Miracle-dz/UniFAR-FTUs huggingface dataset:
Graph Embedding for Multi-Robot Path Planning in Complex Environments

—We propose a graph-embedding approach to approximate continuous multi-robot path planning (MPP) problems as discrete ones, allowing known discrete planning techniques be exploited in realistic, complex continuous environments. We ﬁrst design a special pebble graph with a set of conditions, under which MPP problems have the feasibility guarantee. We then develop a mesh optimization algorithm to embed our pebble graph into arbitrarily complex environments. We show that our feasibility conditions can be converted into differentiable geometric constraints, such that our mesh optimizer can ﬁnd feasible solutions via constrained numerical optimization. Two algorithms can be used to solve MPP problems on our peb-ble graphs. Conﬂict-Based Searches (CBS) are preferred for ﬁnding (near) optimal solutions. We further introduce a space-time parallel scheduling approach to ﬁnd sub-optimal solutions for large swarms of congested robots. We have evaluated the effectiveness of our approach on a set of environments with complex geometries, where our method achieves an average of 99 . 0% free-space coverage and 30 . 3% robot density, ensuring a large solution space of discrete MPP algorithms executed on the graphs.
Super Riemann surfaces, metrics, and gravitinos

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of non-linear super symmetric sigma models, the action functional underlying string theory, can be obtained from a geometric action functional on super Riemann surfaces. All invariances of the super symmetric action functional are explained in super geometric terms and the action functional is a functional on the moduli space of super Riemann surfaces.
Persistence of small noise and random initial conditions

Abstract The effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certain random initial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.
Retrospective Voting and Natural Disasters that Cause No Damage: Accounting for the Selective Reporting of Weather Damage

The decision on how to handle missing data is critical for the reliability of a model’s results. This paper makes three contributions. First, we show that a popular weather damage database suﬀers from a nonrandom missing data problem. Second, we follow the applied statistics literature and demonstrate an imputation procedure that relies on an instrument to estimate missing values under a nonrandom selection mechanism. Third, we apply the instrument-based imputation procedure to account for missing damage and reevaluate a seminal study on retrospective voting following a natural disaster. The original ﬁndings are mostly reversed once accounting for the missing data.
Evidence of short-lived SN Ia progenitors

We use the VESPA algorithm and spectra from the Sloan Digital Sky Survey to investigate the star formation history of the host galaxies of 257 Type Ia supernovae. We find 5$\sigma$ evidence for a short-lived population of progenitors with lifetimes of less than 180 Myr, indicating a Type Ia supernova channel arising from stars in the mass range $\sim$3.5-8 $M_\odot$. As standardizeable candles, Type Ia supernovae play an important role in determining the expansion history of the Universe, but to be useful for future cosmological surveys, the peak luminosity needs to be free of uncorrected systematic effects at the level of 1-2%. If the different progenitor routes lead to supernovae with even moderately small differences in properties, then these need to be corrected for separately, or they could lead to a systematic bias in future supernovae surveys, as the prompt route is likely to increase in importance at high redshift. VESPA analysis of hosts could be a valuable tool in this, by identifying which progenitor route is most likely.
Concentration inequalities and moment bounds for sample covariance operators

Let $X,X_1,\dots, X_n,\dots$ be i.i. d. centered Gaussian random variables in a separable Banach space $E$ with covariance operator $\Sigma:$ $$ \Sigma:E^{\ast}\mapsto E,\ \ \Sigma u = {\mathbb E}\langle X,u\rangle, u\in E^{\ast}. $$ The sample covariance operator $\hat \Sigma:E^{\ast}\mapsto E$ is defined as $$ \hat \Sigma u := n^{-1}\sum_{j=1}^n \langle X_j,u\rangle X_j, u\in E^{\ast}. $$ The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm $\|\hat \Sigma-\Sigma\|$ of the deviation of the sample covariance operator from the true covariance operator. In particular, it is shown that $$ {\mathbb E}\|\hat \Sigma-\Sigma\|\asymp \|\Sigma\|\biggl(\sqrt{\frac{{\bf r}(\Sigma)}{n}}\bigvee \frac{{\bf r}(\Sigma)}{n}\biggr), $$ where $$ {\bf r}(\Sigma):=\frac{\Bigl({\mathbb E}\|X\|\Bigr)^2}{\|\Sigma\|}. $$ Moreover, under the assumption that ${\bf r}(\Sigma)\lesssim n,$ it is proved that, for all $t\geq 1,$ with probability at least $1-e^{-t}$ \begin{align*} \Bigl|\|\hat\Sigma - \Sigma\|-{\mathbb E}\|\hat\Sigma - \Sigma\|\Bigr| \lesssim \|\Sigma\|\biggl(\sqrt{\frac{t}{n}}\bigvee \frac{t}{n}\biggr). \end{align*}
Solitary-wave solutions in binary mixtures of Bose-Einstein condensates under periodic boundary conditions

We derive solitary-wave solutions within the mean-field approximation in quasi-one-dimensional binary mixtures of Bose-Einstein condensates under periodic boundary conditions, for the case of an effective repulsive interatomic interaction. The particular gray-bright solutions that give the global energy minima are determined. Their characteristics and the associated dispersion relation are derived. In the case of weak coupling, we diagonalize the Hamiltonian analytically to obtain the full excitation spectrum of "quantum" solitary-wave solutions.
Hybrid Attitude Observer on SO(3) with Global Asymptotic Stability

This paper presents a deterministic hybrid observer for the attitude dynamics of a rigid body that guarantees global asymptotical stability. Any smooth attitude observer suffers from the inherent topological restriction that it is impossible to achieve global attractivity, and as such, attitude observers have been developed with almost global asymptotic stability. We demonstrate that such attitude observer may yields very slow initial convergence rates, and motivated by this, we propose a hybrid attitude observer that guarantees global asymptotic stability on the special orthogonal group. We illustrate that the proposed observer exhibits substantially improved convergence rate uniformly via numerical examples and experiments.
Dyson indices and Hilbert–Schmidt separability functions and probabilities

A confluence of numerical and theoretical results leads us to conjecture that the Hilbert–Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions Bν(a, b), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit–qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.
TWO-DIMENSIONAL MAPPING OF YOUNG STARS IN THE INNER 180 pc OF NGC 1068: CORRELATION WITH MOLECULAR GAS RING AND STELLAR KINEMATICS

We report the first two-dimensional mapping of the stellar population and non-stellar continua within the inner 180 pc (radius) of NGC 1068 at a spatial resolution of 8 pc, using integral field spectroscopy in the near-infrared. We have applied the technique of spectral synthesis to data obtained with the instrument NIFS and the adaptive optics module ALTAIR at the Gemini North Telescope. Two episodes of recent star formation are found to dominate the stellar population contribution: the first occurred 300 Myr ago, extending over most of the nuclear region; the second occurred just 30 Myr ago, in a ring-like structure at ≈100 pc from the nucleus, where it is coincident with an expanding ring of H2 emission. Inside the ring, where a decrease in the stellar velocity dispersion is observed, the stellar population is dominated by the 300 Myr age component. In the inner 35 pc, the oldest age component (age ⩾ 2 Gyr) dominates the mass, while the flux is dominated by blackbody components with temperatures in the range 700 K ⩽ T ⩽ 800 K which we attribute to the dusty torus. We also find some contribution from blackbody and power-law components beyond the nucleus which we attribute to dust emission and scattered light.
Spiral galaxies as progenitors of pseudo-bulge hosting S0s

We present observations of pseudobulges in S0 and spiral galaxies using imaging data taken with the Spitzer Infra-Red Array Camera. We have used 2-d bulge-disk-bar decomposition to determine structural parameters of 185 S0 galaxies and 31 nearby spiral galaxies. Using the Sersic index and the position on the Kormendy diagram to classify their bulges as either classical or pseudo, we find that 25 S0s (14%) and 24 spirals (77%) host pseudoblges. The fraction of pseudobulges we find in spiral galaxies is consistent with previous results obtained with optical data and show that the evolution of a large fraction of spirals is governed by secular processes rather than by major mergers. We find that the bulge effective radius is correlated with the disk scale length for pseudobulges of S0s and spirals, as expected for secular formation of bulges from disk instabilities, though the disks in S0s are significantly smaller than those in spirals. We show that early-type pseudobulge hosting spirals can transform to pseudobulge hosting S0s by simple gas stripping. However, simple gas stripping mechanism is not sufficient to transform the late-type pseudobulge hosting spirals into pseudobulge hosting S0s.
A note on degenerate Stirling numbers of the first kind

Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic number arguments. Also, we derive an identity connecting the degenerate Stirling numbers of the first kind and the degenerate derangement numbers by using probabilistic method.
Orbifold Quantum Cohomology

This is a research announcement on a theory of Gromov-Witten invariants and quantum cohomology of symplectic or projective orbifolds. Our project started in the summer of 98 where our original motivation was to study the quantum cohomology under singular ﬂops in complex dimension three. In this setting, we allow our three-fold to have terminal singularities which can be deformed into a symplectic orbifold. We spent the second half of 98 and most of spring of 99 to develop the foundation of Gromov-Witten invariants over orbifolds, including the key conceptual ingredient — the notion of good map. In the April of 99, we were lucky to meet R. Dijkgraaf who explained to us the orbifold string theory and the role of twisted sectors. The twisted sector provides the precise topological framework for our orbifold quantum cohomology. Our theory of orbifold quantum cohomology was virtually completed in the summer of 99. Here, we give an over-look of the foundation of orbifold quantum cohomology while we leave its applications in other ﬁelds such as birational geometry for future research. The ﬁrst part of our work has already appeared in [CR1]. The second part [CR2] will appear shortly. We would like to thank R. Dijkgraaf for bringing orbifold string theory to our attention at the critical moment of our project. We are also beneﬁted from many discussions with E. Zaslow. Finally, the second author would like to thank E. Witten for many stimulating discussions about orbifold string theory.
Nonlinear Electron Heat Conduction Equation and Self-Similar Method for 1-D Thermal Waves in the Laser Heating of Solid Density DT Fuel

Electron heat conduction is one way that energy transports in laser heating of fusible target material. The aim of inertial confinement fusion (ICF) is to show that the thermal conductivity is strongly dependent on temperature and the equation of electron heat conduction is a nonlinear equation. In this article, we solve the one-dimensional (1-D) nonlinear electron heat conduction equation with a self-similar method (SSM). This solution has been used to investigate the propagation of 1-D thermal wave from a deuterium-tritium (DT) plane source, which occurs when a giant laser pulse impinges onto a solid DT target. It corresponds to the physical problem of rapid heating of a boundary layer of material, in which the energy of a laser pulse is released in a finite initial thickness.
Representations with Small $K$ Types

Let $\mathfrak{g}_{\mathbb{R}}$ be a split real, simple Lie algebra with complexification $\mathfrak{g}$. Let $G_{\mathbb{C}}$ be the connected, simply connected Lie group with Lie algebra $\mathfrak{g}$, $G_{\mathbb{R}}$ the connected subgroup of $G_{\mathbb{C}}$ with Lie algebra $\mathfrak{g}_{\mathbb{R}}$, and $G$ a covering group of $G_{\mathbb{R}}$ with a maximal compact subgroup $K$. A complete classification of "small" $K$ types is derived via Clifford algebras, and an analog, $P^{\xi}$, of Kostant's $P^{\gamma}$ matrix is defined for a $K$ type ${\xi}$ of principal series admitting a small $K$ type. For the connected, simply connected, split real forms of simple Lie types other than type $C_n$, a product formula for the determinant of $P^{\xi}$ over the rank one subgroups corresponding to the positive roots is proved. We use these results to determine cyclicity of a small $K$ type of principal series in the closed Langlands chamber and irreducibility of the unitary principal series admitting a small $K$ type.
Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Let $k$ be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over $k$ which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra $A$ over a field $k$, every norm similarity of $A$ is inner modulo scalar multiplications. It is known that $k$-forms of $E_8$ with index $E^{78}_{8,2}$ and anisotropic kernel a strict inner $k$-form of $E_6$ correspond bijectively (via Moufang hexagons) to Albert division algebras over $k$. The Kneser-Tits problem for a form of $E_8$ as above is equivalent to the Tits-Weiss conjecture (see \cite{TW}). Hence we provide a solution to the Kneser-Tits problem for forms of $E_8$ arising from pure first Tits construction Albert division algebras. As an application, we prove that for $G={\bf Aut}(A),~G(k)/R=1$, where $A$ is a pure first construction Albert division algebra over $k$ and $R$ stands for $R$-equivalence in the sense of Manin (\cite{M}).
ON THE PROBLEM OF VACUUM ENERGY IN FLRW UNIVERSES AND DARK ENERGY

We present a (hopefully) novel calculation of the vacuum energy in expanding FLRW spacetimes based on the renormalization of quantum field theory in nonzero backgrounds. We compute the renormalized effective action up to the two-point function and then apply the formalism to the cosmological backgrounds of interest. As an example we calculate for quasi de Sitter spacetimes the leading correction to the vacuum energy given by the tadpole diagram and show that it behaves as $\sim H_0^2\Lambda_{\rm P}$ where H0 is the Hubble constant and ΛP is the Planck constant. This is of the same order of magnitude as the observed dark energy density in the universe.
A two-level distributed algorithm for general constrained non-convex optimization with global convergence

This paper is motivated by the desire to develop distributed algorithms for nonconvex optimization problems with complicated constraints associated with a network. The network can be a physical one, such as an electric power network, where the constraints are nonlinear power flow equations, or an abstract one that represents constraint couplings between decision variables of different agents. Thus, this type of problems are ubiquitous in applications. Despite the recent development of distributed algorithms for nonconvex programs, highly complicated constraints still pose a significant challenge in theory and practice. We first identify some intrinsic difficulties with the existing algorithms based on the alternating direction method of multipliers (ADMM) for dealing with such problems. We then propose a reformulation for constrained nonconvex programs that enables us to design a two-level algorithm, which embeds a specially structured three-block ADMM at the inner level in an augmented Lagrangian method (ALM) framework. Furthermore, we prove the global convergence of this new scheme for both smooth and nonsmooth constrained nonconvex programs. The proof builds on and extends the classic and recent works on ALM and ADMM. Finally, we demonstrate with computation that the new scheme provides convergent and parallelizable algorithms for several classes of constrained nonconvex programs, for all of which existing algorithms may fail. To the best of our knowledge, this is the first distributed algorithm that extends the ADMM architecture to general nonlinear nonconvex constrained optimization. The proposed algorithmic framework also provides a new principled way for parallel computation of constrained nonconvex optimization.
On the propagation of regularity for solutions of
the dispersion generalized Benjamin–Ono equation

In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and Benjamin-Ono equation, satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In this case the nonlocal term present in the dispersive generalized Benjamin-Ono equation is more challenging that the one in BO equation. To deal with this a new approach is needed. The new ingredient is to combine commutator expansions into the weighted energy estimate. This allow us to obtain the property of propagation and explicitly the smoothing effect.
On the mod-ℓ homology of the classifying space
for commutativity

We study the mod-$\ell$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gomez. We also study the relationship between the mod-$\ell$ type of a Lie group $G(\mathbb{C})$ and the locally finite group $G(\bar{\mathbb{F}}_p)$ where $G$ is a Chevalley group. We see that the naive analogue for $B(\mathbb{Z}, G)$ of the celebrated Friedlander--Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a $G$ action on $B(\mathbb{Z}, G)$.
Formal Symplectic Realizations

We study the relationship between several constructions of symplectic realizations of a given Poisson manifold. Our main result is a general formula for a formal symplectic realization in the case of an arbitrary Poisson structure on $\R^n$. This formula is expressed in terms of rooted trees and elementary differentials, building on the work of Butcher, and the coefficients are shown to be a generalization of Bernoulli numbers appearing in the linear Poisson case. We also show that this realization coincides with a formal version of the original construction of Weinstein, when suitably put in global Darboux form, and with the realization coming from tree-level part of Kontsevich's star product. We provide a simple iterated integral expression for the relevant coefficients and show that they coincide with underlying Kontsevich weights.
Fibered Knots and Virtual Knots

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let $K$ be a knot and $J$ a knot in the complement of $K$ with $\text{lk}(J,K)=0$. Suppose there is covering space $\pi_J: \Sigma \times (0,1) \to \bar{S^3\backslash V(J)}$, where $V(J)$ is a regular neighborhood of $J$ satisfying $V(J) \cap \text{im}(K)=\emptyset$ and $\Sigma$ is a connected compact orientable 2-manifold. Let $K'$ be a knot in $\Sigma \times (0,1)$ such that $\pi_J(K')=K$. Then $K'$ stabilizes to a virtual knot $\hat{K}$, called a virtual cover of $K$ relative to $J$. We investigate what can be said about a classical knot from its virtual covers in the case that $J$ is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.
Voter Model Perturbations and Reaction Diﬀusion Equations

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions d � 3. Combining this result with properties of the PDE, some methods arising from a low density superBrownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins [8] and the second confirms a conjecture of Ohtsuki et al [38] in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.