On the density of the splitting of 2 in cubic fields

Summary In this paper we consider all algebraic integers of degree 3 and all possible splittings of the ideal generated by 2 in cubic fields; we determine the density of the 's such that the splitting of 2 inQ() is fixed. We also obtain the density of the integers generating cubic fields of index 2.     

A modified Newton method with cubic convergence: the multivariate case

Recently, a modification of the Newton method for finding a zero of a univariate function with local cubic convergence has been introduced. Here, we extend this modification to the multi-dimensional case, i.e., we introduce a modified Newton method for vector functions that converges locally cubically, without the need to compute higher derivatives. The case of multiple roots is not treated. Per iteration the method requires one evaluation of the function vector and solving two linear systems with the Jacobian as coefficient matrix, where the Jacobian has to be evaluated twice. Since the additional computational effort is nearly that of an additional Newton step, the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the Newton method. This much better convergence is indeed possible as shown by a numerical example. Also, the modified Newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the Jacobian as coefficient matrix. An example for this is given where numerical quadrature is involved. Finally, we discuss shortly possible extensions of the method to make it globally convergent.     

A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation

A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L² and L^∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin–Bona–Mahony) and BBMB (Benjamin–Bona–Mahony–Burgers) equations and the well-known decay estimates are demonstrated for the computed solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 695–716, 1998     

The equationxyz=x+y+z=1 in integers of a cubic field

We determine all cubic number fields such that the title equation has a solution in the ring of integers of the field.I am grateful to H. M. Edgar for suggesting this problem.     

On a cubic cyclic field with discriminant 1632

Let K be the cubic subfield of a field of the 163rd root of unity. It is proved that the class number of K is 4. Moreover its Hilbert class field is given explicitly. also a system of the fundamental units is obtained.     

A splitting algorithm for the wavelet transform of cubic splines on a nonuniform grid

For cubic splines with nonuniform nodes, splitting with respect to the even and odd nodes is used to obtain a wavelet expansion algorithm in the form of the solution to a three-diagonal system of linear algebraic equations for the coefficients. Computations by hand are used to investigate the application of this algorithm for numerical differentiation. The results are illustrated by solving a prediction problem.     

Birational equivalence of tori with a cyclic splitting field

It is proved that an algebraic torus T with a cyclic splitting field is birationally equivalent over the field of definition to the direct product of certain standard tori. Further, torus T is stably rational over the field of definition if and only if the character modules of these standard tori are free modules.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 153–158, 1976.     

EPR study of ZnS:Mn2+ nanocrystals and pyrex glasses

Pyrex glasses with different ZnS: Mn²⁺ contents were prepared by melting method. It has been found that Mn ion may occupy two sites: (Mn²⁺)_sub, and (Mn²⁺)_int from photoluminescene (PL) and photoluminescence excitation (PLE) spectra. The results were confirmed by the further electron panmagnetic resonance (EPR) experiments and the three types of states (Mn²⁺)_sub, (Mn²⁺)_int,and Mn clusters were identified. It was observed that theg-factor and the hyperfine structure (HFS) constant increase with the decreasing size of nanocrystallite. This may result from hybridization of sp³ electron states of ZnS and 3d⁵ electron states of Mn by the effects of quantum confinement and the surface states. Project supported by the National Natural Science Foundation of China and Laboratory of Excited State Processes, Chinese Academy of Sciences.     

Paramagnetic resonance of Mn2+ in KNO3 single crystal: Motional effects

The electron paramagnetic resonance of Mn²⁺ in KNO₃ single crystal is investigated over a temperature cycle through transition temperatures. The hyperfine coupling constant, A, half width,Δ H, and the line intensity, I, are found to show sudden changes at the transition temperature, at whichα-KNO₃ changes intoβ-KNO₃. The lines are much sharper in the high temperatureβ-phase than inα-phase of the crystal. They are explained, qualitatively, in terms of structure change and rotation of NO ₃ ^? ions. The spectra in the two phases,α andβ, are analysed in terms of usual spin-Hamiltonian. A search of metastableγ-phase is also made and probable indications for the same are found.     

Infiniteness of zero modes for the Pauli operator with singular magnetic field

We establish that the Pauli operator describing a spin-1/2 two-dimensional quantum system with a singular magnetic field has, under certain conditions, an infinite-dimensional space of zero modes, possibly, both spin-up and spin-down, moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum. In particular, infiniteness takes place if the field has infinite flux, which settles this previously unknown case of Aharonov-Casher theorem.     

The number of cubic surfaces with 27 lines over a finite field

Journal of Algebraic Combinatorics - We study those multiplicative subgroups of $${\mathbb F}_{2^n}^*$$ which are Sidon sets and/or sum-free sets in the group $$({\mathbb F}_{2^n},+)$$ . These...     

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K ₃/?, i.e. $ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $ , where M(m) denotes the number of integral ideals of the field K ₃ of norm m and l ∈ ?. We improve the previous results for $ S_{2,K_3 } (x) $ and $ S_{3,K_3 } (x) $ .     

Fuzzy semi-orders: The case of t-norms without zero divisors

For crisp relations the concept of a semi-order can be stated in a number of equivalent ways. When trying to extend this concept to the fuzzy setting, we observe that the (generalized) definitions fail to be equivalent. In this contribution, we discuss which is the most natural definition of a fuzzy semi-order, and study the hierarchy among the alternative definitions, in particular when using a t-norm without zero divisors.     

Electron paramagnetic resonance of Mn2+ in NaF single crystals

The electron paramagnetic resonance of Mn²⁺ in NaF single crystals is investigated at different temperatures from 573° K to 93° K. Four different spectra designated as I, II, III₂ and III₄ are observed. Spectrum I consists of a single broad resonance corresponding to precipitated Mn²⁺ ions. Spectrum II is isotropic and centred nearg = 2.00. This spectrum corresponds to substitutional Mn²⁺ ions with remote charge compensating sites and therefore with local cubic symmetry. Spectrum III₂ withg = 2.014 and spectrum III₄ withg = 1.995 are also due to substitutional Mn²⁺ ions but subjected to tetragonal crystalline fields and are the same as those reported by earlier workers. Superhyperfine structure has been observed in spectra II, III₂ and III₄. The analysis of that structure in spectra II and III₂ has been carried out for the first time and the constants A_s, and A_σ are given. The spectra are analysed by the usual spin-Hamiltonian method.     

A regularity criterion for the 2D MHD system with zero magnetic diffusivity

This note proves a regularity criterion ∇b∈L¹(0,T;BMO(R²)) for the 2D MHD system with zero magnetic diffusivity.     

A simplified proof for the limit of a tower over a cubic finite field

Recently Bezerra, Garcia and Stichtenoth constructed an explicit tower F=(Fn)_n?0 of function fields over a finite field Fq³, whose limit λ(F)=limn→∞N(Fn)/g(Fn) attains the Zink bound λ(F)?2(q²−1)/(q+2). Their proof is rather long and very technical. In this paper we replace the complex calculations in their work by structural arguments, thus giving a much simpler and shorter proof for the limit of the Bezerra, Garcia and Stichtenoth tower.     

On a family of diophantine triples {k, A 2 k + 2A, (A + 1)2 k + 2 (A + 1)} with two parameters

Let A and k be positive integers. We study the Diophantine quadruples $$ \{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1),d\} $$ . We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then $$ \begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$ when 3 ≦ A ≦ 10. This extends a theorem obtained by Dujella [7] for A = 1, and also, a classical theorem of Baker and Davenport [2] for A = k = 1.