Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. When attempting to find a polynomial approximation of boundary or initial data, one encounters the Bernstein-Vandermonde matrix, which is found to be highly ill-conditioned. Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices. We also use a nonstandard matrix norm to study the conditioning of the Bernstein-Vandermonde matrix, showing that the conditioning in this case is better than in the standard 2-norm. Additionally, we use properties of multivariate Bernstein polynomials to derive a block $LU$ decomposition of the Bernstein-Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex.

High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points.

In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between "lines" of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments in both two and three dimensions confirm the stability and accuracy of this approach.

Avikainen proved the estimate $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}} $ for $p,q in [1,infty)$, one-dimensional random variables $X$ with the bounded density function and $widehat{X}$, and a function $f$ of bounded variation in $mathbb{R}$. In this article, we will provide multi-dimensional analogues of this estimate for functions of bounded variation in $mathbb{R}^{d}$, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use Hardy-Littlewood maximal estimates and pointwise characterizations of these function spaces. We will apply main statements to numerical analysis on irregular functionals of a solution to stochastic differential equations based on the Euler-Maruyama scheme and the multilevel Monte Carlo method, and to estimates of the $L^{2}$-time regularity of decoupled forward-backward stochastic differential equations with irregular terminal conditions.

We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally-defined "hybrid complete Lyapunov function" exists for every hybrid system in this class. Motivated by mechanics and control settings where physical or engineered events cause abrupt changes in a system's governing dynamics, our results apply to a large class of Lagrangian hybrid systems (with impacts) studied extensively in the robotics literature. Viewed formally, these results generalize those of Conley and Franks for continuous-time and discrete-time dynamical systems, respectively, on metric spaces. However, we furnish specific examples illustrating how our statement of sufficient conditions represents merely an early step in the longer project of establishing what formal assumptions can and cannot endow hybrid systems models with the topologically well characterized partitions of limit behavior that make Conley's theory so valuable in those classical settings.

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.

We identify the structure of the lexicographically least word avoiding 5/4-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $p au(varphi(z) varphi^2(z) cdots)$ where $p, z$ are finite words, $varphi$ is a 6-uniform morphism, and $ au$ is a coding. This description yields a recurrence for the $i$th letter, which we use to prove that the sequence of letters is 6-regular with rank 188. More generally, we prove $k$-regularity for a sequence satisfying a recurrence of the same type.

We propose an augmented Lagrangian preconditioner for a three-field stress-velocity-pressure discretization of stationary non-Newtonian incompressible flow with an implicit constitutive relation of power-law type. The discretization employed makes use of the divergence-free Scott-Vogelius pair for the velocity and pressure. The preconditioner builds on the work [P. E. Farrell, L. Mitchell, and F. Wechsung, SIAM J. Sci. Comput., 41 (2019), pp. A3073-A3096], where a Reynolds-robust preconditioner for the three-dimensional Newtonian system was introduced. The preconditioner employs a specialized multigrid method for the stress-velocity block that involves a divergence-capturing space decomposition and a custom prolongation operator. The solver exhibits excellent robustness with respect to the parameters arising in the constitutive relation, allowing for the simulation of a wide range of materials.

We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure solution has a low regularity, which manifests in sub-optimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations, see [10]. We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptional usefulness of the proposed numerical approach.

We consider generic optimal Bayesian inference, namely, models of signal reconstruction where the posterior distribution and all hyperparameters are known. Under a standard assumption on the concentration of the free energy, we show how replica symmetry in the strong sense of concentration of all multioverlaps can be established as a consequence of the Franz-de Sanctis identities; the identities themselves in the current setting are obtained via a novel perturbation of the prior distribution of the signal. Concentration of multioverlaps means that asymptotically the posterior distribution has a particularly simple structure encoded by a random probability measure (or, in the case of binary signal, a non-random probability measure). We believe that such strong control of the model should be key in the study of inference problems with underlying sparse graphical structure (error correcting codes, block models, etc) and, in particular, in the derivation of replica symmetric formulas for the free energy and mutual information in this context.

An approach is given for solving large linear systems that combines Krylov methods with use of two different grid levels. Eigenvectors are computed on the coarse grid and used to deflate eigenvalues on the fine grid. GMRES-type methods are first used on both the coarse and fine grids. Then another approach is given that has a restarted BiCGStab (or IDR) method on the fine grid. While BiCGStab is generally considered to be a non-restarted method, it works well in this context with deflating and restarting. Tests show this new approach can be very efficient for difficult linear equations problems.

You don’t have to be a mathematician to see beauty in math – you don’t even have to be able to do anything beyond basic algebra. That’s according to new research from Yale University and the U.K.’s Bath University.

Mathematics, or Allah Mathematics, is a DJ and producer who has been part of the Wu-Tang Clan since before there was a Wu-Tang Clan. He DJ’d for GZA in the early 1990s, back when the rapper was known as “The Genius.” At the same time, Mathematics kept seeing a young rapper named Rakeem around town. That guy, Rakeem, would soon become the RZA, and would assemble an all-star team of MCs to form one of the most influential rap groups of all time.

Mathematics DJ’d for almost all of the Wu members at different points, as well as spinoff groups like Red and Meth. But starting in the mid-1990s, he really made his mark as a producer. Mathematics produced some of the best-loved songs from Wu-related projects, including “Mighty Healthy,” “Cobra Clutch,” “Wu Banga 101,” and more. He also designed the crew’s famous “W” logo.

In recent years, he’s been heavily involved with group albums like Iron Flag and A Better Tomorrow. All of this culminated in his new album, Wu-Tang: The Saga Continues. The record features contributions from almost every member of the Clan, and is executive-produced by RZA himself.

See http://theciphershow.com/episode/225/ for full show notes and comments.

There are some things that just feel like they’re true. For example, the idea that people who are gifted musicians are also good at learning math, or vice versa. However, there isn’t any data that suggests that there are any links in the brain between these proclivities. As Dr. Art Markman and Dr. Bob Duke...

Back in 2016, Mathea impressed Yvonne Catterfeld with her rendition of Elle King's Ex's & Oh's. As a result her life has changed. She is no longer an Austrian teenager dreaming of a music career, she is a respected and celebrated artist releasing her debut album. While she may not have won her series of The Voice, but her place of the musical map was firmly won.

Scurvy in modern Canada, snake venom sans snakes and hot food tolerance

Foxx’s third Maths-assisted set is probably his best post-Ultravox LP.

Finance Minister Mathias Cormann says "school time is a time to go to school" ahead of thousands planning to leave the classroom for a global climate strike.

Bright minds are being encouraged to consider jobs in fashion, rather than traditional STEM careers such as engineering and mathematics.

DAILY BRIEFING: Injured formula 3 driver Alex Peroni will return to Hobart in the next few days as questions are raised about why he was allowed to walk from a crash scene in Italy instead of being taken off by stretcher.

Cassandra Smith* found herself in hospital and on an IV line directly to her heart when mould took hold of her body, seeping in after her Townsville home was inundated during major flooding, with an expert saying mould in the city is the most extreme she has ever seen.

Albury supermarket owner Bob Mathews says the scheme has cost his business.

Political pressure is mounting on the Member for Chisholm about her alleged links to the Chinese Government.

Can you use simple maths to figure out your best online dating profile match? Or choose the shortest line in the supermarket? According to mathematician Lily Serna, yes you can.

Ordior Has Signed Patrick Mathias Aka Password For A World Wide Exclusive Publishing And Administration Agreement!

(United States Fourth Circuit) - Affirmed: The juvenile offender of the DC Beltway sniper duo was convicted of capital murder for crimes committed when he was 17 years old. He was sentenced to 4 terms of life imprisonment without parole. After his conviction and sentence the US Supreme Court held that persons who committed serious crimes when under 18 could not be sentenced to death and that they could not be sentenced to life imprisonment unless the offenses reflected permanent incorrigibility and that these rules applied retroactively. Because of the new rules, the district court vacated the life sentences without parole and remanded for re-sentencing. The 4th Circuit affirmed.

Ordior Has Signed Patrick Mathias Aka Password For A World Wide Exclusive Publishing And Administration Agreement!

It was a movie!

It was a movie!

The victors in last week’s crossword challenge were: First place, with a score of 276/276: A tie between Dan Williams and Richard Kennaway. Second place, but heartbreakingly close, with a score of 274/276: Another tie, between Tim Goodwyn and the team of Dan Grayson & Carol Livingstone Third place: Paul Epps Fourth Place: Eric Dinsdale […]

As the new school year begins, elementary school teachers across the United States are now able to access a new, free online tool designed to provide elementary school teachers with targeted math resources for their kindergarten through fifth grade classrooms.

Just a short note that the memorial article “Analysis and applications: The mathematical work of Elias Stein” has just been published in the Bulletin of the American Mathematical Society.  This article was a collective effort led by Charlie Fefferman, Alex Ionescu, Steve Wainger and myself to describe the various mathematical contributions of Elias Stein, who […]

Next quarter, starting March 30, I will be teaching “Math 247B: Classical Fourier Analysis” here at UCLA.  (The course should more accurately be named “Modern real-variable harmonic analysis”, but we have not gotten around to implementing such a name change.) This class (a continuation of Math 247A from previous quarter, taught by my colleague, Monica […]

At the most recent MSRI board of trustees meeting on Mar 7 (conducted online, naturally), Nicolas Jewell (a Professor of Biostatistics and Statistics at Berkeley, also affiliated with the Berkeley School of Public Health and the London School of Health and Tropical Disease), gave a presentation on the current coronavirus epidemic entitled “2019-2020 Novel Coronavirus […]

After some discussion with the applied math research groups here at UCLA (in particular the groups led by Andrea Bertozzi and Deanna Needell), one of the members of these groups, Chris Strohmeier, has produced a proposal for a Polymath project to crowdsource in a single repository (a) a collection of public data sets relating to […]

My student, Jaume de Dios, has set up a web site to collect upcoming mathematics seminars from any institution that are open online.  (For instance, it has a talk that I will be giving in an hour.)   There is a form for adding further talks to the site; please feel free to contribute (or make […]

As COVID-19 spreads worldwide, leaders are relying on mathematical models to make public health and economic decisions. A new model developed by Princeton and Carnegie Mellon researchers improves tracking of epidemics by accounting for mutations in diseases. Now, the researchers are working to apply their model to allow leaders to evaluate the effects of countermeasures to epidemics before they deploy them.

Girls are great at math. But if they’re even better at reading, they might be more motivated to choose a humanities-focused career.

Jonah Mathews had 23 points and Daniel Utomi added 19 points and 10 rebounds as USC beat Arizona State 71-61 for a weekend sweep of the Arizona teams.

When USC guard Jonah Mathews beat UCLA with a last-second shot, he hoped it would be the start of a magical March run. Instead, it was the final game of his career.