Some new sequence spaces derived by the domain of the triple band matrix

Let λ denote any of the classical spaces ?_∞,c,c₀, and ?_p of bounded, convergent, null, and absolutely p-summable sequences, respectively, and let λ(B) also be the domain of the triple band matrix B(r,s,t) in the sequence space λ, where 1<p<∞. The present paper is devoted to studying the sequence space λ(B). Furthermore, the β- and γ-duals of the space λ(B) are determined, the Schauder bases for the spaces c(B), c₀(B), and ?_p(B) are given, and some topological properties of the spaces c₀(B), ?₁(B), and ?_p(B) are examined. Finally, the classes (λ₁(B):λ₂) and (λ₁(B):λ₂(B)) of infinite matrices are characterized, where λ₁∈{?_∞,c,c₀,?_p,?₁} and λ₂∈{?_∞,c,c₀,?₁}.     

A B-spline Galerkin method for the Dirac equation

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y^″=−λ²y, that can also be written as a pair of first-order equations y^′=λz, z^′=−λy. Expanding both y(r) and z(r) in the Bk basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the Bk basis and z(r) in the dBk/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method (Bk,Bk±1) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

A combinatorial algorithm for max csp

We consider the problem max csp over multi-valued domains with variables ranging over sets of size s_i?s and constraints involving k_j?k variables. We study two algorithms with approximation ratios A and B, respectively, so we obtain a solution with approximation ratio max(A,B).The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s^1−k. For k=2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of , where s₁ and s₂ are the set sizes for the two variables.We then give a simple combinatorial algorithm which has approximation ratio B, with B>A/e. The bound is greater than s^1−k/e in general, and greater than s^1−k(1−(s−1)/2(k−1)) for s?k−1, thus close to the s^1−k linear programming bound for large k. For k=2, the bound is if s=2, 1/2(s−1) if s?3, and in general greater than the minimum of 1/4s₁+1/4s₂ over constraints with set sizes s₁ and s₂, thus within a factor of two of the linear programming bound.For the case of k=2 and s=2 we prove an integrality gap of . This shows that our analysis is tight for any method that uses the linear programming upper bound.     

A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: The case of Taihu Lake, China

Accurate estimation of phytoplankton chlorophyll a (Chla) concentration from remotely sensed data is particularly challenging in turbid, productive waters. The objectives of this study are to validate the applicability of a semi-analytical three-band algorithm in estimating Chla concentration in the highly turbid, widely variable waters of Taihu Lake, China, and to improve the algorithm using a proposed four-band algorithm. The improved algorithm is expressed as [Rrs(λ₁)^− 1 − Rrs(λ₂)^− 1][Rrs(λ₄)^− 1 − Rrs(λ₃)^− 1]^− 1. The two semi-analytical algorithms are calibrated and evaluated against two independent datasets collected from 2007 and 2005 in Taihu Lake. Strong linear relationships were established between measured Chla concentration and that derived from the three-band algorithm of [Rrs^− 1(660) − Rrs^− 1(692)]Rrs(740) and the four-band algorithm of [Rrs^− 1(662) − Rrs^− 1(693)][Rrs^− 1(740) − Rrs^− 1(705)]^− 1. The first algorithm accounts for 87% and 80% variation in Chla concentration in the 2007 and 2005 datasets, respectively. The second algorithm accounts for 97% of variability in Chla concentration for the 2007 dataset and 87% of variation in the 2005 dataset. The three-band algorithm has a mean relative error (MRE) of 43.9% and 34.7% for the 2007 and 2005 datasets. The corresponding figures for the four-band algorithm are 26.7% and 28.4%. This study demonstrates the potential of the four-band model in estimating Chla even in highly turbid case 2 waters.     

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.     

Bernstein-type operators which preserve polynomials

In this paper we present the sequence of linear Bernstein-type operators defined for f∈C[0,1] by B_n(f°τ⁻¹)°τ, B_n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [0,1], such that τ(0)=0, τ(1)=1 and τ^′(x)>0 for x∈[0,1]. We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B_n. We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B_n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

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.     

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.     

Positivity properties in pencils of matrices†

If A and B are positive definite self-adjoint matrices, then all the matrices P_i, occurring in the expansion of an exterior power Λ^k(A+ λB) = P ₀ + P ₁λ + … + P _kλ^k are positive definite.     

New models for retrieving and partitioning the colored dissolved organic matter in the global ocean: Implications for remote sensing

Despite the importance of CDOM to upper ocean biogeochemical processes and optics, our current understanding of its spatial and temporal distributions and the factors controlling these distributions is very limited. This eventually prevents an understanding of its relationship to the pool of dissolved organic carbon in coastal and open oceans. This work aims to present a new approach for accurate modeling of absorption spectra of CDOM (a_cdom) and deriving information on its composition in global ocean waters. The modeling approach uses measurements (in situ) of the remote sensing reflectances at two wavelengths (denoted ₄₄₃⁵⁵⁵R_rs) to estimate a_cdom(350) and a_cdom(412), applies them to determine two spectral slopes of an exponential curve fit (S) and a hyperbolic curve fit (γ), derives an appropriate parameter (γ^o) for grading the CDOM compositional changes from a_cdom (350) and γ, and finally employs a_cdom(350), S, and γ^o in a modified exponential model to describe a_cdom(λ) as a function of wavelength. The robustness of this model was rigorously tested on three independent datasets, such as NOMAD in situ data, NOMAD SeaWiFS match-ups data and IOCCG simulated data (all of them contain a_cdom(λ) and R_rs(λ)), which represent a variety of waters within coastal and offshore regions around the world. Accuracy of the retrievals found with the new models was generally excellent, with MRE (mean relative error) and RMSE (root mean square error) of − 5.64-3.55% and 0.203-0.318 for the NOMAD in situ datasets, and − 5.63 to −0.98% and 0.136-0.241 for the NOMAD satellite datasets respectively (for λ₄₁₂ to λ₆₇₀). When used with SeaWiFS images collected over the regional and global waters, the new model showed the highest surface abundances of CDOM within the subpolar gyres and continental shelves dominated by terrestrial inputs (and perhaps local production) of colored dissolved materials, and the lowest surface abundances of CDOM in the central subtropical gyres and the open oceans presumably regulated by photobleaching phenomenon, bacterial activity and local processes. Significant interseasonal and interannual seasonal changes in the terrestrially-derived CDOM distributions were noticed from these new products that closely corresponded with the global mean runoff/river discharge induced by climate change/warming scenarios.     

Estimating phytoplankton biomass in coastal waters of Alaska using airborne remote sensing

Empirical airborne remote-sensing relationships were examined to estimate chlorophyll a concentration in the first optical depth (chl_FOD) of coastal waters of Afgonak/Kodiak Islands during July-August 2002. Band-ratio and spectral-curvature models were tested using satellite remote-sensing reflectance (R_rs(λ)) measurements. Additional shipboard and airborne R_rs(λ) data were also analysed to evaluate consistency of proposed chl_FOD-R_rs(λ) relationships. Validation of chlorophyll algorithms was performed using data collected in the northern-part of the Gulf of Alaska and Bering Sea during 1996, 2002, and 2003 cruises. Likewise, oceanographic conditions during the surveys were typified to interpret variability of chl_FOD fields. The SeaWiFS band-ratio algorithm OC2d was the most sensitive R_rs combination (R_rs(509)/R_rs(553)) to detect chl_FOD variability. Conversely, OC2a (R_rs(412)/R_rs(553)) had the lowest performance to derive chl_FOD values. No valid statistical regressions were established for spectral-curvature relationships in the blue spectrum (< 500 nm). Fertile waters (> 5 mg m^− 3) were preferentially located over shallow banks (∼50 m) and at the entrance of the bays. The approach used in this study to derive chl_FOD values could be universal for Alaskan coastal waters. However, chl_FOD-R_rs(λ) relationships must be calibrated locally for a given season.     

Connectivity-preserving transformations of binary images

A binary image I is B_a, W_b-connected, where a, b ∈ {4, 8}, if its foreground is a-connected and its background is b-connected. We consider a local modification of a B_a, w_b-connected image I in which a black pixel can be interchanged with an adjacent white pixel provided that this preserves the connectivity of both the foreground and the background of I. We have shown that for any (a, b) ∈ {(4, 8), (8, 4), (8, 8)}, any two B_a, w_b-connected images I and J each with n black pixels differ by a sequence of Θ(n²) interchanges. We have also shown that any two B₄, W₄-connected images I and J each with n black pixels differ by a sequence of O(n⁴) interchanges.     

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R₁×R₂×?×Rk, where each Ri is a closed interval on the real line of the form [ai,ai+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph.It is known that for a graph G, . Recently it has been shown that for a graph G, cub(G)?4(Δ+1)lnn, where n and Δ are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G=(A∪B,E) with |A|=n₁, |B|=n₂, n₁?n₂, and Δ^′=min{ΔA,ΔB}, where ΔA=maxa∈Ad(a) and ΔB=maxb∈Bd(b), d(a) and d(b) being the degree of a and b in G, respectively, cub(G)?2(Δ^′+2)⌈lnn₂⌉. We also give an efficient randomized algorithm to construct the cube representation of G in 3(Δ^′+2)⌈lnn₂⌉ dimensions. The reader may note that in general Δ^′ can be much smaller than Δ.     

Restricted arc-connectivity of digraphs

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks.The restricted edge-connectivity λ^′(G) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. A connected graph G is called λ^′-connected, if λ^′(G) exists.In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195-199] have shown that each connected graph G of order n?4, except a star, is λ^′-connected and satisfies λ^′(G)?ξ(G), where ξ(G) is the minimum edge-degree of G.If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D−S has a non-trivial strong component D₁ such that D−V(D₁) contains an arc. The restricted arc-connectivity λ^′(D) is the minimum cardinality over all restricted arc-cuts S.We observe that the recognition problem, whether λ^′(D) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Some new results for system decoupling and pole assignment problems

In a related article, we derived a canonical decomposition of the right invertible system {C,A,B} and applied this canonical decomposition to study the Smith form of the matrix pencil and findout the finite zeros and infinite zeros of P(s), the range of the ranks of P(s) for s∈C, and the controllability of the right invertible system.In this paper, we will apply this canonical decomposition of the right invertible system {C,A,B} to deduce the triangular decouple upon to row permutation, provide some new results of the row-by-row decoupling, and associated pole assignment problems.