Advanced comminution modelling: Part 2 - Mills

This second part paper explores rock breakage mechanisms, the life cycle of rocks in mills and the strong influence of end walls on charge motion within mills. We present recent advances in particle-based modelling of mills for comminution focused around wear and the effect of slurry and slurry phase grinding. Three mill scenarios are considered:

• 1.Media flow and the resulting wear evolution of the belly and end wall liners and the resulting change in mill performance for a full industrial scale dry ball mill (modelled using DEM)
• 2.Axial slurry transport and mixing in a wet overflow industrial scale ball mill (modelled using fully coupled DEM and SPH)
• 3.Effect of mill speed on slurry and solid charge motion and the resulting grinding of fine particles in a 1.8 m diameter wet Hardinge pilot mill (modelled using fully coupled DEM and SPH with advection-diffusion-population balance equations solved for the slurry size distribution for each SPH particle)

These demonstrate the nature and level of fidelity that is now possible to include in particle-scale comminution models. They provide insights into the critical importance of curtain flows generated by the end walls of tumbling mills, on wear behaviour on liners, on the structure of slurry pools and mill discharge and on the adverse effect on grinding of increasing mill speed.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

Simulated annealing and tabu search approaches for the Corridor Allocation Problem

In the Corridor Allocation Problem, we are given n facilities to be arranged along a corridor. The arrangements on either side of the corridor should start from a common point on the left end of the corridor. In addition, no space is allowed between two adjacent facilities. The problem is motivated by applications such as the arrangement of rooms in office buildings, hospitals, shopping centers or schools. Tabu search and simulated annealing algorithms are presented to minimize the sum of weighted distances between every pair of facilities. The algorithms are evaluated on several instances of different sizes either randomly generated or available in the literature. Both algorithms reached the optimal (when available) or best-known solutions of the instances with n ? 30. For larger instances with size 42 ? n ? 70, the simulated annealing implementation obtained smaller objective values, while requiring a smaller number of function evaluations.     

Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in (B _X; X) is the supremum of the set

Complete weights andv-peak points of spaces of weighted holomorphic functions

We examine the geometric theory of the weighted spaces of holomorphic functions on bounded open subsets ofC ⁿ ,C ⁿ ,H _v (U) and , by finding a lower bound for the set of weak^*-exposed and weak^*-strongly exposed points of the unit ball of and give necessary and sufficient conditions for this set to be naturally homeomorphic toU. We apply these results to examine smoothness and strict convexity of and . We also investigate whether is a dual space. The second author was supported by MCYT and FEDER Project BFM2002-01423.     

A branch-and-bound algorithm for the singly constrained assignment problem

The singly constrained assignment problem (SCAP) is a linear assignment problem (LAP) with one extra side constraint, e.g., due to a time restriction. The SCAP is, in contrast to the LAP, difficult to solve. A branch-and-bound algorithm is presented to solve the SCAP to optimality. Lower bounds are obtained by Lagrangean relaxation. Computational results show that the algorithm is able to solve different types of SCAP instances up to size n = 1000 within short running times on a standard personal computer.     

Boundary regularity for the [`(?)] b - Neumann\bar \partial _b - Neumann problem, Part 2

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the boundary conditions is an isomorphism. As an application, we establish sharp regularity for a canonical solution to the inhomogenous equation on the unit ball.     

On-line scheduling on a batch processing machine with unbounded batch size to minimize the makespan

We present on-line algorithms to minimize the makespan on a single batch processing machine. We consider a parallel batching machine that can process up to b jobs simultaneously. Jobs in the same batch complete at the same time. Such a model of a batch processing machine has been motivated by burn-in ovens in final testing stage of semiconductor manufacturing. We deal with the on-line scheduling problem when jobs arrive over time. We consider a set of independent jobs. Their number is not known in advance. Each job is available at its release date and its processing requirement is not known in advance. This general problem with infinite machine capacity is noted 1∣p − batch, r_j, b = ∞∣C_max. Deterministic algorithms that do not insert idle-times in the schedule cannot be better than 2-competitive and a simple rule based on LPT achieved this bound [Z. Liu, W. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics 105 (2000) 129–136]. If we are allowed to postpone start of jobs, the performance guarantee can be improved to 1.618. We provide a simpler proof of this best known lower bound for bounded and unbounded batch sizes. We then present deterministic algorithms that are best possible for the problem with unbounded batch size (i.e., b = ∞) and agreeable processing times (i.e., there cannot exist an on-line algorithm with a better performance guarantee). We then propose another algorithm that leads to a best possible algorithm for the general problem with unbounded batch size. This algorithm improves the best known on-line algorithm (i.e. [G. Zhang, X. Cai, C.K. Wong, On-line algorithms for minimizing makespan on batch processing machines, Naval Research Logistics 48 (2001) 241–258]) in the sense that it produces a shortest makespan while ensuring the same worst-case performance guarantee.     

Quasiconformal mappings and singularity of boundary distortion

We extend a well-known theorem of Jones and Makarov [8] on the singularity of boundary distortion of planar conformal mappings. Using a different technique, we recover the previous result and generalize the result to quasiconformal mappings of the unit ball ⊂_∝ ⁿ , n ≥ 2. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere ∂ and show that this estimate is essentially sharp.     

Uniform persistence for a two-species ratio-dependent predator-prey system with diffusion and variable time delays

A two-species ratio-dependent predator-prey diffusion model with variable time delays is investigated. By constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we obtain the permanence of this system. Our results generalize and improve the results of [10] and [11].     

Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in R³${\mathbb{R}}^3$ via two distinct approaches. The first approach invokes the method established in the works Bourgain (1994, 1996) ,  and  based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schrödinger equation on the unit ball in subsequent works Bourgain and Bulut (2013)  and .     

Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem in a Banach space X, where is m-accretive and such that –A generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when A depends on time is also considered.      

Counting Ancestral Reconstructions in a Fixed Phylogeny

We give formulas for calculating in polynomial time the number of ancestral reconstructions for a tree with binary leaf- and root labels for each number of 0 1 and 1 0 arcs. For trees of fixed degree, the corresponding numbers of 0 0 and 1 1 arcs can be deduced. We calculate intervals for the relative cost of 0 1 and 1 0 transitions over which the same labelings remain the cheapest. Received July 27, 2005     

On the security analysis of an image scrambling encryption of pixel bit and its improved scheme based on self-correlation encryption

An image scrambling encryption scheme for pixel bits was presented by Ye [Ye GD. Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit Lett 2010;31:347-54], which can be seen as one kind of typical binary image scrambling encryption considering from the bit-plain of size M × (8N). However, recently, some defects existing in the original image encryption scheme, i.e., Ye’s scheme, have been observed by Li and Lo [Li CQ, Lo KT. Optimal quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext attacks. Signal Process 2011;91:949-54]. In the attack proposed by Li and Lo at least 3 + ⌈log₂(MN)⌉ plain images of size M × N are used to reveal the permutation matrix W = [w(i, k)] (i ∈ {1, 2, … , M}; k ∈ {1, 2, … , 8N}) which can be applied to recover the exact plain image. In the current paper, at first, one type of special plain image/cipher image is used to analyze the security weakness of the original image scrambling scheme under study. The final encryption vectors TM and TN or the decryption vectors TM′ and TN′ are revealed completely according to our attack. To demonstrate the performance of our attack, a quantified comparison is drawn between our attack and the attack proposed by Li and Lo. Compared with Li and Lo’s attack, our attack is more efficient in the general conditions. In particular, when the sizes of images satisfy the condition M = N or M ? 8N, the number of the used plain images/cipher images is at most 9, which is sharply less than 3 + ⌈log₂(MN)⌉ when M and N are of large size. To overcome the weaknesses of the original scheme, in this paper, an improved image scrambling encryption scheme is proposed. In the improved scheme, the idea of the “self-correlation” method is used to resist the chosen-plaintext attack/known-plaintext attack. The corresponding simulations and analyses illustrate that the improved encryption method has good cryptographic properties, and can overcome the weakness of the original image encryption scheme. Finally, farther improvement is briefly presented for the future work.     

Minimizing makespan and total completion time for parallel batch processing machines with non-identical job sizes

This paper considers the scheduling problem of parallel batch processing machines with non-identical job sizes. The jobs are processed in batches and the machines have the same capacity. The models of minimizing makespan and total completion time are given using mixed integer programming method and the computational complexity is analyzed. The bound on the number of feasible solutions is given and the properties of the optimal solutions are presented. Then a polynomial time algorithm is proposed and the worst case ratios for minimizing total completion time and makespan is proved to be 2 and (8/3–2/3 m) respectively. To test the proposed algorithm, we generate different levels of random instances. The computational results demonstrate the effectiveness of the algorithm for minimizing the two objectives.     

Analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods

In this note we show the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods. Taking inspiration from previous papers by Dattoli et al. [4], [5] and [6] about spectral properties of Laguerre derivative, we here generalize some of their results to fractional evolution equations. Besides that, we have two interesting generalized examples. One is about telegraph equation with time dependent coefficient. The other, that could be of some interest for realistic applications, is the fractional diffusion with a space-dependent diffusion coefficient.     

Approximate Inclusion-Exclusion

The Inclusion-Exclusion formula expresses the size of a union of a family of sets in terms of the sizes of intersections of all subfamilies. This paper considers approximating the size of the union when intersection sizes are known for only some of the subfamilies, or when these quantities are given to within some error, or both.In particular, we consider the case when allk-wise intersections are given for everykK. It turns out that the answer changes in a significant way aroundK=n: ifKO(n) then any approximation may err by a factor of (n/K ²), while ifK (n) it is shown how to approximate the size of the union to within a multiplicative factor of .When the sizes of all intersections are only given approximately, good bounds are derived on how well the size of the union may be approximated. Several applications for Boolean function are mentioned in conclusion.Partially supported by NSF 865727-CCR and ARO DALL03-86-K-017. Part of this work was done in U.C. Berkeley, supported by NSF CCR-8411954.     

Singularly perturbed nonlinear elliptic problems on manifolds

Let be a connected compact smooth Riemannian manifold of dimension with or without smooth boundary We consider the following singularly perturbed nonlinear elliptic problem on

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

A strongly polynomial FPTAS for the symmetric quadratic knapsack problem

The symmetric quadratic knapsack problem (SQKP), which has several applications in machine scheduling, is NP-hard. An approximation scheme for this problem is known to achieve an approximation ratio of (1 + ?) for any ? > 0. To ensure a polynomial time complexity, this approximation scheme needs an input of a lower bound and an upper bound on the optimal objective value, and requires the ratio of the bounds to be bounded by a polynomial in the size of the problem instance. However, such bounds are not mentioned in any previous literature. In this paper, we present the first such bounds and develop a polynomial time algorithm to compute them. The bounds are applied, so that we have obtained for problem (SQKP) a fully polynomial time approximation scheme (FPTAS) that is also strongly polynomial time, in the sense that the running time is bounded by a polynomial only in the number of integers in the problem instance.