Dial Gauge Angle Finder
This Dial gauge angle finder can be used to measure the angle of the surface of an object with respect to the horizon. The weighted needle always points up, perpendicular to the horizon, while the side of the angle finder is held against the surface to be measured. The angle of the solar panels aimed at the sun is measured with the angle finder.
- Dial gauge angle measurements
- Measures angle from the horizon
- Measures 0 to 90 degrees
- Accuracy of 0.5 degrees
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Do you realize that all the statistics, facts, figures and even strategies used in your favorite sport, favorite teams and players are all derived using mathematical calculations?
Soccer is a perfect example of using math technique in sport because it's a highly calculated game with plenty of geometry involved with the connections built into the soccer field – rectangular field, an 8-yard by 6-yard goal area, a centre circle with a 10-yard diameter in the middle of the field, and quarter circles with 1-yard diameters around the corner flags. Teams play with eleven players on the field at a time, with ten field players and one goalkeeper. Coaches decide the formation according to the players’ talent. During the actual game, geometry appears as teams apply techniques and perform certain actions using geometrical concepts.
Do Sports and Math Add Up?
March Madness begins with the National Collegiate Athletic Association (NCAA) season, and college basketball fans wait for predictions of sports writers and experts. However, there is one prediction that has a good chance of taking them to the winner’s bracket – that of Associate Professor Joel Sokol (at the School of Industrial and Systems Engineering at the Georgia Institute of Technology). His love of basketball and fondness for math motivated him to perform mathematical analyses of previous basketball games and predict the NCAA winner. His initial research was simple but his later model outperformed all the other standard ways to predict winners of the tournament. For him, the combination of math and sports is more than a way to combine two loves. It's also a great way to motivate his students who are sports fans. They enjoy the challenge while also learning outside the typical curriculum.
Math In Sports Betting
A California State University professor of statistics, Mike Orkin, argues, “You really can't win at gambling unless you understand math. You don't have to have a Ph.D. but you do have to have a mathematical mind." Orkin taught an accredited course on sports betting, horseracing and casino gambling. He notes that to understand odds in sports betting, you have to be able to compute your chances of winning and compare them with their actual payoff odds. People who do well in sports betting know some mathematics to work with a systematic method and process statistical information sensibly.
Turning a Love For Sports Into a Successful Business
Although you may simply call it games, this is big business for many people. Financial and competitive stakes in college and professional athletics is on the rise along with the complexity of a balanced schedule for all the league teams. Consider the Sports Scheduling Group. This is a group of people who are uniquely talented who work alongside intercollegiate athletic conferences and professional sports leagues to get the most value out of their playing schedules. George Nemhauser, a serious academic, mathematician and sports fan turned his love for sports into business. He is one of the world’s experts in discrete optimization (a critical subject related to applied mathematics and computer science) with several hundred publications in the field. He attributes much of their success to “a unique combination of sports business and programming expertise using state-of-the-art technology and computing resources,” which have turned his interest for sports into a vast business empire.
Teaching Math Through Sports
Educators teaching mathematics and statistics face difficulties in keeping students engaged in the subject. However, the widespread interest in sports provides a huge opportunity to grab the students’ attention in math and statistics classes. Textbooks try to motivate students by introducing various applications, but this works only for students who are committed to a particular academic or career field. Educators are building upon the common interests of students by connecting their studies to something that interests them and is relevant to them. Sports appeals to students across the globe and attracts individuals from a widely ranging population.


These modules offer ways of teaching quantitative reasoning and examples of their use. They have been made available through the Pedagogy in Action project:
Mathematical and Statistical Models, Measurement and Uncertainty, Quantitative Writing, Teaching Quantitative Reasoning with the News, Teaching with Data, Teaching with Models, Teaching with Spreadsheets Across the Curriculum
Browse through links to teaching activities collected by NNN. You can browse the collection either by searching for particular words using the search box or by sorting by the particular quantitative skills involved with the activity. You can also share descriptions of QR Websites, Activities, and Courses by clicking on the appropriate link.
Other Sites of Interest
- A selection of curricular materials serving QL needs across a wide variety of disciplines is hosted by The Center for Mathematics and Quantitative Education at Dartmouth (MQED). Several NNN members helped in the creation of modules on QR and personal finance -- on earning, spending, saving and investment, and planning for retirement; these are found on the Money Matters site.
- An NSF grant awarded to Bowdoin College supports the creation of a non-proprietary instrument for assessing QLR (Quantitative Literacy/Reasoning), a pilot study of the instrument, and sharing via the QRLA Portal.
- Examples of assessment questions from Wellesley College's Quantitative Reasoning Program are available here: Download Wellesley QR Booklet (Acrobat (PDF) 871kB Apr4 08)
- The Web site for the City University of New York's NICHE (Numeracy Infusion Course for Higher Education) Project provides information including assessment instruments, QR learning goals, and best practices for QR instruction.
- The SISL Web site (Sustainability Increases Student Learning) provides information about making Sustainability a theme in courses across the curriculum, including QR courses in which energy, climate change, and natural resources would be engaging topics.
- The Mathematics Association of America hosts a downloadable version of Mathematics and Democracy, a collection of essays on the importance of quantitative literacy.
- Mathematics Across the Community College Curriculum provides an extensive list of books and resources on quantitative reasoning and math across the curriculum.


Tipo Seminário: Seminário de Pós-Graduação, Seminário COPGA
Palestrante(s): Bojan Guzina, University of Minnesota, USA
Horário/Local: 14:00-15:30(Auditorio A)
In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency (FW-FF) homogenization is pursued in Rd via second-order asymptotic expansion about the apexes of “wavenumber quadrants” comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and “dense” clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points, that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as “blunted cones”. We illustrate the analytical developments by several examples, including the Green’s function near the edge of a band gap and clusters of nearby dispersion surfaces.


Glitch Machine’s editor allows 16 lines of “code,” and each line can contain up to 16 operators and/or numbers. The separate lines just make editing easier, and also allow you to use the line-muting feature (tap/click a line of code to mute/unmute it).
It doesn’t matter how much or how little code goes on each line… the whole expression behaves as one line of code.
t 2 * t 5 >> |
Is the same as this:
t 2 * t 5 >> |
Which is the same as this:
t 2 * t 5 >> |
The expression is simply calculated left to right, top to bottom, once per time tick (t).
With RPN you enter the numbers first, and then the operator, unlike typical “infix” mathematical notation. So:
1 + 2 * 3
Looks like this in RPN:
1 2 + 3 *
The numbers are placed on a “stack,” which in Glitch Machine has a limit of 256 values. You push numbers onto the stack simply by entering them in the editor:
1 2 3 4 5
This puts five values on the stack. They are pushed onto the stack in order from left to right; 1 is on the bottom of the stack, and 5 is on top. Doing the following:
…adds the top two values on the stack, which in this case results in 9. Thus, the stack now contains the values [1 2 3 9].
Typing in the following will give you 15 as a result:
1 2 3 4 5 + + + +
Here is another way to express the same calculation:
5 4 + 3 + 2 + 1 +
At the end of your expression, Glitch Machine takes the top value of the stack and converts it to audio output. If the status at the top of the screen says “stack:1” then there is one number on the stack, which is enough for Glitch Machine to make sound. Any value for the stack greater than 0 will produce output.
It is OK to leave more than 1 value on the stack, but only the top value will be used. These extra stack values may be used to create a sequence of notes, or read in a future sample update (using PICK) as a form of audio feedback.
If the stack value reports 0, or a negative number, then this is a stack underflow; no sound will be produced.
Some of operators are “bitwise” operators, meaning they operate on the digital representation of the numbers themselves, which are formed from strings of binary “bits” or 1’s and 0’s. Wikipedia’s entry on Bitwise operation.
Glitch Machine uses the following bitwise operators:
>> (shift right)
<< (shift left) The | operator is especially useful for mixing different elements of your expression together. The >> operator is often used to create timed beats and note sequences. For example:
t 4 >>
…shifts the ‘t’ value to the right by 4 bits, which effectively divides it by 16 (right-shifting by 1 bit divides the input value by 2). This creates a sawtooth waveform with a frequency of ~1.95Hz. Now, mix this with a simple sawtooth note (t 2 *), using the | operator:
t 4 >> t 2 * |
Multiple beats can be mixed for more a complex pattern:
t 4 >> t 5 >> | t 2 * |
This tutorial is a work in progress. If you have specific questions you’d like answered in the tutorial, then leave a comment below!


Distributed Storage Papers
The following are my works on distributed storage systems. Both concern regenerating code that have applications in distributed storage systems.
|MoulinAlg20||D W||Multilinear Algebra for Distributed Storage|
|Atrahasis20||D L W||Multilinear Algebra for Minimum Storage Regenerating Codes|
D = Iwan Duursma (Advisor of the time);
L = Xiao Li (academic sister).
A regenerating code consists of
- a file of size $M$ symbols and
- a system of $n$ storage devices, called nodes.
The configuration of the nodes satisfies the following conditions:
- Each node stores $\alpha$ symbols of the file.
- Any $k$ nodes contains sufficient information to recover the file.
- When a node fails, some $d$ other nodes will each send it $\beta$ symbols to repair the failing node.
The code is named regenerating mainly due to the last bullet point—the nodes regenerate themselves.
The theory of regenerating codes concerns the relation among $n, k, d, \alpha, \beta, M$. For example, since any $k$ nodes contain $k\alpha$ symbols and can recover the file, the file size $M$ is at most $k\alpha$. Similarly, since $d\beta$ symbols repair a failing node, the node size $\alpha$ is at most $d\beta$. (Exercise) One can also show that $k - 1$ nodes ($\alpha$) plus $d - k + 1$ help messages ($\beta$) is at least $M$. There is a family of bounds of this type. They are called cut-set bounds and restrict where those parameters can live.
The opposite approach is to construct regenerating codes that aim to achieve low $\alpha$, low $\beta$, and high $M$. MoulinAlg20 utilizes multilinear algebra to do this. We construct a series of regenerating codes which we call Moulin codes. They achieve the best known $\alpha/M$-versus-$\beta/M$ trade-off to date. And it is conjectured that this trade-off is optimal.
See Figure 1 on page 3 in MoulinAlg20 for the $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 3)$ case. Here is another $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 4)$ case. (In a newer version of MoulinAlg20 that I am still working on.) For more general parameters, check out this D3.js plot.
See also Table 2 on page 29 for the relations among some competitive constructions.
Atrahasis20 exploits multilinear algebra to construct MSR codes, which we called Atrahasis codes. Formally, an MSR code is a regenerating code with $M = k\alpha$ and $\beta = \alpha/(d - k + 1)$. From the constraint on $M$ one sees that there is no wastes of storage (hence the name minimum storage regeneration = MSR). Some researchers see MSR codes as the intersection of regenerating codes and MDS codes.
MSR alone attracts significant attentions because people want to minimize node size ($\alpha \geq M/k$), and only then they minimize help messages ($\beta \geq \alpha/(d - k + 1)$ given that $\alpha \geq M/k$). See Table 1 on page 5 in Atrahasis20 for a comparison of some existing contraptions.Scroll to top


Fucking math. No friend of mine.
Trying to teach me anything in regards to numbers is like teaching a stupid old dog how to sit on command. It just doesn’t work and is a waste of time. Unfortunately, math is in my life because my work involves it.
For instance, today I had to do billing. I actually love billing (i dont get it either) but the man decided to make that shit complicated with percentages. See below.
– Only bill Google and Yahoo (70% – 30%)
– Total Media per Month – $45,454.55
– Dart Fees per Month – $54,645.93
– Mgmt Fees – $4641.71 per Month (10% of Gross)
So of course I was like, what the fuck does that mean. Well. Not really. I knew what it meant, I just didn’t know how to do the math. Instead of figuring it out, I text Daniel to find out if he knew math. He said he didn’t, which I knew is a lie. But instead of harassing him re: his filthy lies, I decided that maybe I should take the mans advice and just figure it out. And so I Googled it to find the formula. Instead I found the most amazing tool ever. A percent calculator!!! All you have to do is put numbers in and it does it all for you. Check it out here!
Then I decided to use the mans advice some more and so I went the extra mile. And found a gram to ounce calculator!!! This is useful for things such as baking and drug dealing.
I’m so excited.


Alligator in the Elevator: 1 2 3
“Educational activities at a young age are vital to developing your child’s cognitive abilities.
The multifaceted content in this app is designed so that kids experience the learning to recognize and write the numbers 1-10 through several learning techniques. For instance, one interactive activity uses haptic touch to follow the children’s gestures as they trace the numbers, while another uses shapes to visualize the counting process. Children can play games at their own pace while progressing to increasingly advanced content."
** BONUS: Includes animated cartoon music video for 'Alligator in the Elevator' tune by Rick Charette
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Available now for
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*45 Minutes of iReady due each Friday! :)
My name is Alexandra Miller and I am very excited to be your child's Math and Science teacher this school year. I recently moved here from East TN. I grew up in the area and am happy to be back! I am starting my 7th year teaching and am so blessed that your child gets to experience a fun and educational year in my classroom!
I will be sending homework home as soon as our homeroom finishes the iReady assessment tests. We have been learning about place value (standard form, expanded form, and word form) this week in math. In science we have been learning about photosynthesis!
My goal is to keep in close communicate with you throughout this year! If there is anything you need do not hesitate to let me know!!


Some interesting maths from working on the dissertation. Yes, maths. I have to get used to the Queen's proper manner of speech.
I study a material that is readily exfoliated into sheets. The sides of each sheet are not the same. This dissimilarity causes the sheet to roll up. You might be wanting a figure right about now. Me too. The sheet structure is easy - that's just a matter of getting the crystal structure from some published source (e.g. ICSD). But how to turn this into a scroll? I tried Illustrator's spiral tool to create a path, but got lazy trying to re-create the brush that would represent the structure. Plus, a friend noticed that when the diamond patterns get too close to one another, illustrator puts white space between them (makes one or both of them thinner). That won't do for representations of reality!
So, I turned to the dark, non-artistic side: maths. You can draw a spiral pretty easily using polar coordinates. Check it out: if you have a line of length X, you can make a spiral by setting 2 things:
radius = x+inner radius of spiral
theta(angle) = some range of angles - how many times you want the spiral to make a complete circle.
In python with numpy and matplotlib:
import numpy as np
import matplotlib.pyplot as plt
and you get
2 circles, no offset from 0.
Well, cool, but I need the spacing between layers to be some well-defined value. You can play with starting radii and/or number of times around the circle (and I did, before I got tired of it). Mathematically, you can set the either of these, and determine the other based on the spacing you want.
Spacing = length of line / number of times around.
So, when I want a spiral with 5 unit spacing, and I start with an initial radius of 100, I get 20 spins around the circle:
And you want Cartesian coordinates, because what good molecular modeling software recognizes polar coordinates? The nearest plot already shows this, but here's how I got them. See how bumpy my lines are? Those are atoms! OOOO!
Basic geometry, I know, but I was proud of myself. Especially since I've been intending to figure this out for nearly a year and finally did it. And it only took me a day.


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A shoe company is going to close one of its two stores and combine all the inventory from both stores. These polynomials represent the inventory in each store:
Which expression represents the combined inventory of the two stores?


A few new math anchor charts have graced my classroom walls so I thought I’d share!
Basic operations is by far the toughest concept for my students – mainly in problem-solving. So we refer back to this one a lot!
I wish I’d put all the units of length down the middle of the chart so it was easier to see the how the length increases from millimeter, centimeter, inch, foot, etc. Maybe next year.
We use the Place Value chart a lot! It’s the first unit of the school year so it seems like everything we learn falls right out of their heads.


Ideas, Formulas and Shortcuts for What Is Simplest Form in Math
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Text is among the most naturally compact types of information at about one byte necessary to store each letter. After the button is first pressed, the cursor is situated in the top box ready that you enter the numerator. Consult with the link for details about how to ascertain the best common divisor.
Exclusively by doing a proof can you get to the deep insights that mathematics offersthat inform you why something is correct, not merely buy research papers that it’s true. You also have to concentrate on the new vocabulary (look this up in a math dictionary for another opinion). Each math topic has a lot of unique kinds of math worksheets to cover various forms of problems you may decide to work on.
For example, even when you think of an ideal, objective measure of the caliber of a paper, it’s still hard to judge a journal. If you learn formulas as you go, it can help you to comprehend what’s happening in the new stuff you’re studying. Beyond formulas, the app delivers mathematical concepts, advice, and examples on equations in a very complete method which makes it straightforward to understand for students that are looking to acquire extra understanding.
For https://www.privatewriting.com/essay-writer one-time payments, locking in exchange rates may also be helpful if you ought to make a huge transfer for an upcoming given date ( for instance, making a down payment on a house) and the marketplace is unstable. 1 formula to figure opportunity costs might be the ratio of what you’re sacrificing to what it is you’re gaining. Some benefits made available by FX brokers consist of no cost transfers, online money transfers, better exchange prices, 24-hour support, and access to internet tools like foreign exchange alerts and the capability to set your personal desired exchange rate.
The very last thing that remains is to demonstrate your plot with the aid of the show() function! For instance, if someone is provided a list of randomized numbers ranging from one to ten thousand and is requested to put them in ascending order, odds are it will have a sizable period of time and include some errors. There are normally many methods to address a issue.
What Is Simplest Form in Math Explained
If you enjoy acronyms, consider combining them https://www.missouriwestern.edu/biology/ along with the Memory Palace technique. If you choose to analyze data on a spreadsheet, this is expected to be your go-to tool. Additional users are given the capability to send or email distinctive values back in the calculator instead of being required to memorize long equations and values.
You still remember how a vector was characterized in the prior section as scalar magnitudes that were extended a direction. A correlation indicates that the 2 variables are related somehow. A particle isn’t as easy because I have naively described.
Utilizing these tools in a professional context, however, could require a little more. Fantastic news is that there’s a technique that comes to our rescue. For instance, the successful candidate may need to understand how to establish a company email system or know which social media platforms are best for the provider’s marketing.
Industrial support and training are readily available. Uni-Paper, Think Free’s internet viewer program is extremely handy when you must view any spreadsheet document. The majority of the material is pertinent to any Introductory Business course with concepts that would make an application for a lengthy time.
You can also locate the factorial of a number utilizing recursion. The last sequence is known as the Fibonacci sequence. To truly understand this you require a little quantity of math, but zero math is unfortunately insufficient.
It is possible to add several numbers and locate the total or sum working with this sum of numbers calculator. In order to understand how the variables relate to one another, you create scatterplots. Be aware that the denominator of a fraction may not be 0, as it would produce the fraction undefined.
What Is Simplest Form in Math Secrets That No One Else Knows About
Aimed at advanced users, it delivers an extremely full feature set in addition to an intuitive interface. Authorea is an excellent on-line LaTeX editor, and possesses many of the distinguished features provided by the earlier mentioned tools. If you’re giving a presentation via slideshow, it’s possible to even insert a video in your presentation with no tech abilities.
What Is Simplest Form in Math – Dead or Alive?
With girls, you would like to tie what you’re teaching into real life. For many individuals, math is a really hard subject, and a great deal of teachers aren’t able to give students the one-on-one help they may require as a way to master math. The boys are simply the opposite,” he explained.
A small amount of work on learning the basics will create enormous advantages. As there are many alternatives, it can be difficult to nail down the perfect one for your requirements. This magnificent machine changed our lives in lots of ways.
Zoom in on an object until it will become fuzzy or you can just find a little portion of the overall image. It is likewise an amazing chance to acquire on on the ground floor of some really strong tech. Use a larger chisel to carve the interior of the snowflake if you’ve got one.
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It’s a mature project that has existed for quite a couple of years. You can locate a URL to an illustration of quite good mathematical papers published in this proceedings. See our disclosure policy for more information.
At the previous time of examination you won’t have the ability to refer the entire book. Ultimately, Section 6 provides a overall overview of the key findings of this study. OK, you’re done with the quiz.


Greek mathematics, the study of numbers and their properties, patterns, structure, space, apparent change, and measurement, is said to have originated with Thales of Miletus (l. c. 585 BCE) but was clearly understood during the periods of the Minoan civilization (2000-1450 BCE) and the Mycenaean Civilization (c. 1700-1100 BCE) and was derived from older Mesopotamian and Egyptian mathematical systems.
These systems, however, served practical purposes, as noted by scholar Rosalie David:
Scholar Gwendolyn Leick says the same regarding Mesopotamian mathematics:
Mathematics was an integral part of scribal training and practice tablets from as early as the Early Dynastic Period demonstrate that geometry was essential for the task of dividing up plots of land for farming and irrigation purposes. (116)
Thales, and the other great early mathematician Pythagoras (l. c. 571 to c. 497 BCE, best known for the Pythagorean Theorem), viewed mathematics as a means for apprehending truth, elevating it from the level of a simple pragmatic instrument to the basis for understanding reality. None of Thales' writings have survived, but he is considered the first mathematician of Greece for introducing geometry and, it seems, for how he applied it. Pythagoras, possibly one of his students, made mathematics central to his philosophy which advocated, among other things, reincarnation in the belief that the soul, like numbers, was eternal and returned after death in other and still other forms.
Among the students of Pythagoras was Hippasus of Metapontum (l. c. 530 to c. 450 BCE), credited with discovering irrational numbers, and his work was further developed by Theodorus of Cyrene (l. 5th century BCE). Another Pythagorean, Philolaus of Croton (l. c. 470 to c. 385 BCE) developed the pyrocentric model of the universe and was the teacher of the great mathematician Archytas of Tarentum (l. 5th century BCE). Archytas, in turn, was the teacher of Eudoxus of Cnidus (l. c. 408 to c. 355 BCE), also famous as a brilliant mathematician and astronomer.
These mathematicians influenced the works of Plato (l. 424/423-348/347 BCE) and Aristotle (l. 384-322 BCE), whose philosophies were informed by mathematics, notably geometry. Euclid (l. c. 300 BCE) established geometry through his famous Elements and is known as the "Father of Geometry." The best-known Greek mathematician is the inventor and engineer Archimedes of Syracuse (l. 287-212 BCE), but other notables include Eratosthenes (l. c. 276-195 BCE), Apollonius of Perga (l. c. 240 to c. 190 BCE), Aristarchus of Samos (l. c. 310 to c. 230 BCE), Hipparchus of Nicea (l. c. 190 to c. 120 BCE), Menelaus of Alexandria (l. c. 100 CE), and Claudius Ptolemy (l. 100-170 CE). These thinkers lay the foundation that was developed further by later mathematicians, preserving and advancing their work, which allowed for mathematical progress into the modern era.
Stolen Legacy Claim
The Sumerians of Mesopotamia had already developed sophisticated mathematics by the Early Dynastic Period (2900-2334 BCE) with a sexagesimal place notation of 60. This enabled them to create the concept presently recognized as 'time' based on the concept of 60, which defines a minute as 60 seconds and an hour as 60 minutes. The Sumerian system provides evidence of an understanding of addition, subtraction, multiplication, division, algebra, geometry, reciprocals, squares, and quadratic equations. The Egyptians, more practical and conservative, worked in addition, subtraction, multiplication, and division with elements of calculus and, perhaps, algebra.
As noted, the Mesopotamians and Egyptians used mathematics for practical purposes. Rosalie David elaborates:
The Egyptians certainly possessed an organized system of mathematical knowledge, but they used practical experience rather reasoning skills to solve problems. Although they started with elementary concepts, they soon developed systems that coped with some complicated arithmetical and geometrical problems, and they made use of fractions. (218)
It is therefore inaccurate to claim the Greeks invented mathematics, though this assertion has been advanced for centuries, and more precise to say they developed mathematical concepts further than earlier civilizations. Even though Sumerian mathematics was highly advanced, it was nowhere near as refined as Greek mathematics from the 6th century BCE going forward. The Pythagorean Theorem, for example, was known to the Babylonians long before Pythagoras was even born, but his definition not only preserved but also seems to have refined the concept.
Even so, the claim that Greek mathematics and Greek philosophy were somehow 'stolen' from earlier civilizations is also inaccurate and an oversimplification that ignores actual history. Egyptian hieroglyphics and Mesopotamian cuneiform were not deciphered until the mid-to-late-19th century, and, by that time, a significant body of historical work had been created by European scholars who only knew history through what was available to them: Greek and Latin sources. The 'decorations' noted on ancient Mesopotamian and Egyptian temples, tombs, and inscriptions were not recognized as writing systems – nor was there any way they could have been – as they had not yet been deciphered by c. 1820.
Many of the conclusions the early modern and modern-era scholars came to regarding ancient history now need revision – and there were, no doubt, racialized motives at work in the interpretation of data – but, overall, they did not intentionally sideline earlier civilizations to favor the Greeks; they were simply unaware of Mesopotamian and Egyptian history. If one believes, according to the survey one had done, that one's property covers X number of feet, and is later informed by one's neighbor that he actually owns those feet, one cannot be accused of stealing land; only of being misinformed. If one does not respect and respond to the new information, only then can one be said to be at fault.
Greek mathematics, then, acknowledged as based on earlier systems, dates to the 6th century BCE. The Minoan and Mycenean systems were lost during the Greek Dark Age and have not been reconstructed. Thales of Miletus, therefore, is generally understood as the first to introduce theoretical mathematics to Greece, even though the discipline existed prior to his time in other regions.
Pre-Socratic Philosophers & Pythagoreans
Thales is the first of the so-called Pre-Socratic Philosophers, the term given to the thinkers who lived in Greece before the time of Socrates of Athens (470/469-399 BCE), and so is known as the first philosopher of ancient Greece. According to the works of later Greek writers, he seems to have followed the same paradigm as the Babylonians and Egyptians at first in using mathematics for practical ends. He is said, for example, to have measured the height of the pyramids of Giza by calculating the difference between their shadows and his own and, applying mathematical principles to a logistical problem, diverted the course of the Halys River so the army of King Croesus (r. 560-547 BCE) could cross when he was employed by Croesus as an engineer.
In time, however, his mathematics became more theoretical as he proposed axioms, proofs, defined the meaning of number, developed geometry, proposed the so-called Thales' Theorem (to find the center of a circle), and applied mathematics to astronomy, accurately predicting the solar eclipse of 28 May 585 BCE.
Pythagoras, however (another Pre-Socratic philosopher), developed mathematics further as a course of study. The later writer Proclus notes how "Pythagoras turned geometrical philosophy into a form of liberal education by seeking its first principles in a higher realm of reality" (Baird, 16). To Pythagoras, mathematics was a means to an end but was also the end itself. The early Pre-Socratics were concerned with finding the First Cause of existence, proposing water, air, fire, or otherworldly elements, but Pythagoras claimed that mathematics explained the visible and invisible world because numbers, like creation, have no beginning or end.
Number, to Pythagoras, was the First Cause because it could be proven that the many can come from the one – the plurality observed in the world could, and did, arise from a single source. Pythagoras' concept of number was based on a dot in the center of a circle – the monad, which stood for the concept of One, of unity. From the monad, came the dyad, two circles now sharing a common center and, by drawing a line horizontally across their intersection, an equilateral triangle is formed, shifting the dyad to a triad and so on through the decad. From the One, therefore, came the many and, although numbers might seem limited in sequence running between 1-10, they were actually limitless in the combinations of that sequence running 11-20, then 21-30, and so on forever.
Pythagoras applied the eternal nature of number to the human soul, arguing that one's spirit, like a number, returned again and again after bodily death in different forms – like the sequence of numbers – now appearing as a human being, now as a dog, or a mule, or a housefly. Accordingly, Pythagoras advocated vegetarianism because there was no way of knowing if the animal one slaughtered might not be one's departed uncle or father in the body of a chicken.
Pythagoras' student Philolaus of Croton rejected what he viewed as the nonsense of establishing a first cause and directed his energies to the mathematical examination of the cosmos, eventually concluding that Earth (and all the other planets as well as the sun) revolved around a central, limitless fire, contradicting the accepted model of the day that Earth was the center of the universe. He rejected the Milesian school of Thales and his followers Anaximander and Anaximenes that insisted on the importance of the First Cause in favor of a more practical use of philosophy in recognizing how the observable world works. Scholar Robin Waterfield comments:
[To Philolaus], the true essence of things is accessible only to the gods, or perhaps to a man with divine knowledge; and in the nature of things, we cannot know the infinite. In part, Philolaus is here criticizing the Milesians or similar attempts to divine an ultimate reality behind the things of this world. He is suggesting that this is impossible, and that the best one can do, instead, is to try to say what the necessary preconditions are for the world we are faced with to exist. (93)
Archytas adopted Philolaus' approach and also applied mathematics to understanding the form of the world and how it worked. Archytas, in fact, is said to have been greatly admired by Plato for his ability to combine philosophic idealism with practical application. Archytas was not only a general of Tarentum, elected seven years in a row and considered invincible but is also known as the founder of mathematical mechanics, is famous for solving the geometric problem known as "doubling the cube", and was renowned in his lifetime for his contributions to astronomy and music theory.
Plato & Aristotle
Although Plato is not usually included in lists of Greek mathematicians, the underlying form of his philosophy was mathematics and, specifically, pure geometry. Plato rejected the relativism championed by Sophists like Protagoras of Abdera (l. c. 485-415 BCE), who claimed, "of all things a man is the measure" – that the individual alone defined truth – and insisted on universal, eternal truths that informed what was recognized as 'truth' on earth. To Plato, something was not 'true' just because one believed it so; it was 'true' in so far as it participated in the Form of Truth in a higher realm. Geometry, Plato felt, established these higher truths, calmed the passions, cleared the mind, and freed one from the 'true lie' or 'the lie in the soul' that caused one to believe wrongly about the most important aspects of life.
Aristotle, (Plato's most famous student) on the other hand, is famous for his contributions to Greek mathematics. Scholar Thomas R. Martin writes:
Aristotle regarded science and philosophy not as abstract subjects isolated from the concerns of ordinary existence but rather as the disciplined search for knowledge in every aspect of life. That search epitomized the kind of rational human activity that alone could bring the good life and genuine happiness. (185)
To Aristotle, the purpose of life was happiness and mathematics encouraged one's recognition of the path toward this goal. Whether in his Metaphysics, where he explains the mathematics of symmetry and how it reflects beauty, or in his Nicomachean Ethics, where he advocates for personal balance through his Golden Mean, mathematics informs Aristotle's thought as much as it did Plato's. Aristotle, however, devoted considerably more time to practical mathematics than his teacher, as would many of the mathematicians who came after him.
Euclid, Archimedes, Eratosthenes & Others
Euclid's Elements, easily the bestselling book in the history of the world before the Bible took that title, codified, explained, and expanded upon mathematics in 13 volumes covering Pythagorean constructs, parallelism, the golden ratio, Thales' Theorem, prime numbers, cones, pyramids, optics, spheres, and defined what has come to be known as Euclidian Geometry. His work was so influential and became so popular that Ptolemy I Soter (r. 323-282 BCE) of Egypt invited Euclid to his court to discuss Elements. Scholar Lionel Casson comments:
[Ptolemy I] must have at least dabbled in mathematics, for it was he who, on asking Euclid whether there wasn't a shorter way to learn geometry than through The Elements, got the celebrated reply, "There is no royal road." (32)
Ptolemy I established the Library of Alexandria, which was built during the reign of Ptolemy II Philadelphus (282-246 BCE) and became a beacon for the greatest minds of the age. Among these was Archimedes of Syracuse, the greatest Greek mathematician and recognized today as the most significant in antiquity. Archimedes is known as the "Father of Mathematics " for his contributions to the discipline, including the equilibrium of planes, measurement of circles, mechanical theorems, measurement of spheres and cylinders, equilibrium of planes, the parabola, the ellipse, the area of a spiral, the concept of buoyancy, the invention of the Archimedes Screw based on mathematical principles, and the invention of the pulley and lever using the same. He is said to have died while preoccupied with a mathematical equation and so ignored the command of the Roman soldier who killed him.
Eratosthenes, a polymath, was one of the head librarians at the Library of Alexandria and the first to calculate the circumference of the earth and write a comprehensive geography of the known world. He is even credited with coining the term 'geography.' His astronomical calculations were later used by Hipparchus of Nicea, regarded as the greatest astronomer of his age. Aristarchus of Samos drew on the mathematical developments of his predecessors to arrive at his heliocentric model of the universe, only dismissed by Hipparchus because it contradicted the accepted model of the time as set down by Aristotle, considered the authority on that subject as well as many others. Claudius Ptolemy, in his Almagest, distilled the astronomical data of the past and contributed his own observations, creating a work that would influence Nicolaus Copernicus (l. 1473-1543) and inspire the Scientific Revolution of the Early Modern Era.
Mesopotamia, ancient Egypt, the Indus Valley Civilization, and ancient China, had developed mathematical principles long before the time of Thales of Miletus. The term 'ancient history' does not seem so ancient as applied to Thales' prediction of the eclipse in 585 BCE when one understands that the astronomers of other civilizations had been doing the same over 2,000 years before. Hippocrates, recognized as the "Father of Medicine" was actually a latecomer to the field; Sushruta (l.c. 7th/6th century BCE) of ancient India was practicing surgery and writing medical journals centuries before Hippocrates was even born.
Even so, this does not detract from the contributions of the Greek mathematicians. Scholar Thomas Cahill notes:
It is important…to recall that the underlying question for all [of the Greek philosophers/mathematicians] – "What is the nature of reality?" – remains to this day a fundamental question that each of us must attempt to answer in our lives. When we recall this – and acknowledge how little progress we have made in formulating a satisfactory answer – we gain a measure of sympathy for them and the single-minded spunk with which they approached their daunting task. Because they had no guidelines to follow, they poked their noses into everything in the hopes of finding an adequate answer; and in the process they helped to invent the disciplines of philosophy, theology, the physical sciences, medicine, psychology, political science, and ethics. (151)
The Greeks did not actually invent any of these, but their development of the concepts of each lay the groundwork for others to build on in the progress of Western Civilization. When one uses one's smartphone or searches one's personal computer for a recipe for dinner, one has the Greek mathematicians to thank for that kind of luxury; for that, and so much more.


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Examples of the the word, sum , in a Sentence Context
The word ( sum ), is the 2337 most frequently used in English word vocabulary
Mastering contextual understanding of words and phrases is a vital skill for effective communication and English learning. Enhance your proficiency by practicing with our handpicked collection of 50 English phrases.
At the end of the list you can practice your english pronunciation
- Data1,data2): n = Len (data1) sum 12 = 0 sum 1 = sum (data1) sum 2 =, sum ,(data2) for i in range (n): sum 12 += data*data covariance = ( sum 12 -
- Angle and a right angle is termed the complement of the angle. *Two angles that, sum ,to a straight angle (180°) are called supplementary angles. *: The difference
- His name contains the number 365,the number of the days in the year; i.e. the, sum ,of the numbers denoted by the Greek letters in ΑΒΡΑΣΑΞ according to the rules
- Do not share any interior points are called adjacent angles. *Two angles that, sum ,to one right angle (90°) are called complementary angles. *: The difference
- Any family of nonempty sets is nonempty. **König's theorem: Colloquially,the, sum ,of a sequence of cardinals is strictly less than the product of a sequence of
- To buy Augustus De Morgan using his own name as example;" Great Gun, do us a, sum , " Is attributed to his son William De Morgan, but a family friend John Thomas
- Def naive_covariance (data1,data2): n = Len (data1) sum 12 = 0 sum 1 =, sum ,(data1) sum 2 = sum (data2) for i in range (n): sum 12 += data*data
- X_n, then the arithmetic mean does this best, in the sense of minimizing the, sum ,of squares (xi − X)2 of the residuals. (It follows that the mean is also the
- Function in one variable x, with coefficients in F, can be written as the, sum ,of a polynomial function with rational functions of the form a/ (x − b)n
- Straight angle (180°) is termed the supplement of the angle. *Two angles that, sum ,to one turn (360°) are called elementary angles or conjugate angles. *An
- Fractions with a numerator greater than one, they had to write fractions as the, sum ,of several fractions. For example, they resolved the fraction two-fifths into
- Or arrays as arguments, and derive related functions. For example the ", sum ," function is derived by applying the" reduction" operator to the" addition
- Based on a percentage of construction value, hourly rates or a fixed lump, sum ,fee. Combinations of these structures are also common. Fixed fees are usually
- Arrow \left arrow is displayed by the APL interpreter);: +/N\, \! Prints the, sum ,of N,i.e. 22. The user can save the workspace with all values, programs and
- Def two_pass_covariance (data1,data2): n = Len (data1) mean1 =, sum ,(data1) / n mean2 = sum (data2) / n covariance = 0 for i in range (n): a
- Small signal acoustic wave equation. Then we can express the variables as the, sum ,of the (time averaged) mean field (\angle\dot\range) that varies in
- Be weakened by the effect of other electrons, but still remains attractive in, sum ,). The electron cloud model can only be described by quantum mechanics, in
- Mason. Derleth's papers and comic book collection (valued at a considerable, sum ,upon his death) were donated to the Wisconsin Historical Society in Madison.
- Several fractions. For example, they resolved the fraction two-fifths into the, sum ,of one-third + one-fifteenth. Standard tables of values facilitated this. Some
- The mean is the only single number for which the residuals defined this way, sum ,to zero. *If it is required to use a single number X as an estimate for the
- Skepticism, but the terms agnostic and agnosticism were created by Huxley to, sum ,up his thoughts on contemporary developments of metaphysics about the "
- The sample mean, : \bar x = \display style \ sum _in x_j/n, and then computes the, sum ,of the squares of the differences from the mean, : s^2 = \display style\franc \!
- His allies and began a crusade of plunder in Franconia. Having extorted a large, sum ,of money from the citizens of Nuremberg, he quarrelled with his supporter, the
- Measure of a countable disjoint union of measurable sets is equal to the, sum ,of the measures of the individual sets. **Stone's representation theorem for
- Have negative temperatures, since there is no highest energy state, so that the, sum ,of the probabilities of the states would diverge for negative temperatures.
- Donations. That person may find another person willing to donate a substantial, sum , However, if the person seeking the donations met the person donating, not for
- Frac. It turns out that a more suitable quantity for updating is the, sum ,of squares of differences from the (current) mean, \text style\ sum _in (x_i -
- To handle unequal sample weights, replacing the simple counter n with the, sum ,of weights seen so far. West (1979) suggests this incremental algorithm:
- Associated sum s of squares.: \text_ = \text_ + \text_\, \! See also Lack-of-fit, sum ,of squares. The F-test The F-test is used for comparisons of the components of
- Y_ from experimental unit i when receiving treatment j can be written as the, sum ,of the unit's response y_i and the treatment-effect t_j, that is: y_=y_i+t_j.
- Are computable and therefore definable. The field of algebraic numbers The, sum , difference, product and quotient of two algebraic numbers is again algebraic (
- x)=\ sum _ u_k (x)a_k ex+\ sum _ u_k (x)by\dagger _k ex, \,where the first, sum ,is over positive energy states and the second over those of negative energy.
- Be treated very cautiously by the public. Logic of ANOVA Partitioning of the, sum ,of squares The fundamental technique is a partitioning of the total sum of
- Of larger cardinals. (The reason for the term" colloquially ", is that the, sum ,or product of a" sequence" of cardinals cannot be defined without some aspect
- Agglutinative morphology and SO word order, usually co-occur in languages. In, sum , the idea was that Turkic, Mongolic,and Tunguska form a Sprachbund – the
- As tending to over six acres of land. On May 23, 1845,Abby May was granted a, sum ,from her father's estate which was put into a trust fund, granting minor
- As being built up (in approximation) in an electron configuration that is a, sum ,of simpler hydrogen-like atomic orbitals. In such a configuration, pairs of
- By wishing a Merry Christmas to everyone on Earth. His message appeared to, sum ,up the feelings that all three crewmen had from their vantage point in lunar
- For payment in the form of an advance and royalties. An advance is a lump, sum ,paid in advance of publication. An advance must be earned out before royalties
- Scholar Walter Older has discerned in the early liturgical work Lactates, sum ,(RV ANH 31,written in 1691 at the age of 13) the influence of Legrenzi's
- To decide the sign of the exterior angle measure. *: In Euclidean geometry,the, sum ,of the exterior angles of a simple polygon will be one full turn (360°).
- In its first three weeks at a single New York theater, an almost unheard-of, sum ,at the time. ) This success in turn led to a vogue in America for Japanese
- Of the integers, as the latter are the roots of Monica polynomials for all The, sum , difference and product of algebraic integers are again algebraic integers
- Of the sum of squares The fundamental technique is a partitioning of the total, sum ,of squares S into components related to the effects used in the model. For
- Simply divide by n instead of n − 1 on the last line. Because sum _SQR and, sum ,* mean can be very similar numbers, the precision of the result can be much
- The name of the alkane from" -ANE" to" -YL" * Number the root chain so that, sum ,of the numbers assigned to each side group will be as low as possible * Number
- Data1,data2): n = Len (data1) mean1 = sum (data1) / n mean2 =, sum ,(data2) / n covariance = 0 for i in range (n): a = data - mean1 b =
- By bombarding target atoms of heavy elements with ions, such that the, sum ,of the atomic numbers of the target and ion elements equals the atomic number
- Irreducible element in FX. Then the rational function 1/p can be written as the, sum ,of a polynomial function q with rational functions of the form a/ (x − b)n.
- Of dissection. This involves cutting a shape into pieces, whose areas must, sum ,to the area of the original shape. For an example, any parallelogram can be
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Definition - What does Drilling Formulas mean?
Drilling Formulas are sets of special formulas used by drilling engineers for a wide number of oil well related operations. These formulas form the base of drilling calculations. Also known as drilling formula sheets, these formulas are essential mathematical and engineering factors that help in drilling, managing and controlling an oil well. These formulas are taught in Petroleum engineering and in oil well control schools. The drilling industry makes use of almost 5000 formulas on a regular basis for well control, drilling wells and other operations.
Petropedia explains Drilling Formulas
Petroleum engineering makes use of special formulas to analyze and calculate the various aspects related to the drilling of an oil well and its later operations. Drilling Formulas and drilling calculations play an important part as they help graduate students to learn how to implement these dynamics on real sites. These formulas are regulated by the American Petroleum Institute and are available in several handbooks meant for well control. Nowadays, these formulas are also available in Smartphone applications which help offshore and onshore drilling experts to use them easily. Since all the phases of oil production require the maintenance of the oil well, Drilling Formulas remain pivotal to do so.


A Day in the Life of a YPP Math Literacy Worker
“We help [younger] kids to understand math in a way that we didn’t get to when we were their age. All MLWs have to be leaders. A great MLW is a great team member, someone who comes to the workshop and actually gives energy. Kids look up to teenagers, what they see from us, that’s what they’re going to do.”
The Young People’s Project (YPP) was envisioned and continues to conduct its work in the spirit of Ella Baker. Founded in 1996 by two generations of young people, YPP is an outgrowth of the Algebra Project which Bob Moses founded as part of the legacy of SNCC’s student-led grassroots organizing and voter registration efforts in 1960’s Mississippi. It has since grown into a nationally recognized, math literacy and youth empowerment nonprofit that serves over 600 students and teachers across 14 sites every year.
Through 25 years of partnering with schools and community-based organizations, YPP has demonstrated that it is possible to develop and implement a ground-up education program that engages young people who are most often underrepresented in STEM fields, in learning and teaching math through near-peer mentoring, in a way that increases their interests in math, their persistence, confidence and willingness to more broadly serve their communities. The program shows a path toward improving education for those who are not benefitting from the country’s public educational system. This year (2022) marks YPP’s 25th Anniversary as a nonprofit organization.
YPP is a solution that young people build with each other, under the leadership, support, and guidance of invested adults. MLWs learn pieces of math well enough to teach it and learn to facilitate math activities with middle school students. Together they learn and teach each other math in ways that are meaningful and engaging. As a result, they develop as a community, and become their own support. In doing so they invite and challenge the adults and systems around them to see and treat them differently.
As one middle school principal put it, “YPP may not be the answer for every child, but it is the answer for the community, that we begin to see our children differently, that we begin to believe in them.”
YPP seeks to recruit MLWs from high school students representing a diverse group of academic performers and peer leaders, from the same community as the middle school students. The training of high school students as paid Math Literacy Workers (MLWs) develops the abilities of the middle school students and the high school students themselves, to succeed in school and in life, and in doing so involves them in efforts to eliminate institutional obstacles to their success. MLWs care that the younger students learn and have fun.
YPP is Exploring STEM Literacy
In this video you will see YPP’s emerging Computer Science and STEM literacy
work in action at Excel High School in Boston, MA.
In this video you will see YPP’s National Youth “Flagway” Math Tournaments.
|Gates Foundation Targets Culturally Responsive Math Teaching With New Grants
|Young People's Project Empowers Youth With Math Skills


A wavelet is a kind of mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.
In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of Frame of a vector space (also known as a Riesz basis), for the Hilbert space of square integrable functions.
Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are of continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. The word wavelet is due to Morlet and Grossmann in the early 1980s. They used the French word ondelette, meaning "small wave". Soon it was transferred to English by translating "onde" into "wave", giving "wavelet".
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filterbanks. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a CWT are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency resp. scale to that event. The product of the uncertainties of time and frequency resp. scale has a lower bound. Thus, in the scaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. This is related to Heisenberg's uncertainty principle of quantum physics and has a similar derivation. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.
Wavelet transforms are broadly divided into three classes: continuous, discretised and multiresolution-based.
Continuous wavelet transforms (Continuous Shift & Scale Parameters)In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the Lp function space L^2(\R)). For instance the signal may be represented on every frequency band of the form [f,2f] for all positive frequencies f>0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.
The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function \psi \in L^2(\R), the mother wavelet. For the example of the scale one frequency band [1,2] this function is
The subspace of scale a or frequency band [1/a,\,2/a] is generated by the functions (sometimes called child wavelets)
- \psi_ (t) = \frac1\psi \left( \frac \right),
The projection of a function x onto the subspace of scale a then has the form
- x_a(t)=\int_\R WT_\psi\(a,b)\cdot\psi_(t)\,db
- WT_\psi\(a,b)=\langle x,\psi_\rangle=\int_\R x(t)\overline\,dt.
See a list of some Continuous wavelets.
For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal.
Discrete wavelet transforms (Discrete Shift & Scale parameters)It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the affine system for some real parameters a>1, b>0. The corresponding discrete subset of the halfplane consists of all the points (a^m, n\,a^m b) with integers m,n\in\Z. The corresponding baby wavelets are now given as
A sufficient condition for the reconstruction of any signal x of finite energy by the formula
- x(t)=\sum_\sum_\langle x,\,\psi_\rangle\cdot\psi_(t)
Multiresolution-based discrete wavelet transforms
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. To avoid this numerical complexity, one needs one auxiliary function, the father wavelet \phi\in L^2(\R). Further, one has to restrict a to be an integer. A typical choice is a=2 and b=1. The most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet.
From the mother and father wavelets one constructs the subspaces
- V_m=\operatorname(\phi_:n\in\Z), where \phi_(t)=2^\phi(2^t-n)
- W_m=\operatorname(\psi_:n\in\Z), where \psi_(t)=2^\psi(2^t-n).
- \\subset\dots\subset V_1\subset V_0\subset V_\subset\dots\subset L^2(\R)
From those inclusions and orthogonality relations follows the existence of sequences h=\_ and g=\_ that satisfy the identities
- h_n=\langle\phi_,\,\phi_\rangle and \phi(t)=\sqrt2 \sum_ h_n\phi(2t-n)
- g_n=\langle\psi_,\,\phi_\rangle and \psi(t)=\sqrt2 \sum_ g_n\phi(2t-n).
Mother waveletFor practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the space L^1(\R)\cap L^2(\R). This is the space of measurable functions that are absolutely and square integrable:
- \int_^ |\psi (t)|\, dt and \int_^ |\psi (t)|^2 \, dt .
Being in this space ensures that one can formulate the conditions of zero mean and square norm one:
- \int_^ \psi (t)\, dt = 0 is the condition for zero mean, and
- \int_^ |\psi (t)|^2\, dt = 1 is the condition for square norm one.
For \psi to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform.
For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the identity in the space L^2(\R). Most constructions of discrete WT make use of the multiresolution analysis, which defines the wavelet by a scaling function. This scaling function itself is solution to a functional equation.
In most situations it is useful to restrict \psi to be a continuous function with a higher number M of vanishing moments, i.e. for all integer ''m\int_^ t^m\,\psi (t)\, dt = 0
Some example mother wavelets are:
The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted) by a factor of b to give (under Morlet's original formulation):
- \psi _ (t) = \psi \left( \right).
For the continuous WT, the pair (a,b) varies over the full half-plane \R_+\times\R; for the discrete WT this pair varies over a discrete subset of it, which is also called affine group.
These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat).
Comparisons with Fourier Transform (Continuous-Time)The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multiresolution analysis.
The discrete wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform. This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT.
Definition of a waveletThere are a number of ways of defining a wavelet (or a wavelet family).
Scaling filterThe wavelet is entirely defined by the scaling filter - a low-pass finite impulse response (FIR) filter of length 2N and sum 1. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.
For analysis the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters the time reverse of the decomposition.
Daubechies and Symlet wavelets can be defined by the scaling filter.
Scaling functionWavelets are defined by the wavelet function \psi (t) (i.e. the mother wavelet) and scaling function \phi (t) (also called father wavelet) in the time domain.
The wavelet function is in effect a band-pass filter and scaling it for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html#note7 for a detailed explanation.
For a wavelet with compact support, \phi (t) can be considered finite in length and is equivalent to the scaling filter g.
Meyer wavelets can be defined by scaling functions
Wavelet functionThe wavelet only has a time domain representation as the wavelet function \psi (t).
For instance, Mexican hat wavelets can be defined by a wavelet function. See a list of a few Continuous wavelets.
Applications of Discrete Wavelet TransformGenerally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis. Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.
Wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier Transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations, astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics. This change has also occurred in image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis. In computer vision and image processing, the notion of scale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.
One use of wavelet approximation is in data compression. Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets. This means that although the frame is overcomplete, it is a tight frame (see types of Frame of a vector space), and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression.
A related use is that of smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed.
HistoryThe development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Zweig’s discovery of the continuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound), Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's Harmonic wavelet transform (1993) and many others since.
Wavelet TransformsThere are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below:
Generalized TransformsThere are a number of generalized transforms of which the wavelet transform is a special case. For example, Joseph Segman introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.
Another example of a generalized transform is the chirplet transform in which the CWT is also a two dimensional slice through the chirplet transform.
An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, darkfield electron optical transforms intermediate between direct and reciprocal space have been widely used in the harmonic analysis of atom clustering, i.e. in the study of crystals and crystal defects. Now that transmission electron microscopes are capable of providing digital images with picometer-scale information on atomic periodicity in nanostructure of all sorts, the range of pattern recognition and strain/metrology applications for intermediate transforms with high frequency resolution (like brushlets and ridgelets) is growing rapidly.
List of wavelets
- Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-0692-0
- Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
- A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets, Academic Press, 1992, ISBN 0-12-047140-X
- P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
- Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1-56881-041-5
- Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7
- Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331-371, 1910.
- Ramazan Gençay, Faruk Selçuk and Brandon Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 2001, ISBN 0-12-279670-5
- Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, ISBN 0-5216-8508-7
- Tony F. Chan and Jackie (Jianhong) Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, Society of Applied Mathematics, ISBN 089871589X (2005)
- Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-x
- Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, ISBN 1568810725, ISBN-13 978-1568810720
- Wavelet Digest
- NASA Signal Processor featuring Wavelet methods Description of NASA Signal & Image Processing Software and Link to Download
- 1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)
- Binomial-QMF Daubechies Wavelets
- Wavelets made Simple
- Course on Wavelets given at UC Santa Barbara, 2004
- Wavelet Posting Board
- The Wavelet Tutorial by Polikar (Easy to understand when you have some background with fourier transforms!)
- OpenSource Wavelet C++ Code
- An Introduction to Wavelets
- Wavelets for Kids (PDF file) (Introductory (for very smart kids!))
- Link collection about wavelets
- Wavelet forums (French) Wavelet forum (English)
- Gerald Kaiser's acoustic and electromagnetic wavelets
- A really friendly guide to wavelets
- Wavelet-based image annotation and retrieval
- Very basic explanation of Wavelets and how FFT relates to it
wavelet in German: Wavelet
wavelet in Spanish: Wavelet
wavelet in Finnish: Wavelet-muunnos
wavelet in French: Ondelette
wavelet in Indonesian: Wavelet
wavelet in Italian: Wavelet
wavelet in Polish: Falki
wavelet in Portuguese: Wavelet
wavelet in Russian: Вейвлет
wavelet in Swedish: Vågelement
wavelet in Ukrainian: Вейвлет
wavelet in Chinese: 小波分析
wavelet in Japanese: ウェーブレット


From AP, staff reports
NASHVILLE (AP) — In Tennessee, ACT scores for the high school class of 2012 show a slight improvement in all four subject areas — English, math, reading and science.
All public school students in Tennessee take the ACT in their senior year, which is over 68,000 students statewide, according to ACT results released Wednesday.
The average composite score for Tennessee seniors in 2012 was 19.7, a small increase over the average composite score in 2011 of 19.5. The percentage of Tennessee students earning scores that indicate readiness for college courses also increased in each of the subjects.
• In English, the average score increased to 19.6 this year from 19.4 last year. The percentage of students meeting the English college readiness benchmark was 59 percent this year, compared to 58 last year.
• In reading, the average score increased to 19.9 this year from 19.7 last year. The percentage of students meeting the reading college readiness benchmark was 43 percent this year, up from 41 percent last year
• In mathematics, the average score was 19.1 this year, up from 19.0 last year. The percentage of students meeting the mathematics college readiness benchmark was 29 percent this year, an increase from 27 last year.
• In science, the average score increased to 19.6 this year from 19.4 last year. The percentage of students meeting the science college readiness benchmark was 21 percent this year, up from 20 percent last year.
In the Obion County School System, students from Obion County Central and South Fulton high schools showed a slight increase in both English and reading and on the average composite score, while they held steady on both math and science scores from the previous year, according to Nancy Hamilton, supervisor of instruction for grades 9-12 and assistant director of schools.
OCCHS is now an ACT testing site.
At Union City High School, principal Wes Kennedy said the average ACT for the Class of 2012 was a 20.3.
“We average the highest score that each student makes, not just the last score, and this includes all students of the class, including those with special needs,” he said. “Fifty-one percent qualified for the Lottery Scholarship by making a 21 or higher and 7 percent scored a 30 or higher.”
Across the state line, Kentucky’s public high school seniors scored higher on the ACT college entrance exam than a year ago, but their performance remained below the national average, education officials said Wednesday.
The average composite ACT score rose to 19.5 for Kentucky’s public high school Class of 2012, up from 19.2 in 2011, according to data released by the state Department of Education.
The national average composite score was 21.1, which includes public and non-public-school test-takers.
This year’s senior class in Kentucky showed across-the-board improvement with higher average scores in English, math, reading and science, the data showed. But those scores still trailed the national average in all four subjects.
When adding non-public-school test-takers to the equation, Kentucky’s average 2012 composite score among graduating seniors was 19.8, according to ACT Inc., the Iowa City, Iowa-based not-for-profit that administers the exam nationwide.
This year’s junior class in Kentucky also made slight gains in ACT scores, the data showed.
Published in The Messenger 8.24.12


Canfield student gets highest possible ACT score
A Canfield student has accomplished something that very few people can say they have.
Canfield High School Junior, Tarik Mashqbeh has achieved a composite score of 36 on the June 2023 ACT test. This is the highest possible composite score a student can get on the test.
According to a press release, only about one quarter of 1% of students who take the ACT earn a top score. In the U.S. class of 2022, only 3,376 out of 1.34 million students who took the ACT earned a perfect score.
"Tarik's achievement reflects not only his dedication but also the commitment to academic excellence we value at Canfield Local Schools. We're incredibly proud of his accomplishment and it speaks to the high caliber of students within our district," said Canfield Superintendent Joe Knoll.
The test consists of English, mathematics, reading and science. Each scored on a scale of 1 to 36. A composite score is the average of the four test scores. According to the release, Mashqbeh got a 36 on each individual section.


Here is a secret Blender feature. A calculator. No, there is no calculator window and no, I am not talking about the Interactive Python Console. This secret Blender feature can be found in every numeric input box and every slider button. Every input box can solve arithmetic problems just by entering in the equation and pressing enter. Simply click on the box and enter a problem like .8/3 and press enter.
This may seem like a worthless feature. A calculator? Why would I need that? If you have never used this feature it may seem pointless. However, this feature can dramatically speed up your workflow and once you have used this feature you won’t be able to live with out it! Here’s an example.
Say you are working with a material with a reflection value of .941 and you figure it is about one-tenth too reflective. Now you can guess a value. Somewhere ‘round .85, right? Or you can use the calculator. All you do is just left click on the slider, and then to the right of the number left click and enter the multiplication symbol * and then enter .9 and press enter. That’s it! You have now just lowered the reflection value by exactly 10%.
It can even solve algebraic equations like this: (I have no clue why you would need to do this.)
The Blender “calculator” will always round it’s answer to the nearest one-thousandth (unless your dealing with an integer value like the frame number, in which case it will be rounded to the nearest integer) and the results will not exceed the minimum and maximum values of the input box. Usually 1 is the maximum value and 0 is the minimum ( in some case there are no min’ and max’ values).
Hopefully, you can see the advantages to using mathematical operators in Blender’s “calculator” to perform quick, accurate changes to the value of a parameter. Once you get use to it, it will save a significant amount of time and greatly speed up your workflow. It eliminates the need of a calculator for complex problems.


By Daniel Fleisch, Julia Kregenow
The examine of astronomy bargains an enormous chance for us to achieve a deeper figuring out of our planet, the sunlight procedure, the Milky means Galaxy and the recognized Universe. utilizing the plain-language strategy that has confirmed hugely well known in Fleisch's different Student's publications, this booklet is perfect for non-science majors taking introductory astronomy classes. The authors deal with subject matters that scholars locate so much complex, on topics starting from stars and light-weight to gravity and black holes. Dozens of totally labored examples and over one hundred fifty workouts and homework difficulties support readers become familiar with the ideas in each one bankruptcy. An accompanying site incorporates a host of assisting fabrics, together with interactive ideas for each workout and challenge within the textual content and a chain of video podcasts during which the authors clarify the $64000 techniques of each part of the booklet.
Read Online or Download A Student's Guide to the Mathematics of Astronomy PDF
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Additional info for A Student's Guide to the Mathematics of Astronomy
Write the following numbers in scientific notation with the coefficients and exponents given. (a) Coefficient = 6022; exponent = 20. 91; exponent = −6. (c) Express each of the numbers above in normalized scientific notation. 2 Converting numbers in scientific notation Converting numbers to and from scientific notation is straightforward as long as you pay careful attention to which direction you’re moving the decimal 32 Fundamentals point. Here are the rules for converting numbers from scientific notation (such as 3 × 108 ) into decimal notation (such as 300,000,000).
3 Rate problems 25 before plugging in values. In this case, time is the desired quantity. You can “solve for time” by dividing both sides of the Eq. 9 by speed: ✘ × time ✘ distance ✘ speed = time, = ✘ ✘ speed speed ✘ time = distance . 11) Now that you have time by itself on the left side of the equation, you’re ready to plug in your numerical values. Remember that plugging in numbers should always be the very last step: time = 12 km 3 km hr = 12 3 km km hr ✟ hr = 4 hr. =4 ✟ km ✟ km ✟ One reason that you’re far better off solving for the quantity you’re seeking before plugging in numerical values can be seen by comparing Eq.
Several other instances of inverse proportionality appear in this book. 8) R in which Fg represents the force of gravity and R represents the distance between the centers of the objects. Example: How does the force of gravity between two objects change if the distance between the objects doubles? Writing the proportionality relationship of Eq. 8 as an equation with proportionality constant “z” for both the far and the near distance gives: Fg, f ar =z× Fg,near = z × 1 R 2f ar = z R 2f ar , 1 z = 2 , 2 Rnear Rnear which can be compared by dividing: Fg, f ar = Fg,near z✄ R 2f ar z✄ = 2 Rnear 1 R 2f ar · 2 R2 Rnear = near = 1 R 2f ar Rnear R f ar 2 .
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