Miracle-dz/UniFAR-FTUs huggingface dataset:
Splitting lemmas for the Finsler energy functional on the space of H1‐curves

We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H1 ‐curves around a critical point or a critical R1 orbit of a Finsler isometry‐invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll–Meyer's splitting lemmas on Riemannian manifolds [Gromoll and Meyer, ‘On differentiable functions with isolated critical points’, Topology 8 (1969) 361–369; Gromoll and Meyer ‘Periodic geodesics on compact Riemannian manifolds’, J. Differential Geom. 3 (1969) 493–510]. As an application, we extend to Finsler manifolds a result by Grove and Tanaka [‘On the number of invariant closed geodesics’, Acta Math. 140 (1978) 33–48; Tanaka, ‘On the existence of infinitely many isometry‐invariant geodesics’, J. Differential Geom. 17 (1982) 171–184] about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.
Free boundary regularity for harmonic measures and Poisson kernels

One of the basic aims of this paper is to study the relationship between the geometry of \hypersurface like" subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on R n+1 is close to the doubling constant of the ndimensional Lebesgue measure then its support is well approximated by ndimensional a‐ne spaces, provided that the support is relatively ∞at to start with. Primarily we consider sets which are boundaries of domains in R n+1 . The n-dimensional Hausdorfi measure may not be deflned on the boundary of a domain in R n+1 . Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions (see Section 3). We obtain a new characterization of locally ∞at domains in terms of the doubling properties of their harmonic measure (see Section 3). Along these lines we investigate how the \weak" regularity of the Poisson kernel of a domain determines the geometry of its boundary. Sections 5 and 6 pursue this goal, as in Alt and Cafiarelli’s work (see [AC], [C1], [C2]), and also Jerison’s [J]. In both cases the goal is to prove that, under the appropriate technical conditions at \∞at points" of the boundary, the oscillation of the Poisson kernel controls the oscillation of the unit normal vector. The difierence between our work and the work in [AC] is that we measure the oscillation in an integral sense (BMO estimates)
Dual-wavelength operation of continuous-wave and mode-locked erbium-doped fiber lasers

We study numerically and experimentally multiple-wavelength operation of an erbium-doped figure-eight fiber laser including a multiple-bandpass optical filter formed by two concatenated fiber tapers. Both continuous-wave and pulsed operations are considered. In the continuous-wave regime, stable long-term operation at multiple closely spaced wavelengths is only obtained if fine adjustments of the cavity losses are performed. Under these conditions, simultaneous lasing at up to four wavelengths separated by 1.5 nm was observed experimentally. Tunable single-wavelength operation over more than 20 nm is also observed in the continuous-wave regime. In the passive mode locking regime, numerical simulations indicate that mechanisms involving the filter losses and the nonlinear transmission characteristic of the NOLM contribute in principle to stabilize dual-wavelength operation, allowing less demanding cavity loss adjustments. In this regime, the problem of synchronization between the pulse trains generated at each wavelength adds an additional dimension to the problem. In presence of cavity dispersion, the pulses at each wavelength tend to be asynchronous if the wavelength separation is large, however they can be synchronous in the case of closely spaced wavelengths, if cross-phase modulation is able to compensate for the dispersion-induced walkoff. Experimentally, fundamental and 2nd-order harmonic mode locking was observed, characterized by the generation of noise-like pulses. Finally, a regime of multi-wavelength passive Q-switching was also observed. We believe that this work will be helpful to guide the design of multiple-wavelength fiber laser sources, which are attractive for a wide range of applications including Wavelength Division Multiplexing transmissions, signal processing and sensing.
Efficient Numerical Techniques for Solving Pocklington's Integral Equation Using Multiwavelets

A procedure is presented for solving the electric field integral equation for a thin wire. The method is based upon Alpert multiwavelets functions approximation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Pocklington's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain sparse algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.
Comparison of energy dissipation between N and repeated sawtooth waves with finite amplitude

A theoretical calculation is made on nonlinear attenuation of energy including in an N waveform under comparison with energy dissipation of repeated sawtooth waves. It is found that the attenuation coefficient of energy of N wave estimated is only a quarter of the coefficient of the repeated sawtooth waves, when both waveforms have initially the same pressure amplitude and also the same wavelength at origin located at sound source. Therefore, N wave can propagate longer distance than the sawtooth waves with its pulselength increasing with propagation distance, while the wavelength of the repeated sawtooth waves is invariant. However, the change of the energies of both waves are governed by only a parameter which is given by product of initial slope of straightline between the shock fronts and the propagation distance.
Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

We extend the so-called ‘single ring theorem’ (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the Haagerup–Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.
Hankel Tensor Decompositions and Ranks

Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the the cp rank, symmetric rank, border rank, symmetric border rank, and Vandermonde rank all coincide with each other. In particular, this implies that the Comon's conjecture is true for generic Hankel tensors when the order is even or three. Some open questions are also posed.
Coworker Narcissism: Employee Emotional and Behavioral Reactions as Moderated by Bottom-Line Mentality and Trait Competitiveness

We advance research on narcissism in the workplace by examining the effects of coworker narcissistic rivalry on focal employee emotional states and behavioral intentions. We rely on social function of emotions theory to explain why coworker narcissistic rivalry results in focal employee negative emotions. We then explain that the focal employee is likely to handle their negative emotions arising from coworker narcissistic rivalry differently depending on individual differences of bottom-line mentality (BLM), which captures a defensive competitive posture, or trait competitiveness, which captures an offensive competitive posture. Across three studies, our results generally support our predictions by revealing that the indirect effect of coworker narcissistic rivalry onto focal employees’ social undermining of that coworker (through focal employee negative emotions) is more strongly positive when the focal employee is higher in BLM, whereas the indirect effect of coworker narcissistic rivalry onto focal-employee-intended work-goal progress relative to the coworker (through focal employee negative emotions) is more strongly positive when the focal employee's trait competitiveness is lower.
Nonlinear scalar discrete multipoint boundary value problems at resonance

In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form \begin{equation*} y(t+n)+a_{n-1}(t)y(t+n-1)+\cdots a_0(t)y(t)=g(t,y(t+m-1)) \end{equation*} subject to \begin{equation*} \sum_{j=1}^nb_{ij}(0)y(j-1)+\sum_{j=1}^nb_{ij}(1)y(j)+\cdots+\sum_{j=1}^nb_{ij}(N)y(j+N-1)=0 \end{equation*} for $i=1,\cdots, n$. The existence of solutions will be proved under a mild growth condition on the nonlinearity, $g$, which must hold only on a bounded subset of $\{0,\cdots, N\}\times\mathbb{R}$.
Effects of Acute Cold Stress after Intermittent Cold Stimulation on Immune-Related Molecules, Intestinal Barrier Genes, and Heat Shock Proteins in Broiler Ileum

Simple Summary Animal welfare and health will be negatively impacted by cold stress, resulting in decreased production performance, immune imbalance, and decreased antioxidant capacity. Previous studies focused more on the side effects of low temperature, but animals have the ability to adapt to the environment by regulating metabolic and endocrine processes. The objective of this study was to compare cold resistant of broilers with and without cold stimulation training. By detecting the changes in immunoglobulins, cytokines, toll-like receptors, gut barrier genes, and heat shock proteins gene expression levels before and after acute cold stress, the optimal cold training method was finally determined. The results of our study show that cold stimulation training for 6 h with an interval of one day, at 3 °C lower than the conventional temperature, can change the immune function of broilers. This cold stimulation can also lessen the intestinal damage when subjected to acute cold stress. This research offers a theoretical foundation for the regulation of immune function in broilers raised in cold environments, as well as a scientific basis for the development of cold adaptation in broilers. Abstract Cold stress will have a negative impact on animal welfare and health. In order to explore the effect of intermittent cold stimulation training on the cold resistance of broilers. Immune-related and intestinal barrier genes were detected before and after acute cold stress (ACS), aiming to find an optimal cold stimulation training method. A total of 240 1-day-old Ross broilers (Gallus) were divided into three groups (G1, G2, and G3), each with 5 replicates (16 chickens each replicate). The broilers of G1 were raised at normal temperature, while the broilers of G2 and G3 were treated with cold stimulation at 3 °C lower than the G1 for 3 h and 6 h from 15 to 35 d, respectively, at one-day intervals. At 50 d, the ambient temperature for all groups was reduced to 10 °C for six hours. The results demonstrated that before ACS, IL6, IL17, TLR21, and HSP40 mRNA levels in G3 were apparently down-regulated (p < 0.05), while IL8 and Claudin-1 mRNA levels were significantly up-regulated compared with G1 (p < 0.05). After ACS, IL2, IL6, and IL8 expression levels in G3 were lower than those in G2 (p < 0.05). Compared to G2, Claudin-1, HSP90 mRNA levels, HSP40, and HSP70 protein levels were increased in G3 (p < 0.05). The mRNA levels of TLR5, Mucin2, and Claudin-1 in G2 and IL6, IL8, and TLR4 in G3 were down-regulated after ACS, while IL2, IL6, and IL17 mRNA levels in G2 and HSP40 protein levels in G3 were up-regulated after ACS (p < 0.05). Comprehensive investigation shows that cold stimulation at 3 °C lower than the normal feeding temperature for six hours at one day intervals can enhanced immune function and maintain the stability of intestinal barrier function to lessen the adverse effects on ACS in broilers.
Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

We review and extend the vertex algebra framework linking gauge theory constructions and a quantum deformation of the Geometric Langlands Program. The relevant vertex algebras are associated to junctions of two boundary conditions in a 4d gauge theory and can be constructed from the basic ones by following certain standard procedures. Conformal blocks of modules over these vertex algebras give rise to twisted D-modules on the moduli stacks of G-bundles on Riemann surfaces which have applications to the Langlands Program. In particular, we construct a series of vertex algebras for every simple Lie group G which we expect to yield D-module kernels of various quantum Geometric Langlands dualities. We pay particular attention to the full duality group of gauge theory, which enables us to extend the standard qGL duality to a larger duality groupoid. We also discuss various subtleties related to the spin and gerbe structures and present a detailed analysis for the U(1) and SU(2) gauge theories.
Local symplectic field theory

Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and discuss the resulting algebraic structures.
Gr\"obner Bases: Connecting Linear Algebra with Homological and Homotopical Algebra

The main objective of this paper is to connect the theory of $Gr\"obner$ bases to concepts of homological algebra. $Gr\"obner$ bases, an important tool in algebraic system and in linear algebra help us to understand the structure of an algebra presented by its generators and relations by constructing a basis of its set of relations. In this paper we mainly deal with graded augmented algebras. Given a graded augmented algebra with its generators and relations, it is possible to construct a free resolution from its $Gr\"obner$ basis, known as Anick's resolution. Though rarely minimal, this resolution helps us to understand combinatorial properties of the algebra. The notion of a $K_2$ algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. We compute the Anick's resolution of $K_2$ algebra which shows several nice combinatorial properties. Later we compute some derived functors from Anick's resolution and outline how to construct $A_\infty$-algebra structure on $Ext$-algebra from a minimal graded projective resolution in which we can use Anick's resolution as a tool. Thus this paper provides an unique platform to connect concepts of linear algebra with homological algebra and homotopical algebra with the help of $Gr\"obner$ bases.
A Note on Upper Bounds for Some Generalized Folkman Numbers

Abstract We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including the cases Fe(K3, K4 − e; K5) ≤ 27 and Fe(K4 − e, K4 − e; K5) ≤ 51. The latter bound follows from a construction of a K5-free graph on 51 vertices, for which every edge coloring with two colors contains a monochromatic K4 − e.
Word-representability of subdivisions of triangular grid graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph $K_4$. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any subdivision of boundary triangles in the Sierpi\'{n}ski gasket graph is word-representable.
Analysis and Observer Design for the Parabolic System Weakly Coupled at the Boundary

Abstract. This paper discusses the mathematical properties of a recently developed mathematical model of a direct contact membrane distillation system. The model consists of two-dimensional advection diffusion system coupled at the boundary. A semi-group framework is used to analyze the model. First, the infinitesimal generator operator and its properties are studied. Then, existence and uniqueness of the solutions are investigated using the theory of operators. Some regularity results of the solution are also established. A particular case showing the diagonal property of the principal operator is studied. Moreover, based on this new partial differential model we examined the observer design for the unstable parabolic system coupled at the boundary. One of the main technical tools used in the proof is the backsetteping method. Finally, the lack of the backsetteping technique to design this class of parabolic system is discussed.
On the clique number of the square of a line graph and its relation to Ore-degree

In 1985, Erdős and Nesetřil conjectured that the square of the line graph of a graph $G$, that is $L(G)^2$, can be colored with $\frac{5}{4}\Delta(G)^2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $\omega(L(G)^2)$, is at most $\frac{5}{4}\Delta(G)^2$. In 2015, Śleszynska-Nowak proved that $\omega(L(G)^2)\le \frac{3}{2}\Delta(G)^2$. In this paper, we prove that $\omega(L(G)^2)\le \frac{4}{3}\Delta(G)^2$. This theorem follows from our stronger result that $\omega(L(G)^2)\le \frac{\sigma(G)^2}{3}$ where $\sigma(G) := \max_{uv\in E(G)} d(u) + d(v)$, is the Ore-degree of the graph $G$.
Spectral stability of inviscid columnar vortices

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument, which allows us to concentrate in the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.
Bi-Lipschitz Expansion of Measurable Sets

We show that for $0<\gamma, \gamma' <1$ and for measurable subsets of the unit square with Lebesgue measure $\gamma$ there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least $1-\gamma'$.
Modelling of Raman amplification in silicon-on-insulator optical microcavities

In this paper, the detailed modelling of Raman amplification in silicon-on-insulator (SOI) guided-wave resonant microcavities is developed for the first time. Theoretical results are compared with experiments in literature for either racetrack or microdisk resonators, demonstrating very good agreement. Investigation of resonant microcavity parameters, including pump and Stokes coupling factors and cavity lengths, is presented and their influence on pump enhancement factor is discussed. Design criteria are derived for both CW Raman lasers and pulsed Raman amplifiers.
Numerical techniques on improving computational efficiency of spectral boundary integral method

Numerical techniques are suggested in this paper, in order to improve the computational efficiency of the spectral boundary integral method, initiated by Clamond & Grue [D. Clamond and J. Grue. A fast method for fully nonlinear water‐wave computations. J. Fluid Mech. 2001; 447: 337–355] for simulating nonlinear water waves. This method involves dealing with the high order convolutions by using Fourier transform or inverse Fourier transform and evaluating the integrals with weakly singular integrands. A de‐singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity. An anti‐aliasing technique is developed in this paper to overcome the aliasing problem associated with Fourier transform or inverse Fourier transform with a limited resolution. This paper also presents a technique for determining a critical value of the free surface, under which the integrals can be neglected. Numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques. The tests will demonstrate that the improved method can significantly accelerate the computation, in particular when waves are strongly nonlinear. Copyright © 2015 John Wiley & Sons, Ltd.
Problem of the coexistence of several non-Hermitian observables in PT -symmetric quantum mechanics

During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.