We define a notion of tracial $\mathcal {Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple nonelementary C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal {Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal {Z}$-absorbing C*-algebras which are not $\mathcal {Z}$-absorbing. We use techniques from reduced free products of von Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal {Z}$-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

We show that passively mode-locked lasers, subject to feedback from a single external cavity can exhibit large timing fluctuations on short time scales, despite having a relatively small long-term timing jitter. This means that the commonly used von Linde and Kéfélian techniques of experimentally estimating the timing jitter can lead to large errors in the estimation of the arrival time of pulses. We also show that adding a second feedback cavity of the appropriate length can significantly suppress noise-induced modulations that are present in the single feedback system. This reduces the short time-scale fluctuations of the interspike interval time and, at the same time, improves the variance of the fluctuation of the pulse arrival times on long time scales.

We present a novel multiscale mathematical model of espresso brewing. The model captures liquid infiltration and flow through a packed bed of ground coffee, as well as coffee solubles transport (both in the grains and in the liquid) and solubles dissolution. During infiltration, a sharp interface separates the dry and wet regions of the bed. A matched asymptotic analysis (based on fast dissolution rates) reveals that the bed can be described by four asymptotic regions: a dry region yet to be infiltrated by the liquid, a region in which the liquid is saturated with solubles and very little dissolution occurs, a slender region in which solubles are rapidly extracted from the smallest grains, and region in which slower extraction occurs from larger grains. The position and extent of each of these regions move with time (one being an intrinsic moving internal boundary layer) making the asymptotic analysis intriguing in its own right. The analysis yields a reduced model that elucidates the rate-limiting physical processes. Numerical solutions of the reduced model are compared to those to the full model, demonstrating that the reduced model is both accurate and significantly cheaper to solve.

We consider local and nonlocal Cahn–Hilliard equations with constant mobility and singular potentials including, e.g., the Flory–Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we examine the essential assumptions required for the reaction term to ensure the existence of a weak solution. Also, we explore the scenario involving the nonlocal Cahn–Hilliard equation and provide some illustrative examples that contextualize within our abstract framework.

Let M be a smooth closed oriented surface. Gaussian thermostats on M correspond to the geodesic flows arising from metric connections, including those with non-zero torsion. These flows may not preserve any absolutely continuous measure. We prove that if two Gaussian thermostats on M with negative thermostat curvature are related by a smooth orbit equivalence isotopic to the identity, then the two background metrics are conformally equivalent via a smooth diffeomorphism of M isotopic to the identity. We also give a relationship between the thermostat forms themselves. Finally, we prove the same result for Anosov magnetic flows.

We show that $\alpha $-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for $0<\alpha <1$ and every $\alpha $-stable Lévy motion ${\mathbb {W}}$, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1\leq \alpha <2$ and every symmetric $\alpha $-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1< \alpha <2$ and every $-1\leq \beta \leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_\alpha (\ln (2), \beta ,0)$ random variable.

We give a crystal structure on the set of Gelfand–Tsetlin patterns (GTPs), which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form and are expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between GTPs and semistandard Young tableaux is a crystal isomorphism.

An infinite sequence $\alpha $ over an alphabet $\Sigma $ is $\mu $-distributed with respect to a probability map $\mu $ if, for every finite string w, the limiting frequency of w in $\alpha $ exists and equals $\mu (w)$. We prove the following result for any finite or countably infinite alphabet $\Sigma $: every finite-state selector over $\Sigma $ selects a $\mu $-distributed sequence from every $\mu $-distributed sequence if and only if $\mu $ is induced by a Bernoulli distribution on $\Sigma $, that is, a probability distribution on the alphabet extended to words by taking the product. The primary—and remarkable—consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $\mu $-distributedness. As a consequence, the shift-invariant measures $\mu $ on $\Sigma ^{\omega }$, such that any finite-state selector preserves the property of genericity for $\mu $, are exactly the positive Bernoulli measures.

The Jansen–Rit model of a cortical column in the cerebral cortex is widely used to simulate spontaneous brain activity (electroencephalogram, EEG) and event-related potentials. It couples a pyramidal cell population with two interneuron populations, of which one is fast and excitatory, and the other slow and inhibitory.

Our paper studies the transition between alpha and delta oscillations produced by the model. Delta oscillations are slower than alpha oscillations and have a more complex relaxation-type time profile. In the context of neuronal population activation dynamics, a small threshold means that neurons begin to activate with small input or stimulus, indicating high sensitivity to incoming signals. A steep slope signifies that activation increases sharply as input crosses the threshold. Accordingly, in the model, the excitatory activation thresholds are small and the slopes are steep. Hence, we replace the excitatory activation function with its singular limit, which is an all-or-nothing switch (a Heaviside function). In this limit, we identify the transition between alpha and delta oscillations as a discontinuity-induced grazing bifurcation. At the grazing, the minimum of the pyramidal-cell output equals the threshold for switching off the excitatory interneuron population, leading to a collapse in excitatory feedback.

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

In this paper, we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon :\ C{{-\mathsf{Comod}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta :\ C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\textrm {Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes $a_1$ and $a_2$ mod q, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.

The shimmy oscillations of a truck’s front wheels with dependent suspension are studied to investigate how shimmy depends on changes in inflation pressure, with emphasis on the inclusion of four nonlinear tyre characteristics to improve the accuracy of the results. To this end, a three degree-of-freedom shimmy model is created which reflects pressure dependency initially only through tyre lateral force. Bifurcation analysis of the model reveals that four Hopf bifurcations are found with decreased pressures, corresponding to two shimmy modes: the yaw and the tramp modes, and there is no intersection between them. Hopf bifurcations disappear at pressures slightly above nominal value, resulting in a system free of shimmy. Further, two-parameter continuations illustrate that there are two competitive mechanisms between the four pressure-dependent tyre properties, suggesting that the shimmy model should balance these competing factors to accurately capture the effects of pressure. Therefore, the mathematical relations between these properties and inflation pressure are introduced to extend the initial model. Bifurcation diagrams computed on the initial and extended models are compared, showing that for pressures below nominal value, shimmy is aggravated as the two modes merge and the shimmy region expands, but for higher pressures, shimmy is mitigated and disappears early.