Computer package for microscopic spin Hamiltonian analysis of the 3d4 and 3d6 (spin S = 2) ions at orthorhombic and tetragonal symmetry sites

A FORTRAN package based on the microscopic spin Hamiltonian expressions derived by computer algebra (ALTRAN) for the high spin (S = 2) 3d⁴ and 3d⁶ ions with an orbital singlet ground state at orthorhombic (point groups: C_2v, D₂, D_2h) and tetragonal (point groups: C_4v, D₄, D_4h, D₂d) symmetry sites is presented. The spin—orbit (λ) and the spin—spin (ρ) coupling contributions up to the fourth-order perturbation theory are taken into account within the ⁵D approximation. The package enables efficient numerical calculations of the Zeeman electronic (Ze) parameters and the zero-field splitting (ZFS) parameters for the S = 2 3d⁴ and 3d⁶ ions. The following terms are included: λ and λ² for the Ze parameters, λ², λ³ λ⁴, ρ, ρ² and λ²ρ for the second-rank ZFS ones, and λ⁴, ρ² and λ²ρ for the fourth-rank ZFS ones. The program is applicable to all possible energy level schemes with a ground orbital singlet arising from the ⁵D multiplet due to orthorhombic or tetragonal symmetry crystal fields. The input parameters are λ, ρ, the energy levels Δ_j (j = 1, 2, 3, 4) and the mixing coefficient s, which can be obtained from other spectroscopic data. The ZFS parameters output in the extended Stevens notation B_k^q and b_k^q, as well as the conventional notation (D, E; a, F, K), is provided.     

The computation of fourth virial coefficients for pairwise-additive spherically symmetric interaction potentials

A method is given for computing the fourth virial coefficient D(T) for a pairwise additive spherically symmetric interaction potential. Taking one of the four interacting particles as origin and using the appropriate co-ordinate transformations in the usual way the ninefold integrals defining D(T) are reduced to sixfold integrals which are then formally reduced to triple integrals by expanding out those Ursell-Mayer functions in the integrals not involving the origin particle as infinite series in Legendre polynomials P_S(cos?), where ? is the angle between the radius vectors of the interacting particles. The coefficients in these expansions are integrals of highly oscillatory functions, especially for large s, and are evaluated using the Chebyshev polynomial expansion for the Ursell-Mayer functions, thus making explicit use of the oscillatory behaviour of P_S(cos?). The triple integrals are evaluated using a nonproduct integration formula of the seventh degree employed earlier in the computation of the thirdh virial coefficient. The values of D(T) computed by the present method have the same qualitative behaviour as the literature values but appear to be more accurate, particularly at lower temperatures.     

A Distributed Critical Section Protocol for General Topology

We present a new critical section protocol designed for distributed systems with general topologies, where the physical layer is implemented as point-to-point physical links in contrast to shared access physical media. The protocol operates correctly for any topology; however, its time performance is topology dependent. The distributed system can be modeled by a graph G(V, E), where V denotes the set of processors and E is the set of bidirectional communication links. We use n to denote |V|; D(G) is the diameter of G, T(G) is the spanning tree of G, and D(T) is the diameter of T(G). An important measure of the performance of the protocol is the amount of traffic caused by its operation. Let message-hop be the amount of traffic generated by a single message between two adjacent nodes. The proposed protocol generates network traffic of only 3*(n − 1) ∈ Θ(n) [message-hops] per critical section access for any topology which is less than other existing fully distributed protocols. A lower bound on traffic for a single critical section access for a fully distributed protocol is shown to be 2*(n − 1) [message-hops]. Some previously published algorithms generate Θ(n²) [message-hops] of network traffic for some topologies. Another important measure of the performance of the protocol is the cs-access time. It is the time required to access the critical section in the absence of other requests; and it depends on the topology. The high cs-access time performance is achieved by taking a novel approach of distributing the communication and parts of computation functions of the protocol and exploiting the physical topology. For a constant size message, the time to traverse an edge, including the message communication software processing in the source and destination nodes, is called message-hop-time and it is denoted by t_h. For a general graph G (with spanning tree T) the new protocol has the cs-access time performance Θ(max(D(T), max(deg (v_i)))) [t_h], where deg(v_i) is computed in T. For the graphs where G has D(G) ∈ Θ(log₂n) and max(deg(v_i)) in G is O(log₂n), the cs-access time performance is Θ(log₂n) [t_h]. For the class of graphs where G has D(G) ∈ Θ(n), the cs-access time performance is Θ(n) [t_h]. For the Star graphs the cs-access time performance is Θ(n) [t_h]. The worst case time performance occurs for linear and Star graphs. The proposed protocol has a better network traffic performance and (depending on the topology) a better or equal cs-access time performance than previously published fully distributed protocols. The protocol keeps the clock bounded in well-designed systems using a distributed predictive "clock squashing" mechanism.     

Diameter variability of cycles and tori

The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter of a hypercube, Discrete Applied Mathematics 37/38 (1992) 265-274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D^-k(G) is the least number of edges whose addition to G decreases the diameter by (at least) k, D⁺⁰(G) is the maximum number of edges whose deletion from G does not change the diameter, and D^+k(G) is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D^-k(C_m), D^-1(T_m,n), D^-2(T_m,n), D⁺¹(T_m,n), and a lower bound for D⁺⁰(T_m,n) where C_m be a cycle with m vertices, T_m,n be a torus of size m by n.     

Improved approximation of the largest common subtree of two unordered trees of bounded height

We present a polynomial time 1.5h-approximation algorithm for the problem of finding the largest common subtree between two rooted, labeled, and unordered trees of height at most h, where a tree S is called a subtree of a tree T if S is obtained from T by deletion of some nodes in T. This result improves the previous 2h-approximation algorithm.     

Algorithms for multicommodity flows in planar graphs

This paper gives efficient algorithms for the muiticommodity flow problem for two classes C₁₂ and C₀₁ of planar undirected graphs. Every graph in C₁₂ has two face boundaries B₁ and B₂ such that each of the source-sink pairs lies on B₁ or B₂. On the other hand, every graph inC ₀₁ has a face boundaryB ₁ such that some of the source-sink pairs lie onB ₁ and all the other pairs share a common sink lying onB ₁. The algorithms run inO(kn +nT(n)) time if a graph hasn vertices andk source-sink pairs andT(n) is the time required for finding the single-source shortest paths in a planar graph ofn vertices.     

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system (A,B,C,D) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair (A,B) is controllable or (2) matrix pair (A,B) is reachable and (A,B,C,D) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.     

Further results on structural assignment of linear systems via sensor selection

The problem of assigning structural properties of a linear system through sensor selection is, for a given pair (A,B), to find an output pair (C,D) such that the resulting system (A,B,C,D) has the pre-specified structural properties, such as the finite and infinite zero structures and the invertibility properties. In this paper, by introducing the notion of infinite zero assignable sets for the pair (A,B), we establish necessary and sufficient conditions for the assignability of a given set of infinite zeros and a set of structural properties which includes the left invertibility property. In establishing these conditions, we develop a numerical algorithm for the construction of the required (C,D).     

Recovering the initial state of an infinite-dimensional system using observers

Let A be the generator of a strongly continuous semigroup T on the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X, not necessarily admissible). We consider the problem of estimating the initial state z₀∈D(A) (with respect to the norm of X) from the output function y(t)=CT_tz₀, given for all t in a bounded interval [0,τ]. We introduce the concepts of estimatability and backward estimatability for (A,C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z₀ from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell’s principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.     

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form f (χ + h) = E_α (h^αD_χ^α)f(χ), 0 α ≤ 1, where E_α is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.     

Geometric pattern matching for point sets in the plane under similarity transformations

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ>0, find a similarity transformation T (translation, rotation and scale) such that h(T(P),Q)<δ, where h(⋅,⋅) is the directional Hausdorff distance with L_∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L_∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O(mn²logn) time, where m and n are the numbers of points in P and Q respectively. If the L_∞ distance between every pair of points in Q is at least cδ, for some c, 0<c<1, we present a randomized approximate solution with expected runtime O(n²c−4ε−8log⁴mn), where ε>0 controls the approximation. Our approximation is on the size of the subset, B⊆P, such that h(T(B),Q)<δ and |B|>(1−ε)|P| with high probability.     

A comparison of time series similarity measures for classification and change detection of ecosystem dynamics

Time series of remote sensing imagery or derived vegetation indices and biophysical products have been shown particularly useful to characterize land ecosystem dynamics. Various methods have been developed based on temporal trajectory analysis to characterize, classify and detect changes in ecosystem dynamics. Although time series similarity measures play an important role in these methods, a quantitative comparison of the similarity measures is lacking. The objective of this study was to provide an overview and quantitative comparison of the similarity measures in function of varying time series and ecosystem characteristics, such as amplitude, timing and noise effects. For this purpose, the performance was evaluated for the commonly used similarity measures (D), ranging from Manhattan (D_Man), Euclidean (D_E) and Mahalanobis (D_Mah) distance measures, to correlation (D_CC), Principal Component Analysis (PCA; D_PCA) and Fourier based (D_FFT,D_ξ,D_Fk) similarities. The quantitative comparison consists of a series of Monte-Carlo simulations based on subsets of global MODIS Normalized Difference Vegetation index (NDVI) and Enhanced Vegetation Index (EVI) and Leaf Area Index (LAI) data. Results of the simulations reveal four main groups of time series similarity measures with different sensitivities: (i) D_Man, D_E, D_PCA, D_Fk quantify the difference in time series values, (ii) D_Mah accounts for temporal correlation and non-stationarity of variance, (iii) D_CC measures the temporal correlation, and (iv) the Fourier based D_FFT and D_ξ show their specific sensitivity based on the selected Fourier components. The difference measures show relatively the highest sensitivity to amplitude effects, whereas the correlation based measures are highly sensitive to variations in timing and noise. The Fourier based measures, finally, depend highly on the signal to noise ratio and the balance between amplitude and phase dominance. The heterogeneity in sensitivity of each D stresses the importance of (i) understanding the time series characteristics before applying any classification of change detection approach and (ii) defining the variability one wants to identify/account for. This requires an understanding of the ecosystem dynamics and time series characteristics related to the baseline, amplitude, timing, noise and variability of the ecosystem time series. This is also illustrated in the quantitative comparison, where the different sensitivities of D for the NDVI, EVI, and LAI data relate specifically to the temporal characteristics of each data set. Additionally, the effect of noise and intra- and interclass variability is demonstrated in a case study based on land cover classification.     

A study on the cognitive of complexity and difficulty of chinese characters when reading and recognizing

The Index of cognitive Information Complexity (C_info) and the Index of cognitive Information Difficulty (D_info) are measured at the psychological level for the application of learning and design when people are reading and recognizing the characters; thus, how to effectively monitor the information complexity and difficulty for the purpose of educational learning and design becomes an essential issue. This study combines information theory and Fitts’ Law to propose the concept of Information Mass (M_info) and through the test of the C_info and D_info toward the M_info of Chinese characters, the subjects are surveyed for measuring the complexity and difficulty of 218 selected Chinese characters. The results indicate that there is no significant discrepancy in the C_info of genders. The result of regression analysis indicates that the correlation coefficient of logarithm value of M_info and C_info is 0.684 and RSQ equals to 0.468. On the other hand, the Paired-Samples T-test shows the discrepancy of the D_info on Chinese characters between and among different groups. Overall, the M_info threshold value to recognize the D_info on Chinese characters between male and female subjects reaches to 0.017; among it, the M_info threshold value for male and female high school students is 0.06; for male and female university students is 0.01. The average value of D_info between genders found in regression analysis indicates the correlation coefficient of M_info and D_info is 0.861 and RSQ is 0.741. From results of this study, subjects are shown to have closer relations between M_info and D_info than that of M_info and C_info. In the other words, the concept to apply M_info gives reasonable explanation to the difficulty of Chinese characters and the model of linear regression serves the function for the reference of educational learning and design.     

On simplifying assumptions of Runge-Kutta methods for index 2 differential algebraic problems

In this paper we show a result that ensures certain order for the local error Runge-Kutta methods for index 2 differential algebraic problems with the help of the simplifying conditionsB(p),C(q),D(r) andA ₁(s) for the differential component andB(p), C(q), andA ₂(s) for the algebraic component.     

Deriving failures models for nonuniform concurrency from structured operational semantics

Semantic models are studied for concurrent languages which arenonuniform in that they involve individual variables which store values, and possible actions of an agent depend on its current state. First, an operational modelO _L(O) based on afailures domain is defined from alabeled transition system L(T) which is in turn specified by a setT of rules for deriving transitions. A method is the introduced for deriving a denotational failures modelD _T fromT whenT fits a certain syntactical format, called theNonuniform Non-Blocking Copy-Free SOS format (NU-NB-CF-SOS format), which is based on the format due to De Simone with certain additional restrictions specific to the nonuniform, setting. BothO _L(O) andD _T are constructed by applying the methodology of metric semantics, and the equivalence betweenD _T andO _L(T) is established by showing that bothO _L(T) andD _T are fixed-points of a higher-order mapping, which has a unique fixed-point by Banach’s fixed-point theorem.     

Stereoelectronic interaction effects on the conformational properties of hydrogen peroxide and its analogues containing S and Se atoms: An ab initio,hybrid-DFT study and NBO analysis

Ab initio molecular orbital (MP2/6-311+G**//MP2/6-31G+G**) and hybrid-density functional theory (B3LYP/6-311+G**//MP2/6-311+G**) methods and NBO analysis were used to study the stereoelectronic interaction effects on the conformational properties of hydrogen peroxide (1), hydrogen disulfide (2) and hydrogen diselenide (3). The results showed that the Gibbs free energy difference (G_T − G_S) values at 298.15 K and 1 atm between the skew (S) and trans (T) conformations (ΔG_T–S) increase from compound 1 to compound 2 but decrease from compound 2 to compound 3. The C conformations of compounds 1–3 are less stable than their S and T conformations. Based on these results, the racemization processes of the axial symmetrical (C₂ symmetry) conformations of compounds 1–3 take place via their T conformations. Based on the optimized ground state geometries using the MP2/6-311+G** level of theory, the NBO analysis of donor–acceptor (bond–antibond) interactions revealed that the stabilization (resonance) energy associated with LP₂M2 → σ*_M3-H4 electronic delocalization for the S conformations of compounds 1–3 are 1.35, 5.94 and 4.68 kcal mol⁻¹, respectively. There is excellent agreement between the variations of the calculated ΔG_T–S and stabilization (resonance) energies associated with LP₂M2→σ*_M3-H4 electronic delocalization for the S conformations of compounds 1–3. The correlations between resonance energies, orbital integrals, dipole moments, bond orders, structural parameters and conformational behaviors of compounds 1–3 have been investigated. Test were made of complete basis set methods (CBS-QB3, CBS-4 and CBS-Q), the first two gave results essentially indistinguishable from those we used, but the CBS-Q results were in disagreement with experimental and other theoretical results.     

Embedding K-ary Complete Trees into Hypercubes

In this paper, dilated embedding and precise embedding of K-ary complete trees into hypercubes are studied. For dilated embedding, a nearly optimal algorithm is proposed which embeds a K-ary complete tree of height h, T_K(h), into an (h − 1)[log K] + [log (K + 2)]-dimensional hypercube with dilation Max{2, φ(K), φ(K + 2)}. φ(x) = min{λ: Σ^λ_i=0Cⁱ_d ≥ x and d = [log x]}. It is clear that [([log x] + 1)/2] ≤ φ(x) ≤ [log x], for x ≥ 3.) For precise embedding, we show a (K − 1)h + 1-dimensional hypercube is large enough to contain T_K(h) as its subgraph, K ≥ 3.     

Abstract Parametric Classes and Abstract Data Types defined by Classical and Constructive Logical Methods

We introduce a methodology to treat abstract data types (ADT), abstract parametric classes (APC) and subclasses, together with appropriate inheritance properties, by means of first order theories. The notion of a first order theory axiomatizing an ADT is based on the notion of isoinitial model and has been proposed by the authors in previous papers (Bertoni et al. (1979), Bertoni et al. (1983), Bertoni et al. (1984)). A theory formalizing an APC is seen, in this paper, as a theory T incompletely axiomatizing an ADT. Given a class C of ADT's, the class formalized by T can be seen (under suitable soundness conditions on T) as the class of the instances of T over C. An instantiation of T by an ADT I of C completes T into a T′ formalizing an ADT I′, which extends I and inherits the properties of the APC T.We use both classical and constructive methods in the following sense: on the one hand, the semantics is based on classical model theory; on the other hand, the soundness of a consistent axiomatization can be analyzed by purely syntactical methods, in terms of provability within suitable constructive systems.A theory T formalizing an APC (or an ADT) is not given by a list of axioms, but by a suitable "APC-expression", which explicitly or implicitly (but effectively ) defines the axioms of T. We have APC-expressions to define APC's, to extend already defined APC's and to instantiate APC's (into ADT's or subclasses). We allow also "recurrence APC-expressions". At the end of the paper we give some examples showing how the proposed methodology works.     

Biased Range Trees

A?data structure, called a biased range tree, is presented that preprocesses a set S of n points in ?² and a query distribution D for 2-sided orthogonal range counting queries (a.k.a. dominance counting queries). The expected query time for this data structure, when queries are drawn according to?D, matches, to within a constant factor, that of the optimal comparison tree for S and D. The memory and preprocessing requirements of the data structure are? O(nlog?n).