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Abstract: This paper continues the study of linear group actions with no regular orbits where the largest orbit size equals the order of the abelian quotient of the group. In previous work of the first author with Yong Yang it was shown that if G is a finite solvable group and G a finite group and V a finite faithful completely reducible G-module, possibly of mixed characteristic, and M is the largest orbit size in the action of G on V then \( G/G' \le M\) . In a continuation of this work the first author and his student Nathan Jones analyzed the first open case of when equality occurs and proved the following. If G is a finite nonabelian group and V a finite faithful irreducible G-module and \(M= G/G' \) is the largest orbit of G on V and that there are exactly two orbits if size M on V, then \(G=D_8\) and \(V=V(2,3)\) . This paper is concerned with the next case, the one where, under otherwise the same hypotheses as before, we have three orbits of size \(M= G/G' \) . It turns out that again there is exactly one such action, the one where G is the central product of \(D_8\) and \(C_4\) is acting on the vector space of order 25. PubDate: 2022-01-10
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Abstract: Every finite solvable group G has a normal series with nilpotent factors. The smallest possible number of factors in such a series is called the Fitting height h(G). In the present paper, we derive an upper bound for h(G) in terms of the exponent of G. Our bound constitutes a considerable improvement of an earlier bound obtained in Shalev (Proc Am Math Soc 126(12):3495–3499, 1998). PubDate: 2022-01-08
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Abstract: The main purpose of this paper is to study the generalized squeezing functions and Fridman invariants of some special domains. As applications, we give the precise form of generalized squeezing functions and Fridman invariants of various domains such as n-dimensional annuli. Furthermore, we provide domains with non-plurisubharmonic generalized squeezing function or Fridman invariant. PubDate: 2022-01-07
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Abstract: In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant. PubDate: 2022-01-04
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Abstract: We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space \(W^{s,p}(\Omega )\) for an open, bounded set \(\Omega \subset \mathbb {R}^{d}\) . The density property is closely related to the lower and upper Assouad codimension of the boundary of \(\Omega\) . We also describe explicitly the closure of \(C_{c}^{\infty }(\Omega )\) in \(W^{s,p}(\Omega )\) under some mild assumptions about the geometry of \(\Omega\) . Finally, we prove a variant of a fractional order Hardy inequality. PubDate: 2021-12-15
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Abstract: We prove that for antisymmetric vector field \(\Omega \) with small \(L^2\) -norm there exists a gauge \(A \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}^1,GL(N))\) such that $$\begin{aligned} {\text {div}}_{\frac{1}{2}} (A\Omega - d_{\frac{1}{2}} A) = 0. \end{aligned}$$ This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence. PubDate: 2021-12-14
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Abstract: Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples. PubDate: 2021-12-06 DOI: 10.1007/s10231-021-01158-7
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Abstract: In this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on \(\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))\) . We use this characterisation to classify 4-dimensional projective special Kähler Lie groups. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01096-4
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Abstract: We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01084-8
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Abstract: Let K be affine, that is, \(K=\{z=(x,y)\in {\mathbb {R}}^{n+m}: y_{1}=\cdots =y_{m}=0\}\) . We compute the sharp constant of Hardy inequality related to the distance d(z, K) for polyharmonic operator. Moreover, we show that there exists a constant \(C>0\) such that for each \(u\in C^{\infty }_{0}({\mathbb {R}}^{n+m}\setminus K)\) , there holds $$\begin{aligned} \int _{{\mathbb {R}}^{n+m}} \nabla ^{k} u ^{2}\mathrm{d}x\mathrm{d}y-c_{m,k}\int _{{\mathbb {R}}^{n+m}}\frac{u^{2}}{ y ^{2k}}\mathrm{d}x\mathrm{d}y\ge C\left( \int _{{\mathbb {R}}^{n+m}} y ^{\gamma } u ^{p}\mathrm{d}x\mathrm{d}y\right) ^{\frac{n+m-2k}{n+m}}, \end{aligned}$$ where \(2\le k<\frac{m+n}{2}\) , \(2<p\le \frac{2(n+m)}{n+m-2k}\) , \(\gamma =\frac{(n+m-2k)p}{2}-n-m\) and \(c_{m,k}\) is the sharp Hardy constant. These inequalities generalize the result of Maz’ya (case \(k=1\) ) and Lu and the second author (case \(m=1\) for polyharmonic operators). In order to prove the main result, we establish some Poincaré–Sobolev inequalities on hyperbolic space which is of independent interest. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01091-9
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Abstract: In this paper, we consider a pseudo-parabolic equation, which was studied extensively in recent years. We generalize and extend the existing results in the following three aspects. First, we consider the vacuum isolating phenomenon with the initial energy \(J(u_0)\) satisfying \(J(u_0)\le 0\) and \(0<J(u_0)<d\) , respectively, where d is a positive constant denoting the potential well depth. By means of potential well method, we find that there are two explicit vacuum regions which are annulus and ball, respectively. Second, we study the asymptotic behaviors of the solutions and the energy functional. Generally speaking, we establish the exponential decay of the solutions and energy functional when the solutions exist globally, and the concrete decay rate is given. As for the blow-up solutions, we prove that the solutions grow exponentially and obtain the behavior of energy functional as the time t tends to the maximal existence time. We get further two necessary and sufficient conditions for the solutions existing globally and blowing up in finite time, respectively, under the assumption that \(J(u_0)<d\) . Finally, we give a new blow-up condition with eigenfunction method; it should be point out that this initial condition is independent of the initial energy. Under this condition, an upper bound estimation of the blow-up time is obtained and we prove that the solutions grow exponentially; the grow speed is given specifically. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01099-1
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Abstract: We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian \({\mathcal {G}}\) and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01103-8
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Abstract: We study a free transmission problem in which solution minimizes a functional with different definitions in positive and negative phases. We prove some asymptotic regularity results when the jumps of the diffusion coefficients get smaller along the free boundary. At last, we prove a measure-theoretic result related to the free boundary. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01087-5
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Abstract: Let \(1 \le p \le \infty \) . A Banach lattice X is said to be p-disjointly homogeneous or \((p-DH)\) (resp. restricted \((p-DH)\) ) if every normalized disjoint sequence in X (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in X to the unit vector basis of \(\ell _p\) . We revisit DH-properties of Orlicz spaces and refine some previous results of this topic, showing that the \((p-DH)\) -property is not stable under duality in the class of Orlicz spaces and the classes of restricted \((p-DH)\) and \((p-DH)\) Orlicz spaces are different. Moreover, we give a characterization of uniform \((p-DH)\) Orlicz spaces and establish also closed connections between this property and the duality of the DH-property. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01097-3
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Abstract: Given an arbitrary \(C^\infty \) Riemannian manifold \(M^n\) , we consider the problem of introducing and constructing minimal hypersurfaces in \(M\times \mathbb {R}\) which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space \(\mathbb {R}^3=\mathbb {R} ^2\times \mathbb {R}\) . Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in \(M\times \mathbb {R}\) have the same fundamental uniqueness properties of the helicoids in \(\mathbb {R}^3.\) We provide several examples of properly embedded vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on \(\mathrm{Nil}_3\) and \(\mathrm{Sol}_3\) are also presented. We show that vertical helicoids of \(M\times \mathbb {R} \) whose horizontal sections are totally geodesic in M are locally given by a “twisting” of a fixed totally geodesic hypersurface of M. We give a local characterization of hypersurfaces of \(M\times \mathbb {R}\) which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in \(M\times \mathbb {R}\) if and only if M admits families of isoparametric hypersurfaces. If so, properly embedded vertical catenoids can be constructed through the solutions of a certain first-order linear differential equation. Finally, we give a complete classification of the hypersurfaces of \(M\times \mathbb {R}\) whose angle function is constant. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01085-7
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Abstract: In proving his theorem on global maximizers for the sphere adjoint Fourier restriction inequality, D. Foschi studied some quadratic functional and determined its optimal bound. In this note we extend to higher-dimensional spheres his results on this quadratic functional. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01092-8
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Abstract: We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the \(\mathbb {Q}\) -vector space spanned by 1 and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite–Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by Mahler (1968) Compos Math 19: 95–166. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of \(\mathbb {Q}\) . Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of \(\mathbb {Q}\) , and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which essentially goes back to Siegel. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01104-7
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Abstract: In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01088-4
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Abstract: This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached by analyzing three types of hypotheses: a Hörmander condition, logarithmic growth and super-logarithmic growth. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01090-w
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Abstract: We describe the structure of the projective cover of a module \(M_R\) over a local ring R and its relation with minimal sets of generators of \(M_R\) . The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect. PubDate: 2021-12-01 DOI: 10.1007/s10231-021-01095-5
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