A note on finite groups with few values in a column of the character table

Many structural properties of a finite group G are encoded in the set of irreducible character degrees of G. This is the set of (distinct) values appearing in the "first" column of the character table of G. In the current article, we study groups whose character table has a "non-first" column satisfying one particular condition. Namely, we describe groups having a nonidentity element on which all nonlinear characters take the same value

On the regularity of a graph related to conjugacy classes of groups

AbstractGiven a finite group G, denote by Γ(G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ(G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ(G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ(G) is a 3-regular graph if and only if it is a complete graph with four vertices

Conjugacy classes of finite groups and graph regularity

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if $\Gamma(G)$ is a $k$-regular graph with $k\geq 1$, then $\Gamma(G)$ is a complete graph with $k+1$ vertices

Use of dermal-fat grafts in the post-oncological reconstructive surgery of atrophies in the zygomatic region: Clinical evaluations in the patients undergone to previous radiation therapy

Introduction: Grafting of autologous adipose tissue can be recommended in some cases of facial plastic surgery. Rhabdomyosarcoma is a type of cancer that can also affect the orbit. Enucleation of the eye can cause atrophy of the corresponding hemiface and decreased orbital growth.Case report: We report a case of a female patient with a medical history of surgical enucleation of the right eyeball, who had received rhabdomyosarcoma radiation therapy in her youth. The patient presented with a depression in the right zygomatic region. We took a dermal-fat flap from the abdominal region, which had been previously treated.Results: The surgical outcome, 48 hours, and much clearly 31 days after the surgery, revealed that the right zygomatic region had returned to its proper anatomical shape, although there were still signs of postoperative edema.Discussion: Very damaged tissues, like those exposed to radiation therapy, are generally not suitable for grafting of adipose tissue.Conclusions: In the described case, we achieved a technically and aesthetically satisfying result despite the patient's medical history involving several perplexities about the use of autologous dermal-fat tissues, because of prior radiation therapy exposure. The clinical case shows that even a region exposed to radiation therapy can be a valid receiving bed for dermal-fat grafting. © 2012 Inchingolo et al.; licensee BioMed Central Ltd

Equilibrium in a two-agent assignment problem

In this paper we address a particular generalisation of the Assignment Problem (AP) in a Multi-Agent setting, where distributed agents share common resources. We consider the problem of determining Pareto-optimal solutions that satisfy a fairness criterion (equilibrium). We show that the solution obtained is equivalent to a Kalai Smorodinsky solution of a suitably defined bargaining problem and characterise the computational complexity of finding such an equilibrium. Additionally, we propose an exact solution algorithm based on a branch-and-bound scheme that exploits bounds obtained by suitably rounding the solutions of the corresponding linear relaxation, and give the results of extensive computational experiments. Copyright © 2009, Inderscience Publishers

On vanishing class sizes in finite groups

© 2017 Elsevier Inc. Let G be a finite group. An element g of G is called a vanishing element if there exists an irreducible character χ of G such that χ(g)=0; in this case, we say that the conjugacy class of g is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

Two is better than one? Order aggregation in a meal delivery scheduling problem

We address a single-machine scheduling problem motivated by a last-mile-delivery setting for a food company. Customers place orders, each characterized by a delivery point (customer location) and an ideal delivery time. An order is considered on time if it is delivered to the customer within a time window given by the ideal delivery time , where is the same for all orders. A single courier (machine) is in charge of delivery to all customers. Orders are either delivered individually, or two orders can be aggregated in a single courier trip. All trips start and end at the restaurant, so no routing decisions are needed. The problem is to schedule courier trips so that the number of late orders is minimum. We show that the problem with order aggregation is -hard and propose a combinatorial branch and bound algorithm for its solution. The algorithm performance is assessed through a computational study on instances derived by a real-life application and on randomly generated instances. The behavior of the combinatorial algorithm is compared with that of the best ILP formulation known for the problem. Through another set of computational experiments, we also show that an appropriate choice of design parameters allows to apply the algorithm to a dynamic context, with orders arriving over time

Multi instanton tests of holography

Gauge theories living on stacks of D7-branes are holographically related to IIB gravitational backgrounds with a varying axion-dilaton field (F-theory). The axion-dilaton field is generated by D7, O7 and D-instanton sources and can be written in terms of the chiral correlators of the eight dimensional gauge theory living on the D7-branes. Using localization techniques, we prove that the same correlators determine the gauge coupling of the four-dimensional N=2 supersymmetric SU(2) gauge theories living on the elementary D3-brane which probes the F-theory geometries.Comment: 18 page

On solvable groups with one vanishing class size

Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79-83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free