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The Truth Can Be Messy

Writing in the preview issue of Nautilus, Samuel Arbesman discusses mistaken belief in the ubiquity of the golden ratio and our human tendency to erroneously equate truth with beauty or simplicity.

Mathematicians Elected to National Academy of Sciences

On April 30 the National Academy of Sciences announced the election of 84 new members and 21 foreign associates. Eight mathematicians were among those honored for "their distinguished and continuing achievements in original research."

Topology Troubles Allow Teacher to Empathize

Writing in Slate, high school math teacher Ben Orlin recounts his struggles in a Yale topology seminar and argues that the experience of mathematical failure equips him to better understand his students.

Does P=NP?

Writing for the New Yorker, Alexander Nazaryan recaps a discredited 2010 attempt to prove P≠NP, reviews Lance Fortnow's new book The Golden Ticket: P, NP, and the Search for the Impossible, and reflects on the philosophical implications of this problem, "one of mathematics' most tightly tied Gordian knots."  

MathDL Loci Featured Items RSS feed

Parameterized Knots

From: Loci

This article discusses parameterized knots (polynomial and trigonometric) and includes an interactive gallery of selected knots and their equations.

Rethinking Pythagoras

From: Loci

This article explores analogues of the Pythagorean Theorem in non-Euclidean geometries.

Modeling the Mirascope Using Dynamic Technology

From: Loci

This article analyzes the physical and mathematical properties of the mirascope and models the mirascope using dynamic learning technology

More Features:

From: Loci

• Life After Wolfram|Alpha
• Google-opoly
• Gallery of Ray Tracing
• Trisecting a Line Segment

MAA Book Reviews - Read This!

The Moore Method: A Pathway to Learner-Centered Instruction

The Moore Method: A Pathway to Learner-Centered Instruction

A Guide to Real Variables

A Guide to Real Variables

Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics

Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics

Visual Group Theory

Visual Group Theory

math updates on arXiv.org

A Model for Tracking Fronts of Stress-Induced Permeability Enhancement. (arXiv:1305.3618v1 [physics.geo-ph])

Authors: K.C. Lewis, Satish Karra, Sharad Kelkar

Using an analogy to the classical Stefan problem, we construct evolution equations for the fluid pore pressure on both sides of a propagating stress-induced damage front. Closed form expressions are derived for the position of the damage front as a function of time for the cases of thermally-induced damage as well as damage induced by over-pressure. We derive expressions for the flow rate during constant pressure fluid injection from the surface corresponding to a spherically shaped subsurface damage front. Finally, our model results suggest an interpretation of field data obtained during constant pressure fluid injection over the course of 16 days at an injection site near Desert Peak, NV.

General beta Jacobi corners process and the Gaussian Free Field. (arXiv:1305.3627v1 [math.PR])

Authors: Alexei Borodin, Vadim Gorin

We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.

Squares and Covering Matrices. (arXiv:1305.3629v1 [math.LO])

Authors: Chris Lambie-Hanson

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof amd subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between $\square_{\kappa}$ and $\square(\kappa^+)$ for a regular cardinal $\kappa$. We provide a detailed picture of the implications between these square principles.

Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting. (arXiv:1305.3639v1 [cs.NA])

Authors: Andreas Hellander, Michael Lawson, Brian Drawert, Linda Petzold

The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity.

Excitation spectrum of interacting bosons in the mean-field infinite-volume limit. (arXiv:1305.3641v1 [math-ph])

Authors: Jan Dereziński, Marcin Napiórkowski

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean field infinite volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of R. Seiringer (arXiv:1008.5349 [math-ph]) to large volumes.

Constructing Quantum Circuits for Simple Periodic Functions. (arXiv:1305.3642v1 [quant-ph])

Authors: Omar Gamel, Daniel F. V. James

Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for the simplest periodic function, that is one-to-one within a single period, of a given period p. We conjecture that to create the simplest periodic function of period p, where p is an n-bit number, one needs at most n To?ffoli gates.

Stabilizers of Actions of Lattices in Products of Groups. (arXiv:1305.3648v1 [math.DS])

Authors: Darren Creutz

We present a new proof of the result of Stuck and Zimmer [SZ94] that any ergodic probability-preserving action of an irreducible lattice in a semisimple real Lie group, each simple factor of higher-rank, is essentially free, and generalize to the case when only one simple factor is of higher-rank.

We also prove a generalization of a result of Bader and Shalom [BS06] by showing that any probability-preserving action of a product of simple groups, at least one with property $(T)$, which is ergodic for each simple subgroup is either essentially free or essentially transitive.

Our method involves the study of relatively contractive maps and the Howe-Moore property, rather than the relaying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups of independent interest.

Classical Probability and Quantum Mechanics. (arXiv:1305.3649v1 [quant-ph])

Authors: Ehtibar N. Dzhafarov, Janne V. Kujala

Since Bell's celebrated work, we know that correlations of spins measured along various directions in a system of entangled particles cannot be explained within the framework of classical probability theory, provided all spins are modeled as random variables defined on a single probability space, with the identity of each spin determined only by the choice of its direction, irrespective of measurements on other particles. Here we show the same failure for classical probability models of a more general kind, in which different combinations of measurement directions across all particles correspond to different probability spaces. We found that any such model necessarily mismatches quantum mechanics (QM): if it allows for all QM-compliant correlations, then it also allows for some correlations forbidden by QM; if it forbids all QM-forbidden correlations, then it only allows for correlations satisfying Bell-type inequalities (i.e., those within the scope of classical mechanics).

ScienceDaily: Mathematics News

Can math models of gaming strategies be used to detect terrorism networks?

Mathematicians have developed a mathematical model to disrupt the flow of information in a complex real-world network, such as a terrorist organization, using minimal resources.

Fast and painless way to better mental arithmetic? Yes, there might actually be a way

In the future, if you want to improve your ability to manipulate numbers in your head, you might just plug yourself in. So say researchers who report on studies of a harmless form of brain stimulation applied to an area known to be important for math ability.

Most math being taught in kindergarten is old news to students

Kindergarten teachers report spending much of their math instructional time teaching students basic counting skills and how to recognize geometric shapes -— skills the students have already mastered before ever setting foot in the kindergarten classroom, new research finds.

Early math and reading ability linked to job and income in adulthood

Math and reading ability at age 7 may be linked with socioeconomic status several decades later, according to new research. The childhood abilities predict socioeconomic status in adulthood over and above associations with intelligence, education, and socioeconomic status in childhood.

Mathematical model measures hidden HIV

A new mathematical modeling technique reveals HIV virus may be replicating in body even when undetectable in the blood.

Monkey math: Baboons show brain's ability to understand numbers

Opposing thumbs, expressive faces, complex social systems: it's hard to miss the similarities between apes and humans. Now a new study with a troop of zoo baboons and lots of peanuts shows that a less obvious trait -- the ability to understand numbers -- also is shared by humans and their primate cousins.

How to manage motorway tolls through the Game Theory

Sophisticated mathematical methods could be used to improve traffic management.

Modeling disease spread, including flu

A collaborative research network that formed nearly 10 years ago has pioneered the use of computational and mathematical models to prepare for, detect and respond to influenza, pertussis, West Nile disease, dengue fever, cholera and other infectious disease threats.