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.     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ${\varphi}$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}^n}$ . This threshold is defined as the supremum of constants c > 0 such that ${e^{-2c\varphi}}$ is integrable on a neighborhood of 0. We relate ${c(\varphi)}$ to the intermediate multiplicity numbers ${e_j(\varphi)}$ , defined as the Lelong numbers of ${(dd^c\varphi)^j}$ at 0 (so that in particular ${e_0(\varphi)=1}$ ). Our main result is that ${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$ . This inequality is shown to be sharp; it simultaneously improves the classical result ${c(\varphi)\geqslant 1/e_1(\varphi)}$ due to Skoda, as well as the lower estimate ${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.     

Singular Solutions of the Generalized Dhombres Functional Equation

We consider singular solutions of the functional equation ${f(xf(x)) = \varphi (f(x))}$ where ${\varphi}$ is a given and f an unknown continuous map ${\mathbb R_{+} \rightarrow \mathbb R_{+}}$ . A solution f is regular if the sets ${R_f \cap (0, 1]}$ and ${R_f \cap [1, \infty)}$ , where R _f is the range of f, are ${\varphi}$ -invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system ${({R_f}, \varphi|_{R_f})}$ can have strange properties unknown for the regular solutions. In particular, we show that ${\varphi |_{R_f}}$ can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions.     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .     

A note on da Costa-Doria “exotic formalizations”

We analyze N. C. A. da Costa and F. A. Doria’s “exotic formalization” of the conjecture P = NP [3–7]. For any standard axiomatic PA extension T and any number-theoretic sentence ${\varphi }$ , we let ${\varphi ^{\star} := \varphi \vee \lnot \mathsf{Con}\left( \mathsf{T}\right)}$ and prove the following “exotic” inferences 1–3. 1. ${\mathsf{T}+\varphi ^{\star}}$ is consistent, if so is T, 2. ${\mathsf{T}+\varphi}$ is consistent, provided that ${\mathsf{T}+\varphi ^{\star}}$ is ω-consistent, 3. ${\mathsf{T}+\varphi}$ is consistent, provided that T is consistent and has the same provably total recursive functions as ${\mathsf{T}+\left( \varphi \leftrightarrow \varphi ^{\star }\right) }$ . Furthermore we show that 1–3 continue to hold for ${\varphi ^{\star} := \varphi _{S} :=\varphi \vee \lnot S}$ , where ${S=\forall x\exists yR\left( x,y\right)}$ is any ${\Pi _{2}^{0}}$ sentence satisfying: 4. ${\left( \forall n\in \omega \right) \left( \mathsf{T}\vdash S_{x}\left[ \underline{n}\right] \right) }$ , 5. ${\mathsf{Con}\left( \mathsf{T}\right) \Rightarrow \mathsf{T}\nvdash S}$ . We observe that if ${\varphi :=\left[ \mathsf{P}=\mathsf{NP}\right] }$ and ${S:= \left[\digamma total\right] }$ , where ${\digamma=\digamma _{\mathsf{T}}}$ is da Costa-Doria “exotic” function with respect to T, then 4, 5 are satisfied for most familiar (presumably) consistent T in question, while ${\varphi _{S}}$ becomes equivalent to da Costa-Doria “exotic formalization” ${\left[ \mathsf{P}=\mathsf{NP}\right]^{\digamma}}$ . Moreover, the corresponding “exotic” inferences 1–3 generalize analogous da Costa-Doria results. Hence these “exotic” inferences are universal for all number-theoretic sentences and not characteristic to the conjecture P = NP. Nor do they infer relative consistency of P = NP (see Conclusion 15 in the text).     

Homomorphisms with respect to a function

Let X be a realcompact space and ${H:C(X)\rightarrow\mathbb{R}}$ be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is ${\varphi\in C(\mathbb{R})}$ with ${\varphi(r)>\varphi(0)}$ for all ${r\in\mathbb{R}-\{0\}}$ for which ${H\circ\varphi=\varphi\circ H}$ . This extends (and unifies) classical results by Hewitt and Shirota.     

Uniqueness properties of degenerate elliptic operators

Let ?? be an open subset of R ^d and ${ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}$ a second-order partial differential operator with real-valued coefficients ${c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}$ satisfying the strict ellipticity condition ${C=(c_{ij}) >0 }$ . Further let ${H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}$ denote the principal part of K. Assuming an accretivity condition ${C\geq \kappa (c\otimes c^{\,T})}$ with ${\kappa >0 }$ , an invariance condition ${(1\!\!1_\Omega, K\varphi)=0}$ and a growth condition which allows ${\|C(x)\|\sim |x|^2\log |x|}$ as |x| ?? ?? we prove that K is L ₁-unique if and only if H is L ₁-unique or Markov unique.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

A Note on Classical and p-adic Fréchet Functional Equations with Restrictions

Given X,Y two ${\mathbb{Q}}$ -vector spaces, and f : X → Y, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$ , if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$ , then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$ .     

Comparison of two regulatory results on functional equations with few parameters

Two methods to prove regularity properties of the linear functional equation $$f(x)=h_0(x,y)+\sum_{j=1}^n h_j(x,y)f(x+g_j(y)), $$ where ${(x,y) \in D \subset \mathbb{R}^r \times \mathbb{R}^s}$ , ${x \in \mathbb{R}^r}$ and ${y \in \mathbb{R}^s}$ , with few parameters i.e. allowing 1 ?? s < r are examined. It is proved that??under certain conditions, for some class of equations and in some sense??they are equivalent.     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra ${\mathcal{A}}$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that ${\mathcal{A}}$ separates points in the sense that for each distinct pair ${x, y \in X}$ , there exists an ${f \in \mathcal{A}}$ such that ${f(x) \neq f(y)}$ . If ${\mathcal{A}}$ does not separate points, it is known that there exists an algebra ${\widehat{\mathcal{A}}}$ on a compact Hausdorff space ${(\widehat{X}, \widehat{\tau})}$ that does separate points such that the map ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume ${\mathcal{A}}$ separates points. The construction of ${{\widehat{\mathcal{A}}}}$ and ${(\widehat{X}, \widehat{\tau})}$ does not require that ${\mathcal{A}}$ has any algebraic structure nor that ${(X, \tau)}$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family ${\mathcal{A}}$ of bounded, complex-valued, continuous functions on any topological space ${(X, \tau)}$ . We also demonstrate that further structures may be preserved by the mapping ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.     

On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f.     

A strict minimax inequality criterion and some of its consequences

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y ?? R a C ¹ function with locally Lipschitzian derivative, and ${\varphi : Y \to [0, + \infty[}$ a C ¹ convex function with locally Lipschitzian derivative at 0 and ${\varphi^{-1}(0) = \{0\}}$ . Then, for each ${x_0 \in Y}$ for which J??(x ₀)??? 0, there exists ???> 0 such that, for each ${r \in ]0, \delta[}$ , the restriction of J to B(x ₀, r) has a unique global minimum u _r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all ${x \in B(x_0, r)}$ , where ${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$      

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), ${\Lambda}$ a k-dimensional topological manifold with topological orientation ${\eta}$ , and ${f : D \rightarrow X}$ a locally compact map, where D is an open subset of ${X \times \Lambda}$ . We define Fix(f) as the set of points ${{(x, \lambda) \in D}}$ such that ${x = f(x, \lambda)}$ . For an open pair (U, V) in ${X \times \Lambda}$ such that ${{\rm Fix}(f) \cap U \backslash V}$ is compact we construct a homomorphism ${\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}$ in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a ${C^{\infty}}$ -manifold ${\Lambda}$ , these properties uniquely determine ${\Sigma}$ . By passing to the direct limit of ${\Sigma_{(f,U,V)}}$ with respect to the pairs (U, V) such that ${K = {\rm Fix}(f) \cap U \backslash V}$ , we define a homomorphism ${\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}$ in the ?ech cohomologies. Properties of ${\Sigma}$ and ${\sigma}$ are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.