Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method (LBM) with two amending functions for solving partial differential equations (PDEs) arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O(N), where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

A note on linear transformations over integral domains

It is an easy fact from linear algebra that if M is a finite-dimensional vector space over a field R, ϕM→M a diagonalizable linear transformation, and N a ϕ-invariant subspace of M, then ϕ∣N is diagonalizable. We show that an appropriate generalization of this holds for M a torsion-free module over an integral domain R.     

On the stability of the vertical rotation of a solid suspended on a rod

The problem of the motion of a dynamically symmetric solid suspended from a fixed point by a weightless rod and two ball and socket joints one of which is fixed at the fixed point O', and the other is on the body axis of symmetry at the point O is considered. The question of the stability of the uniform body rotation when points O' and O, and the body centre of inertia C lie on the same vertical, and at the same time point O may be either above or below point O', and point C either above or below point O, is discussed. An analysis of the necessary and sufficient conditions for stability is given. The set of all the system's parameters is reduced to three independent dimensionless parameters L, Ω and β, and in the plane (L, Ω), for fixed values of β, the regions for which the unperturbed rotation is steady, or steady to a first approximation, or non-steady are indicated. The regions for which the body rotation is steady to a first approximation when the point O is situated higher than the point O', and the point C lies higher or lower than the point O are determined.

The sufficient conditions for the vertical rotation of a dynamically symmetric body suspended on a filament were obtained in /1/ and investigated for the cases where in non-perturbed motion the point C is below point O, when points C and O coincide, and when the length of the filament is zero (Lagrange gyroscope). In /2/ an analysis is given of the sufficient conditions for stability obtained in /1/, and also the necessary conditions for the cases where in a non-perturbed motion point C is located above point O.     

Changing cofinalities and collapsing cardinals in models of set theory

If a˜cardinal κ₁, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ₀ and its new cofinality, ρ, is less than κ₀, then, under some additional assumptions, each cardinal λ>κ₁ less than cc(P(κ₁)/[κ₁]^<κ₁) is collapsed to κ₀ as well. If in addition N=M[f], where f : ρ→κ₁ is an unbounded mapping, then N is a˜|λ|=κ₀-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.     

Optimal cell flipping to minimize channel density in VLSI design and pseudo-Boolean optimization

Cell flipping in VLSI design is an operation in which some of the cells are replaced with their “mirror images” with respect to a vertical axis, while keeping them in the same slot. After the placement of all the cells, one can apply cell flipping in order to further decrease the total area, approximating this objective by minimizing total wire length, channel width, etc. However, finding an optimal set of cells to be flipped is usually a difficult problem. In this paper we show that cell flipping can be efficiently applied to minimize channel density in the standard cell technology. We show that an optimal flipping pattern can be found in O(p(n/c)^c) time, where n, p and c denote the number of nets, pins and channels, respectively. Moreover, in the one channel case (i.e. when c = 1) the cell flipping problem can be solved in O(p log n) time. For the multi-channel case we present both an exact enumeration scheme and a mixed-integer program that generates an approximate solution very quickly. We present computational results on examples up to 139 channels and 65000 cells.     

Research Problem

We define an arithmetic invariant for the congruence subgroups Γ₀(N), denoted by

b(Γ₀(N)) and pose the problem of finding good asymptotic upper bonds for b(Γ₀(N)) as N approaches X especially when N is prime or the product of two (not necessarily distinct) primes.     

An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

The Ramsey number N(3,3,3,3; 2) is the smallest integer n such that each 4-coloring by edges of the complete graph on n vertices contains monochromatic triangles. It is well known that 51 ≤ N(3, 3, 3, 3; 2) ≤ 65. Here we prove that N(3, 3, 3, 3; 2) ≤ 64.     

Cases of equality for the spectral norm submultiplicativity of the hadamard product

In this note, we characterize those pairs of nonzero r-by-d complex matrices that satisfy N₂(A ∘ B) = N₂(A)N₂(B), in which N₂(·) is the spectral norm and · is the Hadamard product.     

Decoupling of the N-soliton solution for a new discrete MKdV equation

We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t→∞. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Geometry of higher-rank numerical ranges

We consider geometric aspects of higher-rank numerical ranges for arbitrary N  × N matrices. Of particular interest is the issue of convexity and a possible extension of the Toeplitz-Hausdorff Theorem. We derive a number of reductions and obtain partial results for the general problem. We also conduct graphical and computational experiments.

Added in proof: Following acceptance of this paper, our subject has developed rapidly. First, Hugo Woerdeman established convexity of the higher-rank numerical ranges by combining Proposition 2.4 and Theorem 2.12 with the theory of algebraic Riccati equations. See Woerdeman, H., 2007, The higher rank numerical range is convex, Linear and Multilinear Algebra, to appear. Subsequently Chi-Kwong Li and Nung-Sing Sze followed a different approach that not only yields convexity but also provides important additional insights. See Li, C.-K. and Sze, N.-S., 2007, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, preprint. See also Li, C.-K., Poon, Y.-T., and Sze, N.-S., 2007, Condition for the higher rank numerical range to be non-empty, preprint.     

A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

This article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The maximal length of real trivectors of rank seven

Trivectors of rank seven over the complex number field have irreducible length less than or equal to four. Over the reals however this is not true. An example of a real trivector of rank seven and irreducible length five is presented. Then, in the notation of Busemann and Glassco, we have N(R, 7, 3) = 5 since for any field F we always have N(F ,7, 3) ≤ 5. This paper provides the first published example where N(F, n, r) ≠ N(K, n, r) for two different fields F and K.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

C-normality and solvability of groups

A subgroup H is called c-normal in group G if there exists a normal subgroup N and G such that HN = G and H ∩ N ≤ H_G where H_G =: Core(H) = _gG H^g is the maximal normal subgroup of G which is contained in H. We obtain some results about the c-normal subgroups and the solvability of groups.