Pattern avoidance: themes and variations

AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Critical Exponents and Stabilizers of Infinite Words

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold

逐次干渉除去を用いた多元接続システムのパワー割り当てに関する研究

In future wireless communication networks, the number of devices is likely to increase dramatically due to potential development of new applications such as the Internet of Things (IoT). Consequently, radio access network is required to support multiple access of massive users and achieve high spectral efficiency. From the information theoretic perspective, orthogonal multiple access protocols are suboptimal. To achieve the multiple access capacity, non-orthogonal multiple access protocols and multiuser detection (MUD) are required. For the non-orthogonal code-division multiple access (CDMA), several MUD techniques have been proposed to improve the spectrum efficiency. Successive interference cancellation (SIC) is a promising MUD techniques due to its low complexity and good decoding performance. Random access protocols are designed for the system with bursty traffic to reduce the delay, compared to the channelized multiple access. Since the users contend for the channel instead of being assigned by the base station (BS), collisions happen with a certain probability. If the traffic load becomes relatively high, the throughput of these schemes steeply falls down because of collisions. However, it has been well-recognized that more complex procedures can permit decoding of interfering signals, which is referred to as multi-packet reception (MPR). Also, an SIC decoder might decode more packets by successively subtracting the correctly decoded packets from the collision. Cognitive radio (CR) is an emerging technology to solve the problem of spectrum scarcity by dynamically sharing the spectrum. In the CR networks, the secondary users (SUs) are allowed to dynamically share the frequency bands with primary users (PUs) under primary quality-of-service (QoS) protection such as the constraint of interference temperature at the primary base station (PBS). For the uplink multiple access to the secondary base station (SBS), transmit power allocation for the SUs is critical to control the interference temperature at the PBS. Transmit power allocation has been extensively studied in various multiple access scenarios. The power allocation algorithms can be classified into two types, depending on whether the process is controlled by the base station (BS). For the centralized power allocation (CPA) algorithms, the BS allocates the transmit powers to the users through the downlink channels. For the random access protocols, there are also efforts on decentralized power allocation (DPA) that the users select transmit powers according to given distributions of power and probability, instead of being assigned the transmit power at each time slot by the BS. In this dissertation, the DPA algorithms for the random access protocols with SIC are investigated and new methods are proposed. First a decentralized multilevel power allocation algorithm to improve the MAC throughput performance is proposed, for the general SIC receiver that can decode multiple packets from one collision. Then an improved DPA algorithm to maximize the overall system sum rate is proposed, taking into account of both the MAC layer and PHY layer. Finally, a DPA algorithm for the CR secondary random access is proposed, considering the constraint of interference temperature and the practical assumption of imperfect cancellation. An opportunistic transmission protocol for the fading environment to further reduce the interference temperature is also proposed. For the future work, the optimal DPA for the random access with the SIC receiver is still an open problem. Besides, advanced multiple access schemes that aim to approach the multiple access capacity by combining the advantages of the network coded cooperation, the repetition slotted ALOHA, and the SIC receiver are also interesting.電気通信大学201

Conferences WORDS, years 1997-2017: Open Problems and Conjectures

International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017

Ultrasonic propagation in finite-length granular chains

A narrowband ultrasound source has been used to generate solitary wave impulses in finite-length chains of spheres. Once the input signal is of sufficient amplitude, both harmonics and sub-harmonics of the input frequency can be generated as non-linear normal modes of the system, allowing a train of impulses to be established from a sinusoidal input. The characteristics of the response have been studied as a function of the physical properties of the chain, the input waveform and the level of static pre-compression. The results agree with the predictions of a theoretical model, based on a set of discrete dynamic equations for the spheres for finite-length chains. Impulses are only created for very small pre-compression forces of the order of 0.01 N, where strongly non-linear behaviour is expected

Overlap-Free Words and Generalizations

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions. In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area. In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler. In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement. In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps. In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free. In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice. In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares. In Chapter 8 we conclude the work and present some open problems