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Local planning is an optimization process within a mobile robot navigation stack that searches for the best velocity vector, given the robot and environment state. Depending on how the optimization criteria and constraints are defined, some planners may be better than others in specific situations. We consider two conceptually different planners. The first planner explores the velocity space in real-time and has superior path-tracking and motion smoothness performance. The second planner was trained using reinforcement learning methods to produce the best velocity based on its training $"$experience$"$. It is better at avoiding dynamic obstacles but at the expense of motion smoothness. We propose a simple yet effective meta-reasoning approach that takes advantage of both approaches by switching between planners based on the surroundings. We demonstrate the superiority of our hybrid planner, both qualitatively and quantitatively, over the individual planners on a live robot in different scenarios, achieving an improvement of 26% in the navigation time.

Boiling is a prevalent phase-change process that plays a vital role in facilitating efficient heat transfer from a heating surface. While this heat transfer mechanism is generally effective, a rapid increase in surface temperature can lead to hydrodynamic instabilities, resulting in a boiling crisis. Previous studies have shown that surfactants often improve boiling performance and change the boiling crisis behavior. Conventional wisdom in this field attributes that these changes in boiling behavior are tied to the critical micelle concentration (CMC) of the particular surfactant. However, our work reveals that these changes in boiling behavior are independent of the CMC for three nonionic surfactants across a wide range of molar concentrations. In addition, visual snapshots of the bubbling behavior indicate changes in bubble formation, such as bubble size and nucleation site density, influenced by the molar concentration-dependent diffusion timescale of surfactants. Hence, these findings offer compelling evidence that boiling behavior, encompassing both boiling performance and boiling crisis, is governed by the dynamic adsorption of surfactants rather than dictated by the CMC. This becomes evident when quantifying the heat transfer coefficient (HTC) and critical heat flux (CHF) using the logarithm of molar concentration, as predicted by theory. Building upon these findings, we propose insights for controlling when CHF modification occurs in specific scenarios involving any surfactants. These insights hold significant potential for optimizing heat transfer processes and leveraging surfactants in energy-related applications to maximize boiling efficiency., Comment: 21 pages, 7 figures, 2 appendices

We prove quantitative decay estimates of macroscopic quantities generated by the solutions to linear transport equations driven by a general family of Hamiltonians. The associated particle trajectories are all trapped in a compact region of phase-space and feature a non-degenerate elliptic stagnation point. The analysis covers a large class of Hamiltonians generated by the radially symmetric compactly supported equilibria of the gravitational Vlasov-Poisson system. Working in radial symmetry, our analysis features both the 1+2-dimensional case and the harder 1+1-dimensional case, where all the particles have the same value of the modulus of angular momentum. The latter case is also of importance in both the plasma physics case and two dimensional incompressible fluid flows., Comment: 60 pages, added the decay of the macroscopic density to Theorem 1.7

This work addresses the task of modeling spatiotemporal traffic patterns directly from overhead imagery, which we refer to as image-driven traffic modeling. We extend this line of work and introduce a multi-modal, multi-task transformer-based segmentation architecture that can be used to create dense city-scale traffic models. Our approach includes a geo-temporal positional encoding module for integrating geo-temporal context and a probabilistic objective function for estimating traffic speeds that naturally models temporal variations. We evaluate our method extensively using the Dynamic Traffic Speeds (DTS) benchmark dataset and significantly improve the state-of-the-art. Finally, we introduce the DTS++ dataset to support mobility-related location adaptation experiments., Comment: European Conference on Computer Vision (ECCV) 2024

The localization of teeth and segmentation of periapical lesions in cone-beam computed tomography (CBCT) images are crucial tasks for clinical diagnosis and treatment planning, which are often time-consuming and require a high level of expertise. However, automating these tasks is challenging due to variations in shape, size, and orientation of lesions, as well as similar topologies among teeth. Moreover, the small volumes occupied by lesions in CBCT images pose a class imbalance problem that needs to be addressed. In this study, we propose a deep learning-based method utilizing two convolutional neural networks: the SpatialConfiguration-Net (SCN) and a modified version of the U-Net. The SCN accurately predicts the coordinates of all teeth present in an image, enabling precise cropping of teeth volumes that are then fed into the U-Net which detects lesions via segmentation. To address class imbalance, we compare the performance of three reweighting loss functions. After evaluation on 144 CBCT images, our method achieves a 97.3% accuracy for teeth localization, along with a promising sensitivity and specificity of 0.97 and 0.88, respectively, for subsequent lesion detection., Comment: Accepted for VISIGRAPP 2024 (Track: VISAPP), 8 pages

We show the existence of a new class of initially smooth spherically symmetric self-similar solutions to the non-isentropic Euler-Poisson system. These solutions exhibit supersonic gravitational implosion in the sense that the density blows-up in finite time while the fluid velocity remains supersonic. In particular, they occupy a portion of the phase space that is far from the recently constructed isentropic self-similar implosion. At the heart of our proof is the presence of a two-parameter scaling invariance and the reduction of the problem to a non-autonomous system of ordinary differential equations. We use the requirement of smoothness of the flow as a selection principle that constrains the choice of scaling indices. An important consequence of our analysis is that for all the solutions we construct, the polytropic index $\gamma$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case.

We review a series of recent results on global dynamic properties of radially symmetric self-gravitating compressible Euler flows, which naturally arise in the mathematical description of stars. We focus on the role of scaling invariances and how they interact with nonlinearities to generate imploding finite-time singularities as well as expanding star solutions, arising from smooth initial data. This review paper is based on joint works with Y. Guo, J. Jang, and M. Schrecker., Comment: 36 pages, 11 figures, review paper, slightly updated version of the original published in Quart. Appl. Math

We study the training performance of ROS local planners based on Reinforcement Learning (RL), and the trajectories they produce on real-world robots. We show that recent enhancements to the Soft Actor Critic (SAC) algorithm such as RAD and DrQ achieve almost perfect training after only 10000 episodes. We also observe that on real-world robots the resulting SACPlanner is more reactive to obstacles than traditional ROS local planners such as DWA., Comment: Accepted at 2023 IEEE International Conference on Robotics and Automation (ICRA)

We consider a family of isolated inhomogeneous steady states to the gravitational Vlasov-Poisson system with a point mass at the centre. They are parametrised by the polytropic index $k>1/2$, so that the phase space density of the steady state is $C^1$ at the vacuum boundary if and only if $k>1$. We prove the following sharp dichotomy result: if $k>1$ the linear perturbations Landau damp and if $1/2< k\le1$ they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with $k>1$ is the first such result for the gravitational Vlasov-Poisson system. The key step in the proof is to show that no embedded eigenvalues exist in the essential spectrum of the linearised system., Comment: 49 pages, 1 figure

Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. Two subclasses of this family correspond to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate $\lambda(t)_{t\to\infty}\sim t^{\frac23}$ and linear rate $\lambda(t)_{t\to\infty}\sim t$ respectively. We prove two results: any linearly expanding Goldreich-Weber star is nonlinearly stable, while any given self-similarly expanding Goldreich-Weber star is codimension-4 nonlinearly stable against irrotational perturbations. The codimension-4 condition in the latter result is optimal and reflects the presence of 4 unstable directions in the linearised dynamics in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as a codimension-1 nonlinear stability of the moduli space of self-similarly expanding Goldreich-Weber stars against irrotational perturbations., Comment: 78 pages, added a new result on the stability of linearly expanding Goldreich-Weber stars

Though semantic segmentation has been heavily explored in vision literature, unique challenges remain in the remote sensing domain. One such challenge is how to handle resolution mismatch between overhead imagery and ground-truth label sources, due to differences in ground sample distance. To illustrate this problem, we introduce a new dataset and use it to showcase weaknesses inherent in existing strategies that naively upsample the target label to match the image resolution. Instead, we present a method that is supervised using low-resolution labels (without upsampling), but takes advantage of an exemplar set of high-resolution labels to guide the learning process. Our method incorporates region aggregation, adversarial learning, and self-supervised pretraining to generate fine-grained predictions, without requiring high-resolution annotations. Extensive experiments demonstrate the real-world applicability of our approach., Comment: IEEE/CVF Winter Conference on Applications of Computer Vision (WACV) 2023

Skin lesions can be an early indicator of a wide range of infectious and other diseases. The use of deep learning (DL) models to diagnose skin lesions has great potential in assisting clinicians with prescreening patients. However, these models often learn biases inherent in training data, which can lead to a performance gap in the diagnosis of people with light and/or dark skin tones. To the best of our knowledge, limited work has been done on identifying, let alone reducing, model bias in skin disease classification and segmentation. In this paper, we examine DL fairness and demonstrate the existence of bias in classification and segmentation models for subpopulations with darker skin tones compared to individuals with lighter skin tones, for specific diseases including Lyme, Tinea Corporis and Herpes Zoster. Then, we propose a novel preprocessing, data alteration method, called EdgeMixup, to improve model fairness with a linear combination of an input skin lesion image and a corresponding a predicted edge detection mask combined with color saturation alteration. For the task of skin disease classification, EdgeMixup outperforms much more complex competing methods such as adversarial approaches, achieving a 10.99% reduction in accuracy gap between light and dark skin tone samples, and resulting in 8.4% improved performance for an underrepresented subpopulation.

In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of self-similar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a self-similar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to truncate the self-similar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity., Comment: 150 pages, 9 figures

Dynamical systems models for controlling multi-agent swarms have demonstrated advances toward resilient, decentralized navigation algorithms. We previously introduced the NeuroSwarms controller, in which agent-based interactions were modeled by analogy to neuronal network interactions, including attractor dynamics and phase synchrony, that have been theorized to operate within hippocampal place-cell circuits in navigating rodents. This complexity precludes linear analyses of stability, controllability, and performance typically used to study conventional swarm models. Further, tuning dynamical controllers by hand or grid search is often inadequate due to the complexity of objectives, dimensionality of model parameters, and computational costs of simulation-based sampling. Here, we present a framework for tuning dynamical controller models of autonomous multi-agent systems based on Bayesian Optimization (BayesOpt). Our approach utilizes a task-dependent objective function to train Gaussian Processes (GPs) as surrogate models to achieve adaptive and efficient exploration of a dynamical controller model's parameter space. We demonstrate this approach by studying an objective function selecting for NeuroSwarms behaviors that cooperatively localize and capture spatially distributed rewards under time pressure. We generalized task performance across environments by combining scores for simulations in distinct geometries. To validate search performance, we compared high-dimensional clustering for high- vs. low-likelihood parameter points by visualizing sample trajectories in Uniform Manifold Approximation and Projection (UMAP) embeddings. Our findings show that adaptive, sample-efficient evaluation of the self-organizing behavioral capacities of complex systems, including dynamical swarm controllers, can accelerate the translation of neuroscientific theory to applied domains., Comment: 29 pages, 10 figures

This article discusses how to create an interactive virtual training program at the intersection of neuroscience, robotics, and computer science for high school students. A four-day microseminar, titled Swarming Powered by Neuroscience (SPN), was conducted virtually through a combination of presentations and interactive computer game simulations, delivered by subject matter experts in neuroscience, mathematics, multi-agent swarm robotics, and education. The objective of this research was to determine if taking an interdisciplinary approach to high school education would enhance the students learning experiences in fields such as neuroscience, robotics, or computer science. This study found an improvement in student engagement for neuroscience by 16.6%, while interest in robotics and computer science improved respectively by 2.7% and 1.8%. The curriculum materials, developed for the SPN microseminar, can be used by high school teachers to further evaluate interdisciplinary instructions across life and physical sciences and computer science.

In the supercritical range of the polytropic indices $\gamma\in(1,\frac43)$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows-up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case $\gamma=1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof., Comment: 92 pages

Implementation attacks like side-channel and fault attacks pose a considerable threat to cryptographic devices that are physically accessible by an attacker. As a consequence, devices like smart cards implement corresponding countermeasures like redundant computation and masking. Recently, statistically ineffective fault attacks (SIFA) were shown to be able to circumvent these classical countermeasure techniques. We present a new approach for verifying the SIFA protection of arbitrary masked implementations in both hardware and software. The proposed method uses Boolean dependency analysis, factorization, and known properties of masked computations to show whether the fault detection mechanism of redundant masked circuits can leak information about the processed secret values. We implemented this new method in a tool called Danira, which can show the SIFA resistance of cryptographic implementations like AES S-Boxes within minutes., Comment: This is the extended version of the paper published at ATVA 2021

Road transportation is one of the largest sectors of greenhouse gas (GHG) emissions affecting climate change. Tackling climate change as a global community will require new capabilities to measure and inventory road transport emissions. However, the large scale and distributed nature of vehicle emissions make this sector especially challenging for existing inventory methods. In this work, we develop machine learning models that use satellite imagery to perform indirect top-down estimation of road transport emissions. Our initial experiments focus on the United States, where a bottom-up inventory was available for training our models. We achieved a mean absolute error (MAE) of 39.5 kg CO$_{2}$ of annual road transport emissions, calculated on a pixel-by-pixel (100 m$^{2}$) basis in Sentinel-2 imagery. We also discuss key model assumptions and challenges that need to be addressed to develop models capable of generalizing to global geography. We believe this work is the first published approach for automated indirect top-down estimation of road transport sector emissions using visual imagery and represents a critical step towards scalable, global, near-real-time road transportation emissions inventories that are measured both independently and objectively.

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically., Comment: 104 pages; some errors have been corrected

We propose a novel method for enforcing AI fairness with respect to protected or sensitive factors. This method uses a dual strategy performing training and representation alteration (TARA) for the mitigation of prominent causes of AI bias by including: a) the use of representation learning alteration via adversarial independence to suppress the bias-inducing dependence of the data representation from protected factors; and b) training set alteration via intelligent augmentation to address bias-causing data imbalance, by using generative models that allow the fine control of sensitive factors related to underrepresented populations via domain adaptation and latent space manipulation. When testing our methods on image analytics, experiments demonstrate that TARA significantly or fully debiases baseline models while outperforming competing debiasing methods that have the same amount of information, e.g., with (% overall accuracy, % accuracy gap) = (78.8, 0.5) vs. the baseline method's score of (71.8, 10.5) for EyePACS, and (73.7, 11.8) vs. (69.1, 21.7) for CelebA. Furthermore, recognizing certain limitations in current metrics used for assessing debiasing performance, we propose novel conjunctive debiasing metrics. Our experiments also demonstrate the ability of these novel metrics in assessing the Pareto efficiency of the proposed methods., Comment: Accepted for publication in MIT Neural Computation

Using numerical integration, in 1969 Penston [22] and Larson [17] independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state $p=k\varrho$, $k>0$, and subject to Newtonian gravity. We rigorously prove the existence of such a Larson-Penston solution., Comment: 44 pages

We introduce a deep learning approach to perform fine-grained population estimation for displacement camps using high-resolution overhead imagery. We train and evaluate our approach on drone imagery cross-referenced with population data for refugee camps in Cox's Bazar, Bangladesh in 2018 and 2019. Our proposed approach achieves 7.02% mean absolute percent error on sequestered camp imagery. We believe our experiments with real-world displacement camp data constitute an important step towards the development of tools that enable the humanitarian community to effectively and rapidly respond to the global displacement crisis., Comment: Fixed typo in abstract

Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960's Zel'dovich [50] and Wheeler [22] formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity., Comment: 33 pages, 1 figure

Roadway free-flow speed captures the typical vehicle speed in low traffic conditions. Modeling free-flow speed is an important problem in transportation engineering with applications to a variety of design, operation, planning, and policy decisions of highway systems. Unfortunately, collecting large-scale historical traffic speed data is expensive and time consuming. Traditional approaches for estimating free-flow speed use geometric properties of the underlying road segment, such as grade, curvature, lane width, lateral clearance and access point density, but for many roads such features are unavailable. We propose a fully automated approach, RasterNet, for estimating free-flow speed without the need for explicit geometric features. RasterNet is a neural network that fuses large-scale overhead imagery and aerial LiDAR point clouds using a geospatially consistent raster structure. To support training and evaluation, we introduce a novel dataset combining free-flow speeds of road segments, overhead imagery, and LiDAR point clouds across the state of Kentucky. Our method achieves state-of-the-art results on a benchmark dataset.

We study the model of close-packed dimers on planar lattices belonging to the family of modified rectangular (MR) fractals, whose members are enumerated by an integer $p\geq 2$, as well as on the non-planar 4-simplex fractal lattice. By applying an exact recurrence enumeration method, we determine the asymptotic forms for numbers of dimer coverings, and numerically calculate entropies per dimer in the thermodynamic limit, for a sequence of MR lattices with $2\leq p\leq8$ and for 4-simplex fractal. We find that the entropy per dimer on MR fractals is increasing function of the scaling parameter $p$, and for every considered $p$ it is smaller than the entropy per dimer of the same model on $4$-simplex lattice. Obtained results are discussed and compared with the results obtained previously on some translationally invariant and fractal lattices., Comment: 15 pages, 9 figures, 2 tables

Global existence for the nonisentropic compressible Euler equations with vacuum boundary for all adiabatic constants $\gamma > 1$ is shown through perturbations around a rich class of background nonisentropic affine motions. The notable feature of the nonisentropic motion lies in the presence of non-constant entropies, and it brings a new mathematical challenge to the stability analysis of nonisentropic affine motions. In particular, the estimation of the curl terms requires a careful use of algebraic, nonlinear structure of the pressure. With suitable regularity of the underlying affine entropy, we are able to adapt the weighted energy method developed for the isentropic Euler by Had\v{z}i\'c and Jang to the nonisentropic problem. For large $\gamma$ values, inspired by Shkoller and Sideris, we use time-dependent weights that allow some of the top-order norms to potentially grow as the time variable tends to infinity. We also exploit coercivity estimates here via the fundamental theorem of calculus in time variable for norms which are not top-order., Comment: 52 pages, to appear in Nonlinearity. Please see arXiv:2007.03849 for a recent development in this direction

We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the flow., Comment: 51 pages

This paper addresses the task of road safety assessment. An emerging approach for conducting such assessments in the United States is through the US Road Assessment Program (usRAP), which rates roads from highest risk (1 star) to lowest (5 stars). Obtaining these ratings requires manual, fine-grained labeling of roadway features in street-level panoramas, a slow and costly process. We propose to automate this process using a deep convolutional neural network that directly estimates the star rating from a street-level panorama, requiring milliseconds per image at test time. Our network also estimates many other road-level attributes, including curvature, roadside hazards, and the type of median. To support this, we incorporate task-specific attention layers so the network can focus on the panorama regions that are most useful for a particular task. We evaluated our approach on a large dataset of real-world images from two US states. We found that incorporating additional tasks, and using a semi-supervised training approach, significantly reduced overfitting problems, allowed us to optimize more layers of the network, and resulted in higher accuracy., Comment: 9 pages, 8 figures, WACV 2018

The classical model of an isolated selfrgavitating gaseous star is given by the Euler-Poisson system with a polytropic pressure law $P(\rho)=\rho^\gamma$, $\gamma>1$. For any $1<\gamma<\frac43$, we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface, with continuous mass absorption at the origin. The leading order singular behavior is described by an explicit collapsing solution of the pressureless Euler-Poisson system., Comment: 103 pages, 2 figures

We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $\kappa>0$. We prove their linear instability when $\kappa$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich \& Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, \& Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve., Comment: 93 pages; several proofs are revised and some previous errors corrected. In particular, Lemma 4.4, which is an important tool in the proof of Theorem 4.3 but was in part incorrect, also in the published version, has been corrected

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) and pass the obtained formula to a SAT solver for deciding the QBF. State-of-the-art expansion-based solvers process the given formula quantifier-block wise and recursively apply expansion until a solution is found. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques., Comment: To appear in proceedings of FMCAD 2018

We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer $p$ $(2\leq p<\infty)$, and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter $s$, while the polymer conformations are modelled by weighted Hamiltonian walks (HWs). Applying an exact method of recurrence equations we have found the asymptotic behavior of partition function $Z_N$ for closed HWs consisting of $N$ steps. We have established that $Z_N$ scales as $\omega^N \mu^{N^\sigma}$, where the critical exponent $\sigma$ in the stretched exponential term does not depend on $s$, and takes the value 1/2 for each fractal from the family. The constants $\omega$ and $\mu$ depend on both $p$ and $s$, and, in addition, $\mu$ depends on the parity of MR generator. Besides, we have calculated numerically the stiffness dependence of the polymer persistence length and various thermodynamic quantities (such as free and internal energy, specific heat and entropy), for a large set of members of MR family. We have found that semi-flexible compact polymers, on MR lattices, can exist only in the liquid-like (disordered) phase, whereas the crystal (ordered) phase has not appeared. Finally, the behavior of examined system at zero temperature has been discussed., Comment: 21 pages, 15 figures; introduction minorly revised, some references omitted and new ones added

Using recent developments in the theory of globally defined expanding compressible gases, we construct a class of global-in-time solutions to the compressible 3-D Euler-Poisson system without any symmetry assumptions in both the gravitational and the plasma case. Our allowed range of adiabatic indices includes, but is not limited to all $\gamma$ of the form $\gamma=1+\frac1n$, $n\in\mathbb N\setminus\{1\}$. The constructed solutions have initially small densities and a compact support. As $t\to\infty$ the density scatters to zero and the support grows at a linear rate in $t$., Comment: 29 pages

We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem., Comment: To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non Lin\'eaire

We introduce a novel generalization of Counterexample-Guided Inductive Synthesis (CEGIS) and instantiate it to yield a novel, competitive algorithm for solving Quantified Boolean Formulas (QBF). Current QBF solvers based on counterexample-guided expansion use a recursive approach which scales poorly with the number of quantifier alternations. Our generalization of CEGIS removes the need for this recursive approach, and we instantiate it to yield a simple and efficient algorithm for QBF solving. Lastly, this research is supported by a competitive, though straightforward, implementation of the algorithm, making it possible to study the practical impact of our algorithm design decisions, along with various optimizations., Comment: This is a **very** old version of the paper arXiv:1807.08964 and should be taken down. I did not know you could just replace papers, and did not know whether I could change authors and similar, so that is why I made a different submission. Please take it down

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the three-dimensional isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent $\gamma$ belongs to the interval $(1,\frac53]$, then these affine motions are nonlinearly stable. Small perturbations lead to globally-in-time defined solutions that remain in the vicinity of the manifold of affine motions, they remain smooth in the interior of their support, and no shocks are formed in the process. Our strategy relies on two key ingredients. We first provide a new interpretation of the affine motions using an (almost) invariant action of GL$^+(3)$ on the compressible Euler system. This transformation dictates a particular rescaling of time and a change of variables, which in turn exposes a stabilization mechanism induced by the expansion of the background affine motions when $\gamma\in(1,\frac53]$. We then switch to a Lagrangian description of the rescaled Euler system which reflects the geometry of expanding affine motions. We introduce new ideas with respect to the existing well-posedness frameworks and build high-order energy spaces to prove global-in-time stability, thereby making crucial use of the new stabilization effect, Comment: 43 pages, 1 figure, relevant references and one figure added

The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadzic and Shkoller [31,32] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly., Comment: 58 pages, 1 figure

We prove nonlinear stability of compactly supported expanding star-solutions of the mass-critical gravitational Euler-Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expanding rate of such solutions can be either self-similar or non-self-similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global-in-time radially symmetric solutions, which are not homogeneous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free-boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass-critical with respect to an invariant rescaling and the analysis is carried out in similarity variables.

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods., Comment: 43 pages

We consider the two dimensional free boundary Stefan problem describing the evolution of a spherically symmetric ice ball $\{r\leq \lambda(t)\}$. We revisit the pioneering analysis of [20] and prove the existence in the radial class of finite time melting regimes $$ \lambda(t)=\left\{\begin{array}{ll} (T-t)^{1/2}e^{-\frac{\sqrt{2}}{2}\sqrt{|\ln(T-t)|}+O(1)}\\ (c+o(1))\frac{(T-t)^{\frac{k+1}{2}}}{|\ln (T-t)|^{\frac{k+1}{2k}}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to T $$ which respectively correspond to the fundamental stable melting rate, and a sequence of codimension $k\in \Bbb N^*$ excited regimes. Our analysis fully revisits a related construction for the harmonic heat flow in [42] by introducing a new and canonical functional framework for the study of type II (i.e. non self similar) blow up. We also show a deep duality between the construction of the melting regimes and the derivation of a discrete sequence of global-in-time freezing regimes $$ \lambda_\infty - \lambda(t)\sim\left\{\begin{array}{ll} \frac{1}{\log t}\\ \frac{1}{t^{k}(\log t)^{2}}, \ \ k\in \Bbb N^*\end{array}\right. \quad\text{ as } t\to +\infty $$ which correspond respectively to the fundamental stable freezing rate, and excited regimes which are codimension $k$ stable., Comment: 70 pages, a few references added and typos corrected

The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result [28] in which we studied nearly spherical shapes., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1212.1422

Families of steady states of the spherically symmetric Einstein-Vlasov system are constructed, which are parametrized by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov-Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analyzed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrack-free user interaction online. In particular,binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDD-based configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multi-valued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is non-additive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiple-cost configuration is NP-hard in the input MDD. However, for solving two-cost configuration we develop a pseudo-polynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over real-world configuration models and product-catalogue datasets. Response times are generally within a fraction of a second even for very large instances.

We study small perturbations of the well-known family of Friedman-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the spacelike Cauchy hypersurfaces are diffeomorphic to T^3. These solutions model a quiet pressureless fluid in a dynamic spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our analysis takes place relative to a harmonic-type coordinate system, in which the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends the results of [38,44,42], where analogous results were proved for the Euler-Einstein system under the equations of state p = c_s^2 \rho, 0

We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer $b$ ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents $\gamma_1, \gamma_{11}$, and $\gamma_s$ which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method, for $2\le b\le 4$, we have obtained specific values for these exponents, for various type of polymer conformations. We discuss their mutual relations and their relations with other critical exponents pertinent to SAWs on the SG fractals., Comment: 15 pages, 3 figures

The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities., Comment: 50 pages, references added, minor typos corrected, to appear in Comm. Pure Appl. Math, abstract added for UK REF

The stability of static solutions of the spherically symmetric, asymptotically flat Einstein-Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main result is a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state. The estimate shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. This should be an essential step in a fully non-linear stability analysis for the Einstein-Vlasov system. In the present paper it is exploited for obtaining a linearized stability result., Comment: 35 pages

We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension $\sigma$ converge to solutions of the classical Stefan problem as $\sigma \to 0$., Comment: Various typos corrected and references added

We prove nonlinear asymptotic stability of steady spheres in the two-phase Stefan problem with surface tension. Our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high-order energy method., Comment: 25 pages, important references added, two remarks added, typos corrected

We describe a gradient flow structure for the inviscid Langmuir layer Stokesian subfluid model introduced recently by Alexander et al [1]., Comment: 6 pages