Embedding a complete binary tree into the supercube

Consider a complete binary tree with 2ⁿ − 1 nodes and a supercube with the same number of nodes. We present a new embedding method to map the complete binary tree into the supercube with dilation 1. Our simple mapping method is quite competitive with the previous result.     

Integration of polynomials over n-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space ⁿ. Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space ⁿ into a standard n-simplex in ⁿ, while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n − 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems.     

Manufacture and characterization of sol–gel V1−x−yWxSiyO2 films for uncooled thermal detectors

V_1−x−yW_xSi_yO₂ films for uncooled thermal detectors were coated on sodium-free glass slides with sol–gel process, followed by the calcination under a reducing atmosphere (Ar/H₂ 5%). The V_1−x−yW_xSi_yO₂ films as prepared inherit various phase transition temperatures ranging from 20 to 70 °C depending on the dopant concentrations and the fabrication conditions. Compared to the hysteresis loop of plain VO₂ films, a rather steep loop was obtained with the addition of tungsten components, while a relaxed hysteresis loop with the tight bandwidth was contributed by Si dopants. Furthermore, the films with switching temperature close to room temperature were fabricated to one-element bolometers to characterize their figures of merit. Results showed that the V_0.905W_0.02Si_0.075O₂ film presented a satisfactory responsivity of 2600 V/W and detectivity of 9 × 10⁶ cm  Hz^1/2/W with chopper frequencies ranging from 30 to 60 Hz at room temperature. It was proposed that with appropriate amount of silicon and tungsten dopants mixed in the VO₂, the film would characterize both a relaxed hysteresis loop and a fair TCR value, which effectively reduced the magnitude of noise equivalent power without compromising its performance in detectivity and responsivity.     

Distribution of black nodes at various levels in a linear quadtree

The distribution of black leaf nodes at each level of a linear quadtree is of significant interest in the context of estimation of time and space complexities of linear quadtree based algorithms. The maximum number of black nodes of a given level that can be fitted in a square grid of size 2ⁿ × 2ⁿ can readily be estimated from the ratio of areas. We show that the actual value of the maximum number of nodes of a level is much less than the maximum obtained from the ratio of the areas. This is due to the fact that the number of nodes possible at a level k, 0≤k≤n − 1, should consider the sum of areas occupied by the actual number of nodes present at levels k + 1, k + 2, …, n − 1.     

Extended Games–Chan algorithm for the 2-adic complexity of FCSR-sequences

Binary sequences generated by feedback shift registers with carry operation (FCSR) share many of the important properties enjoyed by sequences generated by linear feedback shift registers. We present an FCSR analog of the (extended) Games–Chan algorithm, which efficiently determines the linear complexity of a periodic binary sequence with period length T = 2ⁿ or pⁿ, where p is an odd prime and 2 is a primitive element modulo p². The algorithm to be presented yields an upper bound for the 2-adic complexity, an FCSR analog of the linear complexity, of a pⁿ-periodic binary sequence.     

Gas-sensing property and mechanism of CaxLa1−xFeO3 ceramics

LaFEO₃ and Ca_xLa_1−xFeO₃ ceramic powders have been prepared by the coprecipitation method from La(NO₃)₃, Fe(NO₃)₃ and Ca(NO₃)₂ aqueous solutions. The orthorhombic perovskite phases of LaFeO₃ and Ca_xLa_1−xFeO₃ are characterized by X-ray diffraction patterns. The sensors fabricated with those powders have high sensitivity to alcohol. Partial substitution of La³⁺ in LaFeO₃ with Ca²⁺ can enhance the sensitivity of the materials to reducing gases. The resistance of an LaFeO₃ sensor in air, vacuum and alcohol-containing air has been measured. Complex impedance spectroscopy has been used to try and analyse the gas-sensing mechanism. According to the experimental results, it can be deduced that the surface adsorptive and lattice oxygen govern the sensing properties of LaFeO₃ and Ca_xLa_1−xFeO₃ ceramics.     

Resolvability and the upper dimension of graphs

For an ordered set W = {w₁, w₂,…, w_k} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v | W) = (d(v, w₁), d(v, w₂),…, d(v, w_k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations. A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented.

A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). A resolving set S of G is a minimal resolving set if no proper subset of S is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim⁺(G). The resolving number res(G) of a connected graph G is the minimum k such that every k-set W of vertices of G is also a resolving set of G. Then 1 ≤ dim(G) ≤ dim⁺(G) ≤ res(G) ≤ n − 1 for every nontrivial connected graph G of order n. It is shown that dim⁺(G) = res(G) = n − 1 if and only if G = K_n, while dim⁺(G) = res(G) = 2 if and only if G is a path of order at least 4 or an odd cycle.

The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dim(G) = dim⁺(G) = a and res(G) = b. Also, for every positive integer N, there exists a connected graph G with res(G) − dim⁺(G) ≥ N and dim⁺(G) − dim(G) ≥ N.     

Trace detection of nitroaromatic compounds with layer-by-layer assembled {SBA/PSS}n/PDDA modified electrode

The {SBA/PSS}_n/PDDA films modified electrode was prepared by layer-by-layer (LBL) assembly with mesoporous SiO₂ (SBA), poly(sodium 4-styrene-sulfonate) (PSS) and poly(diallyldimethylammonium chloride) (PDDA) in this paper. SBA is a large pore-size mesoporous material with highly ordered hexagonally arranged mesochannels and high thermal stability etc. The electrochemical characteristics of the {SBA/PSS}_n/PDDA films have been studied by electrochemical impedance spectroscopy in 0.1 M KCl solution containing 5.0 mM Fe(CN)₆³⁻/Fe(CN)₆⁴⁻ at the formal potential of 0.230 V. The ultratrace nitroaromatic compounds (NACs) such as TNT, TNB, DNT and DNB were determined by differential pulse voltammetry (DPV) measurement. The sensitivities for NACs determination with {SBA/PSS}_n/PDDA modified electrode were dependent on the number of layers, pH and ionic strength of electrolyte, based on which a set of optimized conditions for film fabrication was inferred. The current responses were linear with NACs ranging from 10⁻⁹ to 10⁻⁷ mol/l. The results showed that the {SBA/PSS}_n/PDDA modified electrode established a new way for fast, simple and sensitive analysis of NACs.     

Moments of conjugacy classes of binary words

For each nonempty binary word w=c₁c₂c_q, where c_i{0,1}, the nonnegative integer ∑_i=1^q (q+1−i)c_i is called the moment of w and is denoted by M(w). Let [w] denote the conjugacy class of w. Define M([w])={M(u): u[w]}, N(w)={M(u)−M(w): u[w]} and δ(w)=max{M(u)−M(v): u,v[w]}. Using these objects, we obtain equivalent conditions for a binary word to be an -word (respectively, a power of an -word). For instance, we prove that the following statements are equivalent for any binary word w with |w|2: (a) w is an -word, (b) δ(w)=|w|−1, (c) w is a cyclic balanced primitive word, (d) M([w]) is a set of |w| consecutive positive integers, (e) N(w) is a set of |w| consecutive integers and 0N(w), (f) w is primitive and [w]St.     

A parallel residue-to-binary conversion algorithm without trial division

In recent years, the conversion of residue numbers to a binary integer has been intensively studied. The Chinese Remainder Theorem (CRT) is a solution to this conversion problem of a number to the Residue Number System with a general moduli set. This paper presents a new division-free conversion approach for the conversion of residue numbers to a binary integer. The algorithm differs from others employing a great number of division instructions by using shift instructions instead. These simple instructions keep the power consumption lower. This algorithm can also be implemented with a lookup table or upon a vector machine. Both make the conversion process efficient. This division-free algorithm employs the concept of Montgomery multiplication algorithm. There are two variations of Montgomery algorithm proposed, which are algorithms MMA and IMA. The algorithm MMA is to transform the input number into the output presentation of Montgomery algorithm. Algorithm IMA is therefore inverse the computation of Montgomery algorithm to obtain the multiplicand. These two algorithms are in the complexity of O(n), where n is log₂ q_i. q_i is a modulus. The proposed algorithm for converting the residues to a binary integer therefore runs on O(n × log m) times on O(m) processors. There are O(log m) iterations of O(n) complexity. Compared with the traditional conversion algorithm, the advantages of this proposed algorithm are not only in employing simpler operations but also in performing fewer iterations.     

Theoretical calculation of the phase diagram between one-dimensional long-period structures in the quasi binary sections: Pd3xRh3(1−x)V, Pt3xRh3(1−x)V, and Pt3VxTi(1−x)

The relative stabilities of L1₂, D0₂₂, D0₂₃, 21, and 3 structures in the Pd₃V, Pt₃V, Rh₃V, and Pt₃Ti compounds are investigated employing the Vienna ab initio simulation package. In the pseudobinary Pd_3xRh_3(1−x)V, Pt_3xRh_3(1−x)V, and Pt₃V_xTi_(1−x) alloys, the energy differences from L1₂ of D0₂₂, D0₂₃, 21, and 3 structures are assumed to be linear as function of the number of electrons per atom. At T=0K, the resulting energy diagram shows that the equilibrium between the limiting binary phases is the most stable state. At high temperature, the Gibbs energy curves are computed assuming a Bragg and Williams entropy of mixing in the pseudobinary sections. The D0₂₃ and 21 structures are stabilized in the pseudobinary Pd_3xRh_3(1−x)V, Pt_3xRh_3(1−x)V, and Pt₃V_xTi_(1−x) alloys. The phase diagram between the various structures is calculated in each pseudobinary section and compared with the experimental one.     

Boundary matching algorithm for connected component labelling using linear quadtrees

A computationally fast top-down recursive algorithm for connected component labelling using linear quadtrees is presented. The input data structure used is a linear quadtree representing only black leaf nodes. The boundary matching approach used ensures that at most two adjacencies of any black leaf node are considered. Neighbour searching is carried out within restricted subsets of the input quadtree. The time and space complexities of the algorithm are O(Bn) and O(B) respectively for a linear quadtree with B black leaves constructed from a binary array of size 2ⁿ × 2ⁿ. Simulations show the algorithm to be twice as fast as an existing algorithm that uses an identical input data structure. The top-down algorithm presented can also be used to efficiently generate a linear quadtree representing all nodes — ‘grey’, ‘black’ and ‘white’ — in preorder when given an input linear quadtree representing only ‘black’ leaf nodes. The boundary matching algorithm is computationally fast and has low static and dynamic storage costs, making it useful for applications where linear quadtrees are held in main memory.     

Constructing space-filling curves of compact connected manifolds

Let M be a compact connected (topological) manifold of finite- or infinite-dimension n. Let 0 r 1 be arbitrary but fixed. We construct in this paper a space-filling curve f from [0,1] onto M, under which M is the image of a compact set A of Hausdorff dimension r. Moreover, the restriction of f to A is one-to-one over the image of a dense subset provided that 0 r log|2ⁿ/log(2ⁿ + 2). The proof is based on the special case where M is the Hilbert cube [0,1]^ω.     

Size of ordered binary decision diagrams representing threshold functions

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function. In this paper, the size of ordered binary decision diagrams representing threshold functions is discussed. We consider two cases: the case when a variable ordering is given and the case when it is adaptively chosen. We show 1) O(2^n/2) upper bound for both cases, 2) Ω(2^n/2) lower bound for the former case and 3) Ω(n2^√n/2) lower bound for the latter case. We also show some relations between the variable ordering and the size of OBDDs representing threshold functions.     

A Hamiltonian, explicit algorithm with spectral accuracy for the ‘good’ Boussinesq system

We construct an explicit pseudo-spectral method for the numerical solution of the soliton-producing ‘good’ Boussinesq system w_t = u_xxx + u_x + (u²)_x, u_t = w_x. The new scheme preserves a discrete Poisson structure similar to that of the continuous system. The scheme is shown to converge with spectral spatial accuracy. A numerical illustration is given.     

Coupled phase diagram and thermodynamic analysis of the 18 binary systems formed among Li2Co3, K2Co3,Na2Co3, Lioh, Koh, Naoh, Li2So4, K2So4 and Na2So4

Available phase diagram and thermodynamic data on the eighteen binary systems containing Li⁺ , Na⁺, K⁺, S0₄⁼, CO₃⁼, and OH⁻ have been collected and critically assessed. A set of self-consistent thermodynamic parameters describing the free energies of the compounds, solid solutions, and liquid solutions in the binary systems has been formulated. The phase diagrams have been calculated, and estimated error limits of all diagrams are given.     

On canonical number systems

Let be such that d1,p_d=1,p₀2 and . We are proving in this note a new criterion for the pair to be a canonical number system. This enables us to prove that if p₂,…,p_d−1,∑_i=1^dp_i0 and p₀>2∑_i=1^d|p_i|, then is a canonical number system.     

On space-efficient algorithms for certain NP-complete problems

Some recent results claimed the existence of a class of algorithms for certain NP-complete problems, with running time O(n^{1g k} 2^n/2) and storage requirements O(k 2^n/k), for 2 kn. In this note we show that those results do not hold, implying that an algorithm with time O(n 2^n/2) and space O(2^n/4) is still the best-known solution for such class of NP-complete problems.     

Hydrolyzed colloid reaction (HCR) technique for phosphor powder preparation

Undoped and Eu³⁺, Ce³⁺ and Tb³⁺-doped YVO₄ YPO₄, LaPO₄ and YV_xP_1−xO₄ were prepared in H₂O by the recently introduced hydrolyzed colloid reaction (HCR) technique working at low temperature (<100°C) and atmospheric pressure. Two intermediate — partially hydrophobic — complex colloidal mixtures with metastable characteristics can transform into the stable orthovanadate–orthophosphate phase due to intensive hydrolysis. In contrast with the other low temperature reacting processes — like the sol-gel technique, which makes an amorphous structure — the HCR method can produce crystalline structures in nanometer size ranges. The reaction, morphology, incorporation of activators and different luminescent characteristics are surveyed in this letter-type paper selected from our previous results.