"No wonder life (i.e., the thing that my once 10-year old niece referred to as “the thing that isn’t fair”) comes to us as a filigree of ash stories. Walking down the street past a couple in conversation, an overheard morpheme, a mere glance at a wrongly buttoned raincoat, sparks a narrative in our imagination. Ask any question beginning with “why?” and the answer will surely be a story, or it will be embedded in a story. Or, at the very least, it will offer a tempting thread for some story that you yourself will hold onto, embellish even, as you try to absorb the answer. We interpolate between such fragments. This is, for many of us, simply the way we think.
What about the “why questions” in science, in logic, in mathematics? We should acknowledge how they are often “what questions” or “how questions” in disguise. Or how they slide down into such questions, as the ever-elusive, ever-illusory quest for an X that actually causes a Y dissolves. Some of the more satisfying answers to scientific “why” questions involves deft rephrasing. “Why is the sky blue?” is replaced by the question “what is the function that describes scattering amplitude as dependent on wave-length”?"

"Plotting solution sets for particular equations may be complicated by the existence of turning points. Here we describe an algorithm which not only overcomes such problematic points, but does so in the most general of settings. Applications of the algorithm are highlighted through two examples: the first provides verification, while the second demonstrates a non-trivial application. The latter is followed by a thorough run-time analysis. While both examples deal with bivariate equations, it is discussed how the algorithm may be generalized for space curves in $R^{3}$."

"The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for."

"Suppose someone takes a solved 20x20x20 Rubik's cube and makes five moves - can you figure out [from that scrambled state] what those five moves were?" he asks. In other words, can you solve it in five moves? He suspects that you cannot, but has yet to prove it. "We don't know."

"We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a powerful tool to extend algorithms and heuristics for sparse approximation theory to these problems."

"Our 2.546-approximation is quite simple. The performance guarantee is based on a simple area argument. This gives rise to the following question: what is the smallest square that suffices for packing any set of circles of total area 1? We believe the worst-case may very well be shown in Figure 13, which yields a lower bound of 1.471299... We believe there are relatively easy ways to improve the upper bound."

"Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR} and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We introduce and study a class of functions with the following property: Instead of evaluating an extension of a boolean function on a given set of transients, it is possible to get the same value by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated."

"We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witnesses W in the plane, there is an edge ab between two points of P in the witness Gabriel graph GG-(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing."

"We provide several basic results. We obtain an efficient algorithm for determining the heapa- bility of a sequence, and also prove that the question of whether a sequence can be arranged in a complete binary heap is NP-hard. Regarding subsequences we show that, with high probability, the longest heapable subsequence of a random permutation of n numbers has length (1 − o(1))n, and a subsequence of length (1 − o(1))n can in fact be found online with high probability. We similarly show that for a random permutation a subsequence that yields a complete heap of size αn for a constant α can be found with high probability. Our work highlights the interesting structure underlying this class of subsequence problems, and we leave many further interesting variations open for future work."

"We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n^3) time for polygons of n vertices."

seems like something to look at

"Ron Doerfler has created a truly gorgeous 2010 calendar titled The Age of Graphical Computing. Ron has transformed nomography into a form of art."

"The paper is written in the form of introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations of lines in general position in the three-dimensional space the exposition is free from any background restrictions. We have added few new results, fixed few misprints and terminological inaccuracies and expanded the reference list. Notice that some of the results presented in the paper appeared in other papers without appropriate references."

"Motivated by constraint-based CAD software, we develop the foundation for the rigidity theory of a very general model: the body-and-cad structure, composed of rigid bodies in 3D constrained by pairwise coincidence, angular and distance constraints. We identify 21 relevant geometric constraints and develop the corresponding infinitesimal rigidity theory for these structures. The classical body-and-bar rigidity model can be viewed as a body-and-cad structure that uses only one constraint from this new class. As a consequence, we identify a new, necessary, but not sufficient, counting condition for minimal rigidity of body-and-cad structures: nested sparsity. This is a slight generalization of the well-known sparsity condition of Maxwell."

awesome

"We consider the problem of developing an efficient multi-threaded implementation of the matrix-vector multiplication algorithm for sparse matrices with structural symmetry. Matrices are stored using the compressed sparse row-column format (CSRC), designed for profiting from the symmetric non-zero pattern observed in global finite element matrices. Unlike classical compressed storage formats, performing the sparse matrix-vector product using the CSRC requires thread-safe access to the destination vector. To avoid race conditions, we have implemented two partitioning strategies. In the first one, each thread allocates an array for storing its contributions, which are later combined in an accumulation step. We analyze how to perform this accumulation in four different ways.…"

"We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound.…"

"With some modifications, the proof movie can be used as the data structure underlying an encyclopedia that we envisage containing formal proofs together with an informal narrative explanation, and provide a toolbox for using and manipulating such composite “articles”…"

"The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916 [23]; see also [25], [2]. Let C be a convex body in the plane. Following Bagemihl [2], we call a set B an opaque set or a barrier for C, if it meets all lines that intersect C. A barrier may consist of one or more rectifiable arcs. It does not need to be connected and its portions may lie anywhere in the plane, including the exterior of C; see [2], [4].

What is the length of the shortest barrier for a given convex body C? In spite of considerable efforts, the answer to this question is not known even for the simplest instances of C, such as a square, a disk, or an equilateral triangle…"

"We describe a search for primitive trinomials of high degree and its interaction with the Great Internet Mersenne prime search (GIMPS). The search is complete for trinomials whose degree is the exponent of a Mersenne prime, for all 47 currently known Mersenne primes."

"What is it that mathematicians accomplish?"

"Computational Complexity studies the power and limitations of efficient computation. So is efficient computation purely an abstract mathematical object or is it trying to model a real world phenomenon? I would argue the latter. Efficient computation occurs not just in computers but in biological systems, physical systems, chemical systems, economic systems and much more. Physics focuses on the "what", computational complexity on the "how"."

"… The results obtained are in good agreement with numerical simulations and reproduce random networks behavior in the appropriate limits. Finally, it is demonstrated that our model predicts higher epidemic thresholds for clustered structures than for random topologies."

"Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context."

"Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there also is the task of maintaining the repository. Thus it seems obvious to keep a repsoitory as small as possible, in particular each piece of mathematical knowledge should be formalized only once. In this paper, however, we claim that it might be reasonable or even necessary to duplicate knowledge in a mathematical repository. We analyze different situations and reasons for doing so and provide a number of examples supporting our thesis."

"Recent research on multiple kernel learning has lead to a number of approaches for combining kernels in regularized risk minimization. The proposed approaches include different formulations of objectives and varying regularization strategies. In this paper we present a unifying general optimization criterion for multiple kernel learning and show how existing formulations are subsumed as special cases. We also derive the criterion's dual representation, which is suitable for general smooth optimization algorithms. Finally, we evaluate multiple kernel learning in this framework analytically using a Rademacher complexity bound on the generalization error and empirically in a set of experiments."

"One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in math- ematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory."

"This is the beginning of a project1 to record open problems of interest to researchers in computational geometry and related fields. It commenced with the publication of thirty problems in Computational Geometry Column 42 [MO01] (see Problems 1-30), but has grown much beyond that. We encourage correspondence to improve the entries; please send email to TOPP@cs.smith.edu. If you would like to submit a new problem, please fill out this template."

"Some feel powerless because they have no control over their lives and are doing poorly at their own classes and need to demonstrate power (and their self-belief of their superior mathematical skillz) in the only venue they have, their class. Some of the rest were picked on by business majors when they were undergrads. Some of the rest really don't believe that it's possible for an educated person to be as bad at algebra as the students who attend this university."

"GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. See also the overview and the description of the mathematical capabilities. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. The system, including source, is distributed freely. You can study and easily modify or extend it for your special use."

"In the spirit of sparse representation, we chose to use a syntax tree as the internal data structure for our symbolic calculator. A syntax tree is a kind of tree, which in turn is a kind of linked data structure. Briefly, a linked data structure is an object which contains references, or links, to other like objects. A simple example is a linked list, where each element contains the data for it list entry and a link to the next list element. A tree is a linked structure that starts with a single "root" node. One or more "child" nodes are referenced from the root node, and each of these child nodes may in turn have children of their own. This linking pattern produces a branching data structure, as seen in the following diagram; hence the name "tree"."

"Using total variation based energy minimisation we address the recovery of a blurred (convoluted) one dimensional (1D) barcode. We consider functionals defined over all possible barcodes with fidelity to a convoluted signal of a barcode, and regularised by total variation. Our fidelity terms consist of the L^2 distance either directly to the measured signal or preceded by deconvolution. Key length scales and parameters are the X-dimension of the underlying barcode, the size of the supports of the convolution and deconvolution kernels, and the fidelity parameter. For all functionals, we establish regimes (sufficient conditions) wherein the underlying barcode is the unique minimiser. We also present some numerical experiments suggesting that these sufficient conditions are not optimal and the energy methods are quite robust for significant blurring."

"The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we significantly improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we analyze different heuristics for computing graph partitions tailored to adaptive waveform relaxation."

"Created by Cristóbal Vila, this video manages to explain how everything, from snails to flowers, is built according to predictable mathematical principles. Numbers have never been more sublime."

"…We can also consider what happens when instead of only considering a single type of shuffling we consider combining the j! different shuffling rules that come from all the possible rearrangements of the ordering of the labels. And of course, perhaps the most important thing missing right now is a good magic trick that can be performed using this shuffling rule, which was the original motivation of Larry Carter and J.-C. Reyes who first suggested this problem!"

"NArray is an Numerical N-dimensional Array class. Supported element types are 1/2/4-byte Integer, single/double-precision Real/Complex, and Ruby Object. This extension library incorporates fast calculation and easy manipulation of large numerical arrays into the Ruby language. NArray has features similar to NumPy, but NArray has vector and matrix subclasses."

"Mathematics illuminates the patterns and structures all around us. Our dynamic exhibits and programs will stimulate inquiry, spark curiosity, and reveal the wonders of mathematics."

"NArray is an Numerical N-dimensional Array class. Supported element types are 1/2/4-byte Integer, single/double-precision Real/Complex, and Ruby Object. This extension library incorporates fast calculation and easy manipulation of large numerical arrays into the Ruby language. NArray has features similar to NumPy, but NArray has vector and matrix subclasses."

"In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic 3D-space. Here, we get such a cellular automaton with three states only."

"The 'Primer' is a well-known Life pattern used to calculate prime numbers. The pattern expands in two directions, resembles a breeder, and emits a stream of spaceships representing prime numbers. The presence or absence of a spaceship at a particular generation indicates whether the number is prime or composite. It works by testing whether each integer is divisible by any smaller integer, apart from itself and 1. This is similar in principle to the Sieve of Eratosthenes."

"Suppose you are given a simple closed curve in the plane. (“Simple” means the curve does not intersect itself, “closed” means the curve ends at the same point where it begins.) Can you always find four points on the curve which form the vertices of a square?"

Been here, met these people, and laughed in their faces: "This is just like when teachers ask their students to model or code parts of a system that will be used in the teachers own research eventually. this is really bad for academia in general. Never again propose such things, please." I'm looking at you, winning bidder on the Erdös auction

"Most of Gowers recommendations seem to focus on alignment and cohesion, but point 8, for example, is about separation: one member should not jump onto another's idea with a full proof until there is group support for that. It would be worthwhile to examine the list in detail from the perspective of the three rules of flocking, I bet."

"REvolution Computing offers open source products and services for high performance analytics, including REvolution R Enterprise which delivers 100% R and more—optimized, validated and supported."

"There is no required textbook for this class; I will post electronic copies of relevant papers to this web site as the course progresses. Meanwhile, here is a list of background references, primarily surveys and textbooks. Key references for the course are hilighted. Many of the other references focus on material that we will not cover at all in the course; I include them primarily to give some sense of the diversity of the field."

"The dashed circles above represent "forbidden regions" in which one of the angles would be obtuse. As Lindgren and Cassidy and Lord showed, eight triangles is best possible, and there exist alternate solutions with any even number of triangles larger than eight.

Recently, John Tromp added a new twist to the problem by asking on sci.math how to make the angles as acute as possible. For the eight-triangle solution, he found a placement of the vertices in which the maximum angle is only about 85 degrees, and asked if more triangles would achieve even better angles."

"When modeled in the AMPL modeling language, optimization problems may be examined by a set of tools found in the AMPL Library. Dr. Ampl is a meta solver which, by use of the AMPL Library, dissects such models, obtains statistics on their data, is able to symbolically prove or numerically disprove convexity of the functions involved and provides aid in the decision for an appropriate solver. A problem is associated with a number of appropriate solvers available on the NEOS Server for Optimization by means of relational database."

"A synchronizing word w for a given synchronizing DFA is called minimal if no proper prefix or suffix of w is synchronizing. We characterize the class of synchronizing automata having finite language of minimal synchronizing words (such automata are called finitely generated). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n ¿ 5. We also prove that checking whether a given DFA $\mathcal{A}$ is finitely generated is co-NP-hard, and provide an algorithm for this problem which is exponential in the number of states $\mathcal{A}.$"

"WOB combines a database of bifurcation problems with a tutorial on nonlinear phenomena.

WOB is designed to be part of a virtual university. The approach is example-oriented and experimental. The emphasis is on examples that are application-oriented."

"The puzzle: The puzzle is to find the flaw, the erroneous step, in the switching argument above. This includes determining exactly why and under what conditions that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox."

"The purpose of this site is to try to provide information on the various polyforms: poly-ominoes, -iamonds, -hexes, -cubes etc. There is a great deal of information already on the internet and these pages will try to provide links to all (well hopefully most!) relevant sites. Where no information seems available elsewhere as much information as possible will be posted here."

"The following pictures show n points in the plane so that the ratio r of maximum distance (shown in red) to minimum distance (shown in blue) is the smallest known."

"This is the main blog page for the "Deterministic way to find primes" project, which is currently fairly active already, and should be formally launched within a few weeks.
The main aim of the project is to resolve the following conjecture:
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k. You may assume as many standard conjectures in number theory (e.g. the generalised Riemann hypothesis) as necessary, but avoid powerful conjectures in complexity theory (e.g. P=BPP) if possible."

"Oddly enough we would usually prefer a probabilistic over the deterministic method to find primes. Otherwise the adversary can use the same deterministic procedure and factor your number as easily as you put it together."

"Dean Hickerson's original block-deleting 2c/3 termination almost certainly wasn't designed with this in mind, but it happens to absorb a double-length signal in exactly the same way as a standard signal -- the final stable state is the same in either case. This means that communication speeds approaching 2c/3 can be implemented over long distances in any direction, not just diagonally.

In the accompanying diagram, the input Herschel signal is circled in red. The output signal can be any of a number of optional glider outputs in the Herschel circuit at the bottom.

Two elbows in a row will not work (there's no known way to turn a double-length 2c/3 signal). But in the absence of layout constraints, a single elbow is sufficient to send a 2c/3 signal anywhere in the universe."

"Welcome to Al Zimmermann's Programming Contests. You've entered an arena where demented computer programmers compete for glory and for some cool prizes.

I run one or two contests per year. Each contest asks that you come up with your best solutions to a set of related computationally intensive problems. Although I speak of "programming contests", technically you don't need to write a computer program to enter. You can enter whether you use a computer, manual calculations, or tea leaves to solve the problems. You send me solutions, not programs."

Definitely some targets for genetic programming

Would like to train the squares to self-assemble to do the covering "themselves", while minimizing the number needed.

[more raw material for Nudge project]

"After a bit of work we believe we have solved most of the practical problems that have to be taken care of before starting a free journal. This is probably the easy part. Now we have to decide if it is a good idea or not.

The aim is to have a high quality journal for the CG community that is run by the CG community and free to everyone (really free, no cost to publish and no cost to access). Obviously such a journal needs the support of the CG community to be successful. The work should be shared among the community, i.e., the editorial board and editorial manager(s) should be replaced regularly. "

Interesting prospect for Nudge applications

Maarten's challenge to the Nudge language

READ THIS. All of you.

"But I’m actually thinking that there are some lessons here that the mathematics community could learn from..."

Tempting to fire up Ruby right now