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This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$. We prove that the associativity of functions $T$ only depends on the range of $f$, which is used to give a sufficient and necessary condition for the function $T$ being associative when the triangular norm $F$ is an ordinal sum of triangular norms and an ordinal sum of triangular subnorms in the sense of A. H. Clifford, respectively. These results finally are applied for describing classes of triangular norms generated by strictly increasing functions., Comment: 24

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function which satisfies either $f(x)=f(x^{+})$ when $f(x^{+})\in \mbox{Ran}(f)$ or $f(x)\neq f(y)$ for any $y\neq x$ when $f(x^{+})\notin \mbox{Ran}(f)$ for all $x\in[0,1]$ and $f^{(-1)}:[0,\infty]\rightarrow[0,1]$ is a pseudo-inverse of $f$ depends only on properties of the range of $f$. The necessary and sufficient conditions for the $T$ to be associative are presented by applying the properties of the monotone function $f$., Comment: 16

We study real resonances and embedded eigenvalues of the Kramers--Fokker--Planck operator with a long-range potential. We prove that thresholds are only possible accumulation points of eigenvalues and that the limiting absorption principle holds true for energies outside an exceptional set. We also prove that the eigenfunctions associated with discrete eigenvalues decay exponentially and those associated with embedded non-threshold ones decay polynomially.

Label-free metabolic dynamics contrast is highly appealing but difficult to achieve in biomedical imaging. Interference offers a highly sensitive mechanism for capturing the metabolic dynamics of the subcellular scatterers. However, traditional interference detection methods fail to isolate pure metabolic dynamics, as the dynamic signals are coupled with scatterer reflectivity and other uncontrollable imaging factors. Here, we demonstrate active phase modulation-assisted dynamic full-field optical coherence tomography (APMD-FFOCT) that decouples and quantifies the metabolic dynamics by adding a reference movement for all interferential scatterers. This novel technique enables imaging and dynamic analysis of subcellular structures along with their changes during the apoptotic process in tumor tissues. Furthermore, the nucleus-to-cytoplasm dynamic intensity ratio could serve as an optical biomarker for breast tumor grading, enhancing intraoperative diagnosis.

Pathological features are the gold standard for tumor diagnosis, guiding treatment and prognosis. However, standard histopathological process is labor-intensive and time-consuming, while frozen sections have lower accuracy. Dynamic full-field optical coherence tomography (D-FFOCT) offers rapid histologic information by measuring the subcellular dynamics of fresh, unprocessed tissues. However, D-FFOCT images suffer from abrupt shifts in hue and brightness, which is confusing for pathologists and diminish their interpretability and reliability. Here, we present active phase modulation-assisted D-FFOCT (APMD-FFOCT) to improve the imaging stability and enhance the contrast of static tissues. This enables us to further employ an unsupervised deep learning to convert APMD-FFOCT images into virtual hematoxylin and eosin (H&E) stained images for the first time. Three-dimensional (3D) virtual H&E-stained images have been obtained at a scanning rate of 1 frame per second, as demonstrated in cancer diagnosis for human central nervous system and breast. The results prove that this new method will play a unique and important role in intraoperative histology.

This article pays attention to the $\alpha$-migrativity of 2-uninorms with $\alpha\in [0,1]$ deeply. It describes the $\alpha$-migrativity of 2-uninorms completely, which generalizes and unifies some current existing characterizations for $\alpha$-migrativity of triangular norm, triangular conorm, uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively., Comment: 14

In this article, we first give the characterizations of quasi-homogeneous aggregation functions, which show us that quasi-homogeneous aggregation functions are classified into three classes. We then introduce the concept of triple generator of quasi-homogeneous aggregation function, which is applied to construct a quasi-homogeneous aggregation function., Comment: 15

Recently, much attention has been paid to uncovering the influence of dissipation on a quantum system, particularly on how the non-Hermitian (NH) terms modify the band topology of topological materials and reshape the profile of the wavefunctions of a system (or the NH skin effect). In this paper, a specific NH skin effect that induced by local dissipation is studied for chiral edge modes, in which the NH term corresponds to the local imaginary Fermi velocity of the chiral edge modes. By solving the NH Schr\"{o}dinger equation of the non-Hermitian chiral edge modes (nhCEs) with complex Fermi velocity, we uncovered the remarkable complex spectra and the wavefunctions of the nhCEs. We find that the complex spectra of these modes is a straight line in the topological materials, and its chirality can separates the modes with positive energy from those with negative energy, in which they are localized at different positions. We also studied the nhCEs at the boundary of 2-dimensional (2D) topological materials, the 2D $p$-wave superconductor and the Qi-Wu-Zhang model, in which the general law of nhCEs was verified. We expect that our findings will pave the way for researching the transport properties of the chiral edge modes in the non-equilibrium context., Comment: 7 pages, 6 figures

This article focuses on minimax programming problems subject to addition-{\L}ukasiewicz fuzzy relational inequalities. We first establish two necessary and sufficient conditions that a solution of the fuzzy relational inequalities is a minimal one and explore the existence condition of the unique minimal solution. We also supply an algorithm to search for minimal solutions of the fuzzy relational inequalities starting from a given solution. We then apply minimal solutions of the fuzzy relational inequalities to the minimax programming problems for searching optimal solutions. We provide two algorithms to solve a kind of single variable optimization problems, and obtain the greatest optimal solution. The algorithm for finding minimal solutions of a given solution are also used for searching minimal optimal solutions., Comment: 25

This book provides a systematic study of spectral and scattering theory for many-body Schr\"odinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the center of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on the many-body problem is sparse. However our analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue $\lambda_0$. In general we address the question: How does scattering quantities for the many-body atomic and molecular models behave in the limit when the total energy approaches a fixed two-cluster threshold $\lambda_0$? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross-section and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. Our principal tools are the Feshbach-Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics (of independent interest) are the limiting absorption principle, micro-local resolvent estimates, Rellich and Sommerfeld type theorems and asymptotics of the limiting resolvents at thresholds. While all of these features are fairly well-understood for two-body Schr\"odinger operators, they are poorly understood in the many-body case even for two-cluster thresholds. It is the goal of the book to remedy this point. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.

Based on the Rb$^+$-Rb hybrid trap, we investigate the effect of ion-atom elastic collisions on the number and temperature of the remaining atoms. We measured the remaining atomic number and temperature as a function of the wavelength and intensity of the ionization laser, and whether the ion trap was turned on. Fittings with a single exponential decay function plus an offset to the number and radius of the remaining atoms are found to be in good agreement. We found a difference in the exponential factor of different wavelengths of ionization laser with the ion trap on or off. We suppose that the presence of electrons affects ion-atom collisions through disorder-induced heating. Our research contributes to a better understanding of how ultracold neutral plasma evolves, particularly the subsequent kinetics of atomic processes, which also serves as a useful reference for high-energy-density plasma., Comment: 14 figures, 9 pages

Polyatomic cations $^{87}$Rb$_M^+$ ($M$ = 2, 3,$\ldots$) have been produced by cascade two-body ion-atom reactive collisions in the two-step CW-laser photoionization of laser-cooled $^{87}$Rb atoms and accumulated in the ion trap. Using resonant-excitation mass spectrometry and resonant excitation-assisted time-of-flight mass spectrometry, we directly observed and distinguished the charged reaction products. We experimentally verified the cascade generation and cascade dissociation of $^{87}$Rb$_M^+$. The populations of $^{87}$Rb$_M^+$ are quantitatively investigated by solving the rate equations. The $^{87}$Rb$^+$-$^{87}$Rb reaction rate coefficient was derived as 9.10$\times10^{-11}$ cm$^3$/s accordingly. The methods developed here for assembling and detecting homonuclear polyatomic cations can be applied to any experiment in ion-atom hybrid traps. The present study lays the foundation for exploring atomically precise metal clusters and physics from few- to many-body perspective., Comment: 5 figures

In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First, it provides a sufficient and necessary condition for a vector being an eigenvector of a given matrix, and then presents an algorithm for finding all eigenvalues and eigenvectors of a given matrix. It further proposes a sufficient and necessary condition for a vector being a constrained eigenvector of a given matrix and supplies an algorithm for computing all the constrained eigenvectors and eigenvalues of a given matrix. This article finally discusses the supereigenproblem of a given matrix and presents an algorithm for obtaining the maximum constrained supereigenvalue and depicting the feasible region of all the constrained supereigenvectors for a given matrix. It also gives some examples for illustrating the algorithms, respectively., Comment: 21

This paper deals with the resolutions of fuzzy relation equations with addition-min composition. When the fuzzy relation equations have a solution, we first propose an algorithm to find all minimal solutions of the fuzzy relation equations and also supply an algorithm to find all maximal solutions of the fuzzy relation equations, which will be illustrated, respectively, by numeral examples. Then we prove that every solution of the fuzzy relation equations is between a minimal solution and a maximal one, so that we describe the solution set of the fuzzy relation equations completely., Comment: 19

In this article, we first characterize pseudocomplemented inductive modular lattices by using their two 0-sublattices. Then we use two 0-sublattices of a subgroup lattice to describe all locally cyclic abelian groups. In particular, we show that a locally cyclic abelian group can be characterized by its three subgroups., Comment: 9

In this article, we present two methods to construct 2-uninorms on bounded lattices by using additive generators, which are further used for inducing uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively. We also provide some examples for illustrating the constructing methods of 2-uninorms., Comment: 23

Weight and activation binarization can efficiently compress deep neural networks and accelerate model inference, but cause severe accuracy degradation. Existing optimization methods for binary neural networks (BNNs) focus on fitting full-precision networks to reduce quantization errors, and suffer from the trade-off between accuracy and computational complexity. In contrast, considering the limited learning ability and information loss caused by the limited representational capability of BNNs, we propose IR$^2$Net to stimulate the potential of BNNs and improve the network accuracy by restricting the input information and recovering the feature information, including: 1) information restriction: for a BNN, by evaluating the learning ability on the input information, discarding some of the information it cannot focus on, and limiting the amount of input information to match its learning ability; 2) information recovery: due to the information loss in forward propagation, the output feature information of the network is not enough to support accurate classification. By selecting some shallow feature maps with richer information, and fusing them with the final feature maps to recover the feature information. In addition, the computational cost is reduced by streamlining the information recovery method to strike a better trade-off between accuracy and efficiency. Experimental results demonstrate that our approach still achieves comparable accuracy even with $ \sim $10x floating-point operations (FLOPs) reduction for ResNet-18. The models and code are available at https://github.com/pingxue-hfut/IR2Net., Comment: 11 pages, 8 figures