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We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rørdam [Duke Math. J. 77 (1995), pp. 627–656 ] that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rørdam's result, and in typical cases we significantly improve control of the bound when the dimension grows. [ABSTRACT FROM AUTHOR]

The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with n vertices is more than 100 years old [Quart. J. Math. 49 (1920), pp. 1–36]. In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences. [ABSTRACT FROM AUTHOR]

Recently, Kayumov [Lobachevskii J. Math. 38 (2017), pp. 466–468] obtained a sharp estimate for the n-th truncated area functional for normalized functions in the Bloch space for n\le 5 and then, together with Wirths [Lobachevskii J. Math. 40 (2019), pp. 1319–1323], extended the result for n=6. We prove that for the functions with non-negative Taylor coefficients, the same sharp estimate is valid for all n. For arbitrary functions, we obtain an estimate that is asymptotically of the same order but slightly larger (roughly by a factor of 4/e). We also consider related weighted estimates for functionals involving the powers n^t, t>0, and show that the exponent t=1 represents the critical case for the expected sharp estimate. [ABSTRACT FROM AUTHOR]

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved by us earlier [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941] and by Xiaoliang Cheng and Yihong Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180]. [ABSTRACT FROM AUTHOR]

Let R=k[x,y] be a polynomial ring over a field k of prime characteristic p and let E denote the injective hull of k (which is isomorphic to H^2_{(x,y)}(R)). We prove that E is not an injective object in the category of graded F-modules over R. This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578]. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to discuss the density of proper efficient points. As an interesting application of the results in this paper, we want to prove a density theorem of Arrow, Barankin, and Blackwell. [ABSTRACT FROM AUTHOR]

This paper studies the problem of uniqueness of entire functions that share real zeros and real ones. An example is provided to show that a result of Czubiak and Gundersen is not completely correct. Results in this paper correct the result of Czubiak and Gundersen, and also correct a result of Osgood and Yang. [ABSTRACT FROM AUTHOR]

Under the relation of 0-concordance, the set of knotted 2-spheres in S^4 forms a commutative monoid \mathcal {M}_0 with the operation of connected sum. Sunukjian [Int. Math. Res. Not. IMRN 17 (2015), pp. 7950–7978] has recently shown that \mathcal {M}_0 contains a submonoid isomorphic to \mathbb {Z}^{\ge 0}. In this note, we show that \mathcal {M}_0 contains a submonoid isomorphic to (\mathbb {Z}^{\ge 0})^\infty. Our argument relates the 0-concordance monoid to linear independence of certain Seifert solids in the spin rational homology cobordism group. [ABSTRACT FROM AUTHOR]

Let (R,\mathfrak {m}) be a Noetherian local ring of dimension d\geq 2. We prove that if e(\widehat {R}_{red})>1, then the classical Lech's inequality can be improved uniformly for all \mathfrak {m}-primary ideals, that is, there exists \varepsilon >0 such that e(I)\leq d!(e(R)-\varepsilon)\ell (R/I) for all \mathfrak {m}-primary ideals I\subseteq R. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of I. [ABSTRACT FROM AUTHOR]

Let E and F be two complex Hilbert spaces. Let S_{E} and S_{F} be the shift operators on vector-valued Hardy space H_{E}^{2} and H_{F}^{2}, respectively. We show that the cyclic multiplicity of S_{E}\oplus S_{F} ^{\ast } equals 1+\dim E. This result is classical when \dim E=\dim F=1 (see J. A. Deddens [ On A \oplus A^\ast, 1972]; Paul Richard Halmos [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos's book [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S_{E}\oplus S_{F}^{\ast } when \dim E<\infty and \dim F=1. [ABSTRACT FROM AUTHOR]

Let rk(n) denote the number of representations of the positive integer n as the sum of k squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation involving rk(n) and certain Bessel functions. We provide a proof of this identity for the first time, as well as for another identity, which can be regarded as both an analogue of Popov's identity and an identity involving r2(n) from Ramanujan's lost notebook. [ABSTRACT FROM AUTHOR]

Yau [Math. Ann. 207 (1974), pp. 269–270] proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces with curvature bounded below/above, C^{\alpha }-Hölder Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of Finsler metrics using the warped product notion introduced by Chen et al. [Internat. J. Math. 29 (2018), 1850081], with another "warping", one that is consistent with static spacetimes. We will give the PDE characterization for the proposed metrics to be Ricci-flat and explicitly construct two non-Riemannian examples. [ABSTRACT FROM AUTHOR]

In this work, we investigate the limit case p=1 of the classical Stein-Weiss inequality for the Riesz potential. Our main result is a characterization of this inequality for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 877–921]. As application, we recover and improve some fractional inequalities associated to vector fields. [ABSTRACT FROM AUTHOR]

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if X is a regular topological space, with character < \mathfrak {b} and X \rightarrow (top\, \omega + 1)^{1}_{\omega }, then, for all \alpha < \omega _1, X \rightarrow (top\, \alpha)^{1}_{\omega }, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle \clubsuit _{F}, and use it to prove that it is consistent with \neg CH that \mathfrak {b} is the optimal bound for the character of X. In [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using \diamondsuit. [ABSTRACT FROM AUTHOR]

In this short note, we show that the generalized type semigroup \mathcal {W}(X, \Gamma) introduced by Ma [Ergodic Theory Dynam. Systems 41 (2021), pp. 2148–2165] belongs to the category W introduced by Antoine, Perera and Theil [Mem. Amer. Math. Soc. 251 (2018), vii+191]. In particular, we demonstrate that \mathcal {W}(X, \Gamma) satisfies axioms (W1)–(W4) and (W6). When X is zero-dimensional, we also establish (W5) for the semigroup. This supports the analogy between the generalized type semigroup and pre-completed Cuntz semigroup W(\cdot) for C^*-algebras. [ABSTRACT FROM AUTHOR]

Building upon the work of A. Booker and C. Pomerance [Proc. Amer. Math. Soc. 145 (2017), pp. 5035–5042], we prove that for a prime power q \geq 7, every residue class modulo an irreducible polynomial F \in \mathbb {F}_q[X] has a non-constant, square-free representative which has no irreducible factors of degree exceeding \deg F -1. We also give applications to generating sequences of irreducible polynomials. [ABSTRACT FROM AUTHOR]

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of \mathbb {R}, but in subsets of the p-adic field \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

Let s(n)=\sum _{d\mid n,~d

Pursuing the approach of Gentili and Vezzoni [Math. Res. Not. IMRN 12 (2022), pp. 9499–9528] we study the quaternionic Monge-Ampère equation on HKT (hyperkähler with torsion) manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has always a unique solution for every basic datum. This approach includes the study of the equation on \operatorname {SU}(3). [ABSTRACT FROM AUTHOR]

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of \textit{Bounds for the order of supersoluble automorphism groups of Riemann surfaces} (Proc. Amer. Math. Soc. {\bf108} (1990), 587--600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that. [ABSTRACT FROM AUTHOR]

For a real Hilbert space H_{\mathbb {R}} and -1 < q < 1 Bozejko and Speicher introduced the C^\ast-algebra A_q(H_{\mathbb {R}}) and von Neumann algebra M_q(H_{\mathbb {R}}) of q-Gaussian variables. We prove that if \dim (H_{\mathbb {R}}) = \infty and -1 < q < 1, q \not = 0 then M_q(H_{\mathbb {R}}) does not have the Akemann-Ostrand property with respect to A_q(H_{\mathbb {R}}). It follows that A_q(H_{\mathbb {R}}) is not isomorphic to A_0(H_{\mathbb {R}}). This gives an answer to the C^\ast-algebraic part of Question 1.1 and Question 1.2 in raised by Nelson and Zeng [Int. Math. Res. Not. IMRN 17 (2018), pp. 5486–5535]. [ABSTRACT FROM AUTHOR]

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi's method. [ABSTRACT FROM AUTHOR]

Let K and S be locally compact Hausdorff spaces and let T be a real linear isomorphism of an extremely regular subspace A of C_{0}(K) into C_{0}(S) satisfying \|T \| \ \|T^{-1}\|<2. Put L=T \ \|T^{-1}\| and pick 0 \leq \varepsilon <1 such that \|T \| \ \|T^{-1}\|=1+\varepsilon. We prove that there exist a subset S_0 of S, a proper map \varphi of S_{0} onto K and a map \lambda :S_{0} \to \{-1, 1\} such that \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For \varepsilon < 1/2 the maps \varphi and \lambda are continuous, and this approximation of L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where \varepsilon =0, K and S are compact and A=C_{0}(K). The approximation of L in the general case where 0 \leq \varepsilon <1 is an extension of a Benyamini's result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz's theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K is a continuous image not just of a subset of S but of a locally compact subset of S. Moreover, when K is compact, it is a continuous image not just a closed subset of S but of a compact subset of S, namely the set S_0 having the above properties. [ABSTRACT FROM AUTHOR]

We show that the norm in the Hardy space H^p satisfies \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f')(1-r)^{p\left (1-\frac 1q\right)}\,dr+|f(0)|^p\tag {\dagger } \end{equation} for all univalent functions provided that either q\ge 2 or \frac {2p}{2+p}

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], \phi is immediately periodic with period 2. We show that, for every positive integer \pi, there exists a ring for which \phi is immediately periodic with period \pi. [ABSTRACT FROM AUTHOR]

Let F be the function field of a curve over a non-archimedean local field. Let m \geq 2 be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in H^{3}(F, \mu _{m}^{\otimes 2}) is of the form \chi \cup (f) \cup (g), where \chi is in H^{1}(F, \mathbb {Z}/m\mathbb {Z}) and (f), (g) in H^{1}(F, \mu _{m}). This extends a result of Parimala and Suresh [Ann. of Math. (2) 172 (2010), pp. 1391–1405], where they show this when m is prime and when F contains \mu _{m}. [ABSTRACT FROM AUTHOR]

We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov [Trans. Amer. Math. Soc. 324 (1991), pp. 793-821] and Luo-Tian [Proc. Amer. Math. Soc. 116 (1992), pp. 1119-1129] regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval (0, 2\pi) at a given configuration of three or more points. [ABSTRACT FROM AUTHOR]

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general "minimal test for injectivity" for a left Artinian ring A: we show that if every injective morphism \Sigma _k\rightarrow A from the k'th socle of A extends to a morphism A\rightarrow A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N\rightarrow M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N\rightarrow M extend (quasi-injectivity) and obtain further applications. [ABSTRACT FROM AUTHOR]

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits [Ann. of Math. (2) 97 (1973), pp. 499–571]. Our method extends the approach introduced by the first author in [Proc. Lond. Math. Soc. (3) 102 (2011), pp. 951–983] to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation. [ABSTRACT FROM AUTHOR]

We refer to the discussion on different characterizations of the A_\infty class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi [Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A_\infty condition are considered. For cubes in \mathbb {R}^d every two conditions are known to be equivalent, while for general bases we have a trichotomy: equivalence, one-way implication, or no dependency may occur. In most cases the relations between different conditions have already been established. Here all the unsolved cases are treated and, as a result, a full diagram of the said relations is presented. [ABSTRACT FROM AUTHOR]

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., \{0,1\}-valued functions) to Q-valued functions, for a better quasiorder Q. In this article, we consider more general reducibilities on the Q-valued functions and generalise some results of L. Motto Ros [J. Symbolic Logic 74 (2009), pp. 27–49] in the first direction and of T. Kihara and A. Montalbán [Trans. Amer. Math. Soc. 370 (2018), pp. 9025–9044] in the second direction: Our main result states that the structure of the \mathbf {\Delta }^0_\alpha-degrees of \mathbf {\Delta }^0_{\alpha +\gamma }-measurable Q-valued functions is isomorphic to the \mathbf {\Delta }^0_\beta-degrees of \mathbf {\Delta }^0_{\beta +\gamma }-measurable Q-valued functions, and these are isomorphic to the generalized homomorphism order on the \gamma-th iterated Q-labeled forests. [ABSTRACT FROM AUTHOR]

Let G x X → X be an action of the connected algebraic group G on the irreducible, affine variety X. We discuss the relationship between [k[X]G] and k(X)G, where [k[X]G] denotes the quotient field of k[X]G. We are particularly interested in the following three questions. (1) When is the inclusion [k[X]G] ⊆k(X)G a finite extension of fields? (2) What is the role of G-invariant divisors? (3) What is the exact characterization of "observable in codimension one"? [ABSTRACT FROM AUTHOR]

In this work, we establish precise local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \Omega \subset \mathbb {R}^3 with smooth boundary \partial \Omega \neq \emptyset. This result is the improved estimates established by Len Meas [C. R. Math. Acad. Sci. Paris 355 (2017), pp. 161–165]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \Omega have been proved by Blair, Smith, Sogge [Proc. Amer. Math. Soc. 136 (2008), pp. 247–256; Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), pp.1817–1829]. Optimal estimates in strictly convex domains have been obtained by Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380]. Our case of cylindrical domains is an extension of the result of Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions. [ABSTRACT FROM AUTHOR]

We establish a new asymptotic formula for the number of polynomials of degree n with k prime factors over a finite field \mathbb {F}_q. The error term tends to 0 uniformly in n and in q. Previously, asymptotic formulas were known either for fixed q, through the works of Warlimont [Arch. Math. (Basel) 60 (1993), pp. 58–72] and Hwang [Random Structures Algorithms 13 (1998), pp. 17–47], or for small k, through the work of Arratia, Barbour and Tavaré [Math. Proc. Cambridge Philos. Soc. 114 (1993), pp. 347–368]. As an application, we estimate the total variation distance between the number of cycles in a random permutation on n elements and the number of prime factors of a random polynomial of degree n over \mathbb {F}_q. The distance tends to 0 at rate 1/(q\sqrt {\log n}). Previously this was only understood when either q is fixed and n tends to \infty, or n is fixed and q tends to \infty, by results of Arratia, Barbour and Tavaré. [ABSTRACT FROM AUTHOR]

We establish the Gagliardo-Nirenberg inequality, Trudinger-Moser inequality and John-Nirenberg inequality using the Lorentz spaces L^{p,\alpha }, the Marcinkiewicz space L^{q,\infty } and the weak-L^{\infty } space W introduced by Bennett, DeVore and Sharpley [Ann. of Math. (2) 113 (1981), pp. 601–611]. As consequences, we obtain the Gagliardo-Nirenberg type inequality with weak-L^{\infty } norm and BMO norm, Trudinger-Moser type inequality and John-Nirenberg type estimate with BMO norm and weak-L^{1} norm. [ABSTRACT FROM AUTHOR]

We prove that a Toeplitz operator with a complex Borel measure symbol whose total variation is Carleson is strongly localized, thus improving the recent localization results from Sadeghi and Zorboska [J. Math. Anal. Appl. 485 (2020), p. 16]. We also show that as a corollary, we get yet another view of some of the previously known results on Toeplitz operators with BMO symbols. [ABSTRACT FROM AUTHOR]

We prove that the ideal of nowhere Ramsey sets is Tukey reducible to each of the finite-dimensional Silver null ideals. This is in line with an earlier result of the author [Israel J. Math. 211 (2016), pp. 473–480] that the meager ideal is Tukey reducible to the Silver null ideal, and it answers two questions asked by Spinas and Wohofsky [Fund. Math. 254 (2021), pp. 261–303]. We also show that if the homogeneity number \mathfrak {hm} equals 2^{\aleph _0}, then the additivity of the Marczewski ideal is below the \sigma-splitting number \mathfrak {s_\sigma }. [ABSTRACT FROM AUTHOR]

For a large class of piecewise expanding C1,1 maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known 2/ inf ∣τ'∣. Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim's for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps. [ABSTRACT FROM AUTHOR]