The k-fundamental group of a closed k-surface

In this paper, we deal with the problem of computing the digital fundamental group of a closed k-surface by using various properties of both a (simple) closed k-surface and a digital covering map. To be specific, let be a simple closed k_i-curve with l_i elements in Z^n_i, i∈{1,2}. Then, the Cartesian product is not always a closed k-surface with some k-adjacency of Z^n₁+n₂. Thus, we provide a condition for to be a (simple) closed k-surface with some k-adjacency depending on the k_i-adjacency, i∈{1,2}. Besides, even if is not a closed k-surface, we show that the k-fundamental group of can be calculated by both a k-homotopic thinning and a strong k-deformation retract.     

Relationship between the eccentric connectivity index and Zagreb indices

For a (molecular) graph, the first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V(G), then the eccentric connectivity index of G, ξ^C(G), is defined as, ∑_vi∈V(G)d_ie_i, where d_i is the degree of a vertex v_i and e_i is its eccentricity. In this report we compare the eccentric connectivity index (ξ^C) and the Zagreb indices (M₁ and M₂) for chemical trees. Moreover, we compare the eccentric connectivity index (ξ^C) and the first Zagreb index (M₁) for molecular graphs.     

Global wire routing in two-dimensional arrays

We examine the problem of routing wires of a VLSI chip, where the pins to be connected are arranged in a regular rectangular array. We obtain tight bounds for the worst-case “channel-width” needed to route ann×n array, and develop provably good heuristics for the general case. Single-turn routings are proved to be near-optimal in the worst-case. A central result of our paper is a “rounding algorithm” for obtaining integral approximations to solutions of linear equations. Given a matrix A and a real vector x, then we can find an integral x such that for alli, ¦x _i -x _i ¦ <1 and (Ax) _i -(Ax) _i <Δ. Our error bound Δ is defined in terms of sign-segregated column sums of A: $$\Delta = \mathop {\max }\limits_j \left( {\max \left\{ {\sum\limits_{i:a_{ij} > 0} {a_{ij} ,} \sum\limits_{i:a_{ij}< 0} { - a_{ij} } } \right\}} \right).$$      

Maximum number of edges joining vertices on a cube

Let E_d(n) be the number of edges joining vertices from a set of n vertices on a d-dimensional cube, maximized over all such sets. We show that E_d(n)=∑_i=0^r−1(l_i/2+i)2^l_i, where r and l₀>l₁>?>l_r−1 are nonnegative integers defined by n=∑_i=0^r−12^l_i.     

Efficient algorithm for transversal of disjoint convex polygons

Given a set S of n disjoint convex polygons {P_i∣1?i?n} in a plane, each with k_i vertices, the transversal problem is to determine whether there exists a straight line that goes through every polygon in S. We show that the transversal problem can be solved in O(N+nlogn) time, where N=∑_i=1ⁿk_i is the total number of vertices of the polygons.     

Transversal of disjoint convex polygons

Given a set S of n disjoint convex polygons {P_i∣1?i?n} in a plane, each with k_i vertices, the transversal problem is to find, if there exists one, a straight line that goes through every polygon in S. We show that the transversal problem can be solved in O(N+nlogn) time, where N=∑_i=1ⁿk_i is the total number of vertices of the polygons.     

Online Unweighted Knapsack Problem with Removal Cost

In this paper, we study the online unweighted knapsack problem with removal cost. The input is a sequence of items u ₁,u ₂,…,u _n , each of which has a size and a value, where the value of each item is assumed to be equal to the size. Given the ith item u _i , we either put u _i into the knapsack or reject it with no cost. When u _i is put into the knapsack, some items in the knapsack are removed with removal cost if the sum of the size of u _i and the total size in the current knapsack exceeds the capacity of the knapsack. Here the removal cost means a cancellation charge or disposal fee. Our goal is to maximize the profit, i.e., the sum of the values of items in the last knapsack minus the total removal cost occurred. In this paper, we consider two kinds of removal cost: unit and proportional cost. For both models, we provide their competitive ratios. Namely, we construct optimal online algorithms and prove that they are best possible.     

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].)     

A game-theoretic model of TV show “The Voice”

This paper proposes a game-theoretic model of the two-player best-choice problem with incomplete information. The players (experts) choose between objects by observing their quality in the form of two components forming a sequence of random variables (x_i, y_i), i = 1,..., n. By assumption, the first quality component x_i is known to the players and the second one y_i is hidden. A player accepts or declines an object based on the first quality component only. A player with the maximal sum of the components becomes the winner in the game. The optimal strategies are derived in the cases of independent and correlated quality components.     

Global roundings of sequences

For a given sequence a=(a₁,…,an) of numbers, a global rounding is an integer sequence b=(b₁,…,bn) such that the rounding error |_∑i∈I(ai−bi)| is less than one in all intervals I⊆{1,…,n}. We give a simple characterization of the set of global roundings of a. This allows to compute optimal roundings in time O(nlogn) and generate a global rounding uniformly at random in linear time under a non-degeneracy assumption and in time O(nlogn) in the general case.     

The decentralized stabilization and control of a class of unknown non-linear time-varying systems

The following problem is considered in this paper. Suppose a system S consists of a set of arbitrary interconnected subsystems S_i, i = 1, 2, …, Ω; is it possible to stabilize and satisfactorily control the whole system S by using only local controllers about the individual subsystems without a knowledge of the manner of the actual interconnections of the whole system? Sufficient conditions are obtained for such a result to hold true; in particular it is shown that a system S consisting of a number of subsystems S_i connected in an arbitrary way between themselves with finite gains: $\text{S}{}_{\mathrm{i}}\text{:}\text{x}\text{?}{}_{\mathrm{i}}\text{=}\text{A}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}\text{+}\text{b}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)u}{}_{\mathrm{i}}\text{, y}{}_{\mathrm{i}}\text{=}\text{c}\text{′}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}$ where A_i and b_i have a particular structure, may be satisfactorily controlled by applying only local controllers C_i about the individual subsystems: C_i: u_i = K′_i(?)x_i where K_i is a constant gain matrix with the scalar ? appearing as a parameter, provided ? is large enough.     

A bottom‐up algorithm for weight‐ and height‐bounded minimal partition of trees

Let T = (V,E) be a rooted tree with n edges. We associate a non‐negative weight w(v) with each vertex v in V. A q‐partition of T into q connected components T₁,...,T_q is obtained by deleting k = q‐1 edges of T, 1<=q<=n. Let w(T_i) denote the sum of the weights of the vertices in T_i. The height of 7] is denoted by h(T_i).

The following problem is considered: Given W, H>0, find a q‐partition satisfying w(T_i) <=W, height (T_i) <=,H (1<=i<=q) for which q is a minimum.

A bottom‐up polynomial time algorithm is given for this problem which has complexity 0(Hn), or independently of H, 0(height(T)n).     

A model of best choice under incomplete information

This paper suggests two approaches to the construction of a two-player game of best choice under incomplete information with the choice priority of one player and the equal weights of both players. We consider a sequence of independent identically distributed random variables (x _i , y _i ), i = 1..., n, which represent the quality of incoming objects. The first component is announced to the players and the second component is hidden. Each player chooses an object based on the information available. The winner is the player whose object has a greater sum of the quality components than the opponent’s object. We derive the optimal threshold strategies and compare them for both approaches.     

An optimal parallel algorithm for c-vertex-ranking of trees

For a positive integer c, a c-vertex-ranking of a graph G=(V,E) is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. The c-vertex-ranking problem is to find a c-vertex-ranking of a given graph using the minimum number of ranks. In this paper we give an optimal parallel algorithm for solving the c-vertex-ranking problem on trees in O(log₂n) time using linear number of operations on the EREW PRAM model.     

On n-Dimensional Sequences. I

Let R be a commutative ring and let n ≥ 1. We study Γ(s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an n-dimensional sequence over R .We express f(X₁,…,X_n) · Γ(s)(X^-1₁,…,X^-1_n) as a partitioned sum. That is, we give (i) a 2ⁿ-fold "border" partition (ii) an explicit expression for the product as a 2ⁿ-fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is βo(f, s), the "border polynomial" of f and s, which is divisible by X₁ … X_n.We say that s is eventually rectilinear if the elimination ideals Ann(s)∩R[X_i] contain an f_i (X_i) for 1 ≤ i ≤ n. In this case, we show that Ann(s) is the ideal quotient ($\text{∑}{}^{\mathrm{n}}{}_{\mathrm{i}}\text{=1(fi) : βo(f, s)/(X}{}_1\text{… X}{}_{\mathrm{n}}\text{)}$).When R and R[[X₁, X₂ ,…, X_n]] are factorial domains (e.g. R a principal ideal domain or F [X₁,…, X_n]), we compute the monic generator γ_i of Ann(s) ∩ R[X_i] from known f_i ϵ Ann(s) ∩ R[X_i] or from a finite number of 1-dimensional linear recurring sequences over R. Over a field F this gives an O($\text{∏}{}_{\mathrm{n}}{}^{\mathrm{i}}\text{=1 δγ}{}^3{}_{\mathrm{i}}$) algorithm to compute an F-basis for Ann(s).     

Alcuni problemi relativi alle formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the form $$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n {f(x_{n,i} ),} } } $$ (α>?1, β>?1), with realA _i ,B _i ,K _n and real nodesx _n,i in (?1,1), valid for prolynomials of degree ≤2n+2r?1. In the first part we prove that there is validity for polynomials exactly of degree2n+2r?1 if and only if α=β=?1/2 andr=0 orr=1. In the second part we consider the problem of the existence of the formula $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)dx = A_n f( - 1) + B_n f(1) + C\sum\limits_{i = 1}^n {f(x_{n,i} )} }$$ for polynomials of degree ≤n+2. Some numerical results are given when λ=1/2.     

Formule di quadratura di tipo gaussiano con valori delle derivate dell'integrando

The coefficientsA _hi (m,s) and the nodesx _i (m,s) for Gaussian-type quadrature formulae

$$\int\limits_{ - 1}^1 {f(x)dx = \mathop \sum \limits_{h = 0}^{2s} \mathop \sum \limits_{i = 1}^m } A_{hi} \cdot f^{(h)} (x_i )$$     

Synthesis of cascaded multiple-loop feedback systems with large plant parameter ignorance

In a cascaded plant with n sensing points, n independent feedback loops are available to achieve quantitative sensitivity specifications. The optimum design is defined as that which achieves this with minimum net effect, of the n sensor noise sources, at the plant input. A design procedure is presented wherein one works backwards step by step from the system output, designing each loop function L_i almost as if the remaining loops (L_i+j, j>0) are perfect. Even in large plant ignorance problems, only the outer loop L_i may be large over some frequency range. In general |L_i|_max < 1, i ≠ 1 and |L_i+1|_max < |L_i|_max. Each L_i has only one distinct frequency range say at ω_{c, i} requiring trade-off between L_i and L_i+1. Only in this range is significant design effort required. However, ω_{c, i+1} >ω_{c, i} so the primary price paid is in the steadily increasing “bandwidth” of each loop. The design procedure is highly transparent with strong universalistic features, permitting the use of universal design curves.     

The variable weighted functions of combined forecasting

In this paper, we describe a mathematical framework to determine the weighted functions in variable weight combined forecasting (VWCF) problems with continuous variable weights. Due to the polynomial approximation theorem and matrix analysis, the general formula of the variable weighted functions w_i(t) in the VWCF problems is obtained. We put forward the optimal weighted matrix and get the optimal weights by minimizing errors square sum J at any given times.     

Optimal Embedding of Multiple Directed Hamiltonian Rings into d-dimensional Meshes

In this paper, we consider the embedding of multiple directed Hamiltonian rings into d-dimensional meshes M_d. Assuming two adjacent nodes in M_d are connected by two directed links with opposite directions, we aim to embed as many directed Hamiltonian rings as possible in a way that they are link-disjoint. In particular, we construct d link-disjoint directed Hamiltonian rings in d-dimensional N₁×…×N_d mesh, where each N_i⩾2d is even.