Regularity for graphs with bounded anisotropic mean curvature

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.

We prove upper bounds for the first eigenvalue of the Laplacian of hypersurfaces of Euclidean space involving anisotropic mean curvatures. Then, we study the equality case and its stability.

Given an anisotropy ɸ on R

Rigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bezout (m-Bezout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bezout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bezout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bezout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

The relative importance of collisional vs noncollisional orogenesis at continental-margin orogens is an unresolved problem in tectonics. The fundamental problem is that contraction of continental lithosphere in a subduction hanging wall could be due to subcontinental subduction (noncollisional orogenesis) or to arccontinent collision preceding subcontinental subduction initiated by subduction flip (collisional orogenesis). A secondary problem arises in proving that sutures representing ocean-basin closure and arc collision exist in the complex oceanic terranes that typically constitute the internal parts of continental-margin orogens. This problem exists in part because “tectonic flaking” results in replacing the pre-collision subduction-zone fault with a new fault during collision. Tectonic flaking and subsequent thrust imbrication, folding, normal faulting, and strike-slip faulting make arc-continent sutures in collisional orogens difficult to distinguish from deformed marginal arcs in noncollisional orogens. Phanerozoic continental-margin orogens contain evidence for early orogenic arc accretion, and the Mesozoic cordilleran orogen of the western United States, a classic example of a noncollisional, “Andean-style” orogen, fits a collisional model in detail. Tectonic models for contractional deformation in Archean and Proterozoic continental-margin orogens should consider both collisional and noncollisional processes.

In this paper we propose a general logic programming framework allowing the recognition of plans and intentions behind speech acts through abductive reasoning. These inferences enables each agent to have an active participation in dialogues, namely in cooperative informationseeking dialogues. In our framework the possible actions, events, states and world knowledge are represented by extended logic programs (LP with explicit negation) and the abductive inference process is modeled by a framework wich is based on the Well Founded Semantics augmented with explicit negation (WFSX) and contradiction removal semantics (CRSX) ([PAA92]). It will be shown how this framework supports abductive reasoning with Event Calculus ([Esh88]) and some classical examples in the domain of information-seeking dialogues will be shown ([Lit85, Pol86]). Finally, some open problems will be pointed out.KeywordsAbductive ReasoningUser PlanAbductive InferencePlan RecognitionEvent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A game approach to free boundary problems of anisotropic forced mean curvature flow equations

We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.

In 2004 Chambolle proposed an algorithm for mean curvature flow based on a variational problem. Since then, the convergence, extensions and applications of his algorithm have been studied by many people. In this paper we give a proof of the convergence of an anisotropic version of Chambolle’s algorithm by use of the signed distance function. An application of our scheme to an approximation of the nonlocal curvature flow such as crystalline one is also discussed.

<p>Seismic anisotropy beneath Anatolia is complex, with several layers of different anisotropy. <br>The average anisotropy is well constrained by shear-wave splitting measurements [Kaviani et al., 2009], suggesting very strong anisotropy (over 1.5s delay time). However, the vertical layering of anisotropy and the contribution of each layer is still an open question. </p><p>We construct anisotropic phase-velocity maps of fundamental-mode Rayleigh waves for the Anatolia region using records from several regional seismic stations, using both earthquake and ambient noise data. <br>The collision between the Arabia and Eurasia plates leads to the westward extrusion (and EW anisotropy) of the Anatolian crust, consistent with the seismic anisotropy patterns we found in the crust (1%, EW fast axis) and with previous studies [Mutlu et al., 2011; Legendre et al., 2020].<br>The Aegean/Anatolian subduction system with slab tearing and breakoff induces a complex flow pattern and anisotropy in the upper mantle [van Hinsbergen et al., 2010; Kaviani et al., 2018].<br>This is in agreement with the anisotropy we image in the lithosphere (1%, N020E and N100E fast axes) and asthenosphere (1%, N120E). However, the anisotropy in these layers display limited amplitudes.<br>At deeper depth, remnant Bitlis and Tethyan slabs are lying flat above the 660-km discontinuity [Berk Biryol et al., 2011]. </p><p>The uniform pattern of anisotropy from shear-wave splitting observations can not be explained solely by a single anisotropic layer, and is not consistent with the anisotropy observed in the crustl lithospheric and asthenospheric mantle. This suggests that main contribution of the anisotropy likely originates from a deep source around the mantle transition zone [Legendre et al., 2021].</p>

Permanent magnets containing rare earth elements are essential components for the electrification of society. Ce(Co 1-x Cu x ) 5 permanent magnets are a model system known for their substantial coercivity, yet the underlying mechanism remains unclear. Here, we investigate Ce(Co 0.8 Cu 0.2 ) 5.4 magnets with a coercivity of ∼1 T. Using transmission electron microscopy (TEM) and atom probe tomography (APT), we identify a nanoscale cellular structure formed by spinodal decomposition. Cu-poor cylindrical cells (∼5-10 nm in diameter, ∼20 nm long) have a disordered CeCo 5 -type structure and a composition Ce(Co 0.9 Cu 0.1 ) 5.3 . Cu-rich cell boundaries are ∼ 5 nm thick and exhibit a modified CeCo 5 structure, with Cu ordered on the Co sites and a composition Ce(Co 0.7 Cu 0.3 ) 5.0 . Micromagnetic simulations demonstrate that the intrinsic Cu concentration gradients up to 12 at.% Cu/nm lead to a spatial variation in magnetocrystalline anisotropy and domain wall energy, resulting in effective pinning and high coercivity. Compared to Sm 2 Co 17 -type magnets, Ce(Co 0.8 Cu 0.2 ) 5.4 displays a finer-scale variation of conventional pinning with lower structural and chemical contrast in its underlying nanostructure. The identification of nanoscale chemical segregation in nearly single-phase Ce(Co 0.8 Cu 0.2 ) 5.4 magnets provides a microstructural basis for the long-standing phenomenon of "giant intrinsic magnetic hardness" in systems such as SmCo 5-x M x , highlighting avenues for designing rare-earth-lean permanent magnets via controlled nanoscale segregation.

A function f: Fn2 → {0, 1} is odd-cycle-free if there are no x1,..., xk e Fn2 with k an odd integer such that f(x1) =... = f(xk) = 1 and x1 +... + xk = 0. We show that one can distinguish odd-cycle-free functions from those e-far from being odd-cycle-free by making poly(1/e) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.The first problem we study is directly reducing testing linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have been shown previously (Kral-Serra-Vena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linear-invariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question still remains open.

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

To guide improved properties coincident with reduction of critical materials in permanent magnets, we investigate via density functional theory (DFT) the intrinsic magnetic properties of a promising system, $R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti$Z$ with $R$=Y, Ce and interstitial doping ($Z$=H, C, N). The magnetization $M$, Curie temperature $T_\text{C}$, and magnetocrystalline anisotropy energy $K$ calculated in local density approximation to DFT agree well with measurements. Site-resolved contributions to $K$ reveal that all three Fe sublattices promote uniaxial anisotropy in YFe$_{11}$Ti, while competing anisotropy contributions exist in YCo$_{11}$Ti. As observed in experiments on $R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti, we find a complex nonmonotonic dependence of $K$ on Co content, and show that anisotropy variations are a collective effect of MAE contributions from all sites and cannot be solely explained by preferential site occupancy. With interstitial doping, calculated $T_\text{C}$ enhancements are in the sequence of N$>$C$>$H, with volume and chemical effects contributing to the enhancement. The uniaxial anisotropy of $R$(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti$Z$ generally decreases with C and N; although, for $R$=Ce, C doping is found to greatly enhance it for a small range of 0.7$<$$x$$<$0.9.

We present a variational formulation for the evolution of surface clusters in ℝ^3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including computations for regularized crystalline surface energy densities.