Browsing by Subject "Polynomials."

Now showing 1 - 5 of 5

Open Access
Distance constraints on cyclic networks : a new polynomially solvable class
(1997) Emir, Hülya
Distance Constraints Problem is to locate new facilities on a network so that the distances between new and existing facilities as well as between pairs of now facilities do not exceed given upper bounds, 'rhc ])roblem is N F-Complele on cyclic networks. The oidy known polynornially solvable class of distance constraints on cyclic networks is the case when the linkage network, which is an auxiliary graph induced by the distance bounds between new facility pairs, is a tree. In this thesis, we identify a new polynornially solvable class where each new facilit}'^ is restricted to an a priori specified feasible region which is confined to a single edge and where the linkage network is cj^clic with the restriction that there exists a node whose deletion breaks all cycles. We then extend the above class to a more general class where the linkage network has a cut vertex whose blocks fulfill the above assumptions
Open Access
N-tangle Kanenobu knots with the same Jones polynomials
(2010) Kutluay, Deniz
It is still an open question if there exists a non-trivial knot whose Jones polynomial is trivial. One way of attacking this problem is to develop a mutation on knots which keeps the Jones polynomial unchanged yet alters the knot itself. Using such a mutation; Eliahou, Kauffmann and Thistlethwaite answered, affirmatively, the analogous question for links with two or more components. In a paper of Kanenobu, two types of families of knots are presented: a 2- parameter family and an n-parameter family for n ≥ 3. Watson introduced braid actions for a generalized mutation and used it on the (general) 2-tangle version of the former family. We will use it on the n-tangle version of the latter. This will give rise to a new method of generating pairs of prime knots which share the same Jones polynomial but are distinguishable by their HOMFLY polynomials.
Open Access
Planar p-center problem with Tchebychev distance
(1994) Yılmaz, Dilek
The p-center problem is a model for locating p facilities to serve clients so that the distance between a farthest client and its closest facility is minimized. Emergency service facilities such as fire stations, hospitals and police stations are most of the time located in this manner. In this thesis, the planar p-center problem with Tchebychev distance is studied. The problem is known to be NPHard. We identify certain polynomial time solvable cases and give an efficient branching method which makes use of polynomial time methods in subproblem solutions whenever possible. In addition, a dual problem is posed in light of the existing duality theory on tree networks.
Open Access
Polynomial fitting and total variation based techniques on 1-D and 2-D signal denoising
(2010) Yıldız, Aykut
New techniques are developed for signal denoising and texture recovery. Geometrical theory of total variation (TV) is explored, and an algorithm that uses quadratic programming is introduced for total variation reduction. To minimize the staircase effect associated with commonly used total variation based techniques, robust algorithms are proposed for accurate localization of transition boundaries. For this boundary detection problem, three techniques are proposed. In the first method, the 1−D total variation is applied in first derivative domain. This technique is based on the fact that total variation forms piecewise constant parts and the constant parts in the derivative domain corresponds to lines in time domain. The boundaries of these constant parts are used as the transition boundaries for the line fitting. In the second technique proposed for boundary detection, a wavelet based technique is proposed. Since the mother wavelet can be used to detect local abrupt changes, the Haar wavelet function is used for the purpose of boundary detection. Convolution of a signal or its derivative family with this Haar mother wavelet gives responses at the edge locations, attaining local maxima. A basic local maximization technique is used to find the boundary locations. The last technique proposed for boundary detection is the well known Particle Swarm Optimization (PSO). The locations of the boundaries are randomly perturbed yielding an error for each set of boundaries. Pursuing the personal and global best positions, the boundary locations converge to a set of boundaries. In all of the techniques, polynomial fitting is applied to the part of the signal between the edges. A more complicated scenario for 1−D signal denoising is texture recovery. In the technique proposed in this thesis, the periodicity of the texture is exploited. Periodic and non-periodic parts are distinguished by examining total variation of the autocorrelation of the signal. In the periodic parts, the period size was found by PSO evolution. All the periods were averaged to remove the noise, and the final signal was synthesized. For the purpose of image denoising, optimum one dimensional total variation minimization is carried to two dimensions by Radon transform and slicing method. In the proposed techniques, the stopping criterion for the procedures is chosen as the error norm. The processes are stopped when the residual norm is comparable to noise standard deviation. 1−D and 2−D noise statistics estimation methods based on Maximum Likelihood Estimation (MLE) are presented. The proposed denoising techniques are compared with principal curve projection technique, total variation by Rudin et al, total variation by Willsky et al, and curvelets. The simulations show that our techniques outperform these widely used techniques in the literature.
Open Access
Polynomially solvable cases of multifacility distance constraints on cyclic networks
(1993) Yeşilkökçen, Naile Gülcan
Distance Constraints Problem is to locate one or more new facilities on a network so that the distances between new and existing facilities as well as between pairs of new facilities do not exceed given upper bounds. The problem is AfV-Complete on cyclic networks and polynomially solvable on trees. Although theory for tree networks is well-developed, there is virtually no theory for cyclic networks. In this thesis, we identify a special class of instances for which we develop theory and algorithms that are applicable to any metric space defining the location space. We require that the interaction between new facilities has a tree structure. The method is based on successive applications of EXPANSION and INTERSECTION operations defined on subsets of the location space. Application of this method to general networks yields strongly polynomial algorithms. Finally, we give an algorithm that constructs an e-optimal solution to a related minimax problem.