Rough approximations of vague sets in fuzzy approximation space

The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the β-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on β-(complement) operator. Meantime, λ-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the β-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable.     

Matching polynomials for chains of cycles

Došli? and Måløy (2010) [2] obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the h-cactus chains with the most matchings.     

On the spectral characterization of some unicyclic graphs

Let H(n;q,n₁,n₂) be a graph with n vertices containing a cycle C_q and two hanging paths P_n1 and P_n2 attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12;6,1,5) and H(12;8,2,2), no two non-isomorphic graphs of the form H(n;q,n₁,n₂) are A-cospectral. It is proved that all graphs H(n;q,n₁,n₂) are determined by their L-spectra. And all graphs H(n;q,n₁,n₂) are proved to be determined by their Q-spectra, except for graphs with a being a positive even number and with b≥4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.     

Modelling and management of longevity risk: Approximations to survivor functions and dynamic hedging

This paper looks at the development of dynamic hedging strategies for typical pension plan liabilities using longevity-linked hedging instruments. Progress in this area has been hindered by the lack of closed-form formulae for the valuation of mortality-linked liabilities and assets, and the consequent requirement for simulations within simulations. We propose the use of the probit function along with a Taylor expansion to approximate longevity-contingent values. This makes it possible to develop and implement computationally efficient, discrete-time delta hedging strategies using q-forwards as hedging instruments.The methods are tested using the model proposed by Cairns et al. (2006a) (CBD). We find that the probit approximations are generally very accurate, and that the discrete-time hedging strategy is very effective at reducing risk.     

A local prime factor decomposition algorithm

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived.Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known “classical” prime factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products.     

Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces

We extend the notion of L₂-B-discrepancy introduced in [E. Novak, H. Wo?niakowski, L₂ discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359-388] to what we shall call weighted geometric L₂-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L₂-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wo?niakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.     

Duality theory and propagation rules for higher order nets

Higher order nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have higher order nets and sequences of high quality.In this paper we introduce a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. We use this duality theory to prove propagation rules for such nets. This way we can obtain new higher order nets (sometimes with improved quality) from existing ones. We also extend our approach to the construction of higher order sequences.     

A note on the generalized min-sum set cover problem

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin (STOC 2009). Bansal, Gupta, and Krishnaswamy (SODA 2010) give a 485-approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α-point scheduling to obtain our improvements.     

An improved randomized approximation algorithm for maximum triangle packing

This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant ?>0. By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of 0.5257(1−?) for any constant ?>0.     

On the stability index for weighted composition operators

Let ?>0. A continuous linear operator T:C(X)?C(Y) is said to ?-preserve disjointness if ‖(Tf)(Tg)‖_∞≤?, whenever f,g∈C(X) satisfy ‖f‖_∞=‖g‖_∞=1 and fg≡0. In this paper we continue our study of the minimal interval where the possible maximal distance from a norm one operator which ?-preserves disjointness to the set of weighted composition maps may lie. We provide sharp bounds for both the finite and the infinite case, which turn out to be completely different.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Using this we derive several useful characterizations from the one-dimensional case to the d-dimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

The order of hypotraceable oriented graphs

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented graphs of order n for every n≥8 except possibly for n=9,11 and that is the only one of order less than 8.Furthermore, we determine all the hypotraceable oriented graphs of order 8 and explain the relevance of these results to the problem of determining, for given k≥2, the maximum order of nontraceable oriented digraphs each of whose induced subdigraphs of order k is traceable.     

Construction of a high order fluid-structure interaction solver

Accuracy is critical if we are to trust simulation predictions. In settings such as fluid-structure interaction, it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood flows in the cardiovascular system. In this paper, we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time.First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial differential solves using h/p type Galerkin methods. Then, we briefly describe the handling of high order geometry and the structure solver. Next we outline the high-order space-time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem as well as some initial results.     

Duplicate form of Carlitz inversions and summation formulae

By applying the duplicate form of Carlitz inversions to three special cases of the q-Saalschütz theorem respectively, we establish three classes of interesting summation formulae.     

On the demand pooling anomaly in inventory theory

The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.     

Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses?

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach allows a provision for reduction of capital as a result of insurance mitigation of up to 20%. This paper studies different insurance policies in the context of capital reduction for a range of extreme loss models and insurance policy scenarios in a multi-period, multiple risk setting. A Loss Distributional Approach (LDA) for modeling of the annual loss process, involving homogeneous compound Poisson processes for the annual losses, with heavy-tailed severity models comprised of α-stable severities is considered. There has been little analysis of such models to date and it is believed insurance models will play more of a role in OpRisk mitigation and capital reduction in future. The first question of interest is when would it be equitable for a bank or financial institution to purchase insurance for heavy-tailed OpRisk losses under different insurance policy scenarios? The second question pertains to Solvency II and addresses quantification of insurer capital for such operational risk scenarios. Considering fundamental insurance policies available, in several two risk scenarios, we can provide both analytic results and extensive simulation studies of insurance mitigation for important basic policies, the intention being to address questions related to VaR reduction under Basel II, SCR under Solvency II and fair insurance premiums in OpRisk for different extreme loss scenarios. In the process we provide closed-form solutions for the distribution of loss processes and claims processes in an LDA structure as well as closed-form analytic solutions for the Expected Shortfall, SCR and MCR under Basel II and Solvency II. We also provide closed-form analytic solutions for the annual loss distribution of multiple risks including insurance mitigation.     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.