Three-page approach to knot theory. Encoding and local moves

On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

A method of representing knots, links, and singular knots and links by words in a finite alphabet and so-called d-diagrams is given. A representation of the chord diagram algebra by words in a finite alphabet is considered. This method, unlike coding by Gauss diagrams, allows one to consider knots and links simultaneously. An algorithm for recognition of diagrams of classical knots in terms of d-diagrams is given. Bibliography: 9 titles.

The problem of approximating systems with finite input and output alphabets by finite memory systems for verification or certified control has received much deserved attention in the recent past. The present paper is a further step in that direction, building upon a robust control inspired notion of approximation we recently proposed. A constructive algorithm for deriving deterministic finite state machine (DFM) approximations of a given system over finite alphabets is proposed, based on a partitioning of its input/output behavior into equivalence classes of finite length snapshots. The algorithm is analyzed, and the resulting nominal models and corresponding approximation errors are shown to have desirable properties. An algorithm for conservatively quantifying the resulting approximation error in a manner consistent with the objective of control synthesis is also proposed. Several simple illustrative examples are presented to demonstrate the approach.

Abstract. For an equivalence relation E on the words in some finite alphabet, we consider the recognition problem (decide whether two words are equivalent), the invariant problem (calculate a function constant on precisely the equivalence classes), the normal form problem (calculate a particular member of an equivalence class, given an arbitrary member) and the first member problem (calculate the first member of an equivalence class, given an arbitrary member). A solution for any of these problems yields solutions for all earlier ones in the list. We show that, for polynomial time recognizable E, the first member problem is always in the class Az (solvable in polynomial time with an oracle for an NP set) and can be complete for this class even when the normal form problem is solvable in polynomial time. To distinguish between the other problems in the list, we construct an E whose invariant problem is not solvable in polynomial time with an oracle for E (although the first member problem is in NPz fqco-NPZ), and we construct an E whose normal form problem is not solvable in polynomial time with an oracle for a certain solution of its invariant problem. Key words, polynomial time, Turing reducible, NP-complete, equivalence relation, certificate, normal form Introduction. This paper was stimulated by [3], where finite structures (e.g. graphs)

The paleomagnetic peculiarities of bottom sediments of the South Caspian Sea and their geochronological sections have been studied in present paper. Correlations between separate oil-gas bearing structures near mud volcanoes as well as quantitative determinations of age according to paleomagnetic data are also given in present paper. The association of local changes of paleomagnetic trends with tectonic movements has been shown.

I think that the conjecture P + NP is not as widely taught in courses of mathematical logic as it should be, in view of its capital importance for the foundation of mathematics. Therefore I am writing this note in the hope that all logicians will always include it in the introductory courses of their subject although it does not appear yet in the appropriate books. The original paper of S. Cook [1], where the conjecture was formulated, was indeed written from the point of view of logic but it became the domain of computer scientists (see [2]), particularly because of a paper of Karp [3] where the combinatorial or computational aspects of the conjecture were developed in a very suggestive way. Let us assume that the teacher has already presented the concept of a first order theory and the concept of a Turing machine (neither the concepts of a decidable theory nor that of a recursive function are needed). Then he may proceed as follows: By a normal theory we shall mean a theory which is formalized with a finite alphabet in first order logic with equality and is axiomatizable by a finite set of axioms and axiom schemata in which one can prove 3xy [ x + y ]. (In [4] it is proved that every theory which is recursively axiomatizable and contains a minimal amount of arithmetic or set theory is normal.) By a proof in a normal theory we mean a Hilbert style proof from the axioms. Let E be a finite alphabet and E* the set of all words, i.e., finite sequences of elements of E. For any E*, I denotes the length of (. Now we introduce a more abstract concept of a theory which we will call a T7T-theory. A T7T-theory is a set of pairs T c E* x E* such that there exists a polynomial P(x, y) and a Turing machine M such that, for any (T, 1) E E* x E*, M can decide in time < P(ITI, I7TI) if (T, 7T) E T. If (i, 7T) E T, thenT is called a theorem of T and 7 is called a proof of i-in T. Every normal theory defines a Tg-theory since the time necessary to check the correctness of a Hilbert style proof in a normal theory can be estimated from above by a polynomial in the length of that proof. Now, a T7n-theory T will be called amenable (to automatization) iff there exists another polynomial Po(x, y) and another Turing machine Mo such that, given any word XE * and any positive integer n, MO can decide in time < Po(ITi-, n) if there exists a X E =* with Ig I < n and such that (i, s) E T. (Notice that if we replaced the condition < Po(iTI, n) by the condition < Po(ITI, Cn) where c = card E, then the concept would trivialize since every T7T-theory would be amenable. In fact, given a time Po(lTI, cn), the machine can form all sequences of symbols of length n and find out if any of them is a proof of i.) It is clear that, after Godel's 1931 discovery that all sufficiently strong theories are undecidable,

We study here a mesoscopic metasurface made of a randomly distributed set of long vertical metallic rods attached to a thin elastic plate. The ${A}_{0}$ Lamb wave propagation is strongly affected by the local change in apparent stiffness of the plate induced by the low-quality factor resonance of the rods. At the resonance, the plate-plus-rods system is allowed to move freely, and plate waves can penetrate into the metamaterial. At the antiresonance, the plate behaves in terms of waves as if it was clamped by the rods in the metamaterial region, which induces large-frequency band gaps. Between the resonant and antiresonant frequencies, the continuous change in effective rigidity results in a continuous change in reflectivity. In the present paper, we aim at measuring the corresponding complex impedance of the metasurface in terms of amplitude and phase. Experimental data are presented to estimate the effective impedance of a locally resonant metasurface, in agreement with theoretical prediction and numerical simulation.

In the present paper, tides in the port of Vancouver Harbour have been investigated with a high-resolution three-dimensional hydrodynamic model based on FVCOM (Finite Volume Community Ocean Model). The model was evaluated against field observations including tidal elevations and tidal currents, and the evaluation showed that the model was in good agreement with the observational data. Using the model, we first investigated the horizontal distributions of tides, tidal currents, and tidally induced residual circulation, and then investigated the tidal asymmetry and dynamic mechanisms of tidal flows in the harbour. The tidal residual circulation shows a strong spatial pattern, which is associated with the local coastlines and variation of topography. The tidal asymmetry in the harbour is caused by different mechanisms, not only including the traditional factors, such as residual flows, the interaction between M2 and its overtide M4, but also the interaction of principal astronomical tides of O1, K1, and M2. The momentum balance is dominated by terms of the advection and the pressure gradient in First Narrows and Second Narrows, whereas terms of the local acceleration and the Coriolis are also important in the central harbour. The spatial variations of the momentum terms are strongly associated with the local changes in coastline and topography.

Small-angle x-ray and neutrons scattering is a widespread experimental tool for the investigation of the microstructure of random heterogeneous materials. Validation of (computer-generated) model microstructures often requires the numerical computation of the scattering intensity, which must be carried out with great care due to finite size effects. In this paper, a new method for this computation is presented. It is superior to previously existing methods for three reasons: First, it applies to any type of microstructure (not necessarily granular). Second, closed-form expressions of the size effects inherent to the proposed method can be rigorously derived and removed (in this sense, our method is free of size effects). Third, the complexity of the new algorithm is linear and the computation can easily be updated to account for local changes of the microstructure, while most existing algorithms are quadratic and any change of the microstructure requires a full recomputation. The present paper provides full derivation and validation of this method. Application to the computation of the scattering intensity of dense, polydisperse assemblies of spheres is then presented. A new, simple algorithm for the generation of these dense configurations is introduced. Finally, the results are critically reviewed in the perspective of hardened cement pastes.

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating.

An alphabet is a finite set \(\mathcal{A}\), \(|\mathcal{A}| \ge 2\), elements of which are called characters (or symbols). A string (or word) is a sequence of characters over a given finite alphabet \(\mathcal{A}\). The set of all finite strings over the alphabet \(\mathcal{A}\) is denoted by \(W(\mathcal{A})\). Examples of real world applications, using distances and similarities of string pairs, are Speech Recognition, Bioinformatics, Information Retrieval, Machine Translation, Lexicography, Dialectology.

Chapter 11 - Distances on Strings and Permutations

In this paper, we propose an approach exploiting the relationship between the position of the clustering centers of the observed data and the mixing parameters to realize blind separation of multiple sequences, drawn from finite alphabet set, from a single linear mixture. In theory, we prove that the system estimation by the algorithm is optimum in the sense of least square (LS) by mathematical induction method. In the noise case, the simulation results show that the system estimation deteriorates smoothly with the increasing of noise variance.

NF-κB (nuclear factor κB) controls diverse cellular processes and is frequently misregulated in chronic immune diseases or cancer. The activity of NF-κB is regulated by IκB (inhibitory κB) proteins which control nuclear-cytoplasmic shuttling and DNA binding of NF-κB. In the present paper, we describe a novel role for p65 as a critical regulator of the cellular localization and functions of NF-κB and its inhibitor IκBβ. In genetically modified p65-/- cells, the localization of ectopic p65 is not solely regulated by IκBα, but is largely dependent on the NLS (nuclear localization signal) and the NES (nuclear export signal) of p65. Furthermore, unlike IκBα, IκBβ does not contribute to the nuclear export of p65. In fact, the cellular localization and degradation of IκBβ is controlled by the p65-specific NLS and NES. The results of our present study also reveal that, in addition to stimulus-induced redistribution of NF-κB, changes in the constitutive localization of p65 and IκBβ specifically modulate activation of inflammatory genes. This is a consequence of differences in the DNA-binding activity and signal responsiveness between the nuclear and cytoplasmic NF-κB-IκBβ complexes. Taken together, the findings of the present study indicate that the p65 subunit controls transcriptional competence of NF-κB by regulating the NF-κB/IκBβ pathway.

For a memory less channel with finite input alphabet $A$, finite output alphabet $B$, and probability law $p(b\mid a)$, the capacity $C$ is defined as the maximum over all probability distributions $q$ on $A$ of $$\sum_{ab} q(a)p(b\mid a)\log_2(p(b\mid a)/\sum_a q(a)p(b\mid a)).$$ Shannon [1] has obtained the following result. Exponential error bound. For any $C_0 < C$ there is a number $\rho < 1$ such that, for every positive integer $N$, there is a set $S \subset A^{(N)}$ with at least $2^{C_0N}$ elements and a function $g$ from $B^{(N)}$ to $S$, such that, for every $s = (a_1, \cdots, a_N) \varepsilon S$, $$\sum p(b_1|a_1) \cdots p(b_N|a_N) < 2\rho^N,$$ where the sum extends over all sequences $b_1, \cdots, b_N$ for which $g(b_1, \cdots, b_N) \neq s$. Thus if the sender selects any $s \varepsilon S$ and places its letters $a_1, \cdots, a_N$ successively into the channel, and the receiver, on observing the resulting output sequence $b_1, \cdots, b_N$, decides that the input was $g(b_1, \cdots, b_N)$, the probability that he makes an error is less than $2\rho^N$, no matter what $s \varepsilon S$ was chosen. This result may be described as follows: it is possible to transmit at any rate $C_0 < C$, with arbitrarily small probability of error, by using block codes of sufficient length. We wish to draw a slightly stronger conclusion, as follows. We imagine an infinite sequence $x = (x_1, x_2, \cdots)$ of 0's and 1's, which we are required to transmit across the channel. At time $N$, the sender will have observed the first $\lbrack C_0N\rbrack$ coordinates of $x$, and will place the $N$th input symbol in the channel. The receiver, having at this point observed the first $N$ channel outputs, will estimate the first $M(N)$ coordinates of $x$. If $M(N)/C_0N \rightarrow 1$ as $N \rightarrow \infty$ and if, for every $x$, all but a finite number of his estimates are correct (i.e., agree with $x$ in every coordinate estimated) with probability 1, we shall say that the channel is being used at rate $C_0$. Our result is that, in this sense, a (memoryless) channel can be used at any rate $C_0 < C$. The result stated below is exactly this result, for the special case $C_0 = 1$. The general case involves no new ideas, but only more notation, and we shall restrict attention to the case $C_0 = 1$. The function $f_n$ of a code, as defined below, specifies the $n$th channel input symbol, as a function of the first $n$ coordinates of $x$. The number $M(n)$ is the number of $x$ coordinates to be estimated by the receiver after observing the first $n$ output symbols, and the function $g_n$ specifies the estimate. We now state the result precisely. For any finite set $S$, we denote by $S^{(N)}$ the set of all sequences $(s_1, \cdots, s_N)$, where $s_n \varepsilon S$ for $n = 1, 2, \cdots, N$. For a memoryless channel with finite input alphabet $A$, finite output alphabet $B$, an infinite code (for transmitting at rate 1) is defined as consisting of (a) a sequence $\{f_n\}$ of functions, where $f_n$ maps $I^{(n)}$ into $A$, and $I$ consists of the two elements 0 and 1, (b) a nondecreasing sequence $\{M(n)\}$ of positive integers such that $M(n)/n \rightarrow 1$ as $n \rightarrow \infty$, and (c) a sequence $\{g_n\}$ of functions, where $\{g_n\}$ maps $B^{(n)}$ into $I^{(M(n))}$. An infinite sequence $x = (x_1, x_2, \cdots)$ of 0's and 1's, together with an infinite code, defines a sequence of independent output variables $y_1, y_2, \cdots$, with $$\Pr \{y_n = b\} = p(b \mid f_n(x_1, \cdots, x_n)),$$ where $p(b \mid a)$ is the probability that the output symbol of the channel is $b$, given that the corresponding input symbol is $a$, and defines a sequence of estimated messages $t_1, t_2, \cdots$, where $t_n = g_n(y_1, \cdots, y_n)$. We shall say that the code is effective at $x$ if, with probability 1, $$t_n = (x_1, \cdots, x_{M(n)})$$ for all sufficiently large $n$, and shall say that the code is effective if it is effective for every $x$. The result of this note is the THEOREM: For any memoryless channel with capacity $C > 1$, there is an effective code.