Generic Vopěnka’s Principle,remarkable cardinals,and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume \(\Vert M\Vert \) of M is equal to \(\mathrm{Vol}(M)/v_n\), where \(v_n\) is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio \(\mathrm{Vol}(M)/\Vert M\Vert \) is strictly smaller than \(v_n\) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for \(n\ge 4\), which bounds from below the ratio \(\Vert M\Vert /\mathrm{Vol}(M)\) in terms of the ratio \(\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)\). As a consequence, we show that, for \(n\ge 4\), a sequence \(\{M_i\}\) of compact hyperbolic n-manifolds with geodesic boundary satisfies \(\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n\) if and only if \(\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0\). We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$

Let F be an \(L^2\)-normalized Hecke Maaß cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.     

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Resilience of ranks of higher inclusion matrices

Let \(n \ge r \ge s \ge 0\) be integers and \(\mathcal {F}\) a family of r-subsets of [n]. Let \(W_{r,s}^{\mathcal {F}}\) be the higher inclusion matrix of the subsets in \({{\mathcal {F}}}\) vs. the s-subsets of [n]. When \(\mathcal {F}\) consists of all r-subsets of [n], we shall simply write \(W_{r,s}\) in place of \(W_{r,s}^{\mathcal {F}}\). In this paper we prove that the rank of the higher inclusion matrix \(W_{r,s}\) over an arbitrary field K is resilient. That is, if the size of \(\mathcal {F}\) is “close” to \({n \atopwithdelims ()r}\) then \({{\mathrm{rank}}}_{K}( W_{r,s}^{\mathcal {F}}) = {{\mathrm{rank}}}_{K}(W_{r,s})\), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over \({\mathbb {F}}_q\) is also resilient if \(\mathrm{char}(K)\) is coprime to q.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

A note on amicable numbers and their variations

For any positive integer n, let \(\sigma (\mathrm{n})\) and p(n) denote the sum of divisors and the least prime divisor of n respectively. Let a, b be positive integers. In this paper we prove the following two results: (i) If 4 | a and \(\gcd (a, b)=1\), then a and b do not satisfy \(\sigma (a)= \sigma (b)=a+b\). (ii) If \(a>10^{8}\) and \(p(a)>2\log _{2}a+1\), where \(\log _{2}{a}\) is the logarithm of a with base 2, then a and b do not satisfy \(\sigma (a)=\sigma (b)=a+b+\lambda \), where \(\lambda \in \{0,\pm 1\}\).     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Rotations by roots of unity and Diophantine approximation

For a fixed integer n, we study the question whether at least one of the numbers \(\mathfrak {R}X\omega ^k\), \(1\le k\le n\), is \(\varepsilon \)-close to an integer, for any possible value of \(X\in \mathbb {C}\), where \(\omega \) is a primitive nth root of unity. It turns out that there is always an X for which the above numbers are concentrated around \(1/2\,\mathrm{mod}\,1\). The shortest possible interval centered at 1 / 2 containing the fractional parts of all numbers \(\mathfrak {R}X\omega ^k\) depends only on the prime factors of n, rather than its magnitude. This is directly related to the so–called “pyjama” problem which was solved recently.     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.     

A relative <Emphasis Type="Italic">m</Emphasis>-cover of a Hermitian surface is a relative hemisystem

An m-cover of the Hermitian surface \(\mathrm {H}(3,q^2)\) of \(\mathrm {PG}(3,q^2)\) is a set \(\mathcal {S}\) of lines of \(\mathrm {H}(3,q^2)\) such that every point of \(\mathrm {H}(3,q^2)\) lies on exactly m lines of \(\mathcal {S}\), and \(0<m<q+1\). Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 1965) proved that if q is odd, then \(m=(q+1)/2\), and called such a set \(\mathcal {S}\) of lines a hemisystem. Penttila and Williford (J Comb Theory Ser A 118(2):502–509, 2011) introduced the notion of a relative hemisystem of a generalised quadrangle \(\varGamma \) with respect to a subquadrangle \(\varGamma '\): a set of lines \(\mathcal {R}\) of \(\varGamma \) disjoint from \(\varGamma '\) such that every point P of \(\varGamma \setminus \varGamma '\) has half of its lines (disjoint from \(\varGamma '\)) lying in \(\mathcal {R}\). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order \((q^2,q)\) with respect to a subquadrangle and proving that m must be q / 2 when the subquadrangle is doubly subtended. In particular, a relative m-cover of \(\mathrm {H}(3,q^2)\) with respect to a symplectic subgeometry \(\mathrm {W}(3,q)\) is a relative hemisystem.     

Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$-spaces

Let G be a Polish locally compact group acting on a Polish space \({{X}}\) with a G-invariant probability measure \(\mu \). We factorize the integral with respect to \(\mu \) in terms of the integrals with respect to the ergodic measures on X, and show that \(\mathrm {L}^{p}({{X}},\mu )\) (\(1\le p<\infty \)) is G-equivariantly isometrically lattice isomorphic to an \({\mathrm {L}^p}\)-direct integral of the spaces \(\mathrm {L}^{p}({{X}},\lambda )\), where \(\lambda \) ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of \(\mathrm {L}^{p}({{X}},\mu )\) as an \({\mathrm {L}^p}\)-direct integral of order indecomposable representations. If \(({{X}}^\prime ,\mu ^\prime )\) is a probability space, and, for some \(1\le q<\infty \), G acts in a strongly continuous manner on \(\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )\) as isometric lattice automorphisms that leave the constants fixed, then G acts on \(\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })\) in a similar fashion for all \(1\le p<\infty \). Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If \(({{X}}^\prime ,\mu ^\prime )\) is separable, the representation of G on \(\mathrm {L}^p(X^\prime ,\mu ^\prime )\) can then be disintegrated into order indecomposable representations. The notions of \({\mathrm {L}^p}\)-direct integrals of Banach spaces and representations that are developed extend those in the literature.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space \({\mathcal {M}}_{g}\), of genus \(g\ge 2\) closed Riemann surfaces, is a complex orbifold of dimension \(3(g-1)\) which carries a natural real structure, i.e. it admits an anti-holomorphic involution \(\sigma \). The involution \(\sigma \) maps each point corresponding to a Riemann surface S to its complex conjugate \(\overline{S}\). The fixed point set of \(\sigma \) consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside \(\mathrm {Fix}(\sigma )\) is the locus \({\mathcal {M}}_{g}(\mathbb {R})\), the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä, and R. Silhol. The complement \(\mathrm {Fix}(\sigma )-{\mathcal {M}}_{g}(\mathbb {R})\) consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that \(\mathrm {Fix}(\sigma )\) is connected.