Cyclic codes over $${\mathbb {F}}_{2^m}[u]/\langle u^k\rangle $$ of oddly even length

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 and \(R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}\) (\(u^k=0\)) where \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring \(R[x]/\langle x^{2n}-1\rangle \). In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.     

One weight $$\mathbb {Z}_2\mathbb {Z}_4$$ additive codes

We study one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive codes. It is shown that the image of an equidistant \(\mathbb {Z}_2\mathbb {Z}_4\) code is a binary equidistant code and that the image of a one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive code, with nontrivial binary part, is a linear binary one weight code. The structure and possible weights for all one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive codes are described. Additionally, a lower bound for the minimum distance of dual codes of one weight additive codes is obtained.     

An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Let \(\{a_i:i\in I\}\) be a finite set in \({\mathbb{R}}^n\). The illumination problem addressed in this work concerns the optimal location and orientation of a conic light beam

$$ R\big (z,y,s\big )= \left\{ x \in {\mathbb{R}}^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\right\} .$$

The aperture angle \(\vartheta = 2\arccos s\) of the conic light beam is a decreasing function of the sharpness coefficient \( s\in [0,1]\). The problem at hand is to select an apex z in a prescribed compact region \(Z\subseteq {\mathbb{R}}^n\) and a unit vector \(y\in {\mathbb{R}}^n\) so that the conic light beam R(z, y, s) fulfils two conflicting requirements: it captures as many points \(a_i\) as possible and, at the same time, it has a sharpness coefficient s as large as possible.

     

Study of optical properties of potassium permanganate ($$\hbox {KMnO}_{4}$$) doped poly(methylmethacrylate) (PMMA) composite films

We have developed films of pure polymethylmethacrylate (PMMA) (0.5, 1, 2 and 5%) and potassium permanganate \((\hbox {KMnO}_{4})\)-doped PMMA composite films of thickness (\(\sim 100\, \upmu \hbox {m}\)) using the solution-cast technique. To identify the possible change that happen to the PMMA films due to doping, the optical properties were investigated for different concentrations of \(\hbox {KMnO}_{4}\) by recording the absorbance (A) and transmittance (\(T\%\)) spectra of these films using UV–Vis spectrophotometer in the wavelength range of 300–1100 nm. From the data obtained from the optical parameters viz. absorption coefficient (\(\alpha \)), extinction coefficient (\(\kappa \)), finesse coefficient (F), refractive index (\(\eta \)), real and imaginary parts of dielectric constant (\(\varepsilon _{\mathrm{r}}\) and \(\varepsilon _{\mathrm{i}})\) and optical conductivity (\(\sigma \)) were calculated for the prepared films. The indirect optical band gap for the pure and the doped-PMMA films were also estimated.     

Nonparametric density and survival function estimation in the multiplicative censoring model

Consider the multiplicative censoring model given by \(Y_i=X_iU_i\), \(i=1, \ldots ,n\) where \((X_i)\) are i.i.d. with unknown density f on \({\mathbb {R}}\), \((U_i)\) are i.i.d. with uniform distribution \({\mathcal {U}}([0,1])\) and \((U_i)\) and \((X_i)\) are independent sequences. Only the sample \((Y_i)\) is observed. We study nonparametric estimators of both the density f and the corresponding survival function \(\bar{F}\). First, kernel estimators are built. Pointwise risk bounds for the quadratic risk are given, and upper and lower bounds for the rates in this setting are provided. Then, in a global setting, a data-driven bandwidth selection procedure is proposed. The resulting estimator has been proved to be adaptive in the sense that its risk automatically realizes the bias-variance compromise. Second, when the \(X_i\)s are nonnegative, using kernels fitted for \({\mathbb {R}}^+\)-supported functions, we propose new estimators of the survival function which are also adaptive. By simulation experiments, we check the good performances of the estimators and compare the two strategies.     

Structural,dielectric and piezoelectric study of Ca-, Zr-modified $$\hbox {BaTiO}_{3}$$ lead-free ceramics

We prepared a lead-free ceramic (\(\hbox {Ba}_{0.85}\hbox {Ca}_{0.15})(\hbox {Ti}_{1-x}\hbox {Zr}_{x})\hbox {O}_{3}\) (BCTZ) using the conventional mixed oxide technique. The samples were prepared by an ordinary mixing and sintering technique. In this study we investigated how small amounts of \(\hbox {Zr}^{4+}\) can affect the crystal structure and microstructure as well as dielectric and piezoelectric properties of \(\hbox {BaTiO}_{3}\). X-ray diffraction analysis results indicate that no secondary phase is formed in any of the BCTZ powders for \(0 \le x \le 0.1\), suggesting that \(\hbox {Zr}^{4+}\) diffuses into \(\hbox {BaTiO}_{3}\) lattices to form a solid solution. Scanning electron microscopy micrographs revealed that the average grain size gradually increased with \(\hbox {Zr}^{4+}\) content from 9.5 \(\upmu \!\hbox {m}\) for \(x = 0.02\) to 13.5 \(\upmu \!\hbox {m}\) for \(x = 0.1\); Curie temperature decreased due to the small tetragonality caused by \(\hbox {Zr}^{4+}\) addition. Owing to the polymorphic phase transition from orthorhombic to tetragonal phase around room temperature, it was found that the composition \(x = 0.09\) showed improved electrical properties and reached preferred values of \(d_{33} = 148\) pC \(\hbox {N}^{-1}\) and \(K_{\mathrm{p}} = 27\%\).     

High-pressure studies of superconductivity in $$\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}$$

\(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\) crystallizes in tetragonal CeOBiS\(_{2}\) structure (S. G. P4/nmm). We have investigated the effect of pressure on magnetization measurements. Our studies suggest improved superconducting properties in polycrystalline samples of \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\). The \(T_{\mathrm{c}}\) in our sample is 5.3 K, at ambient pressure, which is marginal but definite enhancement over \(T_{\mathrm{c}}\) reported earlier (= 5.1 K). The upper critical field \(H_{\mathrm{c}2}\)(0) is greater than 3 T, which is higher than earlier report on this material. As determined from the M–H curve, both \(H_{\mathrm{c}2}\) and \(H_{\mathrm{c}1}\) decrease under external pressure P (0 \(\le P \le \) 1 GPa). We observe a decrease in critical current density and transition temperature on applying pressure in \(\hbox {BiO}_{0.75}\hbox {F}_{0.25}\hbox {BiS}_{2}\).     

Bootstrap in semi-functional partial linear regression under dependence

This paper deals with the semi-functional partial linear regression model \(Y={{\varvec{X}}}^\mathrm{T}{\varvec{\beta }}+m({\varvec{\chi }})+\varepsilon \) under \(\alpha \)-mixing conditions. \({\varvec{\beta }} \in \mathbb {R}^{p}\) and \(m(\cdot )\) denote an unknown vector and an unknown smooth real-valued operator, respectively. The covariates \({{\varvec{X}}}\) and \({\varvec{\chi }}\) are valued in \(\mathbb {R}^{p}\) and some infinite-dimensional space, respectively, and the random error \(\varepsilon \) verifies \(\mathbb {E}(\varepsilon |{{\varvec{X}}},{\varvec{\chi }})=0\). Naïve and wild bootstrap procedures are proposed to approximate the distribution of kernel-based estimators of \({\varvec{\beta }}\) and \(m(\chi )\), and their asymptotic validities are obtained. A simulation study shows the behavior (on finite sample sizes) of the proposed bootstrap methodology when applied to construct confidence intervals, while an application to real data concerning electricity market illustrates its usefulness in practice.     

Partial permutation decoding for MacDonald codes

We show how to find s-PD-sets of the minimal size \(s+1\) for the \(\left[ \frac{q^n-q^u}{q-1},n,q^{n-1}-q^{u-1}\right] _q \) MacDonald q-ary codes \(C_{n,u}(q)\) where \(n \ge 3\) and \(1 \le u \le n-1\). The construction of [6] can be used and gives s-PD-sets for s up to the bound \(\lfloor \frac{q^{n-u}-1}{(n-u)(q-1)} \rfloor -1\), of effective use for u small; for \(u \ge \lfloor \frac{n}{2} \rfloor \) an alternative construction is given that applies up to a bound that depends on the maximum size of a set of vectors in \(V_u(\mathbb {F}_q)\) with each pair of vectors distance at least 3 apart.     

Synthesis,structure and thermoelectric properties of $$\mathrm{La}_{1-x}\mathrm{Na}_{x}\mathrm{CoO}_{3 }$$ perovskite oxides

Monovalent ion doped lanthanum cobaltate \(\hbox {La}_{1-x}\hbox {Na}_{x}\hbox {CoO}_{3 }\) (\(0 \le x \le 0.25\)) compositions were synthesized by the nitrate–citrate gel combustion method. All the heat treatments were limited to below 1123 K, in order to retain the Na stoichiometry. Structural parameters for all the compounds were confirmed by the Rietveld refinement method using powder X-ray diffraction (XRD) data and exhibit the rhombhohedral crystal structure with space group R-3c (No. 167). The scanning electron microscopy study reveals that the particles are spherical in shape and sizes, in the range of 0.2–0.5 \(\upmu \)m. High temperature electrical resistivity, Seebeck coefficient and thermal conductivity measurements were performed on the high density hot pressed pellets in the temperature range of 300–800 K, which exhibit p-type conductivity of pristine and doped compositions. The X-ray photoelectron spectroscopy (XPS) studies confirm the monotonous increase in \(\hbox {Co}^{4+}\) with doping concentration up to \(x = 0.15\), which is correlated with the electrical resistivity and Seebeck coefficient values of the samples. The highest power factor of \(10~\upmu \hbox {W~mK}^{-2 }\) is achieved for 10 at% Na content at 600 K. Thermoelectric figure of merit is estimated to be \({\sim }1 \times 10^{-2}\) at 780 K for 15 at% Na-doped samples.     

Thermo-transport properties of Zn-substituted layered Li-nickel oxide, $$\hbox {LiNiO}_{2}$$

The layered Li-TM-\(\hbox {O}_{2}\) materials have been investigated extensively due to their application as cathodes in Li batteries. The electrical properties of these oxides can be tuned or controlled either by non-stoichiometry or substitution. Hence the thermo-transport properties of Zn-substituted \(\hbox {LiNi}_{1-x}\hbox {Zn}_{x}\hbox {O}_{2}\) for \(0 \le x \le 0.16\) have been investigated in the temperature range of 300–900 K for potential application as a high-temperature thermoelectric material. For \(x < 0.08\), the compounds were of single phase belonging to the space group R-3mH while for \(x > 0.08\) an additional minority phase, ZnO forms together with the main layered phase. All the compounds exhibit a semiconducting behaviour with electrical resistivity, varying in the range of  \(\sim 10^{-4}\) to \(10^{-2}\,\,\Omega \hbox {m}\) between 300 and 900 K. The electrical resistivity is found to increase with increasing Zn-substitution predominantly due to a decrease in the charge carrier hole mobility. The activation energy remains constant, \(\sim \)10  meV, with Zn-substitution. The Seebeck coefficient of the compounds is found to decrease with increasing temperature and increase with increasing Zn-substitution. The Seebeck coefficient decreases from \(\sim \)95 to \(35\ \upmu \hbox {V K}^{-1}\) and the corresponding power factor is \(\sim \)12\(\ \upmu \hbox {W m}^{-1}\ {\hbox {K}}^{-2}\) for the \(x = 0.16\) compound.     

Characterization and optical properties of $$\hbox {Pr}_{2}\hbox {O}_{3}$$-doped molybdenum lead-borate glasses

\(\hbox {Pr}^{3+}\) doped molybdenum lead-borate glasses with the chemical composition 75PbO?[25–(x \(+\) y)\(\hbox {B}_{2}\hbox {O}_{3}]\)–\(y\hbox {MoO}_{3}\)–\(x\hbox {Pr}_{2}\hbox {O}_{3}\) (where \(x = 0.5\) and 1.0 mol% and \(y = 0\) and 5 mol%) were prepared by conventional melt-quenching technique. Thermal, optical and structural analyses are carried out using DSC, UV and FTIR spectra. The physical parameters, like glass transition \((T_{\mathrm{g}})\), stability factor \((\Delta T)\), optical energy band gap \((E_{\mathrm{gopt}})\), of these glasses have been determined as a function of dopant concentration. The \({T}_{\mathrm{g}}\) and optical energy gaps of these glasses were found to be in the range of 290–350\({^{\circ }}\hbox {C}\) and 2.45–2.7 eV, respectively. Stability of the glass doped with \(\hbox {Pr}^{3+}\) is found to be moderate (\(\sim \)40). The results are discussed using the structural model of Mo–lead-borate glass.     

<10$$ \bar{1} $$0> Dislocation at a {2$$ \bar{1} $$$$ \bar{1} $$0} low-angle grain boundary in LiNbO3

A LiNbO₃ bicrystal that contains a {2\( \bar{1} \) \( \bar{1} \)0} low-angle grain boundary with both of 2° tilt misorientation and a slight twist misorientation was fabricated, and resulting dislocation structure at the boundary was analyzed by using transmission electron microscopy (TEM) and scanning TEM. The observations revealed that two types of dislocations of b = 1/3 <2\( \bar{1} \) \( \bar{1} \)0> and b = <10\( \bar{1} \)0> are formed at the boundary. A 1/3 <2\( \bar{1} \) \( \bar{1} \)0> dislocation, which dissociates into two partial dislocations with a {2\( \bar{1} \) \( \bar{1} \)0} stacking fault in between, compensates only tilt misorientation of the boundary. On the other hand, it was found that a <10\( \bar{1} \)0> dislocation, which dissociates into three equivalent partial dislocations with b = 1/3 <10\( \bar{1} \)0>, has both edge and screw components in total. That is, the <10\( \bar{1} \)0> dislocations are formed to compensate the twist misorientation of the boundary, in addition to the tilt misorientation. It is interesting that the three partial dislocations from a <10\( \bar{1} \)0> dislocation are arranged in a zigzag pattern with left–right asymmetry. This special configuration is suggested to originate from the presence of stable stacking fault structure on the {2\( \bar{1} \) \( \bar{1} \)3} plane in LiNbO₃.     

Stability of the Supercon ducting d-Wave Pairing Towar d the Intersite Coulomb Repulsion in $$\hbox {CuO}_2$$ Plane

Taking into account the real crystalline structure of the \(\hbox {CuO}_2\) plane and the strong spin-fermion coupling, we study the influence of the intersite Coulomb repulsion between holes on the Cooper instability of the spin-polaron quasiparticles in cuprate superconductors. The analysis shows that only the superconducting d-wave pairing is implemented in the whole region of doping, whereas the solutions of the self-consistent equations for the s-wave pairing are absent. It is shown that intersite Coulomb interaction \(V_1\) between the holes located at the nearest oxygen ions does not affect the d-wave pairing, because its Fourier transform \(V_q\) vanishes in the kernel of the corresponding integral equation. The intersite Coulomb interaction \(V_2\) of quasiparticles located at the next-nearest oxygen ions does not vanish in the integral equations, however, but it is also shown that the d-wave pairing is robust toward this interaction for physically reasonable values of \(V_2\).     

Bracelet monoids and numerical semigroups

Given positive integers \(n_1,\ldots ,n_p\), we say that a submonoid M of \(({\mathbb N},+)\) is a \((n_1,\ldots ,n_p)\)-bracelet if \(a+b+\left\{ n_1,\ldots ,n_p\right\} \subseteq M\) for every \(a,b\in M\backslash \left\{ 0\right\} \). In this note, we explicitly describe the smallest \(\left( n_1,\ldots ,n_p\right) \)-bracelet that contains a finite subset X of \({\mathbb N}\). We also present a recursive method that enables us to construct the whole set \(\mathcal B(n_1,\ldots ,n_p)=\left\{ M|M \quad \text {is a} \quad (n_1,\ldots ,n_p)\text {-bracelet}\right\} \). Finally, we study \((n_1,\ldots ,n_p)\)-bracelets that cannot be expressed as the intersection of \((n_1,\ldots , n_p)\)-bracelets properly containing it.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

Inhomogeneous Superconducting Behaviour in $$\hbox {La}_{5}\hbox {Ni}_{2}\hbox {Si}_{3}$$

The superconducting phase transition at \(T_\mathrm{c} = 2.3\) K was observed for the electrical resistivity \(\rho ({T})\) and magnetic susceptibility \(\chi (T)\) measurements in the ternary compound La\(_{5}\hbox {Ni}_{2}\hbox {Si}_{3}\) that crystallizes in the hexagonal-type structure. Although a single-phase character with the nominal stoichiometry of the synthesized sample was confirmed, a small trace of the La–Ni phase was found, being probably responsible for the superconducting behaviour in the investigated compound. The magnetization loop recorded at \({T} = 0.5\) K resembles a star-like shape which indicates that the density of the critical current can be strongly suppressed by a magnetic field. The low-\(T _{\rho }(T)\) and specific heat \({C}_\mathrm{p}({T})\) data in the normal state reveal simple metallic behaviour. No clear evidence of a phase transition to any long- or short-range order was found for \(C_\mathrm{p}(T)\) measurements in the T-range of 0.4–300 K.     

Further Estimates of $$(T-T_{90})$$ Close to the Triple Point of Water

Recent advances in primary acoustic gas thermometry (AGT) have revealed significant differences between temperature measurements using the International Temperature Scale of 1990, \(T_{90}\), and thermodynamic temperature, T. In 2015, we published estimates of the differences \((T-T_{90})\) from 118 K to 303 K, which showed interesting behavior in the region around the triple point of water, \(T_\mathrm{TPW}=273.16\) K. In that work, the \(T_{90}\) measurements below \(T_\mathrm{TPW}\) used a different ensemble of capsule standard platinum resistance thermometers (SPRTs) than the \(T_{90}\) measurements above \(T_\mathrm{TPW}\). In this work, we extend our earlier measurements using the same ensemble of SPRTs above and below \(T_\mathrm{TPW}\), enabling a deeper analysis of the slope \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) around \(T_\mathrm{TPW}\). In this article, we present the results of seven AGT isotherms in the temperature range 258 K to 323 K. The derived values of \((T-T_{90})\) have exceptionally low uncertainties and are in good agreement with our previous data and other AGT results. We present the values \((T-T_{90})\) alongside our previous estimates, with the resistance ratios W(T) from two SPRTs which have been used across the full range 118 K to 323 K. Additionally, our measurements show discontinuities in \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\) which are consistent with the slope discontinuity in the SPRT deviation functions. Since this discontinuity is by definition non-unique, and can take a range of values including zero, we suggest that mathematical representations of \((T-T_{90})\), such as those in the mise en pratique for the kelvin (Fellmuth et al. in Philos Trans R Soc A 374:20150037, 2016. doi: 10.1098/rsta.2015.0037), should have continuity of \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\).     

A first-principle investigation into effect of B- and BN-doped $$\hbox {C}_{60}$$ in lowering dehydrogenation of $$\hbox {MXH}_{4}$$MXH4 (where M $$=$$= Na,Li and X $$=$$= Al,B)

The present paper reports the effect of B- and BN-doped \(\hbox {C}_{60}\) as catalysts for lowering the dehydrogenation energy in \(\hbox {MXH}_{4}\) clusters (M = Na and Li, X = Al and B) using density functional calculations. \(\hbox {MXH}_{4}\) interacts strongly with B-doped \(\hbox {C}_{60}\) and weakly with BN-doped \(\hbox {C}_{60}\) in comparison with pure \(\hbox {C}_{60}\) with binding energy 0.56–0.80 and 0.05–0.34 eV, respectively. The hydrogen release energy \((E_{\mathrm{HRE}})\) of \(\hbox {MXH}_{4}\) decreases sharply in the range of 38–49% when adsorbed on B-doped \(\hbox {C}_{60}\); however, with BN-doped \(\hbox {C}_{60}\) the decrease in the \(E_{\mathrm{HRE}}\) varies in the range of 6–20% as compared with pure \(\hbox {MXH}_{4}\) clusters. The hydrogen release energy of second hydrogen atom in \(\hbox {MXH}_{4}\) decreases sharply in the range of 1.7–41% for BN-doped \(\hbox {C}_{60}\) and decreases in the range of 0.2–11.3% for B-doped \(\hbox {C}_{60}\) as compared with pure \(\hbox {MXH}_{4}\) clusters. The results can be explained on the basis of charge transfer within \(\hbox {MXH}_{4}\) cluster and with the doped \(\hbox {C}_{60}\).