An extension of the (strong) primitive normal basis theorem

An extension of the primitive normal basis theorem and its strong version is proved. Namely, we show that for nearly all \(A = {\small \left( \begin{array}{cc} a&{}b \\ c&{}d \end{array} \right) } \in \mathrm{GL}_2(\mathbb {F}_{q})\) , there exists some \(x\in \mathbb {F}_{q^m}\) such that both \(x\) and \((-dx+b)/(cx-a)\) are simultaneously primitive elements of \(\mathbb {F}_{q^m}\) and produce a normal basis of \(\mathbb {F}_{q^m}\) over \(\mathbb {F}_q\) , granted that \(q\) and \(m\) are large enough.     

Primitive elements with prescribed trace

Let \(q\) be a power of a prime number \(p\) . Let \(n\) be a positive integer. Let \(\mathbb {F}_{q^n}\) denote a finite field with \(q^n\) elements. In this paper, we consider the existence of the some specific elements in the finite field \(\mathbb {F}_{q^n}\) . We get that when \(n\ge 29\) , there are elements \(\xi \in \mathbb {F}_{q^n}\) such that \(\xi +\xi ^{-1}\) is a primitive element of \(\mathbb {F}_{q^n}\) , and \(\mathrm{Tr}(\xi ) = a, \mathrm{Tr}(\xi ^{-1}) = b\) for any pair of prescribed \(a, b \in \mathbb {F}_q^*\) .     

Containerless Measurements of Density and Viscosity for a Cu_{48}Zr_{52} Liquid

The densities of solid and liquid Cu \(_{48}\) Zr \(_{52}\) and the viscosity of the liquid were measured in a containerless electrostatic levitation system using optical techniques. The measured density of the liquid at the liquidus temperature (1223 K) is (7.02 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) and the density of the solid extrapolated to that temperature is (7.15 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) . The thermal expansion coefficients measured at 1223 K are (6.4 \(\pm \) 0.1) \(\,\times \,10^{-5}\) K \(^{-1}\) in the liquid phase and (3.5 \(\pm \) 0.3) \(\,\times \,10^{-5}\) K \(^{-1}\) in the solid phase. The viscosity of the liquid, measured with the oscillating drop technique, is of the form \(A\exp \left[ \left( {{E}_{0}}+{{E}_{1}}\left( 1/T-1/{{T}_{0}} \right) \right) \times \left( 1/T-1/{{T}_{0}} \right) \right] \) , where \({{T}_{0}}=1223\) K, \(A= (0.0254 \pm 0.0004)\) Pa \(\cdot \) s, \({{E}_{0}}\) =  (8.43 \(\pm \) 0.26) \(\,\times \,10^3\) K and \({{E}_{1}}\) =  (1.7 \(\pm \) 0.2) \(\,\times 10^7\) K \(^{2}\) .     

Limit laws for the maxima of stationary chi-processes under random index

Let \(\{\chi _{k}(t), t\ge 0\}\) be a stationary \(\chi \) -process with \(k\) degrees of freedom. In this paper, we consider the maxima \(M_{k}(T)= \max \{\chi _{k}(t), \forall t\in [0,T]\}\) with random index \(\mathcal {T}_{T}\) , where \(\mathcal {T}_{T}/T\) converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of \(M_{k}(\mathcal {T}_{T})\) exists under some additional conditions.     

Effective Thermal-Conductivity Measurement on Germanate Glass–Ceramics Employing the 3\omega Method at High Temperature

It can be noted that the germanate glass–ceramic is a functional material with excellent thermal stability which can be used in optical devices. The temperature-dependent effective thermal conductivities of CaO–BaO–CoO–Al \(_{2}\) O \(_{3}\) –SiO \(_{2}\) –GeO \(_{2}\) glass–ceramics from 295.5 K to 780 K are determined using a \(3\omega \) method. One of the main advantages for the \(3\omega \) method is to diminish radiation errors effectively when the temperature is as high as 1000 K. Thermal conductivities of CaO–BaO–CoO–Al \(_{2}\) O \(_{3}\) –SiO \(_{2}\) –GeO \(_{2}\) increase with a rise in temperature. Effective thermal conductivities of a sample increase from \(1.55~\hbox {W}\cdot \hbox {m}^{-1}\cdot \hbox {K}^{-1}\) at 295.5 K to \(7.64~\hbox {W}\cdot \,\hbox {m}^{-1}\cdot \hbox {K}^{-1}\) at 698.1 K. The effective thermal conductivity of CaO–BaO–CoO–Al \(_{2}\) O \(_{3}\) –SiO \(_{2}\) –GeO \(_{2}\) glass–ceramic increases with a rise of temperature. This investigation can be used as a basis for the measurement of thermal properties of ceramic materials at higher temperature.     

First- and Second-Order Phase Transition in Parallel Triple Quantum Dots: The Roles of Symmetric and Asymmetric Hopping

By means of the numerical renormalization group method, I study the quantum phase transition (QPT) and the electronic transport in parallel triple quantum dot system with symmetric and/or asymmetric hopping. For symmetric hopping \(t_{1} = t_{2}\) and zero magnetic field \(B = 0\) , I find a first order transition between spin quadruplet and doublet as \(t_{1}\) ( \(t_{2}\) ) increases. With increasing \(B\) , a second order QPT between \(S_{z} = 1/2\) of the doublet and \(S_{z} = 3/2\) of the quadruplet is observed. For asymmetric hopping \(t_{1} \ne t_{2}\) , the QPT depends closely on the other hopping. For fixed \(t_{1} < \varGamma \) , where \(\varGamma \) is the hybridization function between the dots and the leads, a first order transition is observed as \(t_{2}\) increases, while for \(t_{1} \ge \varGamma \) , a crossover occurs. In the presence of \(B\) , the transition between \(S_{z} = 1/2\) and \(S_{z} = 3/2\) is a first order QPT for \(t_{1} < \varGamma \) , while a second order for \(t_{1} \ge \varGamma \) .     

Toward Mega-pixel Neutron Imager Using Current-Biased Kinetic Inductance Detectors of Nb Nanowires with ^{10}B Converter

We present not the results but the idea of a superconducting nanowire detector with \(^{10}\) B conversion layer for sensing a single neutron. We use \(^{7}\) Li ion and \(^{4}\) He ion emitted as two independent heat sources, which appear in opposite direction associated with nuclear reaction \(^{10}\) B(n, \(^{4}\) He) \(^{7}\) Li. We probe a change in the kinetic inductance \(L_\mathrm{k}\) coming from inertia of the Cooper pairs. Our detector is different from a conventional kinetic inductance detector (KID), but is named as a current-biased KID. We use two sets of Nb nanowires with superconducting readout taps to monitor the local signal. In between the X meander and the Y meander, we inserted a \(^{10}\) B layer acting as a conversion layer from neutrons to charged particles. We plan to fabricate a mega-pixel neutron imager by coupling 10 bit linear position-sensitive arrays along the X and Y directions with the single flux quantum readout circuits.     

Exploring Solute–Solvent Interactions of \alpha -Amino Acids in Aqueous [\mathrm{EPyBF}_{4}] Arrangements by Volumetric,Viscometric, Refractometric,and Acoustic Approach

Qualitative and quantitative analysis of molecular interaction prevailing in glycine, l-alanine, l-valine, and aqueous solution of ionic liquid (IL) [1-ethylpyridinium tetrafluoroborate ( \(\mathrm{EPyBF}_{4})\) ] have been investigated by thermophysical properties. The apparent molar volume ( \(\phi _{V}\) ), viscosity \(B\) -coefficient, molal refraction ( \(R_{\mathrm{M}}\) ), and adiabatic compressibility ( \(\phi _{ K} )\) of glycine, l-alanine, and l-valine have been studied in 0.001 mol \({\cdot }\, \mathrm{dm}^{-3}\) , 0.003 mol \({\cdot }\, \mathrm{dm}^{-3}\) , and 0.005 mol  \({\cdot } \,\mathrm{dm}^{-3}\) aqueous 1-ethylpyridinium tetrafluoroborate [ \(\mathrm{EPyBF}_{4}\) ] solutions at 298.15 K from the values of densities \((\rho )\) , viscosities ( \(\eta \) ), refractive index ( \(n_{\mathrm{D}})\) , and speed of sound \((u)\) , respectively. The extent of interaction, i.e., the solute–solvent interaction is expressed in terms of the limiting apparent molar volume ( \(\phi _{V}^0 )\) , viscosity \(B\) -coefficient, and limiting apparent molar adiabatic compressibility ( \(\phi _{K}^0)\) . The limiting apparent molar volumes ( \(\phi _{V}^0 )\) , experimental slopes ( \(S_{V}^*)\) derived from the Masson equation, and viscosity \(A\) - and \(B\) -coefficients using the Jones–Dole equation have been interpreted in terms of ion–ion and ion–solvent interactions, respectively. Molal refractions ( \(R_{\mathrm{M}})\) have been calculated with the help of the Lorentz–Lorenz equation. The role of the solvent (aqueous IL solution) and the contribution of solute–solute and solute–solvent interactions to the solution complexes have also been analyzed through the derived properties.     

DEM analysis of the influence of the intermediate stress ratio on the critical-state behaviour of granular materials

The critical-state response of granular assemblies composed of elastic spheres under generalised three-dimensional loading conditions was investigated using the discrete element method (DEM). Simulations were performed with a simplified Hertz–Mindlin contact model using a modified version of the LAMMPS code. Initially isotropic samples were subjected to three-dimensional stress paths controlled by the intermediate stress ratio, \(b=[(\sigma '_{2}-\sigma '_{3})/\) \((\sigma '_{1}-\sigma '_{3})]\) . Three types of simulation were performed: drained (with \(b\) -value specified), constant volume and constant mean effective stress. In contrast to previous DEM observations, the position of the critical state line is shown to depend on \(b\) . The data also show that, upon shearing, the dilatancy post-peak increases with increasing \(b\) , so that at a given mean effective stress, the void ratio at the critical state increases systematically with \(b\) . Four commonly-used three-dimensional failure criteria are shown to give a better match to the simulation data at the critical state than at the peak state. While the void ratio at critical state is shown to vary with \(b\) , the coordination number showed no dependency on \(b\) . The variation in critical state void ratios at the same \(p'\) value is apparently related to the directional fabric anisotropy which is clearly sensitive to \(b\) .     

({p, \rho,T, x}) Properties of \text{ CO}_{2}/Propane Binary Mixtures at 280 K to 440 K and 3 MPa to 200 MPa

The (p, \(\rho \) , T, x) properties of binary mixtures of CO \(_{2}\) (volume fraction purity 0.99999) and propane (mole fraction purity 0.9999) ( \(x_{1}\) CO \(_{2}+x_{2}\) propane; \(x_{1} = 0.1744\) , 0.3863, 0.5837, and 0.7732) were measured in the compressed liquid phase using a metal-bellows variable volumometer. Measurements were conducted from 280 K to 440 K and 3 MPa to 200 MPa. The expanded uncertainties ( \(k = 2\) ) were estimated to be temperature, \(<\) 3 mK; pressure, 1.5 kPa ( \(p\le 7\)  MPa), 0.06 % (7 MPa \(< p\le 50\)  MPa), 0.1 % (50 MPa \(< p\le 150\)  MPa), 0.2 % ( \(p> 150\)  MPa); density, 0.10 %; and composition, \(4.4\times 10^{-4}\) . At \(p >100\)  MPa and 280 K or 440 K, the uncertainties in density measurements increase to 0.14 % and 0.22 %, respectively. The data were compared with available equations of state. The excess molar volumes, \(v_\mathrm{m}^\mathrm{E}\) , of the mixtures were calculated and plotted as a function of temperature and pressure.     

Measurement of the (p, \rho,T) Properties for Pure Hydrocarbons at Temperatures up to 600 K and Pressures up to 200 MPa

The data available for the thermodynamic properties of propane, \(n\) -butane, and isobutane at temperatures above 440 K are outdated and show significant discrepancies with each other. The ambiguity associated with these data could be limiting to the development of any understanding related to the effects of mixing of these substances with other materials such as \(\text{ CO}_{2}\) , ammonia, and non-flammable or lower-flammable HFC refrigerants. In this study, the (p, \(\rho \) , T) properties of propane, \(n\) -butane, and isobutane were measured at temperatures ranging from (360 to 600) K and pressures ranging from (50 to 200) MPa. Precise measurements were carried out using a metal-bellows variable volumometer with a thermostatted air bath. The expanded uncertainties \((k = 2)\) in the temperature, pressure, and density measurements were estimated to be \(<\) 5 mK, 0.02 MPa, and 0.88 kg  \(\cdot \)  m \(^{-3}\) ( \(T\le 423\)  K, \(p<100\)  MPa), 0.76 kg  \(\cdot \)   \(\text{ m}^{-3}\) ( \(T\le 423\)  K, \(p\ge 100\)  MPa), 0.76 kg  \(\cdot \)   \(\text{ m}^{-3}\) ( \(T>423\)  K, \(p < 100\)  MPa), and 2.94 kg  \(\cdot \)   \(\text{ m}^{-3}\) ( \(T>423\)  K, \(p \ge 100\)  MPa), respectively. The data obtained throughout this study were systematically compared with the calculated values derived from the available equations of state. These models agree well with the measured data at higher temperatures up to 600 K, demonstrating their suitability for an effective and precise examination of the mixing effects of potential alternative mixtures.     

An algorithm for computing the error sequence of p^{n}-periodic binary sequences

For binary sequences with period \(p^{n}\) , where \(p\) is an odd prime and 2 is a primitive root modulo \(p^{2}\) , we present an algorithm which computes the minimum number \(k\) so that the \(k\) -error linear complexity is not greater than a given constant \(c\) . An associated error sequence which gives the \(k\) -error linear complexity is also obtained.     

Some results on the differential functions over finite fields

Let \(g\) be a function over \(\mathbb {F}_q\) . If there exist a function \(f\) and \(a\in \mathbb {F}_q^*\) such that \(g(x)=f(x+a)-f(x)\) , then we call \(g\) a differential function and call \(f\) a differential-inverse of \(g\) . We present two criteria to decide whether a given \(g\) is a differential function. The set of the degrees of all differential functions over \({\mathbb {F}}_q\) is determined. Then we give a lower bound and an upper bound on the number of differential functions over \({\mathbb {F}}_q\) . Besides, we show how to construct differential inverses of a given differential function. At last, some applications of our results are introduced.     

Fabrication of Nb/Al Superconducting Tunnel Junction Using Ozone Gas

An ozone (O \(_{3})\) oxidation process was introduced for Nb/Al-based superconducting tunnel junctions (STJs) in order to form defect-free tunnel barriers at high critical current and to improve the energy resolution ( \(\Delta E\) ) for X-rays. The dependence of critical current ( \(J_\mathrm{C})\) and leak current ( \(I_\mathrm{leak})\) on the O \(_{3}\) exposure was measured to optimize the oxidation condition. The 50-square- \(\upmu \) m STJs produced by the O \(_{3}\) oxidation process exhibited an extremely small \(I_\mathrm{leak}\) of less than 50 pA. As expected, the lower or shorter the O \(_{3}\) exposure, the higher \(J_\mathrm{C}\) and the smaller the normal resistance ( \(R_\mathrm{N})\) . However, the maximum \(J_\mathrm{C}\) was 8 A/cm \(^{2}\) at an O \(_{3}\) exposure of 0.72 Pa min, which is much smaller than those of STJs with the conventional O \(_{2}\) oxidation process. It is expected that the high \(J_\mathrm{C}\) of 1,000 A/cm \(^{2}\) , at which a 9-eV-energy resolution for 277 eV photons is predicted, can be reached by an O \(_{3}\) exposure of 3.5 \(\times \) 10 \(^{-4}\) Pa min.     

Thermal Stability of \beta -PtO_2 Investigated by Simultaneous Thermal Analysis and Its Influence on Platinum Resistance Thermometry

We present thermogravimetric and differential scanning calorimetric studies of PtO \(_2\) powders measured in different atmospheres. In synthetic air a mass loss of 11.4 % is found at the decomposition temperature \(T_\mathrm {D}\)  = 595  \(^{\circ }\hbox {C}\) which can be attributed to the reduction of PtO \(_2\) . In a helium atmosphere the mass loss is 12.0 % and is found at 490  \(^{\circ }\hbox {C}\) . Subsequent heating in air leads to another oxidation process above \(T_\mathrm {D}\) and a reduction at 800  \(^{\circ }\hbox {C}\) . The second oxidation and reduction process is strongly suppressed when the powder is heated in He. The remaining mass above \(T_\mathrm {D}\) does not comply with a reduction path PtO \(_2 \rightarrow \) PtO \(\rightarrow \) Pt. Differential scanning calorimetry shows an endothermic reaction at \(T_\mathrm {D}\) in synthetic air as well as in helium which corresponds with the mass loss. These measurements imply that the powder can be assigned to be \(\beta \) -PtO \(_2\) . Furthermore, catalytic activity of the PtO \(_2\) powder is evidenced by mass spectrometry to be present below 460  \(^{\circ }\hbox {C}\) . Finally, the impact of these findings on the stability of platinum resistance thermometers is discussed.     

Temperature Coefficients of the Refractive Index for Complex Hydrocarbon Mixtures

Temperature coefficients of the refractive index ( \(\mathrm{d}n/\mathrm{d}T\) ) in the \(25\,^{\circ }\mathrm{C}\) to \(35\,^{\circ }\mathrm{C}\) temperature interval for hydrocarbon mixtures containing as many as 14 compounds were investigated in this work. The measured \(-\mathrm{d}n/\mathrm{d}T\) of the mixtures were compared with calculations based on the values for each compound and their concentrations. Differences of about 1 % between measured and calculated values were observed for all mixtures. The additivity of \(-\mathrm{d}n/\mathrm{d}T\) for these hydrocarbons enables preparation of surrogate fuels that are formulated to have properties like those of specific diesel fuels.     

VSH and multiplicative modular relations between small primes with polynomial exponents

We study the frequency of the positive integers \(n\) that are products of two primes of the same order of magnitude and such that the congruence $$\begin{aligned} \prod _{i=1}^k p_i^{f_i (n)} \equiv 1 \pmod {n} \end{aligned}$$ holds with some fixed nonzero polynomials \(f_1(X), \ldots , f_k(X) \in {\mathbb {Z}}[X]\) , where \(p_i\) denotes the \(i\) th prime. The question is motivated by collision finding in the so-called Very Smooth Hash function, introduced by Contini et al. (Lecture notes in computer science, vol. 4004. Springer, Berlin, pp 165–182, 2006).     

Toward a Detector/Readout Architecture for the Background-Limited Far-IR/Submm Spectrograph (BLISS)

We have built and tested 32-element linear arrays of absorber-coupled transition-edge sensors (TESs) read out with a time-division SQUID multiplexer. This detector/readout architecture is designed for the background-limited far-IR/submm spectrograph (BLISS) which is a broadband (35–433  \(\upmu \) m), grating spectrometer consisting of six wavebands each with a modest resolution of R \(\sim \) 700. Since BLISS requires the effective noise equivalent power (NEP) of the TESs to equal 1  \(\times \)  10 \(^{-19}\)  W/Hz \(^{1/2}\) , our detectors consist of very long (1–2 mm), narrow (0.4 \(\upmu \) m), and thin (0.25 \(\upmu \) m) Si \(_{x}\) N \(_{y}\) support beams that reduce the thermal conductance G between the substrate and the optical absorber. The thermistors of our lowest noise TESs consist of iridium with \(T_{c}=130\) mK. We have measured the electrical properties of arrays of these Ir TESs with various meander and straight support beams and absorber shapes and found that G is \(\sim \) 30 fW/K (meander) and \(\sim \) 110 fW/K (straight), the electrical NEP is 2–3  \(\times \)  10 \(^{-19}\) W/Hz \(^{1/2}\) (meander and straight), and the response time \(\tau \) is 10–30 ms (meander) and 2–5 ms (straight). To reduce spurious or “dark” power from heating the arrays, we mounted the arrays into light-tight niobium boxes and added custom L/R and L/C low-pass chip filters into these boxes to intercept dark power from the bias and readout circuit. We found the average dark power equals 1.3 and 4.6 fW for the boxes with L/R and L/C chip filters, respectively. We have built arrays with \(T_{c}= 70\)  mK using molybdenum/copper bilayers and are working to lower the dark power by an order of magnitude so we can demonstrate NEP \(~=~1~\times \)  10 \(^{-19}\)  W/Hz \(^{1/2}\) with these arrays. PACS numbers: 85.25.Pb; 95.85.Gn; 95.85.Fm; 63.22. \(+\) m     

An Aboveground Pulse-Tube-Based Bolometric Test Facility for the Validation of the LUMINEU \hbox {ZnMoO}_4 Crystals

The LUMINEU project aims at developing a pilot double beta decay experiment using scintillating bolometers based on ZnMoO \(_4\) crystals enriched in \(^{100}\hbox {Mo}\) . In the next months regular deliveries of large-mass \(\hbox {ZnMoO}_4\) crystals are expected from the Nikolaev Institute of Inorganic Chemistry (Novosibirsk, Russia). It is therefore crucial for the LUMINEU program to test systematically and in real time these samples in terms of bolometric properties, light yield and internal radioactive contamination. In this paper we describe an aboveground cryogenic facility based on a dilution refrigerator coupled to a pulse-tube cooler capable performing these measurements. A 23.8 g \(\hbox {ZnMoO}_4\) crystal was fully characterised in this setup. We show also that macro-bolometers can be operated with high signal-to-noise ratio in liquid-free dilution refrigerators.     

Numerical Study of Thermal Conductivities of Carbon-Based Mesoporous Composites

The thermal conductivities of ordered mesoporous carbon CMK-3 filled with Al nanoclusters were studied in this article. CMK-3 is a typical example of carbon rods which are arranged in a relatively regular two-dimensional hexagonal array. The initial structure of CMK-3 was generated from the amorphous carbon and validated by XRD simulation which is coincident with experimental data. The thermal conductivities of carbon rods in CMK-3 and Al nanoclusters with 133 atoms were simulated by an equilibrium molecular dynamics method. Then, the effective thermal conductivity (ETC) of a mesoporous composite, CMK-3 filled with \(\text {Al}_{133}\) , was obtained via one-dimensional heat conduction analysis. The influences of the substrate porosity, nanocluster filling ratio, and temperature were discussed. As an anisotropic material, ETCs along the \(X\) and \(Y\) directions are extremely poor, due to the overwhelming effect of the air thermal resistance. However, in the \(Z\) direction, the ETC improves almost linearly as the porosity decreases, and the value is much higher than those of \(X\) and \(Y\) directions. In the case of a 70 % filling ratio, when the porosity is below 59.7 %, the ETC in the \(Z\) direction exceeds the thermal conductivity of Al nanoclusters and approaches a peak value around the melting temperature of \(\text {Al}_{133}\) nanoclusters. The results indicate that the carbon-based mesoporous CMK-3 filled with Al nanoclusters might become a promising phase change material.