On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Partial Difference Equations Analogous to the Cauchy-Riemann Equations and Related Functional Equations On Rings and Fields

Let (S, #, *) be an algebraic structure where # and * are binary operations with identities on the set S. Let (G, +) be an abelian group. We consider the functional equation (i) $$f(x * t, y)+ g(x, y\ \sharp\ t) = h(x, y)\ {\rm for\ all}\ x, y, t \in S,$$ where ?,g,h :S × S → G. As an application of (i) we solve $$f(x + t, y)- f(x, y) = -b(f(x, y+t)- f(x,y))\ {\rm for\ all}\ x, y, t \in S,$$ where ? :S × S → K (a field), and b ∈ K is a constant and b ≠ 0, ±1. If b = i, the pure imaginary unit, S = R and K = C, then the above equation may be considered as a discrete analogue of the Cauchy-Riemann equations. When (R, +, ?) is a commutative ring with 1, the functional equation (ii) $$\phi(y+xt)-\phi(xy+xt)=\phi(y+x)-\phi(xy+x)$$ for all x,y,t ∈ R, where ? : R → G, is basic to the general solutions of (i). We solve (ii) on certain rings and fields.     

On a generalization of Minkowski's convex body theorem

Given a lattice $\text{Λ ? R}{}^{\mathrm{n}}$ and a bounded function g(x), x ∈ Rⁿ, vanishing outside of a bounded set, the functions ?(x)$\text{g}\text{?}\text{(x)?}$max_u∈Λg(u +x), ?(x)?Σ_{u∈Λ g(u +x)}, and ?₊(x)?Σ_u∈Λ max_{v∈Λ min {g(v + x); g(u + v + x)}} are defined and periodic mod Λ on Rⁿ. In the paper we prove that ?(x) + ?₊(x) ? 2?(x) ≥ ?(x) + h?₊(x) ? 2?(x) holds for all x ∈ Rⁿ, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved.     

The Dirichlet problem with a volume constraint

For B the open unit disk in R², let W₁(B) denote the Sobolev space of vector functions x: B→R³ such that x and its first partial derivatives are square integrable. For any y∈W₁(B), S(y) is the set of all x in W₁(B) for which x-y∈W₁₀(B), the closure in W₁(B) of C ₀ ^∞ (B). Assume that for all x ∈ S(y) the area functional A(x)>0. For a given constant K, we show that there is an x_o∈S(y) minimizing the “Dirichlet Integral” $$D(x) = \iint_B {(|x_u |^2 } + |x_v |^2 )dudv$$ in the subset of all x ∈ S(y) for which the oriented volume enclosed by y and x, V(y,x)=K. x_o is analytic on B and is a solution to the differential equation Δx=2H(x_u∧x_v) for some constant H.     

Algebraic Dilogarithm Identities

The Rogers L-function satisfies the functional equation .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form where is algebraic and the c _k are integers.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

Every Akivis Algebra is Linear

An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ?2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c₁-construction if there exists (B, m, k)∈ £ ? {(k, (0), k)} and a finitely generated, faithful B-module N such that R,?B?(the idealization of N). (R.J.k) is called a (c_2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ S_B(the socle of B) such that R?B[x]/(mX, X^p- z). Let M_n×n(K) denote the set of all n x n matrices, over k with n≥2. Let .M_n(k) denote the set of all maximal, commutative A;-subalgebras of M_n×n(k). In this paper, we show any (R J, k) ∈£?M_n;(k) with n>5 is a C₁ or C₂ -construction except for one isomorphism class. The one exception occurs when n = 5.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

On Second Order Linear Differential Polynomials

We classify all functions φ(z) meromorphic in R ? ¦z¦ < ∞ such that ?(z) Φ(z) has no zeros there, where Φ = (z)′’ + a₁(z) ?′(z) + a₀(z) ?(z) and a₀(z), a₁(z) are rational at infinity.     

On some alternative quadratic equations

From the quadratic functional equation ?(x + y) + ?(x ? y) ? 2?(x) ? 2f(y) = 0 various alternative equations are derived here by grouping in different ways its terms and then equating norms. Some equivalence results are proved in the class of functionals ?: X → (?, ¦· ¦). Suitable examples concerning operators ?: X → (E,∥· ∥) with values in normed spaces show that in this more general setting such an equivalence can fail to be true.     

On generalized cocycle equations

Summary Motivated by results on the classical cocycle equation, we solved the more general equationF ₁ (x + y, z) + F ₂ (y + z, x) + F ₃ (z + x, y) + F ₄ (x, y) + F ₅ (y, z) + F ₆ (z, x) = 0 for six unknown functions mapping ordered pairs from an abelian group into a vector space over the rationals.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Inequalities for Means in Two Variables

We present various new inequalities involving the logarithmic mean L(x,y)=(x-y)/(logx-logy) L(x,y)=(x-y)/(\log{x}-\log{y}) , the identric mean I(x,y)=(1/e)(x^x/y^y)^1/(x-y) I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} , and the classical arithmetic and geometric means, A(x,y)=(x+y)/2 A(x,y)=(x+y)/2 and G(x,y)=?{xy} G(x,y)=\sqrt{xy} . In particular, we prove the following conjecture, which was published in 1986 in this journal. If M_r(x,y) = (x^r/2+y^r/2)^1/r(r 1 0) M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) denotes the power mean of order r, then $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} with the best possible parameter c=(log2)/(1+log2) c=(\log{2})/(1+\log{2}) .     

On a functional equation of Bruce Ebanks

In the paper Brillouët-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions ${f: \mathbb{R} \rightarrow \mathbb{R}}$ of the functional equation (1).     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

---

Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire