Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Universal point sets for planar three-trees

For every $n\in \mathbb{N}$, we present a set $S_n$ of $O(n^{3/2}\log n)$ points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of $S_n$. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.     

Frames arising from irreducible solvable actions I

In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time–frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}$. Next, let π be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup $P=\exp ?\mathfrak{p}$ of G. Additionally, we assume that $G=P?M$, $M=\exp ?\mathfrak{m}$ is a closed subgroup of G, $d{\mu }_M$ is a fixed Haar measure on the solvable Lie group M and there exists a linear functional $\lambda \in {\mathfrak{p}}^{?}$ such that the representation $\pi ={\pi }_{\lambda }={\mathrm{ind}}_P^G\left({\chi }_{\lambda }\right)$ is realized as acting in $L^2(M,d{\mu }_M)$. Making no assumption on the integrability of ${\pi }_{\lambda }$, we describe explicitly a discrete subset Γ of G and a vector $\mathbf{f}\in L^2(M,d{\mu }_M)$ such that ${\pi }_{\lambda }\left(\mathrm{\Gamma }\right)\mathbf{f}$ is a tight frame for $L^2(M,d{\mu }_M)$. We also construct compactly supported smooth functions s and discrete subsets $\mathrm{\Gamma }?G$ such that ${\pi }_{\lambda }\left(\mathrm{\Gamma }\right)\mathbf{s}$ is a frame for $L^2(M,d{\mu }_M)$.     

Determinacy of refinements to the difference hierarchy of co-analytic sets

In this paper we develop a technique for proving determinacy of classes of the form ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+\mathrm{\Gamma }$ (a refinement of the difference hierarchy on ${\mathrm{\Pi }}_1^1$ lying between ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1$ and $({\omega }^2+1)\text{-}{\mathrm{\Pi }}_1^1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{\mathrm{\Sigma }}_{\alpha }^0$ for all computable α and of ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{{\mathrm{\Delta }}^1}_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.     

Triangular C1-bialgebra defined as the direct sum of matrix algebras

Let $M_{*}(\mathbf{C})$ denote the ${\mathrm{C}}^1$-algebra defined as the direct sum of all matrix algebras $\{M_n(\mathbf{C}):n?1\}$. It is known that $M_{*}(\mathbf{C})$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$. From a certain set of transformations of integers, we construct a universal R-matrix R of the ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi })$ such that the quasi-cocommutative ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi }\text{,}R)$ is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.