Cyclotomic graphs and perfect codes

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[{\zeta }_m]/A$, with connection sets $\{\pm ({\zeta }_m^i+A):0\le i\le m?1\}$ and $\{\pm ({\zeta }_m^i+A):0\le i\le ?(m)?1\}$, respectively, where ${\zeta }_m$ ($m\ge 2$) is an mth primitive root of unity, A a nonzero ideal of $\mathbb{Z}[{\zeta }_m]$, and ? Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by $G_m(A)$ and $G_m^{?}(A)$, respectively. We give a necessary and sufficient condition for $D/A$ to be a perfect t-code in $G_m^{?}(A)$ and a necessary condition for $D/A$ to be such a code in $G_m(A)$, where $t\ge 1$ is an integer and D an ideal of $\mathbb{Z}[{\zeta }_m]$ containing A. In the case when $m=3,4$, $G_m((\alpha ))$ is known as an Eisenstein–Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for $(\beta )/(\alpha )$ to be a perfect t-code in $G_m((\alpha ))$, where $0\ne \alpha ,\beta \in \mathbb{Z}[{\zeta }_m]$ with β dividing α. In the literature such conditions are known to be sufficient when $m=4$ and $m=3$ under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.     

Vertex-primitive s-arc-transitive digraphs of linear groups

We study G-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups G with socle ${\mathrm{PSL}}_n(q)$. We prove that $s?2$ for such digraphs, which provides the first step in determining an upper bound on s for all the vertex-primitive s-arc-transitive digraphs.     

On eigenvectors,approximations and the Feynman propagator

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $〈x_0|K^t|x_1〉$ of the eigenvector $|x_0〉$ of the position operator with eigenvalue $x_0$ and $K^t(|x_1〉)$, where $K^t$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.     

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems $\dot{x}=X(x,\epsilon )$ which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by $T^{\mathrm{\Sigma }}(q,\epsilon )$ the time needed by a perturbed orbit that starts from $q\in \mathrm{\Sigma }$ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of $\dot{x}=X(x,0)$ is a continuable critical orbit in a family of the form $\dot{x}=X(x,\epsilon )$ if, for any $q\in \mathrm{\Gamma }$ and any Σ that passes through q, there exists $q^{\epsilon }\in \mathrm{\Sigma }$ a critical point of $T^{\mathrm{\Sigma }}(?,\epsilon )$ such that $q^{\epsilon }\to q$ as $\epsilon \to 0$. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of $\dot{x}=X(x,0)$ is a continuable critical orbit in any family of the form $\dot{x}=X(x,\epsilon )$. We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center $\dot{x}=X(x,0)$.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

Homomorphisms of planar (m,n)-colored-mixed graphs to planar targets

An $(m,n)$-colored-mixed graph $G=(V,A_1,A_2,\cdots ,A_m,E_1,E_2,\cdots ,E_n)$ is a graph having m colors of arcs and n colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph G to another $(m,n)$-colored-mixed graph H is a morphism $\phi :V(G)\to V(H)$ such that each edge (resp. arc) of G is mapped to an edge (resp. arc) of H of the same color (and orientation). An $(m,n)$-colored-mixed graph T is said to be $P_g^{(m,n)}$-universal if every graph in $P_g^{(m,n)}$ (the planar $(m,n)$-colored-mixed graphs with girth at least g) admits a homomorphism to T.We show that planar $P_g^{(m,n)}$-universal graphs do not exist for $2m+n\ge 3$ (and any value of g) and find a minimal (in the number vertices) planar $P_g^{(m,n)}$-universal graphs in the other cases.     

Arc-transitive digraphs with quasiprimitive local actions

Let Γ be a finite G-vertex-transitive digraph. The in-local action of $(\mathrm{\Gamma },G)$ is the permutation group $L_{?}$ induced by a vertex-stabiliser on the set of in-neighbours of the corresponding vertex. The out-local action$L_{+}$ is defined analogously. Note that $L_{?}$ and $L_{+}$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_{?},L_{+})$ are possible. We prove some general results, but pay special attention to the case when $L_{?}$ and $L_{+}$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.     

Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form

${?}_{t}u(x,t)+\text{P.V.}\underset{{\mathbb{R}}^n}{\int }K(x,y,t)|u(x,t)?u(y,t){|}^{p?2}(u(x,t)?u(y,t))dy=0,$

$(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$, where P.V. means in the principle value sense, $p\in (1,\infty )$ and the kernel obeys $K(x,y,t)\approx |x?y{|}^{n+ps}$ for some $s\in (0,1)$, uniformly in $(x,y,t)\in {\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{R}$.     

Path decompositions of triangle-free graphs

Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then $p(G)\le ?n/2?$. If G is allowed to be disconnected, then the upper bound $?\frac{3}{4}n?$ for $p(G)$ was obtained by Donald [7], which was improved to $?\frac{2}{3}n?$ independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, $?\frac{2}{3}n?$ is reached and so this bound is tight. If triangles are forbidden in G, then $p(G)\le ?\frac{g+1}{2g}n?$ can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that $p(G)\le ?3n/5?$, which improves the above result with $g=4$.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).