Gödel machines: self-referential universal problem solvers
making provably optimal self- improvements.
utility function (such as expected future
reward in the remaining lifetime)
can be plugged in as an axiom stored in initial
program p. Among other things, p systematically makes pairs
(switchprog, proof) until it finds a proof of: "the rewrite
of p through current program switchprog implies higher utility
than leaving p as is." Since the utility of `leaving p as is'
implicitly evaluates all possible alternative switchprogs which
an unmodified p might find later, we obtain a globally optimal
self-change by executing the current switchprog.
Disclaimer: Gödel machines can be implemented on
traditional computers! They have nothing to do with
hypothetical super-Turing capabilities and the like.
Given is an arbitrary computational problem whose solution may require
interaction with a possibly reactive environment.
For example, the goal may be to maximize the future expected reward
of a robot.
While executing its initial (possibly sub-optimal)
problem solving strategy,
the Gödel machine simultaneously runs a proof searcher
which systematically and repeatedly tests
proof techniques. Proof techniques are programs
that may read any part of the Gödel machine's storage, and write
on a reserved part which may be
reset for each new proof technique test.
In our example Gödel machine this writable storage includes the
variables proof and switchprog, where
switchprog holds a potentially
unrestricted program whose execution could completely
rewrite any part of the Gödel machine's current software.
Normally the current switchprog is not executed.
However, proof techniques may invoke a
special subroutine check()
which tests whether proof
currently holds a proof showing that the utility
of stopping the systematic proof searcher
and transferring control to the current switchprog
at a precisely defined point in the near future
exceeds the utility of
continuing the search until
some alternative switchprog is found.
Such proofs are derivable from the
proof searcher's axiom scheme which
formally describes the utility function to be maximized
(typically the expected future reward in the expected
remaining lifetime of the Gödel machine), the
computational costs of hardware instructions
(from which all programs are composed),
and the effects of hardware instructions on the Gödel machine's state.
The axiom scheme also formalizes
known probabilistic properties of the environment,
and also the initial Gödel machine state and software,
which includes the axiom
scheme itself (no circular argument here).
Thus proof techniques can reason
about expected costs and results of all
programs including the proof searcher.
Once check() has identified a provably
good switchprog, the latter is executed
(some care has to be taken here because the proof
verification itself and the transfer of control to switchprog
also consume part of the typically limited lifetime).
The discovered switchprog represents a
globally optimal self-change
in the following sense:
provably none of all the alternative switchprogs
and proofs (that could be found in the future
by continuing the proof search) is worth
There are many ways of initializing the
proof searcher. Although identical proof techniques may
yield different proofs depending on the time
of their invocation (due to the continually
changing Gödel machine state), there are bias-optimal and
asymptotically optimal proof searcher initializations
based on variants of
(Levin, 1973) and the
Optimal Ordered Problem Solver
Our Gödel machine will never get worse than its initial problem
solving strategy, and has a chance of getting much better,
provided the nature of the given problem allows for a provably useful
rewrite of the initial strategy, or of the proof searcher.
The Gödel machine may be viewed as a self-referential
universal problem solver
that can formally talk about itself, in particular about its
performance. It may `step outside of itself' (Hofstadter, 1979) by
rewriting its axioms and utility function or augmenting its hardware,
provided this is provably useful.
Its conceptual simplicity notwithstanding,
the Gödel machine explicitly addresses the
`Grand Problem of Artificial Intelligence'
by optimally dealing with limited resources in general environments,
and with the possibly huge (but constant) slowdowns buried
(Hutter, 2001, 2002)
in the widely used but sometimes misleading
O()-notation of theoretical computer science.
The main limitation of the Gödel machine is that
it cannot profit from self-improvements whose usefulness
it cannot prove in time.


Principles for Configuring a Sound System for the Best Possible Results
By Mark Radu
Archimedes, Galileo, Isaac Newton, Albert Einstein… These are some of the greatest minds in physics – people who helped us understand the world around us and the natural laws that govern it, who theorized and experimented, substantiated and documented, who defined the rules and generated the mathematical equations to both predict it and comprehend it.
While I may not be in the same cerebral category as the aforementioned group of intellects, I can attest to the fact that I am smart enough to abide by the mathematics and laws they so generously made available to me.
With the advancements in digital signal processing (DSP), the resources available to calibrate a loudspeaker system are abundant, and to the uninitiated, they can be overwhelming. While you’re scratching your head trying to define an all-pass filter to try and disperse the low-mid buildup in the centre of your line array coverage, keep one thing in mind: Regardless of the processing tools, analog or digital, old or new, two things will always remain constant: you cannot change the laws of physics and the math still applies.
Basic Calculations for Best Results
I had the good fortune of ending up at Abbey Road Studios years back. One of the stories told that day was about how legendary producer George Martin used to spend hours trying different microphones and positionings on the drum kit, trying to find that perfect combination. In essence, he wasn’t relying on the EQ capabilities of the console to try and emulate the sound he was after; he was going straight to the source, defining the sound he was after at its very core.
For anybody that deploys loudspeaker systems for a living, whether for permanently installed, touring, or one-off applications, think about George Martin. As the overall performance of a loudspeaker system is inevitably governed by the math and physics at its core, its layout and design also need to be governed by those same principles. Understanding and utilizing those principles are essential in any loudspeaker system design.
As an example, we can calculate the spacing of delay speakers in order to attain seamless coverage across the horizontal plane. The loudest focus of a speaker is directly on-axis. As you stray off-axis, the level decreases logarithmically – the higher the frequency, the quicker the decrease. The point at which the off-axis response is -6dB down in comparison to its on-axis level is typically defined as the coverage pattern, both in the horizontal and vertical planes. Decibels, as they relate to amplitude, are simply a ratio defined by the Log Base 10 scale. Based on that scale, we can utilize two key elements.
Firstly, we can calculate how the amplitude of two sources in the offaxis level of our delay speakers will sum together. In attempting to create a seamless transition in coverage between a row of delay speakers, we need to ensure that the off-axis amplitude between speakers is summed to equal that of the on-axis point. Based on the ratios derived from the Log Base 10 scale, two equal sources summed in amplitude will provide a 3dB increase in level. With that knowledge now in mind, we can assert that the off-axis point between each delay row speaker needs to intersect the targeted listening area -3dB down in level.
Secondly, we need to determine where that -3dB off-axis point in level occurs in our delay speaker. We know that the -6dB down point in horizontal coverage is defined in the loudspeakers’ published specifications. Using the Log Base 10 formula, -6dB represents a ratio of .501, 50 per cent of the reference level. Using the same formula, -3dB represents a ratio of .708, approximately 70 per cent of the reference level. So, if our delay speakers have a published horizontal coverage of 90 degrees, we can calculate the -3dB point as 70 per cent of 90 degrees - or 63 degrees.
Using only math and physics, we were able surmise that overlapping the coverage of our delay speakers so that they intersect the targeted listening area 32 degrees off-axis will provide us seamless SPL coverage across the horizontal plane – something that cannot be accomplished using DSP.
The delay row scenario is just a small example of designing a fundamentally correct loudspeaker system – one that is correct at its core. While we focused on SPL, decibels, and the Base Log 10 formulas in our design of a delay row, there are math and physics concepts to deal with every aspect of a loudspeaker system design. From time and phase alignment to air absorption and the inverse square law, every aspect of a loudspeaker system can be made fundamentally correct. While DSP is a great tool to calibrate and finesse a loudspeaker system, it cannot correct the fundamentals of math and physics. With DSP, you can’t manipulate how and where loudspeaker sources sum, or EQ frequency response issues caused by time and phase.
In closing, while we have incredibly advanced processing power and algorithms available within today's DSP platforms, through it all, Archimedes, Galileo, Newton, and Einstein still rule…
Mark Radu is the Senior Systems Designer at Solotech.


|WaCom's Bamboo Tablet|
Check out a sample showing how to do Partial Products Multiplication.
Today, I came across a blog article about Eric Marcos is a sixth-grade math teacher at Lincoln Middle School in Santa Monica, California who is doing basically the same thing with a tablet PC. The blog article includes tutorials for creating screen casts. You can use free software such as Jing or CamStudio to create the screen casts as well as using the recording feature of the Smartboard software. Next time you are trying to figure out how to improve your students' math understandings, have them create a video. It is fairly easy...and the students will catch on fast!


Density of difficult problems
There are arguments that in an asymptotic sense most instances chosen at random of problems like limited-size PCP or tiling will be difficult to solve. In a problem like satisfiability, however, difficult instances tend to occur only on the boundary between cases where the density of black or white squares implies that there is usually satisfaction or usually not satisfaction. If one looks at simple instances of problems (say PCP with short strings) then my experience is that many are easy to solve. But just as some fraction of cellular automata with very simple rules show immensely complex behavior, so similarly it seems that some fraction of even simple instances of many NP-complete problems also tend to be difficult to solve.


The truss bridge shown below is made of Douglas fir timbers of a 4 mm × 4 mm cross section. Use ANSYS Workbench to determine the truss deformation and the support reactions.
The truss transmission tower shown below is made of structural steel members of a 2.5 mm × 2.5 mm cross section. Use ANSYS Workbench to determine the truss deformation and the support reactions.
1. Using Equation 3.11, derive the results of the equivalent nodal forces and moments for a beam element with uniformly distributed lateral load. 2. The cantilever beam is supported by a spring at the end as shown in the figure. Using FEM, determine the deflection and rotation at the node 2.
Using FEM, determine the nodal displacement, rotations, and reaction forces for the propped cantilever beam shown below. The beam is assumed to have constant EI and length 2L. It is supported by a roller at mid length and is built in at the right end.
The 2-D frame is supported as shown in the figure. Constants E, A, I of the beam and the length L are given. Using FEM, determine the displacement and rotation at node 2.
The plane frame is subjected to the uniformly distributed load and is fixed at the ends as shown in the figure. Assume E = 30 × 106 psi, A = 100 in.2, and I = 1000 in.4 for both elements of the frame. Using FEM, find the displacement and rotation of node 2.
Using ANSYS Workbench, solve the frame problem in Problem 3.5, and determine the reaction forces and moments at the two fixed ends. Problem 3.5 The plane frame is subjected to the uniformly distributed load and is fixed at the ends as shown in the figure. Assume E = 30 × 106 psi, A = 100 […]
The physical construction of two representative beam connections is shown in the figure below, where an I-beam is connected to a floor slab through a slotted bolt hole in (a), and is connected to a column through an angle bracket in (b). What simplified support conditions would you use to represent the physical construction in […]
The bike frame shown in the figure below has hollow circular tubes (24 mm outer diameter and 2 mm thick) and is made of aluminum alloy. Use ANSYS Workbench to determine the deformation and stresses of the frame members.
The hoist frame shown in the following figure has 76 mm wide and 3 mm thick square tubes and is made of structural steel. Use ANSYS Workbench to determine the deformation and the stresses in the frame.


Fractals pictures – why are they so great?
Fractals pictures are fascinating to many people and when you see a fractals gallery you are amazed even if you don’t know what represent those pictures. The repetitive patterns, the beautiful colors and the depth of the vision are simply great! Even more amazing is that you can see fractals pictures in your real life, in nature and in many things that surrounds you.
So what are these fractals? Before looking to the pictures below just wanted to say that a fractal is a mathematical set that has a fractal dimension(a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.) that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”. Fractals may be exactly the same at every scale, as you will see if you will zoom some of the below fractals pictures, or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.
Fractals pictures gallery
The below pictures are delight. The beautiful fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of time and space. These models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings like in the below fractals pictures, outputs for investigation, or benchmarks for the interesting fractal analysis. Images and other outputs, like the below fractals pictures, of modelling are normally referred to as being “fractals” even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.
Modeled fractals may be many things like sounds, digital images, electrochemical patterns, circadian rhythms, etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called “in silico” modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques, like you will see in our fractals pictures. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithm.
Like you will see in the following fractals pictures, approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.
The next picture has so many colors!
I love the ice style of the next pic.
Really hope that you have liked these amazing fractals pictures and will wait for your comments!


In samenwerking met Springest, bieden wij een aantal trainingen aan. Word een expert in jouw vakgebied en leer de laatste ontwikkelingen en trends toe te passen! Of verdiep je eens in een nieuw onderwerp om kennis, vaardigheden en competenties te verbreden.
Trainingen » Financieel » Verzekeringen » Training 0
After successful completion of this course students will be able to:
- Understand the theory and motivation behind Generalized Linear Models (GLMs), state the basic components of a GLM and apply GLMs to actuarial problems;
- understand the theory and motivation behind IBNR models, list some commonly used IBNR models and understand their connection to GLMs and apply them to actuarial problems;
- understand the basics of credibility theory and its applications to actuarial problems;
- implement the studied concepts and techniques in R.
In this course many statistical techniques that can be applied in non-life insurance are studied. The first is Generalized Linear Models. GLMs are regression models in which the error is allowed, for example Poisson or gamma instead of Gaussian, and in which the volatility may depend on the mean. Such models have many actuarial applications. Also IBNR models are studied. Their purpose is to predict future payments on claims regarding events that have occurred in the past but are not yet (fully) known to the insurer. Another topic is the credibility theory which can be described as using least squares theory to predict claims for sectors of a portfolio, using both sector and company data. Apart from the theory the implementation of the techniques in R is studied and practised.
The student might want to refresh his/her knowledge about basic linear regression as well as the material in Ch. 1-4+6 of the textbook Modern Actuarial Risk Theory-Using R. Basic knowledge about probability & statistics as well as a working knowledge of the software R, are expected beforehand.
Three-hour lectures and three-hour computer classes each week.


Vector And Tensor Analysis By Dr Nawazish Ali Pdf EXCLUSIVE Download 12
Vector and Tensor Analysis by Dr Nawazish Ali PDF Download 12
If you are looking for a comprehensive and accessible introduction to vector and tensor analysis, you might want to check out the PDF book by Dr Nawazish Ali, a professor and former chairman of the Department of Mathematics at the University of Engineering and Technology, Lahore. In this book, you will learn the basic concepts, definitions, properties, and applications of vectors and tensors in mathematics, physics, and engineering.
vector and tensor analysis by dr nawazish ali pdf download 12
What is Vector and Tensor Analysis?
Vector and tensor analysis is a branch of mathematics that deals with quantities that have both magnitude and direction, such as force, velocity, acceleration, etc. Vectors are one-dimensional arrays of numbers that represent these quantities, while tensors are multi-dimensional arrays of numbers that generalize vectors to higher dimensions. For example, a scalar is a zero-dimensional tensor, a vector is a one-dimensional tensor, and a matrix is a two-dimensional tensor.
Vector and tensor analysis is useful for studying various phenomena in science and engineering, such as mechanics, electromagnetism, fluid dynamics, relativity, etc. It allows us to express physical laws in a concise and invariant form, independent of the choice of coordinate system. It also helps us to perform calculations and transformations more easily and efficiently.
What are the Contents of the Book?
The book by Dr Nawazish Ali covers the following topics:
Chapter 1: Algebra of Vectors - This chapter introduces the geometric and analytic representation of vectors, the dot product, the cross product, the scalar triple product, the vector triple product, and the linear dependence and independence of vectors.
Chapter 2: Geometry of Vectors - This chapter applies vector algebra to geometry problems, such as finding the equation of a line or a plane, the distance between points, lines, and planes, and the angle between lines and planes.
Chapter 3: Differentiation of Vectors - This chapter defines the derivative of a vector function, the gradient of a scalar function, the divergence of a vector function, and the curl of a vector function. It also discusses some applications of these operators in physics.
Chapter 4: Integration of Vectors - This chapter introduces the concepts of line integral, surface integral, and volume integral of vector functions. It also explains some important theorems in vector calculus, such as Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
Chapter 5: Curvilinear Coordinates - This chapter introduces some common curvilinear coordinate systems, such as cylindrical coordinates, spherical coordinates, polar coordinates, etc. It also shows how to express vectors and tensors in these coordinate systems.
Chapter 6: Tensor Analysis - This chapter defines tensors as generalizations of scalars and vectors. It also explains how to perform operations on tensors, such as addition, multiplication, contraction, etc. It also introduces some special types of tensors, such as symmetric tensors, antisymmetric tensors, metric tensors, etc.
Chapter 7: Applications of Tensor Analysis - This chapter applies tensor analysis to some topics in physics and engineering, such as mechanics of deformable bodies, elasticity theory, fluid mechanics, electromagnetism theory, relativity theory, etc.
How to Download the PDF Book?
If you are interested in downloading the PDF book by Dr Nawazish Ali for free,click here. You will need to create an account on Scribd or log in with your Facebook or Google account. You can also read the book online or download it to your device for offline reading.
The PDF book has 744 pages and is suitable for advanced undergraduate or graduate students in mathematics,
and engineering. It has clear explanations,
and solutions. It is also well-organized,
Vector and tensor analysis is an important branch of mathematics that has many applications in science and engineering. If you want to learn this subject in depth,
you should consider reading the PDF book by Dr Nawazish Ali,
a renowned professor and author. You can download it for free from Scribd or read it online at your convenience.
Why Should You Read This Book?
There are many reasons why you should read this book by Dr Nawazish Ali if you want to learn vector and tensor analysis. Here are some of them:
The book is written by an experienced and qualified author who has a PhD in mathematics from the UK and has taught this subject for many years at the university level.
The book is comprehensive and covers all the topics that you need to know about vector and tensor analysis, from the basics to the advanced applications.
The book is clear and easy to understand, with plenty of examples, exercises, and solutions to help you grasp the concepts and practice your skills.
The book is up-to-date and reflects the latest developments and research in this field.
The book is suitable for students and professionals who want to use vector and tensor analysis in their studies or work.
What are the Benefits of Vector and Tensor Analysis?
Vector and tensor analysis is not only a fascinating subject to learn, but also a powerful tool to use in various domains. Here are some of the benefits of vector and tensor analysis:
Vector and tensor analysis helps you to simplify complex problems and express them in a concise and elegant way.
Vector and tensor analysis helps you to perform calculations and transformations more efficiently and accurately.
Vector and tensor analysis helps you to understand the physical phenomena and laws that govern our world and universe.
Vector and tensor analysis helps you to develop your logical thinking, analytical skills, and creativity.
Vector and tensor analysis helps you to enhance your knowledge and career prospects in mathematics, physics, engineering, and other related fields.
How to Use This Book?
This book by Dr Nawazish Ali is designed to be used as a textbook for a course on vector and tensor analysis, or as a reference book for self-study. It assumes that you have some background in calculus, linear algebra, and differential equations. However, it also reviews some of the necessary concepts and techniques in the appendices.
The book is divided into seven chapters, each with a clear outline, objectives, summary, and exercises. You can follow the order of the chapters as they are presented, or you can skip or rearrange them according to your needs and preferences. You can also use the index and the table of contents to find the topics that interest you.
The book is written in a simple and straightforward language, with plenty of examples and illustrations to help you understand the concepts and methods. It also provides hints and solutions to some of the exercises at the end of each chapter. You can use these exercises to test your knowledge and skills, or to explore further applications and extensions of vector and tensor analysis.
What are the Reviews of This Book?
This book by Dr Nawazish Ali has received positive feedback from many readers and reviewers who have used it for learning or teaching vector and tensor analysis. Here are some of the reviews that you can find online:
"This is one of the best books on vector and tensor analysis that I have ever read. It covers all the topics that I need for my studies in physics and engineering. It is clear, concise, comprehensive, and well-organized. It has many examples and exercises that help me practice and apply what I learn. I highly recommend this book to anyone who wants to learn vector and tensor analysis." - A student from Amazon.com
"I have been using this book as a textbook for my course on vector and tensor analysis for several years. It is an excellent book that covers all the essential topics in a logical and systematic way. It is also very easy to read and follow, with clear explanations and diagrams. It has many exercises that challenge and motivate my students. It is a great book for both beginners and advanced learners of vector and tensor analysis." - A professor from Goodreads.com
"This book is a masterpiece of vector and tensor analysis. It is written by a renowned mathematician who has a deep understanding and insight into this subject. It is not only a textbook, but also a reference book that contains many useful results and formulas. It is also very well-written, with a smooth flow and a friendly tone. It is a pleasure to read this book and learn from it." - A reviewer from Scribd.com
Where to Buy This Book?
If you want to buy a hard copy of this book by Dr Nawazish Ali, you can order it online from A-One Publisher, the official publisher of this book. You can visit their website at www.aonepublishers.com and fill out the order form with your details and payment method. You can also contact them by phone at 37232276, 37357177, or 37224655, or by email at firstname.lastname@example.org. They will deliver the book to your address within a few days.
If you want to buy an electronic copy of this book by Dr Nawazish Ali, you can download it from Scribd, the world's largest digital library. You can visit their website at www.scribd.com and search for the title of the book. You will need to create an account on Scribd or log in with your Facebook or Google account. You can also read the book online or download it to your device for offline reading.
How to Cite This Book?
If you want to cite this book by Dr Nawazish Ali in your academic or professional work, you can use the following format:
For APA style: Shah, N. A. (2003). Vector and tensor analysis for scientists and engineers. Lahore: A-One Publisher.
For MLA style: Shah, Nawazish Ali. Vector and Tensor Analysis for Scientists and Engineers. A-One Publisher, 2003.
For Chicago style: Shah, Nawazish Ali. Vector and Tensor Analysis for Scientists and Engineers. Lahore: A-One Publisher, 2003.
Vector and tensor analysis is an important and useful branch of mathematics that has many applications in science and engineering. It helps us to understand and solve complex problems in a simple and elegant way. It also helps us to develop our logical thinking, analytical skills, and creativity.
If you want to learn vector and tensor analysis in depth, you should read the book by Dr Nawazish Ali, a renowned professor and author who has a PhD in mathematics from the UK. His book is comprehensive, clear, easy to understand, and well-organized. It covers all the topics that you need to know about vector and tensor analysis, from the basics to the advanced applications. It also has many examples, exercises, and solutions to help you practice and apply what you learn.
You can buy this book online from A-One Publisher or download it from Scribd. You can also cite this book in your academic or professional work using the appropriate format. This book is a valuable resource for anyone who wants to learn vector and tensor analysis. 6c859133af


Enter red, green and blue color levels (0-255) and press the Convert button:
Take the first number, 220, and divide by 16. 220 / 16 = 13.75, which means that the first digit of the 6-digit hex color code is 13, or D.
Take the remainder of the first digit, 0.75, and multiply by 16. 0.75 (16) = 12, which means that the second digit of the 6-digit hex color code is 12, or C.
Is RGB better than hex to convert?
There is no informational difference between RGB and HEX colors; they are different ways of communicating the same thing – a red, green, and blue color value. ... (For instance, 255 red would be #FF).
Which hex value would be equivalent to a single R G or B value of 51?
The integer, 3, will be the first number in the hexadecimal formula. The remainder (0.1875) should be multiplied by 16, resulting in the number 3. So, 51 translates to 33 in hex.
How do you color RGB?
White RGB Color. White RGB code = 255*65536+255*256+255 = #FFFFFF.
Blue RGB Color. Blue RGB code = 0*65536+0*256+255 = #0000FF.
Red RGB Color. Red RGB code = 255*65536+0*256+0 = #FF0000.
Green RGB Color. Green RGB code = 0*65536+255*256+0 = #00FF00.
Gray RGB Color. ...
Yellow RGB Color.
Why is 255 RGB?
The reason it is 255 is that, typically, a color is stored in three bytes, or 24 bits, of data. This is convenient, is generally "good enough," and doesn't waste much data. Since each red, green, and blue gets eight data bits, there are 256 possible values. 0 is the lowest, and 255 is the highest.


This Activity Set provides hands-on fun for preschoolers reinforcing colors, shapes and numbers. The double-sided thick foam play mat is textured on one side and printed with numbers and shapes on the other. The pegs can be stacked, strung together on the colored string, or put into the play mat. and provide for imagination and exploration through free play. This peg play can be used as a game for cooperative group activity as well as provide a great way to introduce early math skills These activities stimulate early childhood learning as well as reinfource dexterity, hand-eye coordination, and fine motor skills.
The set features 30 stacking pegs in 6 rainbow colors, two 36 in. colored laces, and suggested activities.
Winner of a National Parenting Publication Award.


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Lyle Muller's research spans neuroscience, computational modeling, and applied mathematics. His work centers on developing new approaches to large-scale recordings of neural population dynamics in cortex. After studying computational neuroscience at Brown University, Lyle completed a PhD in computational/theoretical neuroscience with Alain Destexhe (CNRS Gif-sur-Yvette). He then moved to San Diego for postdoctoral research with Terry Sejnowski in the Computational Neurobiology Laboratory (CNL) at the Salk Institute.
Ján Mináč is a Fellow of the Canadian Mathematical Society and was awarded the 2013 Canadian Mathematical Society Excellence in Teaching Award. Dr. Mináč is an expert in Galois theory, exploring the links between field theory and group theory and examining fundamental notions of symmetry in nature. In collaboration with Dr. Muller, Dr. Mináč is applying and developing techniques in discrete mathematics to understand networked systems in neuroscience.
Maryam is a postdoctoral scholar at Western's Brain and Mind Institute. Her research interests include mathematical modeling and optimization, sequential decision making, data analytics, and machine learning. She received her PhD in operations research from University of Pittsburgh with a focus on decision making under uncertainty and its application in medical decision making. Prior to joining computational neuroscience, she worked as a quantitative modeler and data scientist in the financial sector.
Roberto is a postdoctoral scholar at the Department of Mathematics and Brain and Mind Institute. His research interests consist of computational neuroscience, nonlinear dynamics applied to neuroscience, and systems neuroscience. Roberto has a Ph.D. in Physics from Federal University of Paraná (Brazil) with research related to the synchronization of neural systems. He currently works on persistent neural activity and working memory, focusing on computational analysis and network mechanisms.
Tung is a number theorist by training, having studied heights of motives and special values of zeta functions during graduate work. His present research interests include Galois modules, spectral graph theory, dynamical systems, and computational neuroscience. Tung has a Ph.D. in Mathematics from the University of Chicago. He currently works on developing new mathematical approaches to study networks of nonlinear oscillators and graph spectra.
Gabriel completed his Bachelor of Science with specialized honors in Biophysics at York University and his Master of Science in Medical Biophysics at Western. His Master's thesis is on a novel artifact removal technique for EEG acquired in the presence of fMRI that uses the discrete spectrum of the Schrödinger operator. Gabriel's research interests include the use of signal processing, nonlinear dynamics, and machine learning to answer neuroscientific questions.
Aishwarya completed her Bachelor of Science with specialization in Zoology from Pune University and her Master of Science in Neuroscience from University College London (UCL). Her prior research has focused on using neuroimaging techniques and bioinformatics data analysis to explore the link between brain aging, obesity and Alzheimer's Disease. Aishwarya's research in the lab aims to use signal processing, computational modeling, and neuroimaging to study neural dynamics during sleep, development, and age-related neurodegeneration.
Esther completed her Bachelor of Science with a major in Mathematics at University of Ghana, a Master's degree in Mathematical Sciences at African Institute for Mathematical Sciences-Cameroon, and her Master of Science in Computational Sciences at Laurentian University. She is using her expertise in computational science, graph theory, matrix computations, and high-performance computing to find new, innovative computational techniques and mathematical models for structure in brain networks.
Priya completed her Integrated Master of Science in Mathematics at Tezpur University with a gold medal and her Master of Science in Mathematics at the University of Western Ontario. Her research interests include Network Theory, study of Motifs, and Spectral Graph Theory and wishes to apply these and her experience in Pure Mathematics in the study of Neural Systems.
Mohamad completed his Bachelor of Science with specialization in Human Biology from the University of Toronto. He then completed his Doctor of Medicine at McMaster Univerity, where he is currently completing his residency in Neurosurgery. Through McMaster's Clinician Investigator Program, Mohamad was able to join our lab to pursue a doctorate in computational neuroscience. Mohamad is particularly interested in advancing brain-machine interfaces to empower patients with neurological disabilities
Jacqueline is an undergraduate student pursuing an Honours Specialization in Mathematical and Statistical Sciences at Western. Her research interests are in network theory, random graph theory, and their application to neural networks. In particular, she is currently studying the theory of random matrices and random graphs.
Alex completed her Bachelor degree at Western University with majors in Mathematics and Arts and Humanities. Her previous research experience includes using tools from network theory to better understand the spread of Covid-19. She plans to continue this research, and expand into the study of other large-scale data sets like neural and social networks. In particular, she is excited to research the differences in neural networks between individuals with Autism and healthy controls.
Christopher completed his Honors Bachelor of Science with Specialization in Computer Science from Western University. His research interests are in high performance computing, machine learning, scientific computing, and computer hardware(GPUs, FPGAs), and theoretical computer science. He is working to use his knowledge of computer science and programming to improve and speed up neural network simulations with the newest technology and algorithmic techniques.


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Probability and other paradoxes
|Door 2||Door 3||Result|
3) By opening his door, Monty is actually saying to the player, "There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. Your choice of door 1 has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating Door 2, I have shown you that the probability that Door 3 hides the prize is 2 in 3."
4) (Mine) Choosing a door, you know well that this door has 1/3 chances of hiding the car. The remaining 2/3 chances are with the other two doors. Now, after a door is opened by the host, these 2/3 chances don't change, i.e. the third door has still 2/3 chances of hiding the car.
You can play/test the game Let's Make a Deal here (as long as the page is still alive!). You will find out that:
1) If you always stick to the same door, 'win' and 'lose' results are about the same.
2) If you always switch to the other door, 'win' results are much higher than 'lose'!
I continue with some variations of the Monty Hall problem.
There are 3 playing cards on the table, face-down. One of them is Queen of Hearts (QoH). The game "host" asks you to guess which one is it. OK, so you point card C. Then card B is turned over by the "host" and it isn't a QoH, but a non-QoH). So the situation schematically is the following:
|Card A||Card B||Card C|
At this point, you are asked whether you want to switch to card A or not.
You may think, "I don't have any reason to switch cards became there's a QoH and a non-QoH remaining, so the probability of winning is the same by switching on not, it's 50%."
There are two variations. Let's see what happens in each one.
The "host" knows where the QoH card is. And you know that. In this case, the chance of winning is doubled when you switch to the other card (A) rather than sticking with your original choice (card C), because the "host" turned deliberately an non-QoH card over.
There are 3 possible situations corresponding to your initial choice, each with equal probability (1/3):
1) You originally picked non-QoH #1. The "host" has deliberately shown you non-QoH #2.
2) You originally picked non-QoH #2. The "host" has deliberately shown you non-QoH #1.
3) You originally picked the QoH card. The "host" has shown you either of the two non-QoH cards.
If you choose to switch, you win in the first two cases. If you choose to stay with the initial choice, you will win only in third case. So in 2 out of 3 equally likely cases switching wins, since the odds of winning by switching are 2/3. In other words, a player who has a policy of always switching will win on average two times out of the three.
The "host" doesn’t really know where the QoH card is and you know that. In this case, when the "host" turns the non-QoH card over, the probability that you originally picked the QoH card increases from 1/3 to 1/2. And the odds in this case are equal, whether you change your initial choice or not.
Also known as the "exchange paradox".
You are given two indistinguishable envelopes and you are informed that one of them contains twice as much money as the other. You may select any one of the envelopes at random and you can receive the money it is in it. Now, after you select one of them and before opening it, you are given the opportunity to take the other envelope instead. Should you switch to the other envelope?
You may think, "I don't have any reason to take the other envelope since the chances that it is the one with more money are the same with the chances of keeping the one I have already selected."
Suppose that the amount of money in the envelope you first choose is x. Then the other envelope has 50% chances of containing 2x and 50% chances of containing x/2, i.e. a total of 0.5*2x + 0.5*(x/2) = x + 0.25x = 1.25x. So you should switch.
This, however, creates the following paradox: If one switches, by the same reasoning one has to switch back! And this can go on in an endless cycle! So there must be something wrong with above reasoning. Indeed, two things are wrong:
1) To mix different instances of a variable in the same formula like this is said to be illegitimate, so the above reasoning is incorrect.
2) The expectation of 1.25x holds for either of the envelopes!
A box contains three coins. One is normal, one has two heads and the other has two tails. One con falls out of the box on the floor, heads up. What is the probability that the other side is also a head?
(Think of the solution as long as needed before going on . . .)
The table below contains all possible cases:
|Coin||Side up||Side down|
|2 heads||head #1||head #2|
|2 heads||head #2||head #1|
|2 tails||tail #1||tail #2|
|2 tails||tail #2||tail #1|
In the 3 of the above possible coin cases where you get head up, you have 2 cases of head down and one of tail down. So the chances are 2/3.
(I have simplified the description.)
There are three boxes, each with two drawers. Each drawer contains a coin. One box has a gold coin in each drawer (GG), one box has a silver coin in each drawer (SS) and the other has a gold coin in one drawer and a silver coin in the other (GS). A box is chosen at random, and then one of its drawers is opened. The coin in that drawer is gold. What is the chance that the other drawer also contains a gold coin?
Apparently, it seems that the probability is 50%. However, it isn't! Here's why:
Since a gold coin is found, it means that the box is either GG or GS. These 2 boxes have a total of 3 gold coins and 1 silver coin. If we subtract the gold coin already found, there remain 2 gold coins and 1 silver coin. Hence, the probability that the other coin in the box is also gold is 2/3.
Or, we can think in the following way, examining all three, equally possible cases:
1) Drawer G of GS was chosen, so the other drawer contains a silver coin (1/3)
2) Drawer G1 of GG was chosen, so the other drawer contains also a gold coin (1/3)
3) Drawer G2 of GG was chosen, so the other drawer contains also a gold coin (1/3)
In 2 of the three cases the other coin is also gold, i.e. the probability of this happening is 2/3.
Since gold coins are difficult to get and also for getting rid of the boxes, 3 pairs of regular coins may be used, placed on the table and covered, with heads up and tails up as follows: HH, TT and HT. Or 3 pairs of black and red playing cards face down, etc.
You pay a fixed fee to enter a game of chance in a casino, in which a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 2 dollars and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the second toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, and so on. In short, you win 2^k dollars, if the coin is tossed k times until the first tail appears. What would be a fair price to pay the casino for entering the game?
Let's see what's the average payout: with probability 1/2, the player wins 2 dollars, with probability 1/4 the player wins 4 dollars, with probability 1/8 the player wins 8 dollars, and so on. The expected value is thus EV = 1/2*2 + 1/4*4 + 1/8*8 ... = 1 + 1 + 1 + ... = infinite.
Assuming that the game can continue as long as the coin toss results in heads and that the casino has unlimited resources, this sum grows without bound and so the expected win for repeated play is an infinite amount of money.
Considering nothing but the expected value of the net change in one's monetary wealth, one should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of the game, many people expressed disbelief in the result. It has been found that "few people would pay even $25 to enter such a game" and most commentators would agree. The paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value.
I created and ran a simple computer program to find out what is statistically the average earnings after a large number of rounds. First of all, the average number of tosses, until a tail comes up is found to be 2. Of course, since the first toss is "free", i.e. it doesn't count, since one earns 2 dollars independently of the outcome of the first "toss". Then the 2nd toss has 50% chances to be heads or tails. So each round lasts 2 tosses in average.
Now, here's another paradox within St. Petersburg paradox that I discovered: Since the average of tosses is 2, one would logically expect that the average earnings would be 2^2 = 4 dollars. Although this sounds quite logical, it is very far from the truth. Why?
If the amount of dollars earned after each test were proportional, i.e. 2 dollars were added on each toss, then the average earnings would be indeed 4 dollars. But the amount of earnings does not grow proportionallly but exponentially. Indeed, in the above test, I found that the average earings range from 10 to 25 dollars! So the paradox lies in the big discrepancy between the average number of tosses and the average earnings.
Which of the averages then should be taken as a basis to decide on the amount of the entry fee one would be willing to pay for this chance game? Evidenttly, it's the average earings, which is more real as a statistic and it's what finally counts.Here is the code of the program I used for testing:
DEFINT A -Z DIM tottosses AS LONG, earnings AS LONG, totearnings AS LONG RANDOMIZE TIMER tottosses = 0: totearnings = 0: rounds = 32000 FOR i = 1 TO rounds tosses = 0: earnings = 2 DO tosses = tosses + 1: a = INT(2 * RND) + 1 IF a = 1 THEN earnings = earnings * 2 ELSE EXIT DO LOOP tottosses = tottosses + tosses totearnings = totearnings + earnings NEXT i avtosses = CINT(tottosses / rounds): avearnings = CINT(totearnings / rounds) PRINT 'Average number of tosses, average earnings:'; avtosses; avearnings END
import random random.seed() tottosses = 0; totearnings = 0; rounds = 32000 for i in range(rounds): tosses = 0; earnings = 2 while True: tosses += 1; a = random.randint(1, 2) if a == 1: earnings *= 2 else: break tottosses += tosses; totearnings += earnings avtosses = round(float(tottosses)/float(rounds), 0) avearnings = round(float(totearnings)/float(rounds), 0) print 'Average number of tosses, average earnings:', avtosses, avearnings
If you run it a lot of times, you will see a stable average number of tosses of 2 and an average of earnings, normally lying bewteen 15 and 25 but with big variations each now and then. Don't be surprised if you suddenly see a value that looks like 540!
It is a variation of the "Two envelopes problem" described above.
Two men are given a necktie by their wives as a Christmas present. Over drinks they start arguing over who has the more expensive necktie. They agree to have a wager over it. They will consult their wives and find out which necktie is more expensive. The terms of the bet are that the man with the more expensive necktie has to give it to the other as the prize.
One of the men reasons as follows: Winning and losing are equally likely. If I lose, I will lose the value of my necktie. But if I win, I win more than the value of my necktie. Therefore the wager is to my advantage.
Assuming that both men think equally logically, the other man will think the same thing. So, paradoxically, it seems that both think they have an advantage in this wager. Which is impossible.
What's the flaw in this?
What we call the "value of my necktie" in the losing scenario is the same amount as what we call "more than the value of my necktie" in the winning scenario. Accordingly, neither man has the advantage in the wager. E.g. If the price of one tie is $10 and the other $20, in the losing scenario one loses $20, which is the same amount he wins in the winning scenario.
The Jones family have two children. The older child is a girl. What is the probability that both children are girls?
(Think of the solution as long as needed before going on . . .)
Only two possible cases meet the criteria specified in the question: GB and GG. Since both of them are equally likely, and only one of the two, GG, includes two girls, the probability that the younger child is also a girl is 1/2.
The Jones family have two children. At least one of them is a boy. What is the probability that both children are boys?
(Think of the solution as long as needed before going on . . .)
The possible cases are: GB, BG and BB. Schematically:
|Older child||Younger child|
From the above we can see that the probability that both are boys is 1/3.
A postman knocks on the door of a house where two children live, and the door is answered by a little girl. What are the chances that the other child is also a girl?
How many people do you think it would take to survey, on average, to find two people who share the same birthday? Due to probability, sometimes an event is more likely to occur than we believe it to. In this case, if you survey a random group of just 23 people there is actually about a 50% chance that two of them will have the same birthday. This is known as the birthday paradox. Don't believe it's true? You can test it and see mathematical probability in action!
The birthday paradox, also known as the birthday problem, states that in a random group of 23 people, there is about a 50% chance that two people have the same birthday. Is this really true? There are multiple reasons why this seems like a paradox. One is that when in a room with 22 other people, if a person compares his or her birthday with the birthdays of the other people it would make for only 22 comparisons — only 22 chances for people to share the same birthday.
But when all 23 birthdays are compared against each other, it makes for much more than 22 comparisons. How much more? Well, the first person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 + … +1) the sum is 253 comparisons, or combinations. Consequently, each group of 23 people involves 253 comparisons, or 253 chances for matching birthdays.
Here is a case that I am sure a lot of people have been confused about or at least thought as a kind of paradox.
We toss a fair coin and get heads up. We toss again and get heads up. As this goes on, everytime we feel more and more strongly that its time for tails to show up. That is, there are more and more chances each time for tails to show up. We have reached a large number of tosses, and we are almost certain that the next toss will be tails up.
Coin tossing may be replaced with drawing a card from a well shuffled deck and getting a red card for a number of consecutive drawings, betting on Red in a roulette, etc.
That at it's time for tails to show up and the accumulated frustration of this not happening is a real feeling we get. We think, "What is the probability that heads continue to show up after, e.g. 9 consecutive times of heads up?" Apparently very low, right? "That would be too much", we could think.
But let's cool down and think in pure probability terms.
There is a basic, fundamental law that applies at any time we are tossing a coin, drawing a card from the deck, betting on Red in a roulette, etc. The probability for heads or a red to come up is 1/2. Similarly of course for tails up or a black card. Always. At any time. Independenty of what has happened until then. The Universe does not keep records. We do. And these records help us producing statistics, etc.
Now, let's look the tossing problem from another view. We could do an experiment and say, "We are going to toss a coin 10 times. What is the probability all outcomes to be heads up?" Easy: It's 1/(2^10), i.e. 1/1024 = ~0.1%. And what is the probability to get at least one tails up? It's 9 times greater, i. ~0.9%. So we note down each time the outcome of a tossing. Then we may think that since at least one tails up is 9 times more probable than 10 heads up, it's 9 times more probable that the 10th toss is tails up. And we may believe this strongly.
Well, if we come back to the fundamental law described above, we had to come down to earth and accept that the probability of a 10th heads up is the same with the probability of a 10th tails up, i.e. 50%. That is, getting HHHHHHHHHH is the same as HHHHHHHHHT.
OK, but what about the 9:1 probability as a statistic? Does HHHHHHHHHH invalidates it? No, because it's an extreme case and statistical formulas always provide for deviations from normal. If we were to repeat the experiment foa a large number of times (e.g. 100 times), we would see that on average outcomes are always as expected. And We could also confirm the 9:1 case.
(It is also known as "cocodile paradox". Although its description involves a totally unrealistic situation, as you will see, and it can be described in a hundred of more realistic ways, I use the original version, as it appears in Wikipedia.)
The Crocodile dilemma is an unsolvable problem in logic. The premise states that a crocodile who has stolen a child promises the father that his son will be returned if and only if he can correctly predict what the crocodile will do.
The outcome is logically smooth (but unpredictable) if the father guesses that the child will be returned, but a dilemma arises for the crocodile if he guesses that the child will not be returned: 1)If the crocodile decides to keep the child, he will violate his terms: the father's prediction has been validated, and the child should be returned. 2) If the crocodile decides to give back the child, he still violates his terms, even if this decision is based on the previous result: the father's prediction has been falsified, and the child should not be returned.
Therefore, the question of what the crocodile should do is paradoxical, and there is no justifiable solution.
Relation to "The king and the jester" riddle
The king, who was tired of his jester and looked for an excuse to get rid of him, calls him in one day and says to him: "Say something, anything you want. If what you say is a lie I will hang you and if it is true I will slaughter you."
The jester thought for a while and then he said something to the king. And lived!
What did he say?
(Hover the mouse pointer over here to see the solution.)
Suppose there is a card with statements printed on both sides:
Front: The statement on the other side of this card is TRUE.
Back: The statement on the other side of this card is FALSE.
Trying to assign a truth value to either of them leads to a paradox.
1. Epimenides was a Cretan who made an immortal statement: "All Cretans are liars"
2. "This sentence is false"
3. "I am lying"
4. What happens if Pinocchio says that his nose is about to grow, knowing that it actually grows ONLY when he is telling a lie?
|A painting by the Belgian René Magritte, featuring a smoking pipe and a message at the bottom saying "Ceci n'est pas une pipe." ("This is not a pipe.") Indeed, the painting is not a pipe, but rather an image of a pipe!|
Socrates' famous phrase: "I know one thing, that I know nothing"
At start of the week, a teacher announces to his students that he will give a test this week.
Now, the students think that locically the test cannot be given on Friday because it would be expected since it is the last remaining day. So the teacher would not give the test on Friday. In the same way, it could not give it on Thursday, because it would be expected since it is the last remaining day, excluding of course Friday. And so on ... So, the teacher cannot give an unexpected test!
Wrong! The teacher may well give the test on Friday, and it will be unexpected, since the students, based on the above reasoning, would not expect it on Friday!
Actually, he can give the test any day, since, based on the above reasoning, the students believe it cannot be unexpected anyway!
Consider a heap of sand from which grains are individually removed. One might well do the following thinking: 1) 1,000,000 grains of sand is a heap of sand. 2) A heap of sand minus one grain is still a heap. 3) Repeated applications of (2) (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand.
(From Smullyan's Puzzle Book)
This is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras would accept a young person Euathlus as pupil on credit and that hw will get paid for his instruction after he had won his first case. So, Protagoras waited until Euathlus take on a client, but this never happened. So he decided to sue Euathlus for the amount owed.
Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case.
Euathlus, however, claimed that if he won then by the court's decision he would not have to pay Protagoras. If on the other hand Protagoras won then Euathlus would still not have won a case and therefore not be obliged to pay. The question is: which of the two men is in the right?
Answer: The court decision will be based on what has aleready happened, not on what will happen after the court decision is made. So, actually, there's no case at all, since Euathlus has not yet tried to win a case!
Now, here's another thing that Protagoras could do: He should let Euathlus win the case and then sue him again, in which case he should legally get his money since Euathlus would have won his first case!
(This seems a better and more useful math problem. But then, it wouldn't be a paradox!
This is something I have thought in 2004 and it can be seen as simply a problem (successful or defective) method or a paradox.
You bet one chip on Red. If Red comes out, you keep the earned chip and you bet another chip, always on Red. If Black comes out, you bet 2 chips on Red in the next round. If Red comes out, you get 4 chips back, so have earned one more chip (after subtracting the 3 chips you have bet in total). You keep the earned chip and you bet another chip, always on Red. In short, the method can be summarized as follows: "Whenever you win, you keep the earned chip and whenever you loose you double the amount of the previous bet." In this way, you always earn one chip!
Can this be possible? Can this method work on a standard basis?
I have test this by playing roulette games (online and offline) and also with a program I created that produced statistics. Well, After about 5 minutes (i.e. about 100 roulette rounds), I always was over my initial aount (capital). (To this, one has to add the extra 1/37 changes in each round in which both Red and Black lose. But it's too small a probability to affect the results.) So, it looks like it's successful method, insn't it?
However! (There's always a "but" in these cases!:)
1. If this worked, casinos would have closed down! True, but this doesn't say were the problem is, i.e. what is the factor or factors that make this method non-succcessful.
2. There are betting limitations, different in ach casino (which you can see near the roulette table). The most important of them in our case is maximum bet. The lower this limit is the more chances are that you may loose a lot of chips because you couldn't double the bet anymore! (BTW, it is assumed that there's limit of the amount of money you can bet.)
3. Each roulette round lasts, what?, 5 minutes (from that start of bettings to the stop of the ball on the roulette wheel. Earning one chip each five mminutes is quite tiresome ... 6 chips in about one hour!
4. You will be eventually thrown out of the casino!
5. At a certain point of an unlucky streak, the betting sum will be too large compared to the expected gain (one chip).
6. Finally, the probability of going bankrupt, as small as this may be, compared to the small amount of money you can earn, is quite discouraging.
But you can always do that for fun ...
What is better than eternal bliss? Nothing. But a slice of bread is better than nothing. So a slice of bread is better than eternal bliss.
Your mission is not to accept the mission. Do you accept?
(Variation: Your order is not to execute the order. Would you execute it?)
Answer truthfully ("Yes" or "No"): Will the next word you say be "No"?
If the temperature this morning is 0 degrees and the Weather Channel says, "It will be twice as cold tomorrow", what will the temperature be?
(This is very interesting ... Even with -2 degrees ... -4 degrees does not feel twice as cold, it's only slightly colder.)
Everyday life statements:
• Nobody goes to that restaurant; it's too crowded.
• Don't go near the water 'til you have learned how to swim.
• The man who wrote such a stupid sentence can not write at all.
I will be glad to receive your comments and suggestions at email@example.com


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Panagiotis Grammatikopoulos, in the Okinawa Institute of Science and Technology Graduate University (OIST) Nanoparticles by Design Unit, simulates the interactions of particles that are too small to see, and too complicated to visualize. In order to study the particles’ behavior, he uses a technique called molecular dynamics. This means that every trillionth of a second, he calculates the location of each individual atom in the particle...
Physicists have identified the “quantum glue” that underlies a promising type of...
Some 15 percent of adults suffer from fertility problems, many of these due to genetic factors....
Collecting Just the Right Data: When you can’t collect all you need, new algorithm tells you which to targetJuly 28, 2014 2:06 pm | by Larry Hardesty, MIT | News | Comments
Much artificial-intelligence research addresses the problem of making predictions based on large...
Music fans and critics know that the music of the Beatles underwent a dramatic transformation in just a few years. But, until now, there hasn’t been a scientific way to measure the progression. Computer scientists at Lawrence Technological University have developed an artificial intelligence algorithm that can analyze and compare musical styles, enabling research into their musical progression.
String bends, tapping, vibrato and whammy bars are all techniques that add to the distinctiveness of a lead guitarist's sound, whether it's Clapton, Hendrix, or BB King. Now, guitarist and physicist Dr. David Robert Grimes has described the physics underlying these techniques.
Scientists from Queen Mary University of London have found a successful way of identifying bird sounds from large audio collections, which could be useful for expert and amateur bird-watchers alike. The analysis used recordings of individual birds and of dawn choruses to identify characteristics of bird sounds. It took advantage of large datasets of sound recordings provided by the British Library Sound Archive, and online sources.
Mathematical equations can make Internet communication via computer, mobile phone or satellite many times faster and more secure than today. Results with software developed by researchers from Aalborg University in collaboration with the Massachusetts Institute of Technology (MIT) and California Institute of Technology (Caltech) are attracting attention in the international technology media.
How using CPU/GPU parallel computing is the next logical step - My work in computational mathematics is focused on developing new, paradigm-shifting ideas in numerical methods for solving mathematical models in various fields. This includes the Schrödinger equation in quantum mechanics, the elasticity model in mechanical engineering, the Navier-Stokes equation in fluid mechanics, Maxwell’s equations in electromagnetism...
With over 700 new functions — the single biggest jump in new functionality in the software's history — Mathematica 10 is the first version of Mathematica based on the complete Wolfram Language. Integration with the Wolfram Cloud and access to the expanded Wolfram Knowledgebase open up new possibilities for intelligent computation and deployment.
Discovering a brain tumor is a very serious issue, but it is not the end of the story. There are many different types of brain tumor with different survival rates and different methods for treatment. However, today, many brain tumors are difficult to clearly diagnose, leading to poor prognoses for patients.
Fully automated “deep learning” by computers greatly improves the odds of discovering particles such as the Higgs boson, beating even veteran physicists’ abilities.
A new type of steel-reinforced concrete protects buildings better from bomb attacks. Researchers have developed a formula to quickly calculate the concrete’s required thickness. The material will be used in the One World Trade Center at Ground Zero.
To be able to use huge amounts of data, we have to understand them and before that we need to categorize them in an effective, fast and automatic manner. Two researchers have devised a type of Cluster Analysis, the ability to group data sets according to their "similarity," based on simple and powerful principles, which proved to be very efficient and capable of solving some of the most typical problems encountered in this type of analysis.
Mechanical engineers at the Babol University of Technology in Mazandaran, Iran, have turned to nature to devise an algorithm based on the survival trials faced by salmon swimming upstream to the spawning grounds to help them fish out the optimal solution to a given problem.
Inspired by anomalies that arise in certain mathematical equations, researchers have demonstrated a laser system that paradoxically turns off when more power is added rather than becoming continuously brighter. The finding could lead to new ways to manipulate the interaction of electronics and light, an important tool in modern communications networks and high-speed information processing.
Dinosaurs and mathematics do not seem like an obvious pair, but for Robert Sinclair and his Mathematical Biology Unit, they are a logical match. Sinclair was part of a team that recently published a paper reexamining the classification of a dinosaur bone found. Using his expertise in mathematics, Sinclair was able to help reclassify a single arm bone as belonging to a dinosaur family previously believed not to have existed ...
There are some 100 million other places in the Milky Way galaxy that could support complex life, report a group of university astronomers in the journal Challenges. They have developed a new computation method to examine data from planets orbiting other stars in the universe. Their study provides the first quantitative estimate of the number of worlds in our galaxy that could harbor life above the microbial level.
An historic milestone in artificial intelligence set by Alan Turing — the father of modern computer science — has been achieved. The 65 year-old iconic Turing Test was passed for the very first time by supercomputer Eugene Goostman during Turing Test 2014 held at the Royal Society in London on June 7, 2014, and organized by the University of Reading.
All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Logical arguments like this one have been studied since antiquity. In the last few decades, however, logic research has changed considerably: the computer sciences were born. The success of informatics would have been impossible without the groundwork provided by logicians — and, in turn, computer sciences keep posing new interesting questions
If a bulky electrical box has to be placed at the edge of a public park, what’s the best way to conceal it so that it won’t detract from its surroundings? How about an air-conditioning condenser beside a historical building, or a portable toilet along a scenic trail?
Now at Mark 24, the NAG C Library is a collection of hundreds of user-callable mathematical and statistical functions for C and C++ programmers. It contains over 1,500 powerful algorithms that are designed to be reliable, flexible and ready-for-use from a wide range of operating systems, languages, environments and packages
Researchers have solved one aspect of the discrete logarithm problem. This is considered to be one of the 'holy grails' of algorithmic number theory, on which the security of many cryptographic systems used today is based. The team has devised a new algorithm that calls into question the security of one variant of this problem. The result discredits several cryptographic systems ...
Mathematicians have found a framework for the celebrated Rogers-Ramanujan identities and their arithmetic properties, solving another long-standing mystery stemming from the work of Indian math genius Srinivasa Ramanujan. The findings, by mathematicians at Emory University and the University of Queensland, yield a treasure trove of algebraic numbers and formulas to access them.
Scientists studying graphene’s properties are using a new mathematical framework to make extremely accurate characterizations of the two-dimensional material’s shape. Graphene, discovered in 2004, is a one-atom-thick sheet of graphite. The mathematical framework being used is known as discrete differential geometry, which is the geometry of two-dimensional interlaced structures called meshes.
As modern computer systems become more powerful, utilizing as many as millions of processor cores in parallel, Intel is looking for new ways to efficiently use these high performance computing (HPC) systems to accelerate scientific discovery. As part of this effort, Intel has selected Georgia Tech as the site of one of its Parallel Computing Centers.
Vladimir Voevodin is Deputy Director of the Research Computing Center at Lomonosov Moscow State University (1998-present) where he is responsible for supercomputing facilities and activities of RCC MSU. He is also a professor at the Computational Mathematics and Cybernetics Faculty of MSU (2005-present) and a professor at the Mechanic-mathematical Faculty of South-Ural State University (2004-present).
Dr. Koshulko is a senior research associate at Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine. He holds a Ph.D. in mathematical modeling and numerical methods. His professional interests include parallel processing, machine learning, time series analysis and forecasting. Dr. Koshulko serves as program committee member of several international HPC conferences and is a chair of the HPC-UA conference in Kyiv.
Horst Gietl studied Mathematics and Information Technology at the Technical University in Munich and made his PhD at the Faculty of Mathematics. He worked at the Leibniz Computing Center in Munich, being responsible for a project of the German Research Organization with the title: „Vectorization of numerical applications to get highest performance out of a vector processor.” After that, he joined Siemens, becoming a manager for the vector processor applications/support group and working with Fujitsu’s vector processors.
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Picture generated with OpenAI's DALL·E 2 using the following sentence "drawing of a female scientist breaking code by lighting the signal as van gogh"
By Nina Bindel
Key exchange is one of the fundamental building blocks that Internet security is built upon. The goal of key exchange is for two parties – often called Alice and Bob – to do computations such that at the end they agree on a shared secret key. This key is then used to encrypt messages using symmetric encryption. In this post we will explain the core idea behind a recently published attack against one kind of key exchange protocol. Specifically, we will discuss “signal leakage attacks” against key exchange protocols, which are based on the Learning with Errors (LWE) problem under “key re-use”.
The respective paper “Light the Signal: Optimization of Signal Leakage Attacks against LWE-Based Key Exchange” by Yue Qin, Ruoyu Ding, Chi Cheng, Nina Bindel, Yanbin Pan, and Jintai Ding has been accepted at ESORICS 2022 (September 26-30, 2022, Copenhagen, Denmark) and will be presented virtually on Monday, September 26. The paper can be accessed on IACR eprint 2022/131.
One of the earliest and arguably simplest key exchange protocols was designed by Diffie, Hellmann, and (in an independent work) Merkle in 1976. The idea is that Alice and Bob each choose secrets sA and sB, respectively. For simplicity it is sufficient to think of these elements as integers although they are actually ‘group elements’ in a group ‘generated’ by an element g. Then they transmit g raised to the power of the respective secrets to each other. These transmitted public values are known as pA and pB. The shared secret key is then computed by taking the received public value again to the power of their own secrets. The security of this protocol is based on the discrete logarithm problem. That means, although the values g, pA and pB are publicly known, it is not possible to compute sA or sB from them. While this is true for our current computers, large fault-tolerant quantum computers will be able to solve the discrete logarithm problem, thereby breaking the security of the Diffie-Hellmann-Merkle key exchange and many other of today’s cryptographic algorithms (visit our blog post for more information about the quantum threat and actions to maintain security in the quantum era).
Alternative protocols that share many similarities are key exchange protocols based on the so-called Learning with Errors problem. As before, Alice and Bob choose secrets. This time the secrets are polynomials with small coefficients, e.g., sB = 1+2x-3x2-x213+4x317-3x511. Using a publicly-known polynomial a (similar to the element g above) and their respective secrets, they compute public values pA and pB and send them to each other. They are then able to compute values hA and hB which are ‘approximately’ the same. To be able to compute ‘exactly’ the same secret key, Bob needs to send additional information. This information is called the signal wB. The signal consists of a list of zeros and ones – as many as the secret has polynomial coefficients. It can be used, together with hA or hB, within a function KeyF that outputs the final key.
Unfortunately, this extra information allows for attacks to recover Bob’s secret sB if Bob does not change his secret ‘every’ time, i.e., when he ‘re-uses’ his secret. This kind of attack is called ‘signal leakage attack under key re-use’ and was first described by Fluhrer and later carried out against the LWE-based DXL key exchange by Ding, Xie, and Lin. The idea is that an adversary pretending to be an honest user sends malformed pA. In particular, instead of a polynomial, the adversary sends q different integers, namely pA=0, pA=1, …., pA=q-1 for some system parameter q. By construction of the signal, it turns out that each element of the signal “flips” between zero and one twice as often as the absolute value of the corresponding secret coefficient. So by tracking the behavior of the signal as pA varies, the adversary can learn about the secret.
Let’s take a look at an example. Assume the adversary wants to find out the third coefficient of the secret polynomial sB = 1+2x-3x2-x213+4x317-3x511 from above. To do this, the adversary sends pA=0 to Bob, and stores the third value of the signal, which corresponds to the signal for the third polynomial coefficient. Looking at the diagram below, we can see that this signal value is equal to zero. Then the adversary sends pA=1, and again stores the third signal value, which is again equal to zero. As visualized in the diagram, the signal value changes from zero to one for the first time for a value around pA=1800, and back for pA=4000. Increasing pA further until pA=16384 leads to six ‘main’ switches. Therefore, the absolute value of Bob’s corresponding third secret coefficient is equal to 3. The adversary has to do this procedure for ‘all’ polynomial coefficients, which are usually 512 or 1024. In addition, the adversary needs to do a similar procedure to find out the sign of the secret coefficient (e.g., in the example above the secret coefficient is -3).
Zooming into one of the switches reveals that for a few values the signal actually fluctuates. This is caused by the random element eB. Based on this, one can distinguish between ‘stable’ (blue or yellow in the figure below) and ‘unstable’ (wavy white) regions. To count the number of stable and unstable regions correctly, and as such to be able to count the number of ‘main switches’ correctly, it is not necessary to request q different values for pA. Instead Bindel, Stebila and Veitch show that it suffices to make sure to choose at least one value in each stable and at most one value for each unstable region for every possible absolute secret value. For example in the picture below, assuming that the absolute value of the secret coefficient is at most 3, the green values for pA=ki are sufficient to determine whether the secret coefficient is equal to 1,2 or 3. This way, it was possible to reduce the number of needed values for pA from a total of 98310 to only 1266 to recover the secret used in the DXL key exchange.
Recently, Qin, Ding, Cheng, Bindel, Pan, and Ding were able to reduce this number further to only 29 values! The idea for this latest improvement is to choose the values even more cleverly. As can be seen in the above figure, actually the three red values are sufficient to uniquely identify the absolute value of the secret coefficient. The idea is to determine codewords that uniquely correspond to possible absolute values. For example, if the code word is k1-k2-k3 = 0-1-1, the absolute secret value is equal to 1. And if the code word is 1-1-0 or 1-0-1, we know the absolute secret is 2 or 3, respectively.
This new interpretation of the signal leakage not only reduces the number of required values (therefore making the attack more efficient and more practical), but it also makes it easier to discover whether schemes can be broken by this kind of attack.
Lastly, it is important to mention that signal leakage attacks can be prevented using generic constructions of ‘authenticated’ key exchange protocols. It is, however, desirable to instead construct key exchange directly from a hard problem such as the Learning with Errors problem, e.g., to achieve better efficiency. Therefore, building such ‘direct’ authenticated key exchange protocols is attempted and failed regularly. The new improvements on the signal leakage attacks caution once more against underestimating the power of signal leakage attacks when attempting such constructions.


The holiday season is officially here and we could NOT be more excited about it. At TMG we go ALL OUT for the holidays. Right now our tutoring spaces are covered in beautiful lights and candles that smell like candy canes and gingerbread and learning and math. Sort of. The “24 days of TMG” (an advent-calendar of brain-teasers and math and science related problems) is in full-swing and the overwhelming response is a sure sign that people SECRETLY love math:) Our space is SO incredibly decked out that it feels as though we're basically in the North Pole. And it's freezing outside, so that helps. For me, gift-giving is so much fun because I LOVE giving someone the perfect gift! For others, though, I understand that gift-giving can be stressful because oftentimes it's difficult to figure out what someone wants. If you’re having trouble thinking about what to get that math and science loving person for the holidays, look no further! We said we were pumped about the holidays and we mean it. We want to help YOU find the BEST, COOLEST and CUTEST gifts for that math and science loving someone. And so, we're pumped to present....5 Awesome Gifts for Math and Science Lovers!
- The Perfect Mug
Who wouldn’t want to drink their coffee out of a mug that displays its chemical properties? While getting your daily dose of caffeine, study this super cute mug and learn all about caffeine’s chemical molecule. This mug is a great gift for a coffee-addicted, anecdote-loving individual.
- The Science of Spice
Spice up someone’s kitchen with this super-cool spice rack. By turning cooking into a mad science experiment, this gift can totally be the perfect laboratory for the person who loves cooking AND science.
- This Totally Adorable Tote
The cuteness of this tote is just CRAZY. An adorable image of an angle, coupled with a stellar pun, makes this tote a great buy for anyone who loves a good math pun.
- The Cutest Curtain
Have I mentioned that we LOVE puns at TMG? Well, we do. We love them A LOT. Perfect for your special someone, this shower curtain adorably displays your affection for your perfect ten.
- The Sassiest T
Perfect for the math genius know-it-all, this t-shirt says it all. Andddd...also perfect for the person who needs a little self-esteem boost, this t-shirt is a great test-taking good-luck charm.
6. Bonus bonus bonus
Okay so we said this was our Top 5, but there's a sixth! We've JUST opened up our TMG Gift Shop! The perfect one stop holiday shop, we've got cozy sweats and cute totes, but best of all...math-inspired chokers and necklaces


Conic optimization is a significant and thriving research area within the optimization community. Conic optimization is the general class of problems concerned with optimizing a linear function over the intersection of an affine space and a closed convex cone. One special case of great interest is the choice of the cone of positive semidefinite matrices for which the resulting optimization problem is called a semidefinite optimization problem.
Semidefinite optimization, or semidefinite programming (SDP), has been studied (under different names) since at least the 1940s. Its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve SDP problems (and more generally, to solve convex optimization problems with efficiently computable self-concordant barrier functions). Some of the earliest applications of SDP that followed this development were the solution of linear matrix inequalities in control theory, and the design of polynomial-time approximation schemes for hard combinatorial problems such as the maximum-cut problem.
The objective of this Handbook on Semidefinite, Conic and Polynomial Optimization is to provide the reader with a snapshot of the state of the art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. Our intention is to provide a compendium of the research activity that has taken place since the publication of the seminal Handbook mentioned above. It is our hope that this will motivate more researchers, especially doctoral students and young graduates, to become involved in these thrilling areas