IA Scholar Query: Applications of the theory of weakly nondegenerate conditions to zero decomposition for polynomial systems.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 21 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Degenerate Cahn-Hilliard equation: From nonlocal to local
https://scholar.archive.org/work/ihwsblm6mna35buh5zneanvghy
We consider the nonlocal Cahn-Hilliard equation with degenerate mobility and smooth potential. As the scaling parameter related to nonlocality tends to zero, we prove that the equation converges to a local Cahn-Hilliard equation. The proof relies on compactness properties and an adapted result from Bourgain-Brezis-Mironescu and Ponce.Charles Elbar, Jakub Skrzeczkowskiwork_ihwsblm6mna35buh5zneanvghyWed, 21 Sep 2022 00:00:00 GMTA Localization Theorem for Dirac operators
https://scholar.archive.org/work/fbeehg4iunhyfae7oe2xh7ddkm
We study perturbed Dirac operators of the form D_s= D + s :Γ(E^0)→Γ(E^1) over a compact Riemannian manifold (X, g) with symbol c and special bundle maps : E^0→ E^1 for s>>0. Under a simple algebraic criterion on the pair (c, ), solutions of D_sψ=0 concentrate as s→∞ around the singular set Z_ of . We prove a spectral separation property of the deformed Laplacians D_s^*D_s and D_s D_s^*, for s>>0. As a corollary we prove an index localization theorem.Manousos Maridakiswork_fbeehg4iunhyfae7oe2xh7ddkmWed, 21 Sep 2022 00:00:00 GMTSpatial populations with seed-bank: finite-systems scheme
https://scholar.archive.org/work/jhwz7zfhdfeghbkpy5hpf2ssvy
We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type one of two types, live in colonies, and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group endowed with the discrete topology. In earlier work we showed that the system has a one-parameter family of equilibria controlled by the relative density of the two types. Moreover, these equilibria exhibit a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. If the wake-up time has finite mean, then there is a single universality class for the scaling limit. On the other hand, if the wake-up time has infinite mean, then there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space.Andreas Greven, Frank den Hollanderwork_jhwz7zfhdfeghbkpy5hpf2ssvyWed, 21 Sep 2022 00:00:00 GMTPlanarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra
https://scholar.archive.org/work/fjnafby3dbbbbfki2jkr2jpyzq
In this paper we study the role of planarity in generalized scattering amplitudes, through several closely interacting structures in combinatorics, algebraic and tropical geometry. The generalized biadjoint scalar amplitude, introduced recently by Cachazo-Early-Guevara-Mizera (CEGM), is a rational function of homogeneous degree -(k-1)(n-k-1) in nk-n independent variables; its poles can be constructed directly from the rays of the positive tropical Grassmannian. We introduce for each pair of integers (k,n) with 2≤ k≤ n-2 a system of generalized positive roots which arises as a specialization of the planar basis of kinematic invariants. We prove that the higher root polytope ℛ^(k)_n-k has volume the k-dimensional Catalan number C^(k)_n-k, via a flag unimodular triangulation into simplices, in bijection with noncrossing collections of k-element subsets. We also give a bijection between certain positroidal subdivisions, called tripods, of the hypersimplex Δ_3,n and noncrossing pairs of 3-element subsets that are not weakly separated. We show that the facets of the Planar Kinematics (PK) polytope, introduced recently by Cachazo and the author, are exactly the nk-n generalized positive roots. We show that the PK specialization of the generalized biadjoint amplitude evaluates to C^(k)_n-k. Looking forward, we give defining equations and conjecture explicit solutions using (ℂℙ^n-k-1)^× (k-1) via a notion of compatibility degree for noncrossing collections, for a two parameter family of generalized worldsheet associahedra 𝒲^+_k,n. These specialize when k=2 to a certain dihedrally invariant partial compactification of the configuration space M_0,n of n distinct points in ℂℙ^1. Many detailed examples are given throughout to motivate future work.Nick Earlywork_fjnafby3dbbbbfki2jkr2jpyzqTue, 20 Sep 2022 00:00:00 GMTDetecting quantum phase transitions in the quasi-stationary regime of Ising chains
https://scholar.archive.org/work/eeuwc2w3b5dlzhe5q76g64az4y
Recently, single-site observables have been shown to be useful for the detection of dynamical criticality due to an emergence of a universal critically-prethermal temporal regime in the magnetization [arXiv:2105.05986]. Here, we explore the potential of single-site observables as probes of quantum phase transitions in integrable and nonintegrable transverse-field Ising chains (TFIC). We analytically prove the requirement of zero modes for a quasi-stationary temporal regime to emerge at a bulk probe site, and show how this regime gives rise to a non-analytic behavior in the dynamical order profiles. Our t-DMRG calculations verify the results of the quench mean-field theory for near-integrable TFIC both with finite-size and finite-time scaling analyses. We find that both finite-size and finite-time analyses suggest a dynamical critical point for a strongly nonintegrable and locally connected TFIC. We finally demonstrate the presence of a quasi-stationary regime in the power-law interacting TFIC, and extract local dynamical order profiles for TFIC in the long-range Ising universality class with algebraic light cones.Ceren B. Dağ, Philipp Uhrich, Yidan Wang, Ian P. McCulloch, Jad C. Halimehwork_eeuwc2w3b5dlzhe5q76g64az4yTue, 20 Sep 2022 00:00:00 GMTThe relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero
https://scholar.archive.org/work/ikpot3le65fcxcww4ok7syzu3e
We establish the relative minimal model program with scaling for projective morphisms of quasi-excellent algebraic spaces admitting dualizing complexes, formal schemes admitting dualizing complexes, semianalytic germs of complex analytic spaces, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. To do so, we prove finite generation of relative adjoint rings associated to projective morphisms of such spaces using the strategy of Cascini and Lazi\'c and the generalization of the Kawamata-Viehweg vanishing theorem to the scheme setting recently established by the second author. To prove these results uniformly, we prove GAGA theorems for Grothendieck duality and dualizing complexes to reduce to the algebraic case.Shiji Lyu, Takumi Murayamawork_ikpot3le65fcxcww4ok7syzu3eMon, 19 Sep 2022 00:00:00 GMTVibrational and anion photoelectron spectroscopy of transition metal clusters
https://scholar.archive.org/work/xwa3yzfoyngrjjjdfqwbe2da2y
The understanding of chemical bonding and of the resulting atomic ar rangements is a central topic in molecular physics. The bonding mechanisms of transition metal atoms still constitute a challenge in their theoretical de scription due to the massive number of valence electrons. Moreover, small transition metal clusters and their complexes may serve as models for catalytic systems of interest for science and technology. The goal of this thesis is the characterization of the geometric and electronic structures of isolated transition metal clusters in the gas phase and, consequently, a better understanding of their bonding nature. The first part of the thesis encompasses experimental results and conclusions for the anionic platinum trimer (Pt3-) and the tantalum nitride anion (TaN−). The data is obtained through anion photoelectron spectroscopy via velocity map imaging (VMI), which permits the simultaneous measurement of photoelectron spectra and photoelectron angular distributions (PADs). The study on TaN− reports the first photoelectron spectra of this diatomic molecule. The spectroscopic assignments, carried out with the support of previous theoretical and experimental works, provide measurements of the adiabatic electron affinity (EAad) and of the vibrational frequencies of the anion and the neutral molecule. In addition, the analysis of the PADs reveals the existence of two core excited shape resonances and disentangles the hybridization of a key molecular orbital. In the study of Pt3-, the experiment is performed in the slow electron velocity map imaging (SEVI) mode to resolve the low-frequency vibrational structure, characteristic of metal clusters. A plethora of information is obtained with the support of density functional theory (DFT) calculations, which includes the presence of two isomeric forms (triangular and linear) and hints at pseudo Jahn-Teller and Jahn-Teller effects. Some of the PADs reveal an oscillatory energetic dependence that, according to the quantum analogy established by Fano with the Y [...]David Yubero Valdivielso, Technische Universität Berlin, Gerard Meijer, Gert Von Helden, André Fielickework_xwa3yzfoyngrjjjdfqwbe2da2yThu, 15 Sep 2022 00:00:00 GMTReaction Network Analysis of Metabolic Insulin Signaling
https://scholar.archive.org/work/ouhntn7ikvc5dfiwayq4wfffme
Absolute concentration robustness (ACR) and concordance are novel concepts in the theory of robustness and stability within Chemical Reaction Network Theory. In this paper, we have extended Shinar and Feinberg's reaction network analysis approach to the insulin signaling system based on recent advances in decomposing reaction networks. We have shown that the network with 20 species, 35 complexes, and 35 reactions is concordant, implying at most one positive equilibrium in each of its stoichiometric compatibility class. We have obtained the system's finest independent decomposition consisting of 10 subnetworks, a coarsening of which reveals three subnetworks which are not only functionally but also structurally important. Utilizing the network's deficiency-oriented coarsening, we have developed a method to determine positive equilibria for the entire network. Our analysis has also shown that the system has ACR in 8 species all coming from a deficiency zero subnetwork. Interestingly, we have shown that, for a set of rate constants, the insulin-regulated glucose transporter GLUT4 (important in glucose energy metabolism), has stable ACR.Patrick Vincent N. Lubenia, Eduardo R. Mendoza, Angelyn R. Laowork_ouhntn7ikvc5dfiwayq4wfffmeTue, 13 Sep 2022 00:00:00 GMTRoot groupoid and related Lie superalgebras
https://scholar.archive.org/work/fpgaqcmbdrhgtcoqy4tscqx7ue
We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an abstract groupoid the root groupoid has many connected components and we show that to some of them one can associate an interesting family of Lie superalgebras which we call root superalgebras. We classify root superalgebras satisfying some additional assumptions. To each root groupoid component we associate a graph (called skeleton) generalizing the Cayley graph of the Weyl group. We establish the Coxeter property of the skeleton generalizing in this way the fact that the Weyl group of a Kac-Moody Lie algebra is Coxeter.Maria Gorelik, Vladimir Hinich, Vera Serganovawork_fpgaqcmbdrhgtcoqy4tscqx7ueTue, 13 Sep 2022 00:00:00 GMTEquidistribution of Hodge loci II
https://scholar.archive.org/work/ymak3g3aefhmfayrsux2kl4na4
Let 𝕍 be a polarized variation of Hodge structure over a smooth complex quasi-projective variety S. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull-push form. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is 2, the Hodge numbers are (q,p,q) and the dimension of S is least rq, we prove that the typical locus where the Picard rank is at least r is equidistributed in S with respect to the volume form c_q^r, where c_q is the qth Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in 𝒜_g, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in 𝒜_g. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull-push form appear in this greater generality and we provide several tools to determine it and we compute it in many examples.Salim Tayou, Nicolas Tholozanwork_ymak3g3aefhmfayrsux2kl4na4Mon, 12 Sep 2022 00:00:00 GMTMultiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds
https://scholar.archive.org/work/2cx26xe3abd65a4f2ploltpzd4
We prove the existence of multiple solutions to the Allen–Cahn–Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik–Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy Γ-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.João Henrique Andrade and Jackeline Conrado and Stefano Nardulli and Paolo Piccione and Reinaldo Resendework_2cx26xe3abd65a4f2ploltpzd4Mon, 12 Sep 2022 00:00:00 GMTInstantons, special cycles, and knot concordance
https://scholar.archive.org/work/f5izpchdvfdfznm37mhhr6wc34
We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with Chern-Simons filtration. It is demonstrated that many of the concordance invariants defined using instantons in recent years can be recovered from our framework. This relationship allows us to compute Kronheimer and Mrowka's s^♯-invariant and fractional ideal invariants for two-bridge knots, and more. In particular, we prove a quasi-additivity property of s^♯, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer τ-invariant of Oszváth and Szabó and the ε-invariant of Hom. We provide evidence for a precise relationship between these latter two invariants and the s^♯-invariant. Some new topological applications that follow from our techniques are as follows. First, we produce a wide class of patterns whose induced satellite maps on the concordance group have the property that their images have infinite rank, giving a partial answer to a conjecture of Hedden and Pinzón-Caicedo. Second, we produce infinitely many two-bridge knots K which are torsion in the algebraic concordance group and yet have the property that the set of positive 1/n-surgeries on K is a linearly independent set in the homology cobordism group. Finally, for a knot which is quasi-positive and not slice, we prove that any concordance from the knot admits an irreducible SU(2)-representation on the fundamental group of the concordance complement. While much of the paper focuses on constructions using singular instanton theory with the traceless meridional holonomy condition, we also develop an analogous framework for concordance invariants in the case of arbitrary holonomy parameters, and some applications are given in this setting.Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher Scaduto, Masaki Taniguchiwork_f5izpchdvfdfznm37mhhr6wc34Mon, 12 Sep 2022 00:00:00 GMTConfigurations of points in projective space and their projections
https://scholar.archive.org/work/vjfv3ep6qfazzkxvzjslejzmay
We call a set of points Z⊂ℙ^3_ℂ an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had ab=12, 16, 20, 24, 30, 36, 42, 48, 54 or 60. Here, for any 4 ≤ a ≤ b, we construct nongrid nondegenerate (a,b)-geproci sets in a systematic way. We also show that the only such example with a=3 is a (3,4)-geproci set coming from the D_4 root system, and we describe the D_4 configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of d-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.Luca Chiantini, Łucja Farnik, Giuseppe Favacchio, Brian Harbourne, Juan Migliore, Tomasz Szemberg, Justyna Szpondwork_vjfv3ep6qfazzkxvzjslejzmaySun, 11 Sep 2022 00:00:00 GMTEvaluation transversality of contact instantons and proof of Shelukhin's conjecture
https://scholar.archive.org/work/7f6cnpuynvcmncmwifoa63xefq
In this paper, we prove Shelukhin's conjecture on the translated points on any compact contact manifold (M,ξ) which reads that for any choice of function H = H(t,x) and contact form λ the contactomorphism ψ_H^1 carries a translated point in the sense of Sandon, whenever H≤ 2 T(λ,M). This improves the result proved in the author's paper (arXiv:2205.12351) entitled "Geometry and analysis of contact instantons and entanglement of Legendrian links I" by a factor of 2. Main analytical and geometrical tools are the ones employed in the paper . Additional analytical ingredients are the gluing transversality and some dimension counting argument based on the evaluation map transversality which we also establish in the present paper.Yong-Geun Ohwork_7f6cnpuynvcmncmwifoa63xefqThu, 08 Sep 2022 00:00:00 GMTThe local motivic monodromy conjecture for simplicial nondegenerate singularities
https://scholar.archive.org/work/t5kovebx6vc53j22anata4pie4
We prove the local motivic monodromy conjecture for singularities that are nondegenerate with respect to a simplicial Newton polyhedron. It follows that all poles of the local topological zeta functions of such singularities correspond to eigenvalues of monodromy acting on the cohomology of the Milnor fiber of some nearby point, as do the poles of Igusa's local p-adic zeta functions for large primes p.Matt Larson, Sam Payne, Alan Stapledonwork_t5kovebx6vc53j22anata4pie4Thu, 08 Sep 2022 00:00:00 GMTCyclic independence: Boolean and monotone
https://scholar.archive.org/work/tpqsdxbqrvdnldqt7u3j3bcbee
The present paper introduces a modified version of cyclic-monotone independence which originally arose in the context of random matrices, and also introduces its natural analogy called cyclic-Boolean independence. We investigate formulas for convolutions, limit theorems for sums of independent random variables, and also classify infinitely divisible distributions with respect to cyclic-Boolean convolution. Finally, we provide applications to the eigenvalues of the adjacency matrices of iterated star products of graphs and also iterated comb products of graphs.Octavio Arizmendi, Takahiro Hasebe, Franz Lehnerwork_tpqsdxbqrvdnldqt7u3j3bcbeeThu, 08 Sep 2022 00:00:00 GMTLa Logistica del Computer Quantistico e l'Informatica Relativa
https://scholar.archive.org/work/urhevxgklfgp7ny3sttrgnz7xu
Notizie da giornali e siti web riportato sempre più enfaticamente i successi relativi a computer quantistici. È pertanto naturale porsi delle domante circa questi computer, su come essi operano, su dove sono fisicamente, su cosa serve alla loro gestione. Anzi, è doveroso porsi domande, poiché i titoli delle notizie possono indurre ad intenderli come "supercomputer" effettivamente esistenti, capaci di risolvere in pochi secondi dei problemi di calcolo che i computer classici impiegherebbero millenni a processare. Questo lavoro propone quindi una review degli attuali Computer Quantistici, con particolare riguardo, ove possibile, alla relativa logistica. Nel panorama dell'Informatica Quantistica, si intende approfondire come potrà essere il futuro delle infrastrutture ad essi legate e quali saranno i sistemi base di algoritmi relativi al Calcolo Quantistico. In definitiva, si cercherà di comprendere quale sarà la futura logistica, intesa come nuova «arte del computare». Particolare attenzione verrà data al vantaggio quantistico, la strategia di medio termine relativa alla progettazione di algoritmi per le prossime generazioni di computer quantistici. Tali algoritmi sono richiesti da un ventaglio di applicazioni strategiche. Attualmente sono principalmente algoritmi di simulazione di fisica e chimica quantistica. Si illustreranno due metodi di calcolo: quello basato sulla computazione quantistica abiabatica, che si ottiene attraverso il quantum annealing (sistemi D-Wave), e quello che utilizza un array di porte quantistiche (sistemi IBM, Google ed altri). Si vedranno, con maggiore dettaglio, le simulazioni basate sul metodo del Variational Quantum Eigensolver. Alcune nozioni relative ai qubit, alle porte quantistiche ed alle Hamiltoniane ad esse legate verranno proposte. Oltre a parlare di qubit, si parlerà di qumode. L'arte del computare quantistico ha infatti due facce, quella basata sul qubit e quella basata sul qumode. Il quantum computer Borealis di Xanadu è capace di sviluppare un sistema ibrido, con una emulaz [...]Amelia Carolina Sparavignawork_urhevxgklfgp7ny3sttrgnz7xuWed, 07 Sep 2022 00:00:00 GMTPre-threshold fractional susceptibility function: holomorphy and response formula
https://scholar.archive.org/work/gyelpocfyvbkvnbd26fqeobgqi
For certain smooth unimodal families with negative Schwarzian derivative, we construct a set of Collet-Eckmann and subexponentially recurrent parameters Ω, whose complement set has sufficiently fast decaying density, on which exponential mixing with uniform rates occurs. We use this construction to establish holomorphy of the true fractional susceptibility function of the logistic family, in a disk of radius larger than one, for differentiation index 0≤η<1/2, as recently conjectured by Baladi and Smania. We also obtain a fractional response formula.Julien Sedrowork_gyelpocfyvbkvnbd26fqeobgqiFri, 02 Sep 2022 00:00:00 GMTWeak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem
https://scholar.archive.org/work/sfhf6c3i6zeh5mvtv5r2bcxsdq
We discuss tensor categories motivated by CFT, their unitarizability and applications to various models including the affine VOAs. We discuss classification of type A Verlinde fusion categories. We propose an approach to Kazhdan-Lusztig-Finkelberg theorem. This theorem gives a ribbon equivalence between the fusion category associated to a quantum group at a certain root of unity and that associated to a corresponding affine vertex operator algebra at a suitable positive integer level. We develop ideas by Wenzl. Our results rely on the notion of weak-quasi-Hopf algebra of Drinfeld-Mack-Schomerus. We were also guided by Drinfeld original proof, by Bakalov and Kirillov and Neshveyev and Tuset work for a generic parameter. Wenzl described a fusion tensor product in quantum group fusion categories, and related it to the unitary structure. Given two irreducible objects, the inner product of the fusion tensor product is induced by the braiding of U_q(g), with q a suitable root of 1. Moreover, the paper suggests a suitable untwisting procedure to make the unitary structure trivial. Then it also describes a continuous path that intuitively connects objects of the quantum group fusion category to representations of the simple Lie group defining the affine Lie algebra. We study this procedure. One of our main results is the construction of a Hopf algebra in a weak sense associated to quantum group fusion category and of a twist of it giving a wqh structure on the Zhu algebra and a unitary modular fusion category structure on the representation category of the affine Lie algebra, confirming an early view by Frenkel and Zhu. We show that this modular fusion category structure is equivalent to that obtained via the tensor product theory of VOAs by Huang and Lepowsky. This gives a direct proof of FKL theorem.Sebastiano Carpi, Sergio Ciamprone, Marco Valerio Giannone, Claudia Pinzariwork_sfhf6c3i6zeh5mvtv5r2bcxsdqWed, 31 Aug 2022 00:00:00 GMTHop-Spanners for Geometric Intersection Graphs
https://scholar.archive.org/work/2qbr44ozozcg3paxx2pi3rhj3m
A t-spanner of a graph G=(V,E) is a subgraph H=(V,E') that contains a uv-path of length at most t for every uv∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^1+1/k) edges for k≥ 1. This bound is the best possible for 1≤ k≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture. We study t-spanners for t∈{2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is tight up to a factor of loglog n. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog^2 n) edges.Jonathan B. Conroy, Csaba D. Tóthwork_2qbr44ozozcg3paxx2pi3rhj3mWed, 31 Aug 2022 00:00:00 GMT