Polynomials and finite sets connected with an identity

It is shown that the polynomials satisfying the identityf(x) f(x + 1) = f(x ² +x – a), wherea either belongs to a field of characteristic zero or is transcendental over a prime field of characteristic exceeding 2, are precisely those of the form(x ² –a) ⁿ ; thus extending a result proved by Nathanson in the complex case. The result is not, in general, true in characteristic 2. Additionally, a class of finite sets, considered by Nathanson in connection with the identity, is completely determined.     

Jensen's functional equation on groups

Summary A natural extension of Jensen's functional equation on the real line is the equationf(xy) + f(xy ^–1 ) = 2f(x), wheref maps a groupG into an abelian groupH. We deduce some basic reduction formulas and relations, and use them to obtain the general solution on special groups.     

Some approximation theorems

The general theme of this note is illustrated by the following theorem:Theorem 1. Suppose K is a compact set in the complex plane and 0belongs to the boundary ∂K. Let A(K) denote the space of all functions f on K such that f is holo morphic in a neighborhood of K and f(0) = 0.Also for any givenpositive integer m, let A(m, K) denote the space of all f such that f is holomorphic in a neighborhood of K and f(0) =f′(0) = ... =f ^(m)(0) = 0.Then A(m, K) is dense in A(K) under the supre mum norm on K provided that there exists a sector W = re ^iθ; 0≤r≤ δ,α≤ θ≤ β such that W ∩ K = 0. (This is the well- known Poincare’s external cone condition). We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic. Dedicated to Prof. Ashoke Roy on his 62nd birthday     

On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Star-Configurations in ℙ2 Having Generic Hilbert Function and the Weak Lefschetz Property

We prove that a star-configuration 𝕏 in ?² is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ?² defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3 Ahn , J. , Shin , Y. S. ( 2012 ). The minimal free resolution of a star-configuration in ? ⁿ and the weak lefschetz property . J. Korean Math. Soc. 49 ( 2 ): 405 – 417 . [Google Scholar]]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I _𝕏 + I _?) precisely when 𝕏 and ? are two star-configurations in ?² defined by general forms F ₁,…, F _s , and G ₁,…, G _s , L, respectively, with deg F _i  = deg G _i  = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3.     

On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation

We consider the computation of the Cauchy principal value integral by quadrature formulae Q _n ^F [f] of compound type, which are obtained by replacing f by a piecewise defined function F_n[f]. The behaviour of the constants k_{i, n} in the estimates [R _n ^F [f]] |⩽K _i,n ‖f ⁽ⁱ⁾‖_∞ (where R _n ^F [f] is the quadrature error) is determined for fixed i and n→∞, which means that not only the order, but also the coefficient of the main term of k_{i, n} is determined. The behaviour of these error constants k_{i, n} is compared with the corresponding ones obtained for the method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.     

The number of connected sparsely edged graphs

An (n, q) graph has n labeled points, q edges, and no loops or multiple edges. The number of connected (n, q) graphs is f(n, q). Cayley proved that f(n, n^-1) = n^n?2 and Renyi found a formula for f(n, n). Here I develop two methods to calculate the exponential generating function of f(n, n + k) for particular k and so to find a formula for f(n, n + k) for general n. The first method is a recurrent one with respect to k and is well adapted for machine computation, but does not itself provide a proof that it can be continued indefinitely. The second (reduction) method is much less efficient and is indeed impracticable for k greater than 2 or 3, but it supplies the missing proof that the generating function is of a particular form and so that the first method can be continued for all k, subject only to the capacity of the machine.     

A convolution formula for time‐invariant linear communication systems

For every linear and time‐invariant time‐discrete (communication) system T : l^∞ → l^∞, formally, the following convolution formula can be derived: \input amssym.def $$(Tf)(n)=\sum_{k \in {\Bbb Z}} h(n‐k) f(k),\quad n \in {\Bbb Z}, f \in l^{\infty},$$ where h = Tδ is the delta impulse response. This paper is concerned with the question under which assumptions linear and time‐invariant time‐discrete systems T : l^∞ → l^∞ can be characterized by this formula. For this purpose we derive a convolution formula in a more general situation which also leads to a well‐known convolution formula in the time‐continuous case. Copyright © 1999 John Wiley & Sons, Ltd.     

The number of connected sparsely edged graphs. III. Asymptotic results

The number of connected graphs on n labeled points and q lines (no loops, no multiple lines) is f(n,q). In the first paper of this series I showed how to find an (increasingly complicated) exact formula for f(n,n+k) for general n and successive k. The method would give an asymptotic approximation to f(n,n+k) for any fixed k as n → ∞. Here I find this approximation when k = o(n^1/3), a much more difficult matter. The problem of finding an approximation to f(n,q) when q > n + Cn^1/3 and (2 q/n) - log n → - ∞ is open.     

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?_n=0^￥ a_n X_n zⁿf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X _n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a _n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.     

The transformation theorem for two-parameter pure jump martingales

Summary LetM be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps ofM in the plane.M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which forC ⁴-functions f the processf(M) is described by integrals off ^(k) (M),k=1, 2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula.     

Generalized Cauchy equations on groups

It is well known ([1], [3]) that any measurable solution of the Cauchy functional equationf(x+y)=f(x)+f(y) must actually be continuous. The same is true of some other functional equations likef(x+y)=f(x)f(y),f(x+y)f(x–y)=f(x) ² –f(y) ², etc. (cf. [1]). In this note we prove a general result of this type for functional equations on groups.     

Regularity of functional equations on manifolds

Summary. In this paper the regularity properties of the functional equation¶¶ f (t) = h(t, y, f (g₁(t, y)), ... , f (g_n(t,y))) f (t) = h(t, y, f (g_{1}(t, y)), ... , f (g_{n}(t,y))) ¶ on a \Cal C^￥ {\Cal C}^\infty manifold for the unknown function f are treated. Under general conditions it is proved that solutions which are measurable or have the Baire property are in \Cal C^￥ {\Cal C}^\infty .     

Staggered Taylor–Hood and Fortin elements for Stokes equations of pressure boundary conditions in Lipschitz domain

The purpose of this paper is to develop a general theory on how the inf-sup stable and convergent elements of the velocity Dirichlet boundary (VDB)-Stokes problem with no-slip VDB are still inf-sup stable and convergent for the pressure Dirichlet boundary (PDB)-Stokes problem with PDB in Lipschitz domain. The PDB-Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to (H¹(Ω))², sharply in contrast to the VDB-Stokes problem whose velocity solution still belongs to (H¹(Ω))², and unexpectedly, some well-known inf-sup stable and convergent VDB-Stokes elements may or may no longer correctly converge. It turns out that the inf-sup condition of the PDB-Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor–Hood elements with ℓ ≥ 1 (continuous in both velocity and pressure) and staggered Fortin elements with m ≥ 1 (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB-Stokes problem in Lipschitz domain. We show that the two families are inf-sup stable and are correctly convergent for the non-H¹ singular velocity. Numerical results illustrate the proposed elements and the theoretical results.     

Another logarithmic functional equation

Summary. Let f : ]0,￥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x^-1 + y^-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.     

Fourier–Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ₃û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H¹(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H¹(Ωˆ) is proved to be 𝒪(h+N⁻¹) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.     

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The normalizer of

In this paper, the normalizers of , for k = 2 or 3, in the groups Γ^k of modular group Γ are given by using the concept of Γ^k axes of the set of left cosets of , on which Γ^k acts transitively, in Γ^k.