2016
http://publications.mfo.de/handle/mfo/1348
Wed, 29 Mar 2023 23:16:53 GMT2023-03-29T23:16:53ZFinitary Proof Systems for Kozen's μ
http://publications.mfo.de/handle/mfo/1275
Finitary Proof Systems for Kozen's μ
Afshari, Bahareh; Leigh, Graham E.
We present three finitary cut-free sequent calculi for the modal $μ$-calculus. Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof system with names’ (2014) and feature special inferences that discharge open assumptions. The third system is a variant of Kozen’s axiomatisation in which cut is replaced by a strengthening of the $ν$-induction inference rule. Soundness and completeness for the three systems is proved by establishing a sequence of embeddings between the calculi, starting at Stirling’s tableau-proofs and ending at the original axiomatisation of the $μ$-calculus due to Kozen. As a corollary we obtain a completeness proof for Kozen’s axiomatisation which avoids the usual detour through automata or games.
Research in Pairs 2016
Fri, 30 Dec 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12752016-12-30T00:00:00ZAfshari, BaharehLeigh, Graham E.We present three finitary cut-free sequent calculi for the modal $μ$-calculus. Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof system with names’ (2014) and feature special inferences that discharge open assumptions. The third system is a variant of Kozen’s axiomatisation in which cut is replaced by a strengthening of the $ν$-induction inference rule. Soundness and completeness for the three systems is proved by establishing a sequence of embeddings between the calculi, starting at Stirling’s tableau-proofs and ending at the original axiomatisation of the $μ$-calculus due to Kozen. As a corollary we obtain a completeness proof for Kozen’s axiomatisation which avoids the usual detour through automata or games.The Initial and Terminal Cluster Sets of an Analytic Curve
http://publications.mfo.de/handle/mfo/1274
The Initial and Terminal Cluster Sets of an Analytic Curve
Gauthier, Paul Montpetit
For an analytic curve $\gamma : (a,b) \to \mathbb{C}$, the set of values approaches by $\gamma(t)$, as $t ↘a$ and as $t↗b$ can be any two continuua of $\mathbb{C} \cup \{\infty\}$.
Research in Pairs 2016
Wed, 21 Dec 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12742016-12-21T00:00:00ZGauthier, Paul MontpetitFor an analytic curve $\gamma : (a,b) \to \mathbb{C}$, the set of values approaches by $\gamma(t)$, as $t ↘a$ and as $t↗b$ can be any two continuua of $\mathbb{C} \cup \{\infty\}$.Boundary Representations of Operator Spaces, and Compact Rectangular Matrix Convex Sets
http://publications.mfo.de/handle/mfo/1273
Boundary Representations of Operator Spaces, and Compact Rectangular Matrix Convex Sets
Fuller, Adam H.; Hartz, Michael; Lupini, Martino
We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations.
This yields a canonical construction of the triple envelope of an operator space.
OWLF 2016
Tue, 13 Dec 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12732016-12-13T00:00:00ZFuller, Adam H.Hartz, MichaelLupini, MartinoWe initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations.
This yields a canonical construction of the triple envelope of an operator space.The Berry-Keating Operator on a Lattice
http://publications.mfo.de/handle/mfo/1262
The Berry-Keating Operator on a Lattice
Bolte, Jens; Egger, Sebastian; Keppeler, Stefan
We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating.
Research in Pairs 2016
Thu, 17 Nov 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12622016-11-17T00:00:00ZBolte, JensEgger, SebastianKeppeler, StefanWe construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating.On Weak Weighted Estimates of Martingale Transform
http://publications.mfo.de/handle/mfo/1261
On Weak Weighted Estimates of Martingale Transform
Nazarov, Fedor; Reznikov, Alexander; Vasyunin, Vasily; Volberg, Alexander
We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture, which stayed open after Muckenhoupt-Wheeden's conjecture was disproved by Reguera-Thiele.
Research in Pairs 2016
Sat, 12 Nov 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12612016-11-12T00:00:00ZNazarov, FedorReznikov, AlexanderVasyunin, VasilyVolberg, AlexanderWe consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture, which stayed open after Muckenhoupt-Wheeden's conjecture was disproved by Reguera-Thiele.Spherical Arc-Length as a Global Conformal Parameter for Analytic Curves in the Riemann Sphere
http://publications.mfo.de/handle/mfo/1260
Spherical Arc-Length as a Global Conformal Parameter for Analytic Curves in the Riemann Sphere
Gauthier, Paul Montpetit; Nestoridis, Vassili; Papadopoulos, Athanase
We prove that for every analytic curve in the complex plane $\mathbb{C}$, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in $\mathbb{R}^n$ and $\mathbb{C}^n$ and we discuss the situation of curves in the Riemann sphere $\mathbb{C} \cup \{\infty\}.$
Research in Pairs 2016
Fri, 11 Nov 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12602016-11-11T00:00:00ZGauthier, Paul MontpetitNestoridis, VassiliPapadopoulos, AthanaseWe prove that for every analytic curve in the complex plane $\mathbb{C}$, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in $\mathbb{R}^n$ and $\mathbb{C}^n$ and we discuss the situation of curves in the Riemann sphere $\mathbb{C} \cup \{\infty\}.$Killing Tensors on Tori
http://publications.mfo.de/handle/mfo/1259
Killing Tensors on Tori
Heil, Konstantin; Moroianu, Andrei; Semmelmann, Uwe
We show that Killing tensors on conformally at n-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.
Research in Pairs 2016
Thu, 10 Nov 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12592016-11-10T00:00:00ZHeil, KonstantinMoroianu, AndreiSemmelmann, UweWe show that Killing tensors on conformally at n-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.Late-Time Behaviour of Israel Particles in a FLRW Spacetime with Λ>0
http://publications.mfo.de/handle/mfo/1257
Late-Time Behaviour of Israel Particles in a FLRW Spacetime with Λ>0
Lee, Ho; Nungesser, Ernesto
In this paper we study the relativistic Boltzmann equation in a spatially flat FLRW space-time. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of this type where a physically well-motivated scattering kernel is considered for the general relativistic Boltzmann equation.
Research in Pairs 2016
Sat, 01 Oct 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12572016-10-01T00:00:00ZLee, HoNungesser, ErnestoIn this paper we study the relativistic Boltzmann equation in a spatially flat FLRW space-time. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of this type where a physically well-motivated scattering kernel is considered for the general relativistic Boltzmann equation.Getzler rescaling via adiabatic deformation and a renormalized local index formula
http://publications.mfo.de/handle/mfo/1256
Getzler rescaling via adiabatic deformation and a renormalized local index formula
Bohlen, Karsten; Schrohe, Elmar
We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). After introducing a renormalized supertrace on Lie manifolds with spin structure, defined on a suitable class of rapidly decaying functions, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of the rescaled bundle over the adiabatic groupoid and introduce a rescaling of the heat kernel encoded in a vector bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion using the Lichnerowicz theorem on the fibers of the groupoid and the Lie manifold.
OWLF 2015
Sat, 01 Oct 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12562016-10-01T00:00:00ZBohlen, KarstenSchrohe, ElmarWe prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). After introducing a renormalized supertrace on Lie manifolds with spin structure, defined on a suitable class of rapidly decaying functions, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of the rescaled bundle over the adiabatic groupoid and introduce a rescaling of the heat kernel encoded in a vector bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion using the Lichnerowicz theorem on the fibers of the groupoid and the Lie manifold.Alexander r-Tuples and Bier Complexes
http://publications.mfo.de/handle/mfo/1255
Alexander r-Tuples and Bier Complexes
Jojic, Dusko; Nekrasov, Ilya; Panina, Gaiane; Zivaljevic, Rade
We introduce and study Alexander $r$-Tuples $\mathcal{K} = \langle K_i \rangle ^r_{i=1}$ of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins $\mathcal{K}^*_\Delta$ of Alexander $r$-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the $r$-fold deleted join of Alexander $r$-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the $r$-fold deleted join of a collective unavoidable $r$-tuple is $(n - r - 1)$-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander $r$-tuples and Bier complexes.
Research in Pairs 2016
Sat, 01 Oct 2016 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12552016-10-01T00:00:00ZJojic, DuskoNekrasov, IlyaPanina, GaianeZivaljevic, RadeWe introduce and study Alexander $r$-Tuples $\mathcal{K} = \langle K_i \rangle ^r_{i=1}$ of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins $\mathcal{K}^*_\Delta$ of Alexander $r$-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the $r$-fold deleted join of Alexander $r$-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the $r$-fold deleted join of a collective unavoidable $r$-tuple is $(n - r - 1)$-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander $r$-tuples and Bier complexes.