Novel Theorems and Algorithms Relating to the Collatz Conjecture

Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.Comment: 17 pages, 4 figures, 4 tables, 3 algorithms, 10 theorems, 2 corollaries, GitHub cod

The Collatz tree as a Hilbert hotel:a proof of the 3x + 1 conjecture

The Collatz conjecture maintains that each natural number $n$ is a node in the Collatz tree with a root path to $1$ generated by iterations of the Collatz function, which connects even $n$ to $n/2$ and odd $n$ to $3n+1$. The conjecture holds if a binary subtree $T_{\ge 0}$ exists with as nodes $\textit{all}$ branching numbers constituting the congruence classes ${\left[4,16\right]}_{18}$ with density $2/18$. For each branching number, the $\textit{upward}$ and $\textit{rightward}$ function generate arrows to two branching numbers in the directions $2n$ and $(n-1)/3$. In the automorphism graph of tree $T_{\ge 0}$. The iterates and conjugates of these two functions build outer paths from tree $T_{\ge 0}$ at its root node to two infinite sequences of nodes containing nested subtrees and disjoint cotrees. The nodes and conjugate inner arrows within each subtree and within each cotree correspond one-to-one to the nodes and arrows within tree $T_{\ge 0}$. The cumulative density of the congruence classes of numbers in disjoint cotrees of $T_{\ge 0}$ amounts to the density $2/18$ of all branching numbers, thus proving the conjecture.Comment: Collatz paths between branching classes $[4,16]_18$ and partitioning of the binary tree $T_{ge 0}$ into disjoint cotrees now validate in v5 the density proofs in previous versions (v1 31 Aug 2000

Assessing the Socio-economic Impacts of Secure Texting and Anti-Jamming Technologies in Non-Cooperative Networks

Operating securely over 5G (and legacy) infrastructure is a challenge. In non-cooperative networks, malicious actors may try to decipher, block encrypted messages, or specifically jam wireless radio systems. Such activities can disrupt operations, from causing minor inconvenience, through to fully paralyzing the functionality of critical infrastructure. While technological mitigation measures do exist, there are very few methods capable of assessing the socio-economic impacts from different mitigation strategies. This leads to a lack of robust evidence to inform cost-benefit analysis, and thus support decision makers in industry and government. Consequently, this paper presents two open-source simulation models for assessing the socio-economic impacts of operating in untrusted non-cooperative networks. The first focuses on using multiple non-cooperative networks to transmit a message. The second model simulates a case where a message is converted into alternative plain language to avoid detection, separated into different portions and then transmitted over multiple non-cooperative networks. A probabilistic simulation of the two models is performed for a 15 km by 15 km spatial grid with 5 untrusted non-cooperative networks and intercepting agents. The results are used to estimate economic losses for private, commercial, government and military sectors. The highest probabilistic total losses for military applications include US$300, US$150, and US$75, incurred for a 1, 3 and 5 site multi-transmission approach, respectively, for non-cooperative networks when considering 1,000 texts being sent. These results form a framework for deterministic socio-economic impact analysis of using non-cooperative networks and secure texting as protection against radio network attacks. The simulation data and the open-source codebase is provided for reproducibility

Securing instant messages with hardware-based cryptography and authentication in browser extension

Instant Messaging (IM) provides near-real-time communication between users, which has shown to be a valuable tool for internal communication in companies and for general-purpose interaction among people. IM systems and supporting protocols, however, must consider security aspects to guarantee the messages' authenticity, confidentiality, and integrity. In this paper, we present a solution for integrating hardware-based public key cryptography into Converse.js, an open-source IM client for browsers enabled with the Extensible Messaging and Presence Protocol (XMPP). The proposal is developed as a plugin for Converse.js, thus overriding the original functions of the client; and a browser extension that is triggered by the plugin and is responsible for calling the encryption and decryption services for each sent and received message. This integrated artifact allowed the experimental validation of the proposal providing authenticity of IM users with digital certificates and protection of IM messages with hardware-based cryptography. Results also shows the proposed systems is resistent to adversarial attacks against confidentiality and integrity and it is secure when considering cryptrographic tests like the Hamming distance and the NIST SP800-22

Recent Trends in Communication Networks

In recent years there has been many developments in communication technology. This has greatly enhanced the computing power of small handheld resource-constrained mobile devices. Different generations of communication technology have evolved. This had led to new research for communication of large volumes of data in different transmission media and the design of different communication protocols. Another direction of research concerns the secure and error-free communication between the sender and receiver despite the risk of the presence of an eavesdropper. For the communication requirement of a huge amount of multimedia streaming data, a lot of research has been carried out in the design of proper overlay networks. The book addresses new research techniques that have evolved to handle these challenges

The Effect of Malaysia General Election on Financial Network: An Evidence from Shariah-Compliant Stocks on Bursa Malaysia

Instead of focusing the volatility of the market, the market participants should consider on how the general election affects the correlation between the stocks during 14th general election Malaysia. The 14th general election of Malaysia was held on 9th May 2018. This event has a great impact towards the stocks listed on Bursa Malaysia. Thus, this study investigates the effect of 14th general election Malaysia towards the correlation between stock in Bursa Malaysia specifically the shariah-compliant stock. In addition, this paper examines the changes in terms of network topology for the duration, sixth months before and after the general election. The minimum spanning tree was used to visualize the correlation between the stocks. Also, the centrality measure, namely degree, closeness and betweenness were computed to identify if any changes of stocks that plays a crucial role in the network for the duration of before and after 14th general election Malaysia

Responsible AI and Analytics for an Ethical and Inclusive Digitized Society

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