Miracle-dz/UniFAR-FTUs huggingface dataset:
On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is ‘pseudodual’ to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the ‘Euclidean’ current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.
Conformal $\beta$- change in Finsler spaces

We investigate what we call a conformal β - change in Finsler spaces, namely where σ is a function of x only and β ( x, y ) is a given 1- form. This change generalizes various types of changes: conformal changes, Randers changes and β - changes. Under this change, we obtain the relationships between some tensors associated with ( M, L ) and the corresponding tensors associated with ( M, ∗ L ). We investigate some σ invariant tensors . This investigation allows us to give an answer to the question: Are the properties of C-reducibility, S 3 -likeness and S 4 -likeness invariant under a conformal β - change?
Extreme points and saturated polynomials

We consider the problem of characterizing the extreme points of the set of analytic functions f on the bidisk with positive real part and f(0)=1. If one restricts to those f whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such f that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.
Commensurated subgroups in finitely generated branch groups

Abstract A subgroup H ≤ G ${H\leq G}$ is commensurated if | H : H ∩ g H g - 1 | < ∞ $|H:H\cap gHg^{-1}|<\infty$ for all g ∈ G ${g\in G}$ . We show that a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group and many other branch groups of independent interest is either finite or of finite index.
OPTIMUM BOUNDS FOR THE DISTRIBUTIONS OF MARTINGALES IN BANACH SPACES

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.
Prethermal strong zero modes and topological qubits

We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures, the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This nonequilibrium dynamical phenomenon is a form of prethermalization and occurs because of obstructions to the equilibration of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a nonperturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes and on trapped ion chains.
Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.
Dense random regular digraphs: singularity of the adjacency matrix

Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high probability. The proof uses a couplings approach based on the switchings method of McKay and Wormald. We also rely on discrepancy properties for the distribution of edges in $\Gamma$, recently proved by the author, to overcome certain difficulties stemming from the dependencies between the entries of $M$.
On Nonexistence of type II blowup for a supercritical nonlinear heat equation

In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = Δu + |u|p−1u either on ℝN or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, $$p > p_{s} := {N + 2 \over N - 2} . $$ We prove that if ps < p < p*, then blowup is always of type I, where p* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L∞ norm is always the same as that for the corresponding ODE dv/dt = |v|p−1v. Because it is known that “type II” blowup (or, equivalently, “fast blowup”) can occur if p > p*, the above range of exponent p is optimal.
The $\{1,s\}$-weighted Davenport constant in $C_n^k$ and an application in an inverse problem

Let $G$ be a finite abelian group. The $A$-weighted Davenport constant of $G$, $D_A(G)$, is the smallest positive integer $d$ such that every sequence $x_1, \dots, x_d$ of $G$ has a non-empty subsequence $(x_{j_i})_i$ such that $\sum_{i=1}^t \varepsilon_i x_{j_i} = 0$ for some $\varepsilon_i \in A$. In this paper, we obtain both upper and lower bounds for $D_{\{1,s\}}(\mathbb Z_n)$, where $s^2 \equiv 1 \pmod n$ and $s \not\equiv \pm 1 \pmod n$. As an application, we determine all sequences that are free of product-1 subsequences in the group $C_n \rtimes_s C_2$.
Multiple solutions for a free boundary problem arising in plasma physics

In this paper we study the existence of solutions for a free boundary problem arising in the study of the equilibrium of a plasma confined in a tokamak: where p > 2, Ω is a bounded domain in ℝ2, n is the outward unit normal of ∂Ω, α is an unprescribed constant and I is a given positive constant. The set Ω+ = {x ∊ Ω: u(x) > 0} is called a plasma set. Under the condition that the homology of Ω is non-trivial, we show that for any given integer k ≥ 1 there exists λk > 0 such that for λ > λk the problem above has a solution with a plasma set consisting of k components.
Smaransache Multi-Space Theory

Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means theWORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds.
Functional A Posteriori Error Control for Conforming Mixed Approximations of the Reaction-Convection-Diffusion Problem

In this paper we show how to obtain the exact value of the global error for a conforming mixed approximation of the complex valued reaction-convection-diffusion equation. The results hold also for the real valued case. We operate in the framework of functional type a posteriori error control. The error is measured in a combined norm which takes into account both the primal and dual variables. Our main results state that the exact global error value of a conforming mixed approximation is given by an integral quantity which only includes the problem data and the approximation. These error equalities can be used also for the reaction-diffusion equation and for the convection-diffusion equation, provided that the divergence of the convection vector is negative. We also derive two-sided error bounds for the convection-diffusion equation in the case where the convection vector divergence is zero. This result is also applicable to the diffusion equation.
Depletion-mode Ga2O3 metal-oxide-semiconductor field-effect transistors on β-Ga2O3 (010) substrates and temperature dependence of their device characteristics

Single-crystal gallium oxide (Ga2O3) metal-oxide-semiconductor field-effect transistors were fabricated on a semi-insulating β-Ga2O3 (010) substrate. A Sn-doped n-Ga2O3 channel layer was grown by molecular-beam epitaxy. Si-ion implantation doping was performed to source and drain electrode regions for obtaining low-resistance ohmic contacts. An Al2O3 gate dielectric film formed by atomic layer deposition passivated the device surface and significantly reduced gate leakage. The device with a gate length of 2 μm showed effective gate modulation of the drain current with an extremely low off-state drain leakage of less than a few pA/mm, leading to a high drain current on/off ratio of over ten orders of magnitude. A three-terminal off-state breakdown voltage of 370 V was achieved. Stable transistor operation was sustained at temperatures up to 250 °C.
Colliding winds in five WR+O systems of the Southern hemisphere

We present the results of an optical spectroscopic follow-up of five WR+O systems of the Southern hemisphere that have not been studied so far for wind–wind collisions: WR 12 (WN8h), WR 21 (WN5o+O7V), WR 30 (WC6+O7.5V), WR 31 (WN4o+O8) and WR 47 (WN6o+O5). Observations were made over an entire contiguous month using the Garrison spectrograph attached to the 0.6-m Helen Sawyer Hogg telescope at Complejo Astronomico El Leoncito. We obtain improved orbital elements for these five systems and provide an estimation of the orbital inclination and shock-cone opening angle for four of them by analysing excess emission profiles with the geometrical model of Luhrs. We argue that line variability in one of the stars (WR 12) might be dominated by some dynamic instabilities in the shocked plasma, making the model inappropriate in this case. A sixth star, WR 69 (WC9d+OB), was observed in the programme based on its published photometric period of 2.293 ± 0.005 d. While we found a probable spectroscopic period of 2.15 ± 0.04 d compatible with that of Marchenko et al. (1998), the period is unlikely related to orbital effects, rather to (relatively fast) rotation of the WC9d component of a much longer binary system.
Estimating constellations among primes - I. Uniformity

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.
Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set $R\subset \mathbb{N}$ is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every $\unicode[STIX]{x1D716}>0$ there exists a set $B=\bigcup _{i=1}^{r}a_{i}\mathbb{N}+b_{i}$ , where $a_{1},\ldots ,a_{r},b_{1},\ldots ,b_{r}\in \mathbb{N}$ , such that $$\begin{eqnarray}\overline{d}(R\triangle B):=\limsup _{N\rightarrow \infty }\frac{|(R\triangle B)\cap \{1,\ldots ,N\}|}{N}<\unicode[STIX]{x1D716}. \end{eqnarray}$$ Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $\unicode[STIX]{x1D6F7}_{x}:=\{n\in \mathbb{N}:\boldsymbol{\unicode[STIX]{x1D711}}(n)/n<x\}$ , where $x\in [0,1]$ and $\boldsymbol{\unicode[STIX]{x1D711}}$ is Euler’s totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if $R\subset \mathbb{N}$ is a rational set with $\overline{d}(R)>0$ , then the following are equivalent: (a) $R$ is divisible, i.e. $\overline{d}(R\cap u\mathbb{N})>0$ for all $u\in \mathbb{N}$ ; (b) $R$ is an averaging set of polynomial single recurrence; (c) $R$ is an averaging set of polynomial multiple recurrence. As an application, we show that if $R\subset \mathbb{N}$ is rational and divisible, then for any set $E\subset \mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_{i}\in \mathbb{Q}[t]$ , $i=1,\ldots ,\ell$ , which satisfy $p_{i}(\mathbb{Z})\subset \mathbb{Z}$ and $p_{i}(0)=0$ for all $i\in \{1,\ldots ,\ell \}$ , there exists $\unicode[STIX]{x1D6FD}>0$ such that the set $$\begin{eqnarray}\{n\in R:\overline{d}(E\cap (E-p_{1}(n))\cap \cdots \cap (E-p_{\ell }(n)))>\unicode[STIX]{x1D6FD}\}\end{eqnarray}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if ${\mathcal{A}}$ is a finite alphabet, $\unicode[STIX]{x1D702}\in {\mathcal{A}}^{\mathbb{N}}$ is rationally almost periodic, $S$ denotes the left-shift on ${\mathcal{A}}^{\mathbb{Z}}$ and $$\begin{eqnarray}X:=\{y\in {\mathcal{A}}^{\mathbb{Z}}:\text{each word appearing in}~y~\text{appears in}~\unicode[STIX]{x1D702}\},\end{eqnarray}$$ then $\unicode[STIX]{x1D702}$ is a generic point for an $S$ -invariant probability measure $\unicode[STIX]{x1D708}$ on $X$ such that the measure-preserving system $(X,\unicode[STIX]{x1D708},S)$ is ergodic and has rational discrete spectrum.
Data and theory for a new model of the horizontal structure of rain cells for propagation applications

A large population of radar-measured ground rain cells is used to devise and assess a rain cell model for use in some of the future telecommunication applications. The model is based on cells of exponential profile (which is shown to reproduce best the point rain rate CDF); both rotational and biaxial symmetries are considered for the horizontal cross sections. Furthermore, the proposed model contains analytical expressions for the joint probability densities of the parameters which define the cell, i.e., peak rain intensity, cell size and axial ratio. Finally, an algorithm is given for adapting the model to the characteristics of any given site: this algorithm requires as input the local cumulative distribution of point rainfall and provides the spatial number densities (i.e., the average number of cells per square kilometer and per unit range of the parameters) which this distribution would produce. The model offers the possibility of predicting the statistics of many propagation parameters (such as attenuation or interference by rain scattering) which are determined by the rain cell characteristics and their frequency of occurrence.
Moral asymmetries and the semantics of many

We present the results of four experiments concerning the evaluations people make of sentences involving many , showing that two sentences of the form many As are Bs and many As are Cs need not be equivalent when evaluated relative to a background in which B and C have the same cardinality and proportion to A, but in which B and C are predicates with opposite semantic and affective values. The data provide evidence that subjects lower the standard relevant to ascribe many for the more negatively valued predicate, and that judgments involving many are sensitive to moral considerations in a broad sense, namely to expectations involving a representation of the desirability as opposed to the mere probability of an outcome. We relate the results to similar semantic asymmetries discussed in the psychological literature, in particular to the Knobe effect and to framing effects. http://dx.doi. org/10.3765/sp.8. 13 Supplementary materials , BibTeX info
Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

We study Mean Field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient form, which extend the known conditions to this general framework. Furthermore, we suggest a variational approach to study a weak formulation of these control problems. We show a natural connection between this weak formulation and optimal transport on path space, which inspires a novel discretization scheme.
All 3-manifolds are the boundary of exotic 4-manifolds.

In this note we show that any closed, oriented 3-manifold is the boundary of a simply connected 4-manifold that admits infinitely many distinct smooth structures. We also show that any fillable contact 3-manifold is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any contact manifold.
Global well-posedness for the massless cubic Dirac equation

We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru. We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who proved a similar result for n=3 in the massive case.
Transitivity of conservative toral endomorphisms

It is shown that if a non-invertible area preserving local homeomorphism on T 2 ? > is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in any smoothness category, of those homotopy classes of conservative endomorphisms that consist entirely of transitive maps.
ON THE CONSTANTS OF THE BOHNENBLUST-HILLE INEQUALITY AND HARDY{LITTLEWOOD INEQUALITIES

In this paper, among other results, we improve the best known estimates for the constants of the generalized Bohnenblust-Hille inequality. These enhancements are then used to improve the best known constants of the Hardy{Littlewood inequality; this inequality asserts that for 4 2m p 1 and K = R or C there exists a constant C K
Diagonal unitary entangling gates and contradiagonal quantum states

Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.
Supercritical Branching Random Walks with a Single Source

We consider two continuous-time branching random walks on multidimensional lattices with birth and death of particles at the origin. In the first one, the underlying random walk is assumed to be symmetric. In the second one, an additional parameter is introduced to intensify artificially the prevalence of branching or walk at the origin. As a side effect, it violates symmetry of the random walk. Necessary and sufficient conditions for exponential growth for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained. General methods to study the models in the supercritical case are proposed.