Real <Emphasis Type="Italic">K</Emphasis>3 surfaces without real points,equivariant determinant of the Laplacian,and the Borcherds Φ-function

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y, σ) is not the one in algebraic geometry.     

Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm.     

On equivariant indices of 1-forms on varieties

Given a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic G-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a G-variety with an isolated singular point.     

On the arithmetic of the hyperelliptic curve y2 = xn + a

We study the arithmetic properties of hyperelliptic curves given by the affine equation y² = xⁿ+a by exploiting the structure of the automorphism groups. We show that these curves satisfy Lang’s conjecture about the covering radius (for some special covering maps).     

Estimation and detection of high-variable functions from Sloan—Woźniakowski space

The major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality”: the rates of accuracy in estimation and separation rates in detection problems behave poorly when the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Hölder balls.In this paper we consider functional classes of a new type, first introduced by Sloan and Wo?niakowski in 1998. We consider balls F _σ,s in a “Sloan—Wo?niakowski” or “weighted Sobolev” space characterized by two parameters: σ > 0 is a “smoothness” parameter, and s > 0 determines the weight sequence which describes “importance” of the variables. Previously Kuo and Sloan [18] used the spaces of similar structure to address the problem of numerical integration.For the classes F _σ,s we show that in the white Gaussian noise model, the separation rates in detection are similar to those for one-variable functions of smoothness σ* = min(s,σ) regardless of the original problem dimension; thus the curse of dimensionality is “lifted”. Similar results hold for the estimation problem.The studies are based on known results for estimation and detection problems for ellipsoids. Using these results, the asymptotics in the problems are determined by asymptotics of “distribution of coefficients” of ellipsoids. The key point of the paper is the study of these asymptotics for the balls F _σ,s .     

The Shape of Orthogonal Cycles in Three Dimensions

Let σ be a directed cycle whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that σ is a shape cycle. We consider the following problem. Does there exist an orthogonal representation Γ of σ in 3D space such that no two edges of Γ intersect except at common endpoints and such that each edge of Γ has the direction specified in σ? If the answer is positive, we say that σ is simple. This problem arises in the context of extending orthogonal graph drawing techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.     

Groups of <Emphasis Type="Italic">S</Emphasis>-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form \(\sqrt f /d{h^s}\), where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element \(\sqrt f \) is periodic. We also analyze the continued fraction expansion of the key element \(\sqrt f /{h^{g + 1}}\), which defines the set of quasiperiodic elements of a hyperelliptic field.     

Bergman Spaces and Carleson Measures on Homogeneous Isotropic Trees

Hastings studied Carleson measures for non-negative subharmonic functions on the polydisk and characterized them by a certain geometric condition relative to Lebesgue measure σ. Cima & Wogen and Luecking proved analogous results for weighted Bergman spaces on the unit ball and other open subsets of \(\mathbb {C}^{n}\). We consider a similar problem on a homogeneous tree, and study how the characterization and properties of Carleson measures for various function spaces depend on the choice of reference measure σ.     

Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

We consider the stochastic volatility model d S _t = σ _t S _t d W _t ,d σ _t = ω σ _t d Z _t , with (W _t ,Z _t ) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the β=1 (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the n→∞ limit of a very large number of time steps of size τ, at fixed \(\beta =\frac 12\omega ^{2}\tau n^{2}\) and \(\rho ={\sigma _{0}^{2}}\tau \), and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of S _t . Under the Euler-Maruyama discretization for (S _t ,logσ _t ), the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.     

Functions of finite fractional variation and their applications to fractional impulsive equations

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.     

Transformation formulas associated with integrals involving the Riemann Ξ-function

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.     

Power series and analyticity over the quaternions

We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a power series \({\sum\nolimits_{n \in \mathbb{N}} q^n a_n}\) in its domain of convergence, which is a ball B(0, R) centered at 0. At each \({p \in B(0,R)}\), f admits expansions in terms of appropriately defined power series centered at p, namely \({\sum\nolimits_{n \in \mathbb{N}} (q-p)^{*n} b_n}\). The expansion holds in a ball Σ(p, R ? |p|) defined with respect to a (non-Euclidean) distance σ. We thus say that f is σ-analytic in B(0, R). Furthermore, we remark that Σ(p, R ? |p|) is not always a Euclidean neighborhood of p; when it is, we say that f is strongly analytic at p. It turns out that f is strongly analytic in a neighborhood of \({B(0,R) \cap \mathbb{R}}\) that can be strictly contained in B(0, R). We then relate these notions of analyticity to the class of quaternionic functions introduced in Gentili and Struppa (Adv. Math. 216(1):279–301, 2007), and recently extended in Colombo et al. (Adv. Math. 222(5):1793–1808, 2009) under the name of slice regular functions. Indeed, σ-analyticity proves equivalent to slice regularity, in the same way as complex analyticity is equivalent to holomorphy. Hence the theory of slice regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.     

Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators L _2q , q = ?1, 0, 1, 2,.., of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.     

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

On the solvability of the first nonlocal boundary value problem for an elliptic equation

We consider problems of the existence, uniqueness, and sign-definiteness of the classical solutions of the problem

$(Lu)(x) = f(x)(x \in D),u(x) - \beta (x)u(\sigma x) = \psi (x)(x \in S),$

where L is a linear second-order operator elliptic in the closure of a domain D ? R ⁿ and σ is a single-valued continuous mapping of S ≡ ?D into \(\bar D\).

We show that, under natural assumptions on the smoothness of β, σ, and the coefficients of L, this problem is Fredholm provided that either σ has no attractors on S or σ generates an attractor Θ on S and the spectral radius of the operator A defined on η(x) ∈ C(Θ) by the formula (Aη)(x) = |β(x)|η(σx) is less than unity.We obtain semieffective (in terms of a test function) conditions for the unique solvability of the problem.     

Equivariant Classification of <Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-symplectic Surfaces

Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of b^m-Poisson structures which can be also visualized using differential forms with singularities as b^m-symplectic structures. In this paper we extend the classification scheme in [24] for b^m-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct b^m-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.     

Symmetry of zeros of Lerch zeta-function for equal parameters

For most values of parameters λ and α, the zeros of the Lerch zeta-function L(λ, α, s) are distributed very chaotically. In this paper, we consider the particular case of equal parameters L(λ, λ, s) and show by calculations that the nontrivial zeros either lie extremely close to the critical line σ = 1/2 or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.