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We consider the Cahn--Hilliard/Navier--Stokes system with non-degenerate mobility in the space-periodic case, describing the flow of two viscous immiscible and incompressible Newtonian fluids with matched densities. We identify sufficient conditions on the velocity field for weak solutions to satisfy an energy identity, improving previous results on the literature.

In this paper we give a short and self-contained proof of the fact that weak solutions to the Maxwell-Stefan system automatically satisfy an entropy equality, establishing the absence of anomalous dissipation.

This is the second part of the research project initiated in Cleanthous et al (2024). We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from independent observations. The unknown density is assumed to be uniformly bounded by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. In Cleanthous et al (2024) we have proved that the optimally adaptive estimators do no exist in the considered problem and provided with several lower bounds for the adaptive risk. In this part we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators. The proposed estimation procedure as well as the computation of its risk are heavily based on new concentration inequalities for decoupled $U$-statistics of order two established in Section 4. It is also worth noting that all our results are derived from the unique oracle inequality which may be of independent interest.

We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from independent observations. The unknown density is assumed to be uniformly bounded and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. We will show that the optimally adaptive estimators over the collection of considered functional classes do no exist. Also, in the framework of an abstract density model we present several generic lower bounds related to the adaptive estimation of an arbitrary functional of a probability density. These results having independent interest have no analogue in the existing literature. In the companion paper Cleanthous et al (2024) we prove that established lower bounds are tight and provide with explicit construction of adaptive estimators of $\mathbb{L}_2$-norm of the density.

The Hayashi-Yoshida (\HY)-estimator exhibits an intrinsic, telescoping property that leads to an often overlooked computational bias, which we denote,formulaic or intrinsic bias. This formulaic bias results in data loss by cancelling out potentially relevant data points, the nonextant data points. This paper attempts to formalize and quantify the data loss arising from this bias. In particular, we highlight the existence of nonextant data points via a concrete example, and prove necessary and sufficient conditions for the telescoping property to induce this type of formulaic bias.Since this type of bias is nonexistent when inputs, i.e., observation times, $\Pi^{(1)} :=(t_i^{(1)})_{i=0,1,\ldots}$ and $\Pi^{(2)} :=(t_j^{(2)})_{j=0,1,\ldots}$, are synchronous, we introduce the (a,b)-asynchronous adversary. This adversary generates inputs $\Pi^{(1)}$ and $\Pi^{(2)}$ according to two independent homogenous Poisson processes with rates a>0 and b>0, respectively. We address the foundational questions regarding cumulative minimal (or least) average data point loss, and determine the values for a and b. We prove that for equal rates a=b, the minimal average cumulative data loss over both inputs is attained and amounts to 25\%. We present an algorithm, which is based on our theorem, for computing the exact number of nonextant data points given inputs $\Pi^{(1)}$ and $\Pi^{(2)}$, and suggest alternative methods. Finally, we use simulated data to empirically compare the (cumulative) average data loss of the (\HY)-estimator., Comment: submitted. 15 pages, 1 appendix, 4 figures, 1 table, 1 algo

We lay down the foundation of the theory of spaces of distributions on the product $X_1\times X_2$ of doubling metric measure spaces $X_1$, $X_2$ in the presence of non-negative self-adjoint operators $L_1$, $L_2$, whose heat kernels have Gaussian localization and the Markov property. This theory includes the development of two-parameter functional calculus induced by $L_1, L_2$, integral operators with highly localized kernels, test functions and distributions associated to $L_1, L_2$, spectral spaces accompanied by maximal Peetre and Nikolski type inequalities. Hardy spaces are developed in this two-parameter product setup. Two types of Besov and Triebel-Lizorkin spaces are introduced and studied: ordinary spaces and spaces with dominating mixed smoothness, with emphasis on the latter. Embedding results are obtained and spectral multiplies are developed.

The derivation of an approximate Class-I model for nonisothermal multicomponent systems of fluids, as the high-friction limit of a Class-II model is justified, by validating the Chapman-Enskog expansion performed from the Class-II model towards the Class-I model. The analysis proceeds by comparing two thermomechanical theories via relative entropy.

We give conditions that guarantee uniqueness of renormalized solutions for the Maxwell-Stefan system. The proof is based on an identity for the evolution of the symmetrized relative entropy. Using the method of doubling the variables we derive the identity for two renormalized solutions and use information on the spectrum of the Maxwell-Stefan matrix to estimate the symmetrized relative entropy and show uniqueness. We then show that weak solutions for the Maxwell-Stefan system have sufficient regularity to produce renormalized solutions. Combining the two results yields a uniqueness result for weak solutions of the Maxwell-Stefan system with bounded fluxes.

We study the problem of efficiently answering strong connectivity queries under two vertex failures. Given a directed graph $G$ with $n$ vertices, we provide a data structure with $O(nh)$ space and $O(h)$ query time, where $h$ is the height of a decomposition tree of $G$ into strongly connected subgraphs. This immediately implies data structures with $O(n \log{n})$ space and $O(\log{n})$ query time for graphs of constant treewidth, and $O(n^{3/2})$ space and $O(\sqrt{n})$ query time for planar graphs. For general directed graphs, we give a refined version of our data structure that achieves $O(n\sqrt{m})$ space and $O(\sqrt{m})$ query time, where $m$ is the number of edges of the graph. We also provide some simple BFS-based heuristics that seem to work remarkably well in practice. In the experimental part, we first evaluate various methods to construct a decomposition tree with small height $h$ in practice. Then we provide efficient implementations of our data structures, and evaluate their empirical performance by conducting an extensive experimental study on graphs taken from real-world applications., Comment: Conference version to appear in the proceedings of ALENEX 2024

We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier-Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier-Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.

Spatial join processing techniques that identify intersections between complex geometries (e.g.,polygons) commonly follow a two-step filter-and-refine pipeline; the filter step evaluates the query predicate on the minimum bounding rectangles (MBRs) of objects and the refinement step eliminates false positives by applying the query on the exact geometries. We propose a raster intervals approximation of object geometries and introduce a powerful intermediate step in pipeline. In a preprocessing phase, our method (i) rasterizes each object geometry using a fine grid, (ii) models groups of nearby cells that intersect the polygon as an interval, and (iii) encodes each interval by a bitstring that captures the overlap of each cell in it with the polygon. Going one step further, we improve our approach to approximate each object by two sets of intervals that succintly capture the raster cells which (i) intersect with the object and (ii) are fully contained in the object. Using this representation, we show that we can verify whether two polygons intersect by a sequence of joins between the interval sets that take linear time. Our approximations can effectively be compressed and can be customized for use on partitioned data and polygons of varying sizes, rasterized at different granularities. Finally, we propose a novel algorithm that computes the interval approximation of a polygon without fully rasterizing it first, rendering the computation of approximations orders of magnitude faster. Experiments on real data demonstrate the effectiveness and efficiency of our proposal over previous work., Comment: 34 pages

Space-time memory (STM) network methods have been dominant in semi-supervised video object segmentation (SVOS) due to their remarkable performance. In this work, we identify three key aspects where we can improve such methods; i) supervisory signal, ii) pretraining and iii) spatial awareness. We then propose TrickVOS; a generic, method-agnostic bag of tricks addressing each aspect with i) a structure-aware hybrid loss, ii) a simple decoder pretraining regime and iii) a cheap tracker that imposes spatial constraints in model predictions. Finally, we propose a lightweight network and show that when trained with TrickVOS, it achieves competitive results to state-of-the-art methods on DAVIS and YouTube benchmarks, while being one of the first STM-based SVOS methods that can run in real-time on a mobile device., Comment: Accepted to ICIP 2023

Muscle health is a critical component of overall health and quality of life. However, current measures of skeletal muscle health take limited account of microstructural variations within muscle, which play a crucial role in mediating muscle function. To address this, we present a physics-inspired, machine learning-based framework for the non-invasive and in vivo estimation of microstructural organization in skeletal muscle from diffusion-weighted MRI (dMRI). To reduce the computational expense associated with direct numerical simulations of dMRI physics, a polynomial meta-model is developed that accurately represents the input/output relationships of a high-fidelity numerical model. This meta-model is used to develop a Gaussian process (GP) model to provide voxel-wise estimates and confidence intervals of microstructure organization in skeletal muscle. Given noise-free data, the GP model accurately estimates microstructural parameters. In the presence of noise, the diameter, intracellular diffusion coefficient, and membrane permeability are accurately estimated with narrow confidence intervals, while volume fraction and extracellular diffusion coefficient are poorly estimated and exhibit wide confidence intervals. A reduced-acquisition GP model, consisting of one-third the diffusion-encoding measurements, is shown to predict parameters with similar accuracy to the original model. The fiber diameter and volume fraction estimated by the reduced GP model is validated via histology, with both parameters within their associated confidence intervals, demonstrating the capability of the proposed framework as a promising non-invasive tool for assessing skeletal muscle health and function., Comment: 11 pages, 5 figures, 2 tables

Traumatic axonal injury occurs when loads experienced on the tissue-scale are transferred to the individual axons. Mechanical characterization of axon deformation especially under dynamic loads however is extremely difficult owing to their viscoelastic properties. The viscoelastic characterization of axon properties that are based on interpretation of results from in-vivo brain Magnetic Resonance Elastography (MRE) are dependent on the specific frequencies used to generate shear waves with which measurements are made. In this study, we aim to develop a fractional viscoelastic model to characterize the time dependent behavior of the properties of the axons in a composite white matter (WM) model. The viscoelastic powerlaw behavior observed at the tissue level is assumed to exist across scales, from the continuum macroscopic level to that of the microstructural realm of the axons. The material parameters of the axons and glia are fitted to a springpot model. The 3D fractional viscoelastic springpot model is implemented within a finite element framework. The constitutive equations defining the fractional model are coded using a vectorized user defined material (VUMAT) subroutine in ABAQUS finite element software. Using this material characterization, representative volume elements (RVE) of axons embedded in glia with periodic boundary conditions are developed and subjected to a relaxation displacement boundary condition. The homogenized orthotropic fractional material properties of the axon-matrix system as a function of the volume fraction of axons in the ECM are extracted by solving the inverse problem., Comment: Accepted for publication at the 12th International Conference on Mathematical Modeling in Physical Sciences

We are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general H\"{o}lder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.

The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell-Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modeling. Furthermore, the weak-strong uniqueness property is shown by using the relative entropy method.

Document shadow removal is an integral task in document enhancement pipelines, as it improves visibility, readability and thus the overall quality. Assuming that the majority of practical document shadow removal scenarios require real-time, accurate models that can produce high-resolution outputs in-the-wild, we propose Laplacian Pyramid with Input/Output Attention Network (LP-IOANet), a novel pipeline with a lightweight architecture and an upsampling module. Furthermore, we propose three new datasets which cover a wide range of lighting conditions, images, shadow shapes and viewpoints. Our results show that we outperform the state-of-the-art by a 35% relative improvement in mean average error (MAE), while running real-time in four times the resolution (of the state-of-the-art method) on a mobile device., Comment: ICASSP 2023

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is either $(u,v)$ or $(v,u)$. The problem of finding an orientation of a mixed graph that makes it strongly connected is well understood and can be solved in linear time. Here we introduce the following orientation problem in mixed graphs. Given a mixed graph $G$, we wish to compute its maximal sets of vertices $C_1,C_2,\ldots,C_k$ with the property that by removing any edge $e$ from $G$ (directed or undirected), there is an orientation $R_i$ of $G\setminus{e}$ such that all vertices in $C_i$ are strongly connected in $R_i$. We discuss properties of those sets, and we show how to solve this problem in linear time by reducing it to the computation of the $2$-edge twinless strongly connected components of a directed graph. A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. A $2$-edge twinless strongly connected component (2eTSCC) of $G$ is a maximal subset of vertices $C$ such that any two vertices $u, v \in C$ are in the same twinless strongly connected component of $G \setminus e$, for any edge $e$. These concepts are motivated by several diverse applications, such as the design of road and telecommunication networks, and the structural stability of buildings.

A type-I model of non-isothermal multicomponent systems of gases describing mass diffusive and heat conductive phenomena is presented. The derivation of the model and a convergence result among thermomechanical theories in the smooth regime are discussed. Furthermore, the global-in-time existence of weak solutions and the weak-strong uniqueness property are established for the corresponding system with zero barycentric velocity., Comment: Submitted to the Proceedings of XVIII International Conference on Hyperbolic Problems: Theory, Numerics, Applications (HYP2022)

Topographic feature maps are low dimensional representations of data, that preserve spatial dependencies. Current methods of training such maps (e.g. self organizing maps - SOM, generative topographic maps) require centralized control and synchronous execution, which restricts scalability. We present an algorithm that uses $N$ autonomous units to generate a feature map by distributed asynchronous training. Unit autonomy is achieved by sparse interaction in time \& space through the combination of a distributed heuristic search, and a cascade-driven weight updating scheme governed by two rules: a unit i) adapts when it receives either a sample, or the weight vector of a neighbor, and ii) broadcasts its weight vector to its neighbors after adapting for a predefined number of times. Thus, a vector update can trigger an avalanche of adaptation. We map avalanching to a statistical mechanics model, which allows us to parametrize the statistical properties of cascading. Using MNIST, we empirically investigate the effect of the heuristic search accuracy and the cascade parameters on map quality. We also provide empirical evidence that algorithm complexity scales at most linearly with system size $N$. The proposed approach is found to perform comparably with similar methods in classification tasks across multiple datasets., Comment: 11 Pages, 8 Figures

We show how to find and efficiently maintain maximal k-edge-connected subgraphs in undirected graphs. In particular, we provide the following results. (1) A general framework for maintaining the maximal k-edge-connected subgraphs upon insertions of edges or vertices, by successively partitioning the graph into its k-edge-connected components. This defines a decomposition tree, which can be maintained by using algorithms for the incremental maintenance of the k-edge-connected components as black boxes at every level of the tree. (2) As an application of this framework, we provide two algorithms for the incremental maintenance of the maximal $3$-edge-connected subgraphs. These algorithms allow for vertex and edge insertions, interspersed with queries asking whether two vertices belong to the same maximal $3$-edge-connected subgraph. The first algorithm has $O(m\alpha(m,n) + n^2\log^2 n)$ total running time and uses $O(n)$ space, where $m$ is the number of edge insertions and queries, and $n$ is the total number of vertices inserted. The second algorithm performs the same operations in faster $O(m\alpha(m,n) + n^2\alpha(n,n))$ time in total, using $O(n^2)$ space. (3) We provide efficient constructions of sparse subgraphs that have the same maximal k-edge-connected subgraphs as the original graph. These are useful in speeding up computations involving the maximal k-edge-connected subgraphs in dense undirected graphs. (4) We give two deterministic algorithms for computing the maximal k-edge-connected subgraphs in undirected graphs, with running times $O(m+k^{O(1)}n\sqrt{n}\mathrm{polylog}(n))$ and $O(m+k^{O(k)}n\sqrt{n}\log{n})$, respectively. (5) A fully dynamic algorithm for maintaining information about the maximal k-edge-connected subgraphs for fixed k. Our update bounds are $O(n\sqrt{n}\log{n})$ worst-case time, and we achieve constant time for maximal k-edge-connected subgraph queries.

Bokeh effect highlights an object (or any part of the image) while blurring the rest of the image, and creates a visually pleasant artistic effect. Due to the sensor-based limitations on mobile devices, machine learning (ML) based bokeh rendering has gained attention as a reliable alternative. In this paper, we focus on several improvements in ML-based bokeh rendering; i) on-device performance with high-resolution images, ii) ability to guide bokeh generation with user-editable masks and iii) ability to produce varying blur strength. To this end, we propose Adaptive Mask-based Pyramid Network (AMPN), which is formed of a Mask-Guided Bokeh Generator (MGBG) block and a Laplacian Pyramid Refinement (LPR) block. MGBG consists of two lightweight networks stacked to each other to generate the bokeh effect, and LPR refines and upsamples the output of MGBG to produce the high-resolution bokeh image. We achieve i) via our lightweight, mobile-friendly design choices, ii) via the stacked-network design of MGBG and the weakly-supervised mask prediction scheme and iii) via manually or automatically editing the intensity values of the mask that guide the bokeh generation. In addition to these features, our results show that AMPN produces competitive or better results compared to existing methods on the EBB! dataset, while being faster and smaller than the alternatives., Comment: ECCV 2022 Advances in Image Manipulation Workshop. See the workshop website for posters and recordings