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Let \mathcal M be a semifinite factor with a faithful normal semifinite tracial weight \tau, and \mathscr P the set of all projections in \mathcal M. Denote by \mathscr P_{c} the Grassmann space of all projections in \mathscr P with trace c, where c is a positive real number. A map \psi : \mathscr P_c\rightarrow \mathscr P_c is called an ortho-isomorphism if \psi is a bijection of \mathscr P_c onto \mathscr P_c satisfying, for all P,Q\in \mathscr P_c, P\perp Q if and only if \psi (P)\perp \psi (Q). The aim of this paper is to establish a version of Uhlhorn's theorem in the setting of semifinite factors. We give a complete characterization of ortho-isomorphisms on Grassmann space \mathscr P_c in a semifinite factor. And we show that an ortho-isomorphism \psi : \mathscr P_c\rightarrow \mathscr P_c can be extended to a Jordan *-isomorphism \rho of \mathcal M onto \mathcal M. As an application, we obtain the structure of surjective isometries on \mathscr P_c with respect to strictly increasing unitarily invariant norms. [ABSTRACT FROM AUTHOR]

Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an appropriate reduction mechanism to rule out the possibility of completeness by simply invoking Tarski's Undefinability of Truth theorem. We also use the proof strategy of Theorem A to obtain other incompleteness results. [ABSTRACT FROM AUTHOR]

We prove that if Y is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in \ell _1, or a normalized sequence with upper p-estimates for some p>1, then Y does not satisfy Lindenstrauss' property B. [ABSTRACT FROM AUTHOR]

This article is to develop a method to solve the boundary value problems of the first-order measure differential equations in the space of bound-ed variation functions. Firstly, we obtain the solution and Green's function by applying the integration by parts. Secondly, the criterion for the existence of solution is given by using fixed point theorem and regularization theory. Finally, an example is provided to validate these conclusions. [ABSTRACT FROM AUTHOR]

By employing a quadratic summation formula due to Gasper and Rahman [ Basic hypergeometric series , 2nd ed, vol. 96, Cambridge University Press, Cambridge, 2004] and the creative microscoping method developed by Guo and Zudilin [Adv. Math. 346 (2019), pp. 329–358], we establish some new parametric q-supercongruences, the corresponding supercongruences of which can be deemed the variants of Van Hamme's (J.2) and (L.2) supercongruences. Moreover, we obtain three families of Ramanujan-type formulas on \pi and propose a challenging conjecture. [ABSTRACT FROM AUTHOR]

The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel character theory approach. We first present some general symmetric function expressions for the number of products of two permutations respectively from two arbitrary, but fixed, conjugacy classes indexed by \alpha and \gamma that produce a permutation with m cycles. Our next objective is to derive explicit formulas for the cases where \alpha corresponds to full cycles, i.e., one-face bipartite maps. We prove a far-reaching explicit formula, and show that the number for any \gamma can be iteratively reduced to that of products of two full cycles, which implies an efficient dimension-reduction algorithm for building a database of all these numbers. Note that the number for products of two full cycles can be computed by the Zagier-Stanley formula. Also, in a unified way, we easily prove the celebrated Harer-Zagier formula and Jackson's formula, and we may obtain explicit formulas for several new families as well. [ABSTRACT FROM AUTHOR]

We prove that minimizers of the L^{d}-norm of the Hessian in the unit ball of \mathbb {R}^d are locally of class C^{1,\alpha }. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate with the Hölder regularity theory available for elliptic equations in double-divergence form. [ABSTRACT FROM AUTHOR]

Consider L^0, the F-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin's classical result on the factorization of sublinear continuous operators from a Banach space to L^0, we prove a theorem that characterizes those maps from any quasi-metric space into L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces. [ABSTRACT FROM AUTHOR]

In this note, we prove that the compactly supported mapping class group of a surface containing a genus 3 subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order 6 symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set. [ABSTRACT FROM AUTHOR]

Let H^2(S) be the Hardy space on the unit sphere in \mathbf {C}^n. We show that there are Toeplitz operators T_f and T_g on H^2(S) such that the product T_fT_g is not compact and yet \|T_fT_gk_z\| tends to 0 as |z| \rightarrow 1. Consequently, the Berezin transform \langle T_fT_gk_z,k_z\rangle tends to 0 as |z| \rightarrow 1. [ABSTRACT FROM AUTHOR]

In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas [Ann. Probab. 46 (2018), pp. 1441–1454]. [ABSTRACT FROM AUTHOR]

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [ABSTRACT FROM AUTHOR]

We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure. [ABSTRACT FROM AUTHOR]

We study general elliptic equations with singular/degenerate matrix weights and measurable nonlinearities on nonsmooth bounded domains to obtain a global Calderón-Zygmund type estimate under possibly minimal assumptions that the logarithm of the matrix weight has a small bounded mean oscillation (BMO) norm, the nonlinearity is allowed to be merely measurable in one variable but has a small BMO norm in the other variables and that the boundary of the domain is sufficiently flat in Reifenberg sense. [ABSTRACT FROM AUTHOR]

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel–Darboux formula. These results should be viewed as the direct analogue of the nearest neighbour recurrence relations from the theory of multiple orthogonal polynomials on the real line. [ABSTRACT FROM AUTHOR]

The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

We prove that the (k+d)-th Neumann eigenvalue of the biharmonic operator on a bounded connected d-dimensional (d\ge 2) Lipschitz domain is not larger than its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the (k+d+1)-th Neumann eigenvalue of the biharmonic operator does not exceed its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. In particular, in two dimensions, this special class consists of domains having an axis of symmetry. [ABSTRACT FROM AUTHOR]

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]

In terms of the creative microscoping method, El Bachraoui's lemmas, and the Chinese remainder theorem for coprime polynomials, we establish some q-supercongruences for double and triple basic hypergeometric series. Several related conjectures are also proposed for further research. [ABSTRACT FROM AUTHOR]

The Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [ABSTRACT FROM AUTHOR]

We find sharp constants in the symmetric integral form of the John–Nirenberg inequality. The result is based upon the computation of a new interesting Bellman function. [ABSTRACT FROM AUTHOR]

We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G^{r}_{d}(C) parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d\leq g(C)-1. [ABSTRACT FROM AUTHOR]

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gâteaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler. [ABSTRACT FROM AUTHOR]

We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

Let \mathcal {N}=L_{\infty }(G_{\mathbf {m}})\bar {\otimes }\mathcal {M} where \mathcal {M} is a semifinite von Neumann algebra and G_{\mathbf {m}} is a bounded Vilenkin group. Considering the partial sums S_n(f) of the Vilenkin-Fourier series of f\in L_1(\mathcal {N}), we prove that there is a universal constant c independent of f, n and \mathcal {M} such that \begin{equation*} \|S_n(f)\|_{L_{1,\infty }(\mathcal {N})}\leq c\|f\|_{L_1(\mathcal {N})}. \end{equation*} Consequently, by transference argument, we obtain Scheckter and Sukochev's result in the noncommutative bounded Vilenkin system setting. We also show the H_1-L_1 boundedness for modified partial sums. [ABSTRACT FROM AUTHOR]

Using the Caffarelli–Silvestre extension, we show for a general open set \Omega \subset \mathbf {R}^n that a boundary point x_0 is regular for the fractional Laplace equation (-\Delta)^su=0, 0

We give the definition of extremal lengths of conformal Riemannian metrics on Riemann surfaces. And we obtain the extremal lengths of conformal negatively curved Riemannian metrics on Riemann surfaces. [ABSTRACT FROM AUTHOR]

The Mabuchi constant is a holomorphic invariant of Fano manifolds, which obstructs the existence of Mabuchi's generalized Kähler-Einstein metrics and relative Ding semistability. In this study, we give a formula for the Mabuchi constant of produtcs of Fano manifolds. As an application, we present examples of Fano manifolds which admit Calabi's extremal Kähler metrics in every Kähler class, but are relatively Ding unstable. [ABSTRACT FROM AUTHOR]

Over a field of characteristic 0, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it. [ABSTRACT FROM AUTHOR]

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in L^p-type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying P'(\bar {\rho })>0 is allowed. To achieve it, a new composition estimate is established in the L^2-L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies. [ABSTRACT FROM AUTHOR]

We consider a time varying delayed Cucker-Smale system in a harmonic potential field and analyze its long time collective behavior. Under appropriate assumptions on the initial data, we show the asymptotic collective behavior when the time varying delay is uniformly bounded by a sufficiently small constant. Our strategies are based on the Lyapunov functional approach, forward-backward estimate and the continuity argument. Finally, some numerical tests are performed to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

We prove a strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

This document is a retraction notice published in the Proceedings of the American Mathematical Society. The author, Khadime Salame, apologizes to the mathematical community and withdraws three papers related to fixed point theory. The papers attempted to address the question of whether left amenable semitopological semigroups have a certain fixed point property, but each paper contains errors. The author acknowledges these errors and suggests that Lau's conjecture should be regarded as an open question. The author hopes that this question will be resolved in the future. [Extracted from the article]

In Theorem 0.1 of the paper "A Severi type theorem for surfaces in \mathbb {P}^6" [Proc. Amer. Math. Soc. 149 (2021), pp. 591–605], we claimed to have given a complete classification of smooth surfaces in \mathbb {P}^6 with one 4-secant plane through the general point of \mathbb {P}^6, but the classification is still incomplete. [ABSTRACT FROM AUTHOR]

The (co)homological dimension of a homomorphism \phi :G\to H is the maximal number k such that the induced homomorphism in k-th (co)homology groups is nonzero for coefficients in some H-module. It is known that for geometrically finite groups G, cd(G)=hd(G) and cd(G\times G)=2cd(G). We prove analogous theorems for homomorphisms of geometrically finite groups. The analogy stops working on the Eilenberg-Ganea equality cd(G)=gd(G) where cd(G)>2 and gd(G) is the geometric dimension of G. We show that for every k>2 there is a group homomorphism \phi _k:\pi _k\to \mathbb {Z}^k with cd(\phi _k)

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form. [ABSTRACT FROM AUTHOR]

We present a simple intrinsic geometrical proof of the Bismut-Zhang embedding formula of Atiyah-Patodi-Singer reduced eta invariants [Math. Ann. 295 (1993), pp. 661–684] by making use of geometric K-homology. [ABSTRACT FROM AUTHOR]

For each \alpha \in (0, 1), we construct a bounded monotone deterministic sequence (c_k)_{k \geqslant 0} of real numbers so that the number of real roots of the random polynomial f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k is n^{\alpha + o(1)} with probability tending to one as the degree n tends to infinity, where (\varepsilon _k) is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when (\varepsilon _k) is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of f_n, including the asymptotic behavior of the variance and a central limit theorem. [ABSTRACT FROM AUTHOR]

The p-negative type gap of a finite metric space is a useful tool in studying isometric embedding properties of the space. Wolf gave a versatile formula for computing the p-negative type gap, which relies on properties of the inverse of the matrix D_{p}=(d_{X}(x_{i},x_{j})^{p})_{i,j=0}^{n}. In this article we provide a simplification of Wolf's formula in terms of the Gramian matrix G_{p}=((1/2)(d_{X}(x_{i},x_{0})^{p}+d_{X}(x_{j},x_{0})^{p}-d_{X}(x_{i},x_{j})^{p}))_{i,j=1}^{n}. We also study slight variations of the p-negative type gap and show that that some of these are much easier to compute than the original p-negative type gap. Finally, we provide explicit expressions for the p-negative type gap and some of these variations for the bipartite graph K_{n,1}. [ABSTRACT FROM AUTHOR]

We study varieties X \subset \mathbb {P}^r such that N_X^*(k) is an Ulrich vector bundle for some integer k. We first prove that such an X must be a curve. Then we give several examples of curves with N_X^*(k) an Ulrich vector bundle. [ABSTRACT FROM AUTHOR]

A problem of completing a linear map on C^*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some very general conditions a completely positive map almost everywhere equivalent to a quasi-pure map is actually equal to that map. [ABSTRACT FROM AUTHOR]

Let m\ge 1 be a rational integer. We give an explicit formula for the mean value \begin{equation*} \frac {2}{\phi (f)}\sum _{\chi (-1)=(-1)^m}\vert L(m,\chi)\vert ^2, \end{equation*} where \chi ranges over the \phi (f)/2 Dirichlet characters modulo f>2 with the same parity as m. We then adapt our proof to obtain explicit means values for products of the form L(m_1,\chi _1)\cdots L(m_{n-1},\chi _{n-1})\overline {L(m_n,\chi _1\cdots \chi _{n-1})}. [ABSTRACT FROM AUTHOR]

Let T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let \mathcal {M}^T be the non-empty set of all proper metrics d on T compatible with its topology, and equip \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T^2. We prove that the set \mathcal {A}^{T,1} of metrics d\in \mathcal {M}^T for which the Lipschitz-free space \mathcal {F}(T,d) has the metric approximation property is a dense set in \mathcal {M}^T, and is furthermore residual in \mathcal {M}^T when T is zero-dimensional. We also prove that if T is uncountable then the set \mathcal {A}^T_f of metrics d\in \mathcal {M}^T for which \mathcal {F}(T,d) fails the approximation property is dense in \mathcal {M}^T. Combining the last statement with a result of Dalet, we conclude that for any 'properly metrizable' space T, \mathcal {A}^T_f is either empty or dense in \mathcal {M}^T. [ABSTRACT FROM AUTHOR]

We give a topological proof (using directly some known results about infinite-dimensional spaces) of the following strengthening of a remarkable Hausdorff theorem: each separable metrizable space admitting a continuous map onto [0,1] is the union of a sequence G_1 \subset \ldots \subset G_{\xi } \subset \ldots, {\xi } < {\omega }_1, G_{\xi } \not = X, of zero-dimensional G_{\delta }-sets in X such that for any finite Borel measure \mu on X, \mu (X \setminus G_{\xi }) = 0, for some {\xi }. Our approach yields also an analogous result in a more general setting of ccc \sigma-ideals. Proofs of the Hausdorff theorem in the literature are based on subtle combinatorial reasonings. [ABSTRACT FROM AUTHOR]

We improve the large sieve inequality with k^{\mathrm {th}}-power moduli, for all k\ge 4. Our method relates these inequalities to a variant of Waring's problem with restricted k^{\mathrm {th}}-powers. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two k^{\mathrm {th}}-powers. Secondly, we apply Marmon's bound on the number of representations of a positive integer as a sum of four k^{\mathrm {th}}-powers. Thirdly, we use Wooley's Vinogradov mean value theorem with arbitrary weights. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of k+1 k^{\mathrm {th}}-powers. [ABSTRACT FROM AUTHOR]

We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston's conjecture predicts that every M-bundle over a manifold B where \operatorname {dim}(B)\leq \operatorname {dim}(M) is cobordant to a flat M-bundle. In particular, we study the bordism class of flat M-bundles over low dimensional manifolds, comparing a finite dimensional Lie group G with \mathrm {Diff}_0(G). [ABSTRACT FROM AUTHOR]

For a finite, positive Borel measure \mu on (0,1) we consider an infinite matrix \Gamma _\mu, related to the classical Hausdorff matrix defined by the same measure \mu, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When \mu is the Lebesgue measure, \Gamma _\mu reduces to the classical Hilbert matrix. We prove that the matrices \Gamma _\mu are not Hankel, unless \mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H^p, \, 1 \leq p < \infty, and we study their compactness and complete continuity properties. In the case 2\leq p<\infty, we are able to compute the exact value of the norm of the operator. [ABSTRACT FROM AUTHOR]

Recently, Widmer introduced a new sufficient criterion for the Northcott property on the finiteness of elements of bounded height in infinite algebraic extensions of number fields. We provide a simplification of Widmer's criterion when the extension is abelian, and use this to exhibit fields with the Northcott property that do not satisfy Widmer's new criterion. We also show how the construction of pseudo algebraically closed fields with the Northcott property, carried out in previous work using the first version of Widmer's criterion, can be simplified. [ABSTRACT FROM AUTHOR]