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Phenomenological implications of the Mimetic Tensor-Vector-Scalar theory (MiTeVeS) are studied. The theory is an extension of the vector field model of mimetic dark matter, where a scalar field is also incorporated, and it is known to be free from ghost instability. In the absence of interactions between the scalar field and the vector field, the obtained cosmological solution corresponds to the General theory of Relativity (GR) with a minimally-coupled scalar field. However, including an interaction term between the scalar field and the vector field yields interesting dynamics. There is a shift symmetry for the scalar field with a flat potential, and the conserved Noether current, which is associated with the symmetry, behaves as a dark matter component. Consequently, the solution contains a cosmological constant, dark matter and a stiff matter fluid. Breaking the shift symmetry with a non-flat potential gives a natural interaction between dark energy and dark matter. Comment: Accepted in Phys. Dark Uni

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various non-linearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems. Comment: 11 pages

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex $v$ has its own valuation of the proposal; we say that $v$ is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex $v$ is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested in establishing necessary and sufficient conditions for the class $\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions. Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied Mathematics

The number of studies on deep learning for medical diagnosis is expanding, and these systems are often claimed to outperform clinicians. However, only a few systems have shown medical efficacy. From this perspective, we examine a wide range of deep learning algorithms for the assessment of glioblastoma - a common brain tumor in older adults that is lethal. Surgery, chemotherapy, and radiation are the standard treatments for glioblastoma patients. The methylation status of the MGMT promoter, a specific genetic sequence found in the tumor, affects chemotherapy's effectiveness. MGMT promoter methylation improves chemotherapy response and survival in several cancers. MGMT promoter methylation is determined by a tumor tissue biopsy, which is then genetically tested. This lengthy and invasive procedure increases the risk of infection and other complications. Thus, researchers have used deep learning models to examine the tumor from brain MRI scans to determine the MGMT promoter's methylation state. We employ deep learning models and one of the largest public MRI datasets of 585 participants to predict the methylation status of the MGMT promoter in glioblastoma tumors using MRI scans. We test these models using Grad-CAM, occlusion sensitivity, feature visualizations, and training loss landscapes. Our results show no correlation between these two, indicating that external cohort data should be used to verify these models' performance to assure the accuracy and reliability of deep learning systems in cancer diagnosis. Comment: 12 pages, 10 figures, MedIA

An upper bound for the Lebesgue constant (the supremum norm) of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to n is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to n for an arbitrary d-dimensional simplex was established by the author. The explicit upper bound proved in this article refines this result for d=2.

In principal-agent models, a principal offers a contract to an agent to perform a certain task. The agent exerts a level of effort that maximizes her utility. The principal is oblivious to the agent's chosen level of effort, and conditions her wage only on possible outcomes. In this work, we consider a model in which the principal is unaware of the agent's utility and action space: she sequentially offers contracts to identical agents, and observes the resulting outcomes. We present an algorithm for learning the optimal contract under mild assumptions. We bound the number of samples needed for the principal to obtain a contract that is within $\eps$ of her optimal net profit for every $\eps>0$. Our results are robust even when considering risk-averse agents. Furthermore, we show that when there are only two possible outcomes or the agent is risk-neutral, the algorithm's outcome approximates the optimal contract described in the classical theory.

We address static and dynamical properties of one-dimensional (1D) quantum droplets (QDs) under the action of local potentials in the form of narrow wells and barriers. The QDs are governed by the 1D Gross-Pitaevskii equation including the mean-field cubic repulsive term and the beyond-mean-field attractive quadratic one. In the case of the well represented by the delta-functional potential, three exact stable solutions are found for localized states pinned to the well. The Thomas-Fermi approximation for the well and the adiabatic approximation for the collision of the QD with the barrier are developed too. Collisions of incident QDs with the wells and barriers are analyzed in detail by means of systematic simulations. Outcomes, such as fission of the moving QD into transmitted, reflected, and trapped fragments, are identified in relevant parameter planes. In particular, a counter-intuitive effect of partial or full rebound of the incident QD from the potential well is studied in detail and qualitatively explained. Comment: 13 pages, 32 figures; To be published in Communions in Nonlinear Science and Numerical Simulation

Existing online knowledge distillation approaches either adopt the student with the best performance or construct an ensemble model for better holistic performance. However, the former strategy ignores other students' information, while the latter increases the computational complexity during deployment. In this paper, we propose a novel method for online knowledge distillation, termed FFSD, which comprises two key components: Feature Fusion and Self-Distillation, towards solving the above problems in a unified framework. Different from previous works, where all students are treated equally, the proposed FFSD splits them into a leader student and a common student set. Then, the feature fusion module converts the concatenation of feature maps from all common students into a fused feature map. The fused representation is used to assist the learning of the leader student. To enable the leader student to absorb more diverse information, we design an enhancement strategy to increase the diversity among students. Besides, a self-distillation module is adopted to convert the feature map of deeper layers into a shallower one. Then, the shallower layers are encouraged to mimic the transformed feature maps of the deeper layers, which helps the students to generalize better. After training, we simply adopt the leader student, which achieves superior performance, over the common students, without increasing the storage or inference cost. Extensive experiments on CIFAR-100 and ImageNet demonstrate the superiority of our FFSD over existing works. The code is available at https://github.com/SJLeo/FFSD. Comment: Accepted by IEEE Transactions on Neural Networks and Learning Systems (IEEE TNNLS)