• Volume 124, Issue 1

February 2014,   pages  1-126

• Alexander Duals of Multipermutohedron Ideals

An Alexander dual of a multipermutohedron ideal has many combinatorial properties. The standard monomials of an Artinian quotient of such a dual correspond bijectively to some 𝜆-parking functions, and many interesting properties of these Artinian quotients are obtained by Postnikov and Shapiro (Trans. Am. Math. Soc. 356 (2004) 3109–3142). Using the multigraded Hilbert series of an Artinian quotient of an Alexander dual of multipermutohedron ideals, we obtained a simple proof of Steck determinant formula for enumeration of 𝜆-parking functions. A combinatorial formula for all the multigraded Betti numbers of an Alexander dual of multipermutohedron ideals are also obtained.

• Real Parabolic Vector Bundles over a Real Curve

We define real parabolic structures on real vector bundles over a real curve. Let $(X, 𝜎_X)$ be a real curve, and let $S\subset X$ be a non-empty finite subset of 𝑋 such that $𝜎_X(S) = S$. Let 𝑁 ≥ 2 be an integer. We construct an 𝑁-fold cyclic cover 𝑝: $Y→ X$ in the category of real curves, ramified precisely over each point of 𝑆, and with the property that for any element 𝑔 of the Galois group 𝛤, and any $y\in Y$, one has $𝜎_Y(gy) = g^{-1}𝜎_Y(y)$. We established an equivalence between the category of real parabolic vector bundles on $(X,𝜎_X)$ with real parabolic structure over 𝑆, all of whose weights are integral multiples of 1/𝑁, and the category of real 𝛤-equivariant vector bundles on $(Y, 𝜎_Y)$.

• A Note on Conjugacy Classes of Finite Groups

Let 𝐺 be a finite group and let $x^G$ denote the conjugacy class of an element 𝑥 of 𝐺. We classify all finite groups 𝐺 in the following three cases:

• Each non-trivial conjugacy class of 𝐺 together with the identity element 1 is a subgroup of 𝐺,
• union of any two distinct non-trivial conjugacy classes of 𝐺 together with 1 is a subgroup of 𝐺, and
• union of any three distinct non-trivial conjugacy classes of 𝐺 together with 1 is a subgroup of 𝐺.
• On Approximation of Lie Groups by Discrete Subgroups

A locally compact group 𝐺 is said to be approximated by discrete sub-groups (in the sense of Tôyama) if there is a sequence of discrete subgroups of 𝐺 that converges to 𝐺 in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if 𝐺 has a rational structure. On the other hand, if 𝛤 is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group 𝐺 then 𝐺 is approximated by discrete subgroups $𝛤_n$ containing 𝛤. The proof of the above result is by induction on the dimension of 𝐺, and gives an algorithm for inductively determining $𝛤_n$. The purpose of this paper is to give another proof in which we present an explicit formula for the sequence $(𝛤_n)_{n≥ 0}$ in terms of 𝛤. Several applications are given.

• Maximal Saddle Solution of a Nonlinear Elliptic Equation Involving the 𝑝-Laplacian

A saddle solution is called maximal saddle solution if its absolute value is not smaller than those absolute values of any solutions that vanish on the Simons cone $\mathcal{C} = \{s = t\}$ and have the same sign as 𝑠 - 𝑡. We prove the existence of a maximal saddle solution of the nonlinear elliptic equation involving the 𝑝-Laplacian, by using the method of monotone iteration,

$$-𝛥_{p^u}=f(u) \quad \text{in} \quad R^{2m},$$

where $2m≥ p > 2$.

• Positive Solutions for System of 2𝑛-th Order Sturm-Liouville Boundary Value Problems on time Scales

Intervals of the parameters 𝜆 and 𝜇 are determined for which there exist positive solutions to the system of dynamic equations

\begin{align*}(-1)^n u^{𝛥^{2n}}(t)+𝜆 p(t) f(𝜐(𝜎(t)))=0, & t\in[a, b],\\ (-1)^n𝜐^{𝛥^{2n}} (t) + 𝜇 q(t)g (u(𝜎(t))) = 0, & t\in [a, b],\end{align*}

satisfying the Sturm–Liouville boundary conditions

\begin{align*}& 𝛼_{i+1}u^{𝛥^{2i}}(a)-𝛽_{i+1}u^{𝛥^{2i+1}}(a)=0, 𝛾_{i+1}u^{𝛥^{2i}}(𝜎(b))+𝛿_{i+1}u^{𝛥^{2i+1}}(𝜎(b))=0,\\ & 𝛼_{i+1}𝜐^{𝛥^{2i}}(a)-𝛽_{i+1}𝜐^{𝛥^{2i+1}}(a)=0,𝛾_{i+1}𝜐^{𝛥^{2i}}(𝜎(b))+𝛿_{i+1}𝜐^{𝛥^{2i+1}}(𝜎(b))=0,\end{align*}

for $0≤ i≤ n-1$. To this end we apply a Guo–Krasnosel’skii fixed point theorem.

• Boundedness of Composition of Operators Associated with Different Homogeneities on Weighted Besov and Triebel-Lizorkin Spaces

In this paper, we introduce weighted Besov spaces and weighted Triebel–Lizorkin spaces associated with different homogeneities and prove that the composition of two Calderón–Zygmund operators is bounded on these spaces. This extends a recent result in Han et al, Revista Mat. Iber.

• Vector-Valued almost Convergence and Classical Properties in Normed Spaces

In this paper we study the almost convergence and the almost summability in normed spaces. Among other things, spaces of sequences defined by the almost convergence and the almost summability are proved to be complete if the basis normed space is so. Finally, some classical properties such as completeness, reflexivity, Schur property, Grothendieck property, and the property of containing a copy of 𝑐0 are characterized in terms of the almost convergence.

• General 𝐿𝑝-Harmonic Blaschke Bodies

Lutwak introduced the harmonic Blaschke combination and the harmonic Blaschke body of a star body. Further, Feng and Wang introduced the concept of the 𝐿𝑝-harmonic Blaschke body of a star body. In this paper, we define the notion of general 𝐿𝑝-harmonic Blaschke bodies and establish some of its properties. In particular, we obtain the extreme values concerning the volume and the 𝐿𝑝-dual geominimal surface area of this new notion.

• There is no Monad Based on Hartman-Mycielski Functor

We show that there is no monad based on the normal functor 𝐻 introduced earlier by Radul which is a certain functorial compactification of the Hartman–Mycielski construction 𝐻𝑀.

• # Proceedings – Mathematical Sciences

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