Permanents,Doty coalgebras and dominant dimension of Schur algebras

A (hidden) multiplication on _AZ(n,r)$A_{\mathbb{Z}}(n,r)$, the Z$\mathbb{Z}$-dual of the integral Schur algebra _SZ(n,r)$S_{\mathbb{Z}}(n,r)$ is explicitly constructed, possibly without a unit. The image of the multiplication map is shown to be spanned by bipermanents. Let k   be any field of characteristic p>0$p>0$. The image of the induced multiplication on _Ak(n,r)=_AZ(n,r)_⊗Zk$A_k(n,r)=A_{\mathbb{Z}}(n,r){\otimes }_{\mathbb{Z}}k$ turns out to coincide with the Doty coalgebra _Dn,r,p$D_{n,r,p}$ of truncated symmetric powers. Combined with a new straightening formula for bipermanents, it is proved that such a multiplication induces an isomorphism _Ak(n,r)_⊗Sk(n,r)Ak(n,r)≅_Ak(n,r)$A_k(n,r){\otimes }_{S_k(n,r)}{A}_k(n,r)\cong A_k(n,r)$ as _Sk(n,r)$S_k(n,r)$-bimodules if and only if r≤n(p−1)$r\le n(p-1)$, if and only if _Dn,r,p=_Ak(n,r)$D_{n,r,p}=A_k(n,r)$. As a result, _Sk(n,r)$S_k(n,r)$ is a gendo-symmetric algebra, and its dominant dimension is at least two and admits a combinatorial characterization as long as r≤n(p−1)$r\le n(p-1)$.     

Generalized cover ideals and the persistence property

Let I   be a square-free monomial ideal in R=k[_x1,…,_xn]$R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes Ass(I^s)$\mathrm{Ass}(I^s)$ for all integers s?1$s?1$. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G   is a tree, we explicitly determine Ass(I^s)$\mathrm{Ass}(I^s)$ for all s?1$s?1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.     

Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and _FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))$F_S(t,x,y)=\det (tI+x\Re (S)+y\Im (S))$ be a ternary form associated with S  . We investigate the number of singular points of the curve _FS(t,x,y)=0$F_S(t,x,y)=0$, and show that the number of singular points of _FS(t,x,y)=0$F_S(t,x,y)=0$ associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2$n(n-3)/2$. Furthermore, we verify the upper bound n(n−3)/2$n(n-3)/2$ is sharp for 4?n?7$4?n?7$.     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

S-Noetherian properties on amalgamated algebras along an ideal

Let A and B   be commutative rings with identity, f:A→B$f:A\to B$ a ring homomorphism and J an ideal of B  . Then the subring A?^fJ:={(a,f(a)+j)|a∈A and j∈J}$A{?}^{f}J:=\{(a,f(a)+j)|a\in A\text{and}j\in J\}$ of A×B$A\times B$ is called the amalgamation of A with B along with J with respect to f. In this paper, we investigate a general concept of the Noetherian property, called the S  -Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A?^fJ$A{?}^{f}J$ for a multiplicative subset S   of A?^fJ$A{?}^{f}J$. As particular cases of the amalgamation, we also devote to study the transfers of the S  -Noetherian property to the constructions D+(_X1,…,_Xn)E[_X1,…,_Xn]$D+(X_1,\dots ,X_n)E[X_1,\dots ,X_n]$ and D+(_X1,…,_Xn)E?_X1,…,_Xn?$D+(X_1,\dots ,X_n)E?X_1,\dots ,X_n?$ and Nagata?s idealization.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

The inverse inertia problem for the complements of partial k-trees

Let F$\mathbb{F}$ be an infinite field with characteristic not equal to two. For a graph G=(V,E)$G=(V,E)$ with V={1,…,n}$V=\{1,\dots ,n\}$, let S(G;F)$S(G;\mathbb{F})$ be the set of all symmetric n×n$n\times n$ matrices A=[_ai,j]$A=[a_{i,j}]$ over F$\mathbb{F}$ with _ai,j≠0$a_{i,j}\ne 0$, i≠j$i\ne j$ if and only if ij∈E$ij\in E$. We show that if G is the complement of a partial k  -tree and m?k+2$m?k+2$, then for all nonsingular symmetric m×m$m\times m$ matrices K   over F$\mathbb{F}$, there exists an m×n$m\times n$ matrix U   such that U^TKU∈S(G;F)$U^{T}KU\in S(G;\mathbb{F})$. As a corollary we obtain that, if k+2?m?n$k+2?m?n$ and G is the complement of a partial k-tree, then for any two nonnegative integers p and q   with p+q=m$p+q=m$, there exists a matrix in S(G;R)$S(G;\mathbb{R})$ with p positive and q negative eigenvalues.     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}$\mathrm{Int}(A)=\{f\in B[x]|f(A)\subseteq A\}$. Restricting the coefficients to elements of k  , we obtain the commutative ring _Intk(A)={f∈k[x]|f(A)⊆A}${\mathrm{Int}}_k(A)=\{f\in k[x]|f(A)\subseteq A\}$; this makes Int(A)$\mathrm{Int}(A)$ a left _Intk(A)${\mathrm{Int}}_k(A)$-module. Previous researchers have noted instances when a D-module basis for A   is also an _Intk(A)${\mathrm{Int}}_k(A)$-basis for Int(A)$\mathrm{Int}(A)$. We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)$\mathrm{Int}(A)$, its localizations at primes of D, and finite residue rings of A.