arXiv subjects

Tommy Hofmann

Publications and source records attributed to Tommy Hofmann.

At least 37 records · Page 2Linked to original sources

arXiv2020

Computing isomorphisms between lattices

Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left) Lambda-lattice X either computes a free basis of X over Lambda or shows that X is not free over Lambda. In the present article, we generalise this by showing that, under weaker hypotheses on A, there exists an algorithm that for two given Lambda-lattices X and Y either computes an isomorphism X -> Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A=Q[G], Lambda=Z[G] and Lambda-lattices X and Y contained in Q[G], where G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q with Galois group isomorphic to Q_8 x C_2, the direct product of the quaternion group of order 8 and the cyclic group of order 2.

math.NT↗

arXiv2019

Self-Assembly of Liquid Crystals in Nanoporous Solids for Adaptive Photonic Metamaterials

Nanoporous media exhibit structures significantly smaller than the wavelengths of visible light and can thus act as photonic metamaterials. Their optical functionality is not determined by the properties of the base materials, but rather by tailored, multiscale structures, in terms of precise pore shape, geometry, and orientation. Embedding liquid crystals in pore space provides additional opportunities to control light-matter interactions at the single-pore, meta-atomic scale. Here, we present temperature-dependent 3D reciprocal space mapping using synchrotron-based X-ray diffraction in combination with high-resolution birefringence experiments on disk-like mesogens (HAT6) imbibed in self-ordered arrays of parallel cylindrical pores 17 to 160 nm across in monolithic anodic aluminium oxide (AAO). In agreement with Monte Carlo computer simulations we observe a remarkably rich self-assembly behaviour, unknown from the bulk state. It encompasses transitions between the isotropic liquid state and discotic stacking in linear columns as well as circular concentric ring formation perpendicular and parallel to the pore axis. These textural transitions underpin an optical birefringence functionality, tuneable in magnitude and in sign from positive to negative via pore size, pore surface-grafting and temperature. Our study demonstrates that the advent of large-scale, self-organised nanoporosity in monolithic solids along with confinement-controllable phase behaviour of liquid-crystalline matter at the single-pore scale provides a reliable and accessible tool to design materials with adjustable optical anisotropy, and thus offers versatile pathways to fine-tune polarisation-dependent light propagation speeds in materials. Such a tailorability is at the core of the emerging field of transformative optics, allowing, e.g., adjustable light absorbers and extremely thin metalenses.

cond-mat.soft↗

arXiv2019

On the computation of overorders

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper we consider the related problems of computing all minimal overorders as well as all overorders of a given order. We use techniques from algorithmic representation theory and the theory of minimal integral ring extensions to obtain efficient and practical algorithms, whose implementation is publicly available.

math.NT↗

arXiv2019

Characterization of the soft X-ray spectrometer PEAXIS at BESSY II

The performance of the recently commissioned spectrometer PEAXIS for resonant inelastic soft X-ray scattering (RIXS) and X-ray photoelectron spectroscopy (XPS) and its hosting beamline U41-PEAXIS at the BESSY II synchrotron are characterized. The beamline provides linearly polarized light from 180 eV - 1600 eV allowing for RIXS measurements in the range of 200 eV - 1200 eV. The monochromator optics can be operated in different configurations for the benefit of either high flux, providing up to $10^{12}$ photons/s within the focal spot at the sample, or high energy resolution with a full width at half maximum of <40meV at an incident photon energy of ~400 eV. This measured total energy resolution of the RIXS spectrometer is in very good agreement with the theoretically predicted values by ray-tracing simulations. PEAXIS features a 5 m long RIXS spectrometer arm that can be continuously rotated about the sample position by 106° within the horizontal photon scattering plane, thus enabling the study of momentum-transfer-dependent excitations. To demonstrate the instrument capabilities, d-d excitations and magnetic excitations have been measured on single-crystalline NiO. Measurements employing a fluid cell demonstrate the vibrational Progression in liquid acetone. Planned upgrades of the beamline and the RIXS spectrometer that will further increase the energy resolution by 20 - 30% to ~100meV at 1000 eV incident photon energy are discussed.

physics.ins-det↗

arXiv2019

Efficient Gröbner Bases Computation over Principal Ideal Rings

In this paper we present a new efficient variant to compute strong Gröbner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gröbner basis computations to Gröbner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gröbner basis algorithm pretending $R/nR$ to be field. If we discover a non-invertible leading coefficient $c$, we use this information to try to split $n = ab$ with coprime $a, b$. If no such $c$ is discovered, the returned Gröbner basis is already a strong Gröbner basis for the input ideal over $R/nR$.

math.AC↗

arXiv2019

The conjugacy problem in $GL(n,Z)$

We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in $GL(n,Z)$ of a matrix in $GL(n,Q)$. We do this by reducing these problems respectively to the isomorphism and automorphism group problems for certain modules over rings of the form $\mathcal O_K[y]/(y^l)$, where $\mathcal O_K$ is the maximal order of an algebraic number field and $l \in N$, and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.

math.GR↗

arXiv2018

On Integral Forms of Specht Modules Labelled by Hook Partitions

We investigate integral forms of simple modules of symmetric groups over fields of characteristic $0$ labelled by hook partitions. Building on work of Plesken and Craig, for every odd prime $p$, we give a set of representatives of the isomorphism classes of $\mathbb{Z}_p$-forms of the simple $\mathbb{Q}_p \mathfrak{S}_n$-module labelled by the partition $(n-k,1^k)$, where $n\in\mathbb{N}$ and $0\leq k\leq n-1$. We also settle the analogous question for $p=2$, assuming that $n\not\equiv 0\pmod{4}$ and $k\in\{2,n-3\}$. As a consequence this leads to a set of representatives of the isomorphism classes of $\mathbb{Z}$-forms of the simple $\mathbb{Q}\mathfrak{S}_n$-modules labelled by $(n-2,1^2)$ and $(3,1^{n-3})$, again assuming $n\not\equiv 0\pmod{4}$.

math.RT↗

arXiv2018

Computing tropical points and tropical links

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical varieties, using intersections with affine hyperplanes to reduce the dimension to zero. We use the algorithms to show that the tropical Grassmannians $\mathcal G_{3,8}$ and $\mathcal G_{4,8}$ are not simplicial.

math.AG↗

arXiv2018

Quantized Self-Assembly of Discotic Rings in a Liquid Crystal Confined in Nanopores

Disklike molecules with aromatic cores spontaneously stack up in linear columns with high, one-dimensional charge carrier mobilities along the columnar axes making them prominent model systems for functional, self-organized matter. We show by high-resolution optical birefringence and synchrotron-based X-ray diffraction that confining a thermotropic discotic liquid crystal in cylindrical nanopores induces a quantized formation of annular layers consisting of concentric circular bent columns, unknown in the bulk state. Starting from the walls this ring self-assembly propagates layer by layer towards the pore center in the supercooled domain of the bulk isotropic-columnar transition and thus allows one to switch on and off reversibly single, nanosized rings through small temperature variations. By establishing a Gibbs free energy phase diagram we trace the phase transition quantization to the discreteness of the layers' excess bend deformation energies in comparison to the thermal energy, even for this near room-temperature system. Monte Carlo simulations yielding spatially resolved nematic order parameters, density maps and bond-forientational order parameters corroborate the universality and robustness of the confinement-induced columnar ring formation as well as its quantized nature.

cond-mat.soft↗

arXiv2017

A ferroelectric liquid crystal confined in cylindrical nanopores: Reversible smectic layer buckling, enhanced light rotation and extremely fast electro-optically active Goldstone excitations

The orientational and translational order of a thermotropic ferroelectric liquid crystal (2MBOCBC) imbibed in self-organized, parallel, cylindrical pores with radii of 10, 15, or 20 nm in anodic aluminium oxide monoliths (AAO) are explored by high-resolution linear and circular optical birefringence as well as neutron diffraction texture analysis. The results are compared to experiments on the bulk system. The native oxidic pore walls do not provide a stable smectogen wall anchoring. By contrast, a polymeric wall grafting enforcing planar molecular anchoring results in a thermal-history independent formation of smectic C* helices and a reversible chevron-like layer buckling. An enhancement of the optical rotatory power by up to one order of magnitude of the confined compared to the bulk liquid crystal is traced to the pretransitional formation of helical structures at the smectic-A*-to-smectic-C* transformation. A linear electro-optical birefringence effect evidences collective fluctuations in the molecular tilt vector direction along the confined helical superstructures, i.e. the Goldstone phason excitations typical of the para-to-ferroelectric transition. Their relaxation frequencies increase with the square of the inverse pore radii as characteristic of plane-wave excitations and are two orders of magnitude larger than in the bulk, evidencing an exceptionally fast electro-optical functionality of the liquid-crystalline-AAO nanohybrids.

cond-mat.soft↗

arXiv2017

Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic.

cs.MS↗

arXiv2017

Valuations of $p$-adic regulators of cyclic cubic fields

We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of $p$-adic regulators of cyclic cubic fields is stated.

math.NT↗

arXiv2016

On the computation of the HNF of a module over the ring of integers of a number field

We present a variation of the modular algorithm for computing the Hermite normal form of an $\mathcal O_K$-module presented by Cohen, where $\mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.

math.NT↗

arXiv2016

Zeta Functions of Lattices of the Symmetric Group

The symmetric group $\mathfrak S_{n+1}$ of degree $n+1$ admits an $n$-dimensional irreducible $\mathbf Q \mathfrak S_n$-module $V$ corresponding to the hook partition $(2,1^{n-1})$. By the work of Craig and Plesken we know that there are $σ(n+1)$ many isomorphism classes of $\mathbf Z \mathfrak S_{n+1}$-lattices which are rationally equivalent to $V$, where $σ$ denotes the divisor counting function. In the present paper we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the $\mathbf Z \mathfrak S_{n+1}$-lattice defined by the Specht basis.

math.RT↗

arXiv2016

Computing in quotients of rings of integers

We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.

math.NT↗

arXiv2016

Protein structure prediction guided by cross-linking restraints - A systematic evaluation of the impact of the cross-linking spacer length

Recent development of high-resolution mass spectrometry (MS) instruments enables chemical cross-linking (XL) to become a high-throughput method for obtaining structural information about proteins. Restraints derived from XL-MS experiments have been used successfully for structure refinement and protein-protein docking. However, one formidable question is under which circumstances XL-MS data might be sufficient to determine a protein's tertiary structure de novo? Answering this question will not only include understanding the impact of XL-MS data on sampling and scoring within a de novo protein structure prediction algorithm, it must also determine an optimal cross-linker type and length for protein structure determination. While a longer cross-linker will yield more restraints, the value of each restraint for protein structure prediction decreases as the restraint is consistent with a larger conformational space. In this study, the number of cross-links and their discriminative power was systematically analyzed in silico on a set of 2,055 non-redundant protein folds considering Lys-Lys, Lys-Asp, Lys-Glu, Cys-Cys, and Arg-Arg reactive cross-linkers between 1 Å and 60 Å. Depending on the protein size a heuristic was developed that determines the optimal cross-linker length. Next, simulated restraints of variable length were used to de novo predict the tertiary structure of fifteen proteins using the BCL::Fold algorithm. The results demonstrate that a distinct cross-linker length exists for which information content for de novo protein structure prediction is maximized. The sampling accuracy improves on average by 1.0 Å and up to 2.2 Å in the most prominent example. XL-MS restraints enable consistently an improved selection of native-like models with an average enrichment of 2.1.

physics.bio-ph↗

arXiv2012

Molecular dynamics of n-hexane: A quasi-elastic neutron scattering study on the bulk and spatially nanochannel-confined liquid

We present incoherent quasi-elastic neutron scattering measurements in a wavevector transfer range from 0.4 AA^{-1} to 1.6AA^{-1} on liquid n-hexane confined in cylindrical, parallel-aligned nanochannels of 6 nm mean diameter and 260 micrometer length in monolithic, mesoporous silicon. They are complemented with, and compared to, measurements on the bulk system in a temperature range from 50K to 250K. The time-of-flight spectra of the bulk liquid can be modeled by microscopic translational as well as fast localized rotational, thermally-excited, stochastic motions of the molecules. In the nano-confined state of the liquid, which was prepared by vapor condensation, we find two molecular populations with distinct dynamics, a fraction which is immobile on the time scale of 1ps to 100ps probed in our experiments and a second component with a self-diffusion dynamics slightly slower than observed for the bulk liquid. No hints of an anisotropy of the translational diffusion with regard to the orientation of the channels' long axes have been found. The immobile fraction amounts to about 5% at 250K, gradually increases upon cooling and exhibits an abrupt increase at 160K (20K below bulk crystallization), which indicates pore freezing

cond-mat.soft↗

arXiv2009

Capillary rise of water in hydrophilic nanopores

We report on the capillary rise of water in three-dimensional networks of hydrophilic silica pores with 3.5nm and 5nm mean radii, respectively (porous Vycor monoliths). We find classical square root of time Lucas-Washburn laws for the imbibition dynamics over the entire capillary rise times of up to 16h investigated. Provided we assume two preadsorbed strongly bound layers of water molecules resting at the silica walls, which corresponds to a negative velocity slip length of -0.5nm for water flow in silica nanopores, we can describe the filling process by a retained fluidity and capillarity of water in the pore center. This anticipated partitioning in two dynamic components reflects the structural-thermodynamic partitioning in strongly silica bound water layers and capillary condensed water in the pore center which is documented by sorption isotherm measurements.

physics.flu-dyn↗