In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If a , b , c are elements in a unital associative ring R satisfying a b a = a c a , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal I of R ) of 1 − a c is converted into that of 1 − b a . (ii) If A , B , C are

We establish the higher fractional differentiability of the solutions of a problem in divergence form of the type div A ( x , D u ) = div | F | p ( x ) − 2 2 F in Ω , u = 0 on ∂ Ω . The main features consist in assuming that the partial map ξ → A ( x , ξ ) has p ( x ) ‐growth, the datum F is Besov regular and both the partial map x → A ( x , ξ ) and the function x → p ( x ) are Orlicz–Besov regular

We obtain a new bound on the average value of the error term in the asymptotic formula for the number of k ‐free numbers in arithmetic progressions. In particular, we improve the results of J. Gibson (2014) and C. Hooley (1975).

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C , every holomorphic map from C to X is algebraic. We use the

In this article we classify isometric immersions f : M n ( c ) → S n + p × R of Riemannian manifolds of dimension n and constant sectional curvature c ≤ 1 into the product space S n + p × R , p ≤ n − 3 , where S m denotes the m ‐dimensional unit sphere.

We obtain upper and lower Gaussian density estimates for the laws of each component of the solution to a one‐dimensional fully coupled forward‐backward SDE. Our approach relies on the link between FBSDEs and quasilinear parabolic PDEs, and is fully based on the use of classical results on PDEs rather than on manipulation of FBSDEs, compared to other papers on this topic. This essentially simplifies

In this paper we study 4‐dimensional affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure ω. We prove that the rank of the shape operator is at most one if R k · ω = 0 or ∇ k ω = 0 for some positive integer k . This result is the final step in a classification of Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms

The geometric and algebraic properties of smooth projective varieties with 1‐regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2‐regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable

With a closed symmetric operator A in a Hilbert space H a triple Π = { H , Γ 0 , Γ 1 } of a Hilbert space H and two abstract trace operators Γ0 and Γ1 from A ∗ to H is called a generalized boundary triple for A ∗ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M ( · ) are investigated. The most

We consider semilinear elliptic equations of the form Δ u + g ( u ) = 0 on R N . Given a suitable positive root z of g, we discuss the existence of positive radial solutions u with u ( ∞ ) = z .

The following problem was originally posed by B. H. Neumann and H. Neumann. Suppose that a group G can be generated by n elements and that H is a homomorphic image of G. Does there exist, for every generating n‐tuple ( h 1 , … , h n ) of H, a homomorphism ϑ : G → H and a generating n‐tuple ( g 1 , … , g n ) of G such that ( g 1 ϑ , … , g n ϑ ) = ( h 1 , … , h n ) ?

We introduce new integrals (called packing R and R ∗ integrals) which combine advantages of integrals developed by Pfeffer [17], Malý [12], Kuncová and Malý [10] and Malý and Pfeffer [13]. We prove Gauss–Green theorem in generality of the new integrals and provide comparison with the integrals mentioned above and some others (like M C α by Ball and Preiss [2]).

In this paper we study modular extendability and equimodularity of endomorphisms and E0‐semigroups on factors, when these definitions are recast to the context of faithful normal semifinite weights and this dependency is analyzed. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. Furthermore

In this paper, we obtain a boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from the unit polydisc U n in C n to irreducible bounded symmetric domains realized as the unit ball B X of an N‐dimensional simple JB*‐triple X. In particular, we obtain a version of the boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from U n into the Euclidean unit ball B N in C N

In the article, we consider the multilinear fraction maximal operators and the multilinear fraction integrals with rough kernels, we introduce a new class for ( m + 1 ) ‐weights tuple ( u , ω ⃗ ) , and prove the optimal constant estimates of the sharp multi‐weighted bounds for both operators. We also show the sharp multi‐weighted bounds for the Cohen–Gosselin type multi‐commutator of the both operators

For Banach spaces X and Y we study the vector‐valued spectrum M ∞ ( B X , B Y ) , that is the set of non null algebra homomorphisms from H ∞ ( B X ) to H ∞ B Y , which is naturally projected onto the closed unit ball of H ∞ ( B Y , X ∗ ∗ ) . The aim of this article is to describe the fibers defined by this projection, searching for analytic balls and considering Gleason parts.

We prove a Desch–Schappacher type perturbation theorem for one‐parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi‐continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.

In [3] it was shown that when a three‐dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non‐vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing

In this paper, we are concerned with the Hardy–Hénon equations − Δ u = | x | a u p and Δ 2 u = | x | a u p with a ∈ R and p ∈ ( 0 , 1 ] . Inspired by Serrin and Zou [25], we prove Liouville theorems for nonnegative solutions to the above Hardy–Hénon equations (Theorem 1.1 and Theorem 1.3), that is, the unique nonnegative solution is u ≡ 0 .

We obtain criteria for the boundedness and compactness of weighted composition operators between different Fock spaces in C n . We also give estimates for essential norm of these operators.

We define Sturm's operator for vector valued Siegel modular forms obtaining an explicit description of their holomorphic projection in case of large absolute weight. However, for small absolute weight, Sturm's operator produces phantom terms in addition. This confirms our earlier results for scalar Siegel modular forms.

In this paper we deal with the following class of quasilinear elliptic problems − Δ p u = λ | u | p − 2 u + u + p ∗ − 1 + g ( x , u + ) + f ( x ) in Ω , u = 0 on ∂ Ω , where 1 ≤ p < N , Ω ⊂ R N is bounded with smooth boundary, g is a nonlinearity with subcritical growth condition and f ∈ L ∞ ( Ω ) . This class of equations is inspired by the famous Ambrosetti–Prodi problems and here we prove existence

In this paper, we determine the type numbers of the pseudo‐hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non‐diagonalizable shape operator in the 3‐dimensional anti‐de Sitter space. Also, we investigate the behavior of type numbers of the pseudo‐hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3‐dimensional

We study instanton bundles E on P 1 × P 1 × P 1 . We construct two different monads which are the analog of the monads for instanton bundles on P 3 and on the flag threefold F (0, 1, 2). We characterize the Gieseker semistable cases and we prove the existence of μ‐stable instanton bundles generically trivial on the lines for any possible c 2 ( E ) . We also study the locus of jumping lines.

We discuss the Borisov–Nuer conjecture in connection with the canonical maps from the moduli spaces M E n , h a of polarized Enriques surfaces with fixed h ∈ U ⊕ E 8 ( − 1 ) to the moduli space F g of polarized K 3 surfaces of genus g with g = h 2 + 1 , and we exhibit a naturally defined locus Σ g ⊂ F g . One direct consequence of the Borisov–Nuer conjecture is that Σ g would be contained in a particular

We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞ , or c 0, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of A ‐hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the

When studying the behavior of autonomous ordinary differential equations under time‐dependent perturbations vanishing for t → ± ∞ , their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on

In this paper we prove sufficient conditions for the Fredholm property of a non‐smooth pseudodifferential operator P which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol‐smoothing.

New oscillation criteria for the second‐order Emden–Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. (Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1861–1875), Li and Rogovchenko (Math. Nachr. 288 (2015)

The paper deals with the oscillatory behavior of the fourth‐order linear three‐terms advanced differential equation of the form y ( 4 ) ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( σ ( t ) ) = 0 . Assuming that all solutions of the auxiliary third‐order differential equation z ′ ′ ′ ( t ) + p ( t ) z ( t ) = 0 are nonoscillatory, two types of easily verifiable oscillation criteria for the studied equation

Riesz decomposition theorem says that superharmonic functions on the punctured unit ball are represented as the sum of generalized (Newtonian) potentials and harmonic functions. In this paper we study growth properties near the origin of spherical means for generalized Riesz potentials of functions satisfying Orlicz conditions in the punctured unit ball.

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy E Ω ( z ) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and z = − 2 dim Ω .

The aim of this paper is to prove the existence of two solutions for a nonlinear elliptic problem involving the ( p , q ) ‐Laplacian operator. The solutions are obtained by using variational methods and critical points theorems. The positivity of the solutions is shown by applying a generalized version of the strong maximum principle.

The Asymptotic Plateau Problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space H n + 1 . The modified mean curvature flow (MMCF) ∂ F ∂ t = ( H − σ ) ν , σ ∈ ( − n , n ) , was firstly introduced by Xiao and the second author a few years back in [15], and it provides a tool using geometric flow

In this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two‐weight norm inequalities for both operators on weighted Morrey spaces.

We consider envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one‐parameter families of Legendrian

The objective of the present work is to provide a well‐posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well‐posedness result for a capillary driven thin film with surfactant is provided in [4] by the

In this article we prove that for p > 2 , there exist pairs of self‐adjoint operators ( A 1 , B 1 ) and ( A 2 , B 2 ) and a function f on the real line in the homogeneous Besov class B ∞ , 1 1 ( R 2 ) such that the differences A 2 − A 1 and B 2 − B 1 belong to the Schatten–von Neumann class Sp but f ( A 2 , B 2 ) − f ( A 1 , B 1 ) ∉ S p . A similar result holds for functions of contractions. We also

We prove an abstract and flexible continuation theorem for zeros of parametrized Fredholm maps between Banach spaces. It guarantees not only the existence of zeros to corresponding equations, but also that they form a continuum for parameters from a connected manifold. Our basic tools are transfer maps and a specific topological degree. The main result is tailor‐made to solve boundary value problems

Based on Cynk–Hulek method from [7] we construct complex Calabi–Yau varieties of arbitrary dimensions using elliptic curves with an automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize the result of [11] to obtain arbitrarily dimensional Calabi–Yau manifolds which are Zariski in any characteristic p ≢ 1 ( mod 12 ) .

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo‐differential equation of order two. In agreement with the classical stability analysis, if the Mach number M satisfies M < 2 , the symbol

We consider the Schrödinger equation with a non‐degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schrödinger equation without loss of derivatives including the endpoint case. In contrast to the Riemannian metric case, we need the additional assumptions for the well‐posedness of our Schrödinger equation and for proving Strichartz estimates without loss.

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.

The classical Hardy–Littlewood inequality asserts that the integral of a product of two functions is always majorized by that of their non‐increasing rearrangements. One of the pivotal applications of this result is the fact that the boundedness of an integral operator acting near zero is equivalent to the boundedness of the same operator restricted to the cone of positive non‐increasing functions

We classify all rotational surfaces in Euclidean space whose principal curvatures κ1 and κ2 satisfy the linear relation κ 1 = a κ 2 + b , where a and b are two constants. As a consequence of this classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces. Finally, we give a variational

We consider the problem Δ 2 u = | u | 8 N − 4 u + ε f ( x ) in Ω , u = Δ u = 0 on ∂ Ω , ( P ε )where Ω is a bounded smooth domain in R N , N ≥ 5 , that exhibits certain symmetries and contains the origin, f ∈ L ∞ ( Ω ) , f ≥ 0 , f ≢ 0 , and ε > 0 is a small parameter. By using the Lyapunov–Schmidt reduction method and topological degree theory, for each sufficiently large k ∈ N , we construct sign‐changing

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left‐invariant metric is generated by a certain naturally defined inner product.

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction‐diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition

In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with ( p , q ) ‐growth. We obtain an optimal global Calderón–Zygmund type estimate for the

Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit‐length Killing vector field. 2) The projection of the image of the immersion is bounded in the base manifold. 3) The hypersurface is stochastically

This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order α ∈ ( 0 , 1 ) . In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo–Galerkin method to study several kinds of partial differential equations with fractional derivative in the

Retraction: Luo D, Liang, H. Oscillatory solutions of n‐th order advanced trinomial differential equations. Mathematische Nachrichten. 2018; 1–13. https://doi.org/10.1002/mana.201700219 The above article, published online on 05 October 2018, has been retracted by agreement between the corresponding author, Demou Luo, the journal's Editors‐in‐Chief, R. Denk, B. Andrews, K. Hulek and F. Klopp, and Wiley‐VCH