A Multi-level Blocking Distinct Degree Factorization Algorithm

We give a new algorithm for performing the distinct-degree factorization of a polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our approach is that a finer level of blocking replaces multiplications by squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval polynomials when P(x) is sparse. As an application we give a fast algorithm to search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2), while producing a certificate that can be checked in less time than the full search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over all trinomials of given degree r. Under a plausible assumption about the distribution of factors of trinomials, the new algorithm has complexity O(r^2 (log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r. Our implementation achieves a speedup of greater than a factor of 560 over the naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our program, we have found two new primitive trinomials of degree 24036583 over GF(2) (the previous record degree was 6972593)

Landau's function for one million billions

Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak S}_n$. If the standard factorization of $M$ into primes is M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}, we define $\ell(M)$ to be q_1^{\al_1}+q_2^{\al_2}+... +q_k^{\al_k}; one century ago, E. Landau proved that $g(n)=\max_{\ell(M)\le n} M$ and that, when $n$ goes to infinity, $\log g(n) \sim \sqrt{n\log(n)}$. There exists a basic algorithm to compute $g(n)$ for $1 \le n \le N$; its running time is \co(N^{3/2}/\sqrt{\log N}) and the needed memory is \co(N); it allows computing $g(n)$ up to, say, one million. We describe an algorithm to calculate $g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\it $\ell$-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function \tau(n)=\sum_{d\dv n} 1

Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format

We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than $10^{-15}$,ulp, and we give the worst ones. In particular, the worst case for $|x| geq 3 imes 10^{-11}$ is $exp(9.407822313572878 imes 10^{-2}) = 1.098645682066338,5,0000000000000000,278ldots$. This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains

Maximal determinants and saturated D-optimal designs of orders 19 and 37

A saturated D-optimal design is a {+1,-1} square matrix of given order with maximal determinant. We search for saturated D-optimal designs of orders 19 and 37, and find that known matrices due to Smith, Cohn, Orrick and Solomon are optimal. For order 19 we find all inequivalent saturated D-optimal designs with maximal determinant, 2^30 x 7^2 x 17, and confirm that the three known designs comprise a complete set. For order 37 we prove that the maximal determinant is 2^39 x 3^36, and find a sample of inequivalent saturated D-optimal designs. Our method is an extension of that used by Orrick to resolve the previously smallest unknown order of 15; and by Chadjipantelis, Kounias and Moyssiadis to resolve orders 17 and 21. The method is a two-step computation which first searches for candidate Gram matrices and then attempts to decompose them. Using a similar method, we also find the complete spectrum of determinant values for {+1,-1} matrices of order 13.Comment: 28 pages, 4 figure

Cross-stream transport of asymmetric particles driven by oscillating shear

We study the dynamics of asymmetric, deformable particles in oscillatory, linear shear flow. By simulating the motion of a dumbbell, a ring polymer, and a capsule we show that cross-stream migration occurs for asymmetric elastic particles even in linear shear flow if the shear rate varies in time. The migration is generic as it does not depend on the particle dimension. Importantly, the migration velocity and migration direction are robust to variations of the initial particle orientation, making our proposed scheme suitable for sorting particles with asymmetric material properties.Comment: 5 pages, 4 figure

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

Some arguments concerning correct rounding of the elementary functions

This text briefly presents the current state of our work on correctly rounded transcendentals, and explains why we feel that the correctly rounded elementary functions should appear in the FP standard (at least with a "should")

Error bounds on complex floating-point multiplication

Given floating-point arithmetic with t-digit base-β significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values z0 and z1 can be computed with maximum absolute error |z0||z1|1/2β 1-t√5. In particular, this provides relative error bounds of 2-24√5 and 2-53√5. for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic

Informationsbeurteilungsfähigkeit - Eine Pilotstudie an Zürcher Gymnasien

Der Beitrag hinterfragt, wie (junge) Menschen mit im WWW gefundenen Informationen umgehen. Des Weiteren werden Fragen aufgeworfen wie: über welche Fähigkeiten verfügen Jugendliche, um im Internet gefundene Informationen sinnvoll bewerten und gegebenenfalls weiterverwenden oder verwerfen zu können? Erkennen die WWW-Nutzer bestimmten Informationen zugrunde liegende Ideologien oder Interessen? Kennen sie ihre Grenzen im Umgang mit ihnen unbekannten Wissenselementen? Mögliche Antworten auf diese Fragen versuchten die Autoren im Rahmen einer Pilotstudie, die an Zürcher Gymnasien durchgeführt wurde, zu finden. Dabei verfolgen sie auch ein pragmatisches Ziel: Sie wollen schließlich Wege aufzeigen, wie Schülerinnen und Schüler sowie Studierende beim Aufbau solcher Fähigkeiten mit einem Lehrmittel unterstützt werden können

Validation of a Generic Non-Swirled Multi-Fuel Burner for the Measurement of Flame Stability Limits for Research of Advanced Sustainable Aviation Fuels

Future aviation concepts should be both CO2-neutral and without other emissions. One approach to reaching both targets is based on sustainably produced synthetic liquid fuels, which may allow very clean, lean premixed prevaporized (LPP) combustion. For that, fuels are needed with much longer ignition delay times and a lower flashback propensity than current jet fuels. We describe an experimental setup to investigate the flashback stability of liquid fuels in a multi-fuel burner. In this work, the measurement procedure and the determination of the experimentally obtained accuracy are in focus with regard to prevaporized and preheated iso-propanol/air flames in an equivalence ratio range of 0.85 to 1.05 involving three preheating levels (573, 673, and 773 K). As the determination of the accurate unburnt gas temperature just ahead of the flame is of strong importance for flashback but not directly possible, a model is implemented to determine it from the measurable quantities. Even with this indirect method, and also regarding the hysteresis of the experimental preheating temperature, it is found that the relevant quantities, namely, measured temperatures, mass flows, and values derived from them, can be determined with accuracy in the range below 1.7%