The theory of universal Taylor series can be extended to the case of Padé approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Padé approximants of some power series. We present the first generic result in this direction, for Padé approximants corresponding to Taylor developments of holomorphic functions in simply connected domains. The universal approximation is required only on compact sets K which lie outside the domain of definition and have connected complement. If the sets K are additionally disjoint from the boundary of the domain of definition, then the universal functions can be smooth on the boundary. Subject Classification MSC2010 : primary 41A21, 30K05 secondary 30B10, 30E10, 30K99, 41A10, 41A20.

We prove the existence of holomorphic functions f defined on any open convex subset Ω ⊂ C n , whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from Ω and on denumerably many convex compact sets in C n \Ω which may meet the boundary ∂Ω. If the universal approximation is only required on convex compact sets disjoint from Ω, then f may be chosen to be smooth on ∂Ω, that is f ∈ A ∞ (Ω). Those are generic universalities.Subject Classification MSC2010: primary 30K05, 32A05, 32A30, secondary 40A05, 41A58, 41A99

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The paper documents, based mainly on [3]-[6] published papers where a consistent mathematical description of cyberspace and various types of Cyber-Attacks and protection measures are presented, a holistic mathematical approach to a rigorous description of Advanced Persistent Threat (APT) actors’ modus operandi through various scenarios and Cyber Kill Chain stages [2]. After referring [6] to the various elements of Cyber-Attacks we propose some techniques (via 5 scenarios) of tracking the modus operandi of the most sophisticated and non-linear cyber actors, the Advanced Persistent Threat actors that are usually nation-state or nation-state backed and usually stay undetected for an extended time in later stages of Cyber Kill Chain in defenders’ networks. Keywords: Valuation of cyber assets, vulnerability of cyber assets, node supervision, sophistication of an attack germ of cyber-attack, cyber defense, proactive cyber protection, Advanced Persistent Threat (APT) actors, Indication of Compromise (IOC), Tactics, Techniques and Procedures (TTPs).