http://publications.mfo.de/handle/mfo/1348 Wed, 08 Apr 2026 15:44:46 GMT 2026-04-08T15:44:46Z 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 GMT http://publications.mfo.de/handle/mfo/1275 2016-12-30T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1274 2016-12-21T00:00:00Z 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 $\mathbb{C} \cup \{\infty\}$. 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 GMT http://publications.mfo.de/handle/mfo/1273 2016-12-13T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1262 2016-11-17T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1261 2016-11-12T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1260 2016-11-11T00:00:00Z 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 $\mathbb{C} \cup \{\infty\}.$ 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 GMT http://publications.mfo.de/handle/mfo/1259 2016-11-10T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1257 2016-10-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1256 2016-10-01T00:00:00Z 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. 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 GMT http://publications.mfo.de/handle/mfo/1255 2016-10-01T00:00:00Z 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.