Cohomological characterisation of monads

Let X be an n‐dimensional smooth projective variety with an n‐block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the form where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V₅. We construct a family of monads on a Segre variety and apply our main result to this family.     

The Radon‐Nikodym property,generalized bounded variation and the average range

A generalized bounded variation characterization of Banach spaces possessing the Radon‐Nikodym property is given in terms of the average range. We prove that a Banach space X has the Radon‐Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .     

Some properties of a class of symmetric functions and its applications

For , the symmetric functions are defined by where , and are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.     

Bicovariograms and Euler characteristic of regular sets

We establish an expression of the Euler characteristic of a r‐regular planar set in function of some variographic quantities. The usual framework is relaxed to a regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non‐connected polyrectangular grains in . Applications to excursions of smooth bivariate random fields are derived in the companion paper 25 , and applied for instance to Gaussian fields, generalising standard results.     

Generalized Cauchy–Hankel matrices and their applications to subnormal operators

It is well‐known that for a general operator T on Hilbert space, if T is subnormal, then is subnormal for all natural numbers . It is also well‐known that if T is hyponormal, then T ² need not be hyponormal. However, for a unilateral weighted shift , the hyponormality of (detected by the condition for all ) does imply the hyponormality of every power . Conversely, we easily see that for a weighted shift is not hyponormal, therefore not subnormal, but is subnormal for all . Hence, it is interesting to note when for some , the subnormality of implies the subnormality of T . In this article, we construct a non trivial large class of weighted shifts such that for some , the subnormality of guarantees the subnormality of . We also prove that there are weighted shifts with non‐constant tail such that hyponormality of a power or powers does not guarantee hyponormality of the original one. Our results have a partial connection to the following two long‐open problems in Operator Theory: (i) characterize the subnormal operators having a square root; (ii) classify all subnormal operators whose square roots are also subnormal. Our results partially depend on new formulas for the determinant of generalized Cauchy–Hankel matrices and on criteria for their positive semi‐definiteness.     

Tsirelson like operator spaces

We construct nontrivial examples of weak‐ operator spaces with the local operator space structure very close to . These examples are non‐homogeneous Hilbertian operator spaces, and their constructions are similar to that of 2‐convexified Tsirelson's space by W. B. Johnson.     

On null sequences for Banach operator ideals,trace duality and approximation properties

Let be a Banach operator ideal and X be a Banach space. We undertake the study of the vector space of ‐null sequences of Carl and Stephani on X , , from a unified point of view after we introduce a norm which makes it a Banach space. To give accurate results we consider local versions of the different types of accessibility of Banach operator ideals. We show that in the most common situations, when is right‐accessible for , behaves much alike . When this is the case we give a geometric tensor product representation of . On the other hand, we show an example where the representation fails. Also, via a trace duality formula, we characterize the dual space of . We apply our results to study some problems related with the ‐approximation property giving a trace condition which is used to solve the remaining case () of a problem posed by Kim (2015). Namely, we prove that if a dual space has the ‐approximation property then the space has the ‐approximation property.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

Duality for frames in Krein spaces

A J‐frame for a Krein space is in particular a frame for (in the Hilbert space sense). But it is also compatible with the indefinite inner‐product of , meaning that it determines a pair of maximal uniformly definite subspaces, an analogue to the maximal dual pair associated with an orthonormal basis in a Krein space. This work is devoted to study duality for J‐frames in Krein spaces. Also, tight and Parseval J‐frames are defined and characterized.     

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (proximally C ¹ sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “k‐dimensional regular part” which is a k‐dimensional “uniform” C ^{1, 1} manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.     

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

The dual space of B ‐valued martingale Orlicz–Hardy space with a concave function Φ, which is associated with the conditional p‐variation of B ‐valued martingale, is characterized. To obtain the results, a new type of Campanato spaces for B ‐valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p‐uniform smoothness and q‐uniform convexity of the underlying Banach space.     

A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

A Trudinger–Moser inequality in a weighted Sobolev space and applications

We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation where , f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.     

The derivative of the parallel volume difference

In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in 8 and 9 . Our main result is that the derivative of the difference between the parallel volume of the convex hull of a planar body and the parallel volume of the body itself tends to 0 for at order . We will use this result to examine Brownian paths and Boolean models.     

Some kind of Bishop–Phelps–Bollobás property

In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces X and Y : property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair satisfies property 2. Positive results of pairs of Banach spaces satisfying property 1 are given and concrete pairs of Banach spaces failing both properties are exhibited. A complete characterization of property 1 for the pairs is also provided.     

Moment representations of the exceptional X1‐Laguerre orthogonal polynomials

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so‐called) Type I X₁‐Laguerre polynomial sequence , and , where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. We observe a certain detachedness of the first two moments from the others.     

The mixed BVP for second order nonlinear ordinary differential equation at resonance

Efficient sufficient conditions are established for the solvability of the mixed problem where in the case where the homogeneous linear problem has nontrivial solutions.     

Mond's conjecture for maps between curves

A theorem by D. Mond shows that if is finite and has has degree one onto its image (Y , 0), then the ‐codimension is less than or equal to the image Milnor number , with equality if and only if (Y , 0) is weighted homogeneous. Here we generalize this result to the case of a map germ , where (X , 0) is a plane curve singularity.     

Fourier Multipliers and Littlewood‐Paley for modulation spaces

In this paper we have studied Fourier multipliers and Littlewood‐Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space into itself possesses an l₂‐valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical ‐spaces.