On competitive discrete systems in the plane. I. Invariant Manifolds

Let $T$ be a $C^{1}$ competitive map on a rectangular region $R\subset \mathbb{R}^{2}$. The main results of this paper give conditions which guarantee the existence of an invariant curve $C$, which is the graph of a continuous increasing function, emanating from a fixed point $\bar{z}$. We show that $C$ is a subset of the basin of attraction of $\bar{z}$ and that the set consisting of the endpoints of the curve $C$ in the interior of $R$ is forward invariant. The main results can be used to give an accurate picture of the basins of attraction for many competitive maps. We then apply the main results of this paper along with other techniques to determine a near complete picture of the qualitative behavior for the following two rational systems in the plane. $$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{\gamma_{2}y_{n}}{x_{n}},\quad n=0,1,...,$$ with $\alpha_1,A_{1},\gamma_{2}>0$ and arbitrary nonnegative initial conditions so that the denominator is never zero. $$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad y_{n+1}=\frac{y_{n}}{A_{2}+x_{n}},\quad n=0,1,...,$$ with $\alpha_1,A_{1},A_{2}>0$ and arbitrary nonnegative initial conditions.Comment: arXiv admin note: text overlap with arXiv:0905.1772 by other author

A model of growth and development in copepods

We present a model for the growth and development of copepods based on the simple assumption that a constant fraction of assimilated carbon is invested in the formation of a new carapace. The individual molts into the next stage when the weight of the accumulated carapace building bricks is a fixed fraction of the body mass at the beginning of the stage. This simple stage-transition coordinating system is built into an individual growth model. The model predicts decreasing stage duration and increasing body weight with increasing food concentrations. These trends closely agree with published data. Anisochronality, as found in many copepod species, agrees with model predictions. Acartia species usually show isochronal development. The model predicts isochronal development for one particular value of a parameter of the feeding process. Literature data show that this parameter value is realistic for Acartia spp. ''Equiproportionality'' is predicted by the model. The model thus shows close agreement with observed growth and development pattern

Single-inclusive jet production in electron-nucleon collisions through next-to-next-to-leading order in perturbative QCD

We compute the ${\cal O}(\alpha^2\alpha_s^2)$ perturbative corrections to inclusive jet production in electron-nucleon collisions. This process is of particular interest to the physics program of a future Electron Ion Collider (EIC). We include all relevant partonic processes, including deep-inelastic scattering contributions, photon-initiated corrections, and parton-parton scattering terms that first appear at this order. Upon integration over the final-state hadronic phase space we validate our results for the deep-inelastic corrections against the known next-to-next-to-leading order (NNLO) structure functions. Our calculation uses the $N$-jettiness subtraction scheme for performing higher-order computations, and allows for a completely differential description of the deep-inelastic scattering process. We describe the application of this method to inclusive jet production in detail, and present phenomenological results for the proposed EIC. The NNLO corrections have a non-trivial dependence on the jet kinematics and arise from an intricate interplay between all contributing partonic channels.Comment: 9 pages, 4 figure

Reliable Packet Streams with Multipath Network Coding

With increasing computational capabilities and advances in robotics, technology is at the verge of the next industrial revolution. An growing number of tasks can be performed by artificial intelligence and agile robots. This impacts almost every part of the economy, including agriculture, transportation, industrial manufacturing and even social interactions. In all applications of automated machines, communication is a critical component to enable cooperation between machines and exchange of sensor and control signals. The mobility and scale at which these automated machines are deployed also challenges todays communication systems. These complex cyber-physical systems consisting of up to hundreds of mobile machines require highly reliable connectivity to operate safely and efficiently. Current automation systems use wired communication to guarantee low latency connectivity. But wired connections cannot be used to connect mobile robots and are also problematic to deploy at scale. Therefore, wireless connectivity is a necessity. On the other hand, it is subject to many external influences and cannot reach the same level of reliability as the wired communication systems. This thesis aims to address this problem by proposing methods to combine multiple unreliable wireless connections to a stable channel. The foundation for this work is Caterpillar Random Linear Network Coding (CRLNC), a new variant of network code designed to achieve low latency. CRLNC performs similar to block codes in recovery of lost packets, but with a significantly decreased latency. CRLNC with Feedback (CRLNC-FB) integrates a Selective-Repeat ARQ (SR-ARQ) to optimize the tradeoff between delay and throughput of reliable communication. The proposed protocol allows to slightly increase the overhead to reduce the packet delay at the receiver. With CRLNC, delay can be reduced by more than 50 % with only a 10 % reduction in throughput. Finally, CRLNC is combined with a statistical multipath scheduler to optimize the reliability and service availability in wireless network with multiple unreliable paths. This multipath CRLNC scheme improves the reliability of a fixed-rate packet stream by 10 % in a system model based on real-world measurements of LTE and WiFi. All the proposed protocols have been implemented in the software library NCKernel. With NCKernel, these protocols could be evaluated in simulated and emulated networks, and were also deployed in several real-world testbeds and demonstrators.:Abstract 2 Acknowledgements 6 1 Introduction 7 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Use Cases and Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Opportunities of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 State of the Art of Multipath Communication 19 2.1 Physical Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Data Link Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Network Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Transport Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Application Layer and Session Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Research Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 NCKernel: Network Coding Protocol Framework 27 3.1 Theory that matters! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1 Socket Buffers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2 En-/Re-/Decoder API . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.3 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 Timers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.5 Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Low-Latency Network Coding 35 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Random Linear Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Low Latency Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 CRLNC: Caterpillar Random Linear Network Coding . . . . . . . . . . . . . . . . . . 38 4.4.1 Encoding and Packet Format . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4.2 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4.3 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5.2 Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5.3 Packet Loss Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5.4 Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5.5 Window Size Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Delay-Throughput Tradeoff 55 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Network Coding with ARQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 CRLNC-FB: CRLNC with Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3.1 Encoding and Packet Format . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.2 Decoding and Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.3 Retransmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4.2 Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.3 Systematic Retransmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.4 Coded Packet Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4.5 Comparison with other Protocols . . . . . . . . . . . . . . . . . . . . . . . . 67 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6 Multipath for Reliable Low-Latency Packet Streams 73 6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.1 Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.2 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.3 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.3.4 Reliability Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4 Multipath CRLNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.4.1 Window Size for Heterogeneous Paths . . . . . . . . . . . . . . . . . . . . . 77 6.4.2 Packet Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5.1 Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5.2 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7 Conclusion 94 7.1 Results and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2 Future Research Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Acronyms 99 Publications 101 Bibliography 10