Miracle-dz/UniFAR-FTUs huggingface dataset:
First Families of Regular Polygons

Every regular polygon can be regarded as a member of a well-defined 'family' of related regular polygons. These families arise naturally in the study of piecewise rotations such as outer billiards. In some cases they exist on all scales and can be used to define the fractal dimension of the 'singularity set'. This is well-documented for regular N-gons such as the pentagon, octagon and dodecagon, whose algebraic complexity is 'quadratic' (EulerPhi[N]/2 = 2). Recent evidence suggests that the geometry of these families is intrinsic to the 'parent' polygon and can be derived independently of any mapping. It is the purpose of this paper to show how the First Family for any regular polygon arises naturally from the geometry of the 'star polygons' first studied by Thomas Bradwardine (1290-1349). The nucleus of the First Family are the S[k] 'tiles'. Each S[k] tile has a corresponding 'star-point' sk = tan(kPi/N) and a matching scale. Based on a 1949 result by C.Siegel and S.Chowla, the primitive scales with gcd(k,N) = 1 form a unit basis for the maximal real subfield of the cyclotomic field of N. Traditionally, this subfield has been the source of scaling and rotational parameters for affine piecewise rational rotations, and our results show that the First Family scaling can serve as a 'natural' basis for such investigations - which include the outer billiards map.
Quasimaps to relative GIT quotients and applications

Brown proved that the I-function of a toric fibration lies on an overruled Lagrangian cone of its genus zero Gromov-Witten theory which is introduced by Givental. In this paper, we will prove a theorem for partial flag variety fibrations. To do so, we will construct new moduli spaces as a generalization of moduli spaces of quasimaps constructed by Ciocan-Fontanine, Kim and Maulik.
Certain Liouville properties of eigenfunctions of elliptic operators

We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schr\"odinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.
Inner tableau translation property of the weak order and related results

Let $SYT_{n}$ be the set of all standard Young tableaux with $n$ cells and $\leq_{weak}$ be Melnikov's the weak order on $SYT_n$. The aim of this paper is to introduce a conjecture, called the {\it property of inner tableau translation} and discuss its significance in the studies of the weak order. We will prove the conjecture for some specific cases
A vanishing theorem for tautological classes of aspherical manifolds

Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.
Non viability of hyperbolic quantum mechanics as a theory of Nature

Quantum and classical mechanics share a common algebraic formalism which is expressed naturally in the language of category theory. A third realization of this formalism is the so-called hyperbolic quantum mechanics where split-complex numbers replace the usual complex numbers. We introduce and explore the corresponding generalization of C*-algebras and prove that hyperbolic quantum mechanics is not a viable candidate for describing Nature. Quantum and classical mechanics are the only acceptable theories of Nature which are invariant under tensorial composition.
Variable Besov Spaces: Continuous Version

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calder\`on reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We characterize these function spaces by so-called Peetre maximal functions and we obtain the Sobolev embeddings for these function spaces. We use these results to prove the atomic decomposition for these spaces.
Existence of an isolated band in a system of three particles in an optical lattice

We prove the existence of two- and three-particle bound states of the Schrödinger operators h &mgr; ( k ) , k ∈ T d ? > and H &mgr; ( K ) , K ∈ T d ? > associated to Hamiltonians h &mgr; ? > and H &mgr; ? > of a system of two and three identical bosons on the lattice Z d , d = 1 , 2 ? > interacting via pairwise zero-range attractive &mgr; < 0 ? > or repulsive &mgr; > 0 ? > potentials. As a consequence, we show the existence of an isolated band in the two- and three-bosonic systems in an optical lattice.
Almost sure global well posedness for the BBM equation with infinite \begin{document}$ L^{2} $\end{document} initial data

We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $\mathbb{T}$ with initial data below $L^{2}(\mathbb{T})$. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the $I$-method to obtain an a priori bound on the growth of the `residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.
On Heegaard splittings of glued 3-manifolds

We introduce a new technique for finding lower bounds on the Heegaard genus of a 3-manifold obtained by gluing a pair of 3-manifolds together along an incompressible torus or annulus. We deduce a number of inequalities, including one which implies that $t(K_1# K_2)\geq \max {t(K_1),t(K_2)}$, where $t(-)$ denotes tunnel number, $K_1$ and $K_2$ are knots in $S^3$, and $K_1$ is $m$-small. This inequality is best possible. We also provide an interesting collection of examples, similar to a set of examples found by Schultens and Wiedmann, which show that Heegaard genus can stay persistently low under the kinds of gluings we study here.
Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator

In this work, we proposed a new method called Laplace–Padé–Caputo fractional reduced differential transform method (LPCFRDTM) for solving a two-dimensional nonlinear time-fractional damped wave equation subject to the appropriate initial conditions arising in various physical models. LPCFRDTM is the amalgamation of the Laplace transform method (LTM), Padé approximant, and the well-known reduced differential transform method (RDTM) in the Caputo fractional derivative senses. First, the solution to the problem is gained in the convergent power series form with the help of the Caputo fractional-reduced differential transform method. Then, the Laplace–Padé approximant is applied to enlarge the domain of convergence. The advantage of this method is that it solves equations simply and directly without requiring enormous amounts of computational work, perturbations, or linearization, and it expands the convergence domain, leading to the exact answer. To confirm the effectiveness, accuracy, and convergence of the proposed method, four test-modeling problems from mathematical physics nonlinear wave equations are considered. The findings and results showed that the proposed approach may be utilized to solve comparable wave equations with nonlinear damping and source components and to forecast and enrich the internal mechanism of nonlinearity in nonlinear dynamic events.
Exploring factors driving macrophytes in rivers—A case study in Middle Atlas Morocco

Macrophytes represent a functional compartment that plays multiple and fundamental ecological roles in aquatic environments. They are linked with the other compartments of hydrosystems, both abiotic and biotic, which gives them the ability to integrate environmental conditions and reflect the trophic state of watercourses. Nevertheless, macrophytes in Moroccan rivers have been rarely studied. Thus, the main objective of the present work is to study these plants by determining the environmental factors that control their presence and distribution in the watercourses of the upper Oum Er‐rbia basin (Middle Atlas Morocco). From June 2018 to June 2019, 14 sites spread over five rivers were monitored monthly to describe macrophytic communities and investigate their ecological determinism. We used a principal component analysis to establish each river's typology and a detrended correspondence analysis to explore macrophyte assemblages. Canonical correspondence analysis was used to determine the hydrochemical parameters of water that govern the development and distribution of macrophyte species in the upper Oum Er‐rbia basin. The Amengous river stands out from the other surveyed rivers for its exceptional richness of macrophytic species. This richness is often associated with the enrichment of the aquatic environment by nitrogen and phosphate inputs. Furthermore, in some rivers, such as Oum Er‐rbia, nutrient enrichment plays a secondary function in comparison to the hydromorphological descriptors that tend to be more important for macrophyte composition and distribution.
Effect of a Multifunctional Biosurfactant Extract Obtained from Corn Steep Liquor on Orange and Apple Juices

Biosurfactant extracts are multifunctional ingredients composed of natural polymers that can be used in the food industry as stabilizing and antimicrobial agents, although their inclusion in food matrices has been scarcely explored. In this work, a biosurfactant extract, with antimicrobial properties, obtained from a fermented stream of the corn wet-milling industry was introduced into an apple and orange juice matrix to evaluate the changes produced in the sugar consumption, pH, and biomass formation at different temperatures (4–36 °C) and storage time (1–7 days). It was observed that the addition of biosurfactant extract reduced the hydrolysis rate of polymeric sugars, decreasing the concentration of soluble sugars from 85.4 g/L to 49.0 g/L in apple juice after 7 days at 20 °C in the absence and presence of biosurfactant extract, respectively. In general, soluble sugars increased in juices for 5–6 days and after those sugars decreased at different rates depending on the temperature of storage. Differences in sugar solubilization and degradation were more significant in apple juice than in orange juice at 20 °C and 7 days of storage, achieving for orange juice values of 101 and 102 g/L in the absence and presence of biosurfactant extract, respectively. Biomass growth was almost unaffected by the biosurfactant extract concentration and the optimal conditions for biomass production were detected at intermediated temperatures after 6–7 days of storage for both apple and orange juices, obtaining maximum concentrations of 1.68 g/L and 1.54 g/L for apple juice and orange juice, respectively, in the absence of biosurfactant extract. The pH during storage was kept in the range of 3.35–3.48 for apple juice and of 3.40–3.77 for orange juice.
CLOSURE OPERATORS, FRAMES AND NEATEST REPRESENTATIONS

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$ , we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$ -closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$ -closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$ , the lattice of ${\mathcal{U}}$ -ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$ -distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$ . From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$ , then it has an $(\unicode[STIX]{x1D714},3)$ -representation.
A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map. ” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic.
Different Effects of Perceived Social Support on the Relationship between Perceived Stress and Depression among University Students with Borderline Personality Disorder Symptoms: A Multigroup Mediation Analysis

Background: While perceived social support can mediate the relationship between perceived stress and depression, little is known about the differences between individuals with high and low borderline personality disorder symptoms (BPDS). This study aimed to investigate the associations among perceived stress, perceived social support, and depression, and compare low and high levels of BPDS. Methods. This cross-sectional analysis was a secondary analysis of data from the SI-Bord study. University students across Thailand completed a screening instrument for borderline personality disorder, the Perceived Stress Scale (PSS), the Revised Thai Multi-dimensional Scales of Perceived Social Support (MSPSS), and the Patient-Health Questionnaire (PHQ)-9. Mediation analysis using PROCESS was applied to test the direct and indirect effects of perceived stress on depression. Multigroup mediational analysis was adopted to compare low and high levels of BPDS. Results. The mean age of the 330 participants was 20.27 (SD, 1.4) and 80% were female. Significant correlations were observed between the PSS, MSPSS, and PHQ scores, with greater magnitude among the high-level BPDS group (p < 0.001). A significant direct effect on perceived stress and a significant indirect effect on depression through perceived social support were noted. Of all the sources of social support, only the significant others variable significantly differed between the two groups (p < 0.05). Conclusion. Perception of social support had a significant mediating role in perceived stress and depression. The magnitude of associations was remarkably high for individuals with high BPDS compared to those with low BPDS. Unlike those with low BPDS, all sources of social support were significant mediators between the two groups.
The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives

Let $ p_n(x) $ be a random polynomial of degree $n$ and $\{Z^{(n)}_j\}_{j=1}^n$ and $\{X^{n, k}_j\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $\frac{1}{a_n} \left[ f\left( \frac {Z^{(n)}_1}{b_n} \right)+ \cdots + f \left(\frac {Z^{(n)}_n}{b_n} \right) \right]$ associated with $\{Z^{(n)}_j\}$ has a limit as $n\to\infty$ at some mode of convergence, the linear statistics associated with $\{X^{n, k}_j\}$ converges to the same limit at the same mode. Similar statement also holds for the centered linear statistics associated with the zeros of $p_n$ and $p_n^{(k)}$, provided the zeros $\{Z^{(n)}_j\}$ and the sequences $\{a_n\}$ and $\{b_n\}$ of positive numbers satisfy some mild conditions.
Spectral Dyadic Green’s Function for Multilayer Periodic Bi-Anisotropic Media

A computational technique for evaluating the spectral dyadic Green’s function utilized in the analysis of multilayer structures composed of periodic bi-anisotropic layers is presented. The Green’s function is achieved with the help of a fully vectorial rigorous matrix formulation based on a generalized transmission line (TL) formulation of Maxwell’s equations. This matrix formulation simplifies the mathematics involved in the modeling of the coupled diffraction orders within a periodic bi-anisotropic layer. To examine the soundness of this modeling, the extracted Green’s function is used in an integral equation for the analysis of metallic gratings on homogeneous/inhomogeneous anisotropic and chiral materials. The resulting integral equation is then solved by the Method of Moments (MoM) formulated in terms of sub-domain basis and Galerkin’s test functions. As verification, several examples are analyzed and the results obtained by this method are compared with those available in the literature or obtained by EM-solvers. The efficiency assessment of this method is carried out by considering its convergence rate and cost-time in terms of truncation orders. It is revealed not only the metallic grating with inhomogeneous periodic bi-anisotropic layers can be analyzed by this method but also analysis time, memory, and CPU occupancies are decreased in comparison with commercial EM-solvers.
Symplectic rational $G$-surfaces and equivariant symplectic cones

We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.
Equivariant Picard groups of $C^*$-algebras with finite dimensional $C^*$-Hopf algebra coactions

Let $A$ be a $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(\rho, u)$ be a twisted coaction of $H^0$ on $A$. We shall define the $(\rho, u, H)$-equivariant Picard group of $A$, which is denoted by $\Pic_H^{\rho, u}(A)$, and discuss basic properties of $\Pic_H^{\rho, u}(A)$. Also, we suppose that $(\rho, u)$ is the coaction of $H^0$ on the unital $C^*$-algebra $A$, that is, $u=1\otimes 1^0$. We investigate the relation between $\Pic(A^s )$, the ordinary Picard group of $A^s$ and $\Pic_H^{\rho^s}(A^s )$ where $A^s$ is the stable $C^*$-algebra of $A$ and $\rho^s$ is the coaction of $H^0$ on $A^s$ induced by $\rho$. Furthermore, we shall show that $\Pic_{H^0}^{\widehat{\rho}}(A\rtimes_{\rho, u}H)$ is isomorphic to $\Pic_H^{\rho, u}(A)$, where $\widehat{\rho}$ is the dual coaction of $H$ on the twisted crossed product $A\rtimes_{\rho, u}H$ of $A$ by the twisted coaction $(\rho, u)$ of $H^0$ on $A$.
Random Function Iterations for Consistent Stochastic Feasibility

Abstract We study the convergence of iterated random functions for stochastic feasibility in the consistent case (in the sense of Butnariu and Flåm [Numer. Funct. Anal. Optimiz., 1995]) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point mappings. The iterations are Markov chains and, for the purposes of this study, convergence is understood in very restrictive terms. We show that sufficient conditions for geometric (linear) convergence in expectation of stochastic projection algorithms presented in Nedić [Math. Program, 2011], are in fact necessary for geometric (linear) convergence in expectation more generally of iterated random functions.
A class of semipositone p-Laplacian problems with a critical growth reaction term

Abstract We prove the existence of ground state positive solutions for a class of semipositone p-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform C1,α a priori estimates and by concentration compactness arguments. Our results are new even in the semilinear case p = 2.
Second-order constrained variational problems on Lie algebroids: applications to optimal control

The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost functional which depends on higher-order derivatives of admissible curves on a Lie algebroid. Extending the classical Skinner and Rusk formalism for the mechanics in the context of Lie algebroids, for second-order constrained mechanical systems, we derive the corresponding dynamical equations. We find a symplectic Lie subalgebroid where, under some mild regularity conditions, the second-order constrained variational problem, seen as a presymplectic Hamiltonian system, has a unique solution. We study the relationship of this formalism with the second-order constrained Euler-Poincar\'e and Lagrange-Poincar\'e equations, among others. Our study is applied to the optimal control of mechanical systems.
Invariant valuations on quaternionic vector spaces

Abstract The spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.
Solving large-scale nonlinear eigenvalue problems by rational interpolation approach and resolvent sampling based Rayleigh-Ritz method

Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by matrix structures, property of eigen-solutions, size of the problem, etc. This paper aims to break those limitations and to develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach (CIA), the RIA provides the possibility to select sampling points in more general regions and has advantages in improving accuracy and reducing computational cost. Second, a resolvent sampling scheme using the RIA is proposed for constructing reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, denoted by RSRR, is developed for solving general NEPs. RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of RSRR are demonstrated by a variety of benchmark and practical problems.