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1D symmetry for semilinear pdes from the limit interface of the solution
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Farina, Alberto; Valdinoci, Enrico
We study bounded, entire, monotone solutions of the Allen-Cahn equation. We prove that under suitable assumptions on the limit interface and on the energy growth, the solution is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.
2-Minimal subgroups in classical groups: Linear and unitary groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Parker, Chris; Rowley, Peter
[no abstract available]
A 3-Local characterization of Co2
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Parker, Christopher; Rowley, Peter
Conway's second largest simple group, Co2, is characterized by the centralizer of an element of order 3 and certain fusion data.
A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Parker, Christopher; Rowley, Peter
In this article we give 3-local characterizations of the alternating and symmetric groups of degree 8 and use these characterizations to recognize the sporadic simple group discovered by McLaughlin from its 3-local subgroups.
3D boundary recovery by constrained Delaunay tetrahedralization
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Si, Hang; Gärtner, Klaus
Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a it constrained Delaunay tetrahedralization (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples
3D electrothermal simulations of organic LEDs showing negative differential resistance
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Liero, Matthias; Fuhrmann, Jürgen; Glitzky, Annegret; Koprucki, Thomas; Fischer, Axel; Reineke, Sebastian
Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.
3D numerical simulations of THz generation by two-color laser filaments
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Bergé, Luc; Skupin, Stefan; Köhler, Christian; Babushkin, Ihar; Herrmann, Joachim
Terahertz (THz) radiation produced by the filamentation of two-color pulses over long distances in argon is numerically investigated using a comprehensive model in full spacetime resolved geometry. We show that the dominant physical mechanism for THz generation in the filamentation regime at clamping intensity is based on quasi-dc plasma currents. The calculated THz spectra for different pump pulse energies and pulse durations are in agreement with previously reported experimental observations. For the same pulse parameters, near-infrared pump pulses at 2 m are shown to generate a more than one order of magnitude larger THz yield than pumps centered at 800 nm.
The 3D transient semiconductor equations with gradient-dependent and interfacial recombination
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Disser, Karoline; Rehberg, Joachim
We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.
4 = 2 × 2, or the Power of Even Integers in Fourier Analysis
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Negro, Giuseppe; Oliveira e Silva, Diogo
We describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic analysis with surprising applications to number theory and geometric measure theory.
4-dimensional Manifolds
(Oberwolfach-Walke : Mathematisches Forschungsinstitut Oberwolfach, 2001) Kronheimer, Peter B.; Stern, Ronald J.
[no abstract available]
A Cheeger Type Inequality in Finite Cayley Sum Graphs
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Biswas, Arindam; Saha, Jyoti Prakash
Let G be a finite group and S be a symmetric generating set of G with |S|=d. We show that if the undirected Cayley sum graph CΣ(G,S) is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from −1. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval (−1+h(G)4η,1−h(G)22d2], where h(G) denotes the (vertex) Cheeger constant of the d-regular graph CΣ(G,S) and η=29d8. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph C(G,S).
A Deformed Quon Algebra
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Randriamaro, Hery
The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a_(i,k), (i,k)∈N^∗ × [m], on an infinite dimensional vector space satisfying the deformed q-mutator relations aj,a^(\dag)_(i,k) = q^(\dag)_(i,k)aj,l + q^(β−k,l)δ_(i,j). We prove the realizability of our model by showing that, for suitable values of q, the vector space generated by the particle states obtained by applying combinations of ai,k's and a^(\dag)_(i,k)'s to a vacuum state |0⟩ is a Hilbert space. The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group.
A Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Falcó, Javier; Gauthier, Paul Montpetit; Manolaki, Myrto; Nestoridis, Vassili
For compact sets K⊂Cd, we introduce a subalgebra AD(K) of A(K), which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
A Gentle Introduction to Interpolation on the Grassmann Manifold
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Ciaramella, Gabriele; Gander, Martin J.; Vanzan, Tommaso
[no abstract available]
A McKay Correspondence for Reflection Groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Buchweitz, Ragnar-Olaf; Faber, Eleonore; Ingalls, Colin
We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A=S∗G. If G is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring SG/(Δ) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen--Macaulay modules over the coordinate ring SG/(Δ). These maximal Cohen--Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over SG/(Δ). We identify some of the corresponding matrix factorizations, namely the so-called logarithmic co-residues of the discriminant.
A Note on Endpoint Bochner-Riesz Estimates
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Beltran, David; Roos, Joris; Seeger, Andreas
We revisit an $\varepsilon$-removal argument of Tao to obtain sharp $L^p \to L^r(L^p)$ estimates for sums of Bochner-Riesz bumps which are conditional on non-endpoint bounds for single scale bumps. These can be used to obtain sharp conditional sparse bounds for Bochner-Riesz multipliers at the critical index, refining the conditional weak-type $(p,p)$ estimates of Tao.
A Quantitative Analysis of the “Lion-Man” Game
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Kohlenbach, Ulrich; López-Acedo, Genaro; Nicolae, Adriana
In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an ε-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.
A tale of three curves
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Balakrishnan, Jennifer S.
In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.
A Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Danchin, Raphaël; Fanelli, Francesco; Paicu, Marius
We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in th L^∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d≥2 and for general pressure laws.
Abelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Belomestny, Denis; Panov, Vladimir
In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models $(X, V)$, where both the state process $X$ and the volatility process $V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy processes driving the jumps in $X$ and $V$ . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process $X$. We derive the convergence rates for the corresponding estimator and prove that these rates can not be improved in general.

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