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).     

Symmetries of Fundamental Solutions and Their Application in Continuum Mechanics

This paper is a brief survey of the recent results in problems of approximating functions by solutions of homogeneous elliptic systems of PDEs on compact sets in the plane in the norms of C^m spaces, m ≥ 0. We focus on general second-order systems. For such systems the paper complements the recent survey by M. Mazalov, P. Paramonov, and K. Fedorovskiy (2012), where the problems of C^m approximation of functions by holomorphic, harmonic, and polyanalytic functions as well as by solutions of homogeneous elliptic PDEs with constant complex coefficients were considered.     

Polynomial inequalities on sets with <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>-concave weighted measures

Let μ be a measurewith a k-concave density W on an open convex set V in R^m, that is, W is an integrable weight satisfying the condition

$$W(ax + (1 - a)y) \geqslant {(a{W^K}(x) + (1 - a){W^K}(y))^{1/k}},k \in ( - 1/m,\infty ]$$

for all x ∈ V, y ∈ V, and α ∈ [0, 1]. In this paper, we first show that the Fradelizi μ-distributional inequalities for polynomials P of m variables are sharp for each m and k ∈ (?1/m,∞]. Classes of extremal sets V, weights W, and polynomials P for these inequalities are presented. Sharpness of the Bobkov-Nazarov-Fradelizi dilation-type inequalities is established as well. Second, we find efficient conditions for k-concavity of a weight W and obtain new sharp polynomial inequalities.

     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

Complexity of function systems over a finite field in the class of polarized polynomial forms

The Shannon complexity of a function system over a q-element finite field which contains m functions of n variables in the class of polarized polynomial forms is exactly evaluated: L _q ^PPF (n,m) = q ⁿ for all n ≥ 1, m ≥ 2, and all possible odd q. It has previously been known that L₂^PPF (n,m) = 2 ⁿ and L₃^PPF (n,m) = 3 ⁿ for all n ≥ 1 and m ≥ 2.     

Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Σ = {0} × S ¹ of operators of the form \({L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}\), \({b\not\equiv0}\) and a(0) ≠ 0, defined on \({\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}\), \({\epsilon >0 }\), where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon)}\) and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of \({C^\infty(\Omega_\epsilon)}\) there is a solution \({u\in L^\infty}\) of the equation Lu = f in a neighborhood of Σ; moreover, the L ^∞ regularity is sharp.     

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P _n,m of algebraic polynomials of a given degree n on the unit sphere \(\mathbb{S}^{m - 1}\) of the Euclidean space ? ^m of dimension m ≥ 2. (1) Find the norm of the functional \(F\left( \eta \right) = F_h P_n = \int_{\mathbb{G}\left( \eta \right)} {P_n (x)dx}\), which is the integral over the spherical layer \(\mathbb{G}\left( \eta \right) = \left\{ {x = \left( {x_1 , \ldots ,x_m } \right) \in \mathbb{S}^{m - 1} :h' \leqslant x_m \leqslant h''} \right\}\) defined by a pair of real numbers η = (h′, h″), ?1 ≤ h′ < h″ ≤ 1, on the set P _n,m with the norm of the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of functions summable on the sphere. (2) Find the best approximation in \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) of the characteristic function χ _η of the layer \(\mathbb{G}\left( \eta \right)\) by the subspace P _n,m ^⊥ of functions from \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) that are orthogonal to the space of polynomials P _n,m . (3) Find the best approximation in the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of the function χ _η by the space of polynomials P _n,m . We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L ₁ ^φ (?1, 1) of functions summable on the interval (?1, 1) with ultraspherical weight φ(t) = (1 ? t ²) ^α , α = (m ? 3)/2. We study the respective one-dimensional problems in the space of functions summable on (?1, 1) with an arbitrary not necessarily ultraspherical weight.     

Convergence analysis of the direct extension of ADMM for multiple-block separable convex minimization

Recently, the alternating direction method of multipliers (ADMM) has found many efficient applications in various areas; and it has been shown that the convergence is not guaranteed when it is directly extended to the multiple-block case of separable convex minimization problems where there are m ≥ 3 functions without coupled variables in the objective. This fact has given great impetus to investigate various conditions on both the model and the algorithm’s parameter that can ensure the convergence of the direct extension of ADMM (abbreviated as “e-ADMM”). Despite some results under very strong conditions (e.g., at least (m ? 1) functions should be strongly convex) that are applicable to the generic case with a general m, some others concentrate on the special case of m = 3 under the relatively milder condition that only one function is assumed to be strongly convex. We focus on extending the convergence analysis from the case of m = 3 to the more general case of m ≥ 3. That is, we show the convergence of e-ADMM for the case of m ≥ 3 with the assumption of only (m ? 2) functions being strongly convex; and establish its convergence rates in different scenarios such as the worst-case convergence rates measured by iteration complexity and the globally linear convergence rate under stronger assumptions. Thus the convergence of e-ADMM for the general case of m ≥ 4 is proved; this result seems to be still unknown even though it is intuitive given the known result of the case of m = 3. Even for the special case of m = 3, our convergence results turn out to be more general than the existing results that are derived specifically for the case of m = 3.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].     

Complex vector measure and integral over manifolds with locally finite variations

It is well known that any compactly supported continuous complex differential n-form can be integrated over real n-dimensional C¹ manifolds in C^m (m ≥ n). For n = 1, the integral along any locally rectifiable curve is defined. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞ differential forms). The topic of the article is the integration of measurable complex differential (n, 0)-forms (containing no \(d{\bar z_j}\)) over real n-dimensional C⁰ manifolds in C^m with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimensions n > 1). The last result is that a real n-dimensional manifold C¹ embedded in C^m has locally finite variations, and the integral of a measurable complex differential (n, 0)-form defined in the article can be calculated by a well-known formula.     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

Existence of saddle solutions of a nonlinear elliptic equation involving <Emphasis Type="Italic">p</Emphasis>-Laplacian in more even-dimensional spaces

We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p > 2, -Δ _p u = f(u) in R^2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m ≥ 2p.     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

Analysing the Wu metric on a class of eggs in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> – I

We study the Wu metric on convex egg domains of the form

$$E_{2m} = \big\{ z \in \mathbb{C}^{n}: \vert{z_{1}} \vert^{2m} + \vert {z_{2}} \vert^{2} + {\cdots} + \vert {z_{n-1}} \vert^{2} + \vert {z_{n}} \vert^{2} <1 \big\}, $$

where m ≥ 1/2,m ≠ 1. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be C ²-smooth. Overall however, the Wu metric is shown to be continuous when m = 1/2 and even C ¹-smooth for each m > 1/2, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such E _2m .     

Generalized <Emphasis Type="Italic">n</Emphasis>-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Let n ≥ 2 and let Ω ? ? ⁿ be an open set. We prove the boundedness of weak solutions to the problem

$$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$

where ? is a Young function such that the space W ₀ ¹ L ^Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L ^Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ? ⁿ .

     

On Semisimple Hopf Algebras of Dimension 2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>, II

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 2²ⁿ⁺¹ for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to \(\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}\). Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 ^m?1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 ^m which have this group as their group of group-like elements.     

Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.     

On extensions of characteristic functions

Here we deal with the following question: Is it true that, for any closed interval on the real line ? that does not contain the origin, there exists a characteristic function f such that f(x) coincides with the normal characteristic function \( {\mathrm{e}}^{-{x}^2/2} \) on this interval but f(x) ? \( {\mathrm{e}}^{-{x}^2/2} \) on ?? The answer to this question is positive. We study a more general case of an arbitrary characteristic function g of a continuous probability density, instead of \( {\mathrm{e}}^{-{x}^2/2} \).     

On a functional equation related to derivations in prime rings

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(x ^n+m+1) = (m + n + 1)(x ^m D(x)x ⁿ + x ⁿ D(x)x ^m ) for all \({x\in R}\). In this case R is commutative and D is a derivation.