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Polymers near the glass transition temperature Tg often exhibit a breakdown of time-temperature-superposition (TTS), with chain relaxation times and viscosity exhibiting a weaker temperature dependence than segmental relaxation times. The origin of this onset of thermorheological complexity has remained unsettled and a matter of debate. Here we extend the Heterogeneous Rouse Model (HRM), which generalizes the Rouse model to account for dynamic heterogeneity, to make predictions for the relaxation modulus G(t) and complex modulus G*($\omega$) of unentangled polymers near Tg. The HRM predicts that G(t) and G*($\omega$) exhibit enhanced effective scaling exponents in the Rouse regime in the presence of dynamic heterogeneity, with a more rapid decay from the glassy plateau emerging as the system becomes more dynamically heterogeneous on cooling. This behavior is predicted to emerge from a strand-length dependence of the moment of the segmental mobility distribution probed by chain dynamics. We show that the HRM predictions are in good accord with experimental complex modulus data for polystyrene, poly(methyl methacrylate), and poly(2-vinyl pyridine). The HRM also predicts the onset of distinct temperature dependences among chain scale quantities such as terminal relaxation time and viscosity in our experimental systems, apparently resolving one of the most significant standing objections to a heterogeneity-based origin of TTS-breakdown. The HRM thus provides a generalized theory of the chain-scale linear rheological response of unentangled polymers near Tg, accounting for the origin of TTS-breakdown at a molecular mechanistic level. It also points towards a new strategy of inferring the dynamic heterogeneity of glass-forming polymeric systems based on the temperature-evolution of modified scaling in the Rouse regime.

The temperature at which supercooled liquids turn into solid-like glasses ($T_g$) can change at the free surface, affecting the properties of nanostructured glasses and their applications. However, inadequate experimental resolution to determine the $T_g$ gradient and a longstanding debate over the role of nonequilibrium effects have hindered fundamental understanding of this phenomenon. Using spatially resolved $T_g$ measurements and molecular dynamics simulations, we reveal a crossover from equilibrium behavior to a new regime of near-surface nonequilibrium glass physics on cooling. This crossover causes the form of the nonequilibrium $T_g$ gradient to change, highlighting the need to include these physics for rational understanding of the properties of realistic nanostructured glass-forming materials. They also potentially recast the interpretation of decades of experimental data on nanoconfined glasses., Comment: This PDF file includes two documents: The first is the main manuscript, which consists of 13 pages and 4 figures. The second document is the supplementary materials, which contains 25 pages, 8 figures, and 3 tables

Nanoparticles can greatly enhance the mechanical response of elastomeric polymers essential to a wide range of applications, yet their precise molecular mechanisms of high-strain reinforcement remain largely unresolved. Here we show, based on molecular dynamics simulations, that high-strain reinforcement emerges from an interplay between granular nanoparticulate compressive behavior in the normal direction and polymer incompressibility. This feedback loop, which is initiated by a mismatch in the Poisson ratios of nanofiller and polymer, invokes a contribution from the polymer's bulk modulus to the elongational stress, while the tendency of the polymer to contract in the normal direction maintains a near-jammed filler state. This effect persists even once the direct filler elongational contribution becomes dissipative after the 'Payne effect' yield. These results indicate that direct particle-particle contact effects, even in the absence of potential augmenting mechanisms such as glassy polymer bridges, can drive the mechanical reinforcement effects typical of experimental systems.

We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$ $set$, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi(r) = r^\delta \log^{-2\delta}\log(1/r)$, where $\delta$ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

We introduce a novel loss function to minimize the outage probability of an ML-based resource allocation system. A single-user multi-resource greedy allocation strategy constitutes our application scenario, for which an ML binary classification predictor assists in selecting a resource satisfying the established outage criterium. While other resource allocation policies may be suitable, they are not the focus of our study. Instead, our primary emphasis is on theoretically developing this loss function and leveraging it to train an ML model to address the outage probability challenge. With no access to future channel state information, this predictor foresees each resource's likely future outage status. When the predictor encounters a resource it believes will be satisfactory, it allocates it to the user. Our main result establishes exact and asymptotic expressions for this system's outage probability. These expressions reveal that focusing solely on the optimization of the per-resource outage probability conditioned on the ML predictor recommending resource allocation (a strategy that appears to be most appropriate) may produce inadequate predictors that reject every resource. They also reveal that focusing on standard metrics, like precision, false-positive rate, or recall, may not produce optimal predictors. With our result, we formulate a theoretically optimal, differentiable loss function to train our predictor. We then compare predictors trained using this and traditional loss functions namely, binary cross-entropy (BCE), mean squared error (MSE), and mean absolute error (MAE). In all scenarios, predictors trained using our novel loss function provide superior outage probability performance. Moreover, in some cases, our loss function outperforms predictors trained with BCE, MAE, and MSE by multiple orders of magnitude.

The immense dependence of the glass transition temperature $T_g$ on molecular weight $M$ is one of the most fundamentally and practically important features of polymer glass formation. Here, we report on molecular dynamics simulation of three model linear polymers of substantially different complexity demonstrating that the 70-year-old canonical explanation of this dependence (a simple chain end dilution effect) is likely incorrect at leading order. Our data shows that end effects are present only in relatively stiff polymers and, furthermore, that the magnitude of this end effect diminishes on cooling. Instead, we find that $T_g(M)$ trends are instead dominated by shifts in $T_g$ throughout the entire polymer chain rather than through a chain end effect. We show that these data are consistent with a generic two-barrier model of $T_g$ and its $M$-dependence, motivated by the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory. More broadly, this work indicates both a need to reassess the canonical understanding of $T_g(M)$ in linear polymers (and macromolecules at large) and an opportunity to reveal new glass formation physics with renewed study of $M$ effects on $T_g$., Comment: Discussion expanded with some SI content brought into main manuscript

Device-to-Device (D2D) communication propelled by artificial intelligence (AI) will be an allied technology that will improve system performance and support new services in advanced wireless networks (5G, 6G and beyond). In this paper, AI-based deep learning techniques are applied to D2D links operating at 5.8 GHz with the aim at providing potential answers to the following questions concerning the prediction of the received signal strength variations: i) how effective is the prediction as a function of the coherence time of the channel? and ii) what is the minimum number of input samples required for a target prediction performance? To this end, a variety of measurement environments and scenarios are considered, including an indoor open-office area, an outdoor open-space, line of sight (LOS), non-LOS (NLOS), and mobile scenarios. Four deep learning models are explored, namely long short-term memory networks (LSTMs), gated recurrent units (GRUs), convolutional neural networks (CNNs), and dense or feedforward networks (FFNs). Linear regression is used as a baseline model. It is observed that GRUs and LSTMs present equivalent performance, and both are superior when compared to CNNs, FFNs and linear regression. This indicates that GRUs and LSTMs are able to better account for temporal dependencies in the D2D data sets. We also provide recommendations on the minimum input lengths that yield the required performance given the channel coherence time. For instance, to predict 17 and 23 ms into the future, in indoor and outdoor LOS environments, respectively, an input length of 25 ms is recommended. This indicates that the bulk of the learning is done within the coherence time of the channel, and that large input lengths may not always be beneficial.

Let $\psi:\mathbb R_{+}\to \mathbb R_{+}$ be a nonincreasing function. A pair $(A,\mathbf b),$ where $A$ is a real $m\times n$ matrix and $\mathbf b\in\mathbb R^{m},$ is said to be $\psi$-Dirichlet improvable, if the system $$\|A\mathbf q +\mathbf b-\mathbf p\|^m<\psi(T), \quad \|\mathbf q\|^n

The origin of stretched exponential relaxation in supercooled glass-forming liquids is one of the central questions regarding the anomalous dynamics of these fluids. The dominant explanation for this phenomenon has long been the proposition that spatial averaging over a heterogeneous distribution of locally exponential relaxation processes leads to stretching. Here we perform simulations of model polymeric and small-molecule glass-formers in the isoconfigurational ensemble to show that stretching instead emerges from a combination of spatial averaging and locally nonexponential relaxation. Results indicate that localities in the fluid exhibiting faster-than-average relaxation tend to exhibit locally stretched relaxation, whereas slower-than-average relaxing domains exhibit compressed exponential relaxation. We show that local stretching is predicted by loose local caging, as measured by the Debye-Waller factor, and vice versa. This phenomenology in the local relaxation of in-equilibrium glasses parallels the dynamics of out of equilibrium under-dense and over-dense glasses, which likewise exhibit an asymmetry in their degree of stretching vs compression. On the basis of these results, we hypothesize that local stretching and compression in equilibrium glass-forming liquids results from evolution of particle mobilities over a single local relaxation time, with slower particles tending towards acceleration and vice versa. In addition to providing new insight into the origins of stretched relaxation, these results have implications for the interpretation of stretching exponents as measured via metrologies such as dielectric spectroscopy: measured stretching exponents cannot universally be interpreted as a direct measure of the breadth of an underlying distribution of relaxation times., Comment: 11 pages, 12 figures. This article has been submitted to the Journal of Chemical Physics. After it is published, it will be found at https://aip.scitation.org/journal/jcp

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set \[ R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets., Comment: Comments welcome

We study almost-minimizers of anisotropic surface energies defined by a H\"older continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally H\"older continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism \`a la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarn\'ik and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.

We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game and measure, and then use the game to prove fundamental measure theoretic results such as Fubini's theorem, the Borel-Cantelli lemma, and a general unfolding result for the game which gives, for example, the measurability of $\boldsymbol{\Sigma}^1_1$ sets. We also use the game to give a new, more constructive, proof of a strong form of the R\'{e}nyi-Lamperti lemma, an important result in probability theory with many applications to number theory. The proofs we give are all direct combinatorial arguments using the game, and do not depend on known measure theoretic arguments.

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions -- (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect -- which resolve questions posed by Chousionis, Leykekhman, and Urba\'nski, and go on provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.