Classical Dirichlet Research Articles

Convergence of linear combinations of iterates of an inner function

Let f be an inner function with f(0)=0 which is not a rotation and let fn be its n-th iterate. Let {an} be a sequence of complex numbers. We prove that the series ∑anfn(ξ) converges at almost every point ξ of the unit circle if and only if ∑|an|2<∞. The main step in the proof is to show that under this assumption, the function F=∑anfn has bounded mean oscillation. We also prove that F is bounded on the unit disc if and only if ∑|an|<∞. Finally we describe the sequences of coefficients {an} such that F belongs to other classical function spaces, as the disc algebra and the Dirichlet class.

Solvability of nonlocal Dirichlet problem for generalized Helmholtz equation in a unit ball

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found.

Resistance scaling on 4N-carpets

Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) &gt; 1 {\rho=\rho(N)&gt;1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

An alternative capacity in metric measure spaces

A new condenser capacity \( {\mathrm{Cap}}_{\mathrm{P}}^{\mathrm{M}}\left(E,G\right) \) is introduced as an alternative to the classical Dirichlet capacity in a metric measure space X. For p > 1, it coincides with the Mp-modulus of the curve family Γ(E, G) joining 𝜕G to an arbitrary set E ⊂ G and, for p = 1, it lies between AM1(Γ(E, G)) and M1(Γ(E, G)). Moreover, the \( {\mathrm{Cap}}_{\mathrm{P}}^{\mathrm{M}}\left(E,G\right) \)-capacity has good measure theoretic regularity properties with respect to the set E. The \( {\mathrm{Cap}}_{\mathrm{P}}^{\mathrm{M}}\left(E,G\right) \)-capacity uses Lipschitz functions and their upper gradients. The doubling property of the measure μ and Poincaré inequalities in X are not needed.

Scientific Value Weights more than Being Open or Toll Access: An analysis of the OA advantage in Nature and Science

Abstract Purpose We attempt to find out whether OA or TA really affects the dissemination of scientific discoveries. Design/methodology/approach We design the indicators, hot-degree, and R-index to indicate a topic OA or TA advantages. First, according to the OA classification of the Web of Science (WoS), we collect data from the WoS by downloading OA and TA articles, letters, and reviews published in Nature and Science during 2010–2019. These papers are divided into three broad disciplines, namely biomedicine, physics, and others. Then, taking a discipline in a journal and using the classical Latent Dirichlet Allocation (LDA) to cluster 100 topics of OA and TA papers respectively, we apply the Pearson correlation coefficient to match the topics of OA and TA, and calculate the hot-degree and R-index of every OA-TA topic pair. Finally, characteristics of the discipline can be presented. In qualitative comparison, we choose some high-quality papers which belong to Nature remarkable papers or Science breakthroughs, and analyze the relations between OA/TA and citation numbers. Findings The result shows that OA hot-degree in biomedicine is significantly greater than that of TA, but significantly less than that of TA in physics. Based on the R-index, it is found that OA advantages exist in biomedicine and TA advantages do in physics. Therefore, the dissemination of average scientific discoveries in all fields is not necessarily affected by OA or TA. However, OA promotes the spread of important scientific discoveries in high-quality papers. Research limitations We lost some citations by ignoring other open sources such as arXiv and bioArxiv. Another limitation came from that Nature employs some strong measures for access-promoting subscription-based articles, on which the boundary between OA and TA became fuzzy. Practical implications It is useful to select hot topics in a set of publications by the hot-degree index. The finding comprehensively reflects the differences of OA and TA in different disciplines, which is a useful reference when researchers choose the publishing way as OA or TA. Originality/value We propose a new method, including two indicators, to explore and measure OA or TA advantages.

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps.

Uncertainty principles of Heisenberg type on Dirichlet space

In this paper, we introduce a family of Dirichlet spaces $$\{{\mathcal {D}}_n\}_{n\in {\mathbb {N}}}$$ . This family satisfies the continuous inclusions $${\mathcal {D}}_{n}\subset \cdots \subset {\mathcal {D}}_{2}\subset {\mathcal {D}}_{1}\subset {\mathcal {D}}_0={\mathcal {D}}$$ , where $${\mathcal {D}}$$ is the classical Dirichlet space. Next, we define and study the operator $$Xf(z):=f'(z)-f'(0)$$ and its adjoint operator $$Yf(z)=z^2f'(z)$$ on the Dirichlet space $${\mathcal {D}}$$ , and we establish an uncertainty inequality of Heisenberg type for this space. A more general uncertainty inequality for the space $${\mathcal {D}}_n$$ is also given when we considered the operators $$X_n=X^n$$ and $$Y_n=Y^n$$ .

Holomorphic function spaces on homogeneous Siegel domains

We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results from the more particular cases of the upper half-plane, Siegel domains of tube type over irreducible cones, or symmetric, irreducible Siegel domains of type II.

Radial growth of the derivatives of analytic functions in Besov spaces

Abstract For 1 &lt; p &lt; ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ 𝔺 : |z| &lt; 1} and satisfy ∫𝔻(1 − |z|2) p −2|f ′(z)| p dA(z) &lt; ∞. The space B 2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ )| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 &lt; p &lt; ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.

Blow-up phenomena in nonlocal eigenvalue problems: When theories of L1 and L2 meet

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu=λu+f involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators.Without singular boundary datum and when λ is not an eigenvalue of the operator, we construct an L2-projected theory of solutions, which we extend to the optimal space of data for the operator L. We present a Fredholm alternative as λ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function.We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value λ, there exist “large eigenfunctions” that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as λ approaches the values of the spectrum.We also obtain a maximum principle for weighted L1 solutions when the operator is L2-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below.Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green functions. Thus “large eigenfunctions” represent a purely nonlocal phenomenon.

On the treatment of boundary conditions for bond-based peridynamic models

In this paper, we propose two approaches to apply boundary conditions for bond-based peridynamic models. There has been in recent years a renewed interest in the class of so-called non-local models, which include peridynamic models, for the simulation of structural mechanics problems as an alternative approach to classical local continuum models. However, a major issue, which is often disregarded when dealing with this class of models, is concerned with the manner by which boundary conditions should be prescribed. Our point of view here is that classical boundary conditions, since applied on surfaces of solid bodies, are naturally associated with local models. The paper describes two methods to incorporate classical Dirichlet and Neumann boundary conditions into bond-based peridynamics. The first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points. The second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the so-called horizon vanishes. The approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions. We present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a one-dimensional bar.

A Unified Bayesian Model for Generalized Community Detection in Attribute Networks

Identification of community structures and the underlying semantic characteristics of communities are essential tasks in complex network analysis. However, most methods proposed so far are typically only applicable to assortative community structures, that is, more links within communities and fewer links between different communities, which ignore the rich diversity of community regularities in real networks. In addition, the node attributes that provide rich semantics information of communities and networks can facilitate in-depth community detection of structural information. In this paper, we propose a novel unified Bayesian generative model to detect generalized communities and provide semantic descriptions simultaneously by combining network topology and node attributes. The proposed model is composed of two closely correlated parts by a transition matrix; we first apply the concept of a mixture model to describe network regularities and then adjust the classic Latent Dirichlet Allocation (LDA) topic model to identify community semantically. Thus, the model can detect broad types of network structure regularities, including assortative structures, disassortative structures, and mixture structures and provide multiple semantic descriptions for the communities. To optimize the objective function of the model, we use an effective Gibbs sampling algorithm. Experiments on a number of synthetic and real networks show that our model has superior performance compared with some baselines on community detection.

Gaussian analytic functions and operator symbols of Dirichlet type

Let M={mj,k}j,k=1+∞ be an infinite complex-valued matrix which acts contractively on ℓ2. For the weighted short diagonal sumsSM(l):=∑j,k:j+k=l(ljk)12mj,k, we obtain the estimate∑l=2+∞sll|SM(l)|2≤2slog⁡e1−s,0≤s<1. Expressed more vaguely, |SM(l)|2⪅2 holds in the sense of averages. Concerning the optimality of the above bound, a construction due to Zachary Chase shows that the statement does not hold if the number 2 is replaced by the smaller number 1.72. In the construction, M is a permutation matrix. We interpret our bound in terms of the correlation EΦ(z)Ψ(z) of two copies of a Gaussian analytic function with possibly intricate Gaussian correlation structure between them. The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally Möbius invariant. The study of the correlations EΦ(z)Ψ(z) leads us to introduce a new space, the mock-Bloch space (or Blochish space), which is slightly bigger than the standard Bloch space. Our bound has an interpretation in terms of McMullen's asymptotic variance, originally considered for functions in the Bloch space. Finally, we show that the correlations EΦ(z)Ψ(w) may be expressed as Dirichlet symbols of contractions on L2(D), and show that the Dirichlet symbols of Grunsky operators associated with univalent functions find a natural characterization in terms of a nonlinear wave equation.

Division algorithm and Kronecker’s approximation theorem

Using division algorithm and basic notions of convergence of sequences in real–line, we prove that a real number $$\theta$$ is irrational if and only if there is an eventually nonconstant sequence $$\{p_n\theta +q_n\}$$ converging to 0, where $$p_n$$ and $$q_n$$ are integers for each natural number n. This approach leads to alternative proofs of weaker versions of the classical Dirichlet and Kronecker approximation theorems in number theory.

A note on the Monge–Ampère type equations with general source terms

In this paper we consider numerical approximation to the generalised solutions to the Monge–Ampère type equations with general source terms. We first give some important propositions for the border of generalised solutions. Then, for both the classical and weak Dirichlet boundary conditions, we present well-posed numerical methods for the generalised solutions with general source terms. Finally, we prove that the numerical solutions converge to the generalised solution.

On a nonlocal boundary value problem with an oblique derivative

The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the Rn space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique.

Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.

Blaschke Products and Zero Sets in Weighted Dirichlet Spaces

In this paper, we deal with superharmonically weighted Dirichlet spaces $\mathcal {D}_{\omega }$ . First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for $\mathcal {D}_{\omega }$ . Our conditions improve Shapiro-Shields condition for $\mathcal {D}_{\alpha }$ , when α ∈ (0,1).

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Abstract Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d &gt; 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d &gt; 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Homogenization of nonlinear equations with mixed boundary conditions

In this paper, we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed Dirichlet-Robin boundary conditions. Thanks to the smoothing operator as well as homogenization tools, we could handle the different boundary conditions in a uniform fashion. As a consequence, we establish the sharp rates of convergence in H1 and L2, which may be regarded as an extension from the classical linear equations Dirichlet or Neumann problems to a nonlinear case with the mixed boundary settings.