Search

The aim of this investigation is to study the effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam. The harmonic differential quadrature (HDQ) method is used to compute the natural frequency of Timoshenko beam. In this regards, three different types of meshing are used namely uniform meshing, normalized meshing and non-uniform. The results obtained using these meshing are compared with results obtained in previous published work under different types of boundary conditions. It is found that non-uniform meshing gives better convergence as well as good results as compare to results using uniform and normalized meshing under all boundary conditions.

This paper proposes a particular type of Stokes-Dirac structure for describing a Laplacian used in Poisson’s equations on topologically non-trivial manifolds, i.e., not Euclidian. The operator matrix representation of the structure includes not only exterior differential operators, but also codifferential operators in the sense of the dual of the pairing between differential forms. Since the successive operation of the matrix is equivalent to the Laplace-Beltrami operator, we call it a Stokes-Dirac operator. Furthermore, the Stokes-Dirac operator is augmented by harmonic differential forms that reflect the topological geometry of manifolds. The extension enable us to describe a power balance of particular boundary energy flows on manifolds with a non-trivial shape.

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture., Comment: 28 pages. Supercedes arXiv:1908.00196

A highly accurate method for obtaining static deflections of a thin annular plate is helpful to effectively design the complicated structures with these plates. There have been numerous methods to achieve such a target. However, there is no direct technical literature for comparing these methods comprehensively. Therefore, the current study aims at performing comparison of three methods, optimization method (OM), finite element method (FEM), and harmonic differential quadrature (HDQ) method. Combining an instance, the comparisons give us insight into high accuracy and consistency of each other, showing high accuracy of the methods in this field. Compared with the results of FEM, the maximum error, less than 1%, demonstrates that the accuracy of the OM is high enough. Combining the small errors, the excellent stability of those brings a reliable method in this field. The maximum error and fluctuation drawn from the HDQ are evidently larger than those of the OM, and it is difficult to obtain results with higher accuracy based on the HDQ. Finally, the work described here suggests that the OM can be utilized to deal with such a complex problem in view of engineering and theory, and the HDQ method is more suitable to study the method for solving very complex partial differential systems of high order.

The priority of this paper is to explore the computational characteristics of Harmonic Differential Quadrature (HDQ) method for free flexural vibration analysis of functionally graded piezoelectric material (FGPM) beam. The modified Timoshenko beam theory is used in this study where electric potential is assumed to have sinusoidal variation across the depth. The equations of motion and boundary conditions are derived by employing Hamilton‘s principle. The material properties are assumed to have a power law or sigmoid law variation across the depth. The available equations of motion are then solved using the Harmonic Differential Quadrature (HDQ) method to obtain the natural frequencies of the FGPM beam. The efficacy of the present method is validated by comparing the results with the previous published work. It is observed that the results obtained by HDQ method are at the same level of accuracy than those of previous analysis using GDQ method.

A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.

Consider the bi-harmonic differential expression of the form $ A=\triangle _{M}^{2}+q\ $ on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = ΔE2 + q to be separated in the space L2 (M) on a complete Riemannian manifold (M,g) with metric g, where ΔE is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L2 (M) if for all u ∈ L2 (M) such that Au ∈ L2 (M) we have ΔE2u ∈ L2 (M) and qu ∈ L2 (M).

Generalising a conjecture of Singerman, it is shown that there are orientably regular chiral hypermaps (equivalently regular chiral dessins d'enfants) of every non-spherical type. The proof uses the representation theory of automorphism groups of Riemann surfaces acting on homology and on various spaces of differentials. Some examples are given.

International audience; For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška –Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. In three dimensions, Velte [9] studied a generalization of the notion of conjugate harmonic functions and corresponding generalizations of Friedrich's inequality, and he showed for sufficiently smooth simply-connected domains the equivalence with inf-sup conditions for the divergence and for the curl. For this equivalence, Zsuppán [10] observed that our proof can be adapted, proving the equality between the corresponding constants without regularity assumptions on the domain. Here we formulate a generalization of the Friedrichs inequality for conjugate harmonic differential forms on bounded open sets in any dimension that contains the situations studied by Horgan–Payne and Velte as special cases. We also formulate the corresponding inf-sup conditions or Babuška –Aziz inequalities and prove their equivalence with the Friedrichs inequalities, including equality between the corresponding constants. No a-priori conditions on the regularity of the open set nor on its topology are assumed.

Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional (2D) elasticity theory without any pre-assumption on the displacement field. The top and bottom faces act like the elastic supports on the top and bottom edges of the core. The differential equations of the 2D core are then solved by the harmonic differential quadrature method (HDQM). To circumvent the difficulties in dealing with the locally distributed load by point discrete methods such as the HDQM, a general and rigorous way is proposed to treat the locally distributed load. Detailed formulations are provided. The static behavior of sandwich panels under different locally distributed loads is investigated. For verification, results are compared with data obtained by ABAQUS with very fine meshes. A high degree of accuracy on both displacement and stress has been observed.

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham–Hodge–Kodaira decomposition as well as Weitzenböck and Clark–Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms.

We present a framework for the construction of Weitzenbock and Clark-Ocone formulae for differential forms on the probability space of a normal martingale. This approach covers existing constructions based on Brownian motion, and extends them to other normal martingales such as compensated Poisson processes. It also applies to the path space of Brownian motion on a Lie group and to other geometries based on the Poisson process. Classical results such as the de Rham-Hodge-Kodaira decomposition and the vanishing of harmonic differential forms are extended in this way to finite difference operators by two distinct approaches based on the Weitzenbock and Clark-Ocone formulae.

This study reports the results of an investigation into the static analysis of microbeams based on nonlocal thermal elasticity theory by differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. To show the accuracy of the method, the results of present work are compared with those of other works. Different parameters such as temperature, length-to-thickness ratio, length-to-width ratio and boundary conditions are studied, too. From the knowledge of author, it is the first time that results for static analysis of microbeams by nonlocal thermal elasticity theory in thermal environment using two types of differential quadrature method are investigated and the results may be used as benchmarks for the future works.

In this paper we express the theory of harmonic differential forms on conformal Euclidean manifolds in terms of the so called Clifford multivector fields. The aim is to give good definitions for d and d* operators in Clifford multivector case. Using these definitions we derive a formula for the Laplace operator. Three fundamental examples are included in the end of the paper and connections to existing theory is discussed.

In this paper, the differential quadrature (DQ) method is presented for easy and effective analysis of isotropic functionally graded (FG) and functionally graded coated (FGC) thin plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction. The bending of FG and FGC plates under transverse loading has been studied using the polynomial differential quadrature (PDQ) and the harmonic differential quadrature (HDQ) methods. A three-dimensional elasticity solution for a moderately thick FG plate with exponential Young’s modulus is used as the benchmark. Two examples, including a thin FG rectangular plate and a thin FGC rectangular plate with sigmoidal Young’s modulus, are investigated. The numerical results of PDQ and HDQ methods reveal good agreement with other solutions. Also, it is shown that the formulations for thin FG plates and homogeneous plates are similar, except that the plane strain components of the middle surface in FG plates are not zero.

Given a complete ortho-normal system ϕ = (ϕ0,ϕ1,ϕ2,…) of L2H(\( \mathcal{D} \)), the space of holomorphic and absolutely square integrable functions in the bounded domain \( \mathcal{D} \) of ℂn, we construct a holomorphic imbedding \( \iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty ) \), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \( \mathfrak{F}(n,\infty ) \) there are n closed (μ, μ)-forms τμ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \( \mathfrak{F}(n,\infty ) \) and generate algebraically all the harmonic differential forms of \( \mathfrak{F}(n,\infty ) \). So we obtain in \( \mathcal{D} \) a set of (μ, μ)-forms ιϕ*τμ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of \( \mathcal{D} \). Especially the differential metric ds21 associated to the Kahler form ιϕ*τ1 is a Kahler metric which differs from the Bergman metric ds2 of \( \mathcal{D} \) in general, but in case that the Bergman metric is an Einstein metric, ds12 differs from ds2 only by a positive constant factor.

Let Fr (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of \({\mathbb{R}}^{m+1}\) satisfying \(\partial F_r = 0\) in Ω, where ∂ is the Dirac operator in \({\mathbb{R}}^{m+1}\) . Then it is proved that there exists Hr, an r-vector valued harmonic function in Ω, such that Fr = \(\partial H_r\partial\). Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon primitivation of monogenic functions.

The space HFk(Ω) of harmonic multi-vector fields in a domain \(\Omega \subset \mathbb{R}^{n}\) as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HFk(E) being \(\mathbf{E} \subset \mathbb{R}^{n}\) a compact set. It is proved that HFk(E)* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity.

The geometrically nonlinear static and dynamic analysis of thin rectangular plates resting on elastic foundation has been studied. Winkler–Pasternak foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse and an N-wave, are investigated and the results are presented graphically. The accuracy of the proposed HDQ–FD coupled methodology is demonstrated by the numerical examples.

An examination of the accuracy and convergence behaviors of polynomial basis function differential quadrature (PDQ) and harmonic basis function differential quadrature (HDQ) for free vibration analysis of variable thickness thick skew plates will be carried out. The plate governing equations are based on the first-order shear deformation theory including the effects of rotary inertia. Arbitrary thickness variations will be assumed yielding a system of equations with nonlinear spatial dependent coefficients. Differential quadrature (DQ) analogs of the equations are obtained by transforming the governing equations and boundary conditions into the computational domains. Studies are carried out to examine the effects of different types of boundary conditions, skew angles, and thickness-to-length ratios for thin as well as moderately thick plates. The thickness is simulated by bilinear or nonlinear functions. The results are compared with those of other numerical schemes. It is concluded that both PDQ and HDQ yield accurate solutions for natural frequencies both at low and high modes of vibration. Some new results are presented for skew plates with bilinearly varying thickness and for different sets of boundary conditions, which can be used as benchmarks for future works.

The non-linear static and dynamic response of doubly curved thin isotropic shells has been studied for the step and sinusoidal loadings. Dynamic analogues Von Karman-Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by the numerical examples. Numerical examples demonstrate the satisfactory accuracy, efficiency and versatility of the presented approach.

The main purpose of this study is to compare the methods of differential quadrature (DQ) and harmonic differential quadrature (HDQ). For this purpose, DQ and HDQ methods are presented for buckling, bending, and free vibration analysis of thin isotropic plates and columns. Plates of different shapes such as rectangular, circular, square, skew, trapezoidal, annular, and sectorial plate subjected to different boundary conditions are selected to demonstrate the accuracy of the method. Four different support conditions are taken into consideration for columns. Numerical results are presented to illustrate the method and demonstrate its efficiency. It is emphasized that the HDQ method gives more accurate results and needs less grid points than the DQ method. (C) 2003 Elsevier Ltd. All rights reserved.

Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type., Comment: 15 pages, LaTeX, 4 figures

The concept of behavior spaces introduced by Shiba plays an important role of systematic investigation of abelian differentials on an open Riemann surface. A Shiba behavior space consists of harmonic differentials which satisfy a certain period condition and boundary behavior. In this paper, for any open Riemann surface of infinite genus we construct Shiba behavior spaces with arbitrarily prescribed period condition and with specific boundary behavior.

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three-dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.

A variant of the classical theorem of Runge is established for harmonic differenrial forms on an open subset of ℝn. It generalizes the case of analytic functions for n=2. Harmonic forms with point singularities are introduced, and a theorem of displacement of poles is proved. An integral representation analogous to the Cauchy formula is constructed. Bibliography: 5 titles.

In this paper we consider harmonic differential forms and Clifford multi-vector functions on the hyperbolic upper half-space. We see how the operators and their solutions are related and present a Moisil-Theodorescu-type system related to the harmonic multi-vectors.

Analytical stress analyses are presented for orthotropic composite materials containing a through crack under uniaxial normal loads (mode I). A harmonic differential equation has been established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry by introducing new complex variables. The complex theory was employed to find stress functions to satisfy the equilibrium equation, compatibility equation and boundary condition at infinite and crack surfaces. An analytical solution was examined in the case of isotropic materials. It is demonstrated that the analytical solution obtained is correct for the orthotropic composite plates.

The structure of weighting coefficient matrices of Harmonic Differential Quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric depending on the order of the corresponding derivatives. The properties of both matrices are briefly discussed in this paper. It is noted that the computational effort of the harmonic quadrature for some problems can be further reduced up to 75 per cent by using the properties of the above-mentioned matrices., 12 pages, 1 table, Original MS. Word format, published in Common. Numer. Methods. Engrg

Stress analyses for orthotropic composite materials containing a through crack under remote shear loads (Mode II) are conducted. By employing the complex theory, a harmonic differential equation was established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry. An analytical complex function was introduced by following the Westergaard approach. Stress around a mode II crack in the orthotropic composite plate is deduced to have an analytical form. In addition, the analytical solution for a mode II crack was examined in the case of isotropic materials. It demonstrated that the analytical solution obtained is correct for the mode II cracked orthotropic composite plates.

A layer potential based approach for boundary value problems for harmonic differential forms in nonsmooth domains is developed. This allows a complete and unified treatment of several fundamental problems in potential theory.

A global method of generalised differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.

A classical subject in the theory of Riemann surfaces is the study of harmonic and analytic differentials. In particular it is well-known that the dimensions of the spaces of harmonic and analytic differentials with respect to the corresponding fields are 2g, where g is the topological genus of the surface. A further study of this classical setting about the connection between the topological and conformal structure with different spaces of differential forms is due to A. Pfluger, who obtained some important decomposition theorems in this connection derived from the Dirichlet principle. Here we shall consider some of these questions for Klein surfaces, that is, we allow the surface to be non-orientable or to have a boundary.

Harmonic differential quadrature method is developed for the free vibration analysis of linear elastic beams. In the method of differential quadrature, partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The weighting coefficients are treated as the unknowns. Applying this concept to the governing differential equation of beam gives a set of linear simultaneous equations, which are solved for the unknown weighting coefftcients by accounting for the boundary conditions. Beams of different support combinations such as clamped, simply supported, guided, and free are selected to demonstrate the accuracy of the method. Flexural vibration case is taken into consideration. First two frequencies are obtained in the applications. Numerical results are presented to illustrate the method and demonstrate its efficiency.

Recently, Bert, Wang and Striz [1, 2] applied the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods to analyze static and dynamic behaviors of anisotropic plates. Their studies showed that the methods were conceptually simple and computationally efficient in comparison to other numerical techniques. Based on some recent work by the present author [3, 4], the purpose of this note is to further simplify the formulation effort and improve computing efficiency in applying the DQ and HDQ methods for these cases.

We study a generalization of the Hadamard theorem on three circles to harmonic differential forms. An inequality for the L 2-norms of a harmonic form over concentric spheres is proved. Also, we obtain an estimate for the L ∞-norms.

This work is an attempt to find a multidimensional generalization of the planar theory of uniform rational approximation. Harmonic differential forms in ℝn are considered as analogues of analytic functions in ℂ, whereas rational functions of a complex variable are replaced by the so-called Biot-Savard forms (with singularities on appropriate cycles instead of points). A Runge-like theorem is proved. Theorems by Hartogs-Rosenthal and Rao (on approximation by harmonic gradients in ℝ3) are generalized to any dimension and any degree of forms.

This paper contains the proof of the assertion in its title. My motivation for considering this problem was the following. Let M be a compact differentiable manifold and let D be an elliptic differential operator between the spaces of C ~ sections of two bundles on M. In [4] M. F. Atiyah proves that if index D>0, then, for every Galois covering AT/---~M, the operator / ) induced by D on ~ /has nontrivial L 2 solutions/)u=0. Atiyah goes on to ask whether the same is true for coverings which are not Galois. His guess is that the answer is negative in general but that the counter-example will be difficult to construct. The simplest situation to consider in this connection is that of a surface and the operator whose index is the Euler characteristic. For infinite coverings, since every L 2 harmonic function would be constant and nonzero constants are not in L 2, Atiyah's question reduces to the question of existence of L 2 harmonic forms of degree one. It is shown here that a counter-example will not be found in this simple setting. From now on S will denote an oriented, compact surface of genus g=g(S)>l equipped with a smooth Riemannian metric. S--%S will be an arbitrary (usually infinite) covering of S, and A = A will be the Laplace operator on S with respect to the pull back metric. A differential (exterior form of degree one) on S is harmonic, i.e. Ao)=0, and in L 2 if and only if (cf. [15], Theorem 26)

Every open Riemann surface Rof finite genus can be continued to a closed Riemann surface of the same genus. This classical result is usually proved by a local argument: one considers only a planar neighborhood of the ideal boundary of Rand applies the generalized uniformization theorem of Koebe. In the present paper we prove a continuation theorem of a global character: Let there be given a meromorphic function f on R with a special boundary behavior. Im(df shall be a distinguished harmonic differential of Ahlfors. Then there exists a closed Riemann surface R∗ of the same genus as Rand a meromorphic extension f ∗ of f onto R ∗ such that (i) R ∗\Rhas a vanishing area, (ii) f ∗ is holomorphic on R ∗\R, and (iii) Im f ∗ assumes a constant value on each boundary component of R with respect to R ∗ Since ƒ describes a hydrodynamic phenomenon on R, we call R ∗ a hydrodynamic continuation of R with respect to f. The ideal boundary Ris then realized on R ∗as a set of arcs on the streamlines of f with a total vanis...

Generalizing harmonic differential forms (rot ω=0, div ω=0 in M, M being a smooth riemannian manifold with boundary) of first and second kind (ω=0 and *ω=0 on ∂M resp.) within the framework of Hilbert space notation, it is possible to extend the meaning of the boundary conditions to non-smooth boundaries. It turns out that in this case the classical result is still valid for certain open subregions G of M: The dimension of the space of harmonic differential forms of second kind is given by the q-th Betti number of G; *-duality leads to the respective result for harmonic differential forms of first kind.

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811) . In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.

On finite surfaces, the class of harmonic functions which are constant on each contour is a finite-dimensional vector space of functions with finite Dirichlet norm. This paper considers the corresponding class of functions on bordered surfaces of class SOg and generalizes some of the properties of harmonic measures on finite surfaces. In particular, for generalized harmonic measures, we investigate the level curves and their orthogonal trajectories. The principal results, Theorems 4.1 and 4.4, state, in a sense made precise, that almost all of the level curves of a generalized harmonic measure are analytic Jordan curves and almost all of their orthogonal trajectories begin and end on the border given in the definition of the surface. These results have application to the level curves of a Green's function via a theorem of Kuramochi. We also consider the question on a parabolic surface as to when a harmonic differential with finite norm and integral periods is a weak limit of period reproducing differentials.

1. Throughout this paper we denote by R an open Riemann surface and by Ro a relatively compact subdomain of R with the relative boundary aRo consisting of a finite number of mutually disjoint closed analytic Jordan curves. The open set R, = R Po can be considered to be a neighborhood of the ideal boundary 3 of R. For the sake of simplicity, we denote by a the common relative boundary aRo = aR and we fix the orientation of a positively with respect to the domain Ro. A harmonic differential a defined on R1 = R1 u a is called a harmonic singularity

There has been new interest developing in the study of the Hodge theory and p -harmonic differential forms in recent years, largely pertaining to applications in quasi-conformal analysis, electromagnetic theory, relativity theory, mathematical physics, nonlinear potential theory, etc. In this paper, we develop an effective method to find conjugate p -harmonic differential forms in R n by applying Hodge theory in calculations with the p -harmonic equation. As applications of the method, we obtain some interesting examples of conjugate p -harmonic differential forms.

The purpose of the present paper is to construct harmonic and holomorphic differentials with suitably prescribed periods on a compact Riemann surface of genus g. We use these constructions to investigate the variation of the period matrices along a curve W tµ in Teichmuller space given by a linear family tµ of Beltrami differentials. In the case of a Teichmuller geodesic, i.e., when tµ is a Teichmuller differential \(t\bar{Q}/|Q|\), we obtain some convexity properties for the real period matrix. The formulae we derive for the variation of the harmonic differential with given periods and for the period matrix are exact. We thus have the variations of all orders, and it is by looking at the second variation that we obtain our convexity result.

During the past fifteen years the theory of form-calculus has been mainly responsible for the development of differential geometry. The roots of the subject are to be found in E. Cartan’s famous paper of 1934 on Finster spaces, and later on the subject flourished at the hands of S. Chern, J. A. Schouten, K. Kodiara, S. Bochner, A. Hilt, S. Hseiung, and several others.