Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.     

The Schwarzian derivative and the Wiman-Valiron property

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative S_f of f is transcendental, then f has infinitely many multiple points, the inverse function of S_f does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of S_f. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.     

Tangential polynomials and matrix KdV elliptic solitons

Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {π _i } of the fiber π ^?1(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P¹ with a double pole at each p _i , then these solutions are doubly periodic solutions of the matrix KdV equation U _t = 1/4(3UU _x + 3U _x U + U _xxx ). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {p _i }; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.     

Quasisymmetric geometry of the Julia sets of McMullen maps

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m + λ/z^?, where λ ∈ ? {0} and ? and m are positive integers satisfying 1/?+1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a suffcient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

The barycenter property for robust and generic diffeomorphisms

Let f:M~d→M~d(d≥2) be a diffeomorphism on a compact C~∞ manifold on M.If a diffeomorphism f belongs to the C~1-interior of the set of all diffeomorphisms having the barycenter property,then f is Ω-stable.Moreover,if a generic diffeomorphism f has the barycenter property,then f is Ω-stable.We also apply our results to volume preserving diffeomorphisms.     

A discrete version of Koldobsky’s slicing inequality

Let #K be a number of integer lattice points contained in a set K. In this paper we prove that for each d ∈ N there exists a constant C(d) depending on d only, such that for any origin-symmetric convex body K ? R ^d containing d linearly independent lattice points

$$\# K \leqslant C\left( d \right)\max \left( {\# \left( {K \cap H} \right)} \right)vo{l_d}{\left( K \right)^{\frac{{d - m}}{d}}},$$

where the maximum is taken over all m-dimensional subspaces of R ^d . We also prove that C(d) can be chosen asymptotically of order O(1) ^d d ^d?m . In particular, we have order O(1) ^d for hyperplane slices. Additionally, we show that if K is an unconditional convex body then C(d) can be chosen asymptotically of order O(d) ^d?m .

     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

One-element differential standard bases with respect to inverse lexicographical orderings

We give a simplified proof of the following fact: for all nonnegative integers n and d the monomial y _n ^d forms a differential standard basis of the ideal [y _n ^d ]. In contrast to Levi’s combinatorial proof, in this proof we use the Gröbner bases technique. Under some assumptions we prove the converse result: if an isobaric polynomial f forms a differential standard basis of [f], then f = y _n ^d .     

Eigenvalue estimates for submanifolds with bounded <Emphasis Type="BoldItalic">f</Emphasis>-mean curvature

In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the f-Laplacian on complete noncompact submanifolds of the weighted Riemannian manifold (H ^m (?1),e^?f dv) with respect to the f-mean curvature. In particular, our results generalize those of Cheung and Leung in Math. Z. 236 (2001) 525–530.     

On an Extremal Problem for Poset Dimension

Let f(n) be the largest integer such that every poset on n elements has a 2-dimensional subposet on f(n) elements. What is the asymptotics of f(n)? It is easy to see that f(n) = n ^1/2. We improve the best known upper bound and show f(n) = O (n ^2/3). For higher dimensions, we show \(f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )\), where f _d (n) is the largest integer such that every poset on n elements has a d-dimensional subposet on f _d (n) elements.     

Stability of Cubic and Quartic Functional Equations in Non-Archimedean Spaces

We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kx?y)=k[f(x+y)+f(x?y)]+2(k ³?k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kx?y)=k ²[f(x+y)+f(x?y)]+2k ²(k ²?1)f(x)?2(k ²?1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces.     

On some arithmetic properties of Mahler functions

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ? ≥ 2 such that f(x), f(x^?),..., \(f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})\) are linearly dependent over ?(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(x^?) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.     

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise \(u_{t}=\frac{1}{2}\Delta u+u\dot{W}\) in ?₊×? ^d , whose solution is interpreted in the mild sense. The noise \(\dot{W}\) is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).