• Canada
• 0101 mathematics close

For the Hermite matrix polynomials H"n(@g, A) in which the parameter A is a matrix whose eigenvalues lie in the open right half plane, a bound on @?H"n(@g, A)@?"2 is developed. It has the form of a simple expression in the integer n, multiplied by exp ((5/2)@[email protected]?"[email protected]^2), for any real @g. The bound is exploited to construct an algorithm for computing any matrix exponential to within an arbitrary tolerance which may be prescribed a priori. Numerical effectiveness of the computational process is illustrated.

AbstractWe study the well-posedness and regularity of the generalized Navier–Stokes equations with initial data in a new critical space Qα;∞β,−1(Rn)=∇⋅(Qαβ(Rn))n, β∈(12,1), which is larger than some known critical homogeneous Besov spaces. Here Qαβ(Rn) is a space defined as the set of all measurable functions withsup(l(I))2(α+β−1)−n∫I∫I|f(x)−f(y)|2|x−y|n+2(α−β+1)dxdy<∞ where the supremum is taken over all cubes I with edge length l(I) and edges parallel to the coordinate axes in Rn. In order to study the well-posedness and regularity, we give a Carleson measure characterization of Qαβ(Rn) by investigating a new type of tent spaces and an atomic decomposition of the predual for Qαβ(Rn). In addition, our regularity results apply to the incompressible Navier–Stokes equations with initial data in Qα;∞1,−1(Rn).

This paper extends decision making under risk and uncertainty to group theory via representations of invariant behavioural space for prospect theory. First, we predict that canonical specifications for value functions, probability weighting functions, and stochastic choice maps are homomorphic. Second, we derive a continuous singular matrix operator T for affine transformation of a vector space of skewed S-shape value functions V isomorphic to a vector spaceW of inverted S-shaped probability weighting functions. To characterize the transformation, we decompose the operator into shear, scale and translation components. In that milieu, Moore-Penrose psuedoinverse transformation recovers value functions from probability weighting functionals. Removal of 0 from the point spectrum induces nonsingular operators that support group representation of stochastic choice maps in an invariant subspace of the general linear group GL(V). Third, we demonstrate how group theoretic operations on a gamble provide mathematical foundations of probability weighting functions that subsume the Prelec class. Fourth, we predict that a gamble is isomorphic to an invariant cyclic sub-group in weighted probablity space. This result implies that probability weighting functions [and value functions] fluctuate near their extremes, and explain violation of transitivity axioms in decision theory. Moreover, representations include the special unitary group SU(2) and orthogonal group Θ*3. The former includes Pauli’s spin matrices and accounts for skewness. It also provides microfoundations for construction of (1) behavioural stochastic processes from group character in the frequency domain; and (2) Schrodinger-Pauli Hamiltonian to compute, inter alia, time dependent probabilities in decision field theory.

A Black-Scholes market is considered in which the underlying economy, as modeled by the parameters and volatility of the processes, switches between a finite number of states. The switching is modeled by a hidden Markov chain. European options are priced and a Black-Scholes equation obtained. The approximate valuation of American options due to Barone-Adesi and Whaley is extended to this setting.

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M V \mathcal {M}_V of the multiplier algebra M \mathcal {M} of Drury-Arveson space to a holomorphic subvariety V V of the unit ball B d \mathbb {B}_d . We find that M V \mathcal {M}_V is completely isometrically isomorphic to M W \mathcal {M}_W if and only if W W is the image of V V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d > ∞ d>\infty every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V V and W W are each a finite union of irreducible varieties and a discrete variety, when d > ∞ d>\infty , an isomorphism between M V \mathcal {M}_V and M W \mathcal {M}_W determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- ∗ * continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold—particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

We show how to construct an explicit Hamilton cycle in the directed Cayley graph C → ({σ n , σ n -1 }: S n ), where σ k is the rotation (1 2 … k ). The existence of such cycles was shown by Jackson [1996] but the proof only shows that a certain directed graph is Eulerian, and Knuth [2005] asks for an explicit construction. We show that a simple recursion describes our Hamilton cycle and that the cycle can be generated by an iterative algorithm that uses O ( n ) space. Moreover, the algorithm produces each successive edge of the cycle in constant time; such algorithms are said to be loopless . Finally, our Hamilton cycle can be used to construct an explicit universal cycle for the ( n -1)-permutations of a n -set, or as the basis of an efficient algorithm for generating every n -permutation of an n -set within a circular array or linked list.

AbstractWe consider valued fields with a value-preserving automorphism and improve on model-theoretic results by Bélair, Macintyre and Scanlon on these objects by dropping assumptions on the residue difference field. In the equicharacteristic 0 case we describe the induced structure on the value group and the residue difference field.

We study a generalized inverse eigenvalue problem (GIEP),Ax=λBx, in whichAis a semi-infinite Jacobi matrix with positive off-diagonal entriesci>0, andB= diag (b0,b1,…), wherebi≠0fori=0,1,…. We give an explicit solution by establishing an appropriate spectral function with respect to a given set of spectral data.

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G associated to the constraints in the sum. Comment: 20 pages