$${\cal K}$$-Convexity1 in $$\Re^{n}$$

We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of -convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of -convex functions. We conclude the paper with some suggestions for future research. Support from Columbia University and University of Texas at Dallas is gratefully acknowledged. Helpful comments from Qi Feng are appreciated.     

Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Scheme-Theoretically

We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\).     

Continued fractions arising from $${\mathcal F}_{1,3}$$

We calculate the Jacobi Eisenstein series of weight \(k \ge 3\) for a certain representation of the Jacobi group, and evaluate these at \(z = 0\) to give coefficient formulas for a family of modular forms \(Q_{k,m,\beta }\) of weight \(k \ge 5/2\) for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for \(Q_{k,m,\beta }\) and the Eisenstein series of Bruinier and Kuss p-adically to get an efficient algorithm. The main application is in constructing automorphic products.     

Random self-affine multifractal Sierpinski sponges in $${\mathbb{R}^d}$$

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({\mathbb{R}^{d}}\).     

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Note on $$ \mathcal{I}R $$-closed and $$ A_{\mathcal{I}R} $$-sets

We characterize and discuss the properties of $ \mathcal{I}R $ \mathcal{I}R -closed sets and $ A_{\mathcal{I}R} $ A_{\mathcal{I}R} -sets. Also, we give characterizations of weakly $ \mathcal{I} $ \mathcal{I} -locally closed sets and $ \mathcal{I} $ \mathcal{I} -submaximal spaces. A characterization of codense ideals in terms of $ \mathcal{I}R $ \mathcal{I}R -closed sets is also given.     

The rational cohomology of $${\overline{\mathcal{M}_4}}$$

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space of stable complex curves of genus 4, with its Hodge structure.     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

Large $${\varvec{p}}^{\prime }$$-orbits for $${\varvec{p}}$$p-nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.     

On the Rao modules of minimal curves in $${\mathbb{P}^N}$$

We study the Hartshorne-Rao modules M _C of minimal curves C in \mathbbP^N{\mathbb{P}^N} , with N ≥ 4, lying in the same liaison class of curves on a smooth rational scroll surface. We get a free minimal resolution of M _C for some of such curves and an upper bound for Betti numbers of M _C , for any C.     

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

The $${\mathbb {Q}}$$ Q -Korselt set of $$\mathrm {pq}$$ pq

Periodica Mathematica Hungarica - Let N be a positive integer, $${\mathbb {A}}$$ be a nonempty subset of $${\mathbb {Q}}$$ and $$\alpha =\dfrac{\alpha _{1}}{\alpha _{2}}\in {\mathbb {A}}{\setminus...     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

NONVANISHING OF CONFORMAL BLOCKS DIVISORS ON $$ {\overline{\mathrm{M}}}_{0,n} $$

We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on \( {\overline{\mathrm{M}}}_{0,n} \) is nonzero, solving the problem completely for \( \mathfrak{s}{\mathfrak{l}}_2 \). We give necessary nonvanishing conditions in type A, which are sufficient when theta and critical levels coincide. We also show divisors are subject to additive identities, reflecting a decomposition of the weights and level.     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

Approximation of the Classes C β ψ $$ {C}_{\beta}^{\psi } $$ H�� By Biharmonic Poisson Integrals

Ukrainian Mathematical Journal - We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class...     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.