Graph Realizations Constrained by Connected Local Dimensions and Connected Local Bases

For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.

Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension

Let p be an odd prime and let r be the smallest generator of the multiplicative group Zp∗. We show that there exists a correlation of size Θ(r2) that self-tests a maximally entangled state of local dimension p−1. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. (2019)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set {2,3,5} (D.R. Heath-Brown The Quarterly Journal of Mathematics (1986) and M. Murty The Mathematical Intelligencer (1988)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.

Politics of Memory in Cities Divided by State Borders: Theoretical and Methodological Framework of Research

In the article the author attempts to outline theoretical and methodological framework used in analyzing the phenomenon of specific history politics in cities divided by state borders – places where sometimes radically different history narrations meet or even clash. The article is composed of three parts. The first one specifies the concept of history politics in nationwide dimension (“history politics”). The second part analyzes history politics in local dimension (“local politics of memory”). With reference to these two concepts – levels of history politics and politics of memory – the author indicates: goals, subjects, methods and tools of history politics/politics of memory. The final section of the paper aims at capturing the specificity of history politics in cities divided by state borders and indicating models of relations between different kinds of politics of memory in cities divided by state borders.

1-form symmetries of 4d N=2 class S theories

We determine the 1-form symmetry group for any 4d4d\mathcal{N}=2𝒩=2 class S theory constructed by compactifying a 6d6d\mathcal{N}=(2,0)𝒩=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d4d, modulo screening and flavor charges. Complete specification of a 4d4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.

Through the Eyes of a Mapmaker: Maritime Shrines on Cyprus during the Late Middle Ages

Cyprus acquired special importance, especially from the thirteenth century onwards, on the Eastern Mediterranean’s pilgrimage network. Described by contemporary pilgrims as “Terra christianorum ultima”, the island was considered to be the last Christian land in the south-eastern Mediterranean on the pilgrims’ itinerary on their journey to the Holy Land. This study is concentrated on two maps of Cyprus dated to the fourteenth century and preserved in Milan: Biblioteca Ambrosiana, A95 sup. and Venice: Biblioteca Nazionale Marciana, gr. XI.21. It aims to explore the physical and spiritual mobility and interconnectivity in Cyprus during the late Middle Ages and to consider how these contribute to the development of pilgrimage sites directly related with maritime routes, seamen and travellers. These unique nautical maps captured the sea voyage which had Cyprus as a stopover, bringing to light new insights into fourteenth century Cyprus. The maritime shrines discussed in this article, which are usually “mixed” sacred sites, are directly related with sailors’ needs. They integrate into a wide network of communication, removing them partially from their local dimension.

Bacterial Foraging Algorithm With Potential Field Guidance Mechanism

To address the problems of weak quorum sensing ability and slow convergence speed in bacterial foraging algorithm, a bacterial foraging algorithm with potential field guidance mechanism is proposed. The algorithm combines the sampling guidance mechanism in the artificial potential field algorithm to provide the optimization direction for each bacterium; The original swimming operation of bacterial foraging algorithm is used to realize the local optimization strategy, and the local dimension update is added after swimming, so that the search range of bacteria in chemotaxis operation is wider; In the elimination and dispersal operation of bacterial foraging algorithm, double Gaussian function is introduced to re initialize the location of bacteria, so as to better avoid the algorithm falling into local extremum and improve the optimization ability of the algorithm. The experimental results show that the improved bacterial foraging algorithm has better optimization ability than the basic bacterial foraging algorithm.

Local dimension of trivial extension and amalgamation of rings

Local dimension is an ordinal valued invariant that is in some sense a measure of how far a ring is from being local and denoted [Formula: see text]. The purpose of this paper is to study the local dimension of ring extensions such as homomorphic image, trivial ring extension and the amalgamation of rings.

Self-testing quantum systems of arbitrary local dimension with minimal number of measurements

Bell nonlocality as a resource for device-independent certification schemes has been studied extensively in recent years. The strongest form of device-independent certification is referred to as self-testing, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single d-outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, $${{{\mathrm{log}}}\,}_{2}d$$ log 2 d perfect random bits, while it requires only one random bit to encode the measurement choice.