child is capable of learning and absorbing more information than you can imagine. Advance his training with the use of preschool worksheets. These are very powerful tools to stimulate your child’s brain.
Do choose worksheets that are appropriate for the child`s level, since children can get discouraged easily, if they are unable to perform the activity. Well illustrated worksheets with cartoons are also more likely to appeal to children. Worksheets that use common situations children come across at home, school, in the market place etc and that use common objects known to children would be more relevant.
In choosing a worksheet, it is important to review the source and check the material. Ensure that the material and answers are accurate. Evaluate the worksheet by completing it yourself. The worksheet should provide information clearly and accurately. Make sure it is exactly what you need to homeschool your child.
The math worksheet is not only for the young children in kindergarten and early primary school; they are also used for tutoring high school and university students to keep the students’ math skills sharp. The sites that offer these worksheets have helped a lot and this resource is now a common thing to use for all kinds and levels of educators. The formats for the worksheets differ according to the level and content of the worksheets. For the young kids it is preferable to have the worksheet in large print, while the older students commonly use the small print ones that are simple and uncluttered.
Aside from helping you assess your child’s comprehension of a subject matter, printable home school worksheets also provide something for your child to do while you work on other things. This means that you can be free to run your home while teaching your child at the same time, because the worksheet simplifies the homeschooling job for you.
Writing worksheet is a new way for creation of a document by the teachers to promote the development of the writing skills amongst children. These worksheets are used primarily by teachers and parents. It is an effectual tool for assisting children to know the basic tactics to write.
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I’d like to state in advance that I have negligible familiarity with number representations, so please excuse any inaccuracies. I also enjoy getting rid of legacy systems.
I have recently read an article which pointed out that one of the reasons COBOL is still somewhat prevalent in the banking and governmental sectors is that it used fixed point number representations, which are less prone to accumulating inaccuracies during the typical calculations these sectors use. (There are other factors of course as many languages let you use fixed point numbers with varying levels of difficulty by now.)
This article describes Muller’s recursion formula, which is a good example of this erratic behavior. This series should converge to 5, but using doubles it converges to a 100, after some hectic jumps. I wanted to try this with Julia’s types, so I wrote this code.
module cobol using FixedPointNumbers mullers(y,z) = 108 - (815 - 1500/z)/y function recur(t,n) v = zeros(t, n) v[1:2] .= t[4, 4.25] for ii = 3:n v[ii] = mullers(v[ii-1], v[ii-2]) end v end end # module
Float64 it reproduces the behavior of other languages, as expected. With
BigFloat it properly converges to 5. With
Float64.(recur(Rational,20)) it also converges to 5, but overflows before the double representation would reach it’s fixed point at iteration 29.
What I am interested in (apart from your insights in general) is that by using a fixed point representation
N41f23 it also erroneously converges to 100 (albeit differently than doubles - at least no sign change) - is there a way to do it right with fixed point representation that I haven’t found?


|University of Otago
|Full text PDF:
This thesis shows the level at which reliable geoidal mapping is possible over a small area using the Global Positioning System (GPS), geodetic levelling and appropriate interpolation strategies. The geoid shape over an area 15kmx30km of Dunedin, New Zealand, was determined by the geometric approach (King et al, 1987) using almost 70 control points. Investigations were then made to determine the precision to which orthometric heights can be estimated within this area with various interpolation strategies. This was done by creating subsets of control points with various point densities. Experimental data sets were then generated by estimating the undulations at the points not selected as control. Quantitative analysis of the differences between estimated and known undulations was then carried out. The investigations indicate that: • Where average known undulation point spacing is between one and five kilometres, the mean error in the estimation using linear interpolation will vary by twenty millimetres. Any specific point may vary from the true value by up to fifty millimetres. • Where average known undulation point spacing is between five and ten kilometres, the mean error in the estimation using linear interpolation will vary by sixty millimetres. Any specific point may vary from the true value by up to 150 millimetres. • Low order polynomials (three to five independent variables) are not as good as linear interpolation when average known point spacing is less than ten kilometres.


|Math OER Zoom Room Textbook YouTube Distance Learning Tips|
What playgrounds do is provide kids with a relatively safe way to learn about using their bodies to navigate the world—how to balance, how to get from here to there, what to do when you get stuck. In other words, how to solve problems in the physical world.
As I was watching my daughter, I realized that math too is a playground. But it’s not a playground for our bodies, it's a playground for our minds.
- Jason Marshal
Here are places you can use math to explore, analyze, and estimate without being expected to accomplish any specific goals or reach any specific answers. Just play!
Kids first engage with a physical playground when they see it. They start thinking about the slide, or the tire swing, or whether they could step from one place to another. Math playgrounds are less visual. So these explorations provide a little bit of descriptive guidance to help you "see" the math clearly as you approach.
There are suggestions: questions to think about, and challenges to attempt. But please do not treat these suggestions like homework problems. Approach the exploration in your own way.
You can explore alone. But playgrounds are more fun with friends.
With graph theory we explore math without numbers.
Which kinds of measurements make the best pattern for foot size and height?
Which of ten famous ideas would you pick to win a beauty contest?
Challenge classmates to a math duel in the spirit of the dangerous world of 16th century algebra.
Watch tradgedy unfold as error propagation wrecks estimation!
The start of term reflection helps you start the term with mindfulness and purpose.
The CLO reflection affirms that core learning happens in every class.
These big issues without right answers do not have any step-by-step guidance.
You are on your own, released from the rigor and expectations of the classroom to do real math.
Learn the habit of using math to estimate and explore. Math can help make decisions even if there is no right answer. Issues like these are the "icing on the cake" that sweetens all the math skills and algorthms you have learned.
Practice tests can be playgrounds too.
Do not take them too seriously. Seldom do an entire practice test.
Instead, do the first few problems. Try these again and again on different versions the same practice test. Work on good form. Enjoy affirming your mastery!
Then add another one or two problems. Again, practice your form with a shorter to-do list.
Eventually you will have mastery over the entire practice test. But do not rush it.
Practice mindfulness. You are working on good form by repeating a task with attentiveness to detail. Acknowledge the frustrations and the joys while not focusing on them.
Your actual tests will look just like these practice tests—almost. For some problems, instead of creating answers you will be required to analyze already complete work to explain steps or find errors.
Topic tests happen during class time. They teach you how to take a math test.
The first half of the class time focuses on solving problems using the characteristics of well-written step-by-step problems. Pretend this is a traditional math test where you work by yourself.
The second half of each topic test focuses on group collaboration to finish and fix answers. There will be a lot of valuable learning involving explanations out loud and pointing at each other's papers—two resources that will forever vanish! Capture this learning by making fixes to your scratch work, and by finishing any test problems you did not finish by yourself.
Each group member should write his or her own fixes. Write your fixes and finishings distinctly. Use another color pencil, pencil versus pen, separate pieces of paper, clouds around the fixes, etc. The ability to clearly see which topics were difficult or promoted careless mistakes will help you be properly wary when studying for future tests.
Before the end of the day, text or email your instructor a one sentence study plan.
This study plan should include using helpful study skills and habits and be relevant to the problems you fixed or the problems your group answer inaccurately. It should describe how you will study before the next test. Optionally, it may also include reminders about improving test-taking technique.
The end result is a written record of which problems you need to study and your plan about how to study them. It also has every problem correctly solved in a step-by-step manner to be optimally helpful when studying for the final exam.
The end of term tests are simply larger collections of topic test problems.
A high school math class is a bit like becoming a concert pianist. No one really sees or cares about your hours of practice. They watch you as you get up on stage at the end. You demonstrate your ability, by yourself, under pressure.
In graduate school, more assessment is oral exams. Instead of creating written answers privately, you stand at the chalkboard while your instructors ask you to do work similar to the practice tests. You do fewer problems, but are expected to be smoother. Sometimes this is a social gathering in which a group of students take turns demonstrating their mastery and celebrate afterwards.
In our class, as an undergraduate class, the end of term tests can use either format. The choice of format might be yours to make, or the instructor might assign you a format.


...I have also worked with students with similar issues in my private tutoring. I have successfully helped many students taking the ISEE to get scores required for their choice of private schools in my 15 years as an independent tutor. I am qualified in SSAT, a similar test.
Waltham, MAView Tutor
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Stone Soup by Jan Eliot for August 27, 2013
6 x 8 Gramma Evie: OK, let's do EIGHTS> Holly: @*%# Gramma Evie: Holly, I'm just trying to HELP. Math will be easier for you when you can do some of it IN YOUR HEAD. 6 x 8 Gramma Evie: So don't shoot the messenger. 8 x 6 = Holly: How about if I shoot the messenger IN MY HEAD?


Cristian Bereanu, Jean Mawhin, Departement de Mathematique, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, e-mail: mawhin@ math.ucl.ac.be
Abstract: We use Brouwer degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.
Keywords: nonlinear difference equations, Ambrosetti-Prodi problem, Brouwer degree
Classification (MSC2000): 39A11, 47H11
Full text of the article:


Angular size is a measurement of how large or small something is using rotational measurement. It is useful for measuring things that are so far away that they appear two dimensional.
Images for kids
Angular diameter: the angle subtended by an object
Diagram for the formula of the angular diameter


The purpose of this study was to investigate the relationship between mathematics self-efficacy and mathematics achievement of high school sophomores across the United States, and to examine the effects of gender, ethnicities, and school characteristics on students’ mathematics achievement using hierarchical linear modeling (HLM). The base-year data of the Educational Longitudinal Study (ELS): 2002 were used for analysis. Hierarchical linear models were developed from the one-way random effects ANOVA model, and the unconditional Model with mathematics self-efficacy in level 1, to the contextual models with variables in the both levels. Both fixed effects and random effects were estimated and interpreted for all the models.
Liu, Xing and Koirala, Hari P., "Hierarchical Linear Modeling of Students’ Mathematics Self-Efficacy and School Effects on Mathematics Achievement" (2010). NERA Conference Proceedings 2010. 34.


In this unit of work we link the development of skip-counting patterns to bars on a relationship graph. We also plot our skip-counting patterns on a hundreds board.
- Continue a skip-counting pattern.
- Describe skip-counting patterns.
- Use graphs to illustrate skip-counting patterns.
In this unit we look at skip-counting patterns. These are patterns obtained by adding the same, constant, number to make the next number every time. So the difference between any two terms in a skip-counting pattern is the same. This is a good exercise to help reinforce the various concepts relating to pattern. In particular, it helps us to understand the idea of a recurrence relation between consecutive terms.
But this exercise has two other links. First it links algebra and statistics by using a bar chart to represent the skip-counting pattern. It is worth remembering that there are many links in mathematics.
And second, skip-counting patterns are also called arithmetic progressions. In secondary school these are considered again and expressions for both the general term of the progression and the sum of all of the numbers in the progression are found. These are both reasonably simple algebraic expressions.
Links to Numeracy
This unit provides an opportunity to develop number knowledge in the area of number sequence and order, in particular development of knowledge of skip counting patterns. It can also be used to focus on the development of strategies to solve multiplication problems.
Once students have created bar graphs of the relationships, help them to focus on the number patterns involved by creating tables. For example:
|Number of Beetle Wheels|
|Number of Beetles||
Number of Wheels
As students create tables, focus their attention on the patterns that emerge and pose questions about the continuation of the patterns. Use of a hundreds chart will help students visualise the number patterns more easily and help them to predict which numbers will be part of the patterns. Patterns of 2, 5 and 10 are a good place to start but for students that are coping well you can make it more difficult by using larger numbers. For example, if there were 7 friends in each beetle, how many people would there be in 2 beetles? 3 beetles? What about 10 beetles?
Working with larger numbers of beetles (or other items) will help students develop strategies to solve multiplication and division problems. Encourage students to talk about the way they are solving these problems. Are they using materials, repeated addition or can they derive some of the answers from known multiplication facts?
Questions to develop knowledge / strategy use:
What number comes next in this pattern?
How do you know?
What number will be before 24 in this pattern? (or another number as appropriate)
How do you know?
What is the largest number you can think of in this pattern? How did you work it out?
How many wheels will there be on 5 beetles? 10 beetles? How did you work it out?
If there were 48 wheels in a car park how many beetles would there be? How did you work it out?
- Squared paper for graphing
- Picture of VW beetle
- Pictures of objects for exploration
Today we explore the pattern of 4s by counting the number of wheels on cars. We then use this information to build a relationship graph.
- Ask: How many wheels does a beetle have?
- Share ideas. Hopefully someone will link the beetle to the Volkswagen car rather than the insect or you may have to give a few more hints. Show students a picture of a VW beetle and discuss why it got this nickname (it is shaped like a beetle).
- Using counters begin to develop a chart of the number of wheels to the number of cars.
- Ask: How many wheels are there on 2 beetles?
How did you work that out?
- It is useful for the students to listen to the strategies that others use. More advanced Level 1 students will be be to count on from 4 to find the answer and many may have 4 + 4 as a known fact.
- Repeat the process with 3 and 4 VW beetles. Each time continue to add the information to the chart.
- Ask the students to work out how many wheels there would be on 6 beetles. If some of the students find the answer quickly, ask them to find the answer using another strategy.
- Share solutions. These may include:
- skip-counting with or without the calculator
- counting on using a number line or hundred’s board
- using counters to find 6 groups of 4.
- As the class to complete the chart up to 6 cars.
- Ask: What can you tell me about this chart?
Share ideas. Encourage the students to focus on the relationship between the number of cars and the number of wheels.
- Ask the students how they could record this information using grid paper.
Over the next 2-3 days, the students work in pairs to explore the number patterns of other skip-counts. At the end of each session the students share their charts with the rest of the class.
- Place pictures of items that the students are to investigate in a “hat”. Ask each pair to draw one out and then investigate the pattern up to at least 6. Encourage the more able to students to extend the pattern beyond 6.
- Pictures could include:
- tricycles (3 wheels)
- bicycles (2 wheels)
- hands (5 fingers)
- spiders (8 legs)
- glasses (2 lenses)
- frog (4 limbs)
- stool (3 legs)
- Remind the students that they are to record their explorations on a chart.
- At the end of each session share and discuss charts and number patterns. Ask the students to identify the patterns that are the same.
In today’s session we use calculators to extend our skip-counting into the hundreds. We record our patterns on a hundreds chart.
- As a class look at the chart to show hands (5 fingers). Skip count together in 5s, shading the counts on a hundreds chart.
- As the chart is shaded ask questions which encourage the students to look for patterns in the numbers as they make their predictions.
Which number will be next?
How do you know?
- Give the students (in pairs) a hundred’s chart and ask them to shade in one of the skip counting patterns that they had charted on the previous days.
- Display, share and discuss at the end of the session.


12/8/16 – Dennis Miller and Hubert Ho
Music and Math | Dennis Miller and Hubert Ho
December 8, 2016, 7-8pm
MIT List Visual Arts Center
What is the nature of knowledge? Where does music lead the mind? Dennis Miller and Hubert Ho think about these big questions through the particular lens of music. Music relates to many other fields – to visual art, moving images, to mathematics. In a formal sense they both embrace music composition as a problem solving challenge – aiming to create a unified whole which communicates to the audience. On the one hand a composition defines its own authority. But broader concepts, spanning disciplines, such as form, pattern-recognition, symmetry, and recursion can be identified through music.
Both Miller and Ho address the relationship between mathematics and music, looking at their art form through the lens of another discipline. Incorporating mathematical insights, however, does not preclude the necessity, possibility, or desirability of a spiritual or emotional connection to music. Does mathematics have a role in what they experience internally as composers, and in listeners’ experiences of resulting musical compositions?


This video shows how to calculate the labor rate variance.
The labor rate variance is the difference between:
(1) actual labor hours * actual wage rate and
(2) actual labor hours * standard wage rate
This is sometimes abbreviated as: (AH * AR) – (AH * SR)
If the company spent more than it should have (according to the standard, which is set by management), then the labor rate variance is unfavorable. An unfavorable variance could occur if the company’s employees received a higher wage from working overtime.


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Yes, size matters. The larger your mesh is, the more LODs are going to be ignored by the LI calculation. The radius of the bounding box (think of a sphere encompassing the mesh) is what determines which of the lower LODs is being dropped.
If said radius is larger than 5.43 meters, the lowest LOD will be ignored.
If said radius is larger than 10.86 meters, the low, and lowest LOD will be ignored.
If said radius is larger than 43.44 meters, othe lowest, low, and medium LOD will be ignored.
The formula to get the radius is sqrt(x*x + y*y + z*z) / 2
Also, it's always a good idea to keep larger meshes as lowpoly as possible, and add window frames, and any detail as a separate mesh object.


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What is the common difference for this arithmetic sequence? -6, -2, 2, 6, 10, … What is the common difference for this arithmetic sequence? -6, -2, 2, 6, 10, … January 13, 2023 Others 0 You may also like... A theater sold 85% of the tickets for a play. If 170 tickets were sold, how many tickets Josh has 27 signed baseballs. He places the baseballs equally on 3 shelves. How many baseballs Find the first five multiples of the number write a multiple equation to show their multiple Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Email * Website Save my name, email, and website in this browser for the next time I comment.


Using Computer for Statistical Analysis
Students examine the use for spreadsheets in analyzing data. They make spreadsheets that display and calculate a given data set such as temperature change.
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Collecting and Working with Data
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3rd - 6th Math CCSS: Adaptable
Are Female Hurricanes Deadlier than Male Hurricanes?
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9th - 12th Math CCSS: Designed
Get on your Mark, Get Set, Go! Collect, Interpret, and Represent Data Using a Bar Graph and a Circle Graph
Start an engaging data analysis study with a review of charts and graphs using the linked interactive presentation, which is both hilarious and comprehensive. There are 27 statistics-related vocabulary terms you can use in a word sort....
4th - 7th Math CCSS: Designed


Determine if input is odd
Back to basics 8 - Matrix Diagonals
Remove from a 2-D matrix all the rows that contain at least one element less than or equal to 4
Diagonal & Anti-Diagonal Matrix: Easy
Determine the square root
Find the magnetic field in sollinoid
Cipher and Decipher the message - 3 key
find the index of local minimum in the matrix
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Solving the Target-Value Search Problem
Keywords:Heuristic Search, Combinatorial Search, Target-Value Search
This paper addresses the Target-Value Search (TVS) problem, which is the problem of finding a path between two nodes in a graph whose cost is as close as possible to a given target value, T. This problem has been previously addressed: first, for directed acyclic graphs; second, for general graphs under the assumption that nodes can be revisited given that the same edge can not be traversed twice. In this work we focus on a more restrictive variant of the same problem where nodes can not be revisited. We prove that this variant is NP-complete and discuss novel theoretical properties and provide empirical results to solve this problem optimally.


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Record any names for your feature.
Open up the lunar image and measure the length of a shadow. Just use the x,y coordinate. Convert the pixel to angular distance in the sky and then into kilometers using the distance to the moon and the small angle equation. (Use the Range recorded before.)
To check your conversion from pixel to distance on the lunar surface is correct, pick several craters in the image with sharp, well-defined rims or walls. Measure and record the size of these craters in their longest dimension (which will probably be top-to-bottom). Calculate their sizes. Compare them with a list of known sizes.
If your image covers an area near the center of the lunar disk (i.e. a spot midway between the northern and southern limbs, and midway between the eastern and western limbs), then you should find your crater sizes to be close to that listed, but if your image shows an area far from the middle of the disk, then it will be tilted -- the lunar surface is not face-on to the camera. In that case, you will have to make a correction for the fact you are seeing the crater tilted.
How do we correct for this tilt? Envision the moon and you looking at it:
In the picture below you are viewing two shadows. Each shadow is the same size [42 pixels]. You see the entire 42 pixel length of the one in the center of the moon but the one closer to the edge (limb) of the moon measures only 35 pixels. Thus a picture of center of the lunar disk has little or foreshortening because the lunar surface is roughly face-on to the camera. But a picture of any other area will be tilted with respect to the camera; the closer the area is to the limb of the moon, the larger the tilt.
You will need to correct for this foreshortening.
Use the lunar latitude and longitude of your mountain in the following formula:
tilt correction factor = 1/ (cos(latitude) x cos(longitude))
The correction factor is always larger than 1. Multiply your shadow length (in km) by this factor, to get the corrected shadow length (also in km).
As a check, use this same factor to correct the measured diameters of craters in your image; compare to the known diameters of the craters. How close are your measurements to the listed diameters?
Now you know the size of the shadow you can find the height of the mountain, - if you know the angle of the Sun. See the figure below
The simplest way to the sun angle is to measure the number of degrees between the feature and the terminator (as long as they are less than 90 degrees apart) and multiply by 0.01*(100 - |feature latitude) as a substitute for the solar angle.
Sun angle = (feature longitude -terminator longitude) x .01 x (100 - |feature latitude|)
Shadows are prominent (and most easily measured) when the local solar angle is about 15 degrees or less.
Calculate the height of your feature.
height = shadow length x tan (Sun angle)Can you find another source, which lists the height of your mountain, or other pictures of your mountain? See
The description of your mountain should have its location, name (if you know it) and a run-down on how you go its height. You might even include a sketch.
Be sure to answer all the questions above.
When you find the official values or other information, like the height of Brasstown Bald, you need to say where you got the information - reference, URL, etc.


500lbs pressing down on two gears. Surface area of contact on one gear is 0.03125 inches squared. I calculated pressure. 500lbs / 0.0625 = 8000psi being exerted onto the two gears' surface areas. One gear has 4000psi. The gears' radius is 0.47 inches. I calculated torque for an individual gear. 4000psi (0.47 inches) = 1880 inch pounds. (I think I just found my mistake, but I'll finish posting this with my new results). So the torque being applied to the gears is 1880 inch pounds each. I guess working through the problem for the 3rd time helped. But I would welcome anyone to double check my triple check. I still may have misunderstood something and my calculations are off.


We consider a self-interacting process described in terms of a singleserver system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space-time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit.We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.
|Number of pages||20|
|Journal||Annals of Probability|
|Early online date||5 May 2015|
|Publication status||Published - 2015|
- Greedy policy
- Long-term behavior
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
FingerprintDive into the research topics of 'Greedy walk on the real line'. Together they form a unique fingerprint.
- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics - Professor
Person: Academic (Research & Teaching)


Congratulations to Walford Mathematics Teacher, Joanne Maratos, who was recently included in the Sunday Mail feature, Australia’s Best Teachers.
Joanne has been nominated for her impact as a Maths teacher. Joanne described her approach to teaching:
“As a Mathematics teacher I am highly motivated to develop my students’ skills and confidence so they can understand the problem and then be able to find the solution. I find this incredibly rewarding, especially when teaching girls who may have struggled with Mathematics and felt apprehensive about their own abilities. My key strategies to achieving positive outcomes are to break down Mathematical concepts, see each student as an individual learner, tailor my teaching to their strengths, and use interesting and relevant examples. Recently we celebrated Pi Day by using cookies to illustrate the main concepts relating to circles and the visuals really assisted the students’ understanding.”