Sturm sequences or law of inertia of quadratic forms?

In obtaining the number of eigenvalues greater than a given constant λ₀ of the eigenvalue problem [A]{x} = λ[B]{x} with [A] and [B] real, symmetric, and [B] positive definite, one usually refers to the Sturm sequence established by the leading principal minors of [A–λ₀B], the proof of which is given basically when the associated special eigenvalue problem is tridiagonal. In this work, using the law of inertia of quadratic forms, it is shown that the number of eigenvalues of [A]{x}] = λ[B]{x} greater than a given constant λ₀ (not an eigenvalue) is the number of positive entries of the diagonal matrix [d] in the identity [A–λ₀B] = [u]^T[d][u] where [u] is the upper triangular matrix associated with Crout-Banachievicz type decomposition of [A –λ₀B], without the help of the separation theorem and the Sturm sequence.     

Structural controllability and matrix nets†

We establish structural controllability results for matrix pairs [A, B] where A = A ₀ + Σ μ _iA_i , B = B ₀ + σ μ _i, B_i , with the A_i , B_i , fixed, and the μ _i , free scalar parameters. The results characterize structural controllability in several ways, via tests involving the checking of the rank or the evaluation of the determinant of various constant matrices formed from the A_i , B_i . A number of the results used as intermediate results tests for a full rank property of matrix nets, i.e. tests that check if M = M₀+ Σμ_iM_i prescribed, μ _i variable, has full rank for almost all μ _i .     

The method of solution of equations with coefficients that contain measurement errors,using artificial neural network

This paper presents an algorithm for solving N-equations of N-unknowns. This algorithm allows to determine the solution in a situation where coefficients A _i in equations are burdened with measurement errors. For some values of A _i (where i = 1,…, N), there is no inverse function of input equations. In this case, it is impossible to determine the solution of equations of classical methods.     

Gleichheit von Produkt und Formalprodukt bei Intervallpolynomen

The paper gives necessary and sufficient conditions for the product of two interval polynomials being equal to their formally defined product. For this purpose criteria must be deduced for which intervalsA _i, B _j the equality(A ₁+...+A _m ) (B ₁ +...+B _n )=A ₁ B ₁ +A ₁ B ₂ +...+A _m B _n (relation of distributivity) holds.     

Periodic compensation of a class of decentralized systems with fixed modes

This paper first finds that the high frequency (ω) periodic controller suggested in Lee, Meerkov, and Runolfsson (1987) for fixed mode removal, and pole and zero placement, of a class of decentralized systems (A,B_i,C_i) all the strongly interconnected channels of which satisfy C_iB_i=0, causes large, O(ω), oscillations in the outputs. Next it proposes an alternative controller, a dual of the above, that achieves the compensation with oscillations of O(1) only.     

On relativizing auxiliary pushdown machines

We consider relativizing the constructions of Cook in [4] characterizing space-bounded auxiliary pushdown automata in terms of timebounded computers. LetS(n) ≥ logn be a measurable space bound. LetDT ^A[NT^A] be the class of setsS such that there exists a machineM such thatM with oracleA recognizes the setS andM is a deterministic [nondeterministic] oracle Turing machine acceptor that runs in time 2 ^cS(n) for some constantc. LetDB _i ^A [NB _i ^A ] be the class of setsS such that there exists a machineM such thatM with oracleA recognizes the setS andM is a deterministic [non-deterministic] oracle Turing machine acceptor with auxiliary pushdown that runs in spaceS(n) and never queries the oracle about strings longer than:S(n) ifi = 1, 2 ^cS(n) for some constantc, ifi = 2, + ∞ ifi = 3. Then we prove the following results: These contrast with Cook's (unrelativized) result:DT = NB = DB.     

A fully parallel method for the singular eigenvalue problem

In this paper, a fully parallel method for finding some or all finite eigenvalues of a real symmetric matrix pencil (A, B) is presented, where A is a symmetric tridiagonal matrix and B is a diagonal matrix with b₁ > 0 and b_i ≥ 0, i = 2,3,…,n. The method is based on the homotopy continuation with rank 2 perturbation. It is shown that there are exactly m disjoint, smooth homotopy paths connecting the trivial eigenvalues to the desired eigenvalues, where m is the number of finite eigenvalues of (A, B). It is also shown that the homotopy curves are monotonic and easy to follow.     

On the RAS-algorithm

Given a nonnegative real (m, n) matrixA and positive vectorsu, v, then the biproportional constrained matrix problem is to find a nonnegative (m, n) matrixB such thatB=diag (x) A diag (y) holds for some vectorsx ∈ ? ^m andy ∈ ? ⁿ and the row (column) sums ofB equalu _i (v _j )i=1,...,m(j=1,..., n). A solution procedure (called the RAS-method) was proposed by Bacharach [1] to solve this problem. The main disadvantage of this algorithm is, that round-off errors slow down the convergence. Here we present a modified RAS-method which together with several other improvements overcomes this disadvantage.     

Generalized L systems

This paper proposes the concept of generalized L systems, GL systems for short, which can describe asynchronized concurrent phenomena. We have proved that the GL systems are proper extensions of the traditional L systems. We have also defined a classification of GL systems and proved a sufficient and necessary condition for the equivalence of two subclasses of GL systems: two GPD0L (a class of deterministic GL systems) systems L[m ₁, m ₂, …, m _j ] and L[n ₁, n ₂, …, n _k ] are equivalent, iff k = j and there exists a common divisor g of all m _l and a common divisor h of all n _l such that ? i: m _i/g = n _i/h.     

A Model for Minimizing Active Processor Time

We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job J _i has an integer length ? _i as well as a set T _i of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t so long as it is “active” at t. The goal is to schedule all the jobs in the fewest number of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is non-zero). In other words, subject to ? _i units of each job i being scheduled in its feasible region and at each slot at most B jobs being scheduled, we are interested in minimizing the total time during which the machine is active. We present a linear time algorithm for the case where jobs are unit length and each T _i is a single interval, assuming that jobs are given in sorted order. For general T _i , we show that the problem is NP-complete even for B=3. However when B=2, we show that it can be efficiently solved. In addition, we consider a version of the problem where jobs have arbitrary lengths and can be preempted at any point in time. For general B, the problem can be solved by linear programming. For B=2, the problem amounts to finding a triangle-free 2-matching on a special graph. We extend the algorithm of Babenko et al. (Proceedings of the 16th Annual International Conference on Computing and Combinatorics, pp. 120–129, 2010) to handle our variant, and also to handle non-unit length jobs. This yields an \(O(\sqrt{L} m)\) time algorithm to solve the preemptive scheduling problem for B=2, where L=∑ _i ? _i . We also show that for B=2 and unit length jobs, the optimal non-preemptive schedule has active time at most 4/3 times that of the optimal preemptive schedule; this bound extends to several versions of the problem when jobs have arbitrary length.     

Reflectance characteristics of major land surfaces in slash‐and‐burn ecosystems in Laos

The in situ reflectance spectra in the 400–2500 nm wavelength region were obtained using a portable radiometer over a range of land surfaces including burnt fields, crop canopies, and fallow vegetation at different community ages in slash‐and‐burn ecosystems in Laos. Normalized difference spectral indices (NDSI[i,j] = [R_j ?R_i ]/[R_j +R_i ]) were derived using reflectance R_i and R_j at i and j nm wavelengths for a thorough combination (14 706 pairs) of 172 wavebands (10‐nm resolution). The separability of burnt fields from dry/senescent vegetation was highest at NDSI[1090, 2390], whereas it was highly discriminated from fallow and crop vegetation by NDSI[760, 1970]. NDSIs using 730–760 nm with 1970–1990 nm showed the largest differences between dry/senescent vegetation and fallow or crop vegetation. None of the NDSIs was useful in discriminating between fallow and crop vegetations or between slashed/senescent vegetation and crop residue/abandoned field. Community age and biomass of fallow vegetation could not be inferred directly from spectral information, since no NDSIs showed any significant differences among crop and fallow vegetation that had a large variability in the amount of green vegetation. Results would provide useful information for various applications of optical satellite sensor images especially in assessments of land use or post‐fire regeneration of vegetation.     

Liquid-liquid solubility and critical phenomena in binary mixtures

By assuming a proportionality exists between a difference in Boltzmann factors and a difference in compositions, the following expression is developed for the mole fraction order parameter in symmetric binary liquid mixtures $\text{Δx = [1 ?}\text{exp}\text{E}\text{RT}{}_{\mathrm{c}}\text{(l?T}{}_{\mathrm{c}}\text{/T)]}{}^{\mathrm{\beta }}$ In the critical limit this expression reduces to the form for simple scaling fx253-fig 1 By treating the original components A and B as if they transfer between the phases in the form of aggregates with the empirical formulas A_mB_1?m and B_n, in which 0<m≤Landn>0, coexistence curves in diverse systems become symmetric and are fitted by the above expressions. Beta exponent values obtained from these fits range in value from 0.312±0.002 to 0.333±0.003.     

Traceability of slash‐and‐burn land‐use history using optical satellite sensor imagery: a basis for chronosequential assessment of ecosystem carbon stock in Laos

This study examined the use of satellite sensor imagery for chronosequential assessment of land use and ecosystem carbon stock in slash‐and‐burn (S/B) regions of Laos. The segmentation approach was useful because the boundaries of S/B patches are subject to change due to natural or anthropogenic factors. Polygon‐based classification using six optical bands of Landsat Enhanced Thematic Mapper Plus (ETM+) imagery showed that S/B patches could be discriminated with high accuracy (0.98). Normalized difference spectral indices, NDSI[i, j] = [R_j ?R_i ]/[R_j +R_i ], using reflectances R_j and R_i at j and i nm wavelengths for S/B polygons during four consecutive years (1999–2002) showed that NDSI[2215, 830], NDSI[1650, 830] and NDSI[660, 830] ( = the normalized difference vegetation index, NDVI) values decreased significantly in S/B years compared to those under fallow conditions (by 0.21±0.04, 0.20±0.04 and 0.17±0.03, respectively). Only slight differences were found before and after the S/B year, regardless of fallow length or biomass estimated by the allometry method. Relating reflectance signatures directly to fallow biomass was unsuitable, but these NDSIs were also useful for distinguishing S/B patches. Land‐use history, including the community age of fallow vegetation, can be traced on a pixel basis using a superimposed set of segmented classified images.     

A fast algorithm for computing a longest common increasing subsequence

Let A=〈a₁,a₂,…,am〉 and B=〈b₁,b₂,…,bn〉 be two sequences, where each pair of elements in the sequences is comparable. A common increasing subsequence of A and B is a subsequence 〈ai₁=bj₁,ai₂=bj₂,…,ail=bjl〉, where i₁<i₂<?<il and j₁<j₂<?<jl, such that for all 1?k<l, we have aik<aik+1. A longest common increasing subsequence of A and B is a common increasing subsequence of the maximum length. This paper presents an algorithm for delivering a longest common increasing subsequence in O(mn) time and O(mn) space.     

Mappings of linear vector spaces induced by a transformation matrix and its applications†

The projections of a vector Ax along a certain orthonormal basis are chosen as the n components of a vector ρ. As the vector x sweeps the n–dimensional space, the locus of the point vector ρ is a closed surface, called the ρ locus. Some of the properties of the ρ locus are similar to those of the eigenvalues of the A matrix. These properties are exploited in the analysis of the unforced, linear time-invariant, dynamical system given by [xdot] = Ax and for the forced systems of the [xdot] = Ax + Bu, where u represents an impulse, step, ramp or sinusoidal input.     

Stability of linear infinite-dimensional systems revisited

The paper is devoted to a study of stability questions for linear infinite-dimensional discrete-time and continuous-time systems. The concepts of power stability and l ^p Instability for a linear discrete-time system x _k+1 = Ax _k (where x _k ε X, X is a Banach space, A is linear and bounded) are introduced and studied. Relationships between these concepts and the inequality r(A) < 1, where r(A) denotes the spectral radius of A, are also given. The discrete-time results are used for a simple derivation of some well-known properties of exponentially stable and L^p-stable linear continuous-time systems described by [xdot](t) = Ax(t) (A generates here a strongly continuous semigroup of linear and bounded operators on X). Some remarks on norms related to stable systems are also included.     

Lower bounds for the smallest singular value

Let A = (a_ij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σ_n, satisfies: σ_n ≥ min σ_ij | (i, j) ∈ E, where g_ij ≡ a_i + a_j − [(a_i − a_j)² + (r_i + c_i)(r_j + c_j)]^1/2/2 with a_i ≡ |a_ii| and r_i,c_i are the i^th deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σ_n ≥ min_i≠j σ_ij. (See [1].)     

On generalized language equations

A system of generalized language equations over an alphabet A is a set of n equations in n variables: X_i = G_i(X₁,..., X_n), i = 1,...,n, where the G_i are functions from [P(A^*)]ⁿ into P(A^*), i=1,..., n, P(A^*) denoting the set of all languages over A. Furthermore the G_i are expressible in terms of set-operations, concatenations, and stars which involve the variable X_i as well as certain mixed languages. In this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations. Furthermore we show that a solution is regular if all fixed languages are regular.     

Monodispersed water-in-oil emulsions prepared with semi-metal microfluidic EDGE systems

Monodispersed water-in-oil emulsions were prepared with EDGE (Edge based Droplet GEneration) systems, which generate many droplets simultaneously from one junction. The devices (with plateau height of 1.0 μm) were coated with Cu and CuNi having the same hydrophobicity but different surface roughness. Emulsification was performed by using water as dispersed phase and oils with different viscosities (hexadecane, decane, hexane and sunflower oil) as continuous phases; lecithin, polyglycerol polyricinoleate (PGPR) and span80 were used as emulsifiers. The roughness affected the emulsification behaviour significantly. The smoother Cu surface exhibited droplet formation over the entire length of the droplet formation unit, while the rougher CuNi surface showed non-uniform filling of the plateau and much lower droplet formation frequency. In spite of this different behaviour, monodispersed droplets (CV <10 %) were produced by both systems (with span80 and PGPR), with a size six times the plateau height (d _avg ≈ 6.0 μm). The droplet size decreased with increasing viscosity ratio and remained constant above some critical value. The emulsification process was stable over a wider range of pressures as previously found for silicon-based systems. The amount of PGPR influenced the pressure stability, but the system could be used effectively, while with lecithin and span80 the stable pressure range was very small. The pressure and viscosity stability of these semi-metal systems with rough surfaces show that the EDGE system has potential for practical applications, especially since overall productivity is not affected.