Evidence for One-Gap Superconductivity in Mg(B_{1-x}C_{x})_{2} Single Crystals at x=0.132 by Point-Contact Spectroscopy

We report the results of directional point-contact measurements in Mg(B $_{1-x}$ C $_{x})_{2}$ single crystals. The amplitudes of the gaps, $\Delta_{\pi}$ and $\Delta_{\sigma}$ , were determined for each C content by fitting the experimental low-temperature normalized conductance curves of our “soft” point contacts with the BTK model generalized to the two-band case. We found that, on increasing the carbon content, $\Delta_{\sigma}$ decreases almost linearly with $T_{c}$ and $\Delta_{\pi}$ slightly increases until, at $x=0.132$ (where $T_{c}=19$ K), they assume the same value $\Delta =3.2 \pm 0.9$ meV. This result is confirmed by the temperature and magnetic-field dependence of the conductance curves at this C content, which do not show any evidence of two distinct gap values. In particular, the Δ versus T curve follows very well a standard BCS curve, with a gap ratio $2\Delta /k_{B} T_{c}=3.9$ . These experimental findings are compared to the theoretical predictions of the two-band model in the Eliashberg formulation.     

Computational aspects of gonal maps and radical parametrization of curves

We develop in this article an algorithm that, given a projective curve $C$ , computes a gonal map, that is, a finite morphism from $C$ to $\mathbb P ^1$ of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $\le 4$ .     

Nonlinear correction to Darcy’s law for channels with wavy walls

For low Reynolds numbers ${\mathcal{R}}$ , the flow of a viscous fluid through a channel is described by the well-known Darcy’s law which corresponds to a linear relation between the pressure gradient ${\overline{\nabla p}}$ and the average velocity ${\overline{u}}$ . When the channel is not straight and when the Reynolds number is not negligible, additional terms appear in this relation. Some previous authors investigated the first three coefficients in the expansion of ${|\overline{\nabla p}|}$ in the powers of ${\overline{u}}$ and they showed that the coefficient of ${\overline{u}^2}$ vanishes for moderate ${\mathcal{R}}$ . Other authors demonstrated that this coefficient can be non-zero. This question is addressed and solved. It is demonstrated that both cases occur; Forchheimer’s law has a cubic correction for small ${\mathcal{R}}$ and a quadratic one for large ${\mathcal{R}}$ . Two analytical–numerical algorithms are constructed to prove this property. These algorithms are applied to the Navier–Stokes equations in three-dimensional channels enclosed by two wavy walls whose amplitude is proportional to ${b{\varepsilon}}$ , where 2b is the mean clearance of the channels and ${\varepsilon}$ is a small dimensionless parameter. The first algorithm is applied for small ${\mathcal{R}}$ by representing the velocity and the pressure in terms of a double Taylor series in ${\mathcal{R}}$ and ${\varepsilon}$ . The accuracy ${O(\mathcal{R}^2)}$ and ${O(\varepsilon^6)}$ following Padé approximations yield analytical approximate formulae for Forchheimer’s law. The first algorithm is applied to symmetric channels on the theoretical level (all terms on ${\mathcal{R}}$ and ${\varepsilon}$ are taken into account) to show that ${|\overline{\nabla p}|}$ is an odd function of ${\overline{u}}$ . This observation yields, in particular, a cubic correction to Darcy’s law. Numerical examples for non-symmetrical channels yield the same cubic correction. The second algorithm is based on the analytical–numerical solution to the Navier–Stokes equations for arbitrary ${\mathcal{R}}$ up to ${O(\varepsilon^{3})}$ . This algorithm yields, in particular, a quadratic correction to Darcy’s law for higher ${\mathcal{R}}$ .     

Reparametrizing swung surfaces over the reals

Let $\mathbb {K}\subseteq \mathbb {R}$ be a computable subfield of the real numbers (for instance, $\mathbb {Q}$ ). We present an algorithm to decide whether a given parametrization of a rational swung surface, with coefficients in $\mathbb {K}(\mathtt {i})$ , can be reparametrized over a real (i.e., embedded in $\mathbb {R}$ ) finite field extension of $\mathbb {K}$ . Swung surfaces include, in particular, surfaces of revolution.     

Influence of pH and bath composition on properties of Ni–Fe alloy films synthesized by electrodeposition

Fe?CNi films were electrodeposited on ITO glass substrates from the electrolytes with different molar ratio of Ni $^{\boldsymbol{2+}}$ /Fe $^{\boldsymbol{2+}}$ and different pH values (2 $\boldsymbol{\cdot}$ 1, 2 $\boldsymbol{\cdot}$ 9, 3 $\boldsymbol{\cdot}$ 7 and 4 $\boldsymbol{\cdot}$ 3) at 25 $\boldsymbol{^\circ}$ C. The properties of Fe?CNi alloy films depend on both Ni $^{\boldsymbol{2+}}$ and Fe $^{\boldsymbol{2+}}$ concentrations in electrolyte and pH values. The content of Ni increases from 38% to 84% as the mole ratio of NiSO $_{\boldsymbol{4}}$ /FeSO $_{\boldsymbol{4}}$ increasing from 0 $\boldsymbol{\cdot}$ 50/0 $\boldsymbol{\cdot}$ 50 to 0 $\boldsymbol{\cdot}$ 90/0 $\boldsymbol{\cdot}$ 10 in electrolyte and slightly decreases from 65% to 42% as the pH values increase from 2 $\boldsymbol{\cdot}$ 1 to 4 $\boldsymbol{\cdot}$ 3. The X-ray diffraction analysis reveals that the structures of the films strongly depend on the Ni content in the binary films. The magnetic performance of the films shows that the saturation magnetization ( $\boldsymbol{M}_{\boldsymbol{\rm s}})$ decreases from 1775 $\boldsymbol{\cdot}$ 01 emu/cm $^{\boldsymbol{3}}$ to 1501 $\boldsymbol{\cdot}$ 46 emu/cm $^{\boldsymbol{3}}$ with the pH value increasing from 2 $\boldsymbol{\cdot}$ 1 to 4 $\boldsymbol{\cdot}$ 3 and the saturation magnetization ( $\boldsymbol{M}_{\boldsymbol{\rm s}})$ and coercivity ( $\boldsymbol{H}_{\boldsymbol{\rm c}})$ move up from 1150 $\boldsymbol{\cdot}$ 44 emu/cm $^{\boldsymbol{3}}$ and 58 $\boldsymbol{\cdot}$ 86 Oe to 2498 $\boldsymbol{\cdot}$ 88 emu/cm $^{\boldsymbol{3}}$ and 93 $\boldsymbol{\cdot}$ 12 Oe with the increase of Ni $^{\boldsymbol{2+}}$ concentration in the electrolyte, respectively.     

Classification of Novikov algebras

We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over $\mathbb{R }$ and $\mathbb{C }$ , as well as the classification of the 4-dimensional Novikov algebras over $\mathbb{C }$ whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over $\mathbb{C }$ .     

On the automorphism group of a binary self-dual [120, 60, 24] code

We prove that an automorphism of order 3 of a putative binary self-dual $[120, 60, 24]$ [ 120 , 60 , 24 ] code $C$ C has no fixed points. Moreover, the order of the automorphism group of $C$ C divides $2^a\cdot 3 \cdot 5\cdot 7\cdot 19\cdot 23\cdot 29$ 2 a · 3 · 5 · 7 · 19 · 23 · 29 with $a\in \mathbb N _0$ a ∈ N 0 . Automorphisms of odd composite order $r$ r may occur only for $r=15, 57$ r = 15 , 57 or $r=115$ r = 115 with corresponding cycle structures $3 \cdot 5$ 3 · 5 - $(0,0,8;0), 3\cdot 19$ ( 0 , 0 , 8 ; 0 ) , 3 · 19 - $(2,0,2;0)$ ( 2 , 0 , 2 ; 0 ) or $5 \cdot 23$ 5 · 23 - $(1,0,1;0)$ ( 1 , 0 , 1 ; 0 ) respectively. In case that all involutions act fixed point freely we have $|\mathrm{Aut}(C)| \le 920$ | Aut ( C ) | ≤ 920 , and $\mathrm{Aut}(C)$ Aut ( C ) is solvable if it contains an element of prime order $p \ge 7$ p ≥ 7 . Moreover, the alternating group $\mathrm{A}_5$ A 5 is the only non-abelian composition factor which may occur in $\mathrm{Aut}(C)$ Aut ( C ) .     

Molecular beam epitaxy of semipolar AlN(11\bar{2}2) and GaN(11\bar{2}2) on m-sapphire

We report on the plasma-assisted molecular-beam epitaxy of semipolar $\hbox{AlN}(11\bar{2}2)$ and GaN( $11\bar{2}2$ ) films on $(1\bar{1}00)$ m-plane sapphire. AlN deposited on m-sapphire settles into two main crystalline orientation domains, $\hbox{AlN}(11\bar{2}2)$ and $\hbox{AlN}(10\bar{1}0),$ whose ratio depends on the III/V ratio. Growth under moderate nitrogen-rich conditions enables to isolate the $(11\bar{2}2)$ orientation. The in-plane epitaxial relationships of $\hbox{AlN}(11\bar{2}2)$ on m-plane sapphire are $[11\bar{2}\bar{3}]_{\rm AlN} \vert \vert [0001]_{\rm sapphire}$ and $[1\bar{1}00]_{\rm AlN} \vert \vert [11\bar{2}0]_{\rm sapphire}.$ GaN deposited directly on m-sapphire results in ( $11\bar{2}2$ )-oriented layers with ( $10\bar{1}\bar{3}$ )-oriented inclusions. A ～100 nm-thick AlN( $11\bar{2}2$ ) buffer imposes the ( $11\bar{2}2$ )-orientation for the GaN layer grown on top. By studying the Ga-desorption on GaN( $11\bar{2}2$ ), we conclude that these optimal growth conditions corresponds to a Ga excess of one monolayer on the GaN( $11\bar{2}2$ ) surface.     

Hamming weights and Betti numbers of Stanley–Reisner rings associated to matroids

To each linear code $C$ over a finite field we associate the matroid $M(C)$ of its parity check matrix. For any matroid $M$ one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type $M(C)$ , these weights are the same as those of the code $C$ . In our main result we show how the weights $d_1,\ldots ,d_k$ of a matroid $M$ are determined by the $\mathbb N $ -graded Betti numbers of the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of $M$ , and derive some consequences. We also give examples which give negative results concerning other types of (global) Betti numbers, and using other examples we show that the generalized Hamming weights do not in general determine the $\mathbb N $ -graded Betti numbers of the Stanley–Reisner ring. The negative examples all come from matroids of type $M(C)$ .     

Influence of Magnetic Field and LO Phonon Effects on the Spin Polarization State Energy of Strong-Coupling Bipolaron in a Quantum Dot

On the basis of Lee–Low–Pines unitary transformation, the influence of magnetic field and LO phonon effects on the energy of spin polarization states of strong-coupling bipolarons in a quantum dot (QD) is studied by using the variational method of Pekar type. The variations of the ground state energy $E_0$ and the first excited state the energy $E_1$ of bipolarons in a two-dimensional QD with the confinement strength of QDs $\omega _0$ , dielectric constant ratio $\eta $ , electron–phonon coupling strength $\alpha $ and cyclotron resonance frequency of the magnetic field $\omega _{c}$ are derived when the influence of the spin and external magnetic field is taken into account. The results show that both energies of the ground and first excited states ( $E_0$ and $E_1)$ consist of four parts: the single-particle energy of electrons $E_\mathrm{e}$ , Coulomb interaction energy between two electrons $E_\mathrm{c}$ , interaction energy between the electron spin and magnetic field $E_\mathrm{S}$ and interaction energy between the electron and phonon $E_{\mathrm{e-ph}}$ ; the energy level of the first excited state $E_1$ splits into two lines as $E_1^{(1+1)}$ and $E_1^{(1-1)}$ due to the interaction between the single-particle “orbital” motion and magnetic field, and each energy level of the ground and first excited states splits into three “fine structures” caused by the interaction between the electron spin and magnetic field; the value of $E_{\mathrm{e-ph}}$ is always less than zero and its absolute value increases with increasing $\omega _0$ , $\alpha $ and $\omega _c$ ; the effect of the interaction between the electron and phonon is favorable to forming the binding bipolaron, but the existence of the confinement potential and Coulomb repulsive energy between electrons goes against that; the bipolaron with energy $E_1^{(1-1)}$ is easier and more stable in the binding state than that with $E_1^{(1+1)}$ .     

Specific Heat of K_{0.71}Na_{0.29}Fe_2As_2 at Very Low Temperatures

A commercially available calorimeter has been used to investigate the specific heat of a high-quality K $_{0.71}$ Na $_{0.29}$ Fe $_2$ As $_2$ single crystal. The addenda heat capacity of the calorimeter is determined in the temperature range $0.02 \, \mathrm{K} \le T \le 0.54 \, \mathrm{K}$ . The data of the K $_{0.71}$ Na $_{0.29}$ Fe $_2$ As $_2$ crystal imply the presence of a large $T^2$ contribution to the specific heat which gives evidence of $d$ -wave order parameter symmetry in the superconducting state. To improve the measurements, a novel design for a calorimeter with a paramagnetic temperature sensor is presented. It promises a temperature resolution of $\Delta T \approx 0.1 \, \mathrm{\mu K}$ and an addenda heat capacity less than $200 \, \mathrm{pJ/K}$ at $ T < 100 \, \mathrm{mK}$ .     

Linear complexity and correlation of a class of binary cyclotomic sequences

Let $p_1,p_2,\ldots ,p_n$ be distinct odd primes and let $e_1,e_2,\ldots ,e_n$ be positive integers. Based on cyclotomic classes proposed by Ding and Helleseth (Finite Fields Appl 4:140–166, 1998), a binary cyclotomic sequence of period $p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}$ is defined and denoted by $\underline{s_\Upsilon }$ . The linear complexity of $\underline{s_\Upsilon }$ is determined and is proved to be greater than or equal to $(p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}-1)/2$ . The autocorrelation function of $\underline{s_\Upsilon }$ is explicitly computed. Let $\ell \in \{1,2,\ldots ,n\}$ . We also explicitly compute the crosscorrelation function of $\underline{s_\Upsilon }$ and the Legendre sequence $\underline{L_{p_\ell }}$ with respect to $p_\ell $ . It is shown that $\underline{s_\Upsilon }$ and $\underline{L_{p_\ell }}$ have two-level or three-level crosscorrelation, and all their two-level crosscorrelation functions are determined.     

Irreducible compositions of polynomials over finite fields of even characteristic

This paper presents some results with the constructive theory of synthesis of irreducible polynomials over a Galois field with even characteristic. We prove a theorem that plays an important role for constructing irreducible polynomials. By this theorem two recurrent methods for constructing families of irreducible polynomials of degree $n2^{k}~(k=1,2,\ldots )$ over $\mathbb F _{2^{s}}$ are proposed. It is shown that in this special case, the sequences of irreducible polynomials are N-polynomial of degree $2^{k}$ .     

Measurements of Microstructural, Mechanical, Electrical, and Thermal Properties of an Al–Ni Alloy

The Al–7.5 wt% Ni alloy was directionally solidified upwards with different temperature gradients, $G$ ( $0.86\,\text{ K}~{\cdot }~ \text{ mm}^{-1}$ to $4.24\,\text{ K}~{\cdot }~\text{ mm}^{-1})$ at a constant growth rate, $V$ ( $8.34\,\upmu \text{ m}~{\cdot }~\text{ s}^{-1})$ . The dependence of dendritic microstructures such as the primary dendrite arm spacing ( $\lambda _{1}$ ), the secondary dendrite arm spacing ( $\lambda _{2}$ ), the dendrite tip radius ( $R$ ), and the mushy zone depth ( $d$ ) on the temperature gradient were analyzed. The dendritic microstructures in this study were also compared with current theoretical models, and similar previous experimental results. Measurements of the microhardness (HV) and electrical resistivity ( $\rho $ ) of the directionally solidified samples were carried out. Variations of the electrical resistivity ( $\rho $ ) with temperature ( $T$ ) were also measured by using a standard dc four-point probe technique. And also, the dependence of the microhardness and electrical resistivity on the temperature gradient was analyzed. According to these results, it has been found that the values of HV and $\rho $ increase with increasing values of $G$ . But, the values of HV and $\rho $ decrease with increasing values of dendritic microstructures ( $\lambda _{1}, \lambda _{2}, R,$ and $d$ ). It has been also found that, on increasing the values of temperature, the values of $\rho $ increase. The enthalpy of fusion ( $\Delta {H}$ ) for the Al–7.5 wt%Ni alloy was determined by a differential scanning calorimeter from a heating trace during the transformation from solid to liquid.     

Nonlinear grain–grain forces and the width of the solitary wave in granular chains: a numerical study

Any impulse results in the formation of a solitary wave of time averaged width $W$ in a granular chain. If the grain–grain interaction potential $V\sim \delta ^n$ , where $\delta $ is the distance by which the grains approach each other, then it is well established that $n\ge 2$ . Here we present dynamical simulation based results which suggest that $W-1\propto (n-2)^{-\alpha }$ where $\alpha =0.3283$ or $\approx $ 1/3. While in qualitative agreement, the result is quantitatively different from the formula for $W$ proposed earlier by Nesterenko.     

Study of Glass-Transition Kinetics of Pb-Modified Se_{80}In_{20} System by Using Non-isothermal Differential Scanning Calorimetry

Glass-transition kinetics of $\mathrm{Se}_{80}\mathrm{In}_{20-\mathrm{x}}\mathrm{Pb}_{\mathrm{x}}$ ( $x =$ 0, 5, 10, and 15) chalcogenide glasses have been carried out at different heating rates by using differential scanning calorimeter (DSC) under the non-isothermal condition. The glass-transition temperature $T_{\mathrm{g}}$ and peak glass-transition temperature $T_{\mathrm{pg}}$ have been determined from DSC thermograms. The reduced glass temperature $T_{\mathrm{rg}}$ , total relaxation time $\tau _{T_{g}}$ thermal-stability parameters $K^{l}$ and $S$ , the activation energy of glass transition $E_{\mathrm{g}}$ , the fragility index $F_{\mathrm{i}}$ , and the average coordination number $\langle Z\rangle $ have been calculated on the basis of the experimental results. The temperature differences $(T_{\mathrm{c}}-T_{\mathrm{g}}), K_{\mathrm{gl}}, K^{l}, S$ , and $E_{\mathrm{g}}$ are found to be maxima for $\mathrm{Se}_{80}\mathrm{In}_{10}\mathrm{Pb}_{10}$ glass. This indicates that $\mathrm{Se}_{80}\mathrm{In}_{10}\mathrm{Pb}_{10}$ glass has the highest thermal stability and glass-forming ability in the investigated compositional range. These results could be explained on the basis of modification of the chemical bond formation due to incorporation of Pb in the Se–In glassy matrix.     

1-generator quasi-cyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring with $1\ne 0$ . 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let $\gamma $ be a fixed generator of the maximal ideal of R, $F=R/\langle \gamma \rangle $ and $|F|=q$ . For any positive integers m, n satisfying $\mathrm{gcd}(q,n)=1$ , let $\mathcal{R}_n=R[x]/\langle x^n-1\rangle $ . Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator $\mathcal{R}_n$ -submodules of the module $\mathcal{R}_n^m$ . First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that $\mathrm{gcd}(|q|_n,m)=1$ , where $|q|_n$ denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.     

Construction of cyclic codes over \mathbb F _2+u\mathbb F _2 for DNA computing

We construct codes over the ring $\mathbb F _2+u\mathbb F _2$ with $u^2=0$ for use in DNA computing applications. The codes obtained satisfy the reverse complement constraint, the $GC$ content constraint, and avoid the secondary structure. They are derived from cyclic reverse-complement codes over the ring $\mathbb F _2+u\mathbb F _2$ . We also construct an infinite family of BCH DNA codes.