Strongly Starlike Functions and Related Classes

We consider univalent functions, analytic in the unit disc $ |z|<1$in the complex plane ${\mathbb{C}}$ which map $ |z|<1$ onto a domainwith some nice property. The purpose of this paper is to find somenew conditions for strong starlikeness and some related results.

Any-Precision Deep Neural Networks

We present Any-Precision Deep Neural Networks (Any- Precision DNNs), which are trained with a new method that empowers learned DNNs to be flexible in any numerical precision during inference. The same model in runtime can be flexibly and directly set to different bit-width, by trun- cating the least significant bits, to support dynamic speed and accuracy trade-off. When all layers are set to low- bits, we show that the model achieved accuracy compara- ble to dedicated models trained at the same precision. This nice property facilitates flexible deployment of deep learn- ing models in real-world applications, where in practice trade-offs between model accuracy and runtime efficiency are often sought. Previous literature presents solutions to train models at each individual fixed efficiency/accuracy trade-off point. But how to produce a model flexible in runtime precision is largely unexplored. When the demand of efficiency/accuracy trade-off varies from time to time or even dynamically changes in runtime, it is infeasible to re-train models accordingly, and the storage budget may forbid keeping multiple models. Our proposed framework achieves this flexibility without performance degradation. More importantly, we demonstrate that this achievement is agnostic to model architectures. We experimentally validated our method with different deep network backbones (AlexNet-small, Resnet-20, Resnet-50) on different datasets (SVHN, Cifar-10, ImageNet) and observed consistent results.

A vortex ring on a sphere: the case of total vorticity equal to zero

The stability of a ring of vortices has attracted the interest of researchers for over a century. Recent beautiful observations of polygonal configurations of vortices present in the atmospheres of Jupiter and Saturn, and of polygonal jets in the Earth's atmosphere, have revived the interest in the subject. In the observed cases, the vortex ring is in the presence of a central vortex. We present analytical and numerical results about the linear, spectral and Lyapunov stability of a ring in the presence of polar vortices. Motivated by both atmospheric observations we considered the special case of total vorticity equal to zero. Such a case has also the very nice property of being universal , i.e. not depending on a choice of gauge. We considered the two cases of fixed and non-fixed polar vortices. A ring in the northern (respectively, southern) hemisphere is stabilized by the presence of a northern (respectively, southern) polar vortex of suitable strength, in agreement with what is observed numerically and atmospherically. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Coloring Chains for Compression with Uncertain Priors

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$.

EFFICIENT DATA STRUCTURE FOR REPRESENTING AND SIMPLIFYING SIMPLICIAL COMPLEXES IN HIGH DIMENSIONS

We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference Flag (G)\ K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.

OVERLAP-FREE LANGUAGES AND SOLID CODES

Solid codes have a nice property called synchronization property, which is useful in data transmission. The property is derived from infix-freeness and overlap-freeness of solid codes. Since a code is a language, we look at solid codes from formal language viewpoint. In particular, we study regular solid codes (that are solid codes and regular). We first tackle the solid code decidability problem for regular languages and propose a polynomial time algorithm. We, then, investigate the decidability of the overlap-freeness property and show that it is decidable for regular languages but is undecidable for context-free languages. Then, we study the prime solid code decomposition of regular solid codes and propose an efficient algorithm for the prime solid code decomposition problem. We also demonstrate that a solid code does not always have a unique prime solid code decomposition.

Biometric Security Technology

The word biometrics comes from the Greek words “bios” (life) and “metrikos” (measure). Strictly speaking, it refers to a science involving the statistical analysis of biological characteristics. Thus, we should refer to biometric recognition of people, as those security applications that analyze human characteristics for identity verification or identification. However, we will use the short term “biometrics” to refer to “biometric recognition of people”. Biometric recognition offers a promising approach for security applications, with some advantages over the classical methods, which depend on something you have (key, card, etc.), or something you know (password, PIN, etc.). A nice property of biometric traits is that they are based on something you are or something you do, so you do not need to remember anything neither to hold any token.

ELLIPTIC CURVES RELATED TO CYCLIC CUBIC EXTENSIONS

The aim of this paper is to study certain family of elliptic curves [Formula: see text] defined over a number field F arising from hyperplane sections of some cubic surface [Formula: see text] associated to a cyclic cubic extension K/F. We show that each [Formula: see text] admits a 3-isogeny ϕ over F and the dual Selmer group [Formula: see text] is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve [Formula: see text] with nice property. We also prove that the Shafarevich–Tate group [Formula: see text] coincides with a class group of K as a special case.

A POLYNOMIAL RING CONSTRUCTION FOR THE CLASSIFICATION OF DATA

Drensky and Lakatos (Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181–188) have established a convenient property of certain ideals in polynomial quotient rings, which can now be used to determine error-correcting capabilities of combined multiple classifiers following a standard approach explained in the well-known monograph by Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005)). We strengthen and generalise the result of Drensky and Lakatos by demonstrating that the corresponding nice property remains valid in a much larger variety of constructions and applies to more general types of ideals. Examples show that our theorems do not extend to larger classes of ring constructions and cannot be simplified or generalised.