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Consider the orbit of real points of a maximal Q-torus from identity coset on SL(N,R)/SL(N,Z), it naturally carries a possibly infinite Haar measure. We classify all possible limit measures of it when translated by a sequence of elements from SL(N,R). This is a natural extension of Shapira and Zheng’s work where only Q-split tori are considered.

ITE welcomes the following new members who recently joined our community of transportation professionals: (M) Matthew L. Snyder, E.I.T., LEED AR (M) Bill Starkey (M) Trandon Struck (M) Matt Stubblefield (M) Carl Sundstrom (M) Derek Sunstrom, RE. (M) Tori Winters (M) Viktoriya Wise (M) Josh Workman (M) Grant Wuebben (M) Htet-Htet Wunn (M) Zach Wyche (M) Elizabeth Yarnall (M) Tabitha Yeoh (M) Gary Yeung (M) Md Yousuf, M.A.Sc., REng.

Introduçâo: As exostoses sâo protuberancias ósseas que têm origem da cortical óssea. Na cavidade bucal as formas mais comuns sâo o toras palatino e o toras mandibular. Apresentam etiologia nâo comprovada, mas acredita-se que sejam alteraçôes de desenvolvimento multifatoriais. Objetivo: Apresentar um relato de caso clínico envolvendo toras palatino. Materials e Métodos: Paciente I.A.S., género masculino, 35 anos, procurau a Clínica de Cirargia Buco Maxilo Facial do Curso de Odontología - UFPE, queixando-se de um carroço no céu da boca que impedía o uso de protese. Após anamnese e exame clínico, detectou-se urna exostose óssea no palato que impedia a adaptaçâo de qualquer protese. Foram solicitados exames pré-operatórios, com sua posterior avaliaçâo e realizaçâo da cirargia. Após a profilaxia e degermaçâo das regiôes adjacentes, incisionou-se a fibromucosa palatina até atingir o osso. Os retalhos foram levantados, expondo assim o crescimento ósseo e permitindo sua exérese, em seguida, procedeu-se à irrigaçâo com soluçâo fisiológica, hemostasia, e finalmente a ferida foi suturada com pontos individuáis. Resultados: O toras palatino foi removido, possibilitando a adaptaçâo da futura prótese. Conclusäo: A cirargia pré-protética é um procedimento a ser cogitado quando houver indicaçâo, possibilitando a instalaçâo de proteses que reabilitem o paciente funcional e estéticamente.

Let$f\\;:\\; M\\rightarrow \\mathbb{C}P^{2}$be an isometric immersion of a compact surface in the complex projective plane$\\mathbb{C}P^{2}$. In this paper, we consider the Helfrich-type functional$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)=\\int _{M}(|H|^{2}+\\lambda _{1}+\\lambda _{2} C^{2})\\textrm{d} M$, where$\\lambda _{1}, \\lambda _{2}\\in \\mathbb{R}$with$\\lambda _{1}\\geqslant 0$,$H$and$C$are respectively the mean curvature vector and the Kähler function of$M$in$\\mathbb{C}P^{2}$. The critical surfaces of$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)$are called Helfrich surfaces. We compute the first variation of$\\mathcal{H}_{\\lambda _{1},\\lambda _{2}}(f)$and classify the homogeneous Helfrich tori in$\\mathbb{C}P^{2}$. Moreover, we study the Helfrich energy of the homogeneous tori and show the lower bound of the Helfrich energy for such tori.

We examine a modified Sprott A system, one of the 17 chaotic systems without equilibria introduced by Jafari, Sprott, and Golpayegani (2013). For specific parameter values, the modified system exhibits invariant spheres. Using a stereographic map, we analyze the stability of the equilibria and demonstrate that all orbits, except for the unstable equilibrium, converge to the stable equilibrium. For other parameter values, the system has neither invariant spheres nor equilibria. Instead, the state space is foliated by tori.

Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

We study 31 little red dots (LRD) detected by JADES/NIRCam and covered by the SMILES/MIRI survey, of which ∼70% are detected in the two bluest MIRI bands and 40% in redder MIRI filters. The median/quartiles redshifts are z=6.95.97.7 (55% spectroscopic). The spectral slopes flatten in the rest-frame near-infrared, consistent with a 1.6 μm stellar bump but bluer than direct pure emission from active galactic nuclei (AGN) tori. The apparent dominance of stellar emission at these wavelengths for many LRDs expedites stellar mass estimation: the median/quartiles are logM⋆/M⊙=9.49.19.7 . The number density of LRDs is 10−4.0±0.1 Mpc−3, accounting for 14% ± 3% of the global population of galaxies with similar redshifts and masses. The rest-frame near-/mid-infrared (2–4 μm) spectral slope reveals significant amounts of warm dust (bolometric attenuation ∼3–4 mag). Our spectral energy distribution modeling implies the presence of <0.4 kpc diameter knots, heated by either dust-enshrouded OB stars or an AGN producing a similar radiation field, obscured by A(V) > 10 mag. We find a wide variety in the nature of LRDs. However, the best-fitting models for many of them correspond to extremely intense and compact starburst galaxies with mass-weighted ages 5–10 Myr, very efficient in producing dust, with their global energy output dominated by the direct (in the flat rest-frame ultraviolet and optical spectral range) and dust-recycled emission from OB stars with some contribution from an obscured AGN (in the infrared).

Non-Abelian topological order is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged 1 – 4 . These anyonic excitations are promising building blocks of fault-tolerant quantum computers 5 , 6 . However, despite extensive efforts, non-Abelian topological order and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian topological order. Here we present the realization of non-Abelian topological order in the wavefunction prepared in a quantum processor and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum’s H2 trapped-ion quantum processor, we create the ground-state wavefunction of D 4 topological order on a kagome lattice of 27 qubits, with fidelity per site exceeding 98.4 per cent. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunnelling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon—a peculiar feature of non-Abelian topological order. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices. A trapped-ion quantum processor is used to create ground-states and excitations of non-Abelian topological order on a kagome lattice of 27 qubits with high fidelity.

This is the third in a series of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.