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Let $D$ be a digraph of order $n$ with adjacency matrix $A(D)$. For $\alpha\in[0,1)$, the $A_{\alpha}$ matrix of $D$ is defined as $A_{\alpha}(D)=\alpha {\Delta}^{+}(D)+(1-\alpha)A(D)$, where ${\Delta}^{+}(D)=\mbox{diag}~(d_1^{+},d_2^{+},\dots,d_n^{+})$ is the diagonal matrix of vertex outdegrees of $D$. Let $\sigma_{1\alpha}(D),\sigma_{2\alpha}(D),\dots,\sigma_{n\alpha}(D)$ be the singular values of $A_{\alpha}(D)$. Then the trace norm of $A_{\alpha}(D)$, which we call $\alpha$ trace norm of $D$, is defined as $\|A_{\alpha}(D)\|_*=\sum_{i=1}^{n}\sigma_{i\alpha}(D)$. In this paper, we find the singular values of some basic digraphs and characterize the digraphs $D$ with $\mbox{Rank}~(A_{\alpha}(D))=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_{\alpha}$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_{\alpha}$ matrix attains minimum. We obtain a lower bound for the $\alpha$ spectral norm $\sigma_{1\alpha}(D)$ of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the $\alpha$ trace norm of digraphs and characterize the extremal digraphs., Comment: 19 pages, 1 figure with 11 digraphs

Fractional flow reserve (FFR) is the gold standard for diagnosing coronary artery disease (CAD). FFRCT uses computational fluid dynamics (CFD) to evaluate FFR non-invasively by simulating coronary flow in geometries reconstructed from computed tomography (CT). However, it faces challenges related to the cost of computing and uncertainties in defining patient-specific boundary conditions (BCs). We investigated using time-averaged steady BCs instead of pulsatile ones to reduce computational time and deployed a self-adjusting method for tuning BCs to patient clinical data. 133 coronary arteries were reconstructed from CT images of CAD patients. For each vessel, invasive FFR was measured. Steady BCs for CFD were defined in two steps: i) rest BCs were extrapolated from clinical and image-derived data; ii) hyperemic BCs were computed from resting conditions. Flow rate was iteratively adjusted during the simulation until patient aortic pressure was matched. Pulsatile BCs were defined using the convergence values of steady BCs. Lesion-specific hemodynamic indexes were computed and compared between groups of patients indicated for surgery and those not. The whole pipeline was implemented as a straightforward, fully automated process. Steady and pulsatile FFRCT showed a strong correlation (r=0.988) and correlated with invasive FFR (r=0.797). The per-point difference between the pressure and FFRCT field predicted by the two methods was below 0.01 and 0.02, respectively. Both approaches exhibited good diagnostic performance: accuracy was 0.860 and 0.864, with AUCs of 0.923 and 0.912, for steady and pulsatile cases, respectively. Computational time for steady BCs CFD was approximately 30-fold lower than for the pulsatile case. This work demonstrates the feasibility of using steady BCs CFD for computing FFRCT in coronary arteries and its performance in a fully automated pipeline.

We solve the fifth-order Korteweg-de Vries (fKdV) equation which is a modified KdV equation perturbed by a fifth-order derivative term multiplied by a small parameter $\epsilon^2$, with $0< \epsilon \ll 1$. Unlike the KdV equation, the stationary fKdV equation does not exhibit exactly localized 1-soliton solution, instead it allows a solution which has a well defined central core similar to that of the KdV 1-soliton solution, accompanied by extremely small oscillatory standing wave tails on both sides of the core. The amplitude of the standing wave tail oscillations is $\mathcal{O}(\exp(-1/\epsilon))$, i.e. it is beyond all orders small in perturbation theory. The analytical computation of the amplitude of these transcendentally small tail oscillations has been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections by using the complex method of matched asymptotics. Also the long-standing discrepancy between the $\mathcal{O}(\epsilon^2)$ perturbative result of Grimshaw and Joshi (1995) and the numerical results of Boyd (1995) has been resolved. In addition to the stationary symmetric weakly localized solitary wave-like solutions, we analyzed the stationary asymmetric solutions of the fKdV equation which decay exponentially to zero on one side of the (slightly asymmetric) core and blows up to large negative values on other side of the core. The asymmetry is quantified by computing the third derivative of the solution at the origin which also turns out to be beyond all orders small in perturbation theory. The analytical computation of the third derivative of a function at the origin has also been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections. We use the exponentially convergent pseudo-spectral method to solve the fKdV equation numerically. The analytical and the numerical results show remarkable agreement., Comment: A thesis submitted to the institute of Physics (E\"otv\"os Lor\'and University, Budapest, Hungary) in partial fulfillment of the requirements for the degree of Masters of Science Physics

The recently introduced structured input-output analysis is a powerful method for capturing nonlinear phenomena associated with incompressible flows, and this paper extends that method to the compressible regime. The proposed method relies upon a reformulation of the compressible Navier-Stokes equations, which allows for an exact quadratic formulation of the dynamics of perturbations about a steady base flow. To facilitate the structured input-output analysis, a pseudo-linear model for the quadratic nonlinearity is proposed and the structural information of the nonlinearity is embedded into a structured uncertainty comprising unknown `perturbations'. The structured singular value framework is employed to compute the input-output gain, which provides an estimate of the robust stability margin of the flow perturbations, as well as the forcing and response modes that are consistent with the nonlinearity structure. The analysis is then carried out on a plane, laminar compressible Couette flow over a range of Mach numbers. The structured input-output gains identify an instability mechanism, characterized by a spanwise elongated structure in the streamwise-spanwise wavenumber space at a subsonic Mach number, that evolves into an oblique structure at sonic and supersonic Mach numbers. In addition, the structured input-output forcing and response modes provide insight into the thermodynamic and momentum characteristics associated with a source of instability. Comparisons with a resolvent/unstructured analysis reveal discrepancies in the distribution of input-output gains over the wavenumber space as well as in the modal behavior of an instability, thus highlighting the strong correlation between the structural information of the nonlinearity and the underlying flow physics., Comment: Submitted to the Journal of Fluid Mechanics

Continual self-supervised learning (CSSL) learns a series of tasks sequentially on the unlabeled data. Two main challenges of continual learning are catastrophic forgetting and task confusion. While CSSL problem has been studied to address the catastrophic forgetting challenge, little work has been done to address the task confusion aspect. In this work, we show through extensive experiments that self-supervised learning (SSL) can make CSSL more susceptible to the task confusion problem, particularly in less diverse settings of class incremental learning because different classes belonging to different tasks are not trained concurrently. Motivated by this challenge, we present a novel cross-model feature Mixup (CroMo-Mixup) framework that addresses this issue through two key components: 1) Cross-Task data Mixup, which mixes samples across tasks to enhance negative sample diversity; and 2) Cross-Model feature Mixup, which learns similarities between embeddings obtained from current and old models of the mixed sample and the original images, facilitating cross-task class contrast learning and old knowledge retrieval. We evaluate the effectiveness of CroMo-Mixup to improve both Task-ID prediction and average linear accuracy across all tasks on three datasets, CIFAR10, CIFAR100, and tinyImageNet under different class-incremental learning settings. We validate the compatibility of CroMo-Mixup on four state-of-the-art SSL objectives. Code is available at \url{https://github.com/ErumMushtaq/CroMo-Mixup}., Comment: ECCV 2024

Large Language Models (LLMs) have become ubiquitous in NLP and deep learning. In-Context Learning (ICL) has been suggested as a bridging paradigm between the training-free and fine-tuning LLMs settings. In ICL, an LLM is conditioned to solve tasks by means of a few solved demonstration examples included as prompt. Argument Mining (AM) aims to extract the complex argumentative structure of a text, and Argument Type Classification (ATC) is an essential sub-task of AM. We introduce an ICL strategy for ATC combining kNN-based examples selection and majority vote ensembling. In the training-free ICL setting, we show that GPT-4 is able to leverage relevant information from only a few demonstration examples and achieve very competitive classification accuracy on ATC. We further set up a fine-tuning strategy incorporating well-crafted structural features given directly in textual form. In this setting, GPT-3.5 achieves state-of-the-art performance on ATC. Overall, these results emphasize the emergent ability of LLMs to grasp global discursive flow in raw text in both off-the-shelf and fine-tuned setups.

The dynamics of nonlinear ion-acoustic solitary waves in the presence of kinetic (Landau type) damping have been investigated in a collisionless, non-magnetized electron-ion plasma. A cold ion fluid model, coupled to a Vlasov-type kinetic equation for the electron dynamics, has been adopted as a starting point. The electron population was assumed to be in a kappa-distributed state, in account of the non-Maxwellian behavior of energetic (suprathermal) electrons often observed in Space. A multiscale perturbation technique has led to an evolution equation for the electrostatic potential, in the form of a modified Korteweg-de Vries (KdV) equation, incorporating a non-local term accounting for Landau damping (associated with the electron statistics). Exact analytical solutions have been obtained, representing solitary waves undergoing amplitude decay over time. The combined effect of Landau damping and non-Maxwellian electron statistics (via the kappa parameter) on the characteristics of IASWs has been examined. Numerical integration of the evolution equation has been undertaken, to elucidate the importance of kinetic Landau damping on a shock-shaped initial condition. The results of this investigation aim to improve our understanding of the dynamics of nonlinear electrostatic waves under the influence of Landau damping in various space plasma environments., Comment: 18 pages, 12 figures [Accepted version]

With the utilization of Transformer architecture, large Vision and Language (V&L) models have shown promising performance in even zero-shot settings. Several studies, however, indicate a lack of robustness of the models when dealing with complex linguistics and visual attributes. In this work, we introduce a novel V&L benchmark - ColorFoil, by creating color-related foils to assess the models' perception ability to detect colors like red, white, green, etc. We evaluate seven state-of-the-art V&L models including CLIP, ViLT, GroupViT, and BridgeTower, etc. in a zero-shot setting and present intriguing findings from the V&L models. The experimental evaluation indicates that ViLT and BridgeTower demonstrate much better color perception capabilities compared to CLIP and its variants and GroupViT. Moreover, CLIP-based models and GroupViT struggle to distinguish colors that are visually distinct to humans with normal color perception ability.

The computation of the amplitude, $\alpha$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, $\epsilon^2\ll1$, a new derivation of the ``beyond all orders in $\epsilon$'' amplitude, $\alpha$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $\epsilon$, that the value of $\alpha$ can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $\alpha$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $\epsilon$, which are beyond the capabilities of standard numerical methods., Comment: 25 pages, 6 figures. Two Mathematica notebooks and a C code are provided as ancillary files