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An $n$-mean (also called a ''topological social choice rule'') on a topological space $X$ is a continuous function $p:X^n\to X$ satisfying $p(x,\dots, x)=x$ for every $x\in X$ and $p(x_1,\dots, x_n)=p(x_{\sigma(1)},\dots x_{\sigma(n)})$ for any permutation $\sigma$ of $\{1,\dots, n\}$. If, in addition, $X$ is a $G$-space and $p$ is equivariant with respect to the diagonal action of $G$ on $X^n$, we say that $p$ is an equivariant $n$-mean. In this paper, we continue the work initiated by H. Ju\'arez-Anguiano about conditions on a $G$-space $X$, under which the existence of an equivariant $n$-mean guarantees that $X$ is a $G$-AR. We also explore this problem when we remove the symmetry condition on the definition of an $n$-mean., Comment: 19 pages

Nonreciprocity, which denotes the asymmetric or even unidirectional transmission of light, constitutes the cornerstone of modern photonic circuits. In the realm of photonic devices, it has been widely utilized in isolators, circulators and so on. Recent topology in artificial materials, an unprecedented degree of freedom, has been proposed to solve the effect of impurities on nonreciprocal transmission. However, in view of the bulk-edge correspondence, the spatial width of the transmission channel with uniform field distribution is quite narrow and needs further exploration. In this paper, we proposed a one-way valley-locked waveguide with a large channel in an all-dielectric photonic crystal. Quite different from the topological edge modes, the unidirectional property of our waveguide comes from the bulk modes with valley-lock, which can fully utilize the whole dimension of the structure with an efficiency of 100%. Additionally, the electrical field is uniformly distributed across the entire channel, which opens a new avenue for low-loss nonreciprocity devices., Comment: 16 pages and 5 figures

Facial Expression Recognition (FER) is a machine learning problem that deals with recognizing human facial expressions. While existing work has achieved performance improvements in recent years, FER in the wild and under challenging conditions remains a challenge. In this paper, a lightweight patch and attention network based on MobileNetV1, referred to as PAtt-Lite, is proposed to improve FER performance under challenging conditions. A truncated ImageNet-pre-trained MobileNetV1 is utilized as the backbone feature extractor of the proposed method. In place of the truncated layers is a patch extraction block that is proposed for extracting significant local facial features to enhance the representation from MobileNetV1, especially under challenging conditions. An attention classifier is also proposed to improve the learning of these patched feature maps from the extremely lightweight feature extractor. The experimental results on public benchmark databases proved the effectiveness of the proposed method. PAtt-Lite achieved state-of-the-art results on CK+, RAF-DB, FER2013, FERPlus, and the challenging conditions subsets for RAF-DB and FERPlus., Comment: Copyright 2024 IEEE. Personal use of this material is permitted. IEEE Access 2024

Let $k \geq 3$. If a multiplicative function $f$ satisfies \[ f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, then $f$ is the identity function. The set of positive cubes is said to be a $k$-additive uniqueness set for multiplicative functions. But, the condition for $k=2$ can be satisfied by infinitely many multiplicative functions. Besides, if $k \geq 3$ and a multiplicative function $g$ satisfies \[ g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, then $g$ is the identity function. However, when $k=2$, there exist three different types of multiplicative functions., Comment: 15 pages, several Mathematica codes

Photonic topological phases offering unprecedented manipulation of electromagnetic waves have attracted much research interest which, however, have been mostly restricted to a single band gap. Here, we report on the experimental discovery of hybrid topological photonic crystals which host simultaneously quantum anomalous Hall and valley Hall phases in different photonic band gaps. The underlying hybrid topological phase manifests itself in the edge responses as the coexistence of the chiral edge states and valley Hall edge states in different frequency ranges. We experimentally verify such an emergent phenomenon and show that such a feature enables novel multiplexing of photon transport in the edge channels. Our study reveals a situation with coexisting topology of distinct nature in a single photonic system that may enable frequency-dependent filtering and manipulation of topological edge photons.

Topological phases of matter are featured with exotic edge states. However, the fractional topological numbers at edges, though predicted long ago by Jackiw and Rebbi, remain elusive in topological photonic systems. Here, we report on the observation of fractional topological numbers at the topological edges and corners in one- and two-dimensional photonic crystals. The fractional topological numbers are determined via the measurements of the photonic local density-of-states. In one-dimensional photonic crystals, we witness a rapid change of the fractional topological number at the edges rising from 0 to 1/2 when the photonic band gap experiences a topological transition, confirming the well-known prediction of Jackiw and Rebbi. In two-dimensional systems, we discover that the fractional topological number in the corner region varies from 0 to 1/2 and 1/4 in different photonic band gap phases. Our study paves the way toward topological manipulation of fractional quantum numbers in photonics., Comment: All comments are welcome

If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e., \[ f(x^2+xy+y^2) = f(x)^2 + f(x)\,f(y) + f(y)^2, \] then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form $x^2-xy+y^2$, then $f$ is one of three kinds of functions: the identity function, the constant function, and an indicator function for $\mathbb{N}\setminus p\mathbb{N}$ with a prime $p$., Comment: Under review

This paper introduces a straightforward sieve-based approach for estimating and conducting inference on regression parameters in panel data models with interactive fixed effects. The method's key assumption is that factor loadings can be decomposed into an unknown smooth function of individual characteristics plus an idiosyncratic error term. Our estimator offers advantages over existing approaches by taking a simple partial least squares form, eliminating the need for iterative procedures or preliminary factor estimation. In deriving the asymptotic properties, we discover that the limiting distribution exhibits a discontinuity that depends on how well our basis functions explain the factor loadings, as measured by the variance of the error factor loadings. This finding reveals that conventional ``plug-in'' methods using the estimated asymptotic covariance can produce excessively conservative coverage probabilities. We demonstrate that uniformly valid non-conservative inference can be achieved through the cross-sectional bootstrap method. Monte Carlo simulations confirm the estimator's strong performance in terms of mean squared error and good coverage results for the bootstrap procedure. We demonstrate the practical relevance of our methodology by analyzing growth rate determinants across OECD countries.

We discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. We characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group $G$ on a topological space $X$ induces a continuous action on $CL(X)$.

Most natural and artificial materials have crystalline structures from which abundant topological phases emerge [1-6]. The bulk-edge correspondence, widely-adopted in experiments to determine the band topology from edge properties, however, becomes inadequate in discerning various topological crystalline phases [7-17], leading to great challenges in the experimental classification of the large family of topological crystalline materials [4-6]. Theories predict that disclinations, ubiquitous crystallographic defects, provide an effective probe of crystalline topology beyond edges [18-21], which, however, has not yet been confirmed in experiments. Here, we report the experimental discovery of the bulk-disclination correspondence which is manifested as the fractional spectral charge and robust bound states at the disclinations. The fractional disclination charge originates from the symmetry-protected bulk charge patterns---a fundamental property of many topological crystalline insulators (TCIs). Meanwhile, the robust bound states at disclinations emerge as a secondary, but directly observable property of TCIs. Using reconfigurable photonic crystals as photonic TCIs with higher-order topology, we observe those hallmark features via pump-probe and near-field detection measurements. Both the fractional charge and the localized states are demonstrated to emerge at the disclination in the TCI phase but vanish in the trivial phase. The experimental discovery of bulk-disclination correspondence unveils a novel fundamental phenomenon and a new paradigm for exploring topological materials., Comment: 4 figures

Let $n_0$ be 1 or 3. If a multiplicative function $f$ satisfies $f(p+q-n_0) = f(p)+f(q)-f(n_0)$ for all primes $p$ and $q$, then $f$ is the identity function $f(n)=n$ or a constant function $f(n)=1$., Comment: 7 pages, a variation of Y.-G. Chen, J.-H. Fang, P. Yuan, and Y. Zheng's paper, On multiplicative functions with $f(p + q + n_0) = f(p) + f(q) + f(n_0)$, J. Number Theory 165 (2016), 270--289

With the widespread adoption of information systems, recommender systems are widely used for better user experience. Collaborative filtering is a popular approach in implementing recommender systems. Yet, collaborative filtering methods are highly dependent on user feedback, which is often highly sparse and hard to obtain. However, such issues could be alleviated if knowledge from a much denser and a related secondary domain could be used to enhance the recommendation accuracy in the sparse target domain. In this publication, we propose a deep learning method for cross-domain recommender systems through the linking of cross-domain user latent representations as a form of knowledge transfer across domains. We assume that cross-domain similarities of user tastes and behaviors are clearly observable in the low dimensional user latent representations. These user similarities are used to link the domains. As a result, we propose a Variational Autoencoder based network model for cross-domain linking with added contextualization to handle sparse data and for better transfer of cross-domain knowledge. We further extend the model to be more suitable in cold start scenarios and to utilize auxiliary user information for additional gains in recommendation accuracy. The effectiveness of the proposed model was empirically evaluated using multiple datasets. The experiments proved that the proposed model outperforms the state of the art techniques.

Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise, $f(n)=0$ for all $n \ge 2$ except for $n=3,9,11$., Comment: 6 pages

This paper reports on the re-analysis of solar flares in which the hard X-rays (HXRs) come predominantly from the corona rather than from the more usual chromospheric footpoints. All of the 26 previously analyzed event time intervals, over 13 flares, are re-examined for consistency with a flare model in which electrons are accelerated near the top of a magnetic loop that has a sufficiently high density to stop most of the electrons by Coulomb collisions before they can reach the footpoints. Of particular importance in the previous analysis was the finding that the length of the coronal HXR source increased with energy in the 20 - 30 keV range. However, after allowing for the possibility that footpoint emission at the higher energies affects the inferred length of the coronal HXR source, and using analysis techniques that suppress the possible influence of such footpoint emission, we conclude that there is no longer evidence that the length of the HXR coronal sources increase with increasing energy. In fact, for the 6 flares and 12 time intervals that satisfied our selection criteria, the loop lengths decreased on average by 1.0 +/- 0.2 arcsec between 20 and 30 keV, with a standard deviation of 3.5 arcsec. We find strong evidence that the peak of the coronal HXR source increases in altitude with increasing energy. For the thermal component of the emission, this is consistent with the standard CHSKP flare model in which magnetic reconnection in a coronal current sheet results in new hot loops being formed at progressively higher altitudes. The explanation for the nonthermal emission is not so clear., Comment: Accepted for publication in the Astrophysical Journal. 24 pages, 8 figures

Topology has been revealed to play a fundamental role in physics in the past decades. Topological insulators have unconventional gapless edge states where disorder-induced back-scattering is suppressed. In photonics, such edge states lead to unidirectional waveguides which are useful for integrated photonic chips. Cavity modes, another type of fundamental components in photonic chips, however, are not protected by band topology because of their lower dimensions. Here we demonstrate that concurrent wavevector-space and real-space topology, dubbed as the "dual-topology", can lead to light-trapping in lower-dimensions. The resultant photonic bound state emerges as a Jackiw-Rebbi soliton mode localized on a dislocation in a two-dimensional (2D) photonic crystal, as predicted theoretically and discovered experimentally. Such a strongly-confined 0D localized mode, which is solely due to the topological mechanism, is found to be robust against perturbations. Our study unveils a new mechanism for topological light-trapping in lower-dimensions, which is valuable for fundamental physics and a variety of applications in photonics.

Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$., Comment: 5 pages

This paper shows that compressed sensing realized by means of regularized deconvolution and the Finite Isotropic Wavelet Transform is effective and reliable in hard X-ray solar imaging. The method utilizes the Finite Isotropic Wavelet Transform with Meyer function as the mother wavelet. Further, compressed sensing is realized by optimizing a sparsity-promoting regularized objective function by means of the Fast Iterative Shrinkage-Thresholding Algorithm. Eventually, the regularization parameter is selected by means of the Miller criterion. The method is applied against both synthetic data mimicking the Spectrometer/Telescope Imaging X-rays (STIX) measurements and experimental observations provided by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The performances of the method are compared with the results provided by standard visibility-based reconstruction methods. The results show that the application of the sparsity constraint and the use of a continuous, isotropic framework for the wavelet transform provide a notable spatial accuracy and significantly reduce the ringing effects due to the instrument point spread functions.

Let $\mathtt{PRIMES}$ be the set of all primes. We show that a multiplicative function which satisfies \[ f(p+q-2) = f(p) + f(q) - f(2) \text{ for }p,q \in \mathtt{PRIMES} \] is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n \ge2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies \[ f(a+b) = f(a) + f(b) \text{ for }a,b \in \mathtt{PRIMES}-1 \] is the identity function., Comment: 6 pages

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are involved, then such $f$ is uniquely determined. That is, if a multiplicative function $f$ satisfies \[ f(a_1^2 + a_2^2 + \dotsb + a_k^2) = f(a_1^2) + f(a_2^2) + \dotsb + f(a_k^2) \] for arbitrary positive integers $a_i$, then $f$ is the identity function. In this sense, we call the set of all posotive squares a \emph{$k$-additive uniqueness set} for multiplicative functions., Comment: 7 pages

Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the identity function., Comment: 9 pages

The next grand challenges for society and science are in the brain sciences. A collection of 60+ scientists from around the world, together with 10+ observers from national, private, and foundations, spent two days together discussing the top challenges that we could solve as a global community in the next decade. We eventually settled on three challenges, spanning anatomy, physiology, and medicine. Addressing all three challenges requires novel computational infrastructure. The group proposed the advent of The International Brain Station (TIBS), to address these challenges, and launch brain sciences to the next level of understanding., Comment: 6 pages

Shearlets are a relatively new directional multi-scale framework for signal analysis, which have been shown effective to enhance signal discontinuities such as edges and corners at multiple scales. In this work we address the problem of detecting and describing blob-like features in the shearlets framework. We derive a measure which is very effective for blob detection and closely related to the Laplacian of Gaussian. We demonstrate the measure satisfies the perfect scale invariance property in the continuous case. In the discrete setting, we derive algorithms for blob detection and keypoint description. Finally, we provide qualitative justifications of our findings as well as a quantitative evaluation on benchmark data. We also report an experimental evidence that our method is very suitable to deal with compressed and noisy images, thanks to the sparsity property of shearlets.

Visual representation is crucial for a visual tracking method's performances. Conventionally, visual representations adopted in visual tracking rely on hand-crafted computer vision descriptors. These descriptors were developed generically without considering tracking-specific information. In this paper, we propose to learn complex-valued invariant representations from tracked sequential image patches, via strong temporal slowness constraint and stacked convolutional autoencoders. The deep slow local representations are learned offline on unlabeled data and transferred to the observational model of our proposed tracker. The proposed observational model retains old training samples to alleviate drift, and collect negative samples which are coherent with target's motion pattern for better discriminative tracking. With the learned representation and online training samples, a logistic regression classifier is adopted to distinguish target from background, and retrained online to adapt to appearance changes. Subsequently, the observational model is integrated into a particle filter framework to peform visual tracking. Experimental results on various challenging benchmark sequences demonstrate that the proposed tracker performs favourably against several state-of-the-art trackers., Comment: Pattern Recognition (Elsevier), 2015

If a multiplicative function $f$ satisfies $f(a^2+b^2+c^2) = f(a)^2+f(b)^2+f(c)^2$ for all positive integers $a$, $b$, and $c$, then $f$ is an identity function., Comment: This paper was merged to arXiv:1608.08270

Given relatively prime integers $a_1, \dotsc, a_n$, the Frobenius number $g(a_1, \dotsc, a_n)$ is defined as the largest integer which cannot be expressed as $x_1 a_1 + \dotsb + x_n a_n$ with $x_i$ nonnegative integers. In this article, we give the Frobenius number of primitive Pythagorean triples. That is, \[ g(m^2-n^2, 2mn, m^2+n^2) = (m-1)(m^2-n^2) + (m-1)(2mn) - (m^2 + n^2). \], Comment: 6 papges

In recent years, there are some major changes in the way content is being distributed over the network. The content distribution techniques have recently started to embrace peer-to-peer (P2P) systems as an alternative to the traditional client-server architecture. P2P systemsthat are based on the BitTorrent protocol uses end-users' resources to provide a cost effective distribution of bandwidth intensive content to thousands of users. The BitTorrent protocol system offers a scalable mechanism for distributing a large volume of data to a set of peers over the Internet. With the growing demand for file sharing and content distribution, BitTorrent has become one of the most popular Internet applications and contributes to a signification fraction of the Internet traffic. With the wide usage of the BitTorrent protocol system, it has basically solved one of the major problems where data can be quickly transferred to a group of interested parties. The strength of the BitTorrent protocol lies in efficient bandwidth utilization for the downloading and uploading processes. However, the usage of BitTorrent protocol also causes latency for other applications in terms of network bandwidth which in turn has caused concerns for the Internet Service Providers, who strives for quality of service for all their customers. In this paper, we study the network traffic patterns of theBitTorrent network traffic and investigate its behavior by usingthe time series ARMA model. Our experimental results show that BitTorrent network traffic can be modeled and forecasted by using ARMA models. We compared and evaluated the forecasted network traffic with the real traffic patterns. This modeling can be utilized by the Internet Service Providers to manage their network bandwidth and also detect any abnormality in their network., Comment: 14 pages

We describe an example of learning with multiple representations in an A-level revision lesson on mechanics. The context of the problem involved the motion of a ball thrown vertically upwards in air and studying how the associated physical quantities changed during its flight. Different groups of students were assigned to look at the ball's motion using various representations: motion diagrams, vector diagrams, free-body diagrams, verbal description, equations and graphs, drawn against time as well as against displacement. Overall, feedback from students about the lesson was positive. We further discuss the benefits of using computer simulation to support and extend student learning., Comment: 10 pages, 5 figures, 2 tables http://iopscience.iop.org/0031-9120

Recent trends show that there are swift developments and fast convergence of wireless and mobile communication networks with internet services to provide the quality of ubiquitous access to network users. Most of the wireless networks and mobile cellular networks are moving to be all IP based. These networks are connected through the private IP core networks using the TCP/IP protocol or through the Internet. As such, there is room to improve the mobility support through the Internet and support ubiquitous network access by providing seamless handover. This is especially true with the invention of portable mobile and laptop devices that can be connected almost everywhere at any time. However, the recent explosion on the usage of mobile and laptop devices has also generated several issues in terms of performance and quality of service. Nowadays, mobile users demand high quality performance, best quality of services and seamless connections that support real-time application such as audio and video streaming. The goal of this paper is to study the impact and evaluate the mobility management protocols under micro mobility domain on link layer and network layer handover performance. Therefore, this paper proposes an integration solution of network-based mobility management framework, based on Proxy Mobile IPv6, to alleviate handover latency, packet loss and increase throughput and the performance of video transmission when mobile host moves to new network during handover on high speed mobility. Simulations are conducted to analyze the relationship between the network performances with the moving speed of mobile host over mobility protocols. Based on simulation results, we presented and analyzed the results of mobility protocols under intra-domain traffics in micro mobility domain., Comment: 20 pages, 15 figures, 3 tables; Lee, H., Han, Y. & Min, S. (2010) "Network Mobility Support Scheme on PMIPv6 Networks", International Journal of Computer Networks & Communications (IJCNC), Vol.2, No.5

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\Q{-m}$ which admits even universal binary Hermitian lattices., Comment: 11 Pages

If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs., Comment: 6 pages

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$ and provide a complete list of the (normal) Hermitian lattices., Comment: 26 pages

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13., Comment: 20 Pages

We present millimetre wavelength observations of the black hole candidate X-ray binary Cygnus X-1 which indicate a suppression, or quenching, of the emission as the source switches to a softer X-ray state. Combining the data with those for another black hole candidate, XTE J1118+480, we demonstrate that the millimetre emission shows the same coupling to X-rays as the radio emission, although with a much stronger sensitivity to spectral shape. We therefore confirm the association of the millimetre emission with the jets in low/hard state black hole candidate X-ray binaries., Comment: Accepted for publication in MNRAS