MathDL Math in the News RSS feed

Boas Honored as 2012 Presidential Professor

Mathematics professor Harold P. Boas has been named a Presidential Professor for Teaching Excellence at Texas A&M University.

Jigsaw Puzzles Designed Using Snowflake Math: No Two Are Alike

Boston-based design company Nervous System is using a script based on Laplacian growth to develop what they call “a puzzle for the 21st century.”

A Mathematical Challenge to Obesity

Carson C. Chow talks with New York Times science writer Claudia Dreifus about applying mathematics to understanding the obesity epidemic in the US.

A Milestone Appointment in Mathematics at Howard University

This fall, Talitha Washington will become the second Black woman to hold a tenured position as an associate professor of mathematics at Howard University, an historically Black university in Washington, D.C.

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

Bianchi Type V Universe with Bulk Viscous Matter and Time Varying Gravitational and Cosmological Constants. (arXiv:1205.5265v1 [gr-qc])

Authors: Prashant S. Baghel, J. P. Singh

Spatially homogeneous and anisotropic Bianchi type V space-time with bulk viscous fluid source and time varying gravitational constant $G$ and cosmological term $\Lambda$ are considered. Coefficient of bulk viscosity $\zeta$ is assumed as a simple linear function of Hubble parameter $H$ (i.e. $\zeta=\zeta_0+\zeta_1 H$, where $\zeta_0$ and $\zeta_1$ are constants). The Einstein field equations are solved explicitly by using a law of variation for the Hubble parameter, which yields a constant value of deceleration parameter. Physical and kinematical parameters of the models are discussed. The models are found to be compatible with the results of astronomical observations.

Spectral Norm of Symmetric Functions. (arXiv:1205.5282v1 [cs.CC])

Authors: Anil Ada, Omar Fawzi, Hamed Hatami

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions.

Solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and quasi-periodic parameters using theory of vessels. (arXiv:1205.5285v1 [math-ph])

Authors: Andrey Melnikov

We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this paper quasi-periodic means a finite sum of periodic integrable functions. The solutions for a general s are explicitly constructed in terms of the solutions zn(x), satisfying the SL equation with initial conditions zn(0)=0, zn'(0)=1 for a discrete Levinson set of numbers s=sn, n-natural number. The tau function tau(x) of the corresponding vessel realizes the given potential via the formula q(x)= - 2(ln(tau(x)))".

We also prove an analogue of the inverse scattering theorem in this setting too.

Using the notion of "KdV evolutionary vessel", we construct a solution of the Korteweg-de-Vries (KdV) equation q'_t = - 3/2 q q'_x + 1/4 q"'_{xxx}, which coincides for t=0 with a given (periodic or quasi-periodic) potential.

Tailoring Three-Point Functions and Integrability IV. Theta-morphism. (arXiv:1205.5288v1 [hep-th])

Authors: Nikolay Gromov, Pedro Vieira

We compute structure constants in N=4 SYM at one loop using Integrability. This requires having full control over the two loop eigenvectors of the dilatation operator for operators of arbitrary size. To achieve this, we develop an algebraic description called the Theta-morphism. In this approach we introduce impurities at each spin chain site, act with particular differential operators on the standard algebraic Bethe ansatz vectors and generate in this way higher loop eigenvectors. The final results for the structure constants take a surprisingly simple form. For some quantities we conjecture all loop generalizations. These are based on the tree level and one loop patterns together and also on some higher loop experiments involving simple operators.

The Galois group of random elements of linear groups. (arXiv:1205.5290v1 [math.NT])

Authors: Alexander Lubotzky, Lior Rosenzweig

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of \Gamma (but not on \Gamma).

On the Accuracy of Anisotropic Fast Marching. (arXiv:1205.5300v1 [math.NA])

Authors: J.-M. Mirebeau

The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric. A major challenge for these algorithms is the non-isotropy of the riemannian metric. Applications of the eikonal equation to image processing often involve pronounced anisotropies, which motivated the design of new algorithms.

A recently introduced variant of the fast marching algorithm addresses the problem of large anisotropies using an algebraic tool named lattice basis reduction. The numerical complexity of this algorithm is insensitive to anisotropy, under weak assumptions. We establish in this paper, in the simplified setting of a constant riemannian metric, that the accuracy of this algorithm is also extremely robust to anisotropy : in an average sense, it is independent of the anisotropy ratio. We also extend this algorithm to higher dimension.

On integral well-rounded lattices in the plane. (arXiv:1205.5301v1 [math.NT])

Authors: Lenny Fukshansky, Glenn Henshaw, Philip Liao, Matthew Prince, Xun Sun, Samuel Whitehead

We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form $x^2+Dy^2=z^2$ where $D>0$ is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.

Floquet theory for integral and integro-differential equations. (arXiv:1205.5302v1 [math.CA])

Authors: S. A. Belbas

We study the extension of Hill's method of infinite determinants to the case of integro-differential equations with periodic coefficients and kernels. We develop the analytical theory of such methods, and we obtain certain qualitative properties of the equations that determine the boundaries between regions of dynamic stability and dynamic instability.

ScienceDaily: Mathematics News

Math predicts size of clot-forming cells

Mathematicians have helped biologists figure out why platelets, the cells that form blood clots, are the size and shape that they are. Because platelets are important both for healing wounds and in strokes and other conditions, a better understanding of how they form and behave could have wide implications.

Morphing robots and shape-shifting sculptures: Origami-inspired design merges engineering, art

Researchers have shown how to create morphing robotic mechanisms and shape-shifting sculptures from a single sheet of paper in a method reminiscent of origami, the Japanese art of paper folding.

New twist on ancient math problem could improve medicine, microelectronics

A hidden facet of a math problem that goes back to Sanskrit scrolls has just been exposed by nanotechnology researchers.

It's official: Physics is hard

Scientists have conducted scientific research on the difficulty –- from a computational complexity theory perspective -- of addressing some of the challenges of physics.

First light: Researchers develop new way to generate superluminal pulses

Researchers have developed a novel way of producing light pulses that are "superluminal" -- in some sense they travel faster than the speed of light. The new method could be used to improve the timing of communications signals and to investigate the propagation of quantum correlations.

Study finds twist to the story of the number line: Number line is learned, not innate human intuition

Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins -- notations that map numbers onto space and often represent magnitude -- are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don't stop to wonder: Is it "natural"? Is it cultural? Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned.

Shedding light on southpaws: Sports data help confirm theory explaining left-handed minority in general population

Lefties (only ten percent of the general population) have always been a bit of a puzzle. Researchers have now developed a mathematical model that shows the low percentage of lefties is a result of the balance between cooperation and competition in human evolution. They are the first to use real-world data (from competitive sports, including baseball, boxing and hockey) to test and confirm the hypothesis that social behavior is related to population-level handedness.

Mathematics: First-ever image of a flat torus in 3-D

Just as a terrestrial globe cannot be flattened without distorting the distances, it seemed impossible to visualize abstract mathematical objects called flat tori in ordinary three-dimensional space. However, a team of mathematicians and computer scientists has succeeded in constructing and visually representing an image of a flat torus in three-dimensional space. This is a smooth fractal, halfway between fractals and ordinary surfaces.