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A sublattice of the three-dimensional integer lattice Z 3 is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector v ∈ Z 3 whose squared length is divisible by d 2 , there exists a cubic sublattice containing v with edge length d. This improves one of the main result of a paper of Goswick et al. (J. Number Theory 132(1), 37–53 (2012)), where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices. [ABSTRACT FROM AUTHOR]

This article presents a new network model for IoT that is based on a multi-story building structure. The model locates network nodes in a regular, cubic lattice-like structure, resulting in an equation for the signal-to-noise ratio (SNR). The study also determines the relationship between traffic density, network density, and SNR. In addition, the article explores the potential of percolation theory in characterizing network functionality. The findings offer a new approach to network design and planning, allowing for selecting a network topology that meets criteria and requirements while ensuring connectivity and improving efficiency. The developed analytical apparatus provides valuable insights into the properties of the network and its applicability to specific conditions. [ABSTRACT FROM AUTHOR]

A computer simulation of the phase transitions in a three-dimensional weakly diluted five-state Potts model on a simple cubic lattice is performed. Calculations are carried out for spin systems with periodic boundary conditions at spin concentrations of p = 1.0 and p = 0.90. Systems with linear dimensions L × L × L = N, L = 10–80 are considered. The temperature dependences of the specific heat, susceptibility, and magnetization as a function of the linear dimensions of the studied systems are obtained. Using the fourth-order Binder cumulants and using histogram-data analysis, it is shown that the presence of weak disorder in the form of nonmagnetic impurities of the order c = 10% (c = 1 – p, p is the spin concentration) does not affect the phase transition of the first-order. [ABSTRACT FROM AUTHOR]

Direct simulation was performed for the solar porous receiver having cubic lattice as elementary structure with high porosity (0.80 - 0.98). The cell size was changed in four steps from 0.49 mm to 2.94 mm at three levels of the power over air mass (POM). Silicon carbide (SiC) was assumed as receiver material, and the absorptivity of the wall surface was assigned 0.9. The numerical data for the porous receiver was compared with the honeycomb receiver with low porosity. Such comparison proved the superiority of the highly porous receiver: the receiver-exit temperature and the receiver efficiency surpass those of honeycomb receiver at the same level of POM. Regarding the size effects, the performance of the porous receiver improves as the cell size reduces. When the cell size is 0.98 mm at POM =1000 kJ/kg, the receiver exit temperature increases beyond 1000 K and the receiver efficiency 0.8. When the cell size is 0.49 mm at POM = 2000 kJ/kg, the receiver exit temperature exceeds 1700 K and the efficiency 0.8. The numerical results thus demonstrated that the smaller cell size maintains the efficiency at the higher levels of POM. The mechanism for superiority of the small cell size was investigated through analysis of the heat transfer loss. This investigation revealed that the convective heat transfer enhancement in the case of small cell size decreases the temperature in the inlet region and attenuates the thermal radiation loss. Therefore, the high performance of the small-cell size receiver is attributed to the cooling effect of the air stream with high heat transfer coefficient.

This article presents a new network model for IoT that is based on a multi-story building structure. The model locates network nodes in a regular, cubic lattice-like structure, resulting in an equation for the signal-to-noise ratio (SNR). The study also determines the relationship between traffic density, network density, and SNR. In addition, the article explores the potential of percolation theory in characterizing network functionality. The findings offer a new approach to network design and planning, allowing for selecting a network topology that meets criteria and requirements while ensuring connectivity and improving efficiency. The developed analytical apparatus provides valuable insights into the properties of the network and its applicability to specific conditions.

The effect of the ternary alloys composition and overheating of their melts (at 100–370 K above the liquidus temperature) on the morphology and composition of aluminides in the Al–Sc–Ti, Al–Sc–Zr, Al–Sc–Hf systems were investigated. It was shown that during the crystallization of these melts under certain conditions, the primary precipitated phase are the complex aluminides Al3(ScxZr1−x), Al3(ScxTi1−x), Al3(ScxHf1−x) having a metastable cubic lattice with L12 structure, which matches the α-Al structural type. The variety of growth forms of aluminides is explained by a combination of a number of factors: the magnitude of overheating of the melt, the difference in the diffusion coefficients of transition metals, and the local concentration of transition metals in the respective growth zones. [ABSTRACT FROM AUTHOR]

The conditions of the formation of stable and metastable aluminides Aln(Zr1 – xNbx) formed during the crystallization of superheated melts of the Al‒Zr‒Nb system are considered. Upon a close zirconium content in the alloys of 0.23–0.25 at %, the niobium content varies from 0.05 to 0.21 at %. Alloys are prepared in a resistance furnace at a temperature of 1230°C in an argon atmosphere in graphite crucibles. The alloys are cast into a bronze mold; the cooling rate is estimated as 200°C/s. The structural features, distribution, and morphology; the composition of matrix, dendritic cells, and aluminides; and the structural type of aluminides in the alloys are studied using scanning electron and optical microscopy, X-ray diffraction, and inductively coupled plasma atomic emission spectroscopy. It is shown that the growth forms of stable aluminides (D023 structural type) change from faceted to dendritic and primary metastable aluminides (L12 structural type) begin to form; in the process of the Al‒Zr‒Nb melt, overheating was 360–365°C above the liquidus temperature. Exclusively metastable aluminides with both polyhedral and dendritic growth forms are formed in the Al‒Zr‒Nb alloys when the overheating of their melt is 390–395°C (as well as at higher overheating) above the liquidus temperature. The near-peritectic niobium composition of the Al‒Zr‒Nb alloy and the zirconium content (more than seven times higher than peritectic) are crucial factors in the formation of a large fraction of metastable AlnZr aluminides having a cubic L12 structure. It is shown that, in accordance with the rules of isomorphism, Nb replaces Zr at equivalent positions of the aluminides crystal lattice. The intensity of isomorphism of the formed Al4(Zr0.79Nb0.21) aluminides increases with the increase in temperature of the melt overheating. [ABSTRACT FROM AUTHOR]

Protein folding optimization is a very important and tough problem in computational biology. For solving this problem, a population-based metaheuristic algorithm named chemical reaction optimization (CRO) with HP cubic lattice model has been proposed in this paper. The proposed algorithm is combined with evolution and H&P compliance mechanisms which are responsible for increasing the performance of the algorithm. The evolution mechanism improves the performance of each individual solution. On the other hand, the H&P compliance mechanism tries to place the H monomer close to the center and place the P monomer as far as possible from the center of the related structure. The algorithm also applies four reactant operations of typical CRO algorithm decomposition, on-wall ineffective collision, synthesis and inter-molecular ineffective collision to solve the problem efficiently. The reactants or mechanisms may cause overlapping of the corresponding solutions. The algorithm also includes a repair mechanism which transforms invalid solutions into valid ones by removing overlapping in cubic lattice points. This algorithm has been tested over some sets of sequences and it shows very good performance. [ABSTRACT FROM AUTHOR]

We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most O (n 3 / 4) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions. [ABSTRACT FROM AUTHOR]

Highlights • From the Bogoliubov inequality the mean field approximation is applied in the Ashkin-Teller model. • The stable, metastable and unstable states of a cubic structure are investigated. • From the critical values in the free energy of the model, those phases are classified. • Phase transition lines of first-order and continuous are obtained. • Multicritical points are obtained according to this procedure. Abstract The mean field approximation from the Bogoliubov inequality is applied to study the ferromagnetic Ashkin-Teller model in three dimensions. Phase transitions for the stable, metastable and unstable states is observed. From the critical values in the free energy of the model, those transitions are classified. Besides the free energy, magnetization, critical frontiers and tricritical points are obtained. Phase diagram is presented for a better illustration of the phase transitions and its order. [ABSTRACT FROM AUTHOR]

The mean field theory results are obtained from the Bogoliubov inequality for the spin-1/2 Ashkin–Teller model on a cubic lattice for different cluster sizes. The phase diagram, magnetization and free energy are obtained. From those expressions we observed a new phase in the model. Denoted in the course of this work by Baxter (2) this new phase presents 〈 S 〉 ≠ 〈 σ 〉 ≠ 0 . The phase transitions between the Baxter (2) and the others well known phases for the model are studied and classified. [ABSTRACT FROM AUTHOR]

Microstructural features of new master alloys of the Al-Hf-Sc system with metastable aluminides with a cubic lattice identical to the lattice of a matrix of aluminum alloys are investigated using optical microscopy, scanning electron microscopy, and electron probe microanalysis. Binary and ternary alloys are smelted in a coal resistance furnace in graphite crucibles in argon. Alloys Al-0.96 at % Hf (5.98 wt % Hf) and Al-0.59 at % Hf (3.77 wt % Hf) are prepared with overheating above the liquidus temperature of about 200 and 400 K, respectively. Alloys are poured into a bronze mold, the crystallization rate in which is ~10 K/s. Metastable AlHf aluminides with a cubic lattice are formed only in the alloy overheated above the liquidus temperature by 400 K. Overheating of ternary alloys, in which metastable aluminides Al(HfSc) formed, is 240, 270, and 370 K. Depending on the Hf-to-Sc ratio in the alloy, the fraction of hafnium in aluminides Al(HfSc) varies from 0.46 to 0.71. Master alloys (at %) Al-0.26Hf-0.29Sc and Al-0.11Hf-0.25Sc (wt %: Al-1.70Hf-0.47Sc and Al-0.75Hf-0.42Sc) have a fine grain structure and metastable aluminides of compositions Al(HfSc) and Al(HfSc), respectively. Sizes of aluminides do not exceed 12 and 7 μm. Their lattice mismatch with a matrix of aluminum alloys is smaller than that for AlSc. This makes it possible to assume that experimental Al-Hf-Sc master alloys manifest a high modifying effect with their further use. In addition, the substitution of high-cost scandium with hafnium in master alloys can considerably reduce the consumption of the latter. [ABSTRACT FROM AUTHOR]

This study experimentally explores the effect of permeability as an isolated control parameter on the downstream wake structure and the aerodynamic properties of the porous square cylinder. An addictive manufacturing technique was utilized to fabricate the porous cylinder, comprised of a simple cubic lattice structure. This unique manufacturing technique, coupled with a periodic and scalable lattice structure, allows us to decouple the permeability from the porosity. Based on permeability measurements in this study, Darcy number (D a) of the porous cylinder varies in a range of 2. 44 × 1 0 − 5 − 4. 07 × 1 0 − 4 , while its porosity (Φ) is fixed at 0.7. Interesting features in the downstream wake are observed due to the mutual interaction between longitudinal and lateral bleeding flows. This mutual interplay prevents wake entrainment, relocating a recirculation bubble downstream and making its size smaller with increasing D a. When D a > 2. 0 × 1 0 − 4 , in particular, a steady wake is observed in the near-wake despite the low degree of porosity. Finally, we proposed a new approach to defining the length scale of the flow adjustment that is suitable for the porous square cylinder and comparable to the porous cylinder with a circular cross-section. [Display omitted] • A periodic and scalable porous structure fabricated by 3D printing technique. • Permeability served as an isolated control parameter instead of porosity. • Reduction in drag coefficient with increasing D a in a log-linear relation. • Mutual interplay between two bleedings significantly affects downstream wake structure. • A new approach to defining length scale of the downstream flow adjustment proposed. [ABSTRACT FROM AUTHOR]

In this paper, we explored the impact of surface roughness on ultra-thin dielectrics characteristics. Simple cubic lattice is used as a base for depositing spheres of radius R, designed as a thin film, as the base for simulation of defects generation and deposition. Continuum percolation model, used to generate a rough surface in order to analyse how roughness and temperature influence the conductivity, consists of randomly placed spheres of either one or some distribution of radii, which then form a thin, rough layer in question. It is shown that when the temperature of the substrate is increased material defects tend to form within the film contributing to a faster cluster growth and greater film conductivity. The influence of temperature on average cluster size for homogeneous distribution of defects is assumed according to the applied temperature (depending on the free carrier's diffusion length) is researched. It is evident from the performed simulations that temperature has an important influence on thin film conductivity in several different ways; it can contribute to the increase of the number of spanning clusters or the average size of clusters. This way, defects can either provide starting points for new spanning clusters or can encourage clusters merging. [ABSTRACT FROM AUTHOR]

Soft magnetic composites (SMCs) are a promising alternative to laminated steel in many electrical engineering applications. This is largely owing to their low level of eddy current (EC) losses. The electromagnetic behavior of SMC in electromagnetic devices cannot be easily predicted using standard numerical techniques, such as the finite-element (FE) method, mostly due to the computational cost required to model the material microstructure. Another difficulty lies in the high contrast between matrix and inclusion properties. In this paper, we propose a homogenization strategy to estimate EC losses in SMC. It is based on the calculation of a homogenized complex magnetic permeability. The imaginary part reflects the EC losses, while the real part describes the standard magnetic permeability. The complex permeability approach is applied to SMC with fiber or spherical inclusions. EC losses obtained from the complex permeability approach are compared with FE calculations, with very satisfying agreement. [ABSTRACT FROM PUBLISHER]

This paper presents a genetic algorithm applied to the protein structure prediction in a hydrophobic-polar model on a cubic lattice. The proposed genetic algorithm is extended with crowding, clustering, repair, local search and opposition-based mechanisms. The crowding is responsible for maintaining the good solutions to the end of the evolutionary process while the clustering is used to divide a whole population into a number of subpopulations that can locate different good solutions. The repair mechanism transforms infeasible solutions to feasible solutions that do not occupy the lattice point for more than one monomer. In order to improve convergence speed the algorithm uses local search. This mechanism improves the quality of conformations with the local movement of one or two consecutive monomers through the entire conformation. The opposition-based mechanism is introduced to transform conformations to the opposite direction. In this way the algorithm easily improves good solutions on both sides of the sequence. The proposed algorithm was tested on a number of well-known hydrophobic-polar sequences. The obtained results show that the mechanisms employed improve the algorithm's performance and that our algorithm is superior to other state-of-the-art evolutionary and swarm algorithms. [ABSTRACT FROM AUTHOR]

Although it is beyond doubt that mechanical stimulation is crucial to maintain bone mass, its role in preserving bone architecture is much less clear. Commonly, it is assumed that mechanics helps to conserve the trabecular network since an 'accidental' thinning of a trabecula due to a resorption event would result in a local increase of load, thereby activating bone deposition there. However, considering that the thin trabecula is part of a network, it is not evident that load concentration happens locally on the weakened trabecula. The aim of this work was to clarify whether mechanical load has a protective role for preserving the trabecular network during remodeling. Trabecular bone is made dynamic by a remodeling algorithm, which results in a thickening/thinning of trabeculae with high/low strain energy density. Our simulations show that larger deviations from a regular cubic lattice result in a greater loss of trabeculae. Around lost trabeculae, the remaining trabeculae are on average thinner. More generally, thin trabeculae are more likely to have thin trabeculae in their neighborhood. The plausible consideration that a thin trabecula concentrates a higher amount of strain energy within itself is therefore only true when considering a single isolated trabecula. Mechano-regulated remodeling within a network-like architecture leads to local concentrations of thin trabeculae. [ABSTRACT FROM AUTHOR]

Silver selenide, a phase-changing chalcogenide material, is prepared using electro deposition method for various molarities. X-ray diffraction studies show the cubic lattice of the material. The micro-structural properties such as grain size, strain, dislocation density, and texture coefficient are examined. The lattice constant is calculated using Nelson-Relay function. Morphological studies are done and uniform distributions of grains are observed. High purities of thin films are confirmed by energy dispersive X-ray analysis. The band gap is calculated using UV-vis spectroscopy and photoluminescence technique, and hence, the Stokes's effect is observed in silver selenide thin films. It is the first time that the lattice constant and the Urbach energy for various molarities in the case of silver selenide thin films are reported. [ABSTRACT FROM AUTHOR]

Abstract: Two vectors in are called twins if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers M with the property that each integral vector with length has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible. [Copyright &y& Elsevier]

Abstract: The use of porous structures with high external surface area represents an important breakthrough in several industrial applications. Foam structures have received an increasing scientific and industrial interest since the last decade. Knowledge of pressure drop induced by these foam structures is thus essential for successful design and operation of high performance industrial systems. In this context, an analytical investigation was conducted for the determination of the permeability and the inertial coefficient in foams. The theoretical model is based on modified cubic lattice, which allows to take into account the presence of matter at the junction of struts. The existing model developed in the literature is then modified to incorporate this geometrical approach for determining the tortuosity of the foam. Finally, the permeability and inertial coefficient analysis are performed in order to derive the pressure drop on foams. The modeling procedure is based only on physical principles and geometrical considerations with no adjustable parameters in order to reconcile the theoretical work with the experimental data of the literature. Finally, this model is validated for two marginal cases (i.e. ‘slim’ and ‘fat’ foams). [Copyright &y& Elsevier]

Abstract: A Prussian-blue analogue Mn3[Cr(CN)6]2·12H2O has been prepared and characterized by single crystal X-ray analysis and magnetic measurements. The complex has a disordered face-centered cubic lattice and shows two magnetic transition temperatures of 74 and 61K. [Copyright &y& Elsevier]

An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete knot is the digitalized representation of a knot, which is composed of the minimum number of constant orthogonal straight-line segments and is represented by the knot-number notation. The proposed algorithm for obtaining minimal discrete knots tries to reduce the number of straight-line segments of a given discrete knot preserving its shape. This algorithm is based on two fundamental transformations: U and L. In order to prove the efficiency and rapidity of the algorithm, a great variety of examples of discrete knots are presented: complete families of discrete knots at different orders; random discrete knots; examples of non-trivial and unsolved discrete knots. [ABSTRACT FROM AUTHOR]

Lattice stick number sL(K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. In this paper, we give lattice stick numbers of small knots such as 31 and 41. More precisely we prove that sL(31) = 12 and sL(K) ≥ 14 for any other non-trivial knot K. [ABSTRACT FROM AUTHOR]

A method for generating families of particular 3D curves which differs from random self-avoiding walks is presented. This method is based on a chain code called knot numbers. The knot-number notation describes discrete knots. A discrete knot is the digitalized representation of a knot and is composed of constant orthogonal straight-line segments. The orthogonal direction changes of the contiguous straight-line segments of the knot define the chain elements. There are only five possible orthogonal direction changes for representing any discrete knot. Thus, the chain elements are considered as one base-five integer number (knot number). In this manner, we obtain a unique knot descriptor. This description is invariant under translation, rotation, starting point, and coding direction of the discrete knot. By evaluating all possible combinations of chain elements of curves and considering some restrictions, we obtain interesting families of curves which allow us to find the smallest nontrivial knot. Finally, we present a modest attempt at building the table of minimal discrete knots. [ABSTRACT FROM AUTHOR]

Let e1,…,em be m different symbols, let r1⩾⋯⩾rm be positive integers, and let n=∑i=1mri. The combinohedron, denoted by C(r1,…,rm), is the loopless graph whose vertices are the n-tuples in which the symbol ei appears exactly ri times, and where an edge joins two vertices if and only if one can be transformed into the other by interchanging two adjacent entries. The graph known as permutohedron is a particular case of the combinohedron. Here, we extend to the combinohedron some results on embeddability of the permutohedron. [Copyright &y& Elsevier]

We review mosaics T (tilings of Euclidean plane by regular polygons) with respect to possible embedding, isometric up to a scale, of their skeletons or the skeletons of their duals T*, into some cubic lattice Zn. The main result of this paper is the classification, given in Table 1, of all 58 such mosaics among all 165 mosaics of the catalog, given in Chavey (Comput. Math. Appl. 17 (1989) 147) and including all main classifications of mosaics. [Copyright &y& Elsevier]

The ferromagnetic spin-1 Ashkin-Teller model in three dimensions is investigated by the application of the mean field approximation from the Bogoliubov inequality. Phase diagrams, magnetizations and free energy curves are presented. The states presented by the system are classified according to the critical values in the free energy of the model. Phase transitions for the stable, metastable and unstable states are observed. • The mean field approximation is applied to the spin-1 Ashkin-Teller model. • Stable, metastable and unstable states are investigated. • First order and continuous phase transition lines are obtained. • Magnetization curves are presented for a better illustration of the phase transitions. [ABSTRACT FROM AUTHOR]

We have derived rigorously the generating function for the number of pyramid-like polyhedra g(a,b,c) on the cubic lattice with width a, depth b, and height c. [ABSTRACT FROM AUTHOR]

We study the quantum phase transition in a three-dimensional network of Josephson-coupled junctions including the charging-energy effects in the presence of external magnetic flux f=Φ/Φ0. To capture the effects of quantum and spatial fluctuations we go beyond the mean-field level description and derive an effective quantum non-linear σ-model for the array Hamiltonian with a spherical closure relation. Subsequently, we examine the T=0 and finite temperature superconductor-paracoherent phase boundary as a function of various control parameters. [Copyright &y& Elsevier]