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  • EFFECTIVE VERSIONS OF TWO THEOREMS OF RADO
    Q. J. Math. (IF 0.636) Pub Date : 2020-03-17
    Bell J, Funk D, Du Kim B, et al.

    Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.

    更新日期:2020-03-21
  • GABRIEL–ROITER MEASURE, REPRESENTATION DIMENSION AND REJECTIVE CHAINS
    Q. J. Math. (IF 0.636) Pub Date : 2020-03-17
    Conde T.

    The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and

    更新日期:2020-03-21
  • Presentations for Subrings and Subalgebras of Finite CO-Rank
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-29
    Mayr P, Ruškuc N.

    Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented

    更新日期:2020-03-16
  • Mass distribution for toral eigenfunctions via Bourgain’s de-randomization
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-20
    Sartori A.

    We study the mass distribution of Laplacian eigenfunctions at Planck scale for the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. By averaging over the ball centre, we use Bourgain’s de-randomization to compare the mass distribution of toral eigenfunctions to the mass distribution of random waves in growing balls around the origin. We then classify all possible limiting distributions

    更新日期:2020-03-16
  • Homogeneous Spinor Flow
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-20
    Freibert M, Schiemanowski L, Weiss H.

    We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension three and on almost abelian Lie groups in detail. As a further example, the flag manifold in dimension six is treated.

    更新日期:2020-03-16
  • Banach Spaces of Almost Universal Complemented Disposition
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-10
    Castillo J, Moreno Y.

    We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented

    更新日期:2020-03-16
  • BOGOMOLOV MULTIPLIERS OF P-GROUPS OF MAXIMAL CLASS
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-06
    FernÁndez-Alcober G, Jezernik U.

    Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier ${\operatorname{B}}_0(G)$ is trivial in terms of the lower central series of $G$ and $P_1 = C_G(\gamma _2(G) / \gamma _4(G))$. If in addition $G$ has positive degree of commutativity and $P_1$ is metabelian, we show how understanding ${\operatorname{B}}_0(G)$ reduces to the simpler commutator

    更新日期:2020-03-16
  • NUMBER OF PRIME FACTORS OVER ARITHMETIC PROGRESSIONS
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-05
    Meng X.

    Numerical experiments suggest that there are more prime factors in certain arithmetic progressions than others. Greg Martin conjectured that the function $\sum _{n\leq x, n\equiv 1 \bmod 4} \omega (n)-\sum _{n\leq x, n\equiv 3 \bmod 4} \omega (n)$ will attain a constant sign as $x\rightarrow \infty $, where $\omega (n)$ is the number of distinct prime factors of $n$. In this paper, we prove explicit

    更新日期:2020-03-16
  • LOGARITHMIC INTERPOLATION METHODS AND MEASURE OF NON-COMPACTNESS
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-02
    Besoy B, Cobos F.

    We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with $\theta = 0,1$ between quasi-Banach spaces. Applications are given to operators between Lorentz–Zygmund spaces.

    更新日期:2020-03-16
  • Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-25
    Laustsen N, Troitsky V.

    We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots ,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in{\mathbb{N}}$: • there is a vector lattice homomorphism $\Phi _{\boldsymbol{x}}\colon H_n\to X$ such that $$\be

    更新日期:2020-03-16
  • HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-23
    Lagarias J, Rodgers B.

    The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function

    更新日期:2020-03-16
  • Positivity of Divisor Classes on the Strata of Differentials
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-20
    Chen D.

    Three decades ago Cornalba and Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

    更新日期:2020-03-16
  • SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-27
    Guillou B, May J, Merling M, et al.

    We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric

    更新日期:2020-03-16
  • UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-20
    Gros M, Masaharu K.

    Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants.For a simply connected semisimple algebraic group

    更新日期:2020-03-16
  • EXTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-18
    Funayoshi K, Koda Y.

    An automorphism $f$ of a closed orientable surface $\Sigma $ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma $ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect

    更新日期:2020-03-16
  • GRADIENT AND HESSIAN ESTIMATES FOR DIRICHLET AND NEUMANN EIGENFUNCTIONS
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-10
    Wang F.

    We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

    更新日期:2020-03-16
  • AN EXTENSION OF THE BOURGAIN–SARNAK–ZIEGLER THEOREM WITH MODULAR APPLICATIONS
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-06
    Cafferata M, Perelli A, Zaccagnini A.

    We first prove an extension of the Bourgain–Sarnak–Ziegler theorem, relaxing some conditions and giving quantitative estimates. Then we apply our extension to bound certain exponential sums, where the coefficients come from modular forms and the exponential involves polynomial sequences of any degree.

    更新日期:2020-03-16
  • HOMOLOGICAL STABILITY FOR UNLINKED CIRCLES IN A 3-MANIFOLD
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-05
    Kupers A.

    We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.

    更新日期:2020-03-16
  • ON RELATIVE COMPLETE REDUCIBILITY
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-05
    Attenborough C, Bate M, Gruchot M, et al.

    Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832], gives a purely algebraic description of the closed $K$-orbits in $G^n$, where

    更新日期:2020-03-16
  • Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-25
    Sato S.

    We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.

    更新日期:2020-03-16
  • THE DISCREPANCY OF (nkx)k=1∞ WITH RESPECT TO CERTAIN PROBABILITY MEASURES
    Q. J. Math. (IF 0.636) Pub Date : 2020-03-12
    Technau N, Zafeiropoulos A.

    Let $(n_k)_{k=1}^{\infty }$ be a lacunary sequence of integers. We show that if $\mu $ is a probability measure on $[0,1)$ such that $|\widehat{\mu }(t)|\leq c|t|^{-\eta }$, then for $\mu $-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies $$\begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*}$$for some constant $C>0$. This proves

    更新日期:2020-03-16
  • THE COMPOSITION OPERATION ON SPACES OF HOLOMORPHIC MAPPINGS
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-25
    Acosta M, Galindo P, Moraes L.

    We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of

    更新日期:2020-02-27
  • A SHARP ADAMS-TYPE INEQUALITY FOR WEIGHTED SOBOLEV SPACES
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-07
    do Ó J, Macedo A, de Oliveira J.

    In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

    更新日期:2020-02-07
  • A HYPERFINITE FACTOR WHICH IS NOT AN INJECTIVE C*-ALGEBRA
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-05
    Wright J, Saitô K.

    We exhibit a wild monotone complete C*-algebra which is a hyperfinite factor but is not an injective C*-algebra.

    更新日期:2020-02-07
  • A Sharp Lorentz-Invariant Strichartz Norm Expansion for the Cubic Wave Equation in ℝ1+3
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-03
    Negro G.

    We provide an asymptotic formula for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant for the linear Strichartz estimate. We calculate the constant in the second term, which differs depending on whether the equation is focussing or defocussing. The sign of this coefficient also changes accordingly

    更新日期:2020-02-06
  • ON TANGENCY IN EQUISINGULAR FAMILIES OF CURVES AND SURFACES
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-03
    Giles Flores A, Silva O, Snoussi J.

    We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families

    更新日期:2020-02-06
  • WARING’S PROBLEM FOR RATIONAL FUNCTIONS IN ONE VARIABLE
    Q. J. Math. (IF 0.636) Pub Date : 2020-02-03
    Im B, Larsen M.

    Let $f\in{\mathbb{Q}}(x)$ be a non-constant rational function. We consider ‘Waring’s problem for $f(x)$,’ i.e., whether every element of ${\mathbb{Q}}$ can be written as a bounded sum of elements of $\{f(a)\mid a\in{\mathbb{Q}}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent

    更新日期:2020-02-03
  • QUANTUM COHOMOLOGY AND CLOSED-STRING MIRROR SYMMETRY FOR TORIC VARIETIES
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-11
    Smith J.

    We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety $X$, by showing directly that the Kodaira–Spencer map of Fukaya–Oh–Ohta–Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, $X$ need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings

    更新日期:2020-01-13
  • Varieties of involution monoids with extreme properties
    Q. J. Math. (IF 0.636) Pub Date : 2019-06-22
    Lee E.

    A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These

    更新日期:2020-01-04
  • Differential operators on polar harmonic Maass forms and elliptic duality
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-03
    Bringmann K, Jenkins P, Kane B.

    In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

    更新日期:2020-01-04
  • The Ricci pinching functional on solvmanifolds
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Lauret J, Will C.

    We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group

    更新日期:2020-01-04
  • Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Fino A, Rollenske S, Ruppenthal J.

    It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalizes previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

    更新日期:2020-01-04
  • Fillings and fittings of unit cotangent bundles of odd-dimensional spheres
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Kwon M, Zehmisch K.

    We introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

    更新日期:2020-01-04
  • Hopf invariants for sectional category with applications to topological robotics
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    González J, Grant M, Vandembroucq L.

    We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular, we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological

    更新日期:2020-01-04
  • On Local definability of holomorphic functions
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-16
    Jones G, Kirby J, Le Gal O, et al.

    Given a collection $\mathcal {A}$ of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from $\mathcal {A}$. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from $\mathcal {A}$ in the neighbourhood of a generic point. We prove that this description is

    更新日期:2020-01-04
  • Calculations with graded perverse-coherent sheaves
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-24
    Achar P, Hardesty W.

    In this paper, we carry out several computations involving graded (or ${{\mathbb {G}}_{\textrm {m}}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the ${{\mathbb {G}}_{\textrm {m}}}$-action on certain normalized (or ‘canonical’) simple objects, confirming an old prediction of Ostrik

    更新日期:2020-01-04
  • On the 2-head of the colored Jones polynomial for pretzel knots
    Q. J. Math. (IF 0.636) Pub Date : 2019-08-27
    Beirne P.

    In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

    更新日期:2020-01-04
  • COUNTING INTEGERS WITH A SMOOTH TOTIENT
    Q. J. Math. (IF 0.636) Pub Date : 2019-09-10
    Banks W, Friedlander J, Pomerance C, et al.

    In an earlier paper we considered the distribution of integers $n$ for which Euler’s totient function at $n$ has all small prime factors. Here we obtain an improvement that is likely to be best possible.

    更新日期:2020-01-04
  • Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-04
    Basu S, Lerario A, Natarajan A.

    Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$ such that each manifold $Z_d$ is diffeomorphic to a component of the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case

    更新日期:2020-01-04
  • SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION
    Q. J. Math. (IF 0.636) Pub Date : 2019-09-26
    Heap W, Radziwiłł M, Soundararajan K.

    We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin–Chandee–Radziwiłł.

    更新日期:2020-01-04
  • HYPERSYMPLECTIC MANIFOLDS AND ASSOCIATED GEOMETRIES
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Thakre V.

    We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU$(1,1)$. When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU$(1,1)$ is also proper, then the hypersymplectic manifold fibres over a para-quaternionic Kähler manifold. We conclude the article with some examples

    更新日期:2020-01-04
  • BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Benson D, Kessar R, Linckelmann M.

    Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers

    更新日期:2020-01-04
  • MOMENTS OF SPINOR L-FUNCTIONS AND SYMPLECTIC KLOOSTERMAN SUMS
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Waibel F.

    We compute the second moment of spinor $L$-functions at central points of Siegel modular forms on congruence subgroups of large prime level $N$ and give applications to non-vanishing.

    更新日期:2020-01-04
  • DIOPHANTINE APPROXIMATION WITH GAUSSIAN PRIMES
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-18
    Harman G.

    In this paper we prove that the exact analogue of the author’s work with real irrationals and rational primes (G. Harman, On the distribution of $\alpha p$ modulo one II, Proc. London Math. Soc. (3) 72, 1996, 241–260) holds for approximating $\alpha \in \mathbb{C}\setminus \mathbb{Q}[i]$ with Gaussian primes. To be precise, we show that for such $\alpha $ and arbitrary complex $\beta $ there are infinitely

    更新日期:2020-01-04
  • THE REAL GRADED BRAUER GROUP
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-18
    Karoubi M, Weibel C.

    We introduce a version of the Brauer–Wall group for Real vector bundles of algebras (in the sense of Atiyah) and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer–Wall and Witt groups.

    更新日期:2020-01-04
  • A Unified Factorization Theorem for Lipschitz Summing Operators
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-20
    Botelho G, Maia M, Pellegrino D, et al.

    We prove a general factorization theorem for Lipschitz summing operators in the context of metric spaces that recovers several linear and nonlinear factorization theorems that have been proved recently in different environments. New applications are also given.

    更新日期:2020-01-04
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