Topological spaces compact with respect to a set of filters

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T ₁ topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.     

An additivity formula for the strict global dimension of C(Ω)

Let A be a unital strict Banach algebra, and let K ₊ be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K ₊), the algebra of continuous functions on K ₊. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .     

Filippov Lemma for certain second order differential inclusions

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y??(0) = 0, where the matrix A ?? ? ^d×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y ₀ ?? W ^2,1 be an arbitrary function fulfilling the above initial conditions and such that where p ₀ ?? L ¹[0, 1]. Then there exists a solution y ?? W ^2,1 to the above differential inclusions such that a.e. in [0, 1], .     

Compact differences of composition operators on weighted Dirichlet spaces

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S ², the space of analytic functions whose first derivative is in H ², and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.     

On the asymptotic form of convex hulls of Gaussian random fields

We consider a centered Gaussian random field X = {X _t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? ^m or ? ^m . It is supposed that the distribution of X _t is independent of t. We consider the asymptotic behavior of closed convex hulls W _n = conv{X _t : t ∈ T _n}, where (T _n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b _n ) _n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W _n ), where f is some function, is also studied.     

The one-point Lindelöfication of an uncountable discrete space can be surlindelöf

We prove that the one-point Lindelöfication of a discrete space of cardinality ω ₁ is homeomorphic to a subspace of C _p (X) for some hereditarily Lindelöf space X if the axiom holds.     

Relation modules of infinite groups, II

Let F _n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ? F _m ? G and S ? F _m ? G are presentations of G and let $\bar R$ and $\bar S$ denote the associated relation modules of G. It is well known that $\bar R \oplus (\mathbb{Z}G)^{d(G)} \cong \bar S \oplus (\mathbb{Z}G)^{d(G)}$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations as above such that . Our approach depends on the existence of nonfree stably free modules over certain commutative rings and, in particular, on the existence of certain Hurwitz-Radon systems of matrices with integer entries discovered by Geramita and Pullman. This approach was motivated by results of Adams concerning the number of orthonormal (continuous) vector fields on spheres.     

Syzygies of torsion bundles and the geometry of the level ℓ modular variety over \overline{\mathcal{M}}_{g}

We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ? in its Jacobian. These statements can be viewed an analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space $\mathcal{R}_{g,\ell}$ of twisted level ? curves of genus g and use this to derive results about the birational geometry of $\mathcal{R}_{g, \ell}$ . For instance, we prove that $\mathcal{R}_{g,3}$ is a variety of general type when g>11 and the Kodaira dimension of $\mathcal{R}_{11,3}$ is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.     

The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone

Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.     

A few remarks on the Generalized Vanishing Conjecture

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lambda^m f^m = 0] \Longrightarrow \forall_{m \gg 0}[\Lambda^m (g f^m) = 0]$$ for a fixed differential operator ${\Lambda \in k[\partial]}$ follows from a special case of it, namely that the additional factor g is a power of the radical polynomial f. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of Λ), we may assume that Λ is a linear combination of powers of distinct partial derivatives. At last, we show that the Generalized Vanishing Conjecture holds for products of linear forms in ?, in particular homogeneous differential operators ${\Lambda \in k[\partial_1,\partial_2]}$ .     

On ideal equal convergence

We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów R., Szuca P., Three kinds of convergence and the associated I-Baire classes, J. Math. Anal. Appl., 2012, 391(1), 1–9]. We also solve a few problems posed in the paper by Das, Dutta and Pal.     

Anisotropic inverse harmonic mean curvature flow

We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients Q _H (A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements Q _H (A) ^H and also the classical ring of quotients for A. We introduce a full subcategory of such that the algebras in are integral over its subalgebras of invariants and construct a functor ?? , which is left adjoined to the inclusion ?? .     

Renormalization in the Hénon family,II: the heteroclinic web

We study highly dissipative Hénon maps

$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

     

Weak partition properties on trees

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ ^<ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals.     

The Brill–Noether curve and Prym-Tyurin varieties

We prove that the Jacobian of a general curve C of genus $g=2a+1$ , with $a\ge 2$ , can be realized as a Prym-Tyurin variety for the Brill–Noether curve $W^{1}_{a+2}(C)$ . As consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series $g^{a-1}_{3a-2}$ .     

Proper holomorphic mappings, Bell’s formula, and the Lu Qi-Keng problem on the tetrablock

We consider proper holomorphic maps ${\pi : D\rightarrow G}$ , where D and G are domains in ${\mathbb{C}^{n}}$ . Let ${\alpha\in \mathcal{C}(G,\mathbb{R}_{ > 0})}$ . We show that every π induces some subspace H of ${\mathbb{A}^{2}_{\alpha\circ\pi}(D)}$ such that ${\mathbb{A}^{2}_{\alpha}(G)}$ is isometrically isomorphic to H via some unitary operator Γ. Using this isomorphism we construct the orthogonal projection onto H, and we derive Bell’s transformation formula for the weighted Bergman kernel function under proper holomorphic mappings. As a consequence of the formula, we get that the tetrablock is not a Lu Qi-Keng domain.     

On minimal non- -groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.