Strongly Regular Graphs

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.

CLASSIFICATION OF THE OLBIAN COINS OF THE SECOND HALF OF THE 4TH CENTURY BC.

Since the beginning of the XXI century, the amount of available for the researchers numismatic material has increased significantly, so introducing a new types of coins into the scientific circulation has become an especially relevant in modern numismatics, even when the archaeological context of most of these finds is almost lost. The study of ancient numismatics of Olbia is rapidly gaining in modern Ukraine. At the beginning of the XX century, ancient numismatics already had some significant achievements, but the accumulated material required urgent cataloging and systematization. During last 10 years since the publication of the most important and thorough catalog of ancient coins by Vladlen Opanasovich Anokhin, as well as the results of cataloging Olbia coins by other researchers - Valery Nechitaylo and Grigory Makandarov, numismatics has been enriched by new previously unknown coin types. The aim of the study. The main purpose of the article is to introduce into the scientific circulation new varieties of Olbia coins and to compile the most complete classification of Olbia coins of the IV century BC. Research methodology. In the process of scientific elaboration of the topic general scientific methods were used: analytical, chronological, and topographical, as well as source methods: critical, metrological and iconographic. A systematic approach to the processing of modern finds from private collections and access to the collections of foreign museums was the impetus for writing an expanded classification work. The scientific novelty. The value of the processed materials is that they not only complement this group of coins, but also refine previously published types in unsatisfactory condition, where incorrect reading of the names and trinkets has led to inaccuracies. The Conclusions. The so-called «obol series» covers the period of the Olbia minting around 350-330 BC. The monetary system consists of four denominations: obol (on the coin field depicts Demeter and the eagle on the dolphin), dikhalk (on the coin field depicts Demeter and the eagle on the dolphin), hulk (on the coin field depicts Demeter and the ear, dolphin) and hemihalk (depicts Demeter and dolphin). The die analysis allowed to divide the coins of Olbia of the IV century BC senior denomination for two stylistic groups. According to the results of our own research, we were able to determine the following number of varieties of each of these denominations: obols – 24 types, dikhalks - 6 types, hulks - 6 types, hemihalks - 2 types. We see the prospect of further research in the introduction into scientific circulation of new previously undiscovered varieties of Olbian coins from little-studied sources - materials of museum collections in Ukraine and abroad, among numismatic rarities sold at numismatic auctions and private collections.

Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $$\pm 1$$ ± 1 . The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

Yamabe and gradient Yamabe solitons on real hypersurfaces in the complex quadric

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric [Formula: see text]. In the following, as an application, we show a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex quadric [Formula: see text].

A certain η-parallelism on real hypersurfaces in a nonflat complex space form

In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the η-parallelism of the tensor field h(= (1/2)𝓛 ξ ϕ). In addition we investigate real hypersurfaces whose tensor h is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor h is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting η-parallelism with respect to the tensor h.

Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

The complete classification of classical r-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincar ́e groups such that their Lorentz sector is a quantum sub-group is presented. It is found that there exists three classes of such r-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

Using deep learning method to identify left ventricular hypertrophy on echocardiography

Background Left ventricular hypertrophy (LVH) is an independent prognostic factor for cardiovascular events and it can be detected by echocardiography in the early stage. In this study, we aim to develop a semi-automatic diagnostic network based on deep learning algorithms to detect LVH. Methods We retrospectively collected 1610 transthoracic echocardiograms, included 724 patients [189 hypertensive heart disease (HHD), 218 hypertrophic cardiomyopathy (HCM), and 58 cardiac amyloidosis (CA), along with 259 controls]. The diagnosis of LVH was defined by two experienced clinicians. For the deep learning architecture, we introduced ResNet and U-net++ to complete classification and segmentation tasks respectively. The models were trained and validated independently. Then, we connected the best-performing models to form the final framework and tested its capabilities. Results In terms of individual networks, the view classification model produced AUC = 1.0. The AUC of the LVH detection model was 0.98 (95% CI 0.94–0.99), with corresponding sensitivity and specificity of 94.0% (95% CI 85.3–98.7%) and 91.6% (95% CI 84.6–96.1%) respectively. For etiology identification, the independent model yielded good results with AUC = 0.90 (95% CI 0.82–0.95) for HCM, AUC = 0.94 (95% CI 0.88–0.98) for CA, and AUC = 0.88 (95% CI 0.80–0.93) for HHD. Finally, our final integrated framework automatically classified four conditions (Normal, HCM, CA, and HHD), which achieved an average of AUC 0.91, with an average sensitivity and specificity of 83.7% and 90.0%. Conclusion Deep learning architecture has the ability to detect LVH and even distinguish the latent etiology of LVH.

Holographic symmetry algebras for gauge theory and gravity

All 4D gauge and gravitational theories in asymptotically flat spacetimes contain an infinite number of non-trivial symmetries. They can be succinctly characterized by generalized 2D currents acting on the celestial sphere. A complete classification of these symmetries and their algebras is an open problem. Here we construct two towers of such 2D currents from positive-helicity photons, gluons, or gravitons with integer conformal weights. These generate the symmetries associated to an infinite tower of conformally soft theorems. The current algebra commutators are explicitly derived from the poles in the OPE coefficients, and found to comprise a rich closed subalgebra of the complete symmetry algebra.