The aim of this paper is to study the existence of solution to an equation involving the Bessel operator in R N ( I − Δ ) α u + λ V ( x ) u = γ f ( x , u ) , where λ , γ are real positive parameters, V : R N → R + is a continuous function, 0 < α < 1 with 2 α < N , f is a continuous function on R N × R which does not satisfy the Ambrosetti–Rabinowitz condition. By using the Fountain theorem and Morse

We study the well‐posedness of the fractional degenerate integro‐differential equations ( P α ) : D α ( M u ) ( t ) = A u ( t ) + ∫ − ∞ t a ( t − s ) A u ( s ) d s + ∫ − ∞ t b ( t − s ) B u ( s ) d s + f ( t ) , ( t ∈ T : = [ 0 , 2 π ] ) , in Lebesgue–Bochner spaces L p ( T ; X ) and Besov spaces B p , q s ( T ; X ) , where A , B and M are closed linear operators on a Banach space X satisfying D (

We describe the Aluthge transform of an unbounded weighted composition operator acting in an L2‐space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize p‐hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation

In this paper we prove mixed inequalities for the maximal operator M Φ , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy–Littlewood maximal operator proved by E. Sawyer. We show that given r ≥ 1 , if u , v r are weights belonging to the A1‐Muckenhoupt class and Φ is a Young function as above, then the inequality u v

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p‐adic Kochen operator provides a p‐adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p‐integral elements of K . We use this to formulate and

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half‐planes { s ∈ C | Re s > ε } for each ε > 0 . The properties of the formal inverse of the differentiation are also investigated.

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 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 .

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

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).

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying

We introduce a class of null hypersurfaces of a semi‐Riemannian manifold, namely, screen quasi‐conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical

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

We analyze the effect of nonlinear boundary conditions on an advection‐diffusion equation on the half‐line. Our model is inspired by models for crystal growth where diffusion models diffusive relaxation of a displacement field, advection is induced by apical growth, and boundary conditions incorporate non‐adiabatic effects on displacement at the boundary. The equation, in particular the boundary fluxes

We present quantitative versions of Bohr's theorem on general Dirichlet series D = ∑ a n e − λ n s assuming different assumptions on the frequency λ = ( λ n ) , including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on λ, we give upper bounds for the norm of the partial sum operator S N ( D )

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

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

In this note we study an abstract class of weakly dissipative second‐order systems with finite memory. We establish the polynomial decay of Rivera, Naso and Vegni for the solution of the system under a very weak condition on the relaxation function.

We prove a generalization of the Littlewood–Paley characterisation of the BMO space where the shifts of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel–Lizorkin spaces can be characterized in a similar way.

Let H be a separable Hilbert space with the unit operator I , let A be a bounded linear operator in H with a Schatten–von Neumann Hermitian component ( A − A ∗ ) / 2 i ( A ∗ means the operator adjoint to A ) and let f ( z ) be a function analytic on the spectra of A and A ∗ . For f ( A ) we obtain the representation in the form of the sum of a normal operator and a quasi‐nilpotent Schatten–von Neumann

We give an original representation integral formula for the resolvent families associated to the fractional Cauchy equation with memory effects C D t α u ( t ) − A u ( t ) + ∫ 0 t β ( t − s ) A u ( s ) d s = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , T > 0 , where u ( 0 ) = u 0 ∈ X and A is a sectorial operator on a Banach space X . Moreover, we get spatial bounds for the resolvent families in order to study

The study of symmetries is a well known research topic in differential geometry with relevant physical interpretations. Given a Riemannian manifold ( M , g ) , we consider pseudo‐Riemannian g‐natural metrics G on its tangent bundle T M and characterize conformal, homothetic and Killing vector fields of ( T M , G ) obtained from natural lifts of vector fields of M .

Assume that G is a finite group. For every a , b ∈ N , we define a graph Γ a , b ( G ) whose vertices correspond to the elements of G a ∪ G b and in which two tuples ( x 1 , ⋯ , x a ) and ( y 1 , ⋯ , y b ) are adjacent if and only if ⟨ x 1 , ⋯ , x a , y 1 , ⋯ , y b ⟩ = G . We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and

The global existence and asymptotic behavior for anisotropic parabolic equation in R n are considered. By using the classical Galerkin approximation and suitable conditions on the weighted function, it is obtained the global estimate and the uniformly estimates for the global weak solution. Besides, the L 2 ( R n ) ∩ L r ( R n ) global attractor for anisotropic parabolic p‐Laplacian equation is also

Let G be a simple graph on the vertex set [n ] and let J G be the corresponding binomial edge ideal. Let G = v ∗ H be the cone of v on H . In this article, we compute all the Betti numbers of J G in terms of the Betti numbers of J H and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S / J G in terms of Cohen–Macaulay defect of S H / J H and using

For the spaces D L p , L p and M 1 , we consider the topology of uniform convergence on absolutely convex compact subsets of their (pre‐)dual space. Following the notation of J. Horváth's book we call these topologies κ‐topologies. They are given by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre‐)dual spaces. In many cases it is more convenient to

For perturbed Stark operators H u = − u ′ ′ − x u + q u , the author has proved that lim sup x → ∞ x 1 2 | q ( x ) | must be larger than 1 2 N 1 2 in order to create N linearly independent eigensolutions in L 2 ( R + ) [29]. In this paper, we apply generalized Wigner–von Neumann type functions to construct embedded eigenvalues for a class of Schrödinger operators, including a proof that the bound 1

In this paper, we consider the following Klein–Gordon–Maxwell equations − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) + h ( x ) in R 3 , − Δ ϕ + ϕ u 2 = − ω u 2 in R 3 , where ω > 0 is a constant, u , ϕ : R 3 → R , V : R 3 → R is a potential function. By assuming the coercive condition on V and some new superlinear conditions on f , we obtain two nontrivial solutions when h is nonzero and infinitely

Given B , C and W operators in the algebra L ( H ) of bounded linear operators on the Krein space H , the minimization problem min ( B X − C ) # W ( B X − C ) , for X ∈ L ( H ) , is studied when the weight W is selfadjoint. The analogous maximization and min‐max problems are also considered. Complete answers to these problems and to those naturally associated to trace clase operators on Krein spaces

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

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 ) ?

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.

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

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.

In this paper, we obtain a boundary Schwarz lemma for C 1 (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 C 1 (pluriharmonic, holomorphic) mappings from U n into the Euclidean unit ball B N in C

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]).

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

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 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

We derive a formula for the discriminant of the ring class polynomial of conductor N associated to the j ‐invariant and an imaginary quadratic field of odd discriminant − d using the theory of Borcherds lifts. When d is a prime and N = 1 , the formula recovers the Gross–Zagier discriminant formula.

In this paper, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

Reflexive polytopes have been studied from viewpoints of combinatorics, commutative algebra and algebraic geometry. A nef‐partition of a reflexive polytope P is a decomposition P = P 1 + ⋯ + P r such that each P i is a lattice polytope containing the origin. Batyrev and van Straten gave a combinatorial method for explicit constructions of mirror pairs of Calabi–Yau complete intersections obtained from

There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still true.

We classify totally geodesic and parallel hypersurfaces of four‐dimensional non‐reductive homogeneous pseudo‐Riemannian manifolds.

Let M 2 → S 3 be an umbilic‐free surface. Four basic invariants of M 2 under the Möbius transformation group of S 3 are the Möbius metric g , the Blaschke tensor A , the Möbius second fundamental form B and the Möbius form Φ. A symmetric (0,2) tensor D = A + μ B called Para‐Blaschke tensor, where μ is constant, is also Möbius invariant. In this paper, we classify the surfaces with isotropic Para‐Blaschke

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

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