Search

This is a continuation of our previous works on entanglement entropy (EE) in interacting field theories. In arXiv:2103.05303, we have proposed the notion of $\mathbb{Z}_M$ gauge theory on Feynman diagrams to calculate EE in quantum field theories and shown that EE consists of two particular contributions from propagators and vertices. As shown in the next paper arXiv:2105.02598, the purely non-Gaussian contributions from interaction vertices can be interpreted as renormalized correlation functions of composite operators. In this paper, we will first provide a unified matrix form of EE containing both contributions from propagators and (classical) vertices, and then extract further non-Gaussian contributions based on the framework of the Wilsonian renormalization group. It is conjectured that the EE in the infrared is given by a sum of all the vertex contributions in the Wilsonian effective action., Comment: 29 pages, 10 figures; typos corrected, published version in Symmetry (v2)

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given., MTM2017-89677-P, MINECO/ AEI/FEDER, UE.

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SUq(2) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck’s constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck’s constant and a Hilbert space where natural, densely defined operators act.

Let $M$ be a Riemannian manifold and $\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\Omega$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? In this paper we give an affirmative answer to this question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the $r$-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem., Comment: 41 pages

In this paper, we characterize the pointwise slant submersions from cosymplectic manifolds onto Riemannian manifolds and give several examples. In this paper, we characterize the pointwise slant submersions from cosymplectic manifolds onto Riemannian manifolds and give several examples.

The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions; the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T^3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T^2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T^3-Gowdy symmetry to this list of requirements, we obtain C^0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter., Comment: 72 pages. Revised version to appear in JEMS

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection ∇ with skew-symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar ∇-curvature, it is necessarily compact and it has finite first fundamental group π1. The longest part of the paper is devoted to the systematic construction of large families of examples.We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion.We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein–Sasaki manifold and any seven-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion., FCT, CMAT, PEst-C/MAT/UI0013/2011 and PTDC/MAT/118682/2010.

In this work, we propose to approximate the Gaussian integral, the error function and the cumulative distribution function by using the power series extender method (PSEM). The approximations proposed in this paper present a high accuracy for the complete domain [–∞,∞]. Furthermore, the approximations are handy and easy computable, avoiding the application of special numerical algorithms. In order to show its high accuracy, three case studies are presented with applications to science and engineering.

We study the definition of trace anomalies for models of Dirac and Weyl fermions coupled to a metric and a gauge potential. While in the non-perturbative case the trace anomaly is the response of the effective action to a Weyl transformation, the definition in a perturbative approach is more involved. In the latter case we use a specific formula proposed by M.Duff, of which we present a physical interpretation. The main body of the paper consists in deriving trace anomalies with the above formula and comparing them with the corresponding non-perturbative results. We show that they coincide and stress the basic role of diffeomorphism invariance for the validity of the approach., 30 pages, to be published in "Symmatry: Manifest and Hidden Symmetries in Field and String Theories"

Recently in various theoretical works, path-breaking progress has been made in recovering the well-known Page Curve of an evaporating black hole with Quantum Extremal Islands, proposed to solve the long-standing black hole information loss problem related to the unitarity issue. Motivated by this concept, in this paper, we study cosmological circuit complexity in the presence (or absence) of Quantum Extremal Islands in the negative (or positive) Cosmological Constant with radiation in the background of Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) space-time i.e the presence and absence of islands in anti-de Sitter and the de Sitter spacetime having SO(2, 3) and SO(1, 4) isometries respectively. Without using any explicit details of any gravity model, we study the behaviour of the circuit complexity function with respect to the dynamical cosmological solution for the scale factors for the above-mentioned two situations in FLRW space-time using squeezed state formalism. By studying the cosmological circuit complexity, Out-of-Time Ordered Correlators, and entanglement entropy of the modes of the squeezed state, in different parameter spaces, we conclude the non-universality of these measures. Their remarkably different features in the different parameter spaces suggest their dependence on the parameters of the model under consideration., Comment: 75 pages, 29 figures, 4 tables, Dr. Sayantan Choudhury would like to dedicate this work to his lovable father and prime inspiration Professor Manoranjan Choudhury who recently have passed away due to COVID 19. Updated and revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: Manifest and Hidden Symmetries in Field and String Theories)

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.

In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski space L3 to be f-biharmonic. Then, using these equations, we classify such unit speed curves in L3.

In this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent { - Δ ⁢ u + V ⁢ ( x ) ⁢ u + q 2 ⁢ ϕ ⁢ u = μ ⁢ | u | p - 1 ⁢ u + | u | 4 ⁢ u in ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in ⁢ ℝ 3 , \left\{\begin{aligned} \displaystyle{}{-}\Delta u+V(x)u+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right. where μ > 0 {\mu>0} is a parameter and 2 < p < 5 {2 . Under certain assumptions on V, we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

Let M be an almost cosymplectic 3-h-a-manifold. In this paper, we prove that the Ricci operator of M is transversely Killing if and only if M is locally isometric to a product space of an open interval and a surface of constant Gauss curvature, or a unimodular Lie group equipped with a left invariant almost cosymplectic structure. Some corollaries of this result and some examples illustrating main results are given.

In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.

The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a basic inequality become equality. The next purpose of this paper to find the necessary condition admitting gradient Ricci soliton, that the warped product semi-slant submanifold of Kenmotsu space form, is an Einstein warped product. We also discuss some obstructions to these constructions in more detail.

In this paper, we have investigated Noether symmetries of the Lagrangian of Kantowski&ndash, Sachs spacetime. The associated Lagrangian of the Kantowski&ndash, Sachs metric is used to derive the set of determining equations. Solving the determining equations for several values of the metric functions, it is observed that the Kantowski&ndash, Sachs spacetime admits the Noether algebra of dimensions 5, 6, 7, 8, 9, and 11. A comparison of the obtained Noether symmetries with Killing and homothetic vectors is also presented. With the help of Noether&rsquo, s theorem, we have presented the expressions for conservation laws corresponding to all Noether symmetries. It is observed that the positive energy condition is satisfied for most of the obtained metrics.

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a $P_n$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph., 49 pages. v2: many proofs rewritten and the paper is split into two parts

One-qubit pure states, living on the surface of Bloch sphere, can be mapped onto the usual complex plane by using stereographic projection. In this paper, after reviewing the entanglement of two-qubit pure state, it is shown that the \emph{quaternionic} stereographic projection is related to concurrence measure. This is due to the fact that every two-qubit state, in ordinary complex field, corresponds to the one-qubit state in quaternionic skew field, called quaterbit. Like the one-qubit states in complex field, the stereographic projection maps every quaterbit onto a quaternion number whose complex and quaternionic parts are related to Schmidt and concurrence terms respectively. Rather, the same relation is established for three-qubit state under \emph{octonionic} stereographic projection which means that if the state is bi-separable then, quaternionic and octonionic terms vanish. Finally, we generalize recent consequences to $2 \otimes N$ and $4\otimes N$ dimensional Hilbert spaces ($N\geq 2$) and show that, after stereographic projection, the quaternionic and octonionic terms are entanglement sensitive. These trends are easily confirmed by direct computation for general multi-particle W- and GHZ-states., Published version, 20 pages, 1 figure

As a continuation of the previous paper [7], we shall consider in this paper the problem of prediction given bounded intervals and obtain integral representations of predictors and prediction errors. For that purpose we shall introduce innovation processes well matched with bounded intervals. We follow the notation and terminology in [6].

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

The main purpose of this paper is to give a method to evaluate the actual value of the filtering error which arises in the transmission of a signal process, using the optimal coding, over a Gaussian channel. In his earier papers ([4] and [7]), the author has shown a method to construct an optimal causal coding for which the filtering error is minimized and at the same time the mutual information is maximized.

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

It is known that the difference tensor \(R \cdot C - C \cdot R\) and the Tachibana tensor \(Q(S,C)\) of any semi-Riemannian Einstein manifold \((M,g)\) of dimension \(n \ge 4\) are linearly dependent at every point of \(M\). More precisely \(R \cdot C - C \cdot R = (1/(n-1))\, Q(S,C)\) holds on \(M\). In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies \(R \cdot C = C \cdot R = Q(S,C) = 0\). Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors \(R \cdot C - C \cdot R\) and \(Q(S,C)\) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.

We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2., 31 pages, 1 figure, partially replaces arXiv:1010.0992

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $\sigma$ and $\tau$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics $g$ for the $C^0$-topology., Comment: Proposition 1.6 changed, error in the proof of this result in the previous version of the article

In this paper, we investigate the existence of foliations by constant mean curvature (CMC) hypersurfaces in maximal, globally hyperbolic, spatially compact, spacetimes of constant curvature. ¶ In the non-positive curvature case (i.e. for flat and locally anti-de Sitter spacetimes), we prove the existence of a global foliation of the spacetime by CMC Cauchy hypersurfaces. The positive curvature case (i.e. locally de Sitter spacetimes) is more delicate: in general, we are only able to prove the existence of a foliation by CMC Cauchy hypersurfaces in a neighbourhood of the past (or future) singularity. ¶ Except in some exceptional and elementary cases, the leaves of the foliation we construct are the level sets of a time function, and the mean curvature of the leaves increases with time. In this case, we say that the spacetime admits a CMC time function. ¶ Our proof is based on using the level sets of the cosmological time function as barriers. A major part of the work consists of proving the required curvature estimates for these level sets. One of the difficulties is the fact that the local behaviour of the cosmological time function near one point depends on the global geometry of the spacetime.

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also define the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose Inequality. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the Positive Mass Theorem that allows for certain types of incomplete metrics., 36 pages, 9 figures

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (greater than 280) of a general polarized K3 surface, then the second gaussian map of C is surjective. The resulting bound for the genus g of a general curve with surjective second gaussian map is decreased to g >152., final version, to appear in Nagoya Mathematical Journal

An important and natural question in the analysis of Ricci flow behavior in all dimensions n ≥ 4 is this: What are the weakest conditions that guarantee that a solution remains smooth? In other words, what are the weakest conditions that provide control of the norm of the full Riemann curvature tensor? In this short paper, we show that the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, for all compact solutions in all dimensions n ≥ 3, without any hypotheses on the initial data.

We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

Certain perturbative aspects of two-dimensional sigma models with (0,2) supersymmetry are investigated. The main goal is to understand in physical terms how the mathematical theory of ``chiral differential operators'' is related to sigma models. In the process, we obtain, for example, an understanding of the one-loop beta function in terms of holomorphic data. A companion paper will study nonperturbative behavior of these theories., 58 pp, minor corrections

This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness., 33pp, gap in one proof fixed, exposition improved. To appear in Advances in Math

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

On the half space Rn × R+, it has been known that harmonic Bergman space bp can contain a positive function only if . Thus, for , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.

Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

In this paper we first prove a simple relation between sums of a certain type and quadratic residue symbols. Then we apply this to Dedekind sums introduced by Sczech [5]. In particular one of his conjectures in [6] will be proved.

In this paper, we use the theta function method [10] to give explicit Doi-Naganuma type maps associated to an imaginary quadratic field K, lifting cusp forms on any congruence subgroup of SL(2, Z) to forms on SL(2, C) automorphic with respect to an appropriate arithmetic discrete subgroup. The case of class number one, and form modular with respect to group Γ0(D) and character χ0 = (–D/*), where –D is the discriminant of K has been treated by Asai [1]. In order to complete his discussion, we must first introduce a more general theta function associated to an indefinite quadratic form (here of type (3, 1)), which we regard as a specialization of a symplectic theta function (see also [4]).

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

In the present paper, we are concerned with the problem to know whether two algebraic Riemann manifolds are isometric or not, where we mean Riemann manifolds of class CΩ (real algebraic smoothness or Nash category) simply by algebraic Riemann manifolds.For this purpose we introduce notions of minimal differential polynomials and singular base points, both of which are isometry invariants. With the aid of these invariants, we give the following isometry theorem for algebraic Riemann manifolds

Throughout this paper, K will be a field of characteristic zero. Let K ‹x1,…, xm › be the K-algebra in m variables x1…, xm and Im, n the T-ideal consisting of all polynomial identities satisfied by m n by n matrices. The ring R(n, m) = K ‹x1,…, xm ›/Im, n is called the ring of m generic n by n matrices.

Let C1[0, T] denote (one-parameter) Wiener space; that is the space of continuous functions x on [0, T] such that x(0) = 0. In a recent expository essay [21], Nelson calls attention to some functions on Wiener space which were discussed in the book of Feynman and Hibbs [13, section 3-10] and in Feynman’s original paper [12, section 13]. These functions have the form

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

The aim of this paper is to discuss the multiplicity of the sum of two independent Gaussian processeswhere x1(t) is a Wiener process and is a simple Markov process.

In their paper [12], Toda and the author have concerned themselves in the following Theorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class O HB (O HD , resp.). Then, for any compact subset K of R such that R−K is connected, R−K as an open Riemann surface belongs to the class O AB (O AD , resp .) (Kuramochi [4]). They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.