We study the problems of scheduling jobs, with different release dates and equal processing times, on two types of batching machines. All jobs of the same batch start and are completed simultaneously. On a serial batching machine, the length of a batch equals the sum of the processing times of its jobs and, when a new batch starts, a constant setup time s occurs. On a parallel batching machine,...

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have...

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in `. More precisely, we prove the following. Let R be a real closed field and let P = {P1, . . . , Pm} ⊂ R[Y1, . . . , Y`,X1, . . . , Xk], with degY (Pi) ≤ 2,d...

Proof. Let F be the field of fractions of R. By 6.2.8, we must show that R = Int(R,F). The inclusion R ⊆ Int(R,F) is immediate. Let f/g ∈ Int(R,F) and let it be written in lowest terms. Suppose to the contrary that f/g / ∈ R. Then g = q1 · · · q` with qj prime for 1 ≤ j ≤ `. We can also write f = p1 · · · pk where the pi are primes for 1 ≤ i ≤ k. Note that pi 6∼ qj for any i, j because f/g is i...

We give algorithms for the evaluation of sparse polynomials of the form P = p0 + p1x + p2x 4 + · · ·+ pN−1x 2 , for various choices of coefficients pi. First, we take pi = p i, for some fixed p; in this case, we address the question of fast evaluation at a given point in the base ring, and we obtain a cost quasi-linear in √ N . We present experimental results that show the good behavior of this...

We present a zero structure theorem for a differential parametric system: p1 = 0, · · · , pr = 0, d1 6= 0, · · · , ds 6= 0 where pi and di are differential polynomials in K{u1, · · · , um, x1, · · · , xn} and the u are parameters. According to this theorem we can identify all parametric values for which the parametric system has solutions for the xi and at the same time giving the solutions for...

We find an explicit expression of the Tutte polynomial of an $n$-fan. We also find a formula of the Tutte polynomial of an $n$-wheel in terms of the Tutte polynomial of $n$-fans. Finally, we give an alternative expression of the Tutte polynomial of an $n$-wheel and then prove the explicit formula for the Tutte polynomial of an $n$-wheel.

It is known that any dichotomy of {-1, 1}n can be learned (separated) with a higher-order neuron (polynomial function) with 2n inputs (monomials). In general, less than 2n monomials are sufficient to solve a given dichotomy. In spite of the efforts to develop algorithms for finding solutions with fewer monomials, there have been relatively fewer studies investigating maximum density (Pi(n)), th...

We are given a set of n jobs that have to be executed on a set of m speed-scalable machines that can vary their speeds dynamically using the energy model introduced in [Yao et al., FOCS’95]. Every job j is characterized by its release date rj , its deadline dj , its processing volume pi,j if j is executed on machine i and its weight wj . We are also given a budget of energy E and our objective ...

We study the mixing inequalities which were introduced by Günlük and Pochet (2001). We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a type I mixing inequality and at most n− 1 if it is a type II mixing inequality. We also show that these bounds are tight for n = 2. Given a mixed-integer set PI = P ∩ Z(I) where P is a polyhedron and Z(I) = {x ∈ Rn ...