MATHEMATICA SCANDINAVICA

Power domination in generalized Mycielskian of spiders

2024-08-06T14:04:56+02:00

The power domination problem in the graph was introduced to model the monitoring problem in electric networks. It is to find a set of vertices with minimum cardinality, called a power dominating set (PDS), that monitors the whole set $ V $ after applying the rules, domination and propagation. In this paper, we discuss the power domination problem in generalized Mycielskian of spiders. We characterize spiders with $\gamma _P(\mu _m(T))= 1$. If $\gamma _P(\mu _m(T))\neq 1 $ and $ m $ is even then $ \gamma _P(\mu _m(T))=\frac {m}{2} +1$. We also show that if $ m $ is odd and $ \gamma _P(\mu _m(T))\neq 1 $, then $ \lceil \frac {m+1}{2}\rceil \leq \gamma _P(\mu _m(T))\leq \lceil \frac {m}{2}\rceil +1$.

Cohen-Macaulay permutation graphs

2024-09-05T15:31:36+02:00

In this article, we characterize Cohen-Macaulay permutation graphs. In particular, we show that a permutation graph is Cohen-Macaulay if and only if it is well-covered and there exists a unique way of partitioning its vertex set into $r$ disjoint maximal cliques, where $r$ is the cardinality of a maximal independent set of the graph. We also provide some sufficient conditions for a comparability graph to be a uniquely partially orderable (UPO) graph.

Adjoints and canonical forms of tree amplituhedra

2024-09-24T15:07:34+02:00

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm {Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

$f$-vector inequalities for order and chain polytopes

2024-03-11T13:48:46+01:00

The order and chain polytopes are two $0/1$-polytopes constructed from a finite poset. In this paper, we study the $f$-vectors of these polytopes. We investigate how the order and chain polytopes behave under disjoint unions and ordinal sums of posets, and how the $f$-vectors of these polytopes are expressed in terms of $f$-vectors of smaller polytopes. Our focus is on comparing the $f$-vectors of the order and chain polytope built from the same poset. In our main theorem we prove that for a family of posets built inductively by taking disjoint unions and ordinal sums of posets, for any poset $\mathcal {P}$ in this family the $f$-vector of the order polytope of $\mathcal {P}$ is component-wise at most the $f$-vector of the chain polytope of $\mathcal {P}$.

The space of $r$-immersions of a union of discs in $\mathbb R^n$

2024-08-31T14:03:39+02:00

For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb {R}^n$ is defined to be the space of immersions of $M$ in $\mathbb {R}^n$ such that the preimage of every point in $\mathbb {R}^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb {R}^n$ and the $k^{\text {th}}$ power of the space of injective linear maps from $\mathbb {R}^m$ to $\mathbb {R}^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is “well-known”, but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions.

Unital positive Schur multipliers on $S_n^p$ with a completely isometric dilation

2024-06-08T13:21:56+02:00

Let $1<p\neq 2<\infty $ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible complete isometry acting on a non-commutative $L^p$-space. Then we investigate the infinite dimensional case.

Exact solvable family of discrete Schrödinger operators with long-range hoppings

2024-08-02T10:41:56+02:00

It is shown that one-dimensional discrete Schrödinger operators, with a uniform electric field and long-range hoppings, form an isospectral family with discrete and simple spectrum (and isospectral operators with eigenfunctions having different decay properties). Explicit relations between hopping decay rates and eigenvectors decay rates are obtained. Small perturbations of exponentially decay hopping operators preserve dynamical localization.

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property}

2024-08-01T11:39:46+02:00

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.

Exotic groupoid $C^*$-algebras associated to double groupoids

2024-09-16T13:21:50+02:00

We consider a class of partial action groupoids called double groupoids which are constructed from pairs of subgroups satisfying similar conditions to those of a matched pair of groups. If the double groupoid is étale, then we show that whenever the partially acting group admit exotic ideal completions in the sense of Brown and Guentner, the corresponding double groupoid also admit exotic $C^*$-completions.

Sheaves of measures and KMS-weights on topological graph algebras

2024-04-15T14:27:59+02:00

We show that the collection of regular Borel measures on a second-countable locally compact Hausdorff space has the structure of a sheaf. With this we give an alternate description of the pullback of a regular Borel measure along a local homeomorphism. We are able to use these tools to give a description of the KMS$_\beta $-weights for the gauge-action on the graph $C^*$-algebra of a second-countable topological graph in terms of sub-invariant measures on the vertex space of said topological graph.

Probabilistic aspects of Jacobi theta functions

2024-08-20T15:04:42+02:00

In this note we deduce well-known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

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2024-11-04T10:09:22+01:00

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