Jensen's Formula Research Articles

KRONECKER LIMIT FUNCTIONS AND AN EXTENSION OF THE ROHRLICH–JENSEN FORMULA

Abstract In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$ . In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$ , thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$ , where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

POISSON - JENSEN FORMULAS AND BALAYAGE OF MEASURES

Our main results are certain developments of the classical Poisson--Jensen formula for subharmonic functions. The basis of the classical Poisson--Jensen formula is the natural duality between harmonic measures and Green's functions. Our generalizations use some duality between the balayage of measures and and their potentials.

Lelong–Jensen formula, Demailly–Lelong numbers and weighted degree of positive supercurrents

The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to weakly positive supercurrents on the real superspaces. We generalize then a result of Lagerberg when the supercurrent is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of . The main tool is a variant of the well-known Lelong–Jensen formula in the superformalism case. Moreover, we extend to our setting various interesting theorems in complex analysis such as Demailly and Rashkovskii comparison theorems. We also complete the work begun by Lagerberg on the degree of positive closed supercurrents and we prove a removable singularities result for positive supercurrents.

Characteristics of Variability in the Water Consumption by Forest and Meadow Ecosystems of the Arid Region of Central and South Ningxia

The state and development of vegetation cover is an important criterion for the improvement of the ecological environment of arid regions; therefore, the study of the ecological water use of vegetation has become a pressing problem in ecology and hydrology. This study covered eight counties in the central and southern parts of Ningxia Hui Autonomous Region, which are located from north to south in an arid region in northwestern China. The purpose of the study was to assess the potential evaporation and environmental water consumption of local vegetation based on meteorological data, vegetation-distribution data, data on the state of water resources, etc. This study can help us to understand and assimilate patterns in the spatiotemporal distribution of ecological water consumption and can provide a basis for the planning and cultivation of forest–meadow vegetation in a region. First, the models of Thornthwaite and Penman-Monteith were used to calculate ecological water consumption. Comparison of the calculation results showed that the data obtained from the Penman-Monteith model were more acceptable, since the model uses a number of meteorological variables and geographic location factors. At the same time, the Jensen formula and the regional soil characteristics curve were used in the calculation of the ecological water consumption to determine the factor for soil moisture correction. Second, the potential evaporation and ecological water consumption of local vegetation was estimated month by month based on precipitation. The spatial and temporal variability of potential values was analyzed on this basis. The results showed that th epotential evaporation tends to increase from month to month from January to July and decrease from August to December. Regarding the spatial distribution, the potential evaporation gradually increases from south to north. The spatial variability of the balance between precipitation and the ecological water consumption of vegetation was analyzed; the results showed that the ecological reserves of water in the central region are more substantial than in the southern region, and the largest reserves of water were found in Yanchi, the northernmost district of the central arid region. Conversely, the ecological water consumption in forests was excessive throughout the growing season in the southernmost district of Jingyuan. In addition, the spatiotemporal variability of the relationship between precipitation-dependent ecological water consumption and water resources is discussed. The results showed that there is still enough space for the regional distribution of vegetation in Yanchi, Tongxin, and Haiyuan in the central arid region and relatively dry districts, such as Yuanzhou, Siji and Pengyang in the southern highlands. More land for an increase in vegetation was observed in the Longde districts and Jingyuan, located in the south of a highland where there is a relatively high amount of precipitation.

Log-biharmonicity and a Jensen formula in the space of quaternions

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.

Superharmonic functions associated with hypoelliptic non-Hörmander operators

In this paper, we consider a class of degenerate-elliptic linear operators [Formula: see text] in quasi-divergence form and we study the associated cone of superharmonic functions. In particular, following an abstract Potential-Theoretic approach, we prove the local integrability of any [Formula: see text]-superharmonic function and we characterize the [Formula: see text]-superharmonicity of a function [Formula: see text] in terms of the sign of the distribution [Formula: see text]; we also establish some Riesz-type decomposition theorems and we prove a Poisson–Jensen formula. The operators involved are [Formula: see text]-hypoelliptic but they do not satisfy the Hörmander Rank Condition nor subelliptic estimates or Muckenhoupt-type degeneracy conditions.

Geometry of the scaling site

In this paper we construct the scaling site \({\mathscr {S}}\) by implementing the extension of scalars on the arithmetic site \({\mathscr {A}}\), from the smallest Boolean semifield \({\mathbb B}\) to the tropical semifield \({\mathbb R}_+^\mathrm{max}\). The obtained semiringed topos is the Grothendieck topos \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\), semi-direct product of the Euclidean half-line and the monoid \({\mathbb N}^{\times }\) of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\) coincides with the adele class space of \({\mathbb Q}\) and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime p and obtain a quasi-tropical structure which turns this orbit into a variant \(C_p={\mathbb R}_+^*/p^{\mathbb Z}\) of the classical Jacobi description \({\mathbb C}^*/q^{\mathbb Z}\) of an elliptic curve. On \(C_p\), we develop the theory of Cartier divisors, determine the structure of the quotient \(\mathrm{Div}(C_p)/{\mathcal P}\) of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann–Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of \({\mathscr {S}}\) by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen formula in the complex case.

A note on Jensen formula in Cn

We provide some natural weighted generalizations of certain sharp assertions related with multidimensional Jensen formula which served as a base of recent sharp theorems on generators for spaces of entire functions for several complex variables in Cn.

The Dirichlet problem and the inverse mean‐value theorem for a class of divergence form operators

The aim of this paper is to study some classes of second-order divergence-form partial differential operators ℒ of sub-Riemannian type. Our main assumption is the C∞-hypoellipticity of ℒ, together with the existence of a well-behaved fundamental solution Γ(x, y) for ℒ. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the ℒ-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to Mr; the role of Mr in solving the homogeneous Dirichlet problem related to ℒ in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Ωr(x), superlevel sets of Γ(x,·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for ℒ using Mr, a Poisson–Jensen formula for the ℒ-subharmonic functions and several results concerning the geometry of the sets Ωr(x).

Sur l'existence du nombre de Lelong d'un courant positif psh défini sur une variété presque complexe

The goal of this Note is to extend to an almost complex manifold the existence of the Lelong number of a positive plurisubharmonic (psh) current. In this way, we generalize results of Lelong and Skoda established in the case of an integrable complex structure, and of Haggui in the non-integrable case, but only for a closed positive current. The main point is to establish a Lelong–Jensen formula for a positive psh current defined on an almost complex manifold, which generalizes a formula proved by Demailly when the structure is integrable. To cite this article: F. Elkhadhra, S.K. Mimouni, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The Nevanlinna's theorems for a class§

In an infinite-dimensional Banach space E with a Schauder basis, the Nevanlinna's first and second main theorems are presented using the E-valued generalizing Poisson–Jensen–Nevanlinna formula instead of the classical Poisson–Jensen formula in and using the concept of compact projection in E. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Periodic Points for Good Reduction Maps on Curves

The periodic points of a morphism of good reduction for a smooth projective curve with good reduction over \(\overline {\mathbb{Q}} _p\) form a discrete set. This is used to give an interpretation of the morphic height in terms of asymptotic properties of periodic points, and a morphic analogue of Jensen's formula.

Entropy and the Canonical Height

The height of an algebraic number in the sense of Diophantine geometry is a measure of arithmetic complexity. There is a well-known relationship between the entropy of automorphisms of solenoids and classical heights. We consider an elliptic analogue of this relationship, which involves two novel features. First we consider the introduction of a notion of entropy for sequences of transformations. Second, we consider the recognition of canonical local heights as integrals over the closure of the torsion subgroup of the curve (an elliptic Jensen formula). A sequence of transformations is defined for which there is a canonical arithmetically defined quotient whose entropy is the canonical height and in which the fibre entropy is accounted for by canonical local heights at primes of singular reduction, yielding a dynamical interpretation of singular reduction. This system is related to local systems whose entropy coincides with the canonical local height up to sign. The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula.

A time-varying analogue of Jensen's formula

A generalization of Jensen's formula to operators associated with time-varying systems is presented. Poles and zeros of the transfer function are replaced by elements of the spectrum of two weighted shifts, one associated with the system, the other with its inverse. Exponential dichotomies, together with two operator factorizations play a key role.

On the Elliptic Analogue of Jensen's Formula

On the Elliptic Analogue of Jensen's Formula

On the Zeros of the Riemann Zeta-Function

It is well known that the multiplicity of a complex zero ρ=β+iγ of the zeta-function is O(log∣γ∣). This may be proved by means of Jensen's formula, as in Titchmarsh [7, Chapter 9]. It may also be seen from the formula for the number N(T) of zeros such that 0<γ<T, N ( T ) = T 2 π log T 2 π - T 2 π + S ( T ) + 7 8 + E ( T ) (1) due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and S ( T ) = 1 π arg ζ ( 1 2 + i T ) (2) We assume here that T is not the ordinate of a zero; with appropriate definitions of N(T) and S(T) the formula is valid for all T. We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT), (Cramér [2]), and on the Riemann Hypothesis S ( T ) = O ( log T log log T ) (Littlewood [5]). These results are over 70 years old. Because the multiplicity problem is hard, it seems worthwhile to see what can be said about the number of distinct zeros in a short T-interval. We obtain the following result, which is independent of any unproved hypothesis.

Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity

Three recent papers have studied fourth-order compact discretizations of the streamfunction vorticity equations. They differed primarily in how the no-slip wall boundary conditions were handled. In this paper, these different formulas are compared to one another, as well as to three newly proposed formulas. Special consideration is paid to the truncation errors; in particular, it is shown that many well-known formulations are actually more accurate by O(h) than previously reported, where h is the mesh size. These new theoretical error rates are confirmed with an analytical model problem. The different formulas are then compared with published driven cavity results, both in terms of accuracy and performance, and the newly proposed high-order Jensen formula is judged to have the marginally best combination of these properties. © 1998 John Wiley & Sons, Ltd.

Landau's Inequality via Hadamard's

We give a new definition of the measure of a polynomial. This definition easily leads to a proof of Landau's inequality, M(P) ≤ ‖P‖, just using Hadamard's inequality. In the same way, it gives Jensen's formula for polynomials. It also allows us to produce an algorithm to compute the measure of a polynomial. Moreover, we get new inequalities of some interest.

Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

We present a new, simple way to estimate the rate of exponential growth (Lyapunov exponent) of solutions of the finite-difference Schrodinger equation: $$((H - E)\psi )(n)\mathop = \limits^{def} - [\psi (n + 1) + \psi (n - 1)] + [\lambda f(\alpha n + \theta )]\psi (n).$$ Heref is a non-constant real-analytic function of period 1 and α is irrational. For λ large we prove that the Lyapunov exponent is positive for every energyE in the spectrum ofH and a.e. θ. In particular, the absolutely continuous spectrum ofH is empty. In the continuum we study the quasi-periodic operator onL2(R) $$H = - \frac{{d^2 }}{{dx^2 }} - K^2 [\cos x + \cos (\alpha x + \theta )]$$ for largeK and show that for wide intervals of low energies the Lyapunov exponent is positive. The main idea, which originated from M. Herman's subharmonic argument [11], is to deform the phase θ to the complex plane. This enables us to avoid small denominator problems by moving them off the axis, making estimates much easier to perform. We recover the information for real θ using an elementary extension of Jensen's formula (subharmonicity).

Mahler measure and entropy for commuting automorphisms of compact groups

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR d = ℤ[u 1 ±1 ,...,[d 1 ±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR d -module. Then the dual groupX M ofM is compact, and multiplication onM by each of thed variables corresponds to an action α M of ℤ d by automorphisms ofX M . Every action of ℤ d by automorphisms of a compact abelian group arises this way. Iff∋R d , our main formula shows that the topological entropy of $$\alpha _{R_d /\left\langle f \right\rangle } $$ is given by $$h\left( {\alpha _{R_d /\left\langle f \right\rangle } } \right) = \log M\left( f \right) = \int\limits_0^1 \ldots \int\limits_0^1 {\log \left| {f\left( {e^{2\pi it_1 } , \ldots , e^{2\pi it_d } } \right)} \right|} dt_1 \ldots dt_d $$ , where M(f) is the Mahler measure off. This reduces to the classical result for toral automorphisms via Jensen's formula. While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7ζ(3)/4π2 occur. If p is a non-principal prime ideal, we show $$h\left( {\alpha _{R_d /p} } \right) = 0$$ . Using an analogue of the Yuzvinskii-Thomas addition formula, we computeh(α M ) for arbitraryR d -modulesM, and then the joint entropy for an action of ℤ d on a (not necessarily abelian) compact group. Using a result of Boyd, we characterize those α M which have completely positive entropy in terms of the prime ideals associated toM, and show this condition implies that α M is mixing of all orders. We also establish an analogue of Berg's theorem, proving that if α M has finite entropy then Haar measure is the unique measure of maximal entropy if and only if α M has completely positive entropy. Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy.