7 edition of From geometry to topology found in the catalog.
|Statement||H. Graham Flegg.|
|LC Classifications||QA611 .F48|
|The Physical Object|
|Pagination||xii, 186 p. :|
|Number of Pages||186|
|LC Control Number||74078155|
Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie. Get this from a library! Geometry and topology. [Miles Reid; Balázs Szendrői] -- "Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics. The book introduces the ideas.
A Text-Book of Geometry by G. A. Wentworth; CK Geometry by V. Cifarelli, A. Gloag, D. Greenberg, J. Sconyers, B.; Geometry for Elementary School by Wikibooks; The Elements by Euclid; College Geometry and Beyond. A First Course in Topology: Continuity and Dimension by John McCleary. Download books "Mathematics - Geometry and Topology". Ebook library | B–OK. Download books for free. Find books.
A slim book that gives an intro to point-set, algebraic and differential topology and differential geometry. It does not have any exercises and is very tersely written, so it is not a substitute for a standard text like Munkres, but as a beginner I liked this book because it gave me the big picture in one place without many prerequisites. almost every other branch of mathematics. This makes the study of topology relevant to all who aspire to be mathematicians whether their ﬁrst love is (or willbe)algebra,analysis,categorytheory,chaos,continuummechanics,dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics,File Size: 10MB.
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Of over 4, results for Books: Science & Math: Mathematics: Geometry & Topology: Topology Introducing MATH. Grade 4 by ArgoPrep: + Practice Questions + Comprehensive Overview of Each Topic + Detailed Video Explanations Included | 4th Grade Math Workbook.
Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.
The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the Cited by: From Differential Geometry to Non-commutative Geometry and Topology Teleman, N. () This book aims to provide a friendly introduction to non-commutative geometry.
This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. Available Formats: eBook Softcover.
Book A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry. This textbook provides an introduction to the ideas and techniques of differential geometry and topology. It starts with a brief survey of the physics needed to follow the arguments - including quantum field theory, gauge theory and general relativity - to make sure all readers set off from the same starting point/5.
The intent is to describe the very strong connection between geometry and low- dimensional topology in a way which will be useful and accessible (with some eﬀort) to graduate students and mathematicians working in related ﬁelds, particularly 3-File Size: 1MB.
Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches From geometry to topology book mathematics and physics.
This book introduces the ideas of geometry, and includes a generous supply of simple explanations and examples. The treatment emphasises coordinate systems and the coordinate changes that generate.
e-books in Geometry & Topology category Geometry and the Imagination by Conway, Doyle, Thurston - Rutgers University, Newark, These are notes from an experimental mathematics course entitled Geometry and the Imagination as developed by.
About the Book. Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology uses Mobius transformations to develop hyperbolic, elliptic, and Euclidean geometry - three possibilities for the global geometry of the universe. The text, written for students who have taken vector calculus, also explores the interplay between the shape of 5/5(1).
Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, ) is a considerable expansion of the first few chapters of these notes. Later chapters have not yet appeared in book form.
Please help improve this document by sending to Silvio Levy at [email protected] any useful information such as. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.
The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the. Geometry, Topology, Geometric Modeling. This book is primarily an introduction to geometric concepts and tools needed for solving problems of a geometric nature with a computer.
Topics covered includes: Logic and Computation, Geometric Modeling, Geometric Methods and Applications, Discrete Mathematics, Topology and Surfaces. Author(s): Jean Gallier.
Nakahara - Geometry, Topology and Physics. The go-to book for mathematical prerequisites for e.g. gauge theory, string theory etc. if you ask 90% of physicists. I personally think it's terrible because it doesn't explain anything properly, but I guess it's good to learn buzzwords.
Nash & Sen - Geometry and Topology for Physicists. This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds.
It contains complete proofs Cited by: In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
Sharp distinctions between geometry and. Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc.
such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups, many deeper results require the point space to be (locally) compact and. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.
The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the /5(51). Since hyperbolic geometry is almost the highlight of a first course on geometry, that seems a bit disappointing.
There are however other Generally a good book and useful especially if you need to recall basic facts about metric spaces and point-set topology/5. This note introduces topology, covering topics fundamental to modern analysis and geometry.
It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.
$\begingroup$ Hatcher's book is very well-written with a good combination of motivation, intuitive explanations, and rigorous details. It would be worth a decent price, so it is very generous of Dr.
Hatcher to provide the book for free download. But if you want an alternative, Greenberg and Harper's Algebraic Topology covers the theory in a straightforward and comprehensive manner.
book. • G E Bredon. Topology and Geometry. Springer GTM[$70] — Includes basics on smooth manifolds, and even some point-set topology. • R Bott and L W Tu. Diﬀerential Forms in Algebraic Topology. Springer File Size: 65KB.Geometry/Topology.
From Wikibooks, open books for an open world.Topology. The mathematical study of shapes and topological spaces, topology is one of the major branches of mathematics. We publish a variety of introductory texts as well as studies of the many subfields: general topology, algebraic topology, differential topology, geometric topology, combinatorial topology, knot theory, and more.


Instruct Your Kids to Learn T Versus Mathematics
There’s been quite a bit of discussion over the years about which is the best method to learn math . mathematics.
Most people, however, agree they like to know both mathematics and calculus as early within their own youth as feasible.
The first thing most individuals take into consideration when it has to do with teaching math is utilizing either paper or board and paper and pencils. The idea is always to encourage children to understand by executing the work. It’s not always simply for younger kiddies since their minds don’t have precisely exactly the exact same grip of actuality since grownups. clubessay.com At the same time that you can get good benefits by having them require a project to the class room, most teachers want to get them focus independently. In actuality, the majority of adults discover they are doing better with more hands-on learning.
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Z/n vs. Mathematics isn’t the sole kind of math a youngster should learn. As an example, once your child is first learning how to see, they could want to discover shapes. Even if your kid is not in the point at which they have the ability to count by hand, they need to learn to associate contours with amounts in order they could use this ability using amounts.
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A very excellent means to teach your son or daughter how to learn math vs. math is always to teach them a combination of these sorts. For those who own a child that fights with several characteristics of this issue, then they can benefit from the processes.
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STEM for Tiny Hands
March 15, 2024 @ 10:00 am - 10:30 am
Children with the help of a parent or caregiver will use their tiny hands to explore S.T.E.M. (Science, Technology, Engineering, Mathematics) concepts and activities. This program is filled with hands-on activities that include materials that may be messy or wet. Children ages 2-5 with caregiver welcome.
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The National Mathematical Centre (NMC) Abuja would like to invite you to take part in a consultation that will lead to a national database of experts/resource persons in the Mathematical Sciences. The Centre intends publishing a list of who-is-who in Mathematical Sciences in the second edition of The Mathematician, a publication of the NMC. We would therefore be very grateful indeed if you kindly take part in this short online survey which should last less than 10 minutes.
Many thanks for your kind co-operation. Should you have further questions, please kindly get in touch with us using the following email address: firstname.lastname@example.org
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Critics of ‘Fuzzy’ Methods
Cheer Educators’ Findings;
Drills Without Calculators
The nation’s math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.
In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.
The council’s advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.
New Report Urges Return to Basics In Teaching Math – WSJ.com
Finally getting around to blogging this article from 9/12/06 WSJ – joi
Blogged with Flock


Dynamics, Control, and “Learning”: Cart-Pole
Here we will explore dynamics, modern control methods, and trajectory optimization by implementing various methods to control the canonical underactuated system, the cart-pole example. This is a classic control problem because deriving the dynamics is relatively straightforward, but it still belies some inherent complexity due to its underactuated nature. This means there are fewer actuated degrees of freedom than there are degrees of freedom of the system. In this case, there is only 1 actuated DOF, but 2 system DOF. The concepts explored here can be extended to control any dynamic system, though not all systems are so straightforward to implement.
Part 1. Deriving Inverse Dynamics
The dynamics of the cart pole system is determined by evaluating the system using Lagrangian Mechanics. The Lagrangian of the cart pole system uses the general energy equation L = T – V where L is the Lagrangian term generalizing the total energy in the system, T is the kinetic energy of the system, and V is the potential energy of the system. The kinetic energy of the system is represented by:
The potential energy of the system is represented by the potential energy of the pole. Pole Potential Energy:
The nonlinear dynamics are then found by finding the Euler-Lagrange Equations using:
With j = 2 and q represents generalized coordinate variables of x and theta. This set of partial and total derivatives are computed with the help of the MATLAB symbolic toolbox.
The nonlinear dynamics can then be represented as:
These dynamics need to be linearized in order to implement a linear controller. Linearization is achieved by computing the Jacobian in MATLAB and then inserting the resulting equations into state space form:
The dynamics derived here are implemented through a majority of the simulations. However for trajectory planning, the inertia of the pole is also considered to increase accuracy of the simulation. The primary changes in the dynamics derivation are as follows:
Part 2. Simulator Dynamics
Simscape Multibody in Simulink is used to implement the forward dynamics of the system. This toolbox is an extension to the simulink software provided by MATLAB. It is a physics simulator built around the MATLAB ode45 solver that makes modeling of rigid body dynamics and contact forces relatively straightforward, while letting users harness the power of the MATLAB control systems and matrix algebra tools. Below is a Simscape Multibody diagram modeling the dynamics of the cart-pole system.
The cart is modeled as being attached to the world frame by a prismatic joint, letting it move freely in the x direction while being constrained in all other directions and orientations. On the other hand, the pole is attached to the cart by a revolute joint, letting it spin freely around its base. Here is a video of the system in freefall:
Part 3. Manual Control Gains
As we can see from part 2, this system is unstable when the pole is upright above the cart. In order to keep the pole upright, we must drive the system to its point of unstable equilibrium, which is when the pole is directly above the cart’s center of mass.
We accomplish this by using closed loop full state feedback. Our state vector is taken to be [x, dx, theta, dtheta], which is multiplied by a feedback gain -K and fed back into our plant as the input force to the side of the cart in the positive x direction. In this section we use a tried and true method, hand gain tuning, to find a gain vector K that successfully keeps the pole upright. After some trial and error, a K vector of [-1.5 -2.5 30 7.5] was found to successfully control the cart-pole system. Here is a video showing our results:
As can be seen, decent controlled behavior can be found on a simple system such as this by just using hand-tuned gains. Unfortunately, in any real system the number of states will increase, and the intuition of how the gains affect the system can become much more difficult. Also, there is no guarantee that the gains found are optimal in any way .Wouldn’t it be great if we could somehow automatically compute optimal feedback gains somehow?
Part 4. LQR Control Gains
That’s where LQR (linear quadratic regulator) comes into the picture. LQR poses the control problem as an optimization problem in which we are trying to minimize a quadratic cost function that is a combination of a current state cost, an input cost, and (optionally) a final state cost. This method is guaranteed to give an optimal controller for the cost function provided. The only hiccup in using LQR is that (as the name implies), the system being controlled must be linear. In our case, the cart-pole is a nonlinear system. So how do we control it? By linearizing the system! This is done by taking equations of motion we derived above, and taking the taylor series of each of these equations about some operating point. In the cart-pole example, we will take the operating point as the unstable equilibrium point when the pole is directly above the cart. This gives us with an state vector operating point of [0 0 0 0]. After linearizing the dynamics of the system, we put the dynamics into state space form in order to easily compute the LQR controller. The state space matrices we find are:
Now we have to define a Q and R matrix. The Q matrix is the gain of the cost function on the states of the system, while the R matrix is the gain of the cost function on the input to the system. Because we are primarily concerned with keeping the pole upright (theta=0), and driving the cart back to its starting point (x=0), we give higher cost to these states. In addition, ideally we want the input to be as minimal as possible, so we give a higher cost to R as well. This results in the following LQR gains:
Once we have a linear state space model of our system around our control point and our LQR gains defined, finding the LQR state feedback gain K is as easy as typing lqr(sys) into MATLAB. Having done this, the K vector found is:
This results in the following behavior of the system:
This same control synthesis process can be used to derive a controller for the double inverted pendulum cart-pole system:
Part 5. Nonlinear Control to Optimize Basin of Attraction
Though LQR worked great for stabilizing the cart-pole system around its unstable equilibrium point, it’s basin of attraction is limited. The basin of attraction is defined as the set of initial conditions for which the controller will stabilize the system to an “attractor”, which here is the pole upright position. In the cart-pole case, the most important states for system stability are theta and dtheta. With this in mind, and for more intuitive visualization, we will neglect x and dx when talking about the basin of attraction for this system. For the LQR case the following plots show the basin of attraction for the system:
This plot shows the initial conditions for which the controller successfully stabilized the cart-pole system. As can be seen from this plot, the basin of attraction for the cart-pole with a LQR controller is limited. This is because the controller is derived for the system when it is linearized around the zero operating point. When we stray to far from this 0 linearization point, the linear system approximation to the full nonlinear system becomes poor enough that the controller does not function correctly.
In order to increase the basin of attraction, and thus the robustness of the control policy, we implemented a nonlinear control policy inspired by the LQR-Trees approach. Basically, instead of only linearizing the system about one unstable equilibrium point ([0 0 0 0]), the system was linearized for a range of states. Once again, because theta and dtheta are most important to the stability of the system, the system was systematically linearized around a uniform range of both of these states while keeping the other two states constant. 81 different states were considered and systematically linearized. For each linearization, a state space system was derived, and a LQR feedback gain was computed. These feedback gains were stored in a lookup table matrix of 81 different feedback gain vectors. At any given time, the system will find the linearized feedback gain computed for the operating point closest to the current state of the nonlinear system. Thus, though the system is always still using a linear controller, the linear approximation of the dynamics used to derive that controller will be much closer to the real dynamics of the system at that point. This results in a significant increase in the basin of attraction, which is summarized in the plots below:
Part 6. Trajectory Planning
Up to this point, all the controllers implemented were to drive a system from an unstable initial condition to the zero position where the pole is directly above the cart. But what if we want to swing the pole from its stable hanging position up to its unstable equilibrium point? The traditional feedback controllers do not work well for this purpose, so trajectory optimization is implemented.
Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem.
In our case, we pose the problem as a minimization problem in which we are trying to minimize the control input while satisfying the constraints that the system must follow its dynamics. For this we use the dynamic equations of motion derived in part 1.
The trajectory is discretized into 150 points between the initial state (the stable hanging position), and the target state (pole over cart). These 150 points are then optimized using the direct collocation method. Finally, a trajectory is interpolated between these 150 points to create a continuous trajectory. The problem as a whole is modeled as a boundary-value problem with time as the constrained boundaries. By varying the amount of time we want the trajectory to take, we can get different behaviors. These different swing-up behaviors can be seen below:
The LQR stabilizing controller was then implemented on an Arduino Mega embedded controller. This controller was used to stabilize the Quanser furata inverted pendulum system.
Link to Github Repository:
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"Mom, next Wednesday is pi day and we have to make a pie for math class!" my daughter told me last week. "My teacher's favorite pie is French Silk and I told him we would make one." I sadly told her I had never made one before, but we could look up a recipe and try it over the weekend. This recipe for French Silk Chocolate Pie looked like a winner! I noticed the recommendations to double the filling, picked up the ingredients, and we began baking. This was my daughter's first experience at rolling pie dough, and at holding the mixer for the full five minutes after each egg. Our two trial pies on the weekend turned out delicious! I took one to share with my fellow teachers, and froze a piece for my son in college. Then, on Tuesday night, we began to make the pie for pi day at school. In the middle of the preparations (while waiting for the chocolate to cool), I realized I had only one teaspoon of vanilla, not two! What to do? I looked through my cabinets and found almond extract. I thought with the chocolate, it probably wouldn't even be tasted. The pie was made and in the refrigerator, ready for the final whipped cream topping in the morning. I licked off a little filling from my spatula and just rolled my eyes. That little bit of almond extract was wonderful in that filling! I wished I had made the first two that way! Well, the pie was a hit at school; everyone loved it and the pie pan came home empty.
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"Mom, next Wednesday is pi day and we have to make a pie for math class!" my daughter told me...


Teachers are often encouraged to implement cross-curricular strategies in their classrooms. But what does cross-curricular mean to the teacher who is transitioning from an industry position to education? Not a lot. And what "curriculum" are we supposed to be crossing? The longer I teach the more I hear references to test scores in Math, Science, Social Studies and Writing. We are encouraged to require writing in every class. We are encouraged to reinforce math in each of our courses. We are encouraged to incorporate science and social studies as much as possible - but how?
Integration across subject matter is promoted by teacher educators because it benefits learners of all ages and backgrounds yet new teachers are not specifically taught how to integrate alternative curricular content into their lessons (Cuero, Ruiz & Thornton, 2010). And there is a "need for professional development that assists teachers in changing their conceptual perspectives to integration while also building pedagogical knowledge related to integration of science, mathematics and literacy: (Douville, et. al., 2003, p. 388).
For those of you now lost in the vocabulary: "Pedagogical content knowledge identifies the distinctive bodies of knowledge for teaching. It represents the blending of content and pedagogy into an understanding of how particular topics, problems or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction. Pedagogical content knowledge is the category most likely to distinguish the understanding of the content specialist from that of the pedagogue" (Shulman, 1987, p. 4).
Definition of Pedagogy: the art, science, or profession of teaching; especially education. http://www.merriam-webster.com/dictionary/pedagogy
I always attempt to incorporate cross-curricular information into my lessons. Specifically, I try to incorporate Math, Science and Writing. This week I think I succeeded. Not only did my students see science and math, but they had an "aha" moment.
First, I told students the capacity of a glass jar. ie, this jar holds 4 oz. I explained that they would need to add a liquid to the jar and would need to have room for that liquid to move sufficiently. They should fill the container between 1/3 and 2/3 full. Then I had them do some math! and convert ounces to kitchen measurements so that they could measure the appropriate amount of liquid into their jar. They also had to incorporate ratios - they need to add a certain amount of solid matter to the jar and it had to be in ratio to the liquid (Yes, I'm avoiding giving you the actual measurements on purpose, I don't want to give away any secrets). So, Math... check.
Now for science (and a secret). What happens to cream when you shake it? You incorporate air, you "whip it" making... whipped cream! (imagine that) Now, what if you whip that cream beyond recognition? You obviously don't have whipped cream anymore. You are changing the chemical structure of the cream. By slamming molecules off one another (and adding a little salt) you encourage carbon chains to become saturated. What saturated fat comes from milk? Butter.
After 5-10 minutes of math and science I've taken my students from "Why are we shaking a jar of cream?" to "Hey, is it butter? Can we eat it?"
Yum.... cross-curricular instruction never tasted so good!
Cuero, K. K., Ruiz, E. C., & Thornton, J. S. "Integrating literature in mathematics" a teaching technique for mathematics teachers." School of Science and Mathematics 110.5 (2010): 235+.
Douville, P., Pugalee, D. K., & Wallace, J. D. "Examining instructional practices of elementary science teachers for mathematics and literacy integration." School Science and Mathematics 103 (2003): 388-396.
Shulman, L. ( 1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.


Given two integers L and R, the task is to find the sum of all even numbers in range L and R.
Input: L = 2, R = 5 Output: 6 2 + 4 = 6 Input: L = 3, R = 8 Output: 18
Method-1: Iterate from L to R and sum all the even numbers in that range.
- Find the sum of all even numbers up to R i.e. No. of even numbers up to R will be R/2.
- Find the sum of all even numbers up to L-1 i.e. No. of even numbers up to L-1 will be (L-1)/2.
- Then subtract sumUptoL from sumuptoR.
Sum of all even numbers up to any N will be:
R*(R+1) where R = N/2
Below is the implementation of the above approach:
Sum of Natural numbers from L to R is 6
- Numbers within a range that can be expressed as power of two numbers
- Prime numbers in a given range using STL | Set 2
- Check whether XOR of all numbers in a given range is even or odd
- Sum of all natural numbers in range L to R
- Sum of all odd natural numbers in range L and R
- Count Odd and Even numbers in a range from L to R
- Sum of all numbers divisible by 6 in a given range
- Sum of all the prime numbers in a given range
- Count the numbers divisible by 'M' in a given range
- Print all Good numbers in given range
- Sum of greatest odd divisor of numbers in given range
- Count factorial numbers in a given range
- Find numbers with K odd divisors in a given range
- Sum of all odd length palindromic numbers within the range [L, R]
- Sum of range in a series of first odd then even natural numbers


Communications in Mathematical Research
Editor-in-Chief:Fuquan Fang and Tao Tang
Managing Editors:Yanzhao Cao and Shaoyun Shi
- ISSN Print: 1674-5647
Aims and Scope
Communications in Mathematical Research (CMR; ISSN: 1674-5647) was established in 1985 by Jilin University, with the title "Northeastern Mathematics", and published only articles written in Chinese. Recently the journal was renamed "Communications in Mathematical Research" and only publish articles written in English. In early 2020, the current editorial board is formed aiming to enhance the quality of this old journal. CMR publishes original research and survey papers, in English, of high scientific value in Geometry, Topology, Analysis, Albegra, PDEs, Probability, Statistics, Optimization, Scientific Computing, etc.
5-Year Impact Factor:


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Tom Sawyer Software
Before his time at Tom Sawyer Software, Paulis was a well-respected professor at the University of Latvia, working in the Faculty of Physics and Mathematics. He was also a leading researcher of the university, working in the Institute of Mathematics and Computer Science. Paulis is an expert in graph theory and computational geometry. Paulis began developing graph drawing algorithms as early as 1991. In 1994, he was involved with the GRADE project, which was originally built for Siemens but continued its life as a tool for students at the University of Latvia. The GRADE system was one of the most sophisticated graph layout and diagramming projects ever built. In the 1990s, Paulis got involved in the graph drawing community by publishing papers and winning prizes in graph drawing contests. During his collaboration with Tom Sawyer Software, Paulis built the new Hierarchical Layout, which would remain his main area of work for several years. He also developed the Polyomino Packing algorithm to lay out disconnected components and isolated nodes. Paulis’ work is recognized through numerous publications. Paulis was instrumental in training Tom Sawyer Software’s new generation of layout engineers as well. Retired now, Paulis has returned to the University of Latvia and continues to teach classes.
Paulis earned his university graduation diploma in Mathematics at the Latvia State University. He also has a Candidate degree of Science and a doctorate degree in Mathematics.


Do you understand how to determine the odds of profitable the lottery, including the Florida Lottery? You’ll be able to estimate Just about every list of odds for each various lottery recreation you play. With the help of a small handheld calculator or While using the totally free calculator on the Computer system, you just multiply the quantities alongside one another and add a single division procedure when “the buy” of the decided on quantities will not be essential for a particular lottery recreation.
That which you “want to grasp” is the volume of full balls the profitable quantities are drawn from…..could it be 59, 56, forty two, 49, or 39? When there is a secondary drawing for The only added ball, such as the “red ball” with Powerball or perhaps the Mega Hundreds of thousands’ “gold ball” you need to know the number of balls are Within this team in addition. Are there 49 or 39?
It does not make a difference if it’s the Florida, Ohio, Texas, PA or NJ Lottery. This technique or method provides the real odds. Florida Lottery is 6/fifty three. The big apple Lottery is 6/fifty nine. The Ohio Lottery, Massachusetts Lottery, Wisconsin Lottery, and the Point out of Washington Lottery have a six/49 lottery numbers ratio. Illinois Lottery carries a 6/52.
After getting this details the right way before both you and your calculator in hand, you can start Doing work the formulation. You need to select five normal balls and 1 added ball correctly matched to the winning drawn numbers to gain the multi-million dollar jackpot that Many of us aspiration about winning someday.
In the first case in point there are actually fifty six balls in the very first team and 46 balls during the secondary group. So that you can win the Jackpot you must match these balls (five + 1) accurately, although not automatically in order. The California Lottery’s Tremendous Lotto Additionally is 47/27. The large drum is spinning Along with the initial part of the drawing. You’ve got a one/56 chance to match your amount to this first ball.
With a person ball taken out after the 1st range continues to be drawn, you now have a 1/55 probability of matching A different one within your figures to the second ball drawn. With each drawn amount a ball is eradicated lowering the amount of remaining balls by a total of 1.
The chances of you the right way matching the variety about the third ball to be drawn is now 1/54 from the entire number of balls remaining during the drum. With all the 3rd ball removed from the drum and sitting down with one other two winning quantities, your odds of effectively matching the fourth ball is diminished to 1/fifty three.
As you can see each time a ball is introduced from the drum the odds are decreased by a single. You started off by using a 1/fifty six chance, then with Each and every new profitable quantity it can be lessened to 1/55, one/54, 1/fifty three, and With all the fifth ball you’ve got the percentages of one/fifty two correctly matching this fifth winning variety. This is the to start with Element of the method of the way to estimate your odds of profitable the lottery, such as the Florida Lottery.
Now take these 5 odds symbolizing the 5 successful quantities (one/fifty six, one/55, 1/fifty four, one/53, and 1/fifty two). The “1” on top of the portion signifies your one and only opportunity to correctly match the drawn variety.
Now you take your calculator and multiply all leading quantities (1x1x1x1x1) equivalent a single (1). Following you multiply all the bottom numbers (56x55x54x53x52). Effectively entered and multiplied you learn the entire is 458,377,920. The new fraction becomes one/458,377,920. That is a 458 million to one opportunity to get. If you ended up needed to choose the figures if you want just like They can be drawn, then these could be the chances from you to definitely gain this Decide 5/fifty six ball lottery match.
The good news is or sad to say, You aren’t required to decide the numbers in the exact order they are drawn. The second action with the system will decrease the odds, which lets you match these 5 profitable numbers in any get. In this particular stage you will multiply the volume of balls drawn — 5 (1x2x3x4x5). With calculator in hand you see that the whole equals 120.
To provide you with the correct to choose your five matching figures in almost any purchase, you develop these odds by dividing 120/417,451,320. You surely require a calculator for this one particular. a hundred and twenty/458,377,920 cuts down your odds of winning this lottery to 1/3,819,816. These are generally more than 3.5 million to at least one odds from you of profitable this Pick 5/fifty six ball lottery recreation.
If this had been the Mega Thousands and thousands Lottery, you should increase the “gold ball” to those 5 profitable drawn balls as a way to win the Multi-Million Greenback Jackpot. The only gold ball is calculated as a 1/46 prospect of matching it accurately, and since you are drawing just one amount it should be an exact match. All over again, you have only that “one” chance to get it done right. Now you have to multiply three,819,816 by forty six.
Grab your calculator and do the multiplication. Your final odds in opposition to you profitable the Mega Thousands and thousands Jackpot are 토토사이트 calculated to become 175,711,536 or clearly mentioned one hundred seventy five million, 711 thousand, five hundred 36 thirty-6 to 1 (175,711,536 to one). Now you understand how to calculate the percentages of successful the Mega Millions Lottery.
The Powerball Lottery calculations are determined by a one/59 for the very first five white balls and 1/39 to the “purple” ability ball. The primary set of multipliers is 59x58x57x56x55. This group totals 600,766,320. Now divide 600,766,360 by one hundred twenty (1x2x3x4x5). Your new total is 5,006,386. You will find a one/39 chance to capture the “pink” ball. 39 x five,006,386 provides the true odds of winning the Powerball Jackpot, specifically 195,249,054 to one.
An additional five +1 Lottery that appears to be all over the place in America would be the “Warm Lotto” that has a 39/19 rely. It truly is played in 15 diverse States. DC Lottery, Delaware Lottery, Idaho Lottery, Iowa Lottery, Kansas Lottery, Maine Lottery, Minnesota Lottery, Montana Lottery, New Hampshire Lottery, New Mexico Lottery, North Dakota Lottery, Oklahoma Lottery, South Dakota Lottery, Vermont Lottery, and the West Virginia Lottery. The ultimate odds of winning the minimal $1 Million Jackpot is 10,939,383 to one.
A Select 6/fifty two ball Lottery video game system looks like this: (one/52, one/51, 1/fifty, one/forty nine, 1/48, 1/forty seven) for a total of 14,658,134,four hundred divided by 720 (1x2x3x4x5x6) for the odds of 1/20,358,520. Your chance to gain the 6/fifty two Lottery is in excess of fourteen.five million to 1 to get, like the Illinois Lotto.
The Hoosier Lottery that works by using Indiana Condition’s nickname, carries a 6/forty eight. Michigan Lottery is six/47, Arizona Lottery and Missouri Lottery are six/44, Maryland Lottery is six/forty three, and Colorado Lottery is 6/forty two. Assess this on the Florida Lottery.


Dr. Ziquan Zhuang receives Sloan Research Fellowship
Dr. Ziquan Zhuang, an assistant professor in the Department of Mathematics at Johns Hopkins University, is among the 126 early-career researchers who received a 2023 Sloan Research Fellowship. Zhuang focuses on algebraic geometry, which is a branch of mathematics that studies algebraic varieties, or the geometric manifestations of solutions of systems of polynomial equations. He hopes to continue to research and visit collaborators with the fellowship funds.
Awarded annually since 1955 by the Alfred P. Sloan Foundation, the fellowship honors outstanding early-career researchers from the United States and Canada whose creativity, innovation, and research accomplishments show exceptional promise in the next generation of scientists.
“I feel very honored and excited to be selected as a Sloan Fellow,” Zhuang says. “It is not only an important source of funding for me to start my career, but also a strong community support and recognition of my research work.”
Read more at the Hub.


Education and Experience
I graduated from Loyola College in Maryland in 1993 with a bachelors degree in Management Information Systems. I graduated from Wilmington University in 2006 with a Masters degree in Education.I worked with the Campbell Soup company for 7 years as a computer systems specialist, and computer trainer. I began my teaching career in 2003 at the Pineland Learning Center in Vineland, New Jersey. Pineland Learning Center is a private school for students with many different types of learning disabilities. My teaching experience includes career education and computer science while at Pinelands.This is my ninth year teaching at Glassboro High School. This year I am excited to be teaching Interactive Algebra 3 in an in class support setting and teaching Interactive 5 in a resource room pull out situation. Interactive Algebra and Geometry is an exciting new curriculum that is designed to keep the students engaged in the learning process.


This article is about the Scientific notation calculator as in the present era. The uses of this converter is really important and this is used by everyone, especially by the students.
What is Scientific Notation Calculator?
The calculator is generally used for the calculating the Plus, Minus, Multiply and divide calculations. When we talk about the scientific notation calculator/converter. It is generally stated and referred as a scientific form of the calculation. Or sometimes known as the standard index form. This converter is used to express the big numbers or really small numbers that can’t be handled by the general calculators. Other calculators can’t do the calculations like converting them into decimal forms. This all calculations are easily done by the Scientific Notation Converter and it is generally known as the SCI (Scientific calculator Integration). This converter is generally used by the engineers, scientists or by the mathematician. There are different things that you must know about this, as like the scientific notation, E notations or Estimated Final Digits in the calculator.
Uses of Scientific Notation Calculator or Converter
As I mentioned that, this calculator is generally used by the scientist, engineers or the mathematicians. One of the best examples of the use of the Notation Converter is by the scientists that they have to calculate the smaller values. And sometimes really bigger values that they can’t handle in the general calculator so they use the Scientific calculator. Using this calculator/converter you can get the answer you will get the answer in the three formats as scientific, engineering, and E notation as well. Another use that you will get by using this calculator is that you can use the operations on the integers, whole numbers, or decimal numbers to get the answer in the scientific notation. The other thing that you will get in the scientific notation converter, calculates automatically the number of significant figures when you calculate your answer. To get the answer in the scientific notation you must check the box otherwise you will get the digits more than the sign on the scientific calculator.
Rules of Notation Calculator or Converter
In the normalized scientific calculator the ‘E’ notation and in the engineering notation the space is used to represent the ‘X’ or in front of ‘E’ and it is allowed only in the special accounts. This is less common in front of the alphabetical letters. While converting the numbers, in the first convert the number into the scientific notation form and then convert it into the decimal form or you have to change the exponent part of the equation. Using the scientific calculator and converting the decimal to the science you have to move the decimal separator points sufficient to the desired number and range that you want. When you deal the exponential conversion between the different scientific notation using the calculator. The decimal separator in the significant figures places to the left when you handle the exponential conversions.
Let’s have an example
1.234×103 = 12.34×102 = 123.4×101 = 1234
From this example, you can clearly learn how to handle the exponential values in scientific calculator.
How does it Work
The working of this calculator is different for different notations in the converter. In each example the inputs that we enter is different but they all produce the same result in the scientific notation or E notation in a calculator. First, we discuss the standard notation as it is the usual way to write the numbers and without decimals and commas. The example that we calculate solve the simple additional decimals. E.g. 122500+3655. It is to be noted that the inputs are the standard numbers and it will be calculated in scientific from standard decimals. In the scientific calculator which converts the large decimal number into the equivalent number. And in between 1 to 10 and multiply it with some raised to the power of 10. Very small numbers are converted by the scientific calculator between 1 to 10 and multiplied by 10 raised to some power negative power. When we do the E notation in the converter usually known as the exponential notation as well. E notation is same as like the scientific and lie between the number 1-10 and also works same by multiplying it with some raise power of 10. The number after the ‘e’ in scientific calculator represents the how many powers of tens are there. For example;
1.225e5 + 3.655e3 = 1.26155e5
Few Useful links to Convert Your Values
There are different examples that are used in this converter that can be handled by the calculator easily like.
- A Proton mass is about 0.0000000000000000000000000000000910938356 kg. In scientific notation, this is written 9.10938356×10−32 kg (in system international units).
- The mass of the earth is about 597240000000000000000000000 kg. In scientific converter, this it will be written 5.9724×1026 kg in the system international units.
- The other example like the circumference of the earth is approximately 400000000 m. In scientific converter, this is 4×108 m. while writing it into engineering notation, this is written 40×108 m. In system international unit style, this may be written “40 Mm” 40 megastars.
- Let’s have another example an inch is defined as 25.4 millimeters. Putting a value of 25.400 mm showing that this is correct to the nearest micrometer. An estimated value with only two significant values would be 2.5×101 mm surely. As there are no limitations to the number of significant values, the length of a creep could. If obligatory, be written as (let’s say that) 2.54000000000×101 mm for sure.
This can all be done by the scientific notation calculator.
The benefits of using the scientific notation calculator allow you to calculate the values even smaller and bigger values as well. You can calculate the values and these are the calculation that scientific notation calculator. Does automatically and you never have to calculate by yourself. This is one of the best inventions that are helping the mathematicians, scientists, and students regularly. If you don’t want to do the calculation by yourself or you find it difficult. Then you must use the scientific calculator for the perfect and easy results.
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Escola Massana, Barcelona
video of the non stop 48 hours performance.
In this performance, which lasted 48 hours non-stop, I started by mixing 1000 kilos of clay with a bundle of straw. With the mixture I built an anthropomorphic shape around 2 metres high in the centre of the room, immediately afterwards I divided the shape in two and at the same time I also divided the space, shortly afterwards with each half I built again an anthropomorphic shape in the centre of each space and then I divided again each shape and each space in two, after that I placed another shape in the centre of each space, and so the action is repeated several times, dividing shapes and spaces until the number of 160 figures is obtained, each of which occupies proportionally the same space as the first figure. . in the last part of the action with my feet I flattened the figures until they formed a stratum on which I placed again a large quantity of clay and straw as an invitation to repeat the whole process indefinitely.


In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Tracy Leeper. Her topic, from Geometry:finding the area of a square or rectangle.
How could you as a teacher create an activity of project that involves your topic?
I would start off my lesson by allowing the students to use block manipulatives, and worksheets with different squares and rectangles printed on them, and ask the students to find out how many blocks each square or rectangle takes to fill the space. I would then tell them the space covered by the blocks is called the area, and see if they could find the relationship of . By having the students use the blocks, they can easily see that a square would use 9 blocks. The “hidden” block in the middle of the square would become visible. By doing this, I would allow the students to discover the algorithm for the area of a rectangle / square on their own. This would enable them to remember it better. Using the blocks would also give them a better visual memory of the activity, so later, it should be easier for the students to recall the appropriate formula. Using various sizes of rectangles and squares would also illustrate that the algorithm works every time, regardless of the size of the rectangle or square.
How can technology be used to effectively engage students with this topic.
LearnZillion.com has a great lesson on finding the area of a rectangle. The video starts out by reviewing what a perimeter is, and uses scaffolding to build to finding the area. After the area has been determined to be , the video then goes on to ask questions for finding area on a different rectangle, and then shows that given the area, and one side, by using inverse operations, we can solve for the missing side. The next question is, given a rectangle with a perimeter of 24, what might the area be? Again, the video not only reinforces using the inverse operation, but continues to show the importance of the word “might” by showing that there are multiple solutions. After teaching what mathematical reasoning should be used for this problem, the video then moves to applying the knowledge to a word problem. The video uses proper mathematical terminology, and demonstrates how to apply prior knowledge to help in gaining new knowledge. The video does seem a little dry, and the students might want something flashier to catch their attention, but I feel this video would be a very good tool to use, to reinforce new concepts for students.
How has this topic appeared in the news?
1n 2006, the San Alfonso del Mar resort in Algarrobo, Chile opened the “world’s largest swimming pool” as dubbed by Guinness World Records. It is measured to cover approximately 20 acres, which is 871,200. I would use this as a lead in to an activity for my students. I would show them a picture of the pool and challenge them to find out how many Olympic size pools it would take to cover the same surface area. Since this lesson is on area and not volume, I would give them the measurements of an Olympic Size pool, which are 164 ft. in length and 82 ft. in width. The students would then have to find the total surface area of one Olympic-size pool, which is 13,448. Then the students would have to divide to find out how many Olympic-size pools it would take to cover the same surface area of the San Alfonso del Mar resort pool, which calculates to be approximately 65 Olympic-size pools. I think this would be a good elaborate for the lesson on area of a rectangle or square.


5 edition of Math and literature. found in the catalog.
Math and literature.
Includes bibliographical references (p. 135-136) and index
|Statement||Marilyn Burns, Stephanie Sheffield|
|LC Classifications||QA115 .B957 2004|
|The Physical Object|
|Pagination||xii, 140 p. :|
|Number of Pages||140|
|LC Control Number||2004011479|
Race relations in East Africa [by A. Bharati, James Lewton Brain and George J. Moutafakis.
Gender in campaigns for the U.S. House of Representatives
Harriettes Big Day
Freelance photographers Britain
Collected poems, 1928-1953
Guidebook for student speakers
Lectures and conferences on mathematical statistics
Quick job search for ex-offenders
Problems, cases, and materials on evidence
Wheres it from? When was it issued?
Chats on old silver.
Australia: the dairy country ...
57 rows The Everything Kids’ Joke Book: Side Splitting, Rib–Tickling Fun: Dahl, Michael: 4–3:. Connecting math with literature tips. With younger children generally spend about a week building the connections between the math concepts and skills and the story.
Here are some suggestions. Day 1 – Read the book to the students and enjoy it. Have a sharing circle – those who want to say something about the book have an opportunity.
Math And Literature, Grad has been added to your Cart Add to Cart. Buy Now From Quack and Count to Harry Potter, the imaginative ideas in Math and literature.
book books come to life in math lessons through this unique series. Each resource provides more than 20 classroom-tested lessons that engage children in mathematical problem solving and reasoning/5(8).
54 rows Related Links Advice from Teachers. See advice from Everyday Mathematics teachers on. Connecting math to literature can boost confidence for children who love books but are wary of math.
And students who already love math can learn to appreciate stories in an entirely new way. One book that Math and literature. book read to many first and second graders, always to the delight of the children, is One Is a Snail, Ten Is a Crab by April Pulley Sayre. A math adventure that integrates math and literature.
Gisele, on math board top: Math Literature and Graphing Activities > I will soon be a student teacher. In my method class for > math I was instructed to create a book lesson integrating > math. I want to use Harriet's Halloween Candy by Nancy > Carlson, since it will be. A book list of math literature titles for early elementary ages with links to read alouds on YouTube.
Great as a hook for new skills, to preview books, re-read, or as a supplement your lesson plans. I’ve researched, read, and collected a lot of math books. My oldest loves math, and some of these books have been a great hook for learning. This book, written under the pseudonym A.
Square and published inis the obvious place to start. Flatland is both an exploration of the nature of dimensions and geometry, and of Victorian social hierarchy.
The narrator is a square living in Flatland who dreams of a visit to a one-dimensional world, which leads to the questioning of his (and our) understanding of dimensions and reality. Discovering Math in Literature. You can pull math concepts out of many interesting books.
Use our suggestions below to help you find real-world application of math skills and broaden the educational value of reading in your classroom. Math and Literature: A Match Made in the Classroom. If you are seeking a new way to add relevancy to your classroom math activities, the answer may be right in your school library.
Literature is the ideal vehicle to help your students see the importance of numbers in their daily lives. Math & Literature Home > Math Manipulatives, Supplies & Resources > Math & Literature Bring math concepts to life with engaging literature and picture books.
This book has some great ideas for ways to get your students interested in math but by starting with literature. The story is what brings them to math.
I tried one the first week of school and the class loved it. The lessons are fully scripted and the students are promptly engaged in the lesson. I /5(9). In this series of math book lists, we move on now to math picture books for kindergarteners through second my list of math books for preschoolers I mentioned that there is no shortage of math picture books.
The trick, for me at least, is to find books to teach math concepts that are not boring. I confess that I am partial to math books that incorporate concepts into engaging stories.
Math read-alouds can particularly help these students see math in a new light. A picture book is the perfect low-stress introduction to a new math subject. At the beginning of the year, I read Jon Scieszka and Lane Smith’s Math Curse to my class, and we discussed how we are surrounded by math — and by a proliferation of negative ideas about.
Asking questions and making observations about the math found in picture books can support children’s curiosity and enjoyment of math.
Like many engaging pieces of children’s literature, the math picture books recommended below contain fun and interesting storylines. Connect mathematics, literature, and writing with this elementary book collection.
Marilyn Burns-endorsed reading collections take imaginative fictional books and apply math lessons to the story. This is a fun and charming book collection that students, parents, and teachers will enjoy. A 'must have' fraction read aloud for your math literature collection.
Gator Pie by Louise Matthews In this story two alligators, Alvin and Alice, find a large chocolate-marshmallow pie on a picnic table near the edge of the swamp. Math Literature for Young Adults Search this Guide Search. Curriculum Center: Math Literature for Young Adults.
Curriculum Center Toggle Dropdown. Hours and Location Services; Danielle's Must Read Children's Books; Picture Books without Words; Picture Books.
Exploring Math through Literature, Pre-K–8 is an online teacher resource filled with lessons and activities that teach mathematical concepts while engaging students in a broad assortment of children’s literature, including picture books, stories, poetry, and novels.
The lessons are based on journal articles from NCTM’s Teaching Children Mathematics and Mathematics Teaching in the Middle.
Read this blog post to learn how using math literature books for elementary math instruction is an effective way to teach key concepts and skills in a fun and engaging way for Kindergarten, 1st, 2nd, 3rd, 4th, and 5th grade students.
Check out this post now. Math is at play in every sphere of our lives, from recipes to internet security to the electoral college. But that reality can be hard to convey through the drills, static numbers and strict rules that make up so much of K math education.
Educators have made strides to engage students through way to bring the subject to life, according to a math research organization, is through. Math and Literature, Grades 6–8 contains lessons and ideas based on 30 children’s literature titles.
Children explore mathematical concepts based on lessons derived from titles such as Harry Potter and the Sorcerer’s Stone, by J.
Rowling, and Holes, by Louis book includes a reference chart indicating the mathematical concept each lesson covers, such as number, geometry. Connecting Math to Literature Through Creative Technology Projects A list of children's books about math that make for great student-published books and projects.
Using the ideas in the book, encourage them to make connections to where the shape is found in the real world. This book makes an engaging introduction to the idea of relative size as young students enjoy comparing themselves to the illustrated animals to get a sense of how big or how small they are.
After reading, have students work in pairs to mark and label some of the different animal measurements using various measuring tools (rulers, tape measures. 19 Mar - Useful books to use in primary maths.
See more ideas about Math books, Math literature and Books pins. So sorry for the lack of a kids’ book list last Monday, I hope this selection of Math Chapter Books and Math Story Collections makes the wait worth it.
Regular readers of this blog (Yay!Thanks for sticking around!) may be tired of hearing about my son’s passion for numbers. Recently, I had an opportunity to connect with a new children’s author by the name of Laura Gehl. She wrote a wonderful book entitled, One Big Pair of Underwear. The book infuses counting concepts for primary grades in a way that is fun and humorous.
I eagerly took on the task of creating. Math books Compiled by Dr. Patricia S. Moyer-Packenham–Director of Math Education Center, George Mason University.
Through reading these books, your children will have the opportunity to build their reading and math abilities in meaningful ways. Math-Related Children’s Books. There are many children’s books that have math-related content and themes. When choosing literature, you may want to select books that either have math concepts that drive the story, or that are embedded in the story.
Here are a few examples sorted by concept or skill. Counting, Adding. This is a pdf printable math work book for use in 6 th grade. Read More. 7th Grade Math e-Book Download (90 pages) - USD $ This 7 th grade math e-book download is a pdf printable resource for educators.
Featuring are key seventh grade math topics each. Jun 2, - Explore maryfcarr's board "Books with math themes", followed by people on Pinterest. See more ideas about Math books, Math and Math literature pins.
By now, y’all know that I LOVE living math – and especially math that involves living literature. So many people think math and literature connections only make sense for the elementary years, but I’m here to tell you middle school students still benefit from the connection!.
This list of living books for middle school math is perfect for making those connections and bringing some. The goal of the book is to expose the reader to a wide range of basic problems, some of which emphasize analytic ability, some requiring programming techniques and others focusing on statistical data analysis.
In addition, it covers some areas which are outside the scope of mainstream financial mathematics textbooks.
Measurement Books Home > Math Manipulatives, Supplies & Resources > Math & Literature > Measurement Books Extend measurement concepts and build reading skills. Children's literature or juvenile literature includes stories, books, magazines, and poems that are made for children.
Modern children's literature is classified in two different ways: genre or the intended age of the reader. Children's literature can be traced to stories and songs, part of a wider oral tradition, that adults shared with children before publishing existed.
Math Literature Title Author Math Concept Grade Pigs on the Move Amy Axelrod Time and travel Pigs in the Pantry Amy Axelrod Capacity measurement Pigs Will Be Pigs Amy Axelrod Money Pigs on the Ball Amy Axelrod Geometry Pigs on the Blanket Amy Axelrod Time Pigs go to the Market Amy Axelrod Weight measurement File Size: 55KB.
Math Literature Connections: Geometry & Measurement Featured Book: The Shape of Things. The book also provides suggestions for follow-up measurement activities such as the String-a-Long Game in which students look at (but don't touch) ordinary, everyday objects, then cut a piece of string they think is the same length as the object.
This book is a classic. It includes a wealth of ideas for math activities and games to be done at home. The level of these activities ranges from kindergarten to middle school. The book is a guide for parents and children to work together in active game play, solving problems, experimenting and even discovering.


Matka suggests determining the mathematical sum of the numbers that are drawn, hence this is one of the methods by which people have attempted to guess numbers by using the divination methods. Though it is not the original form of the type of information, there are many variants that have been developed based on the present day practice.
Another factor that's looked at while performing Kalyan Matka guessing is your colour or shape of the cards. While Dpboss supposes, the conventional black-shaped cards are chosen. For instance, if it's a lucky card, then the kalyan matka guessing will involve the usage of the black numbers. Hence, the blessed numbers must also be like the amounts drawn for the divination procedure.
Another important factor that needs to be considered while performing Kalyan Matka guessing is the width of the number line. This is because the width of this line has to match the width of the lines used by the divination procedure.
Hence, if the divination process entails the use of the colored cards along with the lines, then the divination must begin with the divination procedure which has the maximum numbers. Thus, the width of the divination process also has to be matched with the same method utilized for the divination process in the actual world.
Thus, if it is a fortune number card then the width of this line must be bigger than others. For instance, if the divination process is utilizing the black amount card, then the divination process must be launched with the number line bigger than all the other lines.
In conclusion, it can be stated that for the divination procedure in Kalyan Matka guessing, the use of the black number card is preferred. This is due to the fact that the divination method is done in the actual world and also the width of the divination line has to be larger than the divination line in the divination process in the Kalyan Matka guessing.
read here to obtain more information about Satta matka.


1055 Business Instructional Facility
515 East Gregory Drive
Champaign, IL 61820
Phone: (217) 333-2740
Admissions: (217) 244-3890
Fax: (217) 244-9992
Kinesiology: 4 hours of activity courses, numbered 100–110. The same section of a course may not be repeated for credit.
199 Courses: 12 hours maximum.
Foreign Language: May receive credit for courses only 2 levels below highest level taken in high school. For example: 4 years of high school French—no credit below FR 102.
Math: Credit is not given for Math 012, 014, or 016. Once the math requirement is completed, lower level math courses cannot be taken for credit. Preparatory math courses are included in the undergraduate cumulative grade point average.
Military & Naval Science: No credit is allowed for 100 or 200 level courses.
CR/NCR: 18 hours


Exclusive Interview of the Day – Nikolay Andreev
Nikolay Andreev has created the site “Mathematical Etudes”, where non-fiction stories about modern problems of mathematics and various mathematical subjects are visualized; in an original way he continues the work on popularizing modern mathematics of his predecessors, from Perelman to Arnold. In 2010 he received the Prize of the President of the Russian Federation in the field of science and innovation for young scientists “for high results in the creation of innovative educational technologies, popularization and dissemination of scientific knowledge” for his work.
In 2017, he was awarded the Gold Medal of the Russian Academy of Sciences for outstanding achievements in the field of promoting scientific knowledge, particularly for creating the website “Mathematical Etudes”, where non-fiction stories about modern problems of mathematics are posted, and various mathematical subjects are visualized.
Exclusive interview with Nikolay Andreev will be held on 27rd September at 12:00 (UTC) in Zoom.
Title: “Mathematics in perception».
We are used to consider mathematics as a source of beautiful abstraction. But in this lecture the goal is to perceive mathematics not only with our brain but with our eyes also. We will use visual models to discuss mathematical facts and theorems.
To participate, you need to register: https://forms.gle/yDNqYrnJA4RNLx1H6.
An active link to the entry in the Program on the official website of IMO 2020. The registration will end in three hours before the start of the interview.


Pihl Hatch posted an update 3 weeks, 4 days ago
Do you understand how to calculate the chances of winning the lottery, like the Florida Lottery? It is possible to calculate each group of odds for every different lottery game you play. With the help of a small handheld calculator or with the free calculator on your pc, you merely multiply the numbers together and add one division process when "the order" of one’s chosen numbers is not required for a particular lottery game.
What you "have to know" is the number of total balls that the winning numbers are drawn from…..could it be 59, 56, 42, 49, or 39? When there is a second drawing for the single extra ball, like the "red ball" with Powerball or the Mega Millions’ "gold ball" you have to know how many balls are in this group as well. Is there 49 or 39?
It doesn’t matter if it’s the Florida, Ohio, Texas, PA or NJ Lottery. This strategy or formula gives you the true odds. Florida Lottery is 6/53. New York Lottery is 6/59. The Ohio Lottery, Massachusetts Lottery, Wisconsin Lottery, and the State of Washington Lottery carry a 6/49 lottery numbers ratio. Illinois Lottery posesses 6/52.
Once you have this information correctly before you and your calculator at hand, you can start working the formulas.
Hongkong Lottery You should choose five regular balls and one extra ball correctly matched to the winning drawn numbers to win the multi-million dollar jackpot that most of us dream of winning someday.
In the first example you can find 56 balls in the initial group and 46 balls in the secondary group. So that you can win the Jackpot you should match each one of these balls (5 + 1) exactly, but not necessarily in order. The California Lottery’s Super Lotto Plus is 47/27. The big drum is spinning with the initial part of the drawing. You have a 1/56 chance to match your number to this first ball.
With one ball removed following the first number has been drawn, at this point you have a 1/55 potential for matching another one of your numbers to the next ball drawn. With each drawn number a ball is removed lowering the number of remaining balls by way of a total of one.
The odds of you correctly matching the quantity on the 3rd ball to be drawn is now 1/54 from the total number of balls remaining in the drum. With the third ball removed from the drum and sitting with the other two winning numbers, your odds of correctly matching the fourth ball is reduced to 1/53.
As you can see whenever a ball is released from the drum the odds are reduced by one. You started with a 1/56 chance, then with each new winning number it really is reduced to 1/55, 1/54, 1/53, sufficient reason for the fifth ball you have the odds of 1/52 correctly matching this fifth winning number. It is the first the main formula of how exactly to calculate your probability of winning the lottery, including the Florida Lottery.
Now take these five odds representing the five winning numbers (1/56, 1/55, 1/54, 1/53, and 1/52). The "1" along with the fraction represents your only possiblity to correctly match the drawn number.
Now you take your calculator and multiply all top numbers (1x1x1x1x1) equal one (1). After that you multiply all the bottom numbers (56x55x54x53x52). Correctly entered and multiplied you discover the total is 458,377,920. The brand new fraction becomes 1/458,377,920. It is a 458 million to one possiblity to win. If you were necessary to pick the numbers in order just like they are drawn, then these would be the odds against you to win this Pick 5/56 ball lottery game.
Fortunately or unfortunately, you are not required to pick the numbers in the precise order they are drawn. The next step of the formula will certainly reduce the odds, which allows you to match these five winning numbers in any order. In this step you’ll multiply the quantity of balls drawn — five (1x2x3x4x5). With calculator at hand you see that the full total equals 120.
To give you the proper to choose your five matching numbers in any order, you create these odds by dividing 120/417,451,320. You certainly need a calculator because of this one. 120/458,377,920 minimises your odds of winning this lottery to 1/3,819,816. They are over 3.5 million to 1 odds against you of winning this Pick 5/56 ball lottery game.
If this were the Mega Millions Lottery, you need to add the "gold ball" to these five winning drawn balls in order to win the Multi-Million Dollar Jackpot. The single gold ball is calculated as a 1/46 potential for matching it correctly, and since you are drawing just one number it has to be an exact match. Again, you merely have that "1" chance to do it right. Now you need to multiply 3,819,816 by 46.
Grab your calculator and do the multiplication. Your final odds against you winning the Mega Millions Jackpot are calculated to be 175,711,536 or clearly stated 175 million, 711 thousand, 500 36 thirty-six to one (175,711,536 to at least one 1). Now you learn how to calculate the chances of winning the Mega Millions Lottery.
The Powerball Lottery calculations are based on a 1/59 for the first five white balls and 1/39 for the "red" power ball. The first set of multipliers is 59x58x57x56x55. This group totals 600,766,320. Now divide 600,766,360 by 120 (1x2x3x4x5). Your brand-new total is 5,006,386. You will find a 1/39 chance to catch the "red" ball. 39 x 5,006,386 gives you the real probability of winning the Powerball Jackpot, namely 195,249,054 to at least one 1.
Another 5 +1 Lottery that is apparently everywhere in the United States is the "Hot Lotto" that includes a 39/19 count. It really is played in 15 different States. DC Lottery, Delaware Lottery, Idaho Lottery, Iowa Lottery, Kansas Lottery, Maine Lottery, Minnesota Lottery, Montana Lottery, New Hampshire Lottery, New Mexico Lottery, North Dakota Lottery, Oklahoma Lottery, South Dakota Lottery, Vermont Lottery, and the West Virginia Lottery. The ultimate probability of winning the minimum $1 Million Jackpot is 10,939,383 to at least one 1.
A Pick 6/52 ball Lottery game formula looks like this: (1/52, 1/51, 1/50, 1/49, 1/48, 1/47) for a total of 14,658,134,400 divided by 720 (1x2x3x4x5x6) for the odds of 1/20,358,520. Your possiblity to win the 6/52 Lottery has ended 14.5 million to 1 to win, like the Illinois Lotto.
The Hoosier Lottery that uses Indiana State’s nickname, posesses 6/48. Michigan Lottery is 6/47, Arizona Lottery and Missouri Lottery are 6/44, Maryland Lottery is 6/43, and Colorado Lottery is 6/42. Compare this to the Florida Lottery.
A Pick 5/39 ball Lottery game formula looks like this: (1/39, 1/38, 1/37, 1/36, 1/35) for a complete of 69,090,840 divided by 120 (1x2x3x4x5) for the chances of 1/575,757 of winning the Jackpot like the Illinois Little Lotto. Other States with the same 5/39 lottery numbers are the NC Lottery, Georgia and Florida Lottery Fantasy 5, and Tennessee Lottery’s Pick 5. Virginia Lottery’s Cash 5 carries a 5/34 range.


I am excited to show what we've been working on at Evans Homeschool Academy!
We all know that life is full of charts, graphs, cycles, and images. We also know that "reading" information can be one of the most mundane tasks we've ever had to fight through. But what if I told you that you can create some spectacular graphs, awesome data, and incorporate computer skills, mathematical terminology, equations, and conversions in one project?!
Would you be surprised to know the data is as easy as checking the weather?
But how was I to collect all of the data and how in the world would I store it? Then I thought about using excel to create spreadsheets. Then I figured, well, why not just teach the kids how to do it? So I did. And we came up with some pretty awesome graphs!
Check these out!
And here are some more!
We have thoroughly enjoyed using our weather as a way to learn about graphs, formulas, statistics, science, and math in general.
Don't be afraid to get out there and scour up your weather patterns. You'd be surprised at how great those free and natural resources can be!
Until next time,


A poker hand consisting of 5 cards, is dealt from pack of 52... |… When working with the probability of poker hands, we need to know the number of ways a hand can be dealt (this is the denominator) and the numerator is the number of ways to have a certain hand.d) 4 of a kind. we want to pick 1 ordinal from 13. Poker dice: 2 tens rolled, what is the probability of… I presume that this means in a roll of 5 poker dice, 2 of them came up 10. What is the probability that the other 3 came up in such a way that over all, the hand has 3 of a kind, etc.Neither of the other 2 should be a 10 because that would be 4 of a kind, the third question, or 5 of a kind. Poker Math and Probability | Pokerology.com A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep7. What is the probability of a pre-flop pair flopping four of a kind? The probability of being dealt 4 aces in a 5 card poker…
The probability is 0.001441. Four of a kind. This hand has the pattern AAAAB where A and B are from distinct kinds.This consists of the ten, jack, queen, king, and ace of one suit. There are four such hands. The probability is 0.000002.
Math Forum - Ask Dr. Math I have the answers correct for: straight flush, 4 of a kind, full house, flush, and a straight. We were given the answers, but we have to explain the process. I have seen your explanation to 3 of a kind, 2 pair, and 1 pair, and the answers you got are different from what we were given. Poker Overview 06: Odds and Probabilities They were discovered through the use of permutation and combination formulas. The exact number of possible five-card Poker hands in a 52-card deck is 2,598,960. These hands were divided into group (ranks) of : no pair, pair, three of a kind, straight, flush, full house, four of a kind, straight flush and royal flush. What Is The Probability Of Getting Four Of A Kind In Poker ... Poker probability - Wikipedia Poker probability - Wikipedia probability of being dealt four of a kind in poker - Mathematics PROBABILITY: 2005-08-24 Added links to Wikipedia.org and made some further comments to quoted text odds of payout on slot machines from what is the probability of getting four of a kind in poker Encyclopædia Britannica.
Poker Overview 06: Odds and Probabilities - PokerStars School
Video Poker Hand Odds
Poker Math - Wizard of Odds
We draw 5 cards from a French deck. What's the probability we get "four of a kind"? Find Probability of a Pair in Poker Hand of 5 Cards -… Most lucky poker royal flush hands!Probability of Full House 3 of a Kind and a Pair in Poker Hand of 5 Cards - Продолжительность: 6:48 Anil Kumar 4 268 просмотров. QA - Poker Probabilities Q: What is the probability to get a House or Four of a kind if you have Three of a kind, and choose toToday Poker is played all over the world and is extremely popular everywhere, mainly because of internet, and the possibility to play online outside state control which’s very restrictive in most countries. Cracking Probability and Combinatorics: Card Game... |… Poker problems are very common in probability, and are harder than the simple question types mentioned above. The most common type of poker question involves choosing a five cards from the pack and asking theProbability of getting a four of a kind. See why it's a bad idea to gamble?
Probability of Full House 3 of a Kind and a Pair in …
The game of poker has many variants. Common to all is the fact that players get - one way or another - hands of five cards each. The hands are compared according to a predetermined ranking system. Poker Probability Calculator - Slunečnice.cz Poker Probability Calculator 1.0.1 download - In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of… Yahtzee - Wikipedia
Poker Calculator (five card) | Probability of Losing Probability of Four of a Kind losing (percentage) by MichaelBartmess, last modified 11/08/2016 16:44.This calculator can help you make decisions and determine the probabilities of success and failure in the game of poker using a standard 52 card deck. Poker Probabilities - Wizard of Odds - user-casino.com Four of a kind.I've been asked several times about the probabilities of each poker hand in multiple-deck games. Although I strongly feel poker based games should be played with only one deck, I will submit to the will of my readers and present the following tables.


Links to the original proof:
– In French: Fundamental Lemma Proof – Ngo Bao Chau
– In English: bodecoban-ngobaochau
The Fundamental Lemma, abstract:
I will give a gentle overview of the ideas surrounding the Fundamental Lemma and its solution by Ngo Bao-Chau (recently ranked number 7 in Time Magazine‘s Top 10 Scientific Discoveries of 2009). The Fundamental Lemma is a key ingredient in the Arthur-Selberg Trace Formula and the entire Langlands program, with deep implications for number theory. Its conjectural status (to quote Langlands) “rendered progress almost impossible for nearly twenty years”.
Ngo’s solution is a stunning application of the analogy between Riemann surfaces and number fields: it revolves around Hitchin’s integrable system, a construction in the geometry of bundles on surfaces motivated by physics.
The talk will be aimed at a broad graduate audience: http://media.cit.utexas.edu/math-grasp/David_Ben-Zvi_lecture.html.
- Hundreds welcome Fields Medal winner Ngo Bao Chau back (lookatvietnam.com)
- Professor Ngo Bao Chau’s first math teacher (lookatvietnam.com)
- Vietnam maths star prefers life in the shadows (lookatvietnam.com)
- Professor Ngo Bao Chau: “I offer the glory to Vietnam’s young people” (lookatvietnam.com)
- Vietnamese Professor Ngo Bao Chau wins Fields Medal (lookatvietnam.com)
- ICM2010 – Ngo laudatio (gowers.wordpress.com)
- Mathematicians meet begins (thehindu.com)
- Famous mathematic professor returns home (lookatvietnam.com)
- Fields Medals, Other Top Math Prizes, Awarded (news.sciencemag.org)
- Four Are Awarded Medal in Mathematics (nytimes.com)
- MST: It is necessary to have policies to attract brainpower (lookatvietnam.com)
- Education in Vietnam: Low grades for the party (economist.com)


Ø Limit, Continuity, Differentiability.
Ø Method of Differentiation.
Ø Indefinite & Definite Integration.
Ø Area under the Curve.
Ø Differential Equation.
· Questions are based on latest pattern of JEE
· Questions are labeled as level-I to level-IV, matrix match, integer types & Comprehensive based
· Equally use for school syllabus
· Solutions will be made available for purchasing individual subject package also.
· A total of 292 pages
· A total of 233 solved examples
· A total of 1171 number of unsolved examples


Reading Notes - Ch. 6 - Fastbook
When a classifier is being used in practice, it may encounter images that don't have any of the classes it was trained to classify. On the other hand, a single image may have multiple classes in it.
Multi-label classifiers are capable of classifying images that may not contain exactly one type of object. There may be no objects at all or even multiple objects.
Binary Cross Entropy
The cross-entropy loss function can be used to train a model to classify an image into one of several classes. It is not suitable for multi-label classification because:
- It tries to push one activation much ahead of others (due to the softmax function).
- It uses the negative log-likelihood which returns the loss value of just one activation.
The binary cross-entropy loss function allows for training a model that can classify an image into several classes. In essence, we treat each model output as an independent binary classifier that predicts whether a class is present in the image or not.
Let's assume that we want to build a model that can identify the following objects in an image:
The model we build for this will have four outputs - one for each possible class.
Consider the image below that has the objects pen and book.
Let's assume the model's outputs are the following for this image:
When using binary cross entropy, instead of using the softmax function which favors a single output over others, the model outputs are individually passed through a sigmoid activation function which squishes them between 0 and 1.
We now need to push the outputs of the correct classes to 1 and the other outputs to 0.
We can replace the values corresponding to classes that don't exist in the image with
1 - value, so that we just have to focus on bringing all values as close to 1 as possible.
Just like in cross-entropy loss, we take a log of the values to amplify small differences in values and then drop the negative sign (or equivalently, multiply by -1) so that the loss value is maximum when the value we calculated is close to 0. Conversely, the loss value is close to 0 when the value we calculated is close to 1.
PyTorch provides two versions of the binary cross-entropy loss function that can be used:
The first assumes that you have passed the values you are feeding it through a sigmoid activation already. The second passes the values through a sigmoid before calculating the binary cross-entropy.
Since there isn't just one correct class, we can't just pick the output with the highest value as the model's prediction. Instead, we need to pick a threshold beyond which we'll consider the model output (after passing through a sigmoid activation) as 1, and values below this threshold will be considered 0.
If the threshold is too low, the model may pick incorrect classes. However, if the threshold is too high, the model will only pick classes for which it is very confident.
A good way to find the threshold is getting the raw model outputs (passed through sigmoid activation) and plot accuracy against different threshold values. We can then pick the threshold at which the accuracy is maximum.
In a regression problem, the value we are trying to predict is one or more floats (continuous numeric value), for example: the center of a person's face in an image which is given by two values: the row and the column of the center of the face.
A loss function that can be used in regression problems is the
MSELoss - mean squared error loss.


Statistical probability of poker hands
Poker Probability Chart | GGPoker Online Poker
The following table compares the observed proportion of times the various types of hands occurred ( Sample column). Probability column shows the theoretical values. Lower and Upper are the 95% confidence limits. Chinese Poker - Rules, Strategy & Tips - Primedope A pair of queens in the front hand for example will win on average at least 85% of the time, a hand like A-K-Q-high will win at least 50% of the time against an opponent with random cards who sets his hands optimally. Probability in poker – Statistical Data
PROBABILITY: 5-CARD POKER HANDS
Let’s assume that you are playing 5-card poker, and that you don’t get to discard. Then your question is equivalent to asking, “What is the probability of being dealt 5 cards that are a straight flush?” To get this probability, we count the number of possible straight flushes, and then divide by the number of all possible 5-card hands. How to Calculate the probability of hands in poker « Poker ...
May 23, 2018 · For the probability sequence of the core statistics course, one of her assignments is to calculate the probability of single 5 card draw poker hands from a 52-card deck. I well remember this exercise from back in the day, when I computed all such relevant probabilities using basic combination counting techniques for an intro to probability course.
Poker Probabilities - Wizard of Odds I've been asked several times about the probabilities of each poker hand in multiple-deck games. Although I strongly feel poker based games should be played with only one deck, I will submit to the will of my readers and present the following tables. The first table shows the number of raw combinations, and the second the probability. UCI Machine Learning Repository: Poker Hand Data Set Each card is described using two attributes (suit and rank), for a total of 10 predictive attributes. There is one Class attribute that describes the "Poker Hand". The order of cards is important, which is why there are 480 possible Royal Flush hands as compared to 4 (one for each suit - explained in ). Attribute Information: Example: A Poker Hand - Cut-the-Knot The game of poker has many variants. Common to all is the fact that players get - one way or another - hands of five cards each. The hands are compared according to a predetermined ranking system. Below, we shall evaluate probabilities of several hand combinations. The probability of whichever hand ...
5 Apr 2018 ... What is the probability of different poker hands? Find out in this section where we learn how to count combinations of poker cards.
Aug 1, 2010 ... Get the free "Poker Hand Probabilities" widget for your website, blog, Wordpress ... Find more Statistics & Data Analysis widgets in Wolfram|Alpha. Poker hands ranking and probability : educationalgifs - Reddit
5 Card Poker probabilities. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all ...
Statistics or Probabilities? - Poker Copilot User Guide -… To switch between statistics and probabilities, use the HUD configuration menu (the one you see in the top right- hand corner of the HUD while youAfter that hand the HUD will update its probability up or down, based on what the player did. If you’d like to know more about this approach, it is a basic... Math 131 - calculating the probability of poker hands -… Solution to problem 2.32 in Miller - calculates the probabilities of all possible five card hands in poker. SOLUTION: A poker hand consisting of 6 cards is dealt…
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| The British
Sponsored by Data Connection.
|Sample Paper: Solution to Question 3|
This example solution is currently under development.
|A domino is a 2x1 rectangle with 0, 1, 2, 3, 4, 5 or 6 written on each 1x1 half. A set of dominoes consists of the 28 unique dominoes. A domino with the same number on both halves is called a 'double'.|
| You are to
write a program that will fit a set of dominoes onto a
grid which is 7 squares high, and eight squares across.
This is to be subject to restrictions, in the form of
doubles in fixed positions, and to the rule that two
touching squares can only be equal if they belong to the
same domino. A square should only be considered to touch
the four squares directly above, below, left or right.
Dominoes may not overlap.
You should first read in input indicating where the fixed doubles are. For each double this will be in the form of an x co-ordinate and y co-ordinate for one of the halves of the domino, and a R or B to indicate whether the other half is to the right or below. The top left corner of the board is at (1,1). The list of doubles will terminate with the number -1.
Note that not all the doubles may be fixed, and that their value is left for you to decide.
Once you have read the list you should try and fit the other dominoes to the grid. You should then output a valid solution if there is one, or print 'Impossible'. Your program should then terminate.
8 3 B 4 1 R 5 7 R -1 0-1 0 1-1 0-3 0 | | 5-0 2 0-6 1-2 4 1-3 1-4 1-5 1 0 | | 3-2 4-2 5-2 6 0 2-6 3-4 3-5 3-6 4-5 4-6 5-6 5-5 6-6 3-3 2-2 4-4
|Suppose you are given some double restrictions, and a valid solution to the problem. Can you guarantee (or not) that there are any other solutions?|
|Without any restrictions on doubles or the values on touching dominoes, how many ways can 28 dominoes be placed on the 7x8 board?|


Using Better Globe AS tree calculator, you can calculate the expected return on regular savings.
Saving regularly by buying trees is something we really recommend. The regular and annual savings are very valuable to us who are engaged in forestry. At the same time, it benefits you as a tree owner. In the tree calculator below, you can calculate the expected return for each year, and in total, for a period of time. The only thing you need to enter is the number of trees you want to buy.
To learn more about how it works, we recommend reading the following articles: