On solutions of Aczél’s equation and some related equations

Let X be a real linear space and ${M: \mathbb{R}\to\mathbb{R}}$ be continuous and multiplicative. We determine the solutions ${f: X \rightarrow \mathbb{R}}$ of the functional equation $$f(x+M(f(x))y) f(x) f(y) [f(x+M(f(x))y) - f(x)f(y)] = 0$$ that are continuous on rays. In this way we generalize our previous results concerning the continuous solutions of this equation. As a consequence we also obtain some results concerning solutions of a functional equation introduced by J. Aczél.     

Semigroup-valued solutions of some composite equations

Let X be a linear space over the field K of real or complex numbers and (S, °) be a semigroup. We determine all solutions of the functional equation $$f(x+g(x)y)=f(x)\circ f(y)\quad \text{for}\quad x,y\in X$$ in the class of pairs of functions (f,g) such that f : X → S and g : X → K satisfies some regularity assumptions. Several consequences of this result are presented.     

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).     

A criterion of vanishing of the spectral density based on homoclinic sums

Let (X, d) be a compact metric space, let T: X→X be a homeomorphism satisfying a certain suitable hyperbolicity assumption, and let μ be a Gibbs measure on X relative to T. Let λ be a complex number |λ|=1, and let f:X → ? be a Hölder continuous function. It is proved that $\sum\limits_{k \in \mathbb{Z}} {\lambda ^{ - k} } \left( {\int\limits_X {f(T^k x)\bar f(x)\mu (dx) - \left| {\int\limits_X {f(x)\mu (dx)} } \right|^2 } } \right) = 0$ if and only if ∑λ^?k(f(T^ky) ? f(T^kx)) = 0 for all x, y ε X such that $d(T^k x,T^k y)\xrightarrow[{|k| \to \infty }]{}0$ . Bibliography: 11 titles.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X ^* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space.     

Explicit solutions of infinite quadratic programs

LetH be a Hilbert space,X be a real Banach space,A: H→X be an operator withD (A) dense inH, G: H→H be positive definite,x ∈D (A) andb ∈H. Consider the quadratic programming problem: $$\begin{gathered} QP:Minimize \frac{1}{2}\left\langle {p,x} \right\rangle + \left\langle {x,Gx} \right\rangle \hfill \\ subject to Ax = b \hfill \\ \end{gathered} $$ In this paper, we obtain an explicit solution to teh above problem using generalized inverses.     

On initial segments of degrees of constructibility

Let \(\mathfrak{M}\) be a fixed countable standard transitive model of ZF+V=L. We consider the structure Mod of degrees of constructibility of real numbers x with respect to \(\mathfrak{M}\) such that \(\mathfrak{M}\) (x) is a model. An initial segment Q \( \subseteq \) Mod is called realizable if some extension of \(\mathfrak{M}\) with the same ordinals contains exclusively the degrees of constructibility of real numbers from Q (and is a model of Z FC). We prove the following: if Q is a realizable initial segment, then $$[y \in Q \to y< x]]\& \forall z\exists y[z< x \to y \in Q\& \sim [y< z]]]$$ .     

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g 2 over an algebraically closed field k of characteristic p0,and F:X→X(1)the relative Frobenius morphism.Let M s X(r,d)(resp.M ss X(r,d))be the moduli space of(resp.semi-)stable vector bundles of rank r and degree d on X.We show that the set-theoretic map S ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

Behaviour at infinity of solutions of some linear functional equations in normed spaces

Let \({\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}, I = (d, \infty), \phi : I \to I}\) be unbounded continuous and increasing, X be a normed space over \({\mathbb{K}, \mathcal{F} : = \{f \in X^I : {\rm lim}_{t \to \infty} f(t) {\rm exists} \, {\rm in} X\},\hat{a} \in \mathbb{K}, \mathcal{A}(\hat{a}) : = \{\alpha \in \mathbb{K}^I : {\rm lim}_{t \to \infty} \alpha(t) = \hat{a}\},}\) and \({\mathcal{X} : = \{x \in X^I : {\rm lim} \, {\rm sup}_{t \to \infty} \|x(t)\| < \infty\}}\) . We prove that the limit lim _{t → ∞} x(t) exists for every \({f \in \mathcal{F}, \alpha \in \mathcal{A}(\hat{a})}\) and every solution \({x \in \mathcal{X}}\) of the functional equation $$x(\phi(t)) = \alpha(t) x(t) + f(t)$$ if and only if \({|\hat{a}| \neq 1}\) . Using this result we study behaviour of bounded at infinity solutions of the functional equation $$x(\phi^{[k]}(t)) = \sum_{j=0}^{k-1} \alpha_j(t) x (\phi^{[j]}(t)) + f(t),$$ under some conditions posed on functions \({\alpha_j(t), j = 0, 1,\ldots, k - 1,\phi}\) and f.     

ω-Limit sets of smooth cylindrical cascades

Let f(x) be a smooth function on the circle S¹, x mod 1, \(\smallint _{S^1 } f(x)dx = 0\) , α be an irrational number, and q_n be the denominators of convergents of continued fractions. In this note a classification of ω-limit sets for the cylindrical cascade $$T:(x,y) \to (x + \alpha , y + f(x)),$$ x ε S¹, y ε R, is obtained. Criteria for the solvability of the equation g(x +α) — g(x)=f (x) are found. Estimates for the speed of decrease of the function $$h_{q_n } (x) = \sum _{i = 0}^{q_n - 1} f(x + i\alpha )$$ as n → ∞ are obtained.     

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.     

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L ⁰(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T _m : L ^∞(μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca _μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ^∞(μ) → X.     

Orthogonality and Additivity Modulo Z

Let E be a real inner product space of dimension at least 2. Suppose ? : E → ? satisfies If there exist a neighbourhood U of the origin and γ ∈ (0, 1/4) such that ?(U) ? (?γ,γ) + ?, then there exist a real constant c and a continuous linear functional g : E → ? such that Suppose Φ : E → ? satisfies If there exist a neighbourhood U of the origin and β ∈ (0, +∞) such that ¦Φ(x)¦ ≤ β?(Φ(x)) for every x ∈ U, then either Φ vanishes on E? {0} or there exist additive functions a : ? → ? and A : E → ?, a real constant c and a continuous linear functional g : E → ? such that      

Compact complete null curves in complex 3-space

We prove that for any open orientable surface S of finite topology, there exist a Riemann surface M, a relatively compact domain M ? M and a continuous map X: $\overline M $ → ?³ such that:

• M and M are homeomorphic to S, M-M and M ? $\overline M $ contain no relatively compact components in M
• X| _M is a complete null holomorphic curve, $X{|_{\overline M - M}}:\overline M - M \to {{\Bbb C}^3}$ is an embedding and the Hausdorff dimension of X( $\overline M $ ?M) is 1.

Moreover, for any ε > 0 and compact null holomorphic curve Y:N→?³ with non-empty boundary Y (?N), there exist Riemann surfaces M and M homeomorphic to N ^○ and a map X: $\overline M $ → ?³ in the above conditions such that δ ^H (Y(?N),X( $\overline M $ ? M)) < ε, where δ ^H (·,·) means Hausdorff distance in ?³.     

Finite-dimensional quasi-variational inequalities associated with discontinuous functions

In this paper, given a nonempty closed convex setX ? ⁿ , a functionf: X→? ⁿ , and a multifunction Γ:X→2^X, we deal with the problem of finding a point \(\hat x\) ∈X such that $$\hat x \in \Gamma (\hat x) and \langle f(\hat x), \hat x - y\rangle \leqslant 0, for all y \in \Gamma (\hat x).$$ For such problem, we establish a result where, in particular, the functionf is not assumed to be continuous. More precisely, we extend to the present setting a finite-dimensional version of a result by Ricceri on variational inequalities (Ref. 1).     

The equality problem in the class of conjugate means

Let ${I\subset\mathbb{R}}$ be a nonempty open interval and let ${L:I^2\to I}$ be a fixed strict mean. A function ${M:I^2\to I}$ is said to be an L-conjugate mean on I if there exist ${p,q\in{]}0,1]}$ and a strictly monotone and continuous function φ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.     

Markov type and threshold embeddings

For two metric spaces X and Y, say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps ${\{\varphi_{\tau} : X \to Y : \tau > 0\}}$ such that for every ${x,y \in X}$ , $$d_X(x, y) \geq \tau \implies d_Y(\varphi_\tau (x),\varphi_\tau (y)) \geq \|{\varphi}_\tau\|_{\rm Lip}\tau/K,$$ where ${\|{\varphi}_{\tau}\|_{\rm Lip}}$ denotes the Lipschitz constant of ${\varphi_{\tau}}$ . We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset ${X \subseteq L_1}$ threshold-embeds into Hilbert space if and only if X has Markov type 2.